Thin-Walled Structures - Advances and Developments

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THIN-WALLED STRUCTURES ADVANCES AND DEVELOPMENTS

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THIN-WALLED STRUCTURES ADVANCES AND DEVELOPMENTS Third International Conference on Thin-Walled Structures

Edited by

J. Zara~ Technical University of L6d~ Poland

K. Kowal-Michalska Technical University of L6d~ Poland

J. Rhodes University of Strathclyde, UK

e

2001

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9 2001 Elsevier Science Ltd. All rights reserved.

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First edition 2001 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for.

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Transferred to digital printing 2005 Printed and b o u n d by A n t o n y R o w e Ltd, E a s t b o u m e

ORGANISING COMMITTEE Chairman

M. Kr61ak,

Co-Chairmen

J. Rhodes, J. Loughlan, K.P. Chong,

Director Secretary

J. Zara~ M. Marciniak

Members

R. Gr~dzki M. Jaroniek, Z. Kotakowski M. Kote|ko K. Kowal-Michalska T. Kubiak M. Macdonald J. Swiniarski A. 2:eligowski

INTERNATIONAL SCIENTIFIC COMMITTEE Chairman N. E. Shanmugam, Singapore Members W. Abramowicz, Poland M. A. Bradford, Australia Y. K. Cheung, Hong Kong China C. K. Choi, Korea J. M. Davies, United Kingdom P. J. Dowling, United Kingdom D. Dubina, Romania M. Eisenberger, Israel R. Evans, United Kingdom Y. Fukumoto, Japan K. Ghavami, Brazil N. K. Gupta, India G. J. Hancock, Australia J. E. Harding, United Kingdom T. H6glund, Sweden N. Jones, United Kingdom S. Kitipornchai, Australia R. A. LaBoube, USA L. Librescu, USA M. Mahendran, Australia A. Manevich, Ukraine

J. Murzewski, Poland R. Narayanan, India T. Pekfz, USA W. Pietraszkiewicz, Poland K. J. R. Rasmussen, Australia J. Rondal, Belgium J. Spence, United Kingdom J. Stupnicki, Poland C. Szymczak, Poland A. S. Tooth, United Kingdom V. Tvergaard, Denmark S. Ujihashi, Japan T. Usami, Japan G. I. van den Berg, South Africa R. Vaziri, Canada A. C. Walker, United Kingdom F. Werner, Germany T. Wierzbicki, USA H. D. Wright, United Kingdom W. W. Yu, USA J. Zielnica, Poland

This Page Intentionally Left Blank

vii

PREFACE

There has been a substantial growth in knowledge in the field of Thin-Walled Structures over the past few decades. Lightweight structures are in widespread use in the Civil Engineering, Mechanical Engineering, Aeronautical, Automobile, Chemical and Offshore Engineering fields. The development of new processes, new methods of connections, new materials has gone hand-in-hand with the evolution of advanced analytical methods suitable for dealing with the increasing complexity of the design work involved in ensuring safety and confidence in the finished products. Of particular importance with regard to the analytical process is the growth in use of the finite element method. This method, about 40 years ago, was confined to rather specialist use, mainly in the aeronautical field, because of its requirements for substantial calculation capacity. The development over recent years of extremely powerful microcomputers has ensured that the application of the finite element method is now possible for problems in all fields of engineering, and a variety of finite element packages have been developed to enhance the ease of use and the availability of the method in the engineering design process. This volume contains the papers presented at the Third International Conference on Thin-Walled Structures, Cracow, Poland on June 5-7, 2001. This is the third conference in the "Thin-Walled Structures" series, and is a sequel to the first two, which were held in Glasgow in 1996 and in Singapore in 1998. There are 83 papers of which 5 are Keynote papers. They are arranged in 12 sections as follows: Keynote Papers Analysis, Design and Manufacture Bridge Structures Cold-Formed Sections Composites Dynamic Loading (Cyclic, Impact and Vibration)

Finite Element Analysis Laminate and Sandwich Structures Optimization and Sensitivity Analysis Plate Structures Shell Structures Ultimate Load Capacity

The Conference is organised by the Department of Strength of Materials and Structures of the Technical University of L6d~., Poland jointly with the Department of Mechanical Engineering, University of Strathclyde and the College of Aeronautics, Cranfield University, UK and is supported by the State Committee for Scientific Research of Poland. The Editors should like to express their appreciation of the role played by the International Journal "Thin-Walled Structures" in the propagation of research in this field and in providing the impetus for this Conference.

Jan Zara~ Katarzyna Kowal-Michalska Jim Rhodes

This Page Intentionally Left Blank

CONTENTS

Preface

vii

Section I: KEYNOTE PAPERS

Shear Strength of Empty and Infilled Cassettes J.M. Davies and A.S. Fragos Stability and Ductility of Thin High Strength G550 Steel Members and Connections C.A. Rogers, D. Yang and G.J. Hancock

19

Thin-Walled Structural Elements Containing Openings N.E. Shanmugam

37

Sensitivity Analysis of Thin-Walled Members, Problems and Applications C. Szymczak

53

Some Observations on the Post-Buckling Behaviour of Thin Plates and Thin-Walled Members 3. Rhodes

69

Section II" ANALYSIS, DESIGN AND MANUFACTURE

Residual Stresses in Unstiffened Plate Specimens M.R. Bambach and K.J.R. Rasmussen

87

Behaviour and Design of Valley-Fixed Steel Cladding Systems M. Mahendran and D. Mahaarachchi

95

Behaviour of Flexible Thin-Walled Steel Structures for Road and Railway Applications J. Vaslestad, B. Bednarek and L. Janusz

103

A Simplified Computation Model for Arch-Shaped Corrugated Shell Roof Fan Xuewei

109

Fabrication Accuracy and Cost of Thin-Walled Steel Plate Girders in Shop Assembling K. Yoshikawa and S. Shimizu

119

Section III: BRIDGE STRUCTURES

Web Breathing as a Fatigue Problem in Bridge Design U. Kuhlmann and H.-P. Giinther

129

Erosion of the Post-Buckled Reserve of Strength of Thin-Walled Structures due to Cumulative Damage M. ~kaloud and M. Z6rnerov6

137

Evaluation of Strength and Ductility Capacities for Steel Plates in Cyclic Shear T. Usami, P. Chusilp, A. Kasai and T. Watanabe

145

A Rational Model for the Elastic Restrained Distortional Buckling of Half-Through Girder Bridges Z. VrceO"and M.A. Bradford

153

Instability Testing of Steel Plate Girders with Folded Webs T. Yabuki, Y. Arizumi, J.M. Aribert and S. Guezouli

161

Section IV: COLD-FORMED SECTIONS

Modelling of the Behaviour of a Thin-Walled Channel Section Using Beam Finite Elements H. Deg~e

171

Codification of Imperfections for Advanced Finite Analysis of Cold-Formed Steel Members D. Dubina, V. Ungureanu and 1. Szabo

179

Innovative Cold-Formed Steel Structure for Restructuring of Existing RC or Masonry Buildings by Vertical Addition of Supplementary Storey D. Dubina, V. Ungureanu, M. Georgescu and L. Fiilb'p

187

EC.3-Annex Z Based Method for the Calibration of (a) Generalized Imperfection Factor in Case of Thin-Walled Cold-Formed Steel Members in Pure Bending M. Georgescu and D. Dubina

195

Design Aspects of Cold-Formed Portal Frames P. Frti and L. Dunai

203

Local Buckling and Effective Width of Thin-Walled Stainless Steel Members H. Kuwamura, Y. lnaba and A. lsozaki

209

Bifurcation Experiments on Locally Buckled Z-section Columns K.J.R. Rasmussen

217

Buckling Load Capacity of Stainless Steel Columns Subject to Concentric and Eccentric Loading J. Rhodes, M. Macdonald, M. Kotetko and W. McNiff

225

Experimental Behavior of Pallet Racks and Components K. S. (Siva) Sivakumaran

233

Structural Behaviour of Cold-Formed Steel Header Beams S.F. Stephens and R.A. LaBoube

241

Compression Tests of Thin-Walled Lipped Channels with Return Lips J. Yan and B. Young

249

Experimental Investigation of Stainless Steel Circular Hollow Section Columns B. Young and W. Hartono

257

Section V: COMPOSITES

A Model for Ferrocement Thin Walled Sructures D. Abruzzese

269

Effects of Manufacturing Variables on the Service Reliability of Composite Structures A.R.A. Arafath, R. Vaziri, H. Li, R.O. Foschi and A. Poursartip

277

Post-Failure Analysis of Thin-Walled Orthotropic Structural Members M. Kotetko

285

Modal Coupled Instabilities of Thin-Walled Composite Plate and Shell Structures M. Kr6lak, Z. Kotakowski and M. Kotetko

293

Induced Strain Actuation and its Application to Buckling Control in Smart Composite Structures J. Loughlan and S.P. Thompson

301

Failure Analysis of FRP Panels with a Cut-out Under Static and Cyclic Load A. Muc, P. Kcdziora, P. Krawczyk and M. Sikoh

313

Some New Applications of the Theory of Thin-Walled Bars J.B. Obrqbski

321

Buckling Behaviour of Thin-Walled Composite Columns Using Generalised Beam Theory N. Silvestre, D. Camotim, E. Batista and K. Nagahama

329

Shear Connection Between Concrete and Thin Steel Plates in Double Skin Composite Construction H.D. Wright, A. Elbadawy and R. Cairns

339

Section Vl: DYNAMIC LOADING (CYCLIC, IMPACT AND VIBRATION)

Regular and Chaotic Behaviour of Flexible Plates J. Awrejcewicz, V. A. Krysko and A. V. Krysko

349

Seismic Performance of Arc-Spot Weld Deck-to-Frame Connections W.F. Bond, C.A. Rogers and R. Tremblay

357

Dynamic Buckling and Collapse of Rectangular Plates Under Intermediate Velocity Impact S. Cui, H. Hao and H.K. Cheong

365

Structural Optimization of Thin Shells Under Dynamic Loads S.A. Falco, Luiz E. Vaz and S.M.B. Afonso

373

Vibrations of Compressed Sandwich Bars J. Hahkkowiak and F. Roman6w

381

Numerical Simulation of Damaged Steel Pier Using Hybrid Dynamic Response Analysis T. Ikeuchi and N. Nishimura

391

xii Cyclic Response of Metal-Clad Wood-Framed Shear Walls W. Pan and K.S. (Siva) Sivakumaran

399

Vibration of Imperfect Structures J. Ravinger and P. Kleiman

407

Section VII: FINITE ELEMENT ANALYSIS

The Finite Element Method for Thin-Walled Members - Basic Principles M.C.M. Bakker and T. Pek6z

417

Nonlinear Analysis of Locally Buckled I-Section Steel Beam-Columns A.S. Hasham and K.J.R. Rasmussen

427

The Finite Element Method for Thin-Walled Members - Applications A.T. Sarawit, Y. Kim, M.C.M. Bakker and T. PekOz

437

Finite Element Methods for the Analysis of Thin-Walled Tubular Sections Undergoing Plastic Rotation T. Wilkinson and G. Hancock

449

On Finite Element Mesh for Buckling Analysis of Steel Bridge Pier E. Yamaguchi, Y. Nanno, H. Nagamatsu and Y. Kubo

459

Section Vlll: LAMINATE AND SANDWICH STRUCTURES

The Elasto-Plastic Postbuckling Behaviour of Laminated Plates Subjected to Combined Loading R. Grqdzki and K. Kowal-Michalska

469

Axial Post-Buckling of Thin Orthotropic Cylindrical Shells with Foam Core X. Huang and G. Lu

477

Nonlinear Stability Problem of an Elastic-Plastic Sandwich Cylindrical Shell Under Combined Load L. Jaskuta and J. Zielnica

483

Buckling Analysis of Multilayered Angle-Ply Composite Plates H. Matsunaga

491

Optimum Design for Laminated Panel with Cutout: The Genetic Algorithm Approach Z Li and P. W. Khong

499

The Effect of Membrane-Flexural Coupling on the Compressive Stability of Anti-Synunetric Angle-Ply Laminated Plates J. Loughlan

507

Homogeneous and Sandwich Elastic and Viscoelastic Annular Plates Under Lateral Variable Loads D. Pawlus

515

xiii Local Buckling Behaviour of Sandwich Panels N. Pokharel and M. Ma;:endran

523

Section IX: OPTIMIZATION AND SENSITIVITY ANALYSIS

Optimal Design of Steel Telecommunication Towers by Interior Point Algorithms for Non-Linear Programming N.A. Cerqueira, G.S.A. Falcon, J.G.S. da Silva and F.J. da C.P. Soeiro

533

Design Optimisation of Shell Structures with Dimensional Analysis Resources M.P.R. C. Gomes

541

Vector Optimization of Stiffened Plates Subjected to Axial Compression Load Using the Canadian Norm S.A. Falco and K. Ghavami

549

Bicriteria Optimization of Sandwich Cylindrical Panels Aided by Expert System R. Kasperska and M. Ostwald

559

Shaping of Open Cross Section of the Thin-Walled Beam with Flat Web and Multiplate Flange E. Magnucka-Blandzi, R. Krupa and K. Magnucki

567

Two-Criteria Optimization of Thin-Walled Beams-Columns Under Compression and Bending A.1. Manevich and S. V. Raksha

575

Optimization of Volume for Composite Plated and Shell Structures A. Muc and W. Gurba

585

Sensitivity Analysis of Structures Made of Thin-Walled E-Profiles K. Rzeszut, W. Kqkol and A. Garstecki

593

Section X: PLATE STRUCTURES

Buckling Loads of Variable Thickness Plates A. Alexandrov and M. Eisenberger

603

Buckling and Post-Buckling Behavior of Plates on a Tensionless Elastic Foundation A.S. Holanda and P.B. Gonfalves

611

Degree of Wall Joint Work Together with Stiffening Rib in Steel Bunker M.1. Kazakevitch and D.O. Bannikov

619

Pure Distortional Buckling of Closed Cross-Section Columns K. Takahashi, H. Nakamura and K. lmamura

623

Elasto-Plastic Large Deflection of Uniformly Loaded Sector Plates G.J. Turvey and M. Salehi

631

xiv Section XI: SHELL S T R U C T U R E S

Validation of Analytical Lower Bounds for the Imperfection Sensitive Buckling of Axially Loaded Rotationally Symmetric Shells G.D. Gavrylenko and J.G.A. Croll

643

On the Analysis of Cylindrical Tubes Under Flexure F. Guarracino and M. Fraldi

653

Buckling of Aboveground Storage Tanks with Conical Roof L.A. Godoy and J. C. Mendez-Degr6

661

On the Collapse of a Reinforced Concrete Digester Tank L.A. Godoy and S. Lopez-Bobonis

669

Closed Cylindrical Shell Under Longitudinal Self-Balanced Loading V.L. Krasovsky and G. K Morozov

677

Coupled Instability of Cylindrical Shells Stiffened with Thin Ribs (Theoretical Models and Experimental Results) A.1. Manevich

683

Instability Modes of Stiffened Cylindrical Shells J. Murzewski

693

Refined Strain Energy of the Shell R.A. Walentyhski

701

Numerical and Experimental Studies on Generalised Elliptical Barrelled Shells Subjected to Hydrostatic Pressure P. Wang

709

Section XII: ULTIMATE LOAD CAPACITY

Experimental Techniques for Testing Unstiffened Plates in Compression and Bending M.R. Bambach and K.J.R. Rasmussen

719

Effects of Anchoring Tensile Stresses in Axially Loaded Plates and Sections M.R. Bambach and K. J.R. Rasmussen

729

A Probabilistic Approach to the Limit State of Centrally Loaded Thin-Walled Columns Z. Kala, J. Kala, B. Teplf4 M. ,qkaloud

739

Rotational Capacity of I-Shaped Aluminium Beams: A Numerical Study G. De Matteis , V. De Rosa and R. Landolfo

747

Author Index

757

Keyword Index

759

Section I KEYNOTE PAPERS

This Page Intentionally Left Blank

Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

SHEAR STRENGTH OF EMPTY AND INFILLED CASSETTES J M Davies and A S Fragos Manchester School of Engineering, University of Manchester Manchester M13 9PL, UK

ABSTRACT This paper considers the shear buckling of light gauge steel cassette sections both with and without an infilling of relatively rigid thermal insulation. In cassette construction, In-plane shear stresses usually arise as a consequence of stressed skin (diaphragm) action and, in this context, it is local buckling of the wide flange that usually governs the design. Although there are some rudimentary equations for the shear strength of plain cassettes in Part 1.3 of Eurocode 3, this is a subject that is not fully understood and which raises some fundamental questions such as: What are the boundary conditions for plate buckling - is it sufficient to consider individual plate elements or is it necessary to consider the whole section? Is it sufficient to neglect the post-buckling strength or should this be incorporated in the design equations, as it is for plate elements in compression? How can the favourable interaction with a relatively rigid, thermally-insulating infill be incorporated in the design equations? Thus, this Keynote paper addresses some fundamental questions using the title topic to provide an illustration of the principles involved.

KEYWORDS Buckling, cassettes, finite element analysis, insulation, polystyrene, post-buckling, shear, steel, testing.

INTRODUCTION Typical cassette wall construction is illustrated in Figure 1. The first author has been interested in this form of construction for many years and has written a number of papers (eg Davies 1998a, Davies 1998b, Davies 2000a) advocating this method for houses and other low-rise commercial buildings. He has also been responsible for the structural design of a number of projects incorporating this form of load-bearing wall construction. This has led to a detailed study of the design of these structural elements. The design of light gauge steel cassettes (sometimes also known as structural liner trays) subject to axial load, bending or shear, is covered in Part 1.3 of Eurocode 3 (1996) (EC3). Figure 2 shows the most general form of a cassette section. The range of validity of the design procedures in EC3 is stated to be as follows:

0.75 mm 30 mm 60 mm 300 mm

< < < <

tnom bf

~ ~

1.5 mm 60 a n n

h bu I/b. h~ Sl

_< -< < _< <

200 mm 600 turn 10 rlq_/rl4/n'lnl h/8 1000 mm

where I, is the second moment of area of the wide flange about its own centroid as shown in the right hand insert on Figure 2.

L/~ C-shaped ~'~! ~ cassette

l lll

[Floor

. . . .

Figure 1" Cassette wall construction

A b!

Narrowflange

~_F- Lip

Wideflange

L

J Figure 2: Typical cassette section

However, the somewhat simplistic treatment in EC3 conceals a number of aspects of the design of these interesting elements which are only partly understood. In the case of cassettes subject to shear as a consequence of stressed skin (diaphragm) action, local buckling of the wide flange is usually critical. Here, the code offers the well-known "simplified Easley equation" which is safe but the subject is worthy of more detailed investigation bearing in mind that it fails to address the significance of such matters as boundary conditions and post-buckling behaviour. This paper approaches the subject in a more fundamental manner by considering the shear buckling of the basic plate element with different boundary conditions. The results of a number of tests with idealised boundary conditions are compared with alternative analytical approaches. This leads to a more fundamental understanding of the problem. The investigation is initially confmed to plane flanges. Flanges with rolled-in stiffeners will be the subject of a subsequent investigation. Cassettes may be advantageously infilled with rigid insulation (rigid plastic foam or mineral wool). This has the advantage of stabilising the wide flange against all modes of local buckling including shear. Both the test programme and the subsequent analysis included consideration of infilling with foams of different densities.

OVERVIEW OF THE BEHAVIOUR OF CASSETTE WALLS IN SHEAR A cassette wall assembly is a ready-made shear panel or 'diaphragm' for stressed skin construction (Davies and Bryan 1981). Stressed skin design is explicitly allowed in EC3 and appropriate enabling clauses are included in section 9. EC3 also includes somewhat rudimentary provisions for cassettes acting as shear panels. These make it clear that the behaviour of a cassette wall panel in shear is not significantly different from that of a conventional shear panel comprising trapezoidal steel sheeting framed by appropriate edge members so that the procedures described in any of the definitive publications (eg ECCS 1995) may be used. There are three main differences between cassette systems and the trapezoidally profiled roof sheeting and decking for which the calculation procedures were originally devised: 9

There is negligible flexibility due to shear distortion of the profile. This removes a design equation which tends to dominate the deflection calculation for trapezoidal profiles. Here it is possible to make a simplified estimate of deflections based on the assumption that the flexibility arises mainly in the fastenings.

9

The strength calculation tends to be dominated by the tendency of the wide flange to buckle locally in shear before any of the more usual diaphragm failure modes (fastener failure, profile end failure or global shear buckling) are mobilised.

9

There is often no separate longitudinal edge member. This means that there are no longitudinal edge fasteners to check and the web and narrow flange of the outermost cassette act as their own edge member which should be checked for the induced compressive force.

The first two of these considerations lead to the two equations given in EC3 for the ultimate and serviceability limit states respectively. These equations may be derived from the more familiar equations as follows: Baehre (1987) shows that the simplified Easley equation is valid for the determination of the shear flow Tv,sd to cause local buckling:-

_

Tv, Rd -

36 4~D x 3 ~ u2 Dy

where D x = bending stiffness across the wide flange

Dy = bending stiffness along the wide flange =

(1)

E Ix

bu Et 3

=

12(1 - v 2)

Et 3 10.96

I, = second moment of area of the wide flange about its own centroid.

Thus:

Tv, Rd =

4 ~10.963

lit

9 bu

= 6E

I,

(2)

This is the equation in clause 10.3.5 of EC3. It should be noted here that this equation is printed incorrectly in both Baehre's paper and in the original printing of EC3. It is corrected in the Corrigenda

to ENV 1993-1-3 dated 1997-02-25. For the calculation of deflection, Baehre (1987) considered the following four components of shear flexibility (Davies and Bryan 1981, ECCS 1995, etc): 9

c~.2

shear strain

9

c2.1

flexibility of the sheet end fasteners

9

C2.2

flexibility of the seam fasteners

9

C2.3

flexibility of the "shear connector" (longitudinal edge) fasteners

He showed that all are of similar magnitude and stated (without further explanation) that their resultant stiffness could be approximated by:

Sv =

2000 L b u E s(b - bu) (N/mm)

(3)

which is the equation to be found in clause 10.3.5 of the EC3. This rather crude approach to stiffness appears to be justified because, in the absence of the distortion term, the deflections tend to be small and because cassettes tend to be of fairly similar proportions and to have similar fastening systems, the individual fasteners of which have similar flexibilities. However, this simplification is not essential and, if the deflections are at all critical, the more fundamental approach to the calculation of deflections given in the definitive References is to be preferred. More importantly, the wording of the clauses in Eurocode 3, and the above equation for the shear flow to cause local buckling, may lead designers to overlook that fastener strength may also be critical. In addition to considering local buckling of the wide flange, it is essential also to consider the possibility of failure in each of the fastener failure modes considered in conventional stressed skin theory, namely: failure in the seam fasteners between adjacent cassettes failure in the fasteners connecting the ends of the cassettes to the foundation or the primary structure 9

failure in the shear connector (longitudinal edge fasteners)

It appears clear to the author that the excessive simplicity in the approach in EC3 to the design of cassettes subject to diaphragm action is likely to lead to a lack of fundamental understanding and overconfidence. The inevitable result will be serious design errors. However, this paper is primarily concerned with the local buckling phenomenon and the question of the design of fastening systems for cassette wall construction will not be pursued further.

METHODOLOGIES AVAILABLE FOR THE INVESTIGATION Davies (2000b) has reviewed the methodologies available for research into the stability of cold-formed steel structures. The following are available for this study:

Testing: Bearing in mind the influence of uncertain imperfections on buckling phenomena, the uncertainty of the boundary conditions for shear buckling and the interaction between unlike materials, testing will clearly have a major role to play in this research.

The finite element method: This is, of course, the obvious method for this type of research but care will be required when selecting the boundary conditions and dealing with the interface between unlike materials. Solutions of the governing differential equations: Early researchers into the shear buckling of thin plates used energy method solutions based on displacement functions in the form of a double sine series. This approach is clearly worthy of consideration for the more general situation considered here.

TEST P R O G R A M M E The apparatus and the initial test programme were described in Davies and Dewhurst (1997). Further testing using the same apparatus, modified where necessary, was carried out by Fragos (2000). This paper makes reference to the results of both test series, 50 tests in all. Figure 3 shows photographs of the test rig together with a typical buckled specimen and Figure 4 shows a cross-section through it.

Figure 3: View of the test rig showing specimen under test with short ends clamped

Shortage of space precludes a full description of the test procedure and results but the following points may be noted: The boundary conditions are evidently critical. It was considered to be appropriate to attempt to obtain a fully fixed condition along the longitudinal edges in order to allow easier comparison with available theories. The test results suggest that this was achieved, at least as far as initial buckling is concerned.

The short edge boundary condition is also significant. Both the free and restrained conditions were available and both were tested. The restrained condition was obtained by clamping the free edge of the cassette between two metal strips. Tests were carried out with the empty tray and 3 grades of expanded polystyrene foam infllling. The test specimens were produced in-house using a folding press. The width was constant at 400 mm and the length was varied. The majority of the tests were carried out with a length of 2400 mm but lengths of 1600 mm and 800 mm were also used. The metal thickness was varied in the range 0.73 to 1.27 mm.

Figure 4: Cross-section through the test rig

It was found to be rather difficult to identify the experimental buckling load. A number of different shapes of load-deflection curves were obtained and Figure 5 is fairly typical. The construction used to estimate the buckling load is also shown. There is evidently a significant amount of post-buckling strength. The general pattern of the results was the same while the boundary conditions and foam infllling were varied although there were some difficulties with the strongest 250f foam. This was significantly stronger than the alternative SDVB and HDN foams and breakdown of the Bostik D adhesive bond led to a greater scatter of the test results than was observed with the more usual foam insulants. The influence of the foam is considerable, particularly with thinner metal and/or the higher grades of foam. Figure 6 shows a typical family of test results for 2400 mm long cassettes with four edges clamped. The results for two edges clamped and for other lengths were broadly similar.

16

~

i::l ~ i i : : J - ] ~ ~

'~ -.' S

-

iii~ii 6

I

',

',

t.dr",

i-~1

...!. i - - i i ' i " l i

"

0

9

0,~5

'liiiii

i

I ', ', i

',

I ',

} I i. . .i . i i

! ! ! I

i i I

i

[ !

! I : : : : ~

',

',

4] ri'iiii"iii'!",i 2 -it

illiiil

9

"

1,5

!i!!i!i

i

|

2

2,S

"'|

1

i

,

o

'

''

i

3

Deformation z-axis (ram)

Figure 5" Typical test result: Load versus longitudinal shear deflection

Figure 6: Critical buckling loads of cassettes with four edges clamped Figure 6 needs to be interpreted in the light of the properties of the foams used in the tests. These were standard products obtained from a major manufacturer, Vencil Resil Ltd, UK, and the properties, as delivered, were determined using the procedures described in the European Recommendations (CIB 1995). The results are summarised in Table 1 and it is these values that were used in the finite element analyses which are described later in this paper.

Material

Density kg/m3

Compressive modulus N/ram 2

15 21.3 44.4

2.73 3.87 14.8

Tensile modulus N/ram 2

Shear modulus N/mm 2

11.0 8.04 10.3

8.11 4.58 13.6

.....

SDVB

HDN 250f

~,,

Table 1" Material properties of the foams - as used in the numerical modelling

10 The properties of the steel were also measure in both the longitudinal and transverse directions using the standard tensile test. The average measured values were: Transverse

Longitudinal

Yield stress

Thickness

(ram)

Tensile modulus

Yield stress

Tensile modulus

(N/mm2)

(N/ram 2)

fN/mm 2)

0.73 0.80 1.27

188 202 183

144 220 253

189 210 187

155 216 253

(Nlmm 2)

For the theoretical analysis described later, a bi-linear stress-strain curve was assumed with the average values of modulus and yield stress from the longitudinal and transverse directions.

SHEAR BUCKLING OF THIN METAL PLATES Southwell and Skan (1924) obtained the first exact solution for the shear buckling of an infinitely long plate. They showed that the buckling stress is given by: 1:cr =

where

D K b t E v

Et 3

=

~2D

K-b2t

= flexural stiffness of the plate

12(1 - v 2) is is is is is

a non-dimensional buckling stress coefficient the width of the plate the thickness of the plate the elastic modulus of the metal of the plate Poisson's ratio

Figure 7: Local shear buckling of infinitely long thin plates

(4)

It was found that the buckled plate developed a series of inclined waves, as shown in Figure 7, whose wavelength Lcr varied as the boundary condition along the lOngitudinal edge varied from simply supported (Lcr = 1.33b) to fixed (Lc~ = 0.80b). The value of the buckling coefficient K was found to be 5.35 for a simple support and 8.98 for a clamped support. Various authors have continued this work, considering plates of finite length 'a' and alternative boundary conditions. For all situations, including those where the buckled plate element is stiffened by rigid insulation, the basic pattern of inclined waves remains the same. The results are invariably expressed in terms of the buckling coefficient 'K'. Fragos (2000) has reviewed this work and compared it with the results of his own extensive finite element analyses. He has shown that an accurate value for 'K' is given by expressions of the form: K

= tt+e

where ct and 13 depend on the boundary conditions: Boundary condition 1. 2. 3. 4. 5. 6.

All edges simply supported All edges rigidly clamped Two long edges clamped, two short edges simply supported Two long edges simply supported, two short edges clamped One long edge rigidly clamped, other edges simply supported One short edge rigidly clamped, other edges simply supported

tx 4.188 7.868 7.980 4.089 6.209 4.262

1.64 1.92 1.53 2.14 1.56 1.90

The test results for boundary conditions 2 and 3 are in accordance with the above formulae. The finite element analysis referred to above also serves as a partial validation for the more general finite element analysis which follows.

FINITE ELEMENT ANALYSIS FOR INFILLED CASSETTES WITH IDEALISED BOUNDARY CONDITIONS After various trials, the finite element analysis reported in this paper was carried out using the ANSYS 5.4 package using the "SHELL93" 8-node shell element for the metal skin of the cassettes and the "SOLID95" 20-node solid element for the foam infill. Some numerical experiments were carried out using the "CONTA174" 8-node surface-to-surface contact element to represent the surface contact between the foam core and the skin of the cassettes. However, it was found that this was not particularly useful except for some cases with the strong 250f foam where loss of adhesive bond contributed to the failure. A convergence study was carried out in order to determine the minimum number of elements that could be used to obtain an accurate and stable result with a reasonable computation time. It was found that elements measuring 5 cm x 5 cm were adequate, giving 48 x 8 = 384 elements in the wide flange and a further 192 elements in the webs, where these are relevant. Two types of analysis were used in this study. The critical buckling load can be obtained from an eigenvalue bifurcation buckling analysis but it is also interesting to carry out a full non-linear analysis with large deflection effects using Newton-Raphson iteration. This, of course, allows the post-buckling behaviour to be considered. Whenever the full non-linear analysis was used, an initial bifurcation

12 analysis was carried out first. This determined the critical buckling mode and allowed an arbitrary imperfection to be introduced which mobilised the critical mode at the outset of the calculation. The buckled shapes found in this way were invariably similar to those observed in the tests. However, it has to be conceded that the correlation between the experimental and theoretical load deflection curves was not good and that there was much more post-buckling strength in theory than was observed in the tests. The only explanation that can be offered for this is that the longitudinal edge fixity, which certainly approximated to the fully clamped case at initial buckling, broke down as the buckling increased thus leading to a "premature" failure.

Foam None

SDVB m

HDN

250f

Thickness

P o, (test)

Per (FE)

(mm)

(kN)

(kN)

(%)

0.73 0.82 0.95 1.02 1.20 1.27

12.15 16.35

11.96 16.95 26.36 32.62 53.10 62.94

-1.56 +3.67

t

t

+1.46 +4.32 +5.61

38.0 52.0 62.8

28.54 33.80 43.31 49.50 69.40 78.87

+5.70

29.0

t t

t t

-1.89

63.0

0.73 0.82 0.95 1.02 1.20 1.27

t 32.15 50.9 59.6 27.0

t t 50.5

t 81.15

0.82 0.95 1.20 1.27

38.4 49.7

0.82 0.95 1.20 1.27

78.8 107.2

t t

t t

Difference

Pult (test)

P~t (FE)

(kN)

(kN)

12.5 18.0

32.9 44.6 71.6 82.7 126 163

t

t

-2.81

130

40.73 50.31 76.89 86.47

+6.07 +1.23

42.4 57.5

t t

t t

96.03 110.0 142.0 152.9

+21.9 +2.62

81.5 118

t t

t t

78.5 90.0 111 126 191 236 102 128 198 218 244 270 331 387

Note: t indicates that no test was carded out for this arrangement Table 2: Test results: Two long edges clamped, two short edges free It is well-known that the development of post-buckling shear strength in the webs of plate girders requires a fully anchored tension field. It appears evident that, here, the arrangement shown in Figure 4 was not sufficiently strong for this tension field to be fully activated. However, this injects a certain realism into the study and it may be concluded that the full theoretical post-bucking strength cannot be relied upon, even under the "idealised" fully fixed conditions of the tests. However, some postbuckling strength may still be available under more realistic boundary conditions approximating to practice. This, of course, requires further investigation. For the above reasons, although the ultimate load conditions are included for completeness in Tables

13 2 and 3, which summarise the test results and the theoretical comparisons for panels of length 2400 mm and width 400 mm, no detailed consideration is given tO the theoretical ultimate loads and discussion is confined to the buckling values.

Foam

Per (test) (kN)

Per (FE) (kN)

Difference

(%)

Pu~t(test) (kN)

Puk (FE) (kN)

0.73 0.82 0.95 1.02 1.20 1.27

13.20 19.67

13.54 19.19 29.84 36.94 60.14 71.28

+2.58 -2.44

16.0 22.9

-0.43

41.0

37.2 50.3 78.1 89.6 155 176

0.73 0.82 0.95 1.02 1.20 1.27

34.4

Thickness

(mm) None

SDVB

HDN

250f

t 37.1

t 69.8

t t 60.2 84.1

t

0.82 0.95 1.20 1.27

49.6

0.82 0.95 1.20 1.27

92.4

t 90.1

t t 156.9

t

33.47 39.67 50.94 58.30 82.04 93.35 47.44 59.06 90.64 102.1 113.5 129.3 166.6 179.3

t t

t

+2.12

100

-2.70

37.0

t t

t t

-3.16 -2.45

71.2 107

t

t

-4.35

56.0

t

t

+0.60

112

t

t

+22.8

100

t

t

+6.16

161

t

t

92.0 109 140 160 226 256 130 162 249 280 302 336 448 483

Table 3" Test results: All four edges clamped Evidently, the finite element predictions of the buckling load are adequate for all practical purposes. This method, based on the critical buckling load, therefore provides the basis for an accurate design procedure when more realistic boundary conditions are introduced in the next section of this paper.

FINITE ELEMENT ANALYSIS FOR REALISTIC BOUNDARY CONDITIONS Having effectively calibrated the finite element analysis for idealised, clamped longitudinal boundary conditions, the next step is to apply this analysis to more realistic boundary conditions approximating to those likely to arise in practice in a cassette wall or liner tray roof. A number of different boundary conditions were investigated before the one shown in Figure 8 was adopted for a detailed parametric study. This attempts to model the interaction between adjacent trays together with the usual seam fasteners placed somewhere between the middle of the web and the wide upper flange. Thus, at the middle of the web, complete positional and rotational restraint was applied. At the top of the web, only restraint in the transverse x-direction was applied while, at the bottom of the web, the corner had complete positional restraint though no rotational restraint. From the point of view of

14 initial buckling, it is the rotational restraint to the wide flange that is the most critical and, here, this is provided by the remainder of the section. The assumption of x-direction restraint only at the junction between the web and the wide flange may be less conservative when post-buckling is considered.

Figure 8: Boundary conditions used in the parametric study For the purposes of parametric study, the material properties were assumed to be identical to those used previously. The study covered empty cassettes and all three foam insulation materials together with the range of geometries defined in Figure 9. This represents a huge number of analyses and it is not appropriate to provide the detailed results here.

Figure 9: Range of geometries used in the parametric study Figure 10 shows a typical load deflection curve obtained from the full Newton-Raphson non-linear analysis of an HDN-filled cassette of length 1600 mm, width 600 mm and 1.0 mm metal thickness. The vertical axis labelled "time" is in effect the load axis as the value shown multiplied by the load in each sub-step gives the applied load. Evidently the buckling load is clearly defined and, because there was a small but significant difference between the critical buckling loads obtained from the full analysis and those obtained using an eigenvalue bifurcation analysis, the critical buckling loads deduced from the full non-linear analysis were used for the detailed study which follows. The results of many such analyses were processed and it was found that the following equation gave an accurate account of all of the buckling loads obtained: ~=

=

n2D 10 b2t

Ec Ec + 2.9-- - - t at

where

D

=

Et 3 12(I - v 2)

(6)

In Equation 6, the stiffening effect of the foam is described in terms of the compressive modulus Er of the insulant material. Various other core parameters were investigated and this was found to give the most consistent results. A typical comparison between the results given by the above equation and

15 those given by finite element analysis is shown in Figure 11 for 400 mm wide cassettes of different length stiffened by SDVB foam. The cases of empty cassettes and other foams are equally accurate over the range of widths considered. 00

h

700-~.

300 200

-

q~

O, 0,00

!

"|

0,20

I

0,40

9

0,60

'

0,80

1,00

~s~acemom (mm)

Figure 10: Typical load-deflection curve from non-linear analysis

150

r

11o

9

I

~0

-"

1200ram

Finite element

I-- . . . . . .

,38 43 ..... ~

C -2400mm ~

.-. 100 ~

2400ram./

- - I I - - 1600ram

C

-:~,

80

1600ram ~ - 1200ram ~

Equation 6 68,95 ~ ' ~

-

~" 60 40

20 0 0,5

0,7

0,9

11

13

15

1L7

T h i c k n e s s (It-m)

Figure 11" Critical buckling loads from finite element analysis and Equation 6

Equation 6 is clearly suitable for the practical design of infilled cassettes.

ENERGY METHOD OF ANALYSIS The fundamental differential equations for the shear buckling of thin metal plates have been given by Timoshenko and Gere (1961) and these equations have been used by other researchers. The general procedure for plates with simply supported edges is to use a double sine series displacement function:

16

w

= m~| m=ln=l

a

n~tYsm~ " b

(7)

where the x-coordinate is valid between 0 and a and the y-coordinate is valid between 0 and b. Using these equations, it is possible to determine expressions for the strain energy of bending of the buckled metal skin of the plate and the potential energy lost by the load. These equations contain a potentially infinite number of terms in which the unknowns are the constants am and, in order to obtain a satisfactory approximate solution, it is necessary to choose how many of these to use. Thus, for only two terms with constants al~ and a22, the error is approximately 15% and increases as the aspect ratio a/b increases. Increasing the number of terms increases the accuracy but the general characteristic that more terms are needed for higher aspect ratios persists. Fragos (2000) has shown that it is possible to add into the above procedure additional terms which represent the stiffening effect of the core. This is at the cost of considerable complexity in the equations so that the full analysis cannot be presented here. However, the results are encouraging. For a square panel of side 1 metre and metal thickness 8 mm stiffened by 100 mm of foam of shear modulus Ge = 5 N/mm 2 and compression modulus E e = 2.5 N/mm 2, the following results were obtained: Equation 6: Energy method with 2 terms: Energy method with 5 terms:

Per = 6.60 kN Per = 6.46 kN Per = 6.62 kN

However, as the aspect ratio a/b is increased, the accuracy drops quite quickly and it is necessary to use more terms. This has yet to be attempted.

CONCLUSIONS The study reported in this paper is of practical interest. Cassette wall construction has been used for more than 20 actual projects and some of these have used rigid foam infilling, albeit to improve the thermal performance, "feel" and resistance to damage rather than to improve the structural performance. However, as far as the authors are aware, it is also a relatively new research topic and no other investigations of the infilled case in particular are known. This has led to a number of fundamental considerations which have had to be addressed in order to allow the research to proceed and these are addressed in this paper. The authors consider that these have wider implications than to the specific and relatively narrow topic of the paper: The first author has believed for a long time that thin-walled metal members and structures act at their best when they are used in conjunction with other materials. The sandwich panel and the composite floor deck provide good examples of this principle. The rapid expansion in the usage of these components in recent years provides excellent proof that this principle is sound. The significant increase in the shear buckling strength occasioned by infilling cassettes with rigid foam insulant offers a further illustration of this principle. Although not part of this paper, it is clear that this foam infilling is also beneficial with respect to axial and bending loads. The use of testing as a research and development tool has declined dramatically in recent years. The reason for this is clear - testing is expensive and you can solve most problems more

17 cheaply using finite element analysis. You also obtain more quality information. However, there are still some problems which require the use of testing and the problem addressed in this paper is one of these. Although the initial shear buckling load of a cassette is not particularly sensitive to the boundary conditions, the post-buckling behaviour clearly is. Indeed, further testing and analysis is required before any practical appeal can be made to the post-buckling strength that is clearly present in cassette wall and roof construction. As expected, finite element analysis is directly relevant to the problem in question and an extensive parametric study provided the only practical way to obtain useful design equations. However, it is the authors' experience that, when addressing a new situation, most researchers find it difficult to get the boundary conditions right first time. In some cases, the appropriate boundary conditions can only be chosen with certainty after comparison with test results. Here, further research is required before the results given by the non-linear, large deflection analysis can be used with confidence. Direct solution of the governing differential equations is rarely useful in the design of thinwalled structural elements. The main exceptions to this general rule are in global (lateral and lateral-torsional) buckling problems and some simple cases of local buckling. Here, a series solution for local shear buckling in the presence of rigid foam stiffening appears to be possible though the complexity is such as to render it of questionable value in comparison to the results of parametric studies using the finite element method. From the practical point of view, the main outcome of the study reported in this paper is Equation 6 which is offered as a suitable design equation for the shear buckling of empty and infilled cassettes.

REFERENCES

Baehre R. (1987) "Zur Shubfeldwirkung und-bemessung von Kassettenkonstructionen" (On the behaviour and design of cassette assemblies in shear), Stahlbau 7, pp 197-202. CIB (1995) "Preliminary European Recommendations for sandwich panels with additional recommendations for panels with mineral wool core material: Part 1: Design", International Council for Building Research, Studies and Documentation, Publication No 148, Reprinted November 1995. Davies J. M. and Bryan E. R. (1981) "Manual of stressed skin diaphragm design", Granada. Davies J. M. and Dewhurst D. W. (1997) "The shear behaviour of thin-walled cassette sections infilled by rigid insulation", Proc. lnt. Conf. on Experimental Model Research and Testing of Thin-Walled Structures, Academy of Sciences of the Czech Republic, Prague, Sept., pp 209-216. Davies J. M. (1998a) "Light gauge steel cassette wall construction", Nordic Steel Construction Conference 98, Bergen, Sept. 14-16, pp 427-440. Davies J. M. (1998b) "Light gauge steel framing for house construction", 2nd Int. conf. on ThinWalled Structures, Singapore, pp 17-28. Davies J. M. (2000a) "Steel framed house construction", The Structural Engineer, Vol. 78, No. 6, 21 March, pp 17-24.

18 Davies J. M. (2000b) "Recent research advances in cold-formed steel structures", J Constructional Steel Research, Vol. 55, Nos 1-3, July-September, pp 267-288. ECCS (1995) "European recommendations for the application of metal sheeting acting as a diaphragm", European Convention for Constructional Steelwork Publication No. 88. Eurocode 3 (1996): Design of Steel Structures- Part 1.3 : General rules- Supplementary rules for cold formed thin gauge members and sheeting, CEN ENV 1993-1-3, February 1996. Fragos A.S. (2000) "The shear buckling of metal plates and empty and filled C-shaped sections", PhD Thesis, University of Manchester, November 2000. Southwell and Skan (1924) "On the stability under shearing forces of a flat elastic strip", Proc. Royal Society, A, Vol. 105, No. 733. Timoshenko S. P. and Gere J. M. (1961) "Theory of elastic stability", McGraw Hill, New York.

Third InternationalConferenceon Thin-WaUedStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rightsreserved

19

STABILITY AND DUCTILITY OF THIN HIGH STRENGTH G550 STEEL MEMBERS AND CONNECTIONS C.A. Rogers2, D.Yang I and G.J. Hancock 1 IDepartment of Civil Engineering, University of Sydney NSW, Australia, 2006 2Department of Civil Engineering and Applied Mechanics, McGill University Montreal, Quebec, Canada,H3A 2K6

ABSTRACT High strength cold-reduced steel is typically of stress grade G550 (550 MPa nominal yield and tensile strength) and less than 1 mm thick. The steel has been used for many years for sheeting and decking but is now being used for structural members such as roof trusses and stud walls of steel framed houses. The paper summarises a major research program on the stability and ductility of this steel which has been proceeding for several years at the University of Sydney. The paper relates the Sydney research to the work of others being undertaken around the world. KEYWORDS High strength steel, stability, ductility, structural members, structural connections INTRODUCTION

G550 sheet steels (SA, 1993) are manufactured by cold reducing mild sheet steels 0or = 300 MPa) to a thickness which ranges from 0.42 mm to 1.0 mm. Cold reduction produces large deformations of the grain structure which cause an increase in yield stress and ultimate strength and a decrease in ductility. G550 sheet steels (0.48 mm _< t _< 0.75 mm) have recently been introduced for use as structural members in the Australian residential construction industry (Hancock and Murray, 1996), as well as for panel and deck sections in other types of building construction. The use of G550 sheet steels has been restricted to non-structural applications in most countries due to concerns regarding the ductility of members and connections, and possible problems with stability due to low strain hardening. Cold formed structural members are fabricated from sheet steels consisting of various material properties which must meet requirements prescribed in applicable design standards. The Australian/New Zealand Design Standard (SA/NZS, 1996) allows for the use of thin (t < 0.9 mm), high strength (fy = 550 MPa) sheet steels in all structural sections. However, due to the lack of ductility exhibited by sheet steels that are cold reduced in thickness, the engineer must use a yield stress and ultimate strength reduced to 75% of the minimum specified values. The American Iron and Steel Institute (AISI) Design Specification (1997) further limits the use of thin, high strength steels to roofing, siding and floor decking panels. Sheet steels are required to have minimum elongation

20 capability to ensure that members and connections can undergo small displacements without a loss in structural performance, and to reduce the harmful effects of stress concentrations. The ductility criterion specified in the Australian/New Zealand and North American (CSA, 1994; AISI, 1997) Design Standards is based on an investigation of sheet steels by Dhalla and Winter (1974a, b), which did not include the thin high strength G550 steels available today. However, Dhalla and Winter did test commercially available low ductility steel (A446 Grade E (1968) in 0.965 mm thickness- Type Z) which did not satisfy their elongation requirements, but was able to fully plastify for perforated longitudinal test specimens and reach 94% of the net cross-section tensile strength for transverse specimens. Early research by McAdam et al. (1988) on steels with a low fu/fy ratio led to the use of full strength design in Clause A3.3.1 of the AISI Specification for purlins and girts which are principally flexural members. The design of high strength steel compression members was not permitted. However, Daudet and Klippstein (1994) at Dietrich Industries Indiana included work on the stub column strength of sections used for stud walls in residential construction. The material studied had a tensile to yield stress ratio close to 1.0 rather than 1.08 as required by Section A3.3.1 of the AISI Specification. The study showed that steels having an fu/fy as low as 1.01 and elongations as low as 3 percent can be conservatively designed in compression using a reduced yield stress of 0.7fy. A series of research projects on profiled steel roof claddings constructed from G550 steel has been performed by Mahendran at Queensland University of Technology over a period of more than 10 years. These studies have included static behaviour of corrugated roofing under simulated wind uplift (Mahendran 1990), behaviour and design of crest-fixed profiled steel roof claddings under.wind forces (Mahendran, 1994) and pull-over strength of trapezoidal steel roof and wall claddings (Tang and Mahendran, 1999). Design formulae for the pull-over strength of crest fixed cladding have been developed which are directly applicable to G550 steel design. Recent research by Rogers and Hancock at the University of Sydney has included ductility of thin G550 sheet steels in tension including perforations (Rogers and Hancock, 1997), bolted connections in G550 sheet steels (Rogers and Hancock, 1998a), screwed connections in G550 sheet steels (Rogers and Hancock, 1999) and fracture in G550 sheet steel (Rogers and Hancock, 2001). This paper summarises the more important findings and proposed design rules from these studies. Research at the University of Missouri-Rolla by Wu, Yu and LaBoube (1996a, 1996b) on decking sections in flexure composed of ASTM A653 (1997) Structural Grade 80 steel has resulted in an exception clause being added to the latest update of the AISI Specification (AISI, 2000). This exception clause permits a reduced yield point (Rbfy) for computing the nominal flexural strength of multiple web configurations. The reduced yield point depends on the plate slenderness but is generally greater than the 75% discussed previously. Investigations by (Wu, Yu and LaBoube,1997, 1998) of these steels revealed that the full specified minimum yield point can be used to determine web crippling strength and shear strength. However, the latest update to the AISI Specification (AISI, 2000) still requires the reduction in the yield stress to 0.75fy. The compression stability of G550/Grade E steels with low strain hardening has only been investigated in the context of decking sections in bending. However, a research program at the University of Sydney on G550 steel in compression has recently commenced. Test specimens include square box type sections with glued and screwed comer connections. The results of these tests are being compared in the first instance with the RbFyapproach recently adopted in the AISI Specification. The preliminary results of this project are included in this paper.

21 MANUFACTURING PROCESS OF G550 SHEET STEEL G550 Sheet Steels

The usual process for the production of these steels is cold reduction. This process can be used to increase the strength and hardness, as well as to form an accurate thickness for sheet steels and other steel products. Initially the sheet steels are rolled to size in a hot strip mill with finishing and coiling temperatures of approximately 940~ and 670~ respectively. The hot worked coil of steel, typically 2.5 mm in thickness with a minimum specified 300 MPa yield stress, is uncoiled and cleaned in an acid solution to remove surface oxides and scale. The uncoiled strip is then trimmed to size and fed into a cold reduction mill, which may contain any number of stands. High compressive force in the stands and strip tension systematically reduce the thickness of the steel sheet until the desired dimension is reached, e.g. by approximately 85% for the 0.42 mm sheet steels. The milling process causes the grain structure of cold reduced steels to elongate in the rolling direction, which produces a directional increase in the material strength and a decrease in the material ductility. The effects of cold working are cumulative, i.e. grain distortion increases with further cold working as a result of an increase in total dislocation density, however, it is possible to change the distorted grain structure and to control the steel properties through subsequent heat treatment. Various types of heat treatment exist and are used for different steel products. G550 sheet steels are stress relief annealed, although recrystallisation does not occur. Stress relief annealing involves heating the steel to below the recrystallisation temperature, holding the steel until the temperature is constant throughout its thickness, then cooling slowly. Mild sheet steels of similar thicknesses are annealed to a greater extent in comparison with G550 sheet steels, and hence recover their ductile behaviour. Annealing is carried out in a hot dip coating line prior to application of either a zinc or aluminum/zinc coating. Upon final cooling the sheet steel is further processed through a tension levelling mill, e.g. 0.35% extension, to improve the finish quality and the flatness of the coil. The cold-reduced G550 sheet must be differentiated from other sheet steels whose high yield stress and ultimate strength values are obtained by means of an alloying process, i.e. high strength low alloy (HSLA) steels. The material property requirements for G550 or Grade E sheet steels are specified in Australia by AS 1397 (1993) and in North America by the following ASTM Standards; A611, A653, A792 and A875. Material property specifications for HSLA sheet steels can also be found in ASTM Standard A653. G550 DUCTILITY STUDIES

A recent study of G550 steel to AS 1397 in 0.42 mm and 0.60 mm thicknesses was performed by Rogers and Hancock (1997) to ascertain the ductility of G550 steel, and to investigate the validity of Clause 1.5.1.5(b) of AS/NZS 4600 and Section A3.3.2 of the AISI Specification. Three representative elongation distributions are shown in Fig. 1 for unperforated tensile coupons taken from 0.42 mm G550 steel in (a) the longitudinal rolling direction of the steel strip, (b) the transverse direction and (c) the diagonal direction. The coupons were each taken to fracture. In general, G550 coated longitudinal specimens have constant uniform elongation for each 2.5 mm gauge with an increase in percent elongation at the gauge in which fracture occurs as shown in Fig. 1(a). Transverse G550 specimens show almost no uniform elongation, but do have limited elongation at the fracture as shown in Fig. l(b). The diagonal test specimen results in Fig. 1(c) indicate that uniform elongation is limited outside of a 12.5 mm zone around the fracture while local elongation occurs in the fracture gauge as well as in the adjoining gauges. These results show that the G550 steel ductility is dependent upon the

22

direction from which the tensile coupons are obtained. The steel does not meet the Dhalla and Winter requirements described above except for uniform elongation in the longitudinal direction. Rogers and Hancock (1997) also investigated the tensile capacity of perforated coupons to determine whether the full fracture capacity could be developed at the net section. Coupons were taken in the longitudinal, transverse and diagonal directions as described in the preceding paragraph. Circular, square and diamond shaped perforations of varying sizes were placed in the coupons. It was found that despite the low values of elongation measured for this steel, the load carrying capacity of G550 steel as measured in concentrically loaded perforated tensile coupons can be adequately predicted using existing limit states design procedures based on net section fracture without the need to limit the tensile strength to 75% of 550 MPa as specified in Clause 1.5.1.5(b) of AS/NZS 4600 or Section A3.3.2 of the AISI Specification. 40.0 35.0

30.0 1

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Fig. 1 Elongation of 0.42 mm G550 Steel

23 BOLTED CONNECTION STUDIES

Background Recent research has been performed at the University of Sydney by Rogers and Hancock (1998a) on bolted connections of thin G550 and G300 sheet steels in 0.42 mm and 0.60 mm base metal thickness. This is the same steel as was used for the ductility studies described above. The test results indicate that the connection provisions in AS/NZS 4600 and the AISI Specification cannot be used to accurately predict the failure modes of bolted connections from thin G550 and G300 steels. Furthermore, the design rules cannot be used to accurately determine the bearing resistance of bolted test specimens based on a failure criterion for predicted loads.

Proposed Design Provisionsfor Bolted Connections Significantly unconservative predictions of the load resistance obtained for certain bolted connection test specimens have demonstrated a need for a gradated bearing coefficient which is dependent on the stability of the edge of the bolt hole. Unconservative predictions of connection bearing capacity have been recorded for bolted test specimens where thin sheet steels are connected and loaded in shear, as shown for the failure based criterion test-to-predicted results calculated using the AS/NZS 4600, AISI and Eurocode design standards for the 0.42 mm G550, 0.60ram G550, 0.60 mm (3300 test specimens, as well as the 0.80 mm G550 and 1.0 mm G550 test specimens (which all failed in bearing) (Rogers and Hancock, 1998a). A proposed method to accommodate for the change in bearing behaviour, that relies on the ratio of bolt diameter to sheet thickness, d/t, summarised here. This proposed method includes the gross yielding, Eq. 1, and the net section fracture, Eq. 2, failure provisions that are contained in the CAN/CSA-S136 design standard (1994), i.e. no stress reduction factor is used. Calculation of the end pull-out resistance follows the procedure given in the Eurocode design standard, Eq. 3. The recommended equations for gross yielding failure, net section fracture and end pull-out failure are as follows: Nt = Agfy

(I)

where Ag is the area of the gross cross-section andfy is the yield or 0.2% proof stress. Nt = Anfu

(2)

where An is the area of the net cross-section andfu is the ultimate strength. Vf= t ef. / 1.2

(3)

where t is the thickness of the thinnest connected part and e is the distance measured parallel to the direction of applied force from the centre of a standard hole to the nearest edge of an adjacent hole or to the end of the connected part. Modification of the existing bolted connection design provisions was made to the beating formulation. Beating stress ratios, fbu /fu, for some of the bolted connection test specimens that failed by bearing are illustrated in Figure 2. The bearing stress ratios for these specimem decrease as the thickness decreases, hence, a formulation to calculate a bearing coefficient, which is similar to that recommended in the CAN/CSA-S136 design standard, has been proposed and is shown in Fig. 3.

24 4.0 t

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Figure 2 Bearing Stress Ratios for Various Sheet Steel (Full fu used)

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dlt Figure 3 Existing and Proposed Bearing Coefficients for Bolted Connections SCREWED CONNECTION STUDIES

Background Investigations of single overlap connections in G550 steel concentrically loaded in shear have been reported by Rogers and Hancock (1999). The steel ranged in thickness from 0.42 mm to 1.00 mm, and the type, number and placement of screws were varied.

Proposed Design Provisionsfor Screwed Connections A proposed method to accommodate for the change in bearing behaviour with sheet steel thickness, which relies on the ratio of screw diameter to sheet thickness, d/t, was presented in Rogers and Hancock (1999). Unconservative predictions of connection bearing capacity have demonstrated a need for a gradated bearing coefficient which is dependent on the stability of the edge of the screw hole in a similar manner to the bolted connections described above. These unsatisfactory results have been recorded for test specimens where two different thickness sheet steels are connected and loaded in shear. The screwed connection test specimens that have two elements of the same thickness all failed in a combined bearing/tilting mode and have acceptable test-to-predicted ratios. Macindoe and Pham (1996) tested a number of screwed connections where bearing failure was forced to occur because of a large differential in the thickness of the connected sheets. This behaviour differs from that exhibited for the majority of the connections that were tested for by Rogers and Hancock, and by Macindoe and Pham, where failure was caused by a combination of beating and tilting. The connection resistance that was calculated for the screwed connection tests where bearing/tilting failure occurred is

25 reasonably accurate. Hence, the proposed method includes the tilting formulation that is specified in both the AS/NZS 4600 (1996) and AISI (1997) design standards. It also includes the gross yielding (1) and net section fracture (2) failure provisions described for bolts in the preceding sections i.e. no stress reduction factor is used. The material properties for thin G550 sheet steels are not reduced by the 0.75 factor. At present, the bearing coefficient that is contained in the AS/NZS-4600 (1996) and AISI (1997a) design standards is a constant 2.7 for screw connections as shown in Fig. 4. The CAN/CSA-S136 design standard (1994) requires that the bearing coefficient vary depending on the ratio of d/t, as shown in Fig. 4. The proposed method contains a gradated bearing coefficient which is also dependent on d/t, however, the maximum allowed value is lowered to 2.7 and the rate of change of the bearing coefficient is modified accordingly.

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- A S / N Z S 4600, A I S I

(1.5 0.0 0

3

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9

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d/t Figure 4 Proposed bearing coefficient for screwed connections

FRACTURE STUDIES

Background Test results from Rogers and Hancock (1997) reveal that the ability of G550 sheet steels to undergo deformation is dependent on the direction of load within the plane of the sheet, where transverse specimens exhibit the least amount of overall, local and uniform elongation. Typically, transverse specimens fail immediately after exiting the linear elastic phase of deformation with minimal plastification of the net section or the gauge length. The fracture resistance of the material, which in reality contains microscopic cracks, may have influenced this behaviour. The G550 sheet steels that were tested do not meet the Dhalla and Winter (1974b) elongation and ultimate strength to yield stress ratio requirements regardless of direction, except for the uniform elongation of longitudinal coupon specimens. The paper on fracture toughness of G550 sheet steels subject to tension by Rogers and Hancock (2001) reports on the fracture resistance properties of G550 sheet steels that were tested in tension. The fracture resistance of the G550 sheet steels was measured for a range of temperatures, and a numerical study of the effect of cracks on structural performance in the elastic deformation range was completed using the FRANC2D 1995) finite element computer program. A summary of the results from this paper are included here.

26

Measurement of Critical Stress Intensity Factors Stress distribution in a loaded member is greatly affected by the presence of cracks or discontinuities. The classical structural mechanics approach deals with these matters by a numerical multiplier referred to as a stress concentration factor, which can be thought of as the increase in stress caused by a change in geometry such as a notch. Fracture mechanics, however, recognises that the stress intensity at the tip of the crack can be expressed as a stress intensity factor, K, as follows, K = o',pp~

(4)

where travp is the nominal stress applied to the member and a is the size of the crack. Thus K can grow as crsop or a or simply the product of these grows. However, K cannot grow to a value larger than the fracture toughness of the material, Kc, which is the critical stress intensity factor at the tip of a crack and is def'med as follows,

IL = 4Eao

(5)

where E is the elastic constant of the material (Young's Modulus) and Gc is the toughness of the material. As the stress intensity factor at the tip of the crack, K, increases with increased loading, it may reach the value of Kc, when the balance of elastic energy release from the loaded body exceeds the energy requirement for crack extension. At this point a running crack that is known as unstable fracture takes place. The stress intensity factor, K, at the tip of the crack should be kept at a value less than the characteristic Kc of the material under investigation if unstable fracture of the structure is to be avoided. This is analogous to the requirement that the cross-sectional stress, f, must lie below fy if one does not want yielding to occur. A total of 30 notch specimens were tested using the same steels as for the ductility, bolted connection and screwed connection tests described above. The main objective of this experimental testing phase was to determine the critical Mode I, i.e. crack opening as opposed to crack sliding (Mode II) or crack tearing (Mode III), stress intensity factors, Ke, for 0.42 and 0.60 mm G550 sheet steels. Tests were completed to measure the magnitude of the crack tip stress field where ultimate failure was caused by unstable fracture of the notch specimens. All of the test pieces within a material and thickness type were cut from the same sheet, although similar specimens were cut from various locations to avoid localised material properties. The material properties of cold reduced sheet steels have been shown to be anisotropic, hence, specimens were cut from three directions within the sheet; longitudinal, transverse and diagonal with respect to the rolling direction. Critical stress intensity factors were calculated for all of the notch test specimens. Of the 30 notch tests, 18 were completed at a temperature of 21.5~ (room temperature) and the remaining at temperatures which varied from I~ to -21~ The room temperature mean value test results are provided in Table 1, and detailed information for each individual notch specimen can be found in Rogers and Hancock (1998b). All of the sheet steel types that were tested have critical stress intensity factors that exceed 3000 MNm "3r2. A significant decrease in the fracture toughness of the G550 sheet steels is not evident for the transverse direction in comparison to the longitudinal and diagonal directions. However, the transverse Kc values do fall below the longitudinal and diagonal values for both the 0.42 mm and 0.60 mm G550 sheet steels. The measured Kr values are atypically high partially because of the thinness of the G550 sheet steels, which did not allow for plane strain conditions to occur during testing. This does not mean that the measured Kc values are incorrect, but that these values are only valid for the thicknesses tested. Ashby (1981) associates the failure of materials that are found to have Ke values in the range measured with the plastic rupture ductile fracture failure mode.

27 No significant variation in the measured critical stress intensity factors of the G550 sheet steels was observed for the range of temperatures used in testing, i.e. 21.5~ to -21 ~ (see Rogers and Hancock (1998b)). This is an indication that the transition temperature from ductile to brittle fracture behaviour of the G550 sheet steels that were tested lies below the range of temperatures used. Table 1: Mean Measured Kc Values at Room Temperature (21.5~

Material Type & Direction

K~ 3/2

(MNm")

0.42ram G550 Long. 0.42ram G550 Tran. 0.42mm G550 Diag. 0.60ram G550 Long. 0.60mm G550 Tran. 0.60ram G550 Diag. IIIII

3767 3182 3748 3551 3260 3743 I

Evaluation of the Critical Stress Intensity Factorsfor Perforated Coupon Specimens An analytical study of cracked specimens fabricated from G550 sheet steels was completed to determine the design implications of possible failure by unstable fracture in the elastic deformation range. Critical stress intensity factors were computed using a finite element model, and then were compared with the measured critical stress intensity factors obtained from tests. The FRANC2D (1995) finite element computer program, distributed and written by the Comell University Fracture Group, was used because it had been specifically developed for the analysis of crack behaviour. The FRANC2D program allows the user to easily define element meshes at crack tips, extend cracks in any direction and remesh the crack area, as well as calculate stress intensity factors at the crack tip. In this analytical study, it was assumed that for the modelled test specimens rapid unstable fracture resulting in ultimate failure would occur when the stress intensity at the prescribed crack tip reached the critical measured I~ value. This is a conservative assumption that is dependent on the following; 1) once the specimen has reached its ultimate load carrying capacity the maximum load does not decrease as the crack size increases, hence rapid fracture of the specimen is not abated, and 2) that loading occurs over a short period of time so that stable crack growth does not occur prior to ultimate failure, i.e. the length of the fatigue crack is not extended by any further crack growth except at ultimate failure. All of the elements used in the finite element models had elastic-isotropic material properties that were defined using the results obtained from the coupon tests of G550 sheet steels (Rogers and Hancock, 1997). Numerical analyses of representative coupon models, with 1, 2, 5 and 7 mm circular, as well as 5 mm diamond and square perforations, for the two G550 sheet steel types were completed using the FRANC2D (1995) computer program. The nominal size of the previously tested perforated coupons was used (see Rogers and Hancock (1997)). Additional 0.25 mm long non-cohesive cracks were placed on either side of the perforation at the position of the highest stress concentration, perpendicular to the direction of load. The applied ultimate edge stress was calculated based on the assumption that the average stress over the net section, which was calculated using only the perforation dimension and not the crack length, would equal the yield stress of the material (see Rogers and Hancock' 1997). The loaded and boundary edges of the coupon models were defined as found for the notch specimen finite element model.

28 Numerical analysis, using the FRANC2D computer program, of the perforated coupons with additional edge cracks revealed that the applied stress intensity factors, Kapp, did not reach a critical level, Kc, as defined by the tested notch specimens (see Figures 5 and 6 for 0.42 mm G550 steel). These results indicate that failure of the real perforated G550 sheet steel coupon specimens (Rogers and Hancock, 1997) can be attributed to yielding and ultimate rupture of the material at the net section, and not unstable fracture in the elastic deformation range. The results also show that the sharp geometry of a diamond perforation and the presence of a small crack will cause the largest applied stress intensity factors.

Figure 5 Perforated Coupons Kapp versus K~ (0.42 mm G550 Longitudinal)

Figure 6 Perforated Coupons Kapp versus I~ (0.42 mm G550 Transverse)

Evaluation of the Critical Stress Intensity Factorsfor Triple Bolted Specimens

The longitudinal and transverse triple bolt G550 sheet steel connections that are documented in Rogers and Hancock (1998), and which failed through rupture of the net section, were also analysed using the FRANC2D (1995) computer program. For each bolted connection, additional 0.5 mm long noncohesive cracks were placed on either side of the perforation of the innermost bolt hole at the position of the highest stress concentration, perpendicular to the direction of load. Distributed loads were

29 applied to all of the bolt holes along the edge where the bolt and sheet steel were in contact. The boundary edge was defined as previously noted for the notch Specimen finite element model. Analysis of the bolted connection specimens using the FRANC2D computer program revealed that the applied stress intensity factors, Kapp, did not reach a critical level, Kr as defined by the notch specimens that were tested (see Figure 7). These results indicate that failure of the triple bolt connection G550 sheet steel specimens can be attributed to yielding and ultimate rupture of the material at the net section and not unstable fracture in the elastic deformation range. However, the applied stress intensity factors have increased in comparison with the values obtained in the analysis of the perforated coupons. This increase can be attributed to the greater width of the test piece, greater crack length and more localised load distribution. It is most noticeable for the transverse 0.42 mm G550 test specimens where the applied stress intensity factors fall just short of reaching the critical level. Further increases in the crack length would result in elevated applied stress intensity factors and ultimately, unstable fracture of the test specimens in the elastic deformation range.

Figure 7 Triple Bolt Connections Kapp versus Kr

Conclusions on Fracture o f G550 Sheet Steels The Mode I critical stress intensity factor, Kc, for the 0.42 mm G550 and 0.60 mm G550 sheet steels was determined for three directions in the plane of the sheet. Single notch test specimens with fatigue cracks were loaded in tension to determine the resistance of G550 sheet steels to failure by unstable fracture in the elastic deformation range. The critical stress intensity factors obtained by testing were then used in a finite element study to determine the risk of unstable fracture in the elastic deformation zone for a range of different G550 sheet steel structural models. It was determined that the perforated coupon and bolted connection specimens that were previously tested for this research project were not at risk of failure by unstable fracture in the elastic zone. Notch specimens that were tested at different temperatures (1 to -21~ did not provide crack resistance values that significantly varied from those tested at room temperature (21.5~ These results are an indication that the ductile-brittle transition temperature lies below the range of temperatures used in testing.

30 COMPRESSION STUDIES

Background and Geometry of Sections Tested The tests by Wu, Yu and LaBoube (1996a, 1996b) discussed in the introduction to this paper were performed on decking sections in bending and so the stiffened flange elements in compression were supported by 2 webs in bending. In order to produce a section with all elements in compression, box sections were formed from 0.6 mm G550 sheet steel at the University of Sydney. The same steel as used for the earlier ductility tests was used. It was not possible to form a complete box without some type of connection at at least one of the comers. Consequently, boxes were formed from two pieces brake-pressed then connected at two of the comers as shown in Fig. 8. Epoxy was used between the sheet steel sections at the comers. These sections were designed to be as close to a true square hollow section as possible without any welding which would have produced considerable distortion of the very thin sections. The sections ranged from 20 mm to 100 mm fiat widths. The b/t ratios therefore varied from 33.3 to 167. The unstiffened lip elements were kept at 7.5 mm long for all section sizes. The local buckling stresses varied from approximately 30 MPa for the 100 mm section to 710 MPa for the 20 mrn section. The effect of the double thickness comer (lip) elements was to increase the theoretical local buckling coefficients to approximately 4.5 due to slight torsional restraint at 2 of the comers.

Figure 8 G550 Box Column Section- Type 1

Test Results and Comparisons with Design Standards The sections were tested in an MTS Sintech 300 kN testing machine with machined end plates. Pattemstone was used between the end plate located against the top of the specimen and the testing machine top platen to ensure even bearing on the specimens. The first few specimens were found to fracture in the epoxy at loads very close to the ultimate load. It was therefore decided to drill 3 mm holes in the comer lips and to place small diameter bolts and nuts to ensure that the comers did not come apart. Initially, these bolts were only located at the ends, but subsequent testing resulted in increased numbers of bolts with the bolt spacing at approximately 20 mm then 10 ram. A photograph of a box section with the small bolts after test and showing the local buckles is shown in Fig. 9. The measured yield stress of the steel was 711 MPa.

Figure 9 G550 Box Section after Test

The test results have been plotted in Fig. 10 versus the theoretical stub column test strengths computed using AS/NZS 4600 (his) and the AISI Specification (Pn). In Fig. 10, the areas of the holes have not been allowed for. The test results with a square are those for the holes at approximately 20 mm spacing, and those with the crosses are for the tests with the 10 mm hole spacing. It is clear that decreasing the hole spacing decreases the load. Obviously the holes are having an effect. All results lie below the theoretical strength based on R~fy as included in Section A3.3.2 of the latest addendum to the AISI Specification (2000). The results of the test have been plotted in Fig. 11 with the areas of the holes removed from the effective area on the assumption that the lips are fully effective and the holes can simply be removed. In this case, the majority of the test results lie above the revised AISI Section A3.3.2 curve. It is also interesting to note that for more slender sections, the test results are closer to the theoretical strengths assuming no reductions in the yield stress. This is to be expected since the effect of no strain hardening is likely to have a lesser effect for sections which are slender and fail principally by post-local instability and then yielding rather than inelastic local buckling. For the more stocky sections, the results are closer to the revised AISI strengths. At the time of writing this paper, further tests were underway for sections without holes and bolts.

32

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200

Figure 11 Comparison of G550 Stub Box Compression Tests with Design Standards (holes included in calculations)

33 CONCLUSIONS The use of cold reduced G550 coated sheet steel is becoming more common in the construction industry for structural members, particularly in residential construction. A wide range of research projects has been performed mainly at the University of Sydney and the University of Missouri-Rolla to ascertain the strength of these steels when used in load beating applications. The areas of members in compression, bending and tension, and screwed and bolted connections have been investigated. Fracture toughness has been investigated where it has been shown that all specimens and connections tested to date have failed by ductile failure rather than sudden fracture. Further research is required in the area of compression members to ascertain whether sections composed of G550 sheet steels can carry their design capacity without significant reductions in the yield stress being required in the design equations. Ongoing work is being performed in this area at the University of Sydney.

ACKNOWLEDGEMENTS The authors would like to thank the Australian Research Council and BHP Coated Steel Division for their financial support for these projects performed at the University of Sydney. The advise of Associate Professor Kim Rasmussen on the compression tests is gratefully acknowledged. REFERENCES

American Iron and Steel Institute. (1997). "1996 Edition of the Specification for the Design of ColdFormed Steel Structural Members", Washington, DC, USA. American Iron and Steel Institute. (2000). "1996 Edition of the Specification for the Design of ColdFormed Steel Structural Members, Supplement 1, July 1999", Washington, DC, USA.

American Society for Testing and Materials A 611. (1997). "Standard Specification for Steel, Sheet, Carbon, Cold-Rolled, Structural Quality", Philadelphia, PA, USA American Society for Testing and Materials A 653. (1997). "Standard Specification for Steel Sheet, ZincCoated (Galvanized) or Zinc-iron Alloy-Coated (Galvannealed) by the Hot-Dip Process", Philadelphia, PA, USA. American Society for Testing and Materials A 792. (1997). "Standard Specification for Steel Sheet, 55% Aluminum-Zinc Alloy-Coated by the Hot-Dip Process", Philadelphia, PA, USA. American Society for Testing and Materials A 875. (1997). "Standard Specification for Steel Sheet, Zinc5% Aluminium Alloy-Coated by the Hot-Dip Process", Philadelphia, PA, USA. Ashby, M.F.. (1981). Prog. Mat. Sci., Chalmers Anniversary Volume, pp. 1-25. Canadian Standards Association, S 136. (1994). "Cold Formed Steel Structural Members", Etobicoke, Ont, Canada.

34 Dhalla, A.K., Winter, G.. (1974a). "Steel Ductility Measurements", Journal of the Structural Division, ASCE, Vol. 100, No. ST2, pp. 427-444. DhaUa, A.K., Winter, G.. (1974b). "Suggested Steel Ductility Requirements", Journal of the Structural Division, ASCE, Vol. 100, No. ST2, pp. 445-462. Daudet,R., and Klippstein,K.H. (1994), "Stub Column Study using Welded, Cold-Reduced Steel", 12th International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Oct 1994, pp 285 - 302. FRANC2D. (1995). "Tutorial and User's Guide", Version 2.7, ComeU University Fracture Group, Ithaca NY, USA. Hancock, G.J. and Murray, T.M (1996), "Residential Applications of Cold-Formed Stmcaaal Members in Australia, 13th International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Oct 1996, pp 505 - 511. Macindoe, L. and Pham, L., (1995), Test Data from Screwed and Blind Rivetted Connections", CSIRO Division of Building, Construction and Engineering, Document 96/22(M), Higher Victoria, Australia Mahendran, M, (1990), "Static Behaviour of Corrugated Roofing under Simulated Wind Loading",Civil Engg Transactions, Inst. Engrs. Aust., Vo132, No. 4, pp 211-218. Mahendran, M, (1994), "Behaviour and Design of Crest Fixed Profiled Steel Roof Claddings under High Wind Forces",Engg. Struct., Vol. 16 No. 5. McAdam, J.N, Brockenbrough, R.A., LaBoube, R.A., Pekoz, T, and Schneider, E.J., "Low Strain Hardening Ductile Steel Cold-Formed Members", 9~ International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Nov 1988. Rogers, C.A., Hancock, G.J.. (1997). "Ductility of G550 Sheet Steels in Tension", Journal of Structural Engineering, ASCE, Vol. 123, No. 12, 1586-1594. Rogers, C.A., Hancock, G.J.. (1998a). "Bolted Connection Tests of Thin G550 and G300 Sheet Steels", Journal of Structural Engineering, ASCE, Vol. 124, No. 7, pp. 798-808. Rogers, C.A., Hancock, G.J.. (1998b). "Tensile Fracture Behaviour of Thin G550 Sheet Steels", Research Report No. R773, Centre for Advanced Structural Engineering, University of Sydney, Sydney, NSW, Australia Rogers, C.A., Hancock, G.J. (1999). "Screwed Connection Tests of Thin G550 and G300 Sheet Steels", Journal of Structural Engineering, ASCE, Vol. 125, No. 2, pp. 128-136. Rogers, C.A. and Hancock, G.J. (2001). "Fracture Toughness of G550 Sheet Steels subjected to Tension", Journal of Constructional Steel Research, Vo157, pp 71-89. Standards Australia. ( 1 9 9 3 ) . "Steel sheet and strip - Hot-dipped zinc-coated or aluminium/zinc coated - AS 1397", Sydney,NSW, AusUalia Standards Australia / Standards New Zealand. (1996). "Cold-formed steel structures - AS/NZS 4600", Sydney, NSW, Australia

35 Tang, and Mahendmn, M. (1998),"Local Failures in Trapezoidal Steel Claddings", Thin-Walled Stractures Research and Developments, Eds Shanmugam, Liew and Thevendran, Elsevier

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Wu, S., Yu, W.W, and LaBoube, R.A. (1996a), "Strength of Flexural Members using Structural Grade 80 of A653 Steel (Deck Panel Tests)", Second Progress Report, Department of Civil Engineering, University of Missouri-Rolla, November. Wu, S., Yu, W.W, and LaBoube, R.A. (1996b), "Flexural Members using Structural Grade 80 of A653 Steel (Deck Panel Tests)", 13th International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Oct 1996, pp 255-274. Wu, S., Yu, W.W, and LaBoube, R.A. (1997a), "Strength of Flexural Members using Structural Grade 80 of A653 Steel (Web Crippling Tests)", Third Progress Report, Depamnent of Civil Engineering, University of Missouri-Rolla, November. Wu, S., Yu, W.W, and LaBoube, tLA. (1997b), "Web Crippling Strength of Members using High Strength Steels", 14th International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Oct 1998, pp 193-208.

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Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michaiskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

37

THIN-WALLED STRUCTURAL ELEMENTS CONTAINING OPENINGS N.E.Shanmugam Department of Civil Engineering The National University of Singapore 10, Kent Ridge Crescent, Singapore-119260

ABSTRACT This paper is concerned with thin-walled structural elements that contain openings provided to allow for locating services, to facilitate fabrication process and to reduce the self-weight of the structures. Extensive analytical and experimental investigations have been carried out in the past to examine the behavior of thin-walled structures containing openings. However, due to lack of an organized database the results have not been fully utilized by other researchers. As part of this review an endeavor is made to cluster the experimental and analytical results with particular importance on plates, beams and plate girders and compression members and cellular structures containing perforations as an elucidative database using a handy spreadsheet program written in Visual Basic. KEYWORDS Thin-walled, structural elements, local buckling, in-plane loading, shear loading, review, experiments, analyses INTRODUCTION Openings are often introduced in thin-walled structures such as automobiles, aircrafts, bridges, ships and storage racking structures to facilitate access for services and inspection. Presence of these openings also help to lighten the structure and to enhance the constructability of thin-walled sections by allowing easy consolidation of building services, piping, electric-wiring, plumbing etc., within the section depth. Access for repair and maintenance works are provided through openings in webs of plate girders, box girders and ship grillage. In aircraft industry openings are often provided for air passage to the interior. Thin-walled members for some specific tasks are generally manufactured with a regular pattern of multiple holes in order to meet the requisites. Typical openings in thin-walled structure are shown in Figure 1. Presence of openings results in redistribution of the membrane stresses, change in buckling and strength characteristics leading to drop in load-carrying capacity. Geometry of the thin-walled structural element (e.g. angle, channel - plain or lipped, etc.), type of applied stress (e.g. compressive, tensile, shear etc) and shape, size, location and number of the openings have greater influence on the

38

Figure: 1 Some examples of Thin-Walled Structures Containing Openings behaviour of structural members containing openings. Thus, the precise analysis and design of structural steel members with perforated elements are imperative. A considerable amount of research has been directed towards this problem during the last three or four decades and approximate design methods for evaluating the ultimate capacity of thin plate elements containing openings and subjected to in-plane axial or shear loading have been proposed by researchers. A review on the research work to account for the effects of openings on the behaviour of thin-walled elements in steel structures has been presented by Shanmugam (1997). Attention is directed, in the review, to the analytical and experimental work carried out on centrally or eccentrically placed, reinforced or unreinforced circular or rectangular openings in perforated plate elements, stiffened flanges, shear webs and cold-formed steel structural elements. Also, design methods of thin-walled steel structural elements that account for the presence of openings are highlighted. However, detailed information from each of the works is not presented in this paper and, one has to dig out the references listed for such information. The object of this paper is, therefore, to summarize the works and present a data bank in a systematic manner so that future researchers could have one source from which they would be able to get any details of the past work done for their reference. An effort has been made in this paper to provide a comprehensive review and to compile the experimental and analytical results on perforated thin-walled structures in a user-friendly format. The review is presented, for convenience of the readers, under four major topics viz. plates, thin-plated beams, thin-walled columns and cellular structures.

PLATES A number of researchers studied initially perforated flange and web plates subjected to different types of loadings such as uniaxial compression, biaxial compression and shear and investigated the stress concentration around openings. Finite element method was applied to determine the elastic buckling of plate elements with cutouts. The method was extended further to include inelastic and largedeflection behaviour of such plate elements.

Uniaxiai and Biaxial Loadings Pennington Vann (1971) employed finite element method for elastic buckling analysis of uniaxially loaded plates. The theoretical and experimental results presented indicate that unless a central nonflange,d hole is rather large, it will have a small effect on the elastic buckling load of a plate, and that a flanged hole can be expected to make the elastic buckling load greater than that of the corresponding non-perforated plate. Tentative results concerning the ultimate strength of pierced plates indicate that a small non-flanged hole has essentially no effect on the ultimate strength, and a stiffened hole may or may not affect the ultimate strength. The finite element method was extended later (Shanmugam and

39 Narayanan, 1982; Sabir and Chow, 1983)to encompass other forms of loading and support conditions for both square and circular openings. The knowledge of post-buckling behaviour is necessary to determine the ultimate load-carrying capacity. Yu and Davies (1971) and Ritchie and Rhodes (1975) proposed that the effective width concept with slight modification could be extended to the postbuckling analysis of axially compressed perforated plates, taking size and shape of openings into consideration. The proposed method was validated with the experimental results on the buckling and post-buckling behaviour of thin-walled structural elements containing centrally located hole. The finite element formulation, which incorporated the material and geometric non-linearity and the initial plate imperfections, based on variational principles (Azizian and Roberts, 1983 and 1984) could predict the non-linear behaviour of perforated plates, right up to failure. Narayanan and Chan (1985) proposed an approximate method based on energy approach to predict the ultimate load capacity of uniaxially compressed plates under linearly varying load across the width. Plate elements subjected to biaxial loading commonly occur in ship double bottoms, dock gates and multi-cell girders. Only limited information is available on the behaviour of such members in the literature. Elastic buckling of such plate elements containing openings was investigated by employing finite element method (Shanmugam and Narayanan, 1982; Sabir and Chow, 1983). A simplified procedure was proposed to estimate the peak load of biaxially loaded plates containing openings was proposed by Narayanan and Chow (1984a). The proposed procedure has been found to give reasonable predictions compared to the corresponding experimental values. Taking into account of the time and cost factor involved in such methods, Narayanan and Chow (1984b, 1984c, 1987)suggested an approximate design method for evaluating the ultimate capacity of plates containing openings and subjected to uni-axial or biaxial compression. All these methods require a proper understanding of the plate behaviour and the effective width concept. Shanmugam et al. (1999) proposed a design formula using finite element method that does not require any rigorous mathematical computations. The designers need to compute different sets of coefficients and substitute into the general equation to predict the ultimate capacity of the circular and square perforated plates for different shapes of openings, different types of axial loading and different combinations of boundary conditions. Shear Loading

Finite element method was employed by Rockey et al. (1967) to study the buckling of a square plate having a centrally located circular hole when subjected to edge shear load. The introduction of openings was found to result in drop of critical shear stress for square plate panel. Studies by other researchers (Shanmugam and Narayanan, 1982; Sabir and Chow, 1983; Narayanan and Chow, 1984c; Naryanan and Der Avenesian, 1984) of the elastic buckling behaviour of perforated plates under shear loading include different shapes of openings and boundary conditions. It may be necessary to place openings eccentrically in the plate. The effects of such eccentricity on elastic buckling behaviour under shear loading of plates were extensively analysed by Narayanan and Der Avenesian (1984). Reinforcements around rectangular and circular openings were also considered in these studies. For plates with central circular and rectangular cutout, an approximate relationship for the shear-buckling coefficient was suggested based on the finite element results. Narayanan and Chow (1984), based on experimental investigations, concluded that the effect of opening size on buckling is more significant than that of location. Stiffened Plates

It becomes essential for flange plates employed in ship double bottoms, dock gates and box girder bridges to be stiffened longitudinally in order to enhance the efficiency of the structure to resist loads. Such stiffened plates often contain openings in bed-plates in which case the failure load may be determined using a method proposed by Mahendran et al. (1994, 1996). This method has been found

40 to predict the failure load to an acceptable degree of accuracy compared with a number of experimental failure loads (Shanmugam et al. 1985, 1986).

PERFORATED BEAMS Bending Hoglund (1971) concluded that the reduction in bending strength of a thin web I-girder having web openings located at the centre is small since the flanges carry most of the bending moment. Experimental and analytical studies were carried out by Shan (1994) to determine the local buckling characteristics of a flexural member containing web opening and proposed an equation to determine the flexural capacity of such sections. Yu and Davis (1973) studied the behaviour of thin-walled steel members with perforated web elements and concluded that the presence of a circular hole results in reduction of shear capacity; they have proposed an empirical reduction factor to quantify the reduction. An approximate method was proposed by Narayanan and Rockey (1981) for the analysis of perforated plate girders. The method was based on experimental observations on plate girders containing web openings. Equilibrium solutions by Narayanan and Der Avanesian (1983 a-d, 1984 a, b, 1985, 1986) based on further assumptions on tension field action has covered shapes and locations of openings in plate girder webs. Though there is no theoretical basis to confirm the hypothetical tension band around openings, the observed experimental pattern of failure substantiated the concept. The predictions obtained by the equilibrium solutions, compared with test results, have invariably been conservative, however, there has been no systematic study to verify if this would generally be the case. Shan (1994) developed both linear and non-linear reduction factors to determine the nominal shear strength, based on the findings from experimental and analytical studies of the behaviour of web elements of cross sections with web openings subjected to uniform shear force. Schuster (1999)carried out experiments to establish an analytical method for calculating the shear resistance of perforated cold-formed steel Csections subjected to constant shear at the University of Waterloo. Combined effect of bending moment and shear force was investigated at the University of MissouriRolla (Shan, 1994; Shan et al., 1996) on standard C- shaped members containing web openings. The current interaction equation in the AISI Specifications, which adequately predicts the web capacity of the nominal shear and bending strengths, are appropriately modified to account for the presence of web openings based on the test results. In the case of beams and columns subjected to concentrated loads in the steel framing structures provision must be made for the load transfer into the web from the flange. Web crippling occurs when web flange intersection subjected to a large compressive force and, this problem becomes more critical in the presence of openings. This problem of web crippling in coldformed steel members has been addressed in the recent research findings (Sivakumaran, 1988; Sivakumaran et al., 1989; Shan et al., 1993 and 1994). Yu and Davies (1973) have also reported on the reduction in web crippling strength due to presence of web openings. This is based on the experimental results on cold-formed steel members. Recent experimental and theoretical studies by Chung (1995) have shown that for perforated sections with practical shapes and significant sizes, the resistance of the sections to web crippling load resistances is often not significant.

Lateral Buckling The problem of lateral buckling in deep slender beams is an important issue that needs adequate consideration in the design of such beams. However, only limited information could be traced in the published literature on the effect of openings on the buckling capacity of beams (Bower, 1968; ASCE Task Committee Report, 1971; Redwood and Uenoya, 1979). Redwood and Uenoya (1979) have treated the problem of webs as a stability problem of a perforated plate with simplified edge loadings and support conditions. The finite element method has been used to solve the resulting problem. Coull and Alvarez (1980) based on their experimental studies have proposed an empirical method for

41 determining the lateral buckling capacity of beams with a number of openings, either circular or rectangular. Their formulae do not seem to allow for openings Other than those considered by them. Thevendran and Shanmugam (1991, 1992a,b) proposed using the principle of minimum total potential energy a numerical method to predict the elastic lateral buckling load of narrow rectangular and I beams containing web openings and subjected to single concentrated load applied at the centroid of the cross section. The accuracy of the method has been verified with results obtained from experiments similar to those shown in Figure 2. Simply supported and cantilever beams were considered and the method is capable of predicting the effects of size, location and shape of openings on the elastic critical load.

Figure 2 : Experiments on Lateral Buckling of Beams with Openings

Reinforced Web Openings It is an expensive operation to provide reinforcements around openings in order to minimize the strength reduction. However, if the loss of strength implicit in cutting a web hole is unacceptable, the web opening will need to be reinforced around its periphery so that the opening can still be introduced. The method of designing an appropriate circular ring reinforcement to restore the strength lost due to the cutout serves as a useful tool in the design office. Based on their investigation, Narayanan and Der Avenesian (1984 a, b) proposed an equilibrium solution for prediction of the ultimate capacity of girders containing a reinforced circular or rectangular hole. Since the tension field in a non-perforated web is developed predominantly along a diagonal band, it is wise to locate openings away from this band, so that the girder does not suffer any significant drop in strength. The effect of eccentrically placed openings on the behaviour of plate girder webs was investigated by Narayanan et al. (1984,1985). THIN-WALLED COLUMNS Local buckling and post-buckling strength for stiffened and unstiffened doubly symmetric perforated cold-formed steel compression members were first considered by Davis and Yu (1972) and their findings based on experimental investigations formed the basis for further developments in the evaluation of the effective design width of perforated plate elements. Yu and Davis proposed an effective design width equation for plates with either central or square perforation. Because of the intricacy involved in the computation of various constants, this equation is not popular with designers. The effective width equation, in the same form as that of Winter's formula (1947), recommended by Ortiz-Colberg (1981) is now currently practiced in the AISI Specifications (1996). But this equation is only applicable for stiffened plates under uniform compression with certain limitation on slenderness ratio and circular perforation to width ratio. Sivakumaran and Banwait (1987 a, b, c) recommended reassessment of effective design width equations in the design codes CAN3-S 136-M84 and AISI-1986,

42 in which the strength of sections is overestimated. They emphasized on the need for experimental data on the sections with manufacturer's hole. Miller and Pekoz (1994) attempted to modify the unified effective width approach to local buckling in the Specification for Design of Cold-Formed Steel Structural Members (AISI 1986) to model the perforated section. It was concluded that the effect of perforations is negligible unless the perforation is extended to the effective portion of the compressed element. An effective design width equations to determine the ultimate strength of non-perforated and perforated cold-formed steel compression members was proposed (Rahman, 1997; Rahman and Sivakumaran, 1998) based on a proven finite element model (Sivakumaran, Rahman, 1998). It has been found that this equation could predict well for manufacturer's hole.

Multiple Openings In the field of storage racking, an ample proportion of cold-formed steel construction, the upright is generally having many perforations, otten in the form of a repeating pattern to a significant extent as shown in Figure 3. The strut capacity of such member is generally assessed on the basis of tests. A more general, but relatively conservative design approach for perforated wall stud assemblies is given in the Canadian standard S136-94 (CSA 1994), which is not applicable for single members in compression. The CSA $136-1974 equations for non-stiffened plates are not suitable for analyzing perforated studs with short holes similar to those tested. Loov (1984)developed an equation to

Figure 3 : Typical Examples of Multiple Openings determine the effective width of non-stiffened portion of the web beside the holes based on the experiments. The test results support the present equation for the average yield stress in Canadian Standards Association Standard S136-1974 but the present code equations for non-stiffened plates are unduly conservative when applied to the design of the web adjacent to openings of the size considered. Rhodes and Schneider (1996)studied experimentally the effects of multiple circular perforations on the ultimate compressive strength of cold-formed steel channel sections. The application of existing coldformed steel design codes to perforated members is examined on the basis of comparison with the tests, and various modifications to the design codes are considered to take perforations into account. Rhodes and Macdonald (1996) extended the investigation to study the effects of perforation length on the stub column capacity. The study concluded that consideration should be given to the perforation geometry along the member as well as the perforation geometry across the member. Wall structures, especially used in the Nordic Countries, include web-perforated C- or sigma-sections as studs and U-sections as tracks and, e.g. gypsum wallboards attached to the stud flanges to provide

43 quality thermal performance. Jyrki Kesti and Pentti Makelainen (1999)proposed design curves to determine the distortional buckling strength of the section based on the experimental study. The comparison showed that the method used gives reasonable results for C-sections but overestimates the compression capacity of sigma sections. CELLULAR STRUCTURES Multi-cell structures, otten used in bridges, storage tanks, dock gates and ship double bottoms, usually contain large openings in webs (Figure 4). The influence on shear deflection component of web openings resulted in large deflection of this form of structures. Studies (Shanmugam and Evans, 1979 a,b) have also shown that openings have influence on longitudinal stresses too. Empirical curves, based on suitable reduction coefficients for shear and torsional stiffness of the structure, can be used adequately to represent the effect of the web openings in grillage analyses of multi-cellular structures. The analysis was also applied to investigate the elastic behaviour and vibrational characteristics of thin-walled multi-cell structures containing web openings (Shanmugam and Balendra, 1985). Evans and Shanmugam (1981,1985) observed tension field action in the webs of cellular structures also.

Figure 4 : Celluar Structure Models with Web Perforations COMPILATION Extensive research materials on the behavior of perforated thin-walled structures have been reported in the published literature. However, there is no collection of data organized especially for rapid search and retrieval. Review is mandatory to prevent consuming valuable time in replicating the previous work. Compilation is essential to avoid unnecessary time in extracting important information from the other researchers work. An effort is, therefore, made to cluster the experimental and analytical results with particular emphasis on plates, beams, compression members and cellular structures containing perforations as an elucidative database using a handy spreadsheet program. The database in visual basic programming contains main record and sub records preceded by the user-friendly instruction sheet. Main record is carefully organized in such a way that a user can easily view through the sheet and select the research work he needed to retrieve from the sub records. The main record is presented in a table format in Appendix I. The sub records include research group, material properties, experimental and analytical results, design equations critical comments and final results for 59 different sets, categorized under different structures, Plate, Beam and Column. An attempt is made to provide all the necessary data needed in a comprehensible format. So the future researcher can refer this database to get all the information needed about perforated thin-walled members. Sample set is given for each category of plate, beam and column in Appendix II.

44 CONCLUSIONS Significant progress has been made in the investigation of thin walled structures containing openings. Several analytical models have been proposed to predict the effects of cutouts on the behaviour of thinwalled members, particularly on singly and doubly symmetric sections with single and multiple perforations subjected to different types of loadings. A number of experiments have been carried out to verify the proposed models. The investigations have resulted in simplified design methods of thinwalled elements containing openings. An attempt has been made in this paper to consolidate all the research works carried out on perforated thin-walled structural elements and to give a comprehensive review and a data base which could be of use to the future researchers in this field. REFERENCES

Abdel - Rahman, N. (1997). Cold-Formed Steel Compression Members with Perforations, PhD Thesis, Mc Master University, Hamilton, Ont. Azizian, Z. G. and Roberts, T. M. (1983). Buckling and Elasto-Plastic Collapse of Structures,

Proceedings of the International Conference on Instability and Plastic Collapse of Structures, Manchester, 322-328. Balendra, T. and Shanmugam, N.E. (1985). Vibrational Characteristics of Multi-Cellular Structures, Journal of Structural Engineering, ASCE, 111:7, 1449-1459. Banwait, A. S. (1987). Axial Load Behaviour of Thin-Walled Steel Sections with Openings, M.Eng Thesis, McMaster University, Hamilton, Ontario, Canada. Bower, J.E., (1968). Design of Beams with Web Openings, 3'. Struct. Div., ASCE 94, 783-807. Sub-committee on Beams with Web Openings of the Task Committee on Flexural Members of the Structural Division. (1971). Suggested Design Guides for Beams with Web Holes, J. Struct. Div., ASCE 97, 2707-2727. Chung, K. F. (1995). Structural Performance of Cold-Formed Sections with Single and Multiple Web Openings- Part 1" Experimental Investigation, The Structural Engineer, 73:9, 141-149. Chung, K. F. (1995). Structural Performance of Cold-Formed Sections with Single and Multiple Web Openings- Part 2: Design Rules, The Structural Engineer, 73:9, 141-149. Coull, A. and Alvarez, M.C. (1980). Effect of Openings on Lateral Buckling of Beams, J. Struct. Div., ASCE 106, 2553-2560. Davis, C. S. and Yu. W. (1972). The structural Behaviour of Cold-Formed Steel Members with Perforated Elements, Department of Civil Engineering, University of Missouri Rolla, Rolla, Mo. Evans, H. R. and Shanmugam, N. E. (1979). The Elastic Analysis of Cellular Structures containing Web Openings, Proceedings, The Institution of Civil Engineers, 67:2, 035-1063. Evans, H. R. and Shanmugam, N. E. (1981). An Experimental study of the Ultimate Load Behaviour of Small-scale Box Girder Models with Web Openings, Journal of Strain Analysis, 16:4, 251-259. Hoglund, T. (1971). Strength of Thin Plate Girders with Circular or Rectangular Web Holes without Web Stiffeners, Proceedings, Colloquium of the International Association of Bridge and Structural Engineering, London. Jyrki Kesti and Penti Makelainen. (1999). Compression Behaviour of Perforated Steel Wall Studs, Lightweight Steel and Aluminum Structures. Loov, R. (1984). Local Buckling Capacity of C-Shaped Cold-Formed Steel Sections with Punched Webs, Canadian Journal of Civil Engineering, 11, 1-7. Mahendran, M., Shanmugam, N. E. and Richard Liew, J. Y. (1994). Strength of Stiffened Plates with Openings, Proceedings, Twelfth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, 29-40. Mahendran, M., Shanmugam, N. E. and Liew, J. Y. (1996). Design of Stiffened Plates with Openings, Journal of Institution of Engineers, Singapore, 36:2, 15-21.

45 Miller, T. H. and Pekoz, T. (1994). Unstiffened Strip approach for Perforated Wall Studs, ASCE Journal of Structural Engineering, 120:2, 410-421. Nabil Abdel-Rahman and Sivakumaran, K. S. (1998). Effective Design Width for Perforated Cold-Formed Steel Compression Members, Canadian Journal of Civil Engineering, 25, 319-330. Narayanan, R. and Rockey, K. C. (1981). Ultimate Load Capacity of Plate Girders with Webs containing Circular Cut-outs, Proceedings, Institution of Civil Engineers, 71:2, 845-862. Narayanan, R. and Der Avenesian, N. G. V. (1983a). Strength of Webs Containing Circular Cutouts, IABSE Proceedings P-64/83, 141-152. Narayanan, R. and Der Avenesian, N. G. V. (1983b). Equilibrium Solution for Predicting the Strength of Webs with Rectangular Holes, Proceedings of the Institution of Civil Engineers, 75:2, 265-282. Narayanan, R. and Der Avenesian, N. G. V. and Ghannam, M. M. (1983). Small-scale Model Tests on Perforated Webs, The Structural Engineer, 61 (3), 47-53. Narayanan, R. (1983). Ultimate Shear Capacity of Pate Girders with Openings in Webs, Plated Structures-Stability and Strength, ed. R. Narayanan, Applied Science Publishers, London, pp. 3976. Narayanan, R. and Der Avenesian, N. G. V. (1984a). Design of Slender Webs Containing Circular Holes, IABSE Proceedings P-72/84, pp. 25-32. Narayanan, R. and Der Avenesian, N. G. V. (1984b). An Equilibrium Method for assessing the Strength of Plate Girders with Reinforced Web Openings, Proceedings, Institution of Civil Engineers, 77:2, pp. 107-137. Narayanan, R. and Der Avenesian, N. G. V. (1984c). Elastic Buckling of Perforated Plates under Shear, Thin-Walled Structures, 2, 51-73. Narayanan, R. and Der Avenesian, N. G. V. (1984d). Strength of Webs with Comer Openings, The Structural Engineer, 62B, 6-11. Narayanan, R. and Chow, F. Y. (1984a). Strength of Biaxially Compressed Perforated Plates, Proceedings, Seventh International Specialty Conference on Cold-Formed Steel Structures, University of Missouri-Rolla, 55-73. Narayanan, R and Chow, F.Y. (1984b). Ultimate Capacity of Uniaxially Compressed Perforated Plates, Thin-Walled Structures, 2, 241-264,. Narayanan, R and Chow, F.Y. (1984c). Buckling of Plates Containing Openings, Proceedings, Seventh International Specialty Conference on Cold-Formed Steel Structures, University of Missouri Rolla, 39-53. Narayanan, R. and Chow, F. Y. (1984d). Experiments on Perforated Plates Subjected to Shear, Thin-Walled Structures, 2, 51-73. Narayanan, R. and Der Avenesian, N. G. V. (1985). Design of Slender Webs having Rectangular Holes, dournal of Structural Engineering, ASCE, 111:4, 777-787. Narayanan, R. and Darwish, I. Y. S. (1985). Strength of Slender Webs having Non-Central Holes, The Structural Engineer, 63B, 57-62,. Narayanan, R. and Chan, S.L. (1985). Ultimate Capacity of Plates Containing Holes under Linearly Varying Edge Displacements, Computers and Structures, 21:4, 841-849. Narayanan, R. and Der Avenesian, N. G. V. (1986). Analysis of Plate Girders with Perforated Webs, Thin-Walled Structures, 4, 363-380. Narayanan, R. (1987). Simplified Procedures for the Strength Assessment of Axially Compressed Plates with or without Openings, Proceedings, International Conference on Steel and Aluminum Structures, Cardiff, 592-606. Ortiz-Colberg, R. (1981). The Load Carrying Capacity of Perforated Cold-Formed Steel Columns, M.Sc. Thesis, Cornell University, Ithaca, N.Y. Pennington Vann, W. (1971 ) Compressive Buckling of Perforated Plate Elements, Proceedings of the first Specialty Conference on Cold-Formed Structures, University of Missouri Rolla, 52-57. Redwood, R.G. and Uenoya, M. (1979). Critical Loads for Webs with Holes, d. Struct. Div., ASCE 105, 2053-2067.

46 Ritchie, D. and Rhodes, J. (1975). Buckling and Post-Buckling Behaviour of Plates with Holes, Aeronautical Quarterly, 281-296. Roberts, T. M. and Azizian, Z. G. (1984). Strength of Perforated Plates subjected to in-plane Loading, Thin-Walled Structures, 2:2, 153-164. Rockey, K. C., Anderson, R. G. and Cheung, Y. K. (1967). The behaviour of square Shear Webs having Circular Hole, Proceedings of the Swansea Conference on Thin-Walled Structures, Crosby Lockwood and Sons, London, 148-169. Rhodes, J., and Macdonald, M. (1996). The effects of Perforation Length on the behaviour of Perforated Elements in Compression, Proceedings, 13th International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Mo., 91-101. Rhodes, J. and Schneider, F. D. (1996). The Compressional behaviour of Perforated Elements, Proceedings, 12th International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Mo., 11-28. Sabir, A.B. and Chow, F.Y. (1983). Elastic Buckling of Flat Panels Containing Circular and Square Holes, Proceedings, International Conference on Instability and Plastic Collapse of Structures, Manchester, pp. 311-321. Said, A. N. (1995). The behaviour of Thin-Walled Perforated Elements under Compression, Final Year Project Thesis, University of Strathclyde, Glascow. Schuster, R. M. (1999). Perforated Cold-Formed Steel C-Sections subjected to shear (Experimental Results), Light-Weight Steel and Alumnium Structures, 779-788. Shan, M.Y., LaBoube, R.A. and Yu W.W. (1993). Shear behaviour of web elements with openings, Proceedings, Structural Stability Research Council Annual Technical Session. 103-113. Shan, M. Y., Batson, K. D., LaBoube, R. A. and Yu, W. W. (1994). Local Buckling flexural Strength of Webs with Openings, Engineering Structures, 16:5, 317-323. Shan, M. Y. (1994). Behaviour of Web Elements with Openings subjected to Bending, Shear and the combination of Bending and Shear, PhD dissertation, Dept. of Civil Engg., University of Missouri Rolla, Roila, Mo.. Shan, M. Y., LaBoube, R. A. and Yu. W. W. (1966). Bending and Shear Behaviour of Web Elements with Openings, Journal of Structural Engineering, ASCE. Shanmugam, N. E. and Evans, H. R. (1979). An Experimental and Theoretical Study of the effects of Web Openings on the Elastic Behaviour of Cellular Structures, Proceedings, The Institution of Civil Engineers, 67:2, 653-676. Shanmugam, N.E and Narayanan, R. (1982). Elastic Buckling of Perforated Square Plates for Various Loading and Edge Conditions, Proc., Proceedings, International Conference on finite element methods, Shanghai, China, 241-245. Shanmugam, N. E. and Evans, H. R. (1985). Structural Response and Ultimate Strength of Cellular Structures with Perforated Webs, Thin-Walled Structures, 3,255-271. Shanmugam, N.E. and Balendra, T. (1985). Model Studies on Multi-Cell Structures, Proceedings, Institution of Civil Engineers, Part 2: Research and Theory, 79, 55-71. Shanmugam, N. E. and Paramasivam and Lee, S. L. (1985). Ultimate Strength of Axially Compressed Stiffened Plates Containing Openings, Proceedings of the International Conference on Metal Structures, Melbourne, Australia, 48-52. Shanmugam, N. E. and Paramasivam and Lee, S. L. (1986). Stiffened Flanges Containing Openings, Journal of Structural Engineering, ASCE, 112:10, 2234-2246. Shanmugam, N. E. and Thevendran, V. (1992). Critical Loads of Thin-Walled Beams Containing Web Openings, Thin-Walled Structures, 14, 291-305. Shanmugam, N.E. (1997). Openings in Thin-Walled Structures. Thin-Walled Structures, 28:3-4, 355-372. Shanmugam, N. E., Thevendran. V. and Y. H. Tan. (1999). Design Formula for Axially Compressed Perforated Plates, Thin-Walled Structures, 34, 1-20.

47 Sivakumaran, K. S., Banwait, A. S. (1987). Effect of Perforation on the Strength of Axially Loaded Cold-Formed Steel Section, Proceedings, International conference on Steel and Aluminum Structures, Cardiff, UK, 8-10 July 1987, pp. 428-437. Sivakumaran, K. S. (1987). Load Capacity of Uniformly Compressed Cold-Formed Steel Section with Punched Web, Canadian Journal of Civil Engineering, 14, 550-558. Sivakumaran, K. S. and Nabil Abdel-Rahman. (1998). A Finite Element Analysis Model for the Behaviour of Cold-Formed Steel Members, Thin-Walled Structures, 31,305-324. Sivakumaran, K.S. (1988). Some Studies on Cold-Formed Steel Sections with Web Openings, Proceedings, Ninth International Specially Conference on Cold-Formed Steel Structures, St. Louis, 513-527. Sivakumaran, K.S. and Zielonka, K. M. (1989). Web Crippling Strength of Thin-Walled Steel Members with Web Openings, Thin-Walled Structures, 8, 295-319. Thevendran, V and Shanmugam, N. E. (1991). Lateral Buckling of Doubly Symmetric Beams Containing Openings, Jr. Eng. Mech., ASCE, 117:7, 1427-41. Thevendran, V and Shanmugam, N. E. (1992). Lateral Buckling of Narrow Rectangular Beams Containing Openings, Computers and Structures, 43(2), 247-254. Winter, G. (1947). Strength of Thin Steel Compression Flanges, Transactions of the American Society of Civil Engineers, 112, 527-554. Yu, W.W. and Davies, C.S. (1971). Bucking Behaviour and Post-Buckling Strength of Perforated Stiffened Compression Elements, Proceedings, The first Specially Conference on Cold-Formed Structures, University of Missouri Rolla, 58-64. Yu, W. W. and Davies, C. S. (1973). Cold-Formed Steel Members with Perforated Elements, ASCE Journal of Structural Engineering Division, 99(ST 10), 2061-2077. APPENDIX I: SAMPLE OF THE MAIN R E C O R D OF THE DATA B A S E STRUCTURE : WORKSHEET:

No

Year

PLATE PLATE

Title

Source

Author

Status

1971

CompressiveBuckling of Perforated Plate Elements

The first SpecialityConference PenningtonVann, on Cold-FormedStructures W

Platel

1971

BuckingBehaviourand PostBuckling Strength of Perforated Stiffened CompressionElements

The first SpecialityConference Yu, W.W., on Cold-FormedStructures Davies, C.S

Plate2

1975

Bucklingand Post-Buckling Behaviour of Plates with Holes

Aeronautical Quarterly

1982

ElasticBuckling of Perforated Square Plates for Various Loading and Edge Conditions

Proc., International Conference Shanmugam,N.E, on finite element methods Narayanan,R

Plate4

1983

Elastic Buckling of Flat Panels Containing Circular and Square Holes

Proc., International Conference Sabir,A.B., on Instability and Plastic Chow, F.Y Collapse of Structures

Plate5

1983

Buckling and Elasto-Plastic Collapse of Perforated Plates

Proc., International Conference Azizian,Z.G, on Instability and Plastic Roberts, T.M Collapse of Structures

Plate6

1984

Ultimate Capacity of Uniaxially Compressed Perforated Plates

Thin-Walled Structures, (Vol.2) Narayanan,R, Chow, F.Y

Plate7

Ritchie, D., Rhodes, Plate3 J

48

STRUCTURE : BEAM WORKSHEET:

BEAM

1

1967

The Behaviour of Square Shear Webs having Circular Hole

Proc., The Swansea Conference on Thin-Walled Structures

Rockey,K.C., Anderson, R.G., Cheung, Y.K

Beaml

2

1971

Bucking Behaviour and PostBuckling Strength of Perforated Stiffened Compression Elements

The first Speciality Conference on Cold-Formed Structures

Yu, W.W., Davies, C.S

Beam2

3

1971

Strength of Thin Plate Girders with Circular or Rectangular Web Holes without Web Stiffeners

Proc., Coloquium, International Association of Bridge and Structural Engineering, London

Hoglund, T.

Beam3

4

1981

Ultimate Load Capacity of Plate Girders with Webs Containing Circular Cut-Outs

Proc., Institution of Civil Engineers, Part2, Vol.71, pp 845-862

Narayanan, R., Rockey, K.C.

Beam4

5

1983

Design of I beams with Web Perforations

Beams and Beam ColumnsStability Strength

Redwood, R.G.

Beam5

6

1983

Equilibrium Solution for Predicting the Strength of webs with Rectangular Holes

Proc., Institution of Civil Engineers, Part2, Vol.75, pp 265-282

Narayanan, R., Der Avenessian, N.G.V.

Beam6

7

1983

Strength of Webs Containing Circular Cut-Outs

IABSE Proceedings P-64/83, pp 141-152

Narayanan, R., Der Avenessian, N.G.V

Beam7

8

1983

Small-Scale Model Tests on Perforated Webs

The Structural Engineer, Vol. 61B, No.3, pp. 47-53

Narayanan, R., Der Avenessian, N.G.V Ghannam, M.M.

Beam8

STRUCTURE

:

WORKSHEET :

COLUMN COLUMN

1

1971

Compressive Buckling of Perforated Plate Elements

The first Speciality Conference on Cold-Formed Structures

Pennington Vann, W

Columnl

2

1971

Bucking Behaviour and PostBuckling Strength of Perforated Stiffened Compression Elements

The first Speciality Conference on Cold-Formed Structures

Yu, W.W., Davies, C.S.

Column2

3

1973

Cold-Formed Steel Members with Perforated Elements

ASCE Journal of Structural Engineering Division, Vol.99(STI 0), pp. 2061-2077

Yu, W.W., Davies, C.S.

Column3

4

1984

Local Buckling Capacity of CShaped Cold-Formed Steel Sections with Punched Webs

Canadian Journal of Civil Engineering, Vol. 11, pp. 1-7

Loov, R.

Column4

5

1984

Effect of Perforation on the Strength of Axially Loaded ColdFormed Steel Section

Proc. Int. natl. Conf. On Steel Aluminum Structures, Cardiff, UK, 1987, pp. 428-437

Sivakumaran, K.S., Banwait, A.S.

Column5

6

1987

Load Capacity of Uniformly Compressed Cold-Formed Steel Section with Punched Web

Canadian Journal of Civil Engineering, Vol. 14, pp. 550558

Sivakumaran, K.S.

Column6

49

APPENDIX

II: SAMPLE

SET OF DATA BASE

PLATE 7 Year Subject Title Author Structure Boundary Condition

: : : : :

1956 Ultimate Capacity of Uniaxially Compressed Perforated Plates Thin-Walled Structures, (Vol.2) Narayanan, R and Chow, F.Y Plate

:

Simply Supported

An approximate method of evaluating the ultimate capacity has been suggested using simple elastic-plastic concepts. Design Curves for centrally perforated plates have been suggested for designers. The analysis is based on small deflection theory, hence the method is not valid for wide plates with slenderness values in excess of 80 or so

Table I : DETAILS OF TEST SPECIMENS CONTAINING CENTRAL CUTOUTS Group Circular

a (mm)

t (mm)

a/t = b/t

d or a' (mm)

d/a or a'/a

Imper (mm)

lmper/t

YldStress N/mm2

CIR2a CIR2b CIR3a CIR4a CIR4b CIR5a CIR6 CIR7 CIR8 CIR9 CIRI 0 CIR11 CIR12

125.0 125.0 125.0 125.0 125.0 125.0 86.0 86.0 86.0 86.0 86.0 86.0 86.0

1.615 1.615 1.615 1.615 1.615 1.615 2.032 1.615 0.972 0.693 2.032 1.615 0.972

77.40 77.40 77.40 77.40 77.40 77.40 42.30 53.23 88.48 124.10 42.30 53.25 88.48

25.0 25.0 37.5 50.0 50.0 62.5 25.0 25.0 25.0 25.0 40.0 40.0 40.0

0.2 0.2 0.3 0.4 0.4 0.5 0.291 0.291 0.291 0.291 0.465 0.465 0.465

0.229 0.097 0.136 0.304 0.127 0.279 0.254 0.229 0.102 0.051 0.102 0.279 0.152

0.142 0.060 0.084 0.188 0.079 0.173 0.143 0.142 0.105 0.074 0.050 0.173 0.156

323.3 323.3 323.3 323.3 323.3 323.3 334.7 323.3 317.6 322.8 334.7 323.3 317.6

Square SQ2 SQ3 SQ4 SQ5

125.0 125.0 125.0 125.0

1.615 1.615 1.615 1.615

77.40 77.40 77.40 77.40

25.0 37.5 50.0 62.5

0.2 0.3 0.4 0.5

0.097 0.141 0.113 0.209

0.060 0.087 0.070 0.129

323.3 323.3 323.3 323.3

Table 2 : DETAILS OF TEST SPECIMENS CONTAINING ECCENTRICAL CUTOUTS Yield Stress = 317.6 N/mm2 Specimen Circular

a t (mm) (mm)

aJt = b/t

d or a' (mm)

d/a or a'/a

e (mm)

e/a

lmper (mm)

Imper/t

UEC 1 UEC 2

125.0 0.972 125.0 0.972

128.60 128.60

37.5 62.5

0.3 0.5

12.5 12.5

0.1 0.1

0.254 0.102

0.261 0.105

Square UES 1 UES 2 UES 3 UES4

125.0 125.0 125.0 125.0

128.60 128.60 128.60 128.60

37.5 62.5 37.5 62.5

0.3 0.5 0.3 0.5

12.5 12.5 25.0 25.0

0.1 0.1 0.2 0.2

0.127 0.229 0.078 0.132

0.131 0.236 0.080 0.136

0.972 0.972 0.972 0.972

50

Table 3: E X P E R I M E N T A L RESULTS FOR UNIAXIALLY LOADED PLATES HAVING CENTRAL HOLES Specimen

a/t

d/a or a'/a

Observed Values Ku Avg. Failure Load (Pf) (kN)

Circulzr

(mm)

(mm)

Pcr Avg. (kN)

PL CIR2a CIR2b CIIL3a CIR4a CIR4b CIR5a CIR6 CIR7 CIR8 CIR9 CIRI 0 CIR11 CIR12 Square Hole SQ2 SQ3 SQ4 SQ5

77.40 77.40 77.40 77.40 77.40 77.40 77.40 42.30 53.23 88.48 124.10 42.30 53.25 88.48

0.0 0.2 0.2 0.3 0.4 0.4 0.5 0.291 0.291 0.291 0.291 0.465 0.465 0.465

25.064 22.504 23.228 21.311 19.706 18.358 19.482 6.341 2.320 5.926

4.013 3.604 3.720 3.413 3.156 2.940 3.120 3.205 3.235 2.995

77.40 77.40 77.40 77.40

0.2 0.3 0.4 0.5

22.600 20.290 18.230 19.170

3.620 3.250 2.920 3.070

Pf/Psq

Pxh/Psq

Pxh/Pf

39.32 37.46 38.70 33.94 29.57 28.39 27.35 42.17 26.18 12.35 7.33 33.64 22.14 10.89

0.603 0.574 0.593 0.520 0.453 0.435 0.419 0.721 0.583 0.465 0.381 0.575 0.493 0.410

0.610 0.560 0.560 0.510 0.470 0.470 0.420 0.700 0.615 0.480 0.410 0.560 0.510 0.410

1.012 0.976 0.944 0.981 1.038 1.080 1.002 0.971 1.055 1.032 1.076 0.974 1.034 1.000

33.48 28.85 25.52 21.86

0.525 0.460 0.400 0.340

0.525 0.460 0.400 0.340

1.024 1.041 1.023 1.015

BEAM 12

Year : 1984 Subject : Strength of Webs x~ith Con~er Openings Title : The Structural Engineer, Vol. 62B, pp. 6-11 Author : Narayanan, R and Der-Avanessian, N.G Structure: Plate Girder Material: Steel An equilibrium method of predicting the ultiamte capacity of plate girder webs containing comer web openings and loaded in shear is presented and based on the post-critical behaviour of such webs. Ultimate load = elastic critical load on the web + load carried by the tensile membrane stresses developed in the post=critical stages + load carried by the flanges. TABLE 2: E X P E R I M E N T A L VALUES OF U L I T M A T E LOADS C O M P A R E D WITH PREDICTED VALUES Girder

NCP13 NCPI3 NCPl4 NCPI4 NCP 15 NCP15 NCPl 6 NCPl6

Diameter of Cutout (mm) 125 175 250 325 180 270 360 480

h/t (nominal)

b/h (nominal)

d/h (nominal)

250 250 250 250 360 360 360 360

1.5 1.5 1.5 1.5 1 l l l

0.25 0.35 0.5 0.65 0.25 0.375 0.5 0.67

Observed Ultimate load (kN) 176.4 168.6 125.4 85 234.4 218.8 170.2 112

Equilibrium Solution Predicted load Pre/obs (kN) (kN) 162.9 0.923 157.2 0.932 121.7 0.97 79.7 0.937 221.5 0.945 202.1 0.924 158.8 0.933 87.4 0.78

The theory and experiments confirm that it is advantageous to locate the holes in the comers of the web panels in the compression diagonal i.e far away from the tension field.

51

COLUMN 6 Year : Subject: Title : Author : Structure: Material: Internal Radius: Loading : Section : Opening :

1984 Local Buckling Capacity of C-Shaped Cold-Formed Steel Sections with Punched Webs Canadian Journal of Civil Engineering, Vol. 11, pp. 1-7 Loov, R Column Steel 3.2 Stub-column test Lipped Channel Section Each of the studs had a 38.1 x 44.5 mm rectangular hole centrally punched in the web. Additionally, a 22.2mm diameter hole was centered 41.3ram from one edge of the rectangular opening & a 27.7mm diameter hole was centered in 41.3mm on the opposite side along the centerline of the web.

Specimen

Web

Flange

Lip

Hole Width

Fy

Test

1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

(mm) 63.8 63.8 63.8 63.7 63.7 63.5 63.5 63.5 92.3 92.3 92.3 92.1 92.1 92.1 92.4 92.4 92.4 152.8 152.8 152.8 152.5 152.5 152.5 152.6 152.6 152.6 203.7 203.7 203.7 203.0 203.0 203.0 204.0 204.0 204.0

(mm) 42.6 42.6 42.6 42.2 42.2 42.4 42.4 42.4 42.2 42.2 42.2 42.2 42.2 42.2 42.4 42.4 42.4 42.1 42.1 42.1 41.9 41.9 41.9 42.2 42.2 42.2 42.1 42.1 42.1 41.9 41.9 41.9 42.7 42.7 42.7

(mm) 12.5 12.5 12.5 13.0 13.0 12.1 12.1 12.1 12.5 12.5 12.5 13.0 13.0 13.0 12.8 12.8 12.8 12.7 12.7 12.7 12.8 12.8 12.8 12.9 12.9 12.9 12.5 12.5 12.5 12.9 12.9 12.9 12.9 12.9 12.9

(mm) 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38. I 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1

MPa 384.0 384.0 384.0 382.9 382.9 268.5 268.5 268.5 268.5 268.5 268.5 382.9 382.9 382.9 384.0 384.0 384.0 268.5 268.5 268.5 382.9 382.9 382.9 384.0 384.0 384.0 268.5 268.5 268.5 382.9 382.9 382.9 384.0 384.0 384.0

(kN) 103.62 103.92 102.40 78.65 76.00 44.76 43.87 42.80 46.92 48.11 48.22 72.00 76.40 78.90 119.20 118.00 118.20 51.52 52.32 52.46 84.10 83.40 86.70 128.30 130.00 125.20 51.33 48.70 46.02 87.20 81.70 86.70 133.00 133.50 135.40

This Page Intentionally Left Blank

Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

53

SENSITIVITY ANALYSIS OF THIN-WALLED MEMBERS, PROBLEMS AND APPLICATIONS C. Szymczak Department of Civil Engineering, Technical University of Gdafisk, U1. G. Narutowicza 11, 80-340 Gdafisk, Poland

ABSTRACT A review of problems related to sensitivity analysis of thin walled members with open monosymmetric or bisymmetric cross-section is presented. Three different kinds of restraint imposed on angle of crosssection rotation, transverse displacement and cross-section warping are taken into account. The consideration is based upon the classical theory of thin-walled beams with nondeformable cross-section for linear elastic range of the member material. Attention is focused on the members undergoing torsion, because this topic is not so popular as bending and/or compression. The first variations of state variables due to a change of the design variable are investigated. Arbitrary displacement, internal force or reaction of the member subject to static load, critical buckling load, frequency and mode of torsional vibration are assumed to be the state variables. The dimensions of the cross-section, the material constants, the restraints stiffness, and their locations, position of the member ends are taken as the design variables. All problems under consideration are illustrated by numerical examples. Accuracy of the approximation changes of the state variables achieved by sensitivity analysis is also discussed. Finally some problems dealing with generalization of the sensitivity analysis elaborated upon the space frames and grids assembled with thin-walled members are described.

KEYWORDS Thin-walled structures, statics, stability, vibrations, restraints, sensitivity analysis.

INTRODUCTION Behavior of the thin-walled members is described by means of so-called state variable s such as: displacements, internal forces, reactions, critical buckling loads and frequencies and modes of free vibrations. The values of these state variables depend on many parameters of the members, known as design variables d. In many problems of engineering practice it is very useful to know a direct relation between the state variable variation 6s and the design variable variation of 5d. Sensitivity analysis (Haug, Choi & Komkov (1986)) enables to derive such relations. Since the sensitivity analysis of structures undergoing bending and compression or tension is well developed, the present paper deals with the sensitivity analysis of members subjected to torsion. From the mathematical point of view, one can distinguish two kinds of design variables:

54 Continuous variables, for example, the cross-section dimensions and the member material constants, Discrete variables, for instance, the restraints stiffness and their location and the support position. In case of the variation of the continuous design variable the first order variation of the state variable sought can be expressed as follows -

-

1

~Ss= IFsa(z)Sd dz

(1)

0

where the underintegral function Fsd(z)can be considered as the influence line of the state variable variation due to the unit point variation of the design variable. If the discrete design variables are taken into account, then a similar relation between the state variable variation and the vector of the design variable variation 5d is 5s = WsT &l

(2)

where vector Wsd consists of the first order sensitivity coefficient corresponding to the design variable and (...)T denotes transposition of the vector. The usual assumptions of the classical theory of thin-walled members with nondeformable crosssection (Vlasov (1959)) adopted in this paper are: (1) The thin-walled member is prismatic, (2) The static loads are conservative, (3) The member cross-section is not deformed in its plane but it is subject to warping in the longitudinal direction, (4) The shear deformation in the middle surface vanishes, (5) The deformations and the strains are small, (6) The member material is homogeneous, isotropic and obeys Hooke's law. Because a lack of a general theory of thin-walled members with arbitrary variable cross-section, the sensitivity analysis is restricted to the member cross-section with one or double axis of symmetry. It is well known that for the bisymmetric cross-section, torsion of the member can be considered independently of bending, as in case of monosymmetric cross-section bending with respect of the symmetry axis independently of torsion and bending with respect to the second axis. Three types of elastic restraints are considered in this paper: the flexural restraint against the lateral displacement of the member axis, the torsional restraint against the member cross-section rotation, and the warping restraint against the cross-section warping. The behavior of the restraints is modelled by suitable linear elastic supports. The sensitivity analysis problems are investigated only for linearly elastic range of the member material behavior.

SENSITIVITY ANALYSIS OF MEMBERS SUBJECT TO STATIC LOADS

Continuous design variable Consider a thin-walled beam with bisymmetric open cross-section shown in Figure 1. The rectangular coordinate system x,y,z is so chosen that x and y coincide with the principal axis of the cross-section and z coincide with the longitudinal axis of the beam. The beam of length I is subjected to a distributed torque m, and axial end loads P. Moreover, two types of restraint, torsional and warping, are imposed on the member. Following Szewczak, Smith & DeWolf (1983) continuously distributed restraints under consideration are modelled by linear elastic foundation with corresponding stiffness k0, kw. In this case the torsion of the member can be considered independently of bending. A detailed treatment of the member behavior is presented by Trahair and Bild (1990). The first variation of the state variable 5s determining the behavior of the member due to some variations of the design variable 5d is sought.

55

m

P._l'.z/r

kw, k0

:-i

'

k0

\

kw

Figure 1" Thin-walled member with bisymmetric cross-section The angle of cross-section rotation and its first derivative, the internal force at a specified cross-section or reactions at the supports and restraints are assumed to be the state variable. Our attention in this section is concentrated on the continuous design variables, for example,: the cross-section dimensions, the material moduli, and the stiffness of the continuously distributed restraints. Taking advantage of the concept of adjoint structure (Dems & Mr6z (1983)), the first variation of the design variable can be written as (compare Budkowska & Szymczak (1992a)) 1

5s=

),d

+

+kw,d

+k0,d

0 0 where E and G are Young's and shear modulus, respectively; Iw stands for the warping constant, Id denotes the St. Venant torsion constant, r represents the radius of gyration of the cross-section, 0 m

and 0 are the angle of the cross-section rotation for the external loads and adjoint member, respectively. The derivative with respect to z is denoted by prime and ("'),d is the partial derivative with respect to the design variable d. The underintegral function Fsd(Z) is considered to be the influence line of the state variable variation due to point unit variation of the design variable. The static analysis of the member necessary to determine the influence line can be carried out by the FEM. The first numerical example deals with continuous thin-walled I beam presented in Figure 2. The computer program elaborated on the basis of the sensitivity analysis presented enables to obtain the influence line of the torsional moment on the right side of support B due to point torsional restraint with unit stiffness K0 = 1. m = 2 kNm/m for 0< z <7.2

!z

J ...... .......

A

.L

.a_

~

-

9

4x0.8

Kw = 50 kNm 3

S .....

,,~-

..,

3x0.4

3.2m

~

M=2kN

1.2

~.

K0 = 50 kNm

0.8

4x0.4 4m

1.2m 2m

t,b,j ~ , t

h-0.2 ,

t - 0.01 m b = 0.15 m (B-C) v 0.2 m (A-B, C-D)

Figure 2: Thin-walled I-beam

Discrete design variable Now consider the effect of variations of discrete design variable on the behavior of the member. Of course the effect of such discrete variable as the restraint dimensions or the spring support stiffness can be determined using the above presented influence line. If, for example, one dimension di of the torsional restraint at a specified cross-section z = z0 is taken into account, then the variation of the

56

restraint stiffness 5K0 can be expressed by the design variable variation 6di 5K0 = (K0),(t (Sdi,

(4)

and the state variable variation 5s can be calculated by multiplication of the value of the influence line at z = zo by relation (4). 0.0 0E+0 '-": .

.

.

.

1.0 '

.

.

.

.

2.0 |

.

.

.

.

3.0 |

.

.

.

4.0

.

|

.

.

.

5.0

.

,

.

.

.

.

6.0 |

9

i

,

,

7.0 I

,

,

.

-

8.0 |

.

.

.

.

9.0 ,,

9

z[m] -2E-3 -4E-3 -6E-3" -8E-3 Figure 3: Influence line of torsional moment on the right side of support B due to point torsional restraint with unit stiffness K0 = 1 Therefore only the discrete variable related to location of the member support and the restraints is investigated in this section. The member should be divided into some sections from one restraint to another and in order to find the variation of the state variable, at first some shifts of the section ends 5zo and 5zl (see Figure 1) are analysed. Using the rules of calculus of variations for variable boundary (Gelfand & Fomin(1970)) for the total potential energy of the member and taking into account above mentioned concept of the adjoint system one can derive the variation of state variable 5s due to assumed shifts of the member ends (see Iwicki (1997))

(-

+

(-

+

Ms0,- 0. zo

where B is the bimoment, Ms denotes the torsional moment and by overbar the values of the state variable at the adjoint member are represented .Having used relation (5) for each section of the member and after utilising the continuity conditions, some suitable formulae for the state variation for shift of arbitrary support or restraint can be derived. To illustrate the theoretical considerations a clamped beam subject to uniformly distributed torque is given in Figure 4. The relative variation of the bimoment at the beam middle cross-section 5B/B due to the relative shift of the fight support 5zr/1 is determined and shown in Figure 5. The accuracy of the approximation of the bimoment change given by sensitivity analysis is compared with the exact results. in

z

f:

t.:_i

~...r .....-J - " ....~ ' ...... - - .....,,+- ....- - .....

2 in

~

~1 - 4 in

i.\ ,

.] - 5z~

1 0.2

t-0.01 in, E=210GPa, G-80.77 GPa

Figure 4: Clamped thin-walled I-beam with shift of right support

57 0.4

0,2

-

-

!

Sens Exar

itivity analysi~ t results

0 -

8B/B

J

f -0.2

-f

J

f

I

-0.4

-

-0.1

0 0.1 ~Szl/1 Figure 5: Relative variation of bimoment at the middle cross-section vs. relative shift of beam end

SENSITIVITY ANALYSIS OF CRITICAL BUCKLING LOADS

Continuous design variable Consider the flexural-torsional buckling of a thin-walled member of length 1 with open monosymmetric cross-section shown in Figure 6. Three kinds of restraints are imposed on it. Besides the two above mentioned torsional and warping restraints, an additional one namely a flexural restraint of stiffness ku against horizontal displacement u of the member axis is imposed. The member is subjected to axial end loads P, a distributed load q and two end bending moments Ml and M2. It is also assumed that the moments depend on one parameter m. Thus a three-parameter-stability problem is formulated.

m-Ml~-'~--~ g P ~

"z"Z

ko,kx,kw

m.M2 _~0

. . . .

"/ 1

//"

~

y

Figure 6: Thin-walled beam with monosymmetric open cross-section The total potential energy V of the beam in a buckled position may be written as (Trahair & Bild(1990)) V=0.5 S{ElyU.2 +El w0.2 +Gld 0,2 _ p [u,2 + ~ 2 + yo 2 )~,2 + 2you,0,]}dz 0

+0.5 ~[M (20u" + 13x0'2 )+ kuu2+ k002+ ks0'2 + q(yq_ Yo)02 ]dz, (6) 0 where Iy is the moment of inertia of the cross-section about axis y, Y0 stands for the distance of the distributed load to the member axis, yq denotes the place of acting of the distributed load q and 13x represents the coefficient of the cross-section. Using the first variation the functional (6) with respect to the design variable and taking into account that in critical state the potential energy vanishes we arrive

58 at the following relation between variations of the critical values of the applied loads and the design variable variation (Szymczak & Mikulski (2000)) 1{(EIy),d u"2 + (EIc0),d 0"2 + (GId),d 0'2 +ku,d u2 +k0, d 02 + k s , d 0,2}5ddz 0 + f{M(~x ,d 0'2)- P [~2,d +yO2,d)O'2 + 2YO,d u'O']+ q(yq,d-YO,d)02 }Sd dz

(7)

0

+~{~Mcr ~0u,+ ~x0,2 )._ ~SPcr[u,2 + ~2+ y02 )0,2+ 2Y0U,0,]+ &:lcr(yq_ Yo)02 }dz =0 0 Eqn.7 enables to determine variations of the critical loads in all possible interactions between them. The buckling mode described by displacements 0, u and their derivatives can be determined using the analytical method or the FEM in more complex cases. As a numerical example a simply supported beam subject to axial end loads P and two equal bending moments M at the ends is considered (Figure 7). bg= 0.I

pz~ ~Mi c r

~~.~Mcr

t?

I0 .2

k

l=8m

p

~~

Ijl\

..1

"ba = 0.2 " m E = 210GPa, G = 80.769 GPa, t = 0.01 m

Figure 7: Simply supported thin-walled beam with monosymmetric cross-section 250 200 P

150 gr

[kN]

100

/.

. i

~,B

B

.~.

B

M

/

-

50_

-100

-50

0

50

100

M cr

[kNm]

Figure 8: Interaction curves of critical load Per vs. critical moment Mcr for beam with monosymmetric cross-section (M) and bisymmetric cross-section 03) Figure 8 shows the interaction curves of axial load P and bending moments M for beams with monosymmetric cross-section and comparable bisymmetric cross-section. The width of flanges of the bisymmetric cross-section is equal to an average width of monosymmetric beam. The influence lines of the relative variation of the critical buckling moments for positive (M > 0) and negative (M < 0) values due to a point restraints with unit stiffness are established and shown in Figures 9, 10 and 11. The axial end loads are not taken into account (P = 0). The variations of the critical moments are presented with reference to the critical moments of beam without any restraints. Moreover, similar influence lines

59 corresponding to I beam with bisymmetric cross-section are also added, for comparative purposes. More details for the case of members with a bisymmetric cross-section can be found in Szymczak (1999a). 10p

,,_

D M>0 M<0

8Mer/Mer [10 3]

B

5

.1

\

0

2

4

6

8

z[m] Figure 9: Influence lines of relative variations of critical moment 15M~r/ Mcr for monosymmetric and bisymmetric cross-section (B) due to point flexural restraint unit stiffness Kx = 1 _

f

.~

//

P= 0 M>0 M<0

5Mcr/Mcr

[10 -3 ]

-

2.5

B

\

oi 0

2

4

6

8

z[m] Figure 10: Influence lines of relative variations of critical moment ~Mcr / Mer for monosymmetric and bisymmetric cross-section (B) due to point torsional restraint unit stiffness Ko = 1 Moreover, for thin-walled beams with bisymmetric cross-section the effect of residual normal stresses t~oon torsional buckling and post-buckling behavior of axially loaded columns is also investigated (see Szymczak (1998)). The variation of the critical buckling due to the design variable variation with allowance for arbitrary distributed residual stresses can be written as

~JPcr = ~ 0

w ,d

+ GId,d

- R0a,d

+ kw ,d

+ k|

r2

(8)

where the term including the effect of the residual stresses is R0a = Its0(x 2 + y2)dz

(9)

A The results obtained by Szymczak (1998) indicate that the residual stresses can cause an increase or decrease of the critical buckling load depending on the stress distribution over the cross-section. It is also evident that the point of bifurcation of torsional buckling is symmetric and stable independently of the residual stresses, that means insensitivity of the buckling load to inevitable geometrical imperfection of the column axis.

60

12

-'

=0 ---- M > 0 ]kA lVl

JN ~ u

8Mcr/Mcr

[10 -3]

0

2

4

6

8

z [m]

Figure 11" Influence lines of relative variations of critical moment ~Mcr [ Mcr for monosymmetric and bisymmetric cross-section (B) due to point warping restraint unit stiffness Kw = 1 Discrete design variable

Consider the variation of the flexural-buckling load due to the discrete variable variation. As explained above some changes of location of restraints, supports or the member ends are taken into consideration. The derivation of the first variation of the critical load due to the member end shifts 5z0, 5z~ proceeds in a similar manner to that of previous chapter but the final results are very complex to write them in a general way. If a special case of one parameter load M~ = M2 = M for fixed axial and transverse load (P, q- constants) is investigated, then the variation of the critical moment can be written as 5Mcr

= a-l(Wlz=16Zl- Wlz:05Zo)

(10)

where a=

u" + 0.5J3 x 0

W = 0.5{EIy (u " 2 - 2 u ' u ' ) + EI w {0" 2 - 20'0")+ GIde '2 - kx u2 - ko02 + kwe '2 } _ 0.5{P[u,2 + ~2 + yo2)9,2 + 2you,0,]+ Mcr (2u,0,_ 13x0,2)+ q(yq _ yo)02 }

(11)

and by ("')lz=k the expression calculated for z = k is denoted.

I-----

1.5 -

Sensitivity (M>0) Exact (M >0) Sensi!ivity (M <0) F.~tact (u ~n) J

1.3 i

5Mcr/Mcr

1.1

-

0.9

-

0.7 ....

0.8

0) 0.9

1

1.1

1.2

1+ 8zl/1 Figure 12: Comparison of exact relative variation of the critical moment ~Mcr / Mcr vs. relative shift of beam 8Zl / I to its approximation obtained by sensitivity analysis

61 A numerical example illustrating a theoretical consideration deals with the beam presented in Figure 7 The axial loads P are neglected. The critical moment variation ~Mcr due to the beam end shift 5zl is sought. A comparison of the relative variation of the critical moment 5 M J Mcr obtained by means of sensitivity analysis with the exact results is presented in Figure 12 for a positive and negative value of the critical moment. A good agreement of approximation of the exact results by the sensitivity analysis should be noted. It is also very important, from the engineering point of view, to give a more detailed specification of the point of bifurcation. It is well known (Szymczak (1980)) that the point of bifurcation of torsional buckling of thin-walled column with bisymmetric cross-section is symmetric and stable. It means that the critical load is insensitive to geometrical imperfections of the column axis. Szymczak (1999b) gives a contribution to the effect of restraint on the post-buckling behavior and the bifurcation point. It is worthwhile to point out that, for some stiffness of the restraint, the bifurcation point may be symmetric but unstable. In this case a drastic decrease of the critical buckling load can occur.

SENSITIVITY ANALYSIS OF F R E E TORSIONAL VIBRATION

Continuous design variable Consider free torsional vibration of a thin-walled member with bisymmetric open cross-section shown in Figure 1. The member is subject to axial end loads P but only torsional restraints against the crosssection rotation are taken into consideration. The first variation of the eigenvalue, i.e. the square of natural frequency 5~ = 50a2 due to variation of the distributed parameter ~Sd is sought. Making use of equivalence of the extremum values of the kinetic and potential energy and calculus of variations one can derive the variation under consideration (Budkowska & Szymczak (1992c)) 1

5~. = ~{(EIw),d 0"2 + [(GId),d-(pr2),d ]0 '2 + k0, d 0 2 - ~ o

(12)

[(Plo), d 02 +(PIw), d 0'2]}5d d z / ! p(Io0 2 + I w 0,2 )dz where p is the mass density of the member material and I0 stands for the polar moment of inertia of the cross-section. In Eqn.12 0 represents the mode of vibration corresponding to the frequency of free vibration sought. The dynamic analysis of free vibration to determine the frequencies and modes can be camed out analytically in simple cases or in numerical way by the FEM. The latter manner enables to obtain values of the influence line only in nodes of discretization of the member. Moreover, in this way the analysis in the case of discrete distribution of the restraints is more accurate. A simply supported I beam with constant cross-section subject to axial end loads P given in Figure 13 is acknowledged to be a numerical example. The relative variation of the first three eigenvalues 5~i]~,i for i = 1,2,3 due to the relative variation of the beam flanges 5b/b are under investigation. The influence lines obtained from the sensitivity analysis are presented in Figure 14. Flange width variation zone 5b 1/2=2.00 m ~I ~-I P---1' -.§ ...... ' ...... ~

I__~

Y

1=4.00 m

b=0.20

p

0.01 ~

~

Figure 13: Simply supported thin-walled I beam

x--T~,1 0.01m

y

62

1-st

0.8

2-nd 3-rd

~",-~

eigenvalue eigenvalue eigenvalue

/\

ra

0.6

0.4

0.2

0 .0

i

!

'"

0 2 z Im| 4 Figure 14: Influence line of relative variation of three eigenvalues of torsional vibration 8~,i/~,i vs. relative variation of the I beam flange width 5b/b 0.25 exact solution

zl

sensitivity solution

f

I

f

0.00

-0.25

i

-0.50

i

i

i

I

i

0.00

I

I

I

5b/b

I

]

0.50

Figure 15" Approximation accuracy of relative change of the first natural frequency square to the I-beam flange width change in half of the span 0.6

....

0.4

P= 1 M N P=-I MN P-0 ~,,. ~" ~--/ / / / /

~5~L1/~l due

\ \ \

0.2

0.0

J

-0.2 0

2

z [m]

4

Figure 16: Effect of axial loads on the influence line of relative first eigenvalue due to relative variation of the I-beam flange width 5b/b

63 Figure 15 shows accuracy of the approximation of change of the first eigenvalue ~ i / ~ i due to the Ibeam flange width change in half of the span as seen in Figure 13. A good agreement of the sensitivity analysis result with the exact calculation is observed. Moreover, a distinct change of distribution of the influence lines of the first eigenvalue for three values of the axial loads is presented in Figure 16. The torsional vibration analysis is performed by the FEM, using a typical cubic shape function (Szymczak (1978)) Now, the variation of the vibration mode due to the design variable variation is investigated. It is assumed that the member is divided into some elements and Di denotes the eigenvector under consideration. Because the eigenvectors establish a complete set of vectors in the R k space, then the variation of the eigenvector Di can be represented by their linear combination m=k

8D i = 5". aijmDmSd j m=l where coefficients a0k can be written as

(13)

[OK

DIT ~ j - ~ ' i ~ j j aijl =

i fori41

(14)

(~i-~1)

and for i = 1 aiji = - 2 D T ar M D 19

(15)

where K and M are the stiffness and the mass matrices, respectively. Obviously, Eqn. (14) is valid only when ~i ~ Z.1 i.e. for simple (non-repeated) eigenvalues. To illustrate the results of this section, let us consider the same simply supported I-beam with a constant cross-section shown in Figure 13 but stiffened by three nodal torsional restraints of equal stiffness K0~ = 1(02 = 14,o3= K0 = 1000 kNm as shown in Figure 17. Some modifications of the stiffness of the restraints are assumed 814,01= 0.2K0, 81(02= 0.1K0, 8K03 = -0.214,o. The exact change of the first mode of the torsional vibration is determined by means of the FEM analysis and it is compared to the mode established by the sensitivity analysis in Figure 18. A very good approximation of the mode change obtained by means of the sensitivity analysis is worth noticing. b=0.20 m

z

0"01AA~/~ X~'20

' r

L am

~

lm

,.1., lm

J., lm

m

I K0= 1000 kNm

Y Figure 17: Simply supported I beam stiffened by three torsional restraints

Discrete design variable Now consider the variation of the free torsional vibration eigenvalue of the beam due to change of location of its end, the supports and the restraints imposed on the beam. At first, some shifts of the beam end 8z0. 8z~ are taken into account. As in problems discussed above, by use of the calculus of variations the frequency square variation 8X is derived.

64 60 sensitivity analysis exact solution

I

40

20 8Kol--0.2 Kol

0

\

~Ko2=0.1 Ko2 ~5K03=-0.2 K03 .... !

-20

i

0

1

i

!

2

3

z[m]

4

Figure 18: Comparison of exact first mode of torsional vibration with sensitivity analysis result for modified beam restraints 8~, = {[-EI w 0 ~2 + 2(EI w0")'0' - (GI d - [ - E I w 0"2 +2(EIw0")'0'-(GId

-

+ k~ pr2)0 '2 +ko 02

I

- P~'(I~

- Iw 0'2)] z=l

-

-

p~,(Io 02

Iw0'2)]z_ I ~ 6Zo}/

(16)

1

JP(Io 02 + Iw 0'2)dz 0 Having this variation derived, one can divide the member into some sections from one restraint to another and apply Eqn. (16) to each section. Also taking advantage of the continuity conditions in all points, where the sections are connected, it is possible to obtain an appropriate variation of the eigenvalue for any restraint under consideration. A simple example dealing with the simply supported I beam subject to axial end loads (Figure 19) allows to illustrate effectiveness of the sensitivity analysis presented. The variation of the beam length 81 is assumed. The exact first eigenvalue changes are found analytically and compared with the sensitivity analysis results. In order to investigate the effect of the axial loads on accuracy of the approximation a relative eigenvalue change for P = 0 and P = 1.5 MN is provided (Figure 20). In this case the torsional restraints are neglected. Moreover, the effect of continuously distributed torsional restraints is also studied. Figure 21 shows the accuracy of the sensitivity analysis approximation of the relative egenvalue change for the beam with and without restraints of stiffness k0 = 2 kNm. It is observed that in all cases a fairly good approximation is obtained even for twenty per cent change of the beam length. L ~ rPkr P==I

I ._L

_.L_

9

z

--

-K0

~//~

I

I ==:I Pkr I .L_.N_

~ 1-/~rl

.

.

a F I

b

"y

1+61

I"

j fl

E=210GPa, E/G=-2.6 Figure 19: Simply supported I beam

.

.

O.

"I

O01 9 2

65

4.00

. . . . .

k0=0 ~~--

-

3.00 ~l(1+61) -

P=0 P = 1.5MN

~

......

~',,

1.00 ~ ' ~ ~ ' ~ L ~ , ~

.

-0.3

-0.'2 -61i-

0

0.1

61 / 1

().3-

Figure 20: Comparison of exact change of first eigenvalue of torsional vibration with the results of sensitivity analysis (straight lines) for P = 0 and P = 1.5 MN 3.00 -t ~(1+61)

k0=0

~(1)

k0= 2 kNm

2.00

1.00

0

!

-0.3

-0.2

-O.1

0

0.1

!

61 / 1

0.3

Figure 21" Comparison of exact change of first eigenvalue of torsional vibration with the results of sensitivity analysis (straight lines) for k0 = 0 and k0 = 2 kNm

FRAMES AND GRIDS Frames and grids assembled from thin-walled members are very often used in the industry and civil engineering structures. The sensitivity analysis of these structures is possible provided the members are not undergoing torsion. If the members are subject to torques an additional internal force, referred to as bimoment arises. Unfortunately one can not establish an equilibrium condition for the bimoments in the structural joints similar to the well known conditions for other internal forces, therefore up till now in many papers dealing with these structures it has been assumed that the members are pinned or fixed connected in nodes. Obviously one can observe some discrepancies, in many cases, significant, between the results corresponding to these extreme assumptions. An interesting theoretical and experimental investigation of the problem of frame analysis made of I members has been presented by Szmidt (1975). In order to find a better solution to the problem of the bimoment distribution in nodes Yang & McGuire (1984) introduced the warping indicator concept. According to this idea each member is connected to the node by means of warping springs of appropriate stiffness but no rules for

66 the stiffness are given. Krenk & Damklide (1991) point out that in order to fulfill the continuity conditions in the node in which some thin-walled members are connected, deformations of their crosssection is necessary. Of course, the effect of the cross-section deformation on the frame behavior depends on the geometrical and material parameters of the structure. Finally one can state that the problem of static analysis of frames and grids made of thin-walled members, in compliance with onedimensional theory, is not generally solved in successfully way. Thus a direct application of the sensitivity analysis to the frames and grids is possible only by making allowance for above mentioned, simplification of the bimoment distribution.

APPLICATIONS The presented sensitivity analysis of thin-walled members enables us to obtain an approximation of the state variable variations due to some changes of the member parameters without repetition of the analysis of modified structures. This feature can be useful for designing structure, whereas some of its modifications are necessary in order to fulfill constraints related to displacements, critical loads or frequency of vibration. Moreover, it is possible to apply the sensitivity analysis developed in following problems: Analysis of thin-walled members of variable cross-section especially in a situation when a one state variable is of primary significance, for example the critical buckling load or the displacement in a specified cross-section. Consider, for instance, an I beam of variable width of flanges as shown in Figure 22. Then instead of using the FEM for the member of stepped cross-section one can take advantage of the sensitivity analysis. It is necessary to find the influence line for the variation of the state variable specified vs. the width variation for the beam with constant, for example, average width of flanges and next to calculate the change of the state variable from Eqn. 1. -

!

Figure 22: Variable flange widths of I beam An optimal design of thin-walled members by means of the gradient method. The state variable variation established with aid of the sensitivity analysis can be used directly for the optimization algorithm. - Identification of some parameters of the thin-walled members. Many assumptions of the theory of thin-walled beam mentioned in the introduction cause some discrepancies between the results and experiments carried out on real structures or the analysis by means of more general theory, for example, the theory of shells. In order to improve the mathematical model of the structure, the identification of its parameters on the basis of experiments or the theory of shells is advisable. The least square method is most useful for this purpose. Application of the sensitivity analysis to the identification process is very effective. Some variations of the model parameter di for i = 1,2...n in order to improve the mathematical model are sought. The state variable vector s is determined using the theory of thin-walled members but its variation can be obtained by sensitivity analysis as a linear function of the parameter variation 5di. The objective function to be minimized with respect to the variation of the parameter vector variation f~! is

-

minF(g~) = ~

i~lai si + ~ s i - s i

(17)

where s i is the state variable established experimentally or as a result of a more general theory, and ~i denotes the weight coefficient. The method of identification is successfully applied to the settlement of the tranverse stiffener stiffness in I beams (Szymczak & Mikulski (2000b)). The results indicate that

67 the stiffness of the transverse stiffeners recommended by many researchers, for example, Szewczak, Smith & DeWolf (1983)) should be lowered in order to achieve better compatibility with the shell theory. - Analysis of structures with probabilistic parameters, for example, some imperfections with a given distribution of probabilistic density. The method of linearisation, very popular in analysis of this kind, requires knowledge of the linear relation between the state variable and the parameter variations. It is possible to apply the sensitivity analysis for this purpose. Choice of the best sites for the measurements in order to investigate the material properties of the structure, or to check its dimensions. These measurements should be carried out in the cross-section where the sensitivity coefficient is very large because this means a very big influence on the state variable under consideration. It is a very important problem in engineering expert researches, but very often the choice of the measurements is determined by chance. -

CONCLUSIONS A review of problems and applications of the sensitivity analysis of thin-walled members is presented. The members subject to static loads involving torsion, stability, and free torsional vibration analysis are taken into consideration. Many carefully selected numerical examples illustrate effectiveness, efficiency and broad spectrum of applications of the theoretical investigation. Accuracy of approximation of the state variable changes due to some design variable variations is also studied. The results obtained show in the prevailing cases a sufficiently good accuracy of the approximation even for a large twenty per cent change of the design variable. Numerous interesting detailed conclusions regarding the effects of some parameters on the influence lines determined by the sensitivity analysis can be formulated. Further expansion of the application of the sensitivity analysis to frame and grid structures requires an effective solution of the problem of bimoment distribution in nodal connections of the structural members.

AKNOWLEDGEMENTS The author would like to thank Piotr Iwicki and Tomasz Mikulski, Assisant Professors, Faculty of Civil Engineering, Technical University of Gdafisk for assisting in the numerical calculation of some examples. The financial support of the Polish Committee of Scientific Research (KBN) under Grant No. 7 T07E 01519 is gratefully acknowledged.

References Budkowska B.B., Szymczak C. (1992a). Sensitivity Analysis of Thin-Walled I-Beams Resting on Elastic Foundation. J. Engng. Mech. ASCE 118:6, 1239-1248. Budkowska B.B., Szymczak C. (1992b). Sensitivity Analysis of Critical Torsional Buckling Load of Thin-Walled I-Columns Resting on Elastic Foundation. Thin-Walled Struct. 14:1, 37-44. Budkowska B.B., Szymczak C. (1992). Sensitivity Analysis of Free Torsional Vibration Frequencies of Thin-Walled I-Beam Resting on Elastic Foundation, Thin-Walled Struct. 13:3, 399408. Dems K. and Mr6z Z. (1983). Variational Approach by Means of Adjoint Systems to Structural Optimization and Sensitivity Analysis - I, Variation of Material Parameter within Fixed Domain. Int. J. Solids Struct. 19"8, 677-692.

68 Haug E.J., Choi K.K. & Komkov V. (1986). Design Sensitivity Analysis of Structural System. Academic Press, Orlando. Gelfand I.M. and Fomin S.W. (1970). Calculus of Variations. PWN, Warszawa (in Polish). Iwicki P. (1997). Problems of Sensitivity Analysis of Thin-Walled Bars with Bisymmetric Open Cross-Section Subject to Static Loads. Ph.D. Thesis, Faculty of Civil Engineering, Technical University of Gdafisk, Poland, 1-114 (in Polish). Krenk S. and Damklide L. (1991). Warping of Joints in I-Beam Assemblages. J. Engng. Mech. ASCE 117:11, 2457-2474. Rogers L. C. "Derivatives of Eigenx;alues and Eigenvectors, A/AA J., 8, 943-944, 1970. Svensson S.E. and Plum C.M.(1983). Stiffener Effects on Torsional Buckling of Columns. J. Struct. Engng. ASCE 109:3, 758-772. Szewczak R.M., Smith E.A. & DeWolf J.T. (1983). Beams with Torsional Stiffeners. J. Struct. Engng. ASCE 109:7, 1635-1647. Szmidt J.K. (1975). Analysis of Frames made up of Thin-Walled Members. Engrg. Trans., 23:3, 447-472 (in Polish). Szymczak C. (1978). Torsional Vibration of Thin-Walled Bars with Bisymmetric Cross-Section. Engng. Trans., 26:2, 267-274 (in Polish). Szymczak C. (1980). Buckling and Initial Post-Buckling Behavior of Thin-Walled I Columns. Comput. Struct., 11, 481-487. Szymczak C. (1998). Effect of Residual Stresses on Buckling and Initial Post-Buckling Behaviour of Thin-Walled Columns. Arch. Civil Engng., 44:4, 287-297. Szymczak C. (1999a). Sensitivity Analysis of Critical Loads of Flexural-Torsional Buckling of Thin-Walled I Beam. Arch. Civil Engng., 45"3, 401-412. Szymczak C. (1999b). Effect of Elastic Restraints on Buckling and Initial Post-Buckling Behaviour of Thin-Walled Column. Arch. Civil Engng., 45-4, 601-615. Szymczak C. and Mikulski T. (2000a). Selected Problems of Sensitivity Analysis of Thin-Walled Members with Open Cross-Section. Proc. IX Symposium on Stability of Structures, Zakopane, 269276 (in Polish). Trahair N.S. and Bild S. (1990). Elastic Biaxial Bending and Torsion of Thin-Walled Members. Thin-Walled Struct., 9:2, 269-307. Szymczak C. and Mikulski T. (2000b). Identification of Model of Stiffeners of Thin-Walled Bars with Open Cross-Section. Proc. XLVI Scientific Conf. KILiW PAN and KN PZITB Krynica 2000, Wroctaw-Krynica, 1, 169-174 (in Polish). Vlasov V.Z. (1959). Thin-Walled Elastic Beams [in Russian]. Fizmatgiz, Moscow. Yang Y.-B. and McGuire W. (1984). A Procedure for Analysing Space Frames with Partial Warping Restraints. Int. J. Numer. Meth. Engng., 21}:8, 1377-1398.

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rightsreserved

69

SOME OBSERVATIONS ON THE POST-BUCKLING BEHAVIOUR OF THIN PLATES AND THIN-WALLED MEMBERS. J Rhodes Department of Mechanical Engineering, University of Strathclyde, Glasgow, Scotland, UK.

ABSTRACT A brief examination of some of the research on the post-buckling analysis of plates and plate structures is outlined. Only a very superficial examination has been carried out. The behaviour of plates and sections under eccentric load or eccentric compression is considered, and the application of plate analysis to strut, beam and column design is discussed.

KEYWORDS Plates, thin-walled members, struts, beams, columns, buckling, post-buckling, compression, bending

INTRODUCTION

Analysis of thin plates in compression probably began with von Karman et. al.(1) in 1932, in which the concept of "effective width" was developed and explained. Cox (2), two years later, produced an analysis of compressed plate behaviour in which the effects of buckling deflections on the plate membrane strains and stresses were developed on the basis of geometrical analysis. Cox considered "averaged" membrane strains, and neglected some in-plane effects, so that his approach tended to underestimate the plate post-buckling strength and stiffness. Indeed this approach was later sometimes termed the "lower bound method", although it was not in fact a lower bound approach. Cox also developed more rigorous plate analysis methods, e.g in (3), but his original method has been used by many authors as a simple and expedient method of plate analysis. Probably the first researcher to produce a rigorous elastic plate post-buckling analysis was Marguerre (4). Using Marguerre's analysis the in-plane strains and stresses are determined from the exact satisfaction of compatibility conditions with the out-of plane deflections and the Principle of Minimum Potential Energy is used to furnish the final relationships between loading and displacements. Since Marguerre's paper there has been a proliferation of publications on plate buckling behaviour and its effects on the behaviour of beams, struts and columns. From the 1940s through the 1960s many papers on rigorous analysis using the von Karman large deflection equations, and on extensions of Marguerre's

70 approach were published, and it is outwith the scope of this article to examine these publications in depth. In this paper we shall briefly consider some theoretical plate analyses.

POST-BUCKLING BEHAVIOUR OF PLATES AND SECTIONS. Plate b e h a v i o u r at and after buckling.

Plate buckling, and post-buckling behaviour, is illustrated in Figure 1 for a thin-walled section in which local buckling is present in all the plate elements.

1

i

I Figure 1. Thin-walled locally buckled member In the elastic range the buckled portions of the plate shed load, and become ineffective in resisting further loading, while in the portions of plate close to supports the out of plane buckling is diminished, and these parts have post-buckling reserves of strength and stiffness. The plate as a whole sustains increases in load after buckling, but the axial stiffness reduces. This effect is demonstrated in Figure 2, where point A is the

Perfect plate " LoadP

S

S

S

I

~Imperfect _ plate

End displacement u Figure 2. Load - end displacement characteristics of locally buckled section

buckling point. For a plate without imperfections the post-buckling axial stiffness drops immediately upon buckling, and thereafter reduces still further as loading increases. Also because of the highly

71 redistributed stress system the maximum stress grows at an increased rate after buckling, ensuring earlier failure than if the plate remained unbuckled. As the load increases the stresses also increase, and for metallic plates failure generally occurs at a load close to that at which the maximum membrane stresses within a plate become equal to the yield stress. The plastic behaviour of plates and sections is in itself a very large and important field of study, but this will not be considered in this paper.

"Lower

bound"

plate

analysis

method.

Consider a plate of width 'b' and length 'a' as shown in Figure 3. If the plate is initially compressed axially by an amount 'u' and then buckles into a sinusoidal wave along its length with value 'w' in any strip, the average axial strain in an elementary strip of plate as shown due to the interaction of in-plane and out-of plane deflections is:-

fi/

/ ~r

/--..

~X

..--/

Figure 3. Deflected elementary strip of plate -- -

6X=a

-

w2

(1)

4

If we assume that membrane stresses act in the x-direction only, and the variation of deflections w is given by the expression w = A.Y(y) the strain energy of membrane actions can be written as follows:-

VD=I

E 6 2 dV = E t a I ( u ) 2 l(X]4A4~y4dy x . + 2 2 16k, a J 0

_

. ~ . ( u ] ( ~ ) l 2A2by2., dy ] 0

(2)

The total potential energy is the sum of the membrane energy, the strain energy of bending, and that of any external restraints. The strain energy of bending can be written as:

v. =---4-

JolL~

r 12+ 2 ( l - v )

LtayJ +w ay JJ

: OaA2 ......... 4

(3)

Adding the membrane energy to the bending energy to obtain the total potential energy, V, minimising this by differentiating V with respect to A, and rearranging leads to the following expression for the deflection magnitudes A:-

72

! y2.

A2= 4

a2O b ,: 2

--

o~Y4dy

7

(4)

U - x E E t !yEdyJ

When the term inside the second bracket is zero we have the critical value of u to cause local buckling. We can thus replace the second term inside these brackets by u....cR.Substituting Eqn.(4) into (1) gives:a b

f y2dy x y2

e~ =

b

a

-

(5)

~y4 dy 0

The total load on the plate can be obtained by integrating E t ex across the plate, to get

P=Etb{ u-

K, IU - -~-l} where K, =

(!y2. (6)

b

b ~y4dy 0

Prior to buckling the load - strain relationship is simply

dP du

Ebt a

could be written as ---- = ~ .

P = E b t u_. The a

pre-buckling axial stiffness

dP du

The post-buckling axial stiffness can be written as ~

Et b a

=~

(1 - K~).

The ratio of post-buckling to pre-buckling axial stiffness is otten described by the quantity E*/E, where E* is a fictitious elasticity modulus. Using the lower bound method the ratio E*/E is given by:E'/E=

1-K~

(7)

Equation (7) provides a simple approximate evaluation of post-buckling stiffness for different boundary conditions by direct substitution of suitable functions describing the deflection across the plate. For example the deflections for a plate with simply supported unloaded edges can be taken as Y = sin :r__~y. b Substitution into Equation (3) gives E*/E = 1/3. For some other standard boundary conditions suitable functions and resulting E*/E ratios are as follows:Unloaded Edges fixed-fixed:-

y = sin2 :r_.~y b

:- E*/E=18/35

Unloaded Edges simply supported-free:-

Y =--Y b

:- E*/E=4/9

Unloaded Edges free-flee:-

Y = constant :- E*/E=0

Thus for plates supported on both edges the effects of local buckling are to reduce the axial stiffness to around half of its pre-buckling value. It is interesting to observe that the plate theory used shows agreement with simple column theory in predicting zero post-buckling stiffness for plates with no

73 restraint on the unloaded edges. It is also worthy of notice that this simplified approach gives no variation of post-buckling stiffness ratio with variation in buckle half wavelength. It should also be mentioned that perhaps the most widely known approach to dealing with plate post buckling behaviour is the "effective width" approach. In this approach it is assumed that an "effective width" of plating carrying the maximum applied stress takes the load on the complete plate. In the plate examined, under nominal strain u/a the load carried is

P-Etbeu-Etb{u-K'[u-ucR]}a

whence b---~ 1 - =b

KI[ 1 - -~'R-I

(8,

The lower bound method is easily applied and can give quick results and approximations. This method can also be easily computerised, and used in conjunction with numerical or semi-numerical approaches such as the Finite Strip method to tackle a wide range of problems. The presence of imperfections can be easily taken into account, but these can be dealt with using more rigorous analysis without undue difficulty in any case.

More rigorous post-buckling analysis. The elastic post-buckling behaviour of thin plates is governed by two simultaneous non-linear differential equations originally set up by von Karman (5) and modified some time later by Marguerre (6) to take account of the presence of initial imperfections. These equations may be written as follows, in terms of deflections w and stress function F:-

d4F ~ +

ax4

c~w ~X 4

2

d4F ~ +

ax aY d4w

~

d4F

aY4

=E

I(

d2W / 2- d2w d2w

F,axaY)

ax

_

d2Wo

dxdy

/ 2 d2 9 ] q_

Wo d - W o

dx 2 Oy 2

(9)

d4w

+ 2 O x 2 0 y 2 + Oy 4 =

_- -qd + t [ g f l F d 2 ( w + w o )

-2

F d 2(w+wo) d 2 F d 2(w+wo)] OxOy 2 dxcTy + dx 2 3yZ

(10)

J

The first of these equations, sometimes called the "Compatibility Equation", ensures that in an elastic plate the in-plane and out-of-plane displacements are compatible. The second equation is based on equilibrium principles, and is sometimes termed the "Equilibrium Equation". Exact solution of these equations is only possible for the simplest loading and support conditions, but solutions which are within reasonable accuracy are obtainable for a wide range of problems. Perhaps the most widely used approach to rigorous analysis of elastic plate postbuckling behaviour is to postulate a form for the deflections w, to then evaluate the corresponding stress function for the particular boundary and loading conditions using the exact solution of Equation (9), which is amenable to solution for a wide range of conditions. Atter this the required solution can be attained either by approximate solution of the Equilibrium equation (10) or by disregarding this equation and expressing the total potential energy in terms of F and w and applying the Principal of minimum Potential Energy to obtain the solution. If it is assumed that the deflected form in the axial direction is sinusoidal, generally a most reasonable assumption for plate elements of prismatic structural members, then the corresponding stress function

74 obtained on the basis of Equation (9) has two parts, one constant with respect to x and the other varying with twice the frequency of the deflections, i.e. 2~x F = F~ + F 2 c o s ~

(1 I)

/2

The firstterm, F~,, essentiallydescribes the average membrane stressesalong the plate, i.e.those which could be determined on the basis of Equation (1), and the second term describes a secondary membrane stresssystem which arisesto fullysatisfycompatibilityconditions.

The stress system described by the function F2 indicates mainly periodically varying stresses in the axial direction for plate elements of prismatic thin-walled structural members in which loading is carried in the axial direction only. For plate elements of bridge, ship or similar structures which can have a multiplicity of plates arranged together the unloaded edges remain relatively straight, and this straightness is maintained by periodically varying stresses in the direction perpendicular to loading. The stress variations along simply supported plates of both types is illustrated in Figure 4. In the more rigorous analysis the quantity K~, originally defined in Equation (6) can be re-defined as follows

I! 12 y2dy

(12)

It is obvious from the above equation that if all else is equal, K1 calculated from the above equation will be smaller than that calculated from Equation (6). Thus E*/E and be calculated on the basis of the more rigorous approach will always be greater than those obtained from the lower bound approach, provided the same deflected form is assumed.

Unloaded edges held straight

Unloaded edges free to wave in-plane

Figure 4. Stress systems on simply supported plates with different unloaded edge conditions This hypothesis is almost correct, but not completely so. As may be observed from Figure 4Co), for plates with unloaded edges free to wave in-plane, the axial stress on the edge of the plate varies along the plate, and has a maximum value at the crest of a buckle. This induces the requirement to specify closely what is the function of the effective width. In many design specifications a dual function is assumed, simplification of analysis with regard to plate stiffness and simplification with regard to plate strength, although the Eurocode for cold-formed steel does recognise to some extent that there are differences

75 between these two functions. Among researchers in plate analysis it has long been recognised that two different effective widths should be specified, one based on the largest membrane stress being the effective width for strength and the other, based on the average membrane edge stress being the effective width for stiffness calculations.

BEHAVIOUR FAR BEYOND BUCKLING

As loading progresses beyond the buckling load the deflections change in magnitude, and also change in form. Changes in the form, or shape, of the deformations bring corresponding changes in the axial stiffness of a plate. In general for plates supported on both edges the deflected form across the plate flattens, and the axial stiffness reduces as loading progresses. This can be taken into account in analysis by providing the freedom for the buckled form to change. Many authors have examined such behaviour successfully. An investigation by Stein (7), however, in 1951 is worthy of special mention. Stein used the perturbation approach in which the solution is obtained in terms of the power series expansion of a "perturbation parameter". The parameter used by Stein was :fl= P -1+ P~

where

Wo

(12)

w

wo is the initial deflection of the plate.

A complete picture of the plate behaviour could be derived in terms of a power series of this parameter. The first two terms of this power series could effectively detail the plate post buckling behaviour well into the far post buckling range. Essentially this meant that by obtaining analytical solutions at two specific points, one of which could be the buckling point, and utilising the pertubation approach a picture of the complete post-buckling range of behaviour of identical plates with any magnitude of imperfection could be produced. Walker used this approach in 1975 (8) to obtain explicit solutions for square simply supported plates. The results were used in the 1975 edition of the UK specification for the design of cold-formed steel specimens. Williams and Walker (9) extended this study to deal with a wide variety of plate geometries and boundary conditions, and tables of coefficients obtained from a finite difference analysis were given from which the reader could analyse the plate of his choice. It is only a short step to go from this position to fitting expressions to the coefficients so that by solving simple equations the coefficients governing rectangular plates of arbitrary buckle half wavelength and arbitrary boundary restraint conditions can be determined. In Ref (10) slightly modified forms of explicit expression, obtained on the basis of a Marguerre type analysis allied to the perturbation technique, are presented. The explicit expressions are in the following forms:-

/

PcR = ( q -

8cR

1) fl + c 2 r2

! cy'~ PcRP)=(c3-1)fl+c4fl2 --

= C5 ~

+ C6

(13) (14) (15)

76 In the above 13 is as defined in Eqn. (12). The terms CrcR, 6cR and

PcR are the critical values of axial

stress, axial strain and load respectively, i.e. the values of these quantities at the point of local buckling. Expressions for the coefficients Cl to C6 for plates with varying buckle half wavelengths and rotational restraints on the unloaded edges have been tabulated in Ref (10). The explicit expressions shown here have slight modifications built in to eliminate the possibility of ill conditioning affecting the postulated behaviour in the far post-buckling range, and these equations give results in close agreement with existing theory in comparable cases. At very large strains it can be shown that the load/strain and load/maximum stress relationships given by Equations (13) and (14) tend towards P

= 1,1

S

P

c- r n1.1 tax-

(16)

These relationships have similarities with plate relationships derived in much different manners. It is of interest that yon Karman's effective width equation produces these results with C2 = C3 =1 To illustrate the effects of initial imperfections on plate behaviour, load-out of plane deflection curves and load compression curves for simply supported square plates are shown in Figures 5 and 6. Figure 5 shows that for imperfect plates the out of plane deflections grow from the start of loading. As the load increases the deflections tend towards those of a perfect plate. Figure 6 shows the reduction in load at a given strain which results from imperfections. The maximum reductions occur at around a strain equal to the strain at which local buckling occurs. A plate with initial deflection of magnitude equal to the plate thickness sustains less than half the load of a perfect plate at an applied strain of around the critical strain although the same plate will carry something of the order of 90% of the perfect pate load when the strain is about 7 times the critical strain. Also in Figure 6 the reduction in stiffness of the plate as loading increases is evident. Note that this is due to variation in deflected form across the plate.

wo

/ T =o

/_.~0.2

.

P

1 1

,

2

1

I

3

w/t

Figure 5. Load -out of plane deflection curves for simply supported square plates

77

- • Q0-20.4n - ,~ ==0

/ ~...~~O. 8 ~1.0

.

.,

P

ecR I

e---I

Simply supported unloaded edges 0

1

L,

2

4 ~/eCR

/

,,

I

6

8

Figure 6. Load end displacement curves for square simply supported plates. Figure 7 shows the variation in load with end compression (displacement) and with maximum stress for simply-supported plates with unloaded edges either free to wave in plane or restrained from waving. Curves are also shown for plates with fixed tmloaded edges. In this figure the buckle half wavelength is that at which the plate initially buckles - i.e. equal to the plate width for the simply supported plates and about two thirds of the plate width for the fixed edge plates. The effects of allowing the buckle half wavelength to vary are illustrated in Figure 8 in which the same plates as previously examined are considered. As may be observed allowing the buckle half-wavelength to vary to produce the most flexible case leads to somewhat less stiffness in all cases, and tends to bring the results for different cases closer together. Note that the loads were only minimised with regard to end compression, so that the results in Figure 8 for stresses are for half-wavelengths at which the end displacement is maximised with regard to load, not the maximum stress S S - C o m p - Free to w a v e - e = l S S - S t r e s s - F r e e to w a v e - e = l SS - Comp/stress - restrained e=l FF - C o m p - e = 0 . 6 7 ---,--FF - Stress e=0.67 - - Von Karman's Equation ....

9

12 9

P/PcR

6 3

0

!

i

i

!

20

40

60

80

e / CcR or

100

Crn~x/CrcR

Figure 7. Load v Compression. and Load v maximum stress for plates with fixed buckle half wavelength

78 SS- C o m p - Free to w a v e - e varies SS - Stress- Free to wave - e varies -- F~FF- Cornp/stress- restrained e varies L;orr~ - e varies - - - . - - F F - Stress e varies - - - Von Karrnan's Equation

,21 9-1

P/PcR

r

I

I

I

I

I

0

20

40

60

80

100

616cR or trm~xl trcR Figure 8 Load v Compression and Load v maximum stress for plates with varying buckle half wavelength

ECCENTRICITIES IN LOAD OR END DISPLACEMENT APPLICATION So far it has been considered that the plates examined have been subject to uniform compression. This is not always the case. Plate elements of thin-walled members, and indeed thin-walled members, may be subjected to combinations of bending, compression and tension. Indeed the type of loading undergone by a member may be affected by the member's resistance to load. Consider two plates as shown in Figure 9:-

Load

Load

ili~

i

f Fulcra

(a)

(b)

Figure 9. Constant compression eccentricity and constant load eccentricity on plates Both plates are loaded through rigid loading bars. In case (a) the loading bar rotates about a fulcrum which may be at any specified position (For enforced uniform compression the fulcrum is positioned at a point infinitely distant from the plate). In this case the plate is subjected to a specified eccentricity of

79 compression (or end displacement) which is independent of its reaction to the loading. In case (b) the load is applied at a specific location, giving specified load eccentricity. In such a case the plate reaction to the loading must be such that the internal actions within the plate must be in equilibrium with the load acting. In practice it would be expected that the actual loading conditions in a plate element of a structure would be somewhere between these two conditions. Secondary members or secondary elements are often subjected to external actions which are dependant mainly on the reactions of primary members to the applied loading, so that secondary members are generally subjected to constant compression eccentricity while primary members are generally subjected to constant load eccentricity. In order to deal with either of these cases the possibility of eccentricity of compression must be considered. If the applied end displacement on a plate was to vary linearly from u at y=0 to u ( 1 - a ) at y=b then Equation (1) could be rewritten to take the variation in strain into account as follows:-

u uoy

zx =

a

a b

), w2

4

(17)

The membrane strain energy expression becomes:

ta[Iu

~=--F

b

2

b

b

a UaYab]2 + 1-"-(rt~4A4of 4dy - "21( u ] ( ~ ] A2 fYo 2dy +l(u~aa A2 o~Y2Y"~dy

7

]

(18)

Adding the bending energy and applying the Principle of Minimum Potential Energy gives, for A 2

y2@

y2

@ UCR

=

b

--

fly4@

b

a

(19)

~y2@ 0

Note that in Equation (19) the term under uniform compression.

UcR specifies the critical end displacement corresponding to buckling

Substitution for A in Eqn (17) and performing the appropriate steps give expressions for the end load on the plate and the moment about the x-axis caused by the in-plane forces, as follows:-

=

a

...... u- ~

- Kl

UcR) + K 2 ua]

(20)

(21) where K1 is given by Eqn. (6) or (12) and Note that K2 and I<3 are given above for the lower bound case. For more rigorous analysis the denominator in both functions should be replaced by that of Eqn. (12).

80 Equations (20) and (21) permit the examination of plates under combinations of axial forces and moments. Extension of these equations to deal with any number of plates combined to form structural sections is a simple matter. '

[i L"i~~ I UonifpLio n

~

-

~

~

Central loading

u/ucR Figure 10. Load v compression for simply supported -free plates Examination of uniformly compressed plates, or plates with any fixed compression eccentricity using Eqns. (20) and (21) is simply accomplished by setting the quantity ua to the relevant value (0 for uniform compression). The examination of constant load eccentricity can be carried out by setting M=kP, with k defining the load position, using Eqn. (21). It is found that the cases of constant load eccentricity and constant compression eccentricity are not significantly different for plates with both edges supported. For plates with only one edge supported the cases of constant load eccentricity and constant compression eccentricity give very different results (11). This is illustrated in Figure (10) for a plate with one unloaded edge simply supported and the other free. Uniform compression produces a post-buckling axial stiffness of about 4/9 of that before buckling. Loading applied along a specified line of action - halfway across the plate, produces very small post-buckling axial stiffness, although the pre-buckling axial stiffness is the same as that for the uniform compression case. The main reason for this is that under uniform compression the plate resistance is concentrated near the supported edge, while under central loading the plate has to bend in-plane to resist the applied loading concentrated at the plate centre. The variation of membrane strains across the plates are indicated in the small plate sketches in Figure 10. Some post-local buckling properties of a plate can be determined on the basis of Eqns. (20) and (21) by evaluating u in terms of ucx, UCR and P from Eqn. (20) and substituting into Eqn (21) to obtain the following expression:2

M

-

g 2

(1- K,)

The corresponding equation prior to local buckling can be obtained by setting K~, 1(2 and K3 equal to zero to get

M= Eb2t~xuaa x (-1) + Pb2

(23)

81 Now the term u a / a describes the variation of strain across the plate width "b", and it is a simple matter to show that the in-plane curvature in the plate is given by:C .

1

. . R~

.

02v

. . ax2

ua ab

(24)

On this basis we can see that Eqn. (24) simply states that E I

~ = d2v dx 2

-M

Pb +~ 2

(25)

In the above M is the moment about the plate edge, y=0. We have not as yet made any specification regarding M, i.e. we have not stated how or where the axial load is to be applied. If, for example we wish to examine the case of a plate under a central loading we could specify M=Pb/2, in which case Equation (25) would read :- E 1-----= d2v 0 Thus in this case the plate (in the pre-buckling range) would behave as if dx 2

there was no curvature, as would indeed be the case. In a more general situation we could say that M=Pd, where d is the distance from the y axis at which P is applied. In such a case Equation (25) reads:-

-

(26)

Taking the same approach to equation (24) under the same loading, we can write this as:-

(y. _ y)+,(y" In the above I* and y ' , which may be obtained by inspection from Equation (22), are modified values of the second moment of area I and the distance from the y axis to the neutral axis taking the effects of local buckling into account. The ratio of I* to I indicates the effect of local buckling on the in-plane flexural stiffness of a plate or section. The effects of local buckling on flexural stiffness are somewhat more complicated than on axial stiffness, when evaluated on the basis of the method discussed, as the tensile effects introduced by local buckling can stiffen the tension side of a plate or member undergoing bending, leading in some cases to large discrepancies. For example, in the case of a single plate element, simply supported on both unloaded edges, under pure in-plane bending, rigorous analysis (12) suggests that the initial ratio of postbuckling flexural stiffness to pre-buckling flexural stiffness is about 70%. If, however, it is assumed that the buckled form due to in-plane bending is sinusoidal across the plate (exact for pure compression) a value of 100% is determined for this ratio, i.e. local buckling is calculated to have no effect on subsequent flexural stiffness. This erroneous result arises from the symmetrical deflected form, in which stiffening effects on the tensile side counteracted rigifying effects on the compression side. In general this method requires account to be taken of changes buckled form after initial buckling to be taken into account.

APPLICATIONS TO COLUMNS In the derivation of Equations (26) and (27) we assumed M=Pd. This ignores any effects of deflections of the plate or section on the moment. If we consider that the reference axis of the plate or section has

82 displaced from its initial position by a measurable amount "v" then this should be taken into account in the calculations. If we say that M = P(d+v) then Equations (26) and (27) become:-

dx 2

+

d)

(28)

where for a single plate y = b/2 d x2

+,v-

(29)

The first of these is the very well known equation for buckling of a pin-ended column loaded with eccentricity d - y . The second is the post-buckling equivalent. These equations can be derived for structural sections in much the same way as they have been derived here for plates. Solutions for the case of pin-ended columns under eccentric loading are as follows:v=

v=

d-

see ~

- 1

(30)

I(d-y')+-~y-y*)l[sec/2

p~E./-

(31)

11

In the above P~r = zc2 E I / L 2 a n d Pe" = rc 2 E f / L 2 are the Euler load and reduced Euler load for the locally buckled sections respectively. Equation (30) governs while the column has not buckled locally, while Equation (31) governs after local buckling has occurred. This approach, or very similar approaches, has been used to examine a number of column problems in the past (e.g) bridge plating (13), (14), plane channels (15), lipped channels (16), (17), I sections (18). The reduction in Euler load caused by local buckling varies substantially for these sections. For example a plane channel suffering local buckling such that the flange free edges are further compressed by the overall displacements can lose about 85% of its stiffness, while if the dimensions or loading are such that buckling in the opposite direction occurs only about 20% of the stiffness is lost.

#~ 0.8 0.6

-----Initial Postbuckling Far postbuckling

Before local buckling After local buckling

0.4 0.3

b2

O4

.o 0.2 t-

0.4 0.1

0.2 |

0

0.2 0.4 0.6 0.8

1

b2/bl

Figure 11. Post-buckling Flexural stiffness of Channels

0

0.2

i

f

0.4 0.6 b2/bl

J

0.8

1

Figure 12. Neutral axis positions for channels

83 Figure 11 illustrates this behaviour for a plane channel section with flange width b2 and web width b]. In the figure the flexural stiffness, denoted by I*, calculated immediately after buckling and that calculated well into the post-local buckling range are used to illustrate the variation in behaviour observed. In Figure 12 the assymptotic variation in neutral axis position evaluated in the post-buckling range is shown for the channel section. This demonstrates the well known "wandering neutral axis" concept. The applicablity or inapplicability of this concept in the general scheme of things is a matter of some discussion, but there are at least some instances in which it requires to be taken into account. Figure 13 shows the variation in Euler load due to local buckling for several of different cross sections with variation in slenderness ratio 9 For simplicity it is assumed that local buckling has occurred at one fifth of the squash load for all sections, indicating highly ineffective cross sections. It is obvious from these figures that, all else being equal the effects of local buckling are such that a failure curve based directly on the Euler curve for the gross cross section is very likely to overestimate the capacities of columns which undergo substantial local buckling before failure, and this is evident for slender columns, not just for those in which the squash load governs. This does not seem to be taken due account of in the Eurocode for cold-formed steel (19). In this, although the reductions due to local buckling's effect on the squash load are included, the shape of the transition curve from Euler to squash governed failure does not make any allowance for local buckling effects, as is evidenced in Figure 14, which uses Column curves (b) It is of interest to observe that in the European Recommendations for the design Light gauge steel members (20), the weakening effects of local buckling on column behaviour were taken into account using a biased imperfection formula. The effects of this formula are shown in Figure 14, in which the larger the reduction in section effectiveness the larger the difference between European Recommendations and Eurocode predictions.

12

Faler load

12

~ R a i ~

1

....

08

(1)

B ~ .__,Je~__'en

08

I~on

n 06 n 04

~

.

,

Rain ~

Column curve (b) Solid lines-Eurocode Dashed lines- Euro-Recs

1

(2)

06 04

02

.

0

|

0

50

|

100 L/r

.

.

.

!

150

It

0.2

.

Q "

"

" . ,

,.

=

= .

1

i

I

I

i

i

200

0

50

100 L/r

150

200

Figure 13. Buckling interaction curves

Figure 14. Column design curves

[t is obvious that there is dubiety here as to the effects of local buckling on column behaviour, and to the importance of these effects. It is perhaps a good thing that very light gauge members tend to have been used as beams rather than as columns in the past. If it is intended that light gauge members be widely used as columns there are problems which must be tackled with some urgency.

84 SUMMARY This paper has made a somewhat brief and superficial study of plate elements and members subject to local buckling, and has mentioned a few of the interesting problems faced in this field of study. It is obvious that although there has been a great deal of work carded out on plate bcckling and post-buckling behaviour, there is still a substantial amount to be learnt.

REFERENCES von Karman, E. E. Sechler and L. H. Donnel., Strength of thin plates in compression, Trans. ASME, 54, 1932. 2. Cox, H. L., Buckling ofthin plates in compression, ARC R & MNo.1554, 1934. 3. Cox, H. L., The theory of fiat panels buckled in compression, ARC R & A No.2178, 1945. 4. K. Marguerre, The apparent width of the plate in compression, NACA TA No.833, 1937. 5. T. von Karman. Festigheitsprobleme im Maschinenbau. Encyclopaedie der Mathematischen Wissenschafien, 4, p349, 1910 6. K. Marguerre Zur theorie der gekreummter platte grosser formaendenmg. Procfifth lnt. Congress for Applied Mechanics. Cambridge, 1938 7. M. Stein, Loads and deformations in buckled rectangular plates, NASA Teci Rep. R-40, 1959. 8. A.C. Walker. The post-buckling behaviour of simply supported square plates. Aero Quarterly, XX,, 1969 9. D.G. Williams and A. C. Walker. Explicit solutions for the design of initially deformed plates subject to compression. Proc 1. C. E, 59, 1975 10. J Rhodes. Microcomputer design analysis of plate post-buckling behaviour. Jnl of Strain Analysis, 21, 1986 11. J. Rhodes and J. M. Harvey. Effects of eccentricity of load or compression on the buckling and postbuckling behaviour of fiat plates. Int. Jnl of Mech Sci. Vo113. P867, 1971 12. J Rhodes. The post buckling behaviour of bending elements. Proc 6th lnt Specialty Conf. On ColdFormed Steel Structures, St. Louis, 1982 13. W. C. Fok, J. Rhodes and A. C. Walker. Local buckling of outstands in stiffened plates. Aero Quarterly, XXVll, 1976 14. W. C. Fok, A. C. Walker and J. Rhodes. Buckling of locally imperfect stiffeners in plates. Jnl of the Eng. Mech Div. ASCE. Vo1103 No. EM5, 1977 15. J. Rhodes and J. M Harvey. Interaction behaviour of plain channel columns under concentric or eccentric loading. Prelim Report. 2nd lnt. Colloquiumon the Stability of Steel Structures, Liege, 1977 16. J. Loughlan. Mode interaction in lipped channel columns under concentric or eccentric loading. PhD thesis, University of Strathclyde, 1979. 17. J. Rhodes and J. Loughlan. Simple design analysis of lipped channel column. Proc. 5th Int Specialty Conf on Cold Formed Steel Structures, St Louis, 1980. 18. J. Loug~tlhan and M. Navabian. The behaviour of thin-walled I section cohmms a_fter local buckling. Proc. 8t lnt Specialty Conf on Cold Formed Steel Structures, St Louis, 1986. 19. CEN ENV 1993-1-3:1996. Euroeode3:Design of Steel Structures- Part 1.3:General Rulessupplementary rules for cold-formed thin gauge members and sheeting 20. ECCS TC-7. European recommendations for the design of light gauge steel members, 1987. 1.

Section II ANALYSIS, DESIGN AND MANUFACTURE

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Third International Conference on Thin-Walled Structures J. Zarag, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

87

RESIDUAL STRESSES IN UNSTIFFENED PLATE SPECIMENS M. R. Bambach 1 and K. J. R. Rasmussen ~ 1Department of Civil Engineering, Sydney University, Australia

ABSTRACT Rectangular plate specimens have been prepared for testing in a rig that provides simple support to three sides, leaving the remaining (longitudinal) edge free. In many applications such plate elements exist as flange outstands in thin-walled sections whose fabrication process involves welding and possibly flame cutting. In order to accurately predict the behavior of unstiffened elements in this condition, equivalent residual stress profiles must be induced in the plate specimens prior to testing. Details of the processes involved in inducing and measuring residual stresses in the plate specimens are presented. Previously developed models relating voltage, current and arc speed to the stress induced, as a function of plate width are extended, such that residual stress magnitudes and gradients can be predicted that approximate those in the flange outstand of fabricated sections. KEYWORDS

Weld shrinkage, residual stresses, tendon force, fabricated thin-walled sections, plate elements INTRODUCTION It is well known that residual stresses are induced in members as a result of welding and flame cutting processes, and that these stresses may influence the load carrying capacity of thin-walled members, particularly those that contain slender component plates. In the 1970s an extensive program was undertaken at Cambridge University to investigate the nature and magnitude of residual stress profiles induced by a large variety of welding processes common to the fabrication of thin-walled steel members. Reports by Kamtekar (1974) and White (1977a,b,c) give details of welding experiments performed on single plates and sections, and propose equations that predict the profiles and magnitudes of residual stresses. Reports by Moxham (1971), Dwight and Moxham (1971) and Bradfield (1979) detail welding processes on single plates, for the purpose of inducing residual stresses in plate specimens that were subsequently tested in plate testing rigs. The results of these tests provided valuable data on the effect of residual stresses on the load carrying capacity of stiffened plate elements (plates simply supported on all sides).

88 Most research on plates of that period and since then has concentrated on the testing of stiffened plates, giving little attention to inducing residual stress profiles in plates that are to be tested as unstiffened elements, that is simply supported on three sides, with the remaining (longitudinal) edge free. This paper outlines the applicability of the residual stress theory to unstiffened plates.

TENDON FORCE CONCEPT Centre Welded Plate. The research at Cambridge led to the 'tendon force' concept for beadon-thin-plate weld shrinkage, whereby a weld may be considered as a tendon exerting a force F, which is resisted by the full cross-section. If a continuous weld is laid longitudinally on a thin plate along its centreline, a longitudinal stress pattern is induced that consists of a region of tensile yield stress in the immediate vicinity of the weld, and compressive stress elsewhere, the resultant of which resists the tendon force. For all practical welding and plate parameters, the magnitude of the tendon force is given by (White 1977a): where p is the process efficiency and (Q/v) is the heat input per unit length of weld. The process efficiency for a centre weld was found to be 0.8, and the tendon force was shown to be insensitive to plate width and yield stress. This condition is applicable to the weld shrinkage induced in a flange plate when it is welded into an I-section or T-section. The distribution of longitudinal residual stress is shown in Figure la. A small gradient may exist in the compressive zone due to 'wrap up', where small angular distortions may occur about the line of the weld, as shown in Figure la. It is noted that transverse residual stresses were found to be negligible. Tensile Stress 9

(a) Centreline weld

0 - "

(b) One edge welded

|

~

(c) Both edges welded simultaneously

l,

!

Compressive Stress

Figure 1: Residual Stress Profiles for Bead-on-Thin-Plate Welds

Edge Welded Plate. If the weld is laid eccentrically to the centroid of a plate, a bending stress is induced due to the offset of the tendon force. This causes a stress gradient and increases the magnitude of the tendon force (White 1977b), such that for edge welds it is

given by:

[:;edge--" 0.32p{~v / \--/

The process efficiency (p) was found to be less than that for a centre weld, due heat loss from the weld. White (1977b) suggested a value of 0.42. This applicable to the web in a welded T-section or the flange of a welded channel residual stress gradient generally causes the unwelded edge to be in tension, Figure lb.

(2) to increased condition is section. The as shown in

89 Both Edges Welded Simultaneously. If welds are laid symmetrically and simultaneously, the eccentricities of the tendon forces effectively cancel each other out, and constant compressive stress is induced in the central region of the plate as shown in Figure lc. The magnitude of the tendon force is the same as that given for the centre welded plate in Eqn. 1. This condition is applicable to the web in a welded I-section, when the edges are welded simultaneously, such as in an automatic welding process.

Consecutive Edge Welds. If one edge of the plate is welded and the specimen allowed to cool, the residual stress pattem for a single edge weld is induced (Figure l b). If the other edge is then welded, this weld is laid in a region of existing tensile stress. Weld theory (White 1977a) shows that the material around the weld yields in compression during the heating cycle before tensile stresses develop at the weld during the cooling process. Laying a weld in a tensile zone results in a reduced width of material yielding in compression during the heating cycle, and the tendon force produced is less than that for a similar weld laid in stress free material. The converse is true for welds laid in regions of compressive stress. Experiments by White (1977b) showed the magnitude of the tendon force to be the same as that for edge welds (Eqn. 2), but scaled up or down by a factor (m) according to the magnitude of the existing compressive or tensile stresses respectively: If edge welds are laid consecutively then, with the same heat input per unit length, and weld 1 inducing tension at the opposite edge, weld 2 will produce a smaller tendon force than weld 1. The residual compressive stress resisting the tendon force will thus be less, as shown in Figure 2a. The addition of the residual stresses from each weld produces the final stress profile shown in Figure 2b. Tensile Stress ~y

(a) Addition of residual stresses from consecutive edge welds

-,

i-i

I-1 ,I , I

I

I I I

! ! I

j

..I

I I I

i I 0 = =

Weld1 ~E ~.---...=,

Compressive Stress

(b) Final residual stress pattem

,"""

I

I

I I I

I Weld2

,...--

Figure 2: Addition of Residual Stresses from Consecutive Edge Welds

RESIDUAL STRESS PROFILES IN FLANGE OUTSTANDS Two conditions common to the fabrication of I-sections are detailed here, for the purpose of determining the residual stress profiles exisiting in flange outstands of fabricated sections. Firstly an I-section fabricated from as-rolled or cold-cut plates, and secondly one fabricated from flame cut plates. In the first case, it may be assumed that the plates are stress free when they are welded to form the section, such that the two flange plates develop a residual stress pattern similar to Figure la, and the web similar to Figure lc. It is assumed that the welding process is automated, such that the welds are laid simultaneously. An I-section fabricated in this way by Chick and Rasmussen (1999) produced the profile shown in Figure 3a, having a nearly uniform compressive stress with an average of 109MPa in the web, and a small gradient in the flanges, where the average compressive stress is also 109MPa.

90

J 9

=.

Flame cut plate -200

+200

Plate welded into I-section

1

i

J

j

Addition of residual stresses on flange plate

~: i " L..}::::::::::--;~.....':....~2;:::..::--; .....

(a) I-section fabricated from stress free plates

(b) I-sectionfabricated from flame cut plates

Figure 3: Residual Stress Profiles in Fabricated I-Sections I-sections that are fabricated from flame cut plates develop residual stresses from each heating process. Flame cutting the edges (simultaneously) induces residual stresses in the flange plate similar to those in Figure lc, and when the plate is welded into the I-section residual stresses similar to a centre weld are added, as shown in Figure 3b. The resulting residual stress pattems in the flange outstands are shown for each case in Figure 4. Tensile Stress fy

Compressive Stress

o

(a) I-sectionfabricated from stress free plates

~ (b) I-section fabricated from flame cut plates

Figure 4: Residual Stress Profiles in Flange Outstands of Fabricated I-Sections

INDUCING EQUIVALENT RESIDUAL STRESS PROFILES IN PLATE SPECIMENS Plate specimens have been prepared for testing in a plate rig where the plate will be simply supported on three sides, modelling a flange outstand of a fabricated section. Residual stress profiles are thus required that are equivalent to those in Figures 4a,b. A single edge weld produces yield in tension at the welded edge and less than yield in tension at the unwelded edge, as shown in Figure lb. This pattem does not resemble the one shown in Figure 4a for cold-cut plates in regard to the stress at the edge opposite the weld, which is also the unsupported edge of the flange outstand. It is important to induce compressive residual stresses at this edge and at the centre of the plate since they affect the buckling

9] behaviour. It has been considered to shift the weld closer to the centre so as to reduce the residual stress gradient and obtain a residual stress pattern in agreement with that shown in Figure 4a. However, the shift required to produce reasonable agreement exceeds a quarter of the plate width, which is deemed unacceptable. It is concluded that it is not possible to induce a residual stress pattem similar to that shown in Figure 4a by placing a weld on a single plate. The investigation therefore focuses on flame-cut plates, for which the residual stress pattem is similar to that shown in Figure 4b. Two consecutive edge welds are required to produce this pattem, as shown in Figure 2. Since the second weld is laid in a tensile zone, the tendon force is reduced and a stress gradient will arise. Aiming for a constant value of compressive residual stress, more heat is applied in laying the second weld compared to the first. This requires an estimation of the reduction factor 'm' in Eqn. 3, for which the curves given by White (1977b) were used. The target value of compressive residual stress is taken to be 110 MPa, which is a suitable mean of measured values on flanges of I-sections (Hasham and Rasmussen 1998, Chick and Rasmussen 1999). WELDING EXPERIMENTAL PROCESS As the plate specimens are to be used in a plate testing rig, weld deposit along the edges is not desirable. To avoid this an electric-arc welder was used without the addition of weld metal, such that it was operating as a heat source only. The torch was mounted on a motorised, rail-guided trolley that travelled the length of the plates at a constant speed, such that the heat input was constant along the length of the specimen. The magnitude of the heat input per unit length (Q/v) is controlled by the closed circuit voltage (V) and current (!), and the speed (v) of the trolley. The heat input to arc (Q) is given by Q = V x I. The parameters required to induce a compressive stress of 110MPa were determined by substitution into Eqn. 2 and 3. An estimate of the process efficiency of p=0.45, and reduction factor m=0.7 were used. The test setup is shown in Figure 5.

Figure 5: Welding Table Setup

Figure 6: Extensometer

The residual strains from the welding process were measured with the purpose built extensometer (Denston and White 1977), shown in Figure 6. Small indents are made in the central region of the plates by striking lmm ball bearings with a hammer. The extensometer is seated in the indentations before and after the welds are laid, and the distance between the two holes is measured. From these readings the weld shrinkage strain may be deduced, and the stress is calculated by multiplying the strain by the material Young's modulus.

92

Readings are taken on each surface of the plate and averaged to obtain the membrane stresses. An example of the indentation layout is given in Figure 7. The width of plates varies from 60 to 355mm, with all thicknesses nominally 5mm and lengths five times the width. 100mm

5mm

I i I Plate: 80x400xsmm " I I I

Indentations for J strain m e a s u r e m e n t -

s T 9

15mm 50ram

E

oE T-

I cL

15mm

Figure 7" Typical Indentation Layout RESULTS OF WELDING EXPERIMENTS Forty plate specimens were welded along both longitudinal edges consecutively. Table 1 shows a sample of the results for the plates of width 120mm. It is noted that the required current is set on the welder, and the machine runs at a voltage that is affected by the distance of the arc from the plate. The measured voltage was generally in the range of 9-12 Volts, varying slightly along the length of the plate. The voltage values in Table 1 are average values for the weld runs.

Weld1 Weld2 Weld1 Weld2 Weld1 Weld2

Width (mm) 120 120 120 120 120 120

I

Thickness Voltage (V) (mm) 10 5 11 5 10 5 11 5 10 5 10 5

Experiment Parameters Current Speed Q/v (Amps) (mm/sec) kJ/m 208.25 105 5.04 314.75 148 5.17 203.00 105 5.17 317.46 148 5.13 192.50 105 5.45 286.13 148 5.17

Results Pt I (iPa)

i

Pt 5 ( i P a ) ....

m

p

0.717

0.530

-120

-129

0.694

0.499

-111

-117

0.729

0.478

-107

-111

Table 1" Sample of Welding Results for Plate Width 120mm The centreline of the weld from the edge was 5mm for plate widths less than 100mm, and 10mm for those exceeding 100mm. The average process efficiency of the former was found to be 0.416, and the latter 0.507. This is a result of increased heat loss when the weld is placed closer to the edge of the plate. The process efficiency is calculated from Eqn's 2 and 3, where the reduction factor (m) is deduced from the measured stress gradient and the tendon force (F) from the measured stress magnitude. Figure 8 shows the process efficiency to increase slightly with plate width. All the plates were positioned on heat resistant bricks then clamped with steel clamps to minimise bending near the line of the weld, as shown in Figure 5. As the plate width increases, comparitively less of the specimen is seated on the bricks, and therefore less heat is lost through the bricks and clamps. The process efficiency therefore increases. The process efficiency is largely dependant on the experimental setup. As the process efficiency increases, so does the tendon force in Eqn. 2 for the first weld, and consequently the value of tensile stress produced at the other edge by the first weld increases also. The second weld is thus being laid in a more highly stressed material than

93 was expected from the assumption that the process efficiency would remain constant, and consequently as the plate width increases the calculated reduction factor 'm' decreases. This is shown in Figure 9. The average value of m was found to be 0.701, which is very close to the assumed constant value of 0.7. Where the assumed value of m is different from the actual value, a gradient exists in the residual stress profile. The average difference in stress between the two edges of the compressive regions for all specimens is 15%. The average compressive stress is 104MPa, with a standard deviation of 16MPa. The specimens of width greater than 200mm were excluded from the results, due to inconsistencies resulting from the high heat input required in these wide specimens in order to achieve a compressive stress of 110MPa. a. >, 0 " :.~ =:: uJ

0.7 0.6 0.5 0.4 0.3 U) 0.2 o 0.1 2 a. 0 50

E 1 |

9

,=

9

~ 0L L. 6

~' ~, o.4 13 J

i

100 150 PlateWidth (rnm)

rr"

200

Figure 8: Process Efficiency vs Plate Width

0.2 0

~

50

i

!

100 150 Rate Width(mm)

200

Figure 9: Reduction Factor vs Plate Width

CONCLUSIONS The tendon force concept for determining residual stress profiles resulting from welding processes is reviewed, as are the models developed for inducing profiles in plates that approximate those in the stiffened elements of fabricated sections. The applicability of these methods to the unstiffened elements of fabricated sections is presented. It is shown how the models may be applied to unstiffened plate specimens, in order to create residual stress profiles that approximate those existing in the flange outstands of fabricated sections. The paper shows good correlation between the models and residual stress measurements on forty plate specimens of varying widths.

REFERENCES

Bradfield C.D. (1979). Tests on Single Plates under In-Plane Compression with Controlled Residual Stresses and Out-of-Flatness.University of Cambridge,ReportCUED/C-Struct/TR.78 Chick C.G. and Rasmussen K.J.R. (1999). Thin-Walled Beam-Columns. 1:Sequential Loading and Moment Gradient Tests. Journal of Structural Engineering. ASCE, 125:11, pp 1257-1266 Denston R.J and White J.D. (1977). An Electric Demountable Extensometer. University of Cambridge, Report CUED/C-Struct/TR.61 Dwight J.B. and Moxham K.E. (1971). Further tests on Welded Box Columns. University of Cambridge, Report CUED/C-Struct/TR.15 Hasham A.S. and Rasmussen K.J.R., (1998), Section Capacityof Thin-walled I-section Beam-columns, Journal of Structural Engineering, ASCE, 124:4, 351-359. Kamtekar A.G. (1974). An Experimental Study of Welding Residual Stresses. University of Cambridge, Report CUED/C-Struct/TR.39 Moxham K.E. (1971). Buckling Tests on Individual Welded Steel Plates in Compression. Universityof Cambridge, Report CUED/C-Struct/TR.3 White J.D. (1977a). Longitudinal Shrinkage of a Single Pass Weld. Universityof Cambridge, Report CUED/CStruct/TR.57 White J.D. (1977b). Longitudinal Stresses in a Member Containing Non-Interacting Welds. Universityof Cambridge, Report CUED/C-Struct/TR.58 White J.D. (1977c). Longitudinal Stresses in Welded T-Sections. Universityof Cambridge, Report CUED/CStruct/TR.60

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Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

95

BEHAVIOUR AND DESIGN OF VALLEY-FIXED STEEL CLADDING SYSTEMS Mahen Mahendran ! and Dhammika Mahaarachchi 2 1Associate Professor, 2phD Research Scholar, Physical Infrastructure Centre, School of Civil Engineering, Queensland University of Technology, Brisbane, QLD 4000 Australia

ABSTRACT When thin steel cladding systems are subjected to wind uplift/suction loading, local pull-through failures occur prematurely at their screwed connections whether they are crest- or valley-fixed. During high wind events, these local failures then lead to severe damage to the buildings and their contents. In recent times, the use of valley-fixing has increased in Australia. Although valley-fixing has been used for many years in other countries, and relevant design formulae and test methods are available, the applicability of these formulae and test methods to Australian steel claddings is unknown. Therefore a series of small- and large-scale experiments of valley-fixed high strength steel cladding systems was conducted. This paper presents the details of this experimental investigation, and the results.

KEYWORDS Steel cladding systems, Valley-fixed, Wind uplift, Pull-through failures, Experiments

1. INTRODUCTION The profiled steel roof and wall cladding systems in Australia (Figures l a, b) are commonly made of very thin high strength steels, and are crest-fixed with screw fasteners (Figure l c). The thin high strength steels (0.42 mm base metal thickness) have a very high yield stress (G550 - minimum yield strength of 550 MPa) at the expense of reduced ductility (strain at failure < 2 %). During high wind events, these claddings have suffered severely because of local failures of their screwed connections under wind uplift/suction loading. The two common local failures are pull-out and pull-through failures and have been the subject of many research projects in the past (Mahendran, 1994, Mahendran and Tang, 1998, Mahaarachchi and Mahendmn, 2000). The pull-out failure occurs when the screw fastener pulls out of the batten or purlin whereas the pull-through failure occurs when the screw head pulls through by splitting the steel sheeting due to the presence of large stress concentrations around the fastener holes under wind uplift loading (Mahendran, 1994). In some cases, a localised dimpling failure occurs instead of splitting. Figure 2a shows the pull-through failures (splitting and dimpling) in crest-fixed steel cladding systems.

96

Figure 2: Local Pull-through Failures In recent times, the use of valley-fixing has increased in Australia. The steel claddings often suffer from local pull-through failures at their screw connections whether they are crest- or valley-fixed. Figure 2b shows the local pull-through failure that occurs in valley-fixed steel claddings. In the USA and Europe, valley-fixing has been used for many years, and relevant design formulae and test methods are available. However, the applicability of these formulae and test methods to Australian steel cladding systems is unknown. Since the local pull-through failures in the less ductile G550 steel claddings are initiated by transverse splitting at the fastener holes, analytical studies have not been able to determine the pullthrough failure loads. An appropriate splitting criterion is required for valley-fixed claddings. Therefore a series of experiments of valley-fixed high strength steel cladding systems has been conducted using small scale and large scale experiments. This paper presents the details of the experiments of valleyfixed steel claddings, the splitting criterion, and the accuracy of current design formulae and test methods in predicting the local pull-through failure loads.

2. CURRENT DESIGN AND TEST METHODS Currently the American (AISI, 1996) and European (Eurocode, 1996) design provisions recommend detailed design formulae for a range of mechanically fastened connections such as bolts and screws in tension in cold-formed thin-walled steel sheeting and members. The pull-through capacity of screwed connections in tension is calculated as follows: AISI (1996) Fov = 1.5 t dw fa (1 a) Eurocode 3 (1996) For = t dw fa (1 b) where dw = larger value of the screw head or the washer diameter, but limited to 12.7 mm in Eq.la t = steel thickness fu = ultimate tensile strength of steel In contrast, previous Australian design codes do not recommend any design formula, and the design for the local failures of screwed connections in tension has been entirely based on laboratory experiments. However, Equation l a has now been included in the new cold-formed steel structures code AS4600 (SA, 1996). This design formula was developed for conventional fasteners and thicker mild steel and specifically for valley-fixed steel cladding systems used in the USA and Europe. Mahendran and Tang (1998) and Mahaarachchi and Mahendran (2000) have found that the above design formulae were inadequate to predict the pull-through capacity of crest-fixed steel claddings. It is unknown whether these formulae will predict the pull-through strength of Australian valley-fixed steel cladding systems. Therefore there is a need to verify the applicability of these formulae.

97 As an alternative to the design method, the American and European specifications (AISI, 1992, Eurocode, 1996) and the new code AS4600 (SA, 1996) recommend standard test methods using small scale models. They were mainly developed for valley-fixed claddings and cold-formed connections in general. Hence these test methods were not able to predict the pull-through strength of crest-fixed steel cladding systems (Mahendran, 1995). In the absence of suitable simple design and test methods, Australian manufacturers and designers have been using a two-span steel cladding assembly with simply supported ends (Mahendran, 1994). In this investigation on valley-fixed claddings, it is not known whether the standard test methods recommended by AISI (1992) will predict accurate results. Although the two-span cladding test methods using air bags or bricks were the preferred methods, Mahendran (1994) developed a small scale test method to simulate the pull-through failures in crestfixed claddings as these failures are highly localized around the screw holes (Figure 2). He recommended a small scale cladding specimen of approximately 240 mm x 240 mm bolted to a small wooden frame. It is known whether this method will give accurate results for valley-fixed steel claddings. Therefore in this investigation all the above-mentioned methods, that is, the two-span test method, small scale test methods recommended by AISI (1992) and Mahendran (1994) were used. The following section describes the details of this experimental investigation.

3.

EXPERIMENTAL INVESTIGATION

3.1 Two-span cladding Tests Since the pull-through loads of valley-fixed claddings were considerably larger than those of the crestfixed claddings, the required wind uplift pressure to be applied was very high (>7 kPa). When an airbox facility was used, most of the valley-fixed cladding systems failed due to other modes of failure such as section failures at midspan. The use of reduced cladding spans improved this situation, but then it required very high wind uplift pressures (15 kPa). Therefore the use of an airbox facility or an airbag/brick loading method was not possible. Instead a two-span cladding test method with midspan line loading was used to determine the pull-through load of valley-fixed screw fasteners at the critical central support. Figure 3 shows the test set-up using Tinius Olsen Testing Machine. TABLE 1. Test Results using the Two-span Cladding Method Cladding Profile and Span in mm Trapez. Trapez. Trapez. Trapez. Trapez. Trapez.

Type B / 600 Type B / 400 Type B / 400 Type B / 400 Type B / 400 Type B / 400

Steel Thickness in mm and Grade 0.42 - G550 0.42 - G550 0.42 - G550 0.42- G550 0.42- G550 0.42 - G550

dw Imm)

Failure mode

Average Measurd load per Fastener fastener (N)* Load .....2 3 (N) 3560 2373 2410 372012480 3120 4560 i3040 3490 5170 3447 3310 4510 3007 4140 1880 1253 1840 .

Test to Predict. AISI

I4.5 Splitting 0.73 14:5. Splitting 0.95 19 Punching 1.06 22 Splitting 1.00 19'" Splitting 1.25 10 Splitting0.71 longitudinl _Trapez. Type B / 400 0AS - G550 14..5 Splitting 5860 3907 4160 1.10 Corrugated / 600 0.42 - G550 14.5 Splitting 2010 1340 1690 0.51 Corrugated / 400 0.42 - G550 14.5 Splitting 3100 2067 2760 0.84 Corrugated / 400 0.42 - G550 10 Splitting 2850 1900 2380 0.92 Corrugated / 400 0.48 - G550 14.5 Splitting 3750 2500 3030 0.80 Corrugated / 400 0.60 - G300 14.5 Deformed 4060 2707 2550 0.55 Note: dw (mm) - Larger value of screw iaead or washer diameter ** 10 mm screw hole * 2 = Central support reaction/2 3 = Central support reaction/3 ,..

.

.

.

.

.

.

...

Test to Predict. Euro code 0.96 1.24 1.06 0.87 1.26 1.06 .

.

.

.

.

....

1.45 0.67 1.10 1.37 1.06 0.73 .

....

.....

.

.

.

.

.

98

Figure 3: Test Set-up

Figure 4: Typical Load-deflection Curves

The commonly available trapezoidal and corrugated claddings (Figure 1) were used, but with reduced width so that they can be accommodated in the testing machine. Past research has shown that both single-sheet and two-sheet wide specimens gave identical results for crest-fixed steel cladding systems (Mahaarachchi and Mahendran, 2000). Despite the reduced width, the test specimens had 3 screw fasteners at each support. The following parameters were varied: type of sheeting, span, screw shaft diameter, and screw head/washer diameter. The midspan loading was increased until one or more of the central support fasteners pulled through the sheeting. During the tests, the following measurements were taken: deflections of sheeting at critical locations, central support reaction using two load cells placed at the central support ends, and the load on one of the central support fasteners using a special load cell (Figure 3). Table 1 presents the details of the tests and the results. Figures 4 and 5a show the typical load-deflection curves and pull-through failure modes observed when this test method was used.

(b) Standard Test Method Recommended by AISI (1992) Figure 5: Typical Local Failures

99

Figure 5: Typical Local Failures

3.2 Standard Test Method Recommended by AISI (1992) The standard test method recommended by AISI (1992) and Eurocode (1996) was used in this series of tests (see Figure 6). This method uses a very small scale trapezoidal-shaped specimen fastened at the trough/valley. As it uses the same geometry specimen, it does not allow for the variation in the geometry of cladding profile. Therefore only the following parameters were varied in these tests: steel thickness and screw head/washer diameter. The test set-up recommended by AISI (1992) was used as shown in Figure 6, in which the tension load in the screw fastener was increased using the Tinius Olsen Testing Machine until a failure occurred. Table 2 presents the details of the tests and the results. Figure 5b shows the pull-through failure modes observed when this test method was used.

Figure 6: Test Set-up Recommended by AISI (1992) TABLE 2. Test Results using the Standard Test Method Recommended by AISI (1992) Steel thickness in mm and Grade " 0 . 4 0 - G250 0.40 - G250 0.42 - G550 0.42 - G550 0.42 - G550 0.42 - G550 t,,,

.,

....

Screw head/washer diameter dw (ram)

Failure Mode

Failure Load (N)

10 14.5 10 14.5 19 22

Splitting Splitting Punching Punching Punching Punching

2700 3240 2590 2605 1790 1575

Test to Predicted Value for AISI 1.80 1.70 1.00 0.80 0.54 0.47

Test to Predicted Value for Euroeode 2.70 2.23 1.50 1.03 0.57 0.40

100 3.3 Small Scale Test Method Recommended by Mahendran (1994)

The small scale test method developed by Mahendran (1994) was used in this series of tests. Small scale cladding specimens of approximately 240 mm x 240 mm were bolted to a small wooden frame. The width of the specimen between the bolts in transverse direction was the pitch of the cladding profile including the rib. A long screw fastener with a load cell attached to it was located at the centre of the specimen. The specimen was then loaded with a tension force by using a simple hand-tightening procedure. Unlike the AISI standard test method used in the last series of tests, this method models both the longitudinal and transverse membrane and bending deformations around the fastener hole and the tension force in the screw fastener. The tension force in the screw fastener was increased by handtightening until a failure occurred. This simple method enabled a large number of tests to be completed with limited resources in a short period of time. Figure 7 shows the test set-up whereas Table 3 presents the details of the tests and the results. The failure modes observed are shown in Figure 5c.

TABLE 3: Test Results using the Small Scale Test Method Recommended by Mahendran (1994)

Cladding Profile

Steel thickness in mm and Grade

Screw head/washer diameter dw

Cmm) Trapez Type B 0.42 - G550 Trapez Type B 0.42 - G550 Trapez Type B 0.42 - G550 Trapez Type B 0.42 - G550 Trapez Type B 0.42 - G550 . 0.60 - G550 Trapez Type B Trapez Type B 0.60 - G550 0.60 - G550 Trapez Type B 0.40 - G250 Trapez Type B Trapez Type B 0.40 - G250 0.40 - G250 Trapez Type B 0.40-G250 , Trapez Type B Trapez Type B 0.60 - G250 0.60-G250 i Trapez Type B 0.60 - G250 Trapez Type B 0.60 - G250 Trapez Type B 0.80 - G250 Trapez Type B 0.80 - G250 Trapez Type B 0.80 - G250 Trapez Type B 0.42 - G550 Trapez Type A 0.42 - G550 9Trapez Type A 0.42 - G550 Trapez Type A 0.42 - G550 Corrugated 0.42 - G550 Corrugated ' 0.42-G550 i 9 ' 0.42- G550 Corrugated Note: * - For both AISI and Eurocode thickness<0.9 mm i

i

i

14.5 19 22 19 (1Gramhole) 19 (14mmhole) 14.5 19 22 10 14.5 19 22 10 14.5 19 22 14.5 19 22 14.5 19 22 14.5 19

Failure Test to Load i Predict. (N) AISI*

Failure mode

Test to Predict. Euro code*

Splitt!ng 3755 1.14 1.49 Punching. 3715 1.13 .... 1.13 Splitting 4090 1.24 1.07 1.19 Splitting 3915 1 19 i j Splitting 3775 1.14 1.15 Splitting 6445 1.37 1.80 1.71 Punching 7615 1.62 1.11 Punching 6035 1.28 1.18 Splitting 1890 0.79 1.32 Splitting 3055 1.00 1.19 Splitting 3610 1.18 1.25 Splitting 4390 1.44 1.32 Splitting ~ 3160 0.88 1.55 Splitting i 5390 1.18 1.32 Punching ! 6005 1.31 1.17 1.35 Punching, i 6175 1.79 Punching j 8325 1.37 1.31 Punching I 7955 1.30 >1.04 Didn't fail >7330 >1.20 1.69 i Splitting 4250 1.29 1.21 L Punching , 3995 , 1.21 1.39 , Splitting , 5280 , 1.60 1.70 , Splitting, 4265 1.29 1.29 , Splitting 4230 i 1.28 1.05 , Splitting ' 3990 I 1.21 22 1.19 19 tl0mmhole). S p l i t t i n g 3915 I 1.19 predictions, 75% of fu was used for G550 steel with i

m

101

Figure 7: Small Scale Test Set-up Recommended by Mahendran (1994) 4. RESULTS AND DISCUSSION

4.1 Comparison of Results from Different Test Methods In the two-span cladding tests, the fastener load was measured directly and also calculated based on the measured central support reaction. The measured fastener load appears to be between the two calculated values (see Table 1) and was used in the comparions. The test failure loads shown in Tables 1 to 3 are the average of 2 test results. The standard test method recommended by AISI (1992) and Eurocode (1996) was unable to simulate the longitudinal and transverse bending actions including the localised deformations at the fastener hole. The failure mode of punching at the edges of screw head was due to the pure transverse bending action as seen Figure 5b, and was not the pull-through failures observed in the two-span cladding tests (Figure 5a). Therefore this standard test method cannot be used to determine the pull-through load of valley-fixed cladding, particularly for thin high strength steels (G550). On the other hand, the small scale test method (Mahendran, 1994) simulated the pull-through failures observed in the two-span tests (Figures 5a and 5c). The pull-through failure loads compared reasonably well in many cases, but there were considerable, unexpected differences in some cases. Further tests using varying specimen lengths are required to validate the accuracy of this test method.

4.2 Comparison of Test Failure Loads and Current Design Loads In general, current design formulae appear to be conservative (Tables 1 and 3). This may be due to the fact that they were developed to allow for both pull-through failures, namely the transverse splitting and the punching type failures (see Figure 5). The latter occurred when the screw fastener/washer head size (dw) was increased. Most of the test results indicated that the failure load increased with increasing dw, but the AISI formula predicted the same load for dw>12.5 mm. The Eurocode formula does not have this shortcoming, but it is more conservative than the AISI formula for dw
102 TABLE 4: Comparison of Failure Strains Two-span Test Details Trapez Type B 014243550 span = 400 dw'- 14.5 Trapez Type B 0.42-G550_span -"400 dw= 19 Trapez Type B 0.4243550 span = 400 dw- 22 Corrugate d 0.4243550 span = 400 dw= 14.5 Corrugated 0.42-G550 span = 600 dw= lzi.5 Corrugated 0.4843550 sp.an= 400 dw= 14.5 Corrugated 0.60-G300 span = 400 dw= 14.5

Membrane strain Strain

% Strain

3.19 2.57 3.40

83 81 87 74 79 73 58

1.70

2.39 2.59 1.41

, ,

Fiexurai strain Strain % strain 0.64 17 0.60 19 0.50 13 0.56 26 0.66 21 0.95 27 1.03 42 ,

Total Strain 3.83

3.17 3.90 2.28 3.O5 3.54 2.44 no split

Membrane strains at failure indicate that they have to be greater than about 60% for splitting to occur. This observation is similar to that determined by Mahaarachchi and Mahendran (2000) for crest-fixed claddings. However, the total strain at failure was greater than the tensile strain at failure (2% for 0.42 mm G550 steel). This contradicts the criterion for crest-fixed claddings for which the total strain at failure was found to be equal to the strain at failure from tensile coupon tests. Therefore further tests are required to develop an appropriate criterion for valley-fixed claddings. This will allow advanced numerical modelling to be undertaken and thus improve the design methods.

5. CONCLUSIONS This paper has described the behaviour of valley-fixed profiled steel claddings under wind uplift forces using three different test methods. The standard test method recommended by American and European codes was found to be inadequate in predicting the pull-through failure loads of full scale steel cladding systems made of high strength steel. The current design formula appears to be conservative in predicting the pull-through capacity of valley-fixed steel cladding systems. Further research is continuing to improve the design and test methods of high strength valley-fixed steel cladding systems.

6. REFERENCES 1. 2. 3.

4.

5. 6. 7. 8.

American Iron and Steel Institute (AISI) (1996), Specification for the Design of Cold-formed Steel Structural Members, AISI, Washington, DC. American Iron and Steel Institute (AISI) (1992), Test methods for Mechanically Fastened Coldformed Steel Connections, Report CF92-1, AISI, Washington, DC. Eurocode 3 (1996), Design of Steel Structures, Part 1.3 -General Rules-Supplementary Rules for Cold-formed Thin-gauge Members and Sheeting, European Committee for Standardisation, Brussels. Mahaarachchi, D. and Mahendran, M. (2000), Pull-through Failures of Crest-fixed Steel Claddings Initiated by Transverse Splitting, Proc.15 th International Conference on Cold-formed Steel Structures, St. Louise, MO, October, pp.635-646. Mahendran, M. (1994), Behaviour and Design of Crest-fixed Profiled Steel Roof Claddings under High Wind Forces, Engng. Struct., Vo1.16, No.5. pp.368-376. Mahendran, M. (1995), Test Methods for Determination of Pull-through Strength of Screwed Connections in Profiled Steel Claddings, Civil Eng Trans., IEAust., Vol.CE37, No.3, pp.219-227. Mahendran, M. and Tang, R.B. (1998), Pull-out Strength of Steel Roof and Wall Cladding Systems, Journal of Structural. Engineering, ASCE., Vol. 124 No.10, pp. 1192-1201. Standards Australia (SAA) (1996), AS4600 Steel Structures Code.

Third International Conferenceon Thin-Walled Structures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

103

BEHAVIOUR OF FLEXIBLE THIN-WALLED STEEL STRUCTURES FOR ROAD AND RAILWAY APPLICATIONS J. Vaslestad 1, B. Bednarek 2 and L. Janusz 2 1 ViaCon Norway, Strandveien 3B, 2005 Raelingen 2 ViaCon Poland, 64-130 Rydzyna k/Leszna, ul. Przemysiowa 6

ABSTRACT Corrugated steel culverts are increasingly being used in road and railway projects as alternative solutions to concrete bridges and culverts. The construction period is short, and the structures have both technical and economical advantages. Several full scale tests have been performed in the field to validate the long-term performance and great load bearing capacity of these structures, but not very many structures have been tested in controlled conditions in a test facility like the test in Zmigrod, Wysokowski (1997). The testing consists of static testing and dynamic testing. The dynamic and fatigue part consists of 600.000 cycles, Vaslestad and Wysokowski (1999). The full scale testing and finite element analyses of structures will help us to optimize and design more economic structures. It is of importance to have high quality test results when designing the flexible structures with minimum cover for live loads (railway and road). This paper describes three flexible thin-walled steel culverts, two from Poland and one from Norway. The first structure, built in 1997, is a 9,92 m span steel arch railroad tunnel with low cover and main road crossing over. The structure was instrumented with earth pressure cells and strain gages. The second structure, built in 1998, is a 8,9 m span pipe arch with railroad crossing over. The structure was instrumented with nd strain gages and deformation gages. The third structure, built in 1999, is a 11,58 rn span steel arch with road crossing over. The structure was instrumented with deformation gages for long-term monitoring.

KE~ORDS

Corrugated steel structure, pipe arch, soil-structure interaction, bearing capacity

INTRODUCTION Several approaches have been suggested for the design of thin-walled flexible steel structures, Kloppel and Glock (1970), Vaslestad (1990). Many of the design methods are reviewed by Bednarek (2000).

104 It is also recognized that buries flexible culverts undergo changes in deformation and structural response as time progresses after installation. Long-term behaviour of two instrumented corrugated steel culverts in Norway built in 1983 and 1985 are reported by Vaslestad (1990). The full scale testing and finite element analyses of structures will help us to optimize and design more economic structures. It is of importance to have high quality test results when designing the flexible structures with minimum cover for live loads (railway and road). Several full scale tests have been performed in the field to validate the long-term performance and great load beating capacity of these structures, but not very many structures have been tested in controlled conditions in a test facility like the test in Zmigrod, Wysokowski (1997). The testing consisted of static testing and dynamic testing. The dynamic and fatigue part consists of 600.000 cycles. Vaslestad and Wysokowski (1999). Earth pressure, steel stress and deformations were recorded during the static and dynamic testing. The testing were performed with height of cover ranging from 1,0 m and down to 0,3 m. The results show that there are small deformations and stresses. Full scale testing in the field in Poland on instrumented structures with railway loading have confirmed this. The results shows that the corrugated steel stuctures are safe with height of cover down to 0,30 m for both railways and roads, Vaslestad and Wysokowski (1999).

STEEL ARCH AS RAILROAD TUNNELWITH LOW COVER In 1997 a 55 m long corrugated steel tunnel was built as a railroad culvert with a road crossing over. Corrugated steel plates with 150x50 mm corrugation and steel thickness 7 mm were used. The tunnel is constructed as a steel arch on cast-in-place concrete foundations with 2,0 m width and thickness 0,5 m. The width of the tunnel is 9,92 m. The steel arch was erected in 10 working days. The minimum cover above the culvert was 0,45 m below the road and a concrete relieving slab were used to distribute the traffic loads from the road. The steel culvert were instrumented with strain gages and earth pressure cells, because this was a non-standard solution with low cover and reinforced steep slopes. The instrumentation were financed by the owner Public Roads Administration. Long-term behaviour of two instrumented corrugated steel culverts in Norway built in 1983 and 1985 are reported by Vaslestad (1989). Geotextile reinforced steep slopes were used around the opening ends of the steel culvert. The slope angle is 60 degrees and the maximum height is 10 m. A woven multifilament polyester geotextile with a characteristic short time tensile strength of 150 kN/m were used as reinforcement. The wall facing consisted of steel mesh and vegetation mat. The reinforced slope is reported by Vaslestad and Clausen (1998). A concrete relieving slab with thickness 260 mm and width 12 m were used below the road due to the low cover. An alternative with geotextile instead of the concrete slab were considered, but only used outside the road. Model tests at University of Oxford with geotextile above flexible steel culverts, Pearson and Milligan (1990), were shown to give a reduction with 25 % on the bending moment and 50 % on deformations of the culvert. Similar results were shown with model tests from Canada, Kennedy et.al. (1988). The minimum cover above flexible steel culverts can be reduced by using a layer of high strength geotextile or geogrid. In the summer 1999 Public Roads Administration performed a full scale load test on the road above the relieving slab, Braaten et.al. (2000). A vehicle with total weight 246 kN, and axle load 181 kN were placed on the road, and deformations and earth pressure were measured. The maximum measured vertical deformation on top of the steel culvert were less than 0,5 ram. The maximum increase in earth pressure below the relieving slab were 3 kPa..

105 STEEL PIPE-ARCH REPLACING OLD RAILWAY BRIDGE Poznafi structure

Phot. 1" Poznafi Rubiez: structure view

Fig. 1" Poznah Rubiez: vertical deflection- symmetrical load

106 STEEL ARCH AS ROAD TUNNEL

Lublin structure

Phot.2: Lublin: structure view

REFERENCES

Kennedy J.B., Laba J.T. and Shaheen H. (1988). Reinforced soil-metal structures. Journal of Structural Engineering, VoI. 114, No. 6, ASCE, pp. 1372-1389. Kl6ppel K. and Glock D. (1970). Theoretische und experimentelle Untersuchungen zu den Traglastproblemen biegeweicher, in die Erde eingebetteter Rohre. Heft 10, Institut fiar Statik und Stahlbau der Technischen Hochschule Darmstadt. Madaj A., Vaslestad J. and Janusz L. (1999). Full-scale testing of a long-span corrugated steel culvert used to rehabilitate a concrete frame railway viaduct. (in Polish) Proceedings of the IX Polish Bridge Conference, Poznan, 8-9 June 1999, pp. 106-117. Pearson A.E. and Milligan G.W. (1990). Model tests of reinforced soil in conjunction with flexible culverts. Performance of reinforced soil structures, Thomas Telford, pp. 365-372. Vaslestad J. (1989). Long-term behaviour of flexible large-span culverts. Transportation Research Record 1231, pp 14-24, Transportation Research Board, Washington D'C, 1989. Vaslestad J. (1990). Soil structure interaction of buried culverts. Dr.ing thesis, Norwegian Institute of Technology, Trondheim.

107 Vaslestad J. and Clausen B. (1998). Case histories with long span corrugated steel culverts from Norway. 8th seminar: Contemporaries methods of bridges reinforcing and reconstruction. Poznan 910.06.1998. Vaslestad J. and Janusz L. (2000). Application of flexible structures of corrugated steel plates in military engineering. Examples from Poland and abroad. (in Polish). III Technical and Scientific Conference: Military engineering- problems and prospects. Szklarska Poreba, Poland, 22-24 November 2000. Vaslestad J. and Janusz L. (2000). Flexible corrugated steel road tunnel, the biggest in Poland. (in Polish) Proceedings of the VI International Conference: Durable and safe road pavements, Kielce, Poland, 9-10 May 2000, pp. 95-103. Vaslestad J., Janusz L. and Bednarek B. (2000). Corrugated steel structures used for construction of bridges (in Polish). Materiaty Budowlane, 11/2000, pp. 48-49. Vaslestad J. and Wysokowski A. (1999). Full scale testing of MultiPlate corrugated steel culverts including fatigue problems. Archives of Civil Engineering, XLV, 2, 1999, pp. 375-383. Wysokowski A. (1997). Stand for testing of bridges at natural scale - research possibilities. Report from Polish Road and Bridge Research Institute, Zmigrod 1997. Wysokowski A., Vaslestad J. and Pryga A. (2000). Full scale fatigue testing of corrugated steel culverts. (in Polish) Konstrukcje Stalowe 5 (42), August 2000, pp. 45-47. Bednarek B. (2000) Steel corrugated MultiPlate type bridges and overpasses alongside A-2 motorway (on the distance between Swiecko and Poznan). (in Polish), M.A.Thesis, Poznan University of Technology, Poland.

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109

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

A SIMPLIFIED COMPUTATION MODEL FOR ARCH-SHAPED CORRUGATED SHELL ROOF Fan Xuewei School of Civil Engineering and Architecture, Northem Jiaotong University, Beijing, 100044, P. R. China

ABSTRACT Arch-shaped corrugated shell roof is a new type of steel light structure in China. It is widely used in China now for its short construction period, cheap cost and so on. The construction area covers over 2,000,000 m 2 per year at present. Many arch-shaped corrugated shell roofs have been collapsed in recent years nearly every area in China for heavy snow or typhoon. Xu studied the structure carefully and pointed out that the corrugation on the roof is the key factors for the accidents and the corrugation decreases the beating capacity of the roof. Based on this fact, the author put forward a type of FEM model to design the structure, which is nonlinear thin-walled beam consisted of orthotropic plates. The computation example of the article indicates this model has good precision and is easy to converge. The model is competent for the design of the structure. KEYWORDS arch-shaped corrugated shell roof, finite element method, corrugation effect, steel light structure, thin-walled structure, spatial structure

INTRODUCTION Arch-shaped corrugated shell roof (ACSR) is a new type of steel light structure in China. It is widely used in China for its very short construction period, cheap cost and so on. Its construction process is shown in Figl, Fig2 and Fig 3. In 1997 there were six ACSR collapsed for heavy snow in An' Shah city

,,,

Fig 1. Channel section forming

Fig 2. Corrugated channel forming

110 in northeast China. From then on there were many accidents of ACSR collapsed for heavy snow or typhoon in nearly every area in China. So the accidents attracted the industry's great attention. Xu (1998) pointed out that the corrugation on the roof decreases the bearing capacity of the structure while the designers did not consider the corrugation effect. On one hand as is well known, the difference of the stiffness for the corrugated sheet along the two perpendicular principal axis directions is very large. Generally the strong stiffness of the corrugation sheet is used to bear the load in architecture structure. However the weak stiffness direction of the corrugation sheet is the same as the arch axis direction along which there exists very large axis forces and partial moments. This is to say, the bearing load way of ACSR is to use the weak stiffness to bear the load and it is unreasonable. On the other hand from the computation manual it can be seen that the designers do not consider the corrugation effect. Further more Xu (1998) pointed out that the design of ACSR must consider corrugation influence on the bearing capacity of the structure. Based on this view, the author of the article put forward a FEM model --nonlinear thin-walled beam consisted of orthotropic plates. The element stiffness matrix is derived in the article. Finally an example is provided and the computation results indicate the method has the good precision and good convergence character. Seam together Seam

4

Seam

4

Fig 4. Sketch map of the section

Fig 3. Seam the edges together

G

r

E, 1

:xo Ely,

Fig 5. Coordinates of the cross section

Fig 6. Section property

MECHANICS MODEL AND ASSUMPTIONS Designers always assume that the loads on the roof along the longitudinal direction are the same. So the structure is a typical 2D structure. This is to say unit width roof can be separated from the structure and can be treated as the study object. There are generally two kind of method to design the roof. The first

111 one is to use shell element to compute the structure, but it is very complicated. The second one is to use thin-walled beam element. For the beam, the corrugation shows the orthotropic character. So the designers must consider this character. In this article, a thin-walled beam consisted of orthotropic plates is introduced here. Fig 5 shows the section coordinates of the beam. G, P, A is a general point on the cross section, a pole and an arbitrary point on the contour of the bar, respectively. There are three kinds of coordinates in the cross section, which are X - Y - Z coordinates system, r - q - z coordinates system and n - s - z coordinates system. Define the displacement components of the bar in the x and y directions and the rotation about the positive z direction to be U, V, ~ . Let the displacement of point A to be u, v, w in the n , s, z direction. To distinguish between the two displacement schemes, U, V and 9 are referred as bar displacements and u, v, w as the shell displacements. Wang (1999) tested the orthotropic material properties, and the one of the principle axis of the corrugation slab is parallel to the z-axis of the coordinates system. The elastic moduli of the corrugation slab are E1x, E2x and Ely , E2y. The Poisson ratio are ~l.lx, ~2x andlxly, ~t2y. (See Fig 6.)

The assumptions employed in this article are given below. 1. The cross section is assumed to be perfectly rigid in its plane while free to warp out of its plane. 2. The shear strain in the middle surface is very small and can be neglected. 3. Each segment of the cross section behaves as a thin shell, which means Kirchhoff-Love assumptions are valid. 4. The normal stress in the contour direction ( os ) is very small compared to the axial stress. So it could be neglected.

FEM

F O R M U L A S

The Displacement Strain Relationship u, v and ~v can be expressed in the form below: (Gjelsvik, 1981)

u(z,s) =U(z)sinO(s)-V(z)cosO(s)-~(z)q(s) ] v(s,z):U(z),cosO(s)+V(z)sinO(s)+.(z)r(s) ~ w = W ( z ) - U (z)X~(s)-r" (z)r~(s)-a~'(s)fl~(s)j

(1)

where ~"~a = ~(s)r($) dS is the sectoral area. The displacements u, ~, ~ for a point on the contour in the (n, s, z) directions are related to the displacements u, v, w for a general point on the cross section in the same directions, by the following relations (Fluggle, 1966) -

u=u

-

0u

v=v--as

-

0u

w=w-n-~z

The nonlinear relationship between displacements and strains are

(2)

112 Ow ' ( O u ~ 2 l ( O v ) 2 l ( a w ~ 2" ez=&-+2k,&-zJ +-2 ~zz + 2 \ O z J

Ov l ( O u ) 2 l ( O v ) 2

I(o~'] 2

e==N+Tkos) +2t,Os) +2 ~s o%, o% Ou Ou

ezs

Os

Ov Ov

Ow Ow

Oz Oz Os Oz Os

Oz Os

(3)

Thus the linear and nonlinear strains can be written in terms of the beam displacements ez = W ' - XU"- YV"- [)~" ~,2 nz= (U '2 + V ' 2 ) + - ~ [ ( X - X p ) 2 + ( y - y p ) 2 I - U ' O ( Y - yp)+ V ~ ' ( X - X p )

(5)

ezs = 2nO'

(6)

(4)

Constitutive Relations The constitutive relations for the orthotropic plates are

{} L0 Ns = Nzs

(7)

21 A22 0 A33J[ezsJ

(8) Mu

0

D33Jlk~J

Where N , , 2% and N=s are the membrane forces of the corrugated sheet, Mz, Ms and Mm are the bending and twisting moments of the corrugated sheet. A0 is the stiffness coefficient of the membrane. Dv is the stiffness coefficient of the bending and twisting. After the integration over the contour, the relationship between the beams and displacements in the principle axes of the cross section can be gained. N=A*W'

-Mx=I~V "

M r = I ~ U"

(9) (10)

Mco = I~co~" Ts = I* G~' The expressions of A" , I~x, * I ~ , Ioo, " J*G are listed in the appendix.

The Deriviation of The Element Stiffness Matrix For an arbitrary bar, two nodes can be defined 1 and 2. Then the beam displacements are expressed in terms of nodal displacements as

W=(nl){Wi}

U=(n3){Uz}

V =(n3){V~}

~P=(n3){Oi}

(11)

113 where { ~ } = ( W I

{r}=(Vl

V'I v2

(nl)=(1-f3,f3)

{Ut}=(UI

W2) r

v'2) r

U'I U2 U'2)r t

[o1=(ol

col *2 ~"2)r

(n3)=(1-3~ 2 + 2f3s,f3-2~ 2+f33,3f32-2~3,~3-~ 2)

(12) (13)

(14)

(nl) ,(ns) is the interpolation functions. The virtual work principle can be written t

I tSo'Stg.O"tdV + ~ t(~o~)tl"lij'dV= tV tV

t+At

R-I

t(3ij~)te~jtdV

(15)

tV

For the thin-walled beam, the virtual work done by the internal forces is

8W = ~(N81u + MxSg" + MrSU" + McoS~" + TsSO')dz = I(A*WBW' + I'xxV'BV" + I~u*ru" + I*~,oO'SO" + +j*GO'~O')dz + tr

,_tN

X

A

I~

E l--7;-

Y

tMr + - ~ - t M x - - w - t M o ]

{1(U'2, + V '2) + ~

Ixx

(16)

Inn

[ ( X - X p)2 + ( y _ y p)2] _ U'O(Y - y , ) + V~'(X - X p)}dV

After complicate computation, the element stiffness matrix is derived.

[KE] = [KL] + [ K m ]

(17)

The expression of [KL] is the same as in the book by Bao(1991). [KlvL] is listed in the appendix.

The Solution Of The Nonlinear Equations Increment iterative method is the usual method to solve the nonlinear equations. It assumes that the solution is known at the initial discrete moment t, solution at the presumed moment t + At is derived. At is the increment. The fundamental equation in the nonlinear analysis is t+~t{R}-t+ta{F} =0, in which

t+at{R}is exterior load vector on the nodes, t+at{F} is node force vector equivalent to stress of the element. Iterative method is needed because the node force vector t+at{F} depends on the nodes displacement nonlinearly. For i=1,2,3.., the iterative equations are given below.

{A~R}(i-l)=t+At {R} t+~{F}O-l)

(18)

tf KI{AU}(O = {AR}O -1)

(19)

t+~t{U}CO=t+~{u}O-o + {AU} co

(20)

'+~{uiCO)=,iu},

(21)

'§

Compute the no equilibrium load vector in Eq.(18) in each of iteration, and the displacement increment is derived in Eq. (20). Terminate the iteration until {z~} 0-1) or {AU} O) converges.

114 EXAMPLES To verify this FEM model, an example is given below. The principle mechanical properties of the corrugated sheet are measured to be: Elx =0.24e5MPa; Ely = 0.48e5MPa; ~tlx =0.006; ~tly =0.045;

E2x = 0.09e5MPa; E2y = 0.7e5MPa; ~t2x = 0.02; P2y =0.07. Two computational models are provided in this article. The first one is an orthotropic shell model, and the second model is the thin-walled beam model. Fig 8 shows the load-deflection curves of the two models. The maximum percentage error between two models is less than 5%. The comparison results indicate that the thin-walled beam consisted of orthotropic plates is capable of the design work.

0.4 ,[

I

1,

1,

,I,

* o. 3

]-----model I

I= ~o.2

---0.1

0 0

Fig7. Computation model

30

,

,

60

90

120

Fig 8. Di~lac=~ ent,load curves

CONCLUSIONS In this article, a nonlinear FEM model for ACSR was established. The method has the advantages such as good precision, good convergence character, easy to master for designers and so on. The computation software for the ACSR design has adapted this method recently.

ACKNOWLEDGEMENTS: The authors gratefully acknowledge the support for this study provided by the Ministry of Construction of P. R. China.

References Asokendu Samanta, Madhujit Mukhopadhyay, (1999), Finite Element Static and Dynamic Analyses of Folded Plates. Engineering Structures. 21:3, 277-287 B. Omidvar, A. Ghorbanpoor, (1996), Nonlinear FE Solution for Thin-Walled Open-section Composites Beams, J. Structure Engineering, 122:11, 1369-1378 Bao Shihua, Zhou Jian, (1991), Thin-WalledBars Mechanics, China Architecture Industry Press, Beijing, China Fan Xuewei, Xu Guobin, Cui Ling, (2000), The Static Work Character Analysis of Arch-Shaped Metal

115

Corrugated Shell Roof Considering Corrugated Effect. The 6th Asian Pacific Conference on Shell and Spatial Structures, Seoul, Korea Flugge, W. (1966), Stresses In The Shells, Springer-Verlag, N. Y. Gjelsvik, A. (1981), The Theory of Thin Walled Bars, John Wiley & Sons, N. Y. Nelson R. ,Bauld Jr., Lih-shyng Tzeg, (1984), A Vlasov Theory for The Fiber-Reinforced Beams With Thin-Walled Open Cross Section, Int. d. Solids and Structures, 20:3, 277-297 Wang Xiaoping, Jiang Cangru, Li Guiqing, (1999), Simplifying of Calculating Model on Metal Corrugated Arch Roof, Steel Construction, 14:4, 8-10 Xu Guobin, Cui Ling, Pan Yi, (1998), Collapse Analysis of Wrinkled Steel Thin Walled Arch-Shell Roof. Spatial structures. 4:4, 49-55

Appendix Notation: A* 9axial stiffness of beam. E , : modulus of elasticity of corrugated slab in the direction along the arch axis.

E22 : modulus of elasticity of corrugated slab perpendicular to the direction of El,. I ~ , I ~ 9moment of inertia of the thin-walled beam about principal axes X and Y. Ioo 9warping moment of inertia.

J*G " torsion constant. M x , M r 9bending moments about principal axes X and Y. Mo : bimoment

Ts : Saint Venant torsion.

All=

El~l t, .422= E2~2 t, A12=`421= ~t21El~l t= ~t12E22 t, 1-btl2~t21 1-btl2kt21 1-ktl2bt21 1-kt12~t21

1 1 1 1 1 Dll = ~'411 t3, 922 = ~ "412t3, 912 = D21 = ~ "412t3 = ~'421 t3, D33 = ~GI2 t3

`4* = fAlldS

I ~ = ~[`411ya2 (S) + DllCOS20(s)Jds

I ~ = [[AllXa2(S)+Dllsin20(s)]ds

I*oo~=f[All~a2(S)+Dllq2(s)]ds

`433 = G12t

116

J*G=4~D33ds

=rKNLI! [ K NL ]

LK NL21

0

[ K tcLI1 ]

[KNL21]

KNL12]

6N 5L

0

0

0

6cl 5L

0

0

0

~N 10 ci 10

=

=

K NL22 .]

[KlcLI 2 j r

6N 5L 6c2 5L N 10 0

sym 6c3 5L c2 2LN 10 15 0

c2 10

cl 10 c3 10

0

0

0

0

6N 5L

0

0

0

6ci 5L

0

0

0

N 10 ci 10

=

0

2LN 15 2Lc 2 2Lc I 15 15

0

2Lc 3 15

0

0

6c] 0 N 5L 10 6 N 6c 2 N 0 5L 5L 10 6c2 6cz c-g-2 cl 5L 5L 10 10 N c2 LN 0 10 10 30

0

0

0 c2 10

c--L 10 c3 10

0

LN 30 Lcl Lcl 30 30

c1 10 c2 10 c3 10 Lc 2 30 Lc 1 3O Lcz 30

117

6N 5L

0 [ K NL22] =

6c2

5L

5L

N 10 c~ 10

0

sym

5L

6c I

0 0

6N

N 10 0 c2 10

6c3 5L c_2 10 c~ 10 c3 10

2LN 15 0

2LN 15 2Lc 2 2Lc 1 15 15

2Lc 3 15

c I = NYp - M x , c 2 = NY? - M r ,

c~ N( -

A'

+ xp2 + r A ) + 2 m r x p - 2 m x r p -

Mr-;- ( ~[AHX2 ( X 2 + Y f ) + D1 l ( 3 X a - 2 X a cos 2 0 - 2YasinOcosO]ds ) Iyy Mx + i _ ~ ( ~ [ A 1 1 Y f ( X 2a + y 2 ) + O11(3ya _2ya s i n 2 0 _ 2 X a s i n O c o s O l d s ) Mo

2

2

_--;- ( ~ [ A l l D a ( X a + r 2 ) + D l l ( n a - 2qY a c o s O - 2 q X a sinO]ds)

Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All fights reserved

119

FABRICATION ACCURACY AND COST OF THIN WALLED STEEL PLATE GIRDERS IN SHOP ASSEMBLING K.Yoshikawa I and S. Shimizu 2 1 Matsumoto Factory, Miyaji Iron Works Co.Ltd., 1909 Hata, NAGANO 390-1401 JAPAN 2 Department of Civil Engineering, Shinshu University, Wakasato 4-Chome, NAGANO 380-8553 JAPAN

ABSTRACT The authors are making a series of studies on the laborsaving on these processes in purpose to reduce the cost of bridge fabrication. This paper represents a part of the results of authors studies, in particular with highlighting the relation of the magnitude of the initial imperfection of flatness of web plates and the strength of the girder. In this study, at first, measurement survey is made on the initial imperfection (flatness) of web plates of plate girders before and after the flatting works in a bridge fabrication shop. According to this survey, it is found that, in some bridges, the initial out-of-plane deformation is larger than the requirement of the design rule before the repair of the flatting work, and after the flatting work, all bridges satisfy the rule. Next, a large deflection elasto-plastic FEM analysis is made on the plate girder models, to clarify the influence of initial deformation to the strength and behaviour of the girders. In the analysis, the measured out of plane deformation before and after the flatting work, sine shaped initial deformation with the magnitude of the design rule requirement or larger, and the girder with no initial deformation are considered. The numerical result indicates that all models except a model having extremely larger initial imperfection has almost same strength. This result suggests that the excess work on the repair or the flatting is not needed.

KEYWORDS plate girders, fabrication, initial imperfection, cost, repair of defects, labour-saving, strength

INTRODUCTION This paper deals with the relation between the accuracy and the cost of thin walled steel plate girders during the shop assembling. Assembling or fabricating a bridge generally takes much cost, and to reduce the cost,

120 labour-saving technique must be introduced. Recently, some methods or techniques are proposed to reduce the cost for the bridges. For example, to reduce the cost of welding, automatic welding machines or welding robots are introduced. To omit the tentative assembly, the technique applying the photogrammetry is used. On the other hand, it still takes relatively much time in the processes of fabrication, repair of defects, and straightening or flatting of members of bridges. In fact, the fabrication process takes 15-20% in time in the shop assembling of bridge girders, and repair work takes generally 4-10%. The authors are making a series of studies on the labour-saving on these processes in purpose to reduce the cost of bridge fabrication. This series of studies deal with (1) the influences of initial imperfection of flatness of web plates, (2) the influences of the accuracy of assembling (or the influences of miss-assembling) such as the "dis-locations" of stiffeners, error of angle between the flange and the web and so on, and (3) the interview survey on the consciousness of workers. The current paper represents a part of the results of authors studies, in particular with highlighting the relation of the magnitude of the initial imperfection of flatness of web plates and the strength of the girder. The limitations of the out-of-flatness (or out-of-plane) of plates are specified in, for example in Japan, the JSHB (Japanese Specifications for Highway Bridges). However, the foundation or the basis of the magnitude of the limitation is not shown in the Specification, and it is often that the magnitude of the limitation is too severe. In spite of this, sometimes more accuracy than the limitation of the Specification is required. In this study, at first, measurement survey is made on the imperfection (out-of-flatness) of web plates of plate girders before and after the flatting works in a bridge fabrication shop. Then, an FEM analysis is made on the model girders to clarify the relation of the strength and the magnitude of the imperfections. Finally, a consideration shall be made on the requirement of the repair works. In this paper, hereafter, the words "out-of-plane" is abbreviated as "O.O.P".

REPAIR WORKS & MESURED INITIAL IMPERFECTION

Repair Works In the TABLE 1, examples of the ratios of the working time (or the labour time) of steel girder assembling are shown. In this table, "Miscellaneous" includes the drawing, marking, completion, other detail and small works and indirect works. TABLE 1 RATIOSOF LABOUR-q2MEnq SHOP ASSEMBLYOF STEELBRIDGEGIRDERS Tentative Miscella-neo harking Cutting &' Fabrication Welding Repairing Assembly Bridge Types us Drilling Simple Composite 38.4% 6.0% 13.7% 15.7% 16.2% 6.1% 3.9% Plate Girder 2-spans Continuous 37.7% 10.5% 5.3% 13.7% 20.9% 8.2% 3.7% Box Girder 3-spar~ Continuous 26.2% 10.2% 10.7% 18.3% 20.3% 9.9% 4.4% Plate Girder Simple Steel 40.4% 8.5% 6.2% 16.5% 15.8% 7.6% 5.0% Plate Girder

121 This table indicates that the fabrication, welding and tentative assembly have relatively larger ratio in time in assembling steel girders. However, on the welding or on the tentative assembly, recently, the cost is being reduced with the automatic welding by welding robots, or with the newly developed technique to omit the work. Therefore, cost on the repair is considered to be relatively increased. The major part of the repair work is to reduce the initial imperfection of web plates, especially initial out-of-plane deflection (or out-of-flatness) of web plates. According to the Japanese Bridge Design Code (Japanese Specifications for Highway Bridges, JSI-IB), the accuracy of flatness of a web plate is limited as h/250; here h denotes the height of the web plate. However, recently, questions or problems arise on this limitation. The first question is the "adequateness" of the limitation. That is, it is sometimes pointed out that the limitation specified in the JSHB seems to be too rigorous, and the relaxation of the limitation may be possible. The second is that, in spite of the previous problem on the adequateness of the limitation, the more accuracy than the specification is often required by the client. Therefore, in the process of a bridge girder assembling, repair work to keep the flatness of web plates is almost obliged.

Measured Initial Out-of-Flatness In this study, first of all, out-of-flatness of 24 web plates of girders manufactured in the Matsumoto Factory, Miyaji Iron Works is measured to know the actual circumstance of the imperfection. The measurement is carried out before and after the repair work on the same girders. A typical measured girder is a simple composite girder with its span length of 30m, total nominal depth of 1600nun, web thickness of 9nm~ and has a horizontal stiffener at 300ram from its upper flange as illustrated in Fig.1. The width of the panel (i.e. the interval of the vertical stiffeners) of this girder is 1500mm.

1500

,}22

,_-_ _..~

,.._.._...,.....- .

~.__~00

+9 9

-

425

Fig.2 shows the contours of the measured initial out-of-plane deformation of this panel. Within this figure, Fig.2(a) shows the initial deflection contours Fig.1 Measured Web Panel before the repair work is made, and Fig.2(b) shows the after the repair work. This figure shows the web panel subtended by the horizontal stiffener and the lower flange, and the vertical and horizontal lines drawn in the figure indicate the mesh with interval of 10cm. Because the initial O.O.P deflection of the upper panel that is subtended by the horizontal stiffener and the upper flange is very small, contours of the upper panel are not shown in this figure. As found from the figure, before the repair work, the contours of the O.O.P deflection shows the oval concentric circles, and it is found that the sine shaped O.O.P deflection arises before the repair work. The maximum magnitude of the initial O.O.P deflection before the repair work is 7.4ram The limitation of the magnitude of the initial O.O.P deflection of this girder is 6.4ram because the total depth of the girder is 1600ram and the limitation is specified as h/250 by JSHB. Therefore, before the repair work, the magnitude of the measured initial O.O.P deflection exceeds the limitation.

122 After the repair work, the maximum magnitude of the O.O.P deflection is reduced to 2.4nma, and this value satisfies the limitation. ,1

/,/ x.~--

1

,

..____.~___

r

it\\\

\

\

"

'~/

jJ

\/

/

.-' f )

\

\~.,

I

/

r176

k.''-,.._~'-"',--~,.---'-" .-/t Jl "--..._"" ~ --'": - ' - - - - i - / l J

t\

" ~ ' - ,.,._.. ~

.._._

.__.., ,.....-'

2.~

-

.j

~,o---"

A" "

(a) beforethe RepairWork

(b) after the RepairWork

Fig.2 MeasuredInitialOut-of-planeDeflectionContours

NUMERICAL SIMULATION In this study, numerical calculation is made to know the strength of a girder panel having the same nominal dimensions of a web panel used for the measurement of the initial out-of-plane deflection. That is, the numerical calculation is made on the girder panel shown in the Fig.1. The model is assumed to be consist of the grade SMA490 steel which has the nominal yield stress of 323MPa, and the Young's Modulus E and the Poisson's Ratio -v are E=206GPa, v =0.3. In the numerical model, the center of the girder is assumed to be located at the left edge of the panel, and the shear and the bending moment are considered as the loading at the right edge of the panel. The analysis is made by the elasto-plastic large deflection FEM with arc-length increment technique by using the computer program package LUSAS.

~....~aximum

Initial

Loading

0mm

TABLE 2 MODELNAMESwrrH A PARAMETER 7.Smm 6.4ram 10.Tmm 32.0mm (beforeRepair (h/250) (h/150) (h/50) Work)..

2.4mm

(after Repair

S-32

S-07

Work) S-02

M-32

M-07

M-02

,,

Shear Moment

S-00 "' ,,

M-00 .

S-06 .

M-06 .

S-10 .

M-10 .

.

.

In the analysis, the magnitude of the initial O.O.P deflection is adopted as a parameter. The initial deflection patterns considered in this study are as follows; (1) The web panel with no initial O.O.P deflection (2) The web panel having initial O.O.P of the limitation of the JSHB, 6.4mm (1/250 of the girder depth)

123 with the sine shape. (3) The web panel having O.O.P. deflection larger than the JSHB limitation, 10.7mm (1/250 of the girder depth) with the sine shape. (4) The web panel having O.O.P deflection larger than the JSHB limitation, 32.0mm (1/250 of the girder depth) with the sine shape. (5) The web panel having O.O.P. deflection measured before the repair work, 7.8mm. (6) The web panel having O.O.P. deflection measured atter the repair work, 2.4mm. These numerical models are denominated as shown in the TABLE 2.

NUMERICAL RESULTS Models subjected to Shear The TABLE 3 shows the maximum load of the models subjected to the shear. TABLE 3 MAXIMUMLOAD OF MODELS SUBJECTEDTO SHEAR

This table shows that all models expect the model with the largest initial deflection has the same Pmax, and only the model S-32 has the smaller Maximum load by 6%. This obviously indicates that the magnitude of the initial O.O.P. deflection has small (or almost no) influence to the strength of the web plate unless the web has not so large initial deflection. In Figures 3 and 4, load-deflection relations are plotted. Within these figures, Fig.3 shows the downward deflection at the edge of the analyzed model, i.e. the shear deformation of the panel. The all models except the model S-32 have almost same P-delta relation, and have the peak load at the deflection is about 3.5mm. On the model S-32, which has the largest initial deflection, the peak is reached at the deflection of about 4.5mm. Fig.4 is a figure showing the deflection at the point at which the largest O.O.P. is observed. The curve of the model S-00 having no initial deflection shows a slight plateau at the load is reached at about 1300kN. This suggests that the buckling is occurred at this stage. This model has its peak load at the deflection is 18mm. The model S-02 with small initial deflection is seemed to have the very slight plateau, but in the all other models, plateau is not found in the curves. This figure also indicates that models expect S-00 and S-32 have similar curves. In Fig.5, the deformation patterns of 4 models are illustrated at the stage of their peak load. All these models have the clear shear deformation. It is found that the models S-02 and S07 have similar deformation pattern, and in the model S-00, the web plate is deformed for the alternate direction. Although the figure is not shown in this paper, the models S-06 and S-10 have very similar deformation pattern to the models S-02 and S-07, however, the model S-32 has the larger shear deformation than the another models. These results indicate that the all models except the model having the larger initial deflection have very

124 similar shear behaviour, and the magnitude of the initial deflection has very small contribution to the behaviour within the range of the magnitude of the initial deflection which arise under the normal bridge assembling process.

2000

2000

-

1800

1800

fg=

1600

i

1400

1400

1200

1200 Z Q.

9

1000

Z

_

Q.

I

10oo

......

800

8O0 ~//~"

600

models

"

40O 200

m o tim=

1600

s

60O

" - - S-32

,/ ~

1

I

2

,,

I

I

3

4

D o w n w o r d Deflection (mm)

Fig.3 Load-Deflection Relations

I

5

S-00 - - S-06

...... -

2OO I

0

f

4O0

-

0

"--

-

-

-

S-10 -

-

S-32 S-07

--------- S-02

0 5

10

15

20

Out-of-plane Deflection (mm)

Fig.4 Load-O.O.P. deflection Relations

S-07

Fig.5 Deformation Patterns under Shear

125

Models subjected to Bending On the models subjected to the bending, discussion is made with the stress at the lower edge of the web plate caused by the bending moment instead of the magnitude of the load.

400 -

~---550 300 I~.

I

250 1-

~~ ~

200 t~

o')

........ M-O0 M-02

I

15o I

100

/

. . . . . . M-07

~ --M-IO M-32

I

50 v

0

I

I

I

I

5

10

15

20

O.O.P. deflection (mm) Fig. 6 Load(Stress)- O.O.P. Deflection Relations under Bending

M-02 Fig.7 Deformation Patterns under Bending

M-07

126 Fig.6 shows the load (edge stress)-O.O.P, deflection relations of the models. The O.O.P. deflection in this figure denotes the deflection of the point where the largest deflection is observed at the peak. This figure indicates that, in the all models, the deflection increases very rapidly atter the stress reaches to the yield stress, then it is reached to the peak gradually. This figure also indicates that the all models have the peak stress of about 350MPa. However, only the model M-32 shows the different behaviour to the remained model. On the model M-32, the deflection is developed to the alternate direction at first, and then the web plate is deformed to the positive direction. In Fig.7, the deformation patterns of the models at the peak subjected to the bending are illustrated. The O.O.P. deformation due to the bending is found form this figure at the upper part of the web plate, and these three models have very similar deformation pattern. As same to the models subjected to the share, the similar deformation pattern is also observed on the models M-06 and M-10, and only the model M-32 have different deformation pattern.

CONSIDERATION AND CONCLUSION As described at the first of this paper, according to the Japanese Bridge Design Code (JSHB), the accuracy of flatness of a web plate is limited as h/250. In this study, the measurement survey made on the imperfection (out-of-flatness) of web plates of plate girders before and after the flatting works in a bridge fabrication shop. This measurement is carried out on the 24 web panels, and within these, 12 panels exceed the limitation including a web panel which is referred in the numerical analysis in this study. Following the Japanese design coed, these 12 panels must be subjected to the repair work to keep the flatness of the web panel. In addition, it is sometimes required to make the repair work by the clients that the panels with smaller initial O.O.P. deflection than the limitation to have more accuracy of the flatness. The results shown in this paper suggest that all models except the model having extremely large initial deformation have similar behaviour and strength, and that such a repair work is generally not required. The repair work takes about 4-10% of the shop assembly of girders, and these results indicates the possibility that major part of this can be omitted. The behaviour and strength of a girder is contributed by many factors, although the numerical analysis in this study is made under very limited conditions. Therefore, the authors feel more study must be made on this problem. Thus, the authors have continued a series of study on this problem with considering many other conditions, or with making an interview to the workers of a bridge shop. The results of these studies shall be reported in the another opportunity.

REFERENCE Japan Road Association (1996), Specifications for Highway Bridges, Part II Steel Bridges, Maruzen, Tokyo, JAPAN

Section III BRIDGE STRUCTURES

This Page Intentionally Left Blank

Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

129

WEB BREATHING AS A FATIGUE PROBLEM IN BRIDGE DESIGN Ulrike Kuhlmann and Hans-Peter Gttnther Institute for Structural Design, University of Stuttgart Pfaffenwaldring 7, 70569 Stuttgart, Germany

ABSTRACT By web breathing often repeated high ranges of secondary bending stresses due to postbuckling out-ofplane deflections at the boundaries of the web lead to a serious fatigue problem which limits the life of bridge girders. Recent experimental and numerical research deepened the knowledge about the various influence parameters of web breathing and lead to first reliable assessment procedure. A new comprehensive fatigue assessment study of the problem of web breathing taking into account the real fatigue loading as well as latest results about fatigue resistance is presented. First results are given for a typical composite railway bridge.

KEYWORDS Thin-walled structures, bridges, buckling and postbuckling, fatigue, web breathing, railway bridges, linear-damage accumulation, stress range spectra, detail category, secondary bending stresses INTRODUCTION Postcritical plate buckling is characterised by a marked increase of out-of-plane deflections. For very slender web panels e.g. of high bridge girders, these deformations under service loads may be relatively important and even unacceptable, although due to the postcritical buckling resistance for the Ultimate Limit State safety against collapse is adequately guaranteed. For structures subjected to repeated live loads, especially highway and railway bridges, the repeated out-of-plane deflections produce significant secondary bending stresses at the boundaries of the web, i.e. at the web-to-flange, see Figure 1, and web-to-transverse stiffener connections. These often repeated high ranges of secondary bending stresses lead to a serious fatigue problem which limits the life of the girder. This phenomenon is known as web breathing. As fatigue problem related to the postcritical plate buckling behaviour web breathing is a very complex phenomenon depending on numerous parameters. Recent experimental and numerical research work could solve some of the related questions.

130 _

Detail \

~l

, Detail G ~

crack

/'[/I

"Q J__t

Figure 1: Out-of-plane deflection and secondary bending stresses ~•

RECENT STUDIES

Experimental Studies To study the fatigue behaviour of breathing webs an extensive experimental test program has been conducted among others in Prague: about 130 tests on welded plate girders loaded predominately in shear, as well as in Stuttgart: 6 large welded plate girders consisting of four web panels so that the two inner ones were subjected to both, bending and shear, see Skaloud et. al. (1999), Kuhlmann & Spiegelhalder (1999) and Kuhlmann & Gttnther (1999). The following observations can be summarised for both test series: 9 The fatigue cracks always appeared at the toes of the fillet welds between the web and the boundary members. The cracks were initiated at the zones of maximum secondary stress ranges and grew along the fillet welds. 9 The growth of the fatigue cracks was a continuous process. Never during the tests any abrupt increase in the rate of crack propagation had been observed. Thus no unstable crack growth had been indicated. 9 The occurrence of cracks at the test girders in Stuttgart clearly indicated the unfavourable influence of a combination of bending and shear. Most cracks developed in the inner panels of the 4 test girders, due to the combined action of bending and shear, and only a few cracks in the outer panels where shear was predominant. This observation underlines the conclusion, which is also confirmed by numerical calculations of Spiegelhalder (2000), that the combination of bending and shear is more critical than pure shear loading. 9 The initiation and propagation of cracks and the fatigue lives of the girders are significantly affected by the quality of the fillet welds which connect the web to the boundary members and by the form and magnitude of the initial web imperfections. 9 Collapse occurred at a very late state after quite a number of cracks had already appeared without significant effects on the overall deformations. The collapse were mainly due to some secondary stability failure modes like buckling of the unstiffened flange and belonged to relatively high number of load cycles compared to the number of cycles of crack initiation.

Secondary Bending Stresses Theoretical investigations on the web breathing problem focused their studies on the prediction of the fatigue relevant secondary bending stresses. Early investigations by Maeda & Okura (1984) were based on differential equations. During the last decades and the ongoing progress in FE programs a lot of theoretical investigations for example by Roberts (1996) or Duch~ne (1998) were undertaken with the help of FE software. These simulations have highlighted the very important influence of the form and magnitude of the initial out-of-plane deflection on the magnitude of the secondary bending

131 stresses. An extensive survey and profound comparison to own numerical simulations is given by Spiegelhalder (2000). In general the shape and magnitude of secondary bending stresses or• see Figure 2, depend on many parameters that may be summarised as follows: 9 aspect ratio (a = a / b) 9 slenderness (~ = b / t) 9 form (n = number of sine-half-wave) and magnitude of initial imperfection (e0 / t) 9 loading (00 ,x0 and normal stress distribution tV) Oo

,

.I

ri

O0

Xff[~'

li /i~'

1 17

f t,A t,

,,1

'/t_ t

S e0

'I

~

r'-

1 ___11

a

~,

t

--v-"

g~'~o

Figure 2: Web panel, dimensions, loading t~0 and x0, secondary bending stresses or• The FE model developed by Spiegelhalder (2000) was calibrated to the results of the experimental girder tests, Kuhlmann & Spiegelhalder (1999) and Kuhlmann & GOnther (1999). The FE model as shown in Figure 3 consists of one main web panel with 20x20 shell elements. Two adjacent smaller panels were added in order to achieve a tmiform loading condition within the main panel. The FE analysis was performed using the FE program ANSYS 5.5 with a geometric non-linear solution method. The boundary elements were chosen on the safe side in order to give maximum secondary bending stresses.

Figure 3: FE model, dimensions and loading Assessment Procedures

Traditional assessment procedures for web breathing limit the web slenderness b/t to a certain value, e.g. SIA 161 (1990). The recent research results clearly proved this limit to be insufficient as e.g. so called "non-breathing" webs suffered severe fatigue cracks due to web breathing, Kuhlmann & G(inther (1999). Also the web breathing verification proposed by Eurocode 3 Part 2 (1997) dealing the problem only on the basis of plate buckling is unsafe as shown in Duch~ne (1998) and Spiegelhalder (2000). Spiegelhalder (2000) has developed a fatigue assessment based on diagrams giving the fatigue resistance ~cin dependence of Z. which is a plate slenderness value. These results have been generalised

132 into a safe side verification formula based on a simplified comparison between the Ultimate Limit State verification according t o Eurocode 3 Part 1.5 (1997) and the web breathing verification, Kuhlmarm et. al. (2000): Unless a more accurate method is used, for unstiffened web panels the problem of web breathing may be checked with the following simplified criterion:

~ 1.25-p ?fyd

+

XSd k3 "Zw "fyd / ~

(0.6 + ___ 0.8 R)A

where: Gx.Sd , "lTSd stresses in the web panel calculated using the ultimate load combination A = 1.0 for road bridges and A = 0.85 for railway bridges kl = 1 +(1.25-p) 2, k2 = 1 + 1.25.p.Zw2 and k3 = 1.65-13/(275-50o0 and k3>0.64 with P, Zw according to Eurocode 3 Part 1.5, Section 4 13= b / t web slenderness range of validity: r = a / b aspect ratio (o~ < 1.5) R = 6t / Ca = xl / xa stress ratio (0 < R < 0.5) New developments by Gtinther (2001) now try to come back to the easy to handle limiting slenderness -values which are now deduced by a comprehensive fatigue assessment study taking into account the real fatigue loading as well as the knowledge gained for the fatigue resistance and the behaviour of the secondary bending stresses caused by web breathing.

WEB BREATHINGAS FATIGUEASSESSMENTIN BRIDGE DESIGN General Procedure

Fatigue assessment procedures are usually based on the S-N-concept, which relates a nominal stress range S to the number of load cycles N which causes fatigue failure. Figure 4 names three components of the assessment procedure: For the fati_~e loading, e.g. the model of Eurocode 1 Part 3 (1995) for railway bridges is used to determine the load history. Following each time step of this load history the fatigue stress range is calculated. For the problem of web breathing secondary bending stresses form the relevant fatigue stress range Aai. In dependence of the applicable detail category the ratine resistance is given by the belonging S-N-curve. Fatigue stress range spectra summed up by counting methods are compared to the fatigue resistance in form of S-N-curves. The fatigue damage is calculated following the rule of linear-damage-accumulation according to Palmgren-Miner.

~

FatigueAssessmentProcedure [

Fatigue Loading

Fatigue Stress Range

Fatigue Resistance

Fatigue load model for railway bridges Load history

Secondary bending stresses Counting method Stress range spectra

S-N-curves Detail category Damage accumulation

,,,

Figure 4: Fatigue assessment procedure against web breathing in bridge design

133

Fatigue Loading For railway traffic modelling the fatigue load model "standard traffic mix" of Eurocode 1 Part 3 (1995) is used. This model consists of altogether 8 different train types. For each of these trains, the number of train types per day, the weight of each train and the overall yearly traffic volume expressed in 106 tons per year are given in Annex F of Eurocode 1 Part 3. According to that load model axle loads and spacings are simulated resulting in time-dependent shear and bending stresses acting on the considered web panel. This load history of primary shear x0 and bending stress 60 form the basis for the calculation of the fatigue relevant secondary bending stress range A(r•

Fatigue Stress Range Since the secondary bending stresses depend on so many parameters it is almost impossible to derive an analytical relationship between the influence parameters and the secondary bending stresses o• = f(~, [3, o0, x0, ~, e0/t, n) that is precise enough for a fatigue calculation. Additional aspects have to be considered: the secondary bending stresses vary along the boundaries of the web panel. The point of maximum secondary bending stress range can not be defined beforehand. The more severe loading condition of primary shear stresses combined with primary bending stresses which has to be considered does not only involve a variation of the critical web panel along the girder but also the task to find the decisive combination of the different relation R of maximum and minimum stress values of shear and bending. O'o,t/OE

3kR

,tvlij,k 0 2 4 6

"[0,t/O'E

:e

12 14 "--- 16 l!

--,9 20 22 24 26 28 30

o

4

.

i = O0/OE 8 12 15! 20 24 28 32 ;36 40 44 43 52 56 60

iweb plate i---:'

.,.,~II ~ ..I

. . . . . . . . . . . . . . . . . . . . . . .

...~ I I ~"

r-O+~-

(~.l_,k,t

9

w

time

t

Figure 5" Set-up of data matrix Mij,k and determination of secondary bending stresses This complex problem can only be dealt by an intelligent computer based mechanism as developed in Gtinther (2001): Instead of an analytical relationship between the secondary bending stresses and all the influence parameters, the results of FE calculations are stored within a three-dimensional data matrix Mij.k, which relates the primary stresses of shear and bending to the secondary bending stresses calculated for each of 60 chosen nodes at the boundaries of the web panel, see Figure 5. The indices are as follows: i = c0 / trE (i = 0, 4 .... 60) primary bending stress j = x0 / OE (j = 0, 2 .... 30) primary shear stress k (k = 0, 1.....60) node number along the web boundaries According to this set-up the data matrix contains results of 16xl 6 single FE calculations. The primary

134 stresses 60 and x0 as well as the secondary bending stresses 6• in the data matrix are divided by the Euler stress 6E. Thereby the results are independent of the absolute value of the web slenderness [3. The third index k the node number reflects the shape of the secondary bending stresses along the web boundaries. The node number is limited to 60, the lower boundary within the tension part of the web is not taken into account. Each data matrix is valid for one specific aspect ratio ix, a specific form n and magnitude of imperfection e0/t and a specific normal stress distribution V. By means of the data matrix it is possible to determine the secondary bending stresses for every timedependent load condition of the primary stress history 60,t and x0,t. By repeating this process for every time step of primary stresses the stress history of the secondary bending stresses of each single node along the web boundary 6• is derived, see Figure 5. The resulting stress range spectra of each node is calculated using the rainflow method for cycle counting.

Fatigue Resistance The fatigue resistance depends on the geometry of the constructional detail and the corresponding stress range spectra. Regarding the problem of web breathing the constructional detail changes along the web boundaries. The following two different notch details have to be considered, see Figure 6:

,---,Detail B S

,

Detail A:

Detail B:

Web-to-vertical stiffener

Web-to-flange

U

, Detail A

'H'

A~c > 80 N/mm2

Aac > 110 N/mm2

Figure 6: Fatigue resistance of constructional details related to web breathing

Detail A: Web-to-vertical stiffener On the safe side this detail may be assumed to be within category 80 (vertical stiffener welded to a beam or plate girder) according to the classification table of Eurocode 3 Part 1.9 (2000), though the secondary bending stresses are predominant and should increase the category. Detail B: Web-to-flange The detail consists of a T-section under bending load condition which does not exist in the classification table of Eurocode 3 Part 1.9 (2000). However meanwhile a number of fatigue tests have been carried out on this detail. The evaluation of own fatigue results and additional results from literature in Kuhlmann & Gtinther (1999) and Giinther (2001) gives a minimum value of A6c = 110 N/mm 2 for the characteristic fatigue strength. According to the calculated fatigue stress spectra the cumulative damage based on the Palmgren-Miner rule is determined for each node. The damage calculation is performed using a fatigue strength curve with double slope constants (m = 3 and m = 5), changing at the constant amplitude fatigue limit A6o at 5.106 number of cycles. The limit state of fatigue failure due to web breathing is defined when any one of the 60 points along the web boundaries reaches the damage value ofDd = 1.0.

135

Figure 7: Dimensions and cross section of a single track composite railway bridge The above described method is illustrated for a single-span composite railway bridge with one track. Figure 7 shows the cross sectional dimensions. The bridge is designed according to the Ultimate Limit State (ULS) of Eurocode 1 Part 3 (1995), Eurocode 3 Part 2 (1997) and Eurocode 3 Part 1.5 (1997). The web plates with a steel grade of S 235 are chosen according to approximately 90 to 100 % of the ultimate load capacity of Eurocode 3. As a common value in railway bridge design the height of the web is assumed to 1/18 of the span length L. For the verification against fatigue failure due to web breathing the secondary bending stresses within the data matrix Mi,j,k are determined for a square web panel with a = 1.0 under pure shear. The out-ofplane imperfection e0 is assumed as high as the plate thickness t (e0 = 1.0-t) with a single sine-halfwave as shape. Results at the end-support ~.

1.5

d., 250

e9

1.25

,,.+ 225

...

200 9' - '

-,~t//-

,,.

1

"~ 175 "o ~

0.75 0.5 0.25 0 50

100

150

200 s l e n d e r n e s s 13 [-]

20

b)

~ - Carrying capacity according to Eurocode 3 Part 1.5, ULS

30

40

50

60

s p a n lenght L [m]

-o- Fatigueverificationdue to web breathing

Figure 8: Carrying capacity of a single span railway bridge at the end-support The primary stress of the web panel at the end-support is predominately shear. Figure 8 a) shows the verification Sd/Rd for the Ultimate Limit State as well as for fatigue due to web breathing in relation to the web slenderness 13. The results are derived for a span length of L = 40 m. For this special case the Ultimate Limit State is decisive resulting in a maximum allowable slendemess value of max 13= b/t = 142, while the allowable slenderness to prevent fatigue cracks induced by web breathing is about max [3 = 188. Figure 8 b) shows similar results for different span lengths. As it is typical for

136 the fatigue verification of railway bridges, there is a strong dependence on the span length. It can also be stated, that in all the considered cases the Ultimate Limit State is decisive.

OUTLOOK AND ACKNOWLEDGEMENT By the new fatigue assessment study presented in this paper quite a number of case studies with realistic bridge dimensions similar to those given here for a single-span composite railway bridge are carried out at the moment. They consider further important parameters as the aspect ratio, the imperfection or the steel grade and lead to limiting slenderness values by which fatigue failure due to web breathing can be excluded. These slenderness values are more precise than the safe side verification formula also given in this paper and enlarge the range of usage which had to be limited for that formula. All these new assessment procedures for web breathing lead to a more efficient design of slender steel webs so that in future one can more easily take full advantage of the postcritical plate buckling behaviour. The authors gratefully acknowledge the financial support of their research by Deutsche Forschungsgemeinschatt DFG and Arbeitsgemeinschafi industrieller Forschungsvereinigungen "Otto von Guericke" e.V. AiF.

REFERENCES

ENV 1991-3 (1995): Eurocode 1 - Basis of design and actions on structures- Part 3: Traffic loads on bridges. ENV 1993-1-5 (1997): Eurocode 3 - Design of steel structures - Part 1.5: General rules, Supplementary rules for planar plated structures without transverse loading. prEN 1993-1-9 (2000): Eurocode 3 - Design of steel structures - Part 1.9: Fatigue strength of steel structures, 2nd Drain, 30 August 2000. ENV 1993-2 (1997): Eurocode 3 - Design of steel structures - Part 2: Steel bridges SIA 161 (1990): Stahlbauten (Steel Structures), Swiss Code. Duchrne, Y. (1998): l~tude par voie analytique et numrrique des effets de la respiration des ~.mes 61ancres sur la rrsistance ultime des poutres mrtalliques a ~ne pleine ou en caisson, PhD Thesis, MSM, Universit6 de Liege, Lirge. Gianther, H.-P. (2001): Zur Materialermiidung infolge Stegatmung im Briickenbau, PhD Thesis, No. 2001-x, Institute for Structural Design, University of Stuttgart (in preparation) Kuhlmann, U., Gttnther, H.-P. (1999): Zum Nachweis der Ermtidungsfestigkeit geschweiBter Stahltr~iger mit schlanken Stegen (Fatigue resistance of full-scale welded slender I-girders), Report No. 5/1999, Deutscher AusschuB ~ r Stahlbau DASt. Kuhlmann, U., Spiegelhalder, U. (1999): ErmOdungsversuche an vier geschweil3ten Stahltr~igem mit schlanken Stegen infolge Stegatmung (Fatigue tests on four full-scale welded slender I-girders subjected to web breathing), Internal report, Institute for Structural Design I, University of Stuttgart. Kuhlmann, U., Spiegelhalder, U., Gttnther, H.-P. (2000): Proposal of a new web breathing formula, Internal report, Web Breathing Group, Stuttgart. Maeda, Y., Okura, I. (1984): Fatigue Strength of Plate Girder in Bending Considering Out-of-Plane Deformation of Web, Structural Eng.~Earthquake Eng. Vol. 1, No. 2, 35-45. Roberts, T. M. (1996): Analysis of Geometric Fatigue Stresses in Slender Web Plates, Journal of Constructional Steel Research, Vol. 37, No. 1, 33-45. ~kaloud, M., Z6merov~, M., Kuhlmann, U., Spiegelhalder, U. (1999): Prague and Stuttgart Experimental Research on Web Breathing, Eurosteel 99, 2"d European Conference on Steel Structures, Conference Proceedings, Prague. Spiegelhalder, U. (2000): Zur Materialermtidung infolge Stegatmung (Fatigue behaviour due to web breathing), PhD Thesis, No. 2000-2, Institute for Structural Design I, University of Stuttgart.

Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All fights reserved

137

EROSION OF THE POST-BUCKLED RESERVE OF STRENGTH OF THIN-WALLED STRUCTURES DUE TO CUMULATIVE DAMAGE M. Skaloud and M. Ztirnerovfi Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, Proseck~. 76, 190 00 Prague 9, Czech Republic

ABSTRACT The plate elements of steel bridge and similar structures ,,breathe" under the many times repeated loading to which they are subjected. This phenomenon generates cumulative damage in them, and consequently the initiation and propagation of fatigue cracks, this substantially affecting the limit state of the structure. Based on numerous experiments carried out in Prague, two approaches to the definition of the limit state are discussed and a recommendation for design presented.

KEYWORDS Breathing, cumulative damage, erosion, fatigue craks, limit state, post-buckled reserve of strength.

QUASI-CONSTANT LOADING Plenty of attention was paid,during the last decades, to the derivation of an ultimate load approach to the design of the webs of steel plate and box girders. It was found out, using both theoretical and experimental investigations, that the (so-called) critical load, determined via linear buckling analysis, was in no relation to the actual ultimate strength of the web, since in a great majority of cases there always existed a very substantial post-critical reserve of strength. This is demonstrated in Figure 1, where the results of tests on shear girders carried out by (i) K.C. Rockey and one of the authors in Swansea and Cardiff and (ii) by the authors of this paper in Prague are plotted. They are plotted as Vult/Vcr ratios in terms of the flange rigidity parameter If / a3t, because it was found out that flange stiffness very substantially affected the ultimate load of the test girders. On the left-hand vertical axis, the ultimate load is related to the critical load of a web simply supported on all boundaries, while on the right-hand vertical axis the critical load of a clamped web is used in the denominator of the ratio ultimate strength/critical load. But it can be seen in the figure that, whether we consider the former or the latter value of the critical load, the post-buckled reserve of strength is always very great.

138

Figure 1: Swansea, Cardiff and Prague shear girder test results

REPEATED LOADING

The Web Breathing Phenomenon Although a great part of steel plated structures used in building construction can be listed among structures under the action of quasi-constant loading,this cannot be said about steel bridgework, crane runway girders and similar systems. Such structures are exposed to many times repeated loading. For example, it is not rare that the plate elements (e.g. the webs) of an important bridge should every year be subjected to hundreds of thousands of loading cycles. Then, if the webs are slender (as they usually are in modem construction) and considering the very great number of loading cycles to which the plate elements are exposed and because of their unavoidable geometric imperfections (particularly of their initial out-of-flatness), the webs repeatedly buckle out of their plane. This phenomenon, being now usually termed web breathing, induces significant secondary bending stresses at the boundaries of the web, i.e. at the web-to-flange and web-to-transverse stiffener junctions. Of course, then we can ask the obvious question of whether the breathing phenomenon leads to a significant ,,erosion" of the postcritical reserve of strength described above. It is in the nature of the breathing phenomenon that pronounced cumulative (fatigue) damage is generated in the girder, this very substantially influencing the limit state of the girder. Fatigue cracks then usually occur at several places of the breathing web.

The Regime of Fatigue Crack Growth and the Failure Mechanism As the main impact of web breathing is the initiation and propagation of fatigue cracks in the breathing webs, no reliable design procedure can be established without the regime of crack initiation and growth having been thoroughly mapped. And understandably, a suitably designed experimental investigation is an ideal tool for that. That is why an extensive experimental investigation into the problem of web breathing was started in Prague several years ago, which to date comprised 153 tests. Two series of test girders (with various web depths and flange thicknesses) were tested at the Institute of Theoretical and Applied Mechanics of the Czech Academy of Sciences in Prague. This group to date comprised 126 experiments. Two other series of girders were tested (27 experiments) at Klokner Institute of the Czech Technical University in Prague. The depth-to-thickness ratios of the girder web panels were of 175, 200, 250 and 320. Consequently, they ranged from so-called non-breathing webs

139 to very slender ones. The above research was supported by a fruitful co-operation with Cardiff School of Egineering, where another twenty breathing tests were c ~ e d out several years ago, their test girders being similar to the Prague ones. The results of the collaborative Prague and Cardiff research were published in about fifteen papers (see, for example, Skaloud, Ztrnerov~i and Roberts(1997) and Skaloud and Roberts (1998), where also references to other contributions can be found). In all of these tests, the webs of the girders were subjected to many times repeated predominantly shear. This was given by the small length of the test girders, which had two square web panels. That is why it was to the benefit of the experimental research that it could completed by six tests conducted at the University of Stuttgart (see Skaloud, Ztrnerovli, Kuhlmann and Spiegelhalder (1999)) on six large girders having four web panels, whose two outer ones were also subjected to predominantly shear and two inner ones to both bending and shear. Moreover, the experiments undertaken in Stuttgart were performed on large-sized girders (the depth of the web was 1.5 m ), this giving an opportunity to verify results and conclusions obtained through testing smaller girders. It is worth mentioning at this juncture that, unlike the previous Americn tests and the aforementioned Cardiff ones, in which the experiments were stopped when the first observable crack was detected, we conducted all of our Prague tests up to failure. Thus we were able to ,,map" the whole history of all fatigue cracks - from their initiation to the failure of the girder. Thereby we avoided being ,,fascinated" by the very phenomenon of crack appearance, but were able to study the further development of each crack, to see whether it propagated or stopped, and to find out how far away was the initiation of the first fatigue crack from the fatigue failure of the whole girder. The most important results of our research can be summarised as follows: The main impact of the cumulative damage process in the breathing webs of the test girders was the initiation and propagation of fatigue cracks. They developed in crack-prone areas,i.e, at the toes of the fillet welds connecting the flanges and transverse stiffeners with the web and in zones with maximum ranges of secondary stresses. Just in two cases a fatigue crack developed inside the breathing web, this being probably due to a notch in the web material. The regime of crack initiation and growth was not simple : In just about one half of the test girders, only one fatigue crack (or two cracks in opposite corners of the breathing web) developed, at first advancing along the fillet weld and then (in most cases) turning inside the web and perpendicularly to the tension diagonal in the buckled web. In the others cases more fatigue cracks initiated and developed, forming frequently a rather complicated crack system. In it, some of the fatigue cracks joined up or branched out, some of them stopped; and very frequently it was not the first crack that became the main crack and eventually led to the failure of the girder. Whatever the system of the fatigue cracks in the breathing web, the growth of the cracks was a stable phenomenon, with no abrupt change in the rate of crack propagation. Never during our experiments did we observe a kind of the critical crack length phenomenon known from Fracture Mechanics. All of our test girders collapsed earlier, when the fatigue crack (or cracks) were long enough to generate one of the failure mechanisms described herebelow. The process of the crack growth was very substantially affected by web slenderness. If compared with the fatigue performance of less slender webs ( h / tw = 250 ), in the case of breathing tests on webs with h / tw = 320, the initiation of fatigue cracks occurred after substantially lower numbers of load cycles, and the fatigue cracks grew more speedily so that the fatigue failure was a considerably faster phenomenon. This is quite understandable in the light of the fact that more slender webs buckle and, if

140 exposed to repeated loading, breathe more intensively, thus generating sooner large surface stresses in their crack-prone areas. However, when the fatigue lives of the test girders are plotted in terms of the ratio shear force range/critical shear force, thereby indirectly taking account of the difference in web slenderness, the results of both series exhibit a tendency of lying in one band and determining a S-N curve (Figure 2). ZXV

1.4 9 Prague

Vcr

[] Stuttgart

1.2 cO 9

.

%

P O0

9

0.8 ~41~ 99

9

e 9

9

9

0.6

1

0.4

~e

4.9

~e

4.6

,~e

6.8

0.2

0.5

0

1

1.5

2

2.5

NxlO e 3

Figure 2: The fatigue lives of the Prague and Stuttgart test girders; the role of the shear force range Vm~ - Vmi,

~V

2

Prague

Stuttgart

V cr1.8

9 9 Vm~-'-9"n= 0

1.6

Vmox

Vmo, Vmin

0 ~mox= 0,42 Q ~ =

1.2

9 Vrn=.._.~=n 0,318 Vmox

0

13 Vm=---..o.-" = 0,6

Vmin

1.4

Vm~n ...... = 0 , 5 6 Vmox

0,67

Vm.~ Vma-'--~= 0,311 Vmin

~=0,4 Vmox

9 OO0

1.

=o

0.8. 0.6-

9

I

g9 I~

OD

ee

9

,o', []

0o

OB

[]

[]

--.~=

4.9

---.~ =

4.6

--.-~ = 6.8

0.4 0

0.5

1.5

2

Figure 3: The role of the shear force ratio Vmin/Vmax

2.5

NxlO 6 3

141 All the results exhibited a large scatter. The reason for this is a great sensitivity of breathing thinwalled girders to differences in the quality level of fillet weld~ and in the shape and magnitude of initial imperfections; which varied (in a rather complicated and unpredictable manner) from girder to girder. It was found out that whenever snap-throughs occurred during breathing of the web (and this happened in 15 % of the breathing tests), thereby enhancing the dynamic aspect of the breathing phenomenon, this always led to a significant shortening of the life of the test girder. While the shear force range A V = Vmax- Vmin always and very considerably affected the lifetime of the test girders, the shear force ratio Vmin / Vmax did not appear (Figure 3), at least in the domain investigated (i.e. Vmin/ V~x = 0 - 0.67), to play an important role.

The Collapse Mechanism of the Test Girders The failure of the test girders occurred is one of the following two modes: (a) Most girders exhibited a typical shear failure mode, large buckles developing along the tension diagonal and plastic zones becoming manifest in the girder flanges. In the end, when the main fatigue crack cut most of the tension band, the girders behaved, and failed, like ones having a large opening in the web. The cutting off of the tension band was usually materialised by one long fatigue crack, but in several cases (when the cutting crack followed the flange and the transverse stiffener) by a set of short separate cracks. When one of the transverse stiffeners bounding the breathing web panel was subject to a large point load, as was the case with the central vertical stiffeners of both the Prague and Stuttgart girders, additional fatigue cracks sometimes occurred in the web sheet in the close vicinity of the stiffener, then influencing the failure mechanism described above. (b) Some girders collapsed as a result of the compression flange buckling vertically when the flange was separated from the web sheet (supporting the flange) by a long enough fatigue crack. This flange buckling occurs under the action of the compression force existing in the flange, and generated by the effect of a bending moment; therefore, it tends to happen in those web panels (if the test girder has four panels or more) and in those portions of the flange where the effect of the bending moment is largest.

The Definition of the Fatigue Limit State Definition based on fatigue crack initiation Some authors have for some time been preaching that the fatigue limit state of breathing webs should be defined so that the corresponding lifetime of the web is determined in a way such that it immediately precedes the occurrence of fatigue cracks in the web. This means that the limit state is related to the initiation of fatigue cracks in the breathing web. In preferring it, the corresponding authors wish to avoid having - however small - fatigue cracks in the web during the whole life of the girder. If carefully monitoring the whole process of fatigue crack growth from their initiation to the failure of the girder, as we have been doing in the case of all Prague tests, it is easy to obtain a relationship similar to Figure 2, but this time related to the initiation of the fatigue crack (Figure 4). However, in connection with this approach, the following questions can be raised:

142 (i) The initiation of which crack ? In our 140 Prague experiments we saw that in a great part of the test girders more than one crack developed and that frequently it was not the first crack that was decisive and ,,mortal", but another crack or a system of two or three cracks that, through cutting the tension diagonal, evoked the girder failure. (ii) How should we exactly define the initiation of a fatigue crack in a breathing web? In the case of breathing slender webs, it seems reasonable to define the beginning of the crack history by the onset a detectable ,,through" crack. (iii) And even more importantly, what does the appearance of a very small fatigue crack in the breathing web mean for the fatigue failure of the girder? ~V Vcr

1.4

9 Prague

n Stuttgart

1.2

9 o

0.8

i []

r

9

9

,$

0

O

0.6

-'-I~

9 1.69

2.7

0.4 0.2

0

0.2

0.4

0.6

0.8

1

N ~x 10 6 1.2

Figure 4: The number of load cycles to the initiation of the first fatigue crack Usually very little. Just in those cases where the load range is very large (but in practical cases this occurs scarcely), or when the web is very slender and without longitudinal ribs (as was the case with part of our Prague girders, where the web slenderness was of 320 - but such very slender and longitudinally unstiffened web plates are encountered seldom), the initiation of a fatigue crack is followed by a fast crack growth and a speedy failure of the girder. However, in a great majority of cases the load ranges and web depth-to-thickness ratios are not very large, and then there is a very long way from the initiation of the first fatigue fissure in the web to the collapse of the whole structural system. Very frequently (and we saw this in our experiments) is the number of load cycles at crack initiation only 10 % of the fatigue life of the girder. The above trend is for the Prague girders shown in Figure 5. The history of the growth of the main fatigue crack can be seen in Figure 6 - in terms of the ratio shear force range AV / critical load Ver. As usually several tests were conducted for the same AV / Ve~ ratio, the results cannot be distinguisted on one horizontal line and, consequently are grouped into ,,storeys". The large effect of the load range is obvious therefrom. Considering the aforesaid, we can easily realise that, when the design of breathing webs is based on crack initiation, we can never achieve the desired same safety of all plate girders against their failure.

143 In the case of girders with very slender webs and under the action of large load ranges, the reserve of safety after the initiation of the first fatigue crack (i.e. the difference between failure and crack initiation) is small (10, 20 % or so). On the contrary, for ordinary plate girders, whose webs are practically never so excessively slender, subjected to usually encountered load ranges, the post-crackinitiation safety reserve can be very large (even several hundreds of %).

Figure 5: The initiation of fatigue cracks in the webs of the Prague test girders in relation to their whole lifetime

,,~ - - - - - - - - - - - - - - _ - - - - - - -

....................

AV

vo--;"

o,,~|

,co

~

I ~ ',---n,e C -::::: ~,seI ~ ,

~, - l:k',

o EL~,5,I

-'-: t, 0

Crack _ initiation''A "----O ~

~ ~

~

! 1

Crock le~noth= Q,:2b ,o

i

'

~

~,Foilure -

..... .

.

.

.

.

I 2

.

.

.

.

.

.

.

.

.

.

o.

.

.

.

I 3

.

.

.

.

.

.

.

.

.

,

.

.

.

.

I 4

.

.

.

.

.

.

.

.

'

.

.

.

.

.

I 5

.

.

.

.

.

'~' N x l O "6

Figure 6: The history, in terms of AV/Ver, of the growth of the main fatigue crack in the webs of the Prague test girders

144 Definition based on fatigue failure The other possibility is to relate the definition of the fatigue limit state of breathing webs to their failure. This means that their lifetime is then equal to the number of load cycles up to the formation of a failure mechanism in the girder. Or the fatigue limit state can be related to a partial failure, i.e.to a well defined length of the main fatigue crack. We applied this approach in our collaborative research with T.M.Roberts (see ,~kaloud, Z~rnerovfi and Roberts (1997), Skaloud and Roberts (1998)), relating the fatigue limit state to the main crack length of 100 mm and on this basis established S - N curves for the problem of web breathing : For girders with good quality automatic and MIG welds: log N125 = 12.601 - 3 log Ao p log N125 = 16.536 - 5 log Act p

N < 5.106 N > 5.106

For rod welded girders (which always exhibit larger irregularities in the welds and elsewhere): log Nloo = 12.301 - 3 log Ats p log Nloo = 16.036 - 5 log Ats p

N < 5.106 N > 5.106

AGp is the principal surface stress range in the crack-prone areas for which formulae were established.

References Skaloud M. and Roberts T.M. (1998). Fatigue Crack Initiation and Propagation in Slender Web Breathing Under Repeated Loading. Proc. of the 2nd World Conf. on Steel in Construction, 417419. ~kaloud M., Z6rnerovfi M., Kuhlman U. and Spiegelhalder U. (1999). Prague and Stuttgart Experimental Research on Web Breathing. Proc. of the Int. Conf. Eurosteel'99 1, 75-78. Skaloud M., ZSrnerovfi M. and Roberts T.M. (1997). Fatigue Assessment of the ,,Breathing" Webs of Steel Plate Girders. Proc. of the 18th Czech-Slovak International Conference on Steel Structures and Bridges 4, 59-66.

Third InternationalConferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

145

EVALUATION OF STRENGTH AND DUCTILITY CAPACITIES FOR STEEL PLATES IN CYCLIC SHEAR T. Usami i, p. Chusilpl, A. Kasai l, and T. Watanabe I ~Department of Civil Engineering, Nagoya University, Nagoya, 464-8603, Japan

ABSTRACT This study attempts to develop a simple method for evaluation of maximum shear strength and ductility capacity for unstiffened steel plates subjected to predominant shear loading. The following two cases of plates are investigated: web plates without flanges and web-flange assemblages. To model the material nonlinearity, a modified two-surface model is employed. Both residual stresses and initial out-of-plane deflection are taken into account. Elasto-plastic large displacement analyses are conducted with the consideration of monotonic and cyclic loading patterns. From the results, the effects of slenderness parameter of the plate, flange thickness to web thickness ratio, and loading pattern are discussed. The maximum shear strength and ductility capacity obtained are compared with some approximate formulas available in the literature, and design equations for predicting maximum shear strength and ductility capacity are suggested for the web plates and web-flange assemblages. KEYWORDS Unstiffened plate, stability design, shear strength, ductility, shear buckling, cyclic loading INTRODUCTION Steel flat plates subjected to predominant shear loading are commonly found in beams and beam-tocolumn joints in thin-walled steel bridge piers. These plates are characterized by high width-thickness ratio and, thus, susceptible to severe buckling under a catastrophic earthquake. A pictorial view of shear panels buckling at the mid-span of the beam in a bridge pier damaged in the 1995 Kobe Earthquake is presented in Fig. 1. As shown, stability is an important problem that governs the performance (strength and ductility) of slender web plates. In evaluation or design of these plates, maximum shear strength and ductility capacity are ones of major indices that need to be checked with the seismic demands. Prediction methods of the maximum shear strength have been developed so far. A pilot study was carded out by Basler (1961). Since then, various maximum shear models based on the concept of diagonal tension band were proposed. Among them, the model of Porter et al. (1975) seems to be the most comprehensive because of good agreement between predicted and experimental results. Most studies mentioned above were conducted on plate girders with widely spaced transverse stiffeners

146

TABLE 1 BOUNDARYCONDITIONSOF WEB MODEL u

,,

w

o~

o~

o~

x=O

1

0

1

1

0

1

x=a

0

0

1

1

0

1

~y=O

0

0

1

0

1

0

y=b

0

0

1

0

1

0

free = O, fixed = 1 Figure 1. Shear buckling in web plates of a bridge pier

Figure 2: Finite element model of web model

Figure 3" Distribution of residual stresses of web model

under monotonic, predominant shear loading. The maximum shear strength can alternatively be computed from an empirical formula (Nara et al., 1988), which takes an advantage of simpler design method. For the ductility capacity, very few attempts have been made. Experimental investigations of unstiffened webs under cyclic shear by Krawinkler & Popov (1982) and Fukumoto et al. (1999) are examples. However, their work is limited to few cases of plate parameters. In the present study, maximum shear strength and ductility capacity of unstiffened web plates are investigated. Web plates with and without flanges are considered to gain insight into the contribution of the flanges. Key geometrical parameters of the plates are varied over the practical range, and both monotonic and cyclic loading patterns are adopted. A commercial computer program, ABAQUS (1998), including a constitutive law developed at Nagoya University is employed to carry out nonlinear finite element analyses. Finally, the effects of the geometrical parameters are discussed and design equations for the evaluation of shear strength and ductility capacity are suggested. ANALYTICAL METHOD Unstiffened web plates and web-flange assemblages (referred as web model and web-flange model, respectively) considered are those found in practical steel bridge piers having thin-walled box section. All webs have the width, b, of 2000 mm. The length of the web, a, being the spacing of diaphragm, is kept equal to the width b, since preliminary analyses show insignificant effect of the web aspect ratio, ot (equal to a/b), on the maximum shear strength and ductility capacity. These plates are fabricated from the steel type SS400 (Japanese specification) having the modulus of elasticity, E, of 206 GPa, the yield stress in tension, Cy, of 235 MPa, and the Poisson's ratio, v, of 0.3.

147

Figure 4: Web-flange model

Figure 5: Loading Patterns

Web Model The web model represents the case of web-flange assemblage whose flanges are extremely flexible. Its finite element model is shown in Fig. 2. The boundary conditions are assumed in accordance with Table 1, where u, v, and w denote x-, y-, and z-displacement degree-of-freedoms (d.o.f.'s), respectively, and 0~, Oy, and 0., represent rotational d.o.f.'s, respectively. The node (0,0) is hinged, and shear stress on the edges x = 0 and x = a is generated by means of the displacement 8 at the node (a,0) imposing the constraint that y-displacements at nodes on each loaded edge are linearly interpolated from those at the two ends of the edge. The other two edges unloaded are modeled as infinitely rigid bars and, as a result, the in-plane displacements at nodes along each unloaded edge are linearly interpolated from those at the two ends. According to the assumptions mentioned above, the stress state of the plate before initiation of buckling is close to pure shear condition. The web is divided into 14x14 mesh. Such element division is found to be sufficient to obtain accurate solution. A 4-node doubly curved shell element with five integration points along the plate thickness is employed. In the analyses, material and geometrical imperfections are taken into account. The distribution of residual stresses is illustrated in Fig.3. The tensile residual stress, art, and the compressive one, crc, are applied in x-direction with the magnitudes of ay and 0.3ay, respectively. The initial out-of-plate deflection, w, is assumed in Eqn. 1. The deflection amplitude ww is set as b/150. w= w w sin( a~-~)sin(~ )

(1)

Web-flange Model For the web-flange assemblages (box section), the width of the flange is assumed equal to that of the web. Only a half of the box section is modeled due to the symmetry about the plane z = -b/2 (Fig. 4). As a consequence, Ox, Oy, and w of nodes on the symmetric plane are vanished. Boundary conditions and constraints adopted are similar to those of the web model, except at the edges where the flanges are connected. In addition, all displacement d.o.f.'s on the edge x = 0, y = 0 are constrained to represent hinge supports. At all nodes on the plane x = a, the rotation 0x is not allowed and the displacement u is assumed identical throughout the plane. Vertical shear displacement 8 is applied uniformly along the edge x = a, y = 0.

148 Finite element division of the web-flange model is 14x14 mesh for the web and 7x14 mesh for the half of the flange considered. The distribution of the residual stresses similar to that of the web model is applied to the web and the flanges (Fig. 4). The initial deflection is introduced in the form of Eqn. 2, where ww = wf= b/150. - wwsin(nx / a) sin(ny / b) w = ~- w/sin(nx / a) sin(m / b) [w: sin(~ / a) sin(m / b)

for z = 0 for y = 0

(2)

for y = b

The web and web-flange models are characterized by the web's slenderness parameter

R~=

k

=twV

k . n 2E

= I4.00+ 5.34/t~ 2

for ct_<1

L5.34+4.00/ct2

for a>_l

(3)

(4)

Here, tw = web thickness, xy = yield stress in shear (equal to C~y/4~according to the von Mises yield criterion), Xcr = buckling shear stress, and k = elastic buckling coefficient of web plate in shear approximated by Eqn. 4 (DIN 4114, 1953) in which cc = 1.0. Elasto-plastic large displacement analyses are carded out by employing ABAQUS computer program (1998). A modified two-surface model (2SM) developed at Nagoya University (Shcn et al., 1995) is incorporated to trace the material nonlinearity. Two loading patterns: monotonic and cyclic cases shown in Fig. 5 are applied to the web and web-flange models by means of the displacement ~5, in order to investigate the effect of loading history on shear strength and ductility capacity. The magnitude of ~5is increased step-by-step as a multiple of the yield displacement of the plate in pure shear, ~Sy.The internal force and deformation of the two models are determined by means of average shear stress, ~, and average engineering shear strain,~, as follows:

~=P/(btw) and ~ = 8 / a

(5)

where P is the summation of shear stressesalong the loaded edge. All models are loaded up to failure defined at the stateat which the strength drops by 5 % aRer shear buckling. However, there exists some case where shear buckling is not dominant. The average shear stress increases until yielding, yield plateau forms, and strain hardening develops. In this case, the failure is encountered when the shear strength drops to 95% of the yield value.

EVALUATION O F WEB MODEL Parametric study is carried out on the web model with the parameter R~ ranging from 0.3 to 1.3 (b/t = 34 to 147). In this paper, some results are discussed. More detail of the investigation can be found elsewhere (Usami et al., 1999). Analysis results show distinguished difference in the ultimate behavior of thin and thick web models. Relatively thin plates (R~w> 0.3) quickly reach the failure aRer shear buckling, while thick plate (R~ = 0.3) develops yield plateau and strain hardening without any sign of shear buckling. In Fig. 6,

149 1.4

1.2

~

~ ' 1---9 9 I .... ........... ~ ..........i " ' ] - ' -

i'~i ~,

,

Euler curve G u i d e l i n e (1987) N a r a r al. (1988)

i

i

15

i :

,

!

v t~"9

0 n

.............. " ....... " ~ " ! ......

i

1

o

Monotonic One-side

V

Two-side

-

...."-V--

~.~]o lib-

~ 0.8

I o' M..o,o.,. I

0.6

0.4

"I

12

I v ,

,!,

One-side

I".N"

t .............. "~'" .......'""~"~

r,.o-,~e I ,

I

,

,!,

,

0.5

,,

,

I

5

0 0.5

0

Figure 6: Maximum shear strength of web model

I

Figure 7: Ductility capacity of web model

normalized maximum shear strength, Xm/Xy, obtained from monotonic (circular marks) and cyclic (triangular marks) loading are presented with the design curve suggested by the Japanese guideline for stability design (published in 1987 and adopted in Japan Road Association, 1996) and the Euler curve. An empirical formula (Eqn. 6) proposed by Nara et al. (1988) is chosen for comparison, due to its simplicity. The computed results agree very well with Nara et al.'s curve. It is also observed that the effect of the loading history on the shear strength is insignificant. x_..~= [ 0.486 / 0.333 "Cy

R~ )

for 0.486 < R~ _<2.0

(6)

The failure shear strain, Tu, normalized by the yield strain in shear, 7y (equal to Xy.2(1 + v)/E), are plotted against the parameter R~ in Fig. 7. For R~ ___0.6, the ductility capacity is weakly dependent of the parameter Rr~ regarding to the same loading pattern. The values of ?u are approximately 4Ty for the monotonic loading and 37y for the cyclic ease. The limit value of 37y seems to be appropriate for seismic design considering the random nature of earthquakes. For the plates with R~ < 0.6, the ductility capacity is considerably affected by the loading history. The following design formula of the ductility capacity is proposed for unstiffened web plates as a lower bound curve y....u_~={~0-47R~ 3'y

for R~ <0.575 for R~ > 0.575

(7)

EVALUATION OF WEB-FLANGE MODEL The web-flange models considered have the parameter R~ over the range of 0.3 to 1.3. The contribution of the flange thickness is investigated by varying the ratio of flange thickness to web thickness, ~tw, from 0.8 to 5.0. Ultimate Behavior

Plots of shear stress-shear strain curve of monotonically loaded web-flange models having R~ = 0.3, 0.5, 0.7, and 1.3 with various values of t/tw are shown in Fig. 8. The results obtained from the web model are also presented for sake of comparison. The point of failure is indicated by a mark on the curve. The web model, being equivalent to the web-flange model with t/tw = 0, usually performs

150 9

I

"

'

"

I

"

j

9

9

!

o

! I 7

9

~

O

]

~

O

9

.

~

-

-

....

Web-model

----- t~t.~0.8(Re0.TS)

0.5

9

,

,

J

0

------'Web-model

------

tdt,,-- 1 . 0 ( R r

---'--

tdt.=l.0(Rt~l.0

---.--

tdt,,=l.5(Rr~0.40)

.....

tdt,=l.5(Rr

......

tdt,=3.0(Rr

~

tdt,,=3.0(Re~0.34)

j 0

m

20

40

R~w=

(a)

0.5

"

'

m

20

T/Ty

I)

40 V/Yy

(b) R ~ = 0.5

0.3

a~

R w , -- 0.TJ"

I

'

,

9

i

-

"

~I

....w

I _ _ l

0.5

....

Web-model

------

t d t , = 1 . 0 ( R t = 1.41)

'

.

,

.

I

tdtw= 1.5(Rj=0.94)

[

" ......

t d t - = 3 " 0 ( R t = 0 " 4 7....)

o i

40

y/yy

ii

,

,

i|1|~

~ _

. . . .

9- - - - -

20

--

0.5

- - - . - - - t d t w = l . 2 ( R t = l . I 7)

0

i

i

_

Web-model

/"--'--t~'t"-I'~Rel"1'~l / ....... t"t"-3"~176

,

t /--'--tdt'--S'0(Rr~~

0

20

40

y/~y

(d) R~w = 1.3

(c) Rw = 0.7

Figure 8: Shear stress-shear strain curves of web-flange model under monotonic shear

I I

0.5

0

-0

10

20

r-r~/~y

0

..... C r162 0

5

10

_

)

.-.'W/Vy

Figure 9: Shear stress-shear strain curves of web-flange model under cyclic shear weaker than the web-flange model. As expected, strength deterioration due to shear buckling is not apparent for web-flange models with thick web (R~ _ 0.5). The yield plateau and strain hardening develop and render the failure shear strain greater than 507y. For the web-flange models with thin web (R~w > 0.5), deterioration of shear strength can be observed clearly after the first peak. These plates quickly reach the failure and, therefore, produce very low ductility capacity. Although the plates with thin web may exhibit ascending branch after the defined point of failure, such increase of shear strength should not be accounted in design since considerable strength deterioration has been taken place. In general, higher shear strength and ductility capacity can be expected for higher value of t/t~ because of higher constraint provided by the flanges. The effect of the ratio t//t~ on shear strength and ductility capacity of the plate is not pronounced if the web is slender.

151 Figure 9 gives examples of the envelopes of shear stress-shear strain envelope of web-flange models under cyclic shear compared with the results obtained from the monotonic case. Two values of R~ of 0.3 and 1.3 are selected to represent typical eases of thin and thick webs and the ratio t/tw is kept equal to 1.0 (critical and practical value). As shown, the thick plate behaves less ductile under cyclic shear, because of an early initiation of the strain hardening. For the thin plate, the ductility capacity from the monotonic case is slightly smaller than the cyclic one. The shear strength seems to be insensitive to the loading history for all values of R~.

Prediction of Maximum Shear Strength and Ductility Capacity As mentioned earlier, the contribution of the ratio t/tw on the maximum shear strength and ductility capacity is very small for web-flange models with thin web which is of major concern in this study. Only the critical case of t/tw = 1.0 is, therefore, considered in the prediction of the maximum shear strength and ductility capacity. In Fig. 10, the maximum shear strength of the web-flange model under monotonic and cyclic shear is plotted with the results of cyclically loaded web model, as well as those computed from Nara et al.'s formula (Eqn. 6). Good agreement between the analysis and predicted results verifies the validity of the formula for general web-flange assemblages in shear, irrespective to the loading history. The ductility capacity of the web-flange and web models under the two loading patterns is given in Fig. 11. The web-flange model generally gives higher ductility capacity, as compared to the web model, particularly for small value of R~. From the results obtained, an approximate equation for the ductility capacity of the web-flange model is proposed 7.._L 0.142 + 4.0 < 20.0 ?y (R~-0.18) 4 =

(8)

The failure strain predicted by this equation tends toward the value of 47y as the parameter R~w increases. This value is identical to that suggested by Krawinlder & Popov (1982), which is based on an experimental investigation of unstiffened web plates in the beam-column joints predominantly in shear.

~"

~ ~"

2 0 [ 7'. " ! T " " " ~w.~a.=, =~.,,~.-Lo) F----:.--'~A,-.--i--.-,k...-- ............ ........... 9 --I 0 Mo,=o,~

t

i;

~ "I ~

i

ii

.4 "

i

i.lw, b .o~,~

[.71 ..... ~'"'~'""A'"'~ ............ ! ........... ~1

Is b..-" .... .~"

I'': .......... ~'"'"'A~ ............ i ........... !t

10 r

O.S l w . ~ - n , , p

I o u

] ol

0

~

X!

i

c,,,,,

,,0,~,d

9

~1. . . . . . .

c,~,.

rropo~d

=o~.~I

~0.=0, H c~,., ]

]Web model

0

i

"

|

c~,,~= / I

0.5

. . . . RTw

I I

. . . .

"

1.5

Figure 10: Maximum shear strength of web and web-flange models

...........i

.-..... 'i:

oF ....... ~ ......... i .... i..... i ............... i......... i .......... i......... ~ ........ 'i ..... ......... 0.5 1 1.~ R.cw

Figure 11" Ductility capacity of web and web-flange models

152 Conclusions

This paper presented a parametric study of ultimate behavior of unstiffened web plates and web-flange assemblages (box section) subjected to predominant shear loading. Monotonic and cyclic loading pattern were adopted in the analyses. Key geometrical parameters of the plates were considered over the practical range. A summary of this study is as follows: 1. The maximum shear strength of web plates and web-flange assemblages depends strongly on the parameter R~ and is insensitive to the loading history. For the ductility capacity, both effect of the parameter R~ and that of the loading pattern should be considered. As the value of R~w increases, the ductility capacity tends to approach the values of 3~/yfor the web plates and 4~y for the webflange assemblages. 2. Higher shear strength and ductility capacity can be expected for higher value of t/tw because of higher constraint provided by the flanges. The contribution of the ratio ty/tw on shear strength and ductility capacity of the web-flange assemblages is not pronounced if the web in slender. 3. From the results obtained in this study, design equations of the ductility capacity have been proposed for unstiffened web plates and web-flange assemblages in monotonic or cyclic shear. It is suggested that the maximum shear strength can be determined from the simple formula of Nara et al., due to good agreements between the analysis and predicted results.

References ABAQUS/Standard User's Manual, (1998). Ver. 5.7, Habbitt, Karlson & Sorensen, Inc. Basler, K. (1961). Strength of Plate Girders in Shear. Journal of the Structural Division, ASCE, 87:ST7, 151-180. Deutsches Institut ftir Normung (1953). Stahlbau, Stabilitatsfalle (Knickung, Kippung, Beulung), Berechnungsgrundlagen. DIN 4114, Blatt 2, Berlin, Germany (in German). Fukumoto, Y., Uenoya, M., Nakamura, M., and Takaku, T. (1999). Ductility of Plate Girder Panels under Cyclic Shear. Stability and Ductility of Steel Structures, D. Dubina and M. Iv~yi, eds., Elsevier, Romania, 283-290. Japan Road Association (1996). Design Specifications of Highway Bridges Part H Steel Bridges, Tokyo, Japan. Krawinkler, H. and Popov, E.E (1982). Seismic Behavior of Moment Connections and Joints. Journal of the Structural Division, ASCE, 108:ST2, 373-391. Nara, S., Deguchi, Y., and Fukumoto, Y. (1988). Ultimate Strength of Steel Plate Panels with Initial Imperfections under Uniform Shearing Stress. dournal of Structural Mechanics and Earthquake Engineering, JSCE, 392:I-9, 265-271 (in Japanese). Porter, D.M., Rockey, K.C., and Evans, H.R. (1975). The Collapse Behavior of Plate Girders Loaded in Shear. The Structural Engineer, 8:53, 313-325. Shen, C., Mamaghani, I.H.E, Mizuno, E., and Usami, T. (1995). Cyclic Behavior of Structural Steels. II: Theory. dournal of Engineering Mechanics, ASCE, 121:11, 1165-1172. Usami, T., Ge, H.B., and Amano, M. (1999). Strength and Ductility of Plates in Shear. Advances in Steel Structures, S.L. Chan and J.G. Teng, eds., Elsevier, 563-570.

Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All fights reserved

153

A RATIONAL MODEL FOR THE ELASTIC RESTRAINED DISTORTIONAL BUCKLING OF HALF-THROUGH GIRDER BRIDGES Z. Vrcelj and M.b_ Bradford School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia

ABSTRACT This paper presents a generic buckling analysis to investigate the elastic restrained distortional buckling of half-through girder bridges. The bridge girder is assumed to be of doubly-symmetric I-section, simply supported at its ends, and without web stiffeners that would be deployed for stiffening for shearing actions. The bottom flange of the girder is restrained at the deck level fully against translational and lateral rotational buckling deformations, but is restrained elastically against twist rotation by the flexibility of the deck between adjacent girders. By invoking a Ritz-based procedure, a buckling parameter is identified that may be used to quantify this form of distortional buckling. Design curves are presented that may be used to determine the elastic distortional buckling load. KEYWORDS Bridges, distortional buckling, elastic, energy procedure, restraints, stability, through-girders. INTRODUCTION The utilisation of half-through girders in bridge construction is most usually a result of constraints on headroom. They find frequent use in railway bridges over roadways, where the grade of the railway is predetermined and it is difficult to provide a substructure to support the bridge deck. When the superstructure is in the form of I-section girders, the top flange of the girder is subjected to compression and cannot be braced laterally, except by the provision of transverse web stiffeners which may be used to design against buckling in shear. Conservatively, the girder may be designed against lateral bucklmg without bracing of the compression flange. However, this conservatism can be excessive, and advantage must be taken of the restraint provided at the tension flange level by the bridge deck. When a half-through girder (Fig. 1) does not have a braced top (compressive) flange, its buckling mode must necessarily be distortional (Bradford 1992), since the web must distort in the plane of its cross section as it restrains the compressive flange during buckling. Further, because of the restraint provided at the

154 tension flange level, cross-sectional distortion is more profound, and so this buckling mode is referred to herein as 'restrained distortional buckling' (RDB)(Ronagh & Bradford 2001). In structural design, recourse is usually made to the so called U-frame approach (Oehlers & Bradford 1995), in which the compression flange of the girder is modelled as a uniformly-compressed strut on a Winkler-type foundation, whose stiffness is that of the flexural stiffness of the web in the plane of its cross-section. This model is attractive, since a closed form solution exists for the elastic critical load of such a strut, and it is easy to determine the flexural stiffness of a web plate. However, the approach is overly simplistic, since it ignores the effect of moment gradient, and does not include any instability that may also occur in the web. Svensson (1985) presented a useful modification of the U-frame approach, and this model was refined further by Williams and Jemah (1987). The latter modification suggested that a portion of the web should be included in the strut (which becomes a tee section), with the table of the tee strut being the compressive flange and the stem of the tee strut being 15% ofthe depth ofthe girder's web. Further models have been suggested by Bradford (1996), Lindner (1997) and others.

Figure 1: RDB of a half-through girder bridge

This paper addresses the issue of the buckling of half-through girders by developing a generic approach to the problem using a Ritz-based procedure. The model identifies a distortional buckling parameter that may be used to reduce the proliferation of design graphs normally associated with distortional buckling to comparatively few. The buckling parameter also identifies the relative importance of many geometric dimensions, as well as their interactions on this distortional mode of buckling.

RITZ-BASED PROCEDURE The energy approach in the Ritz procedure used herein requires an assumed shape for the buckled configuration of the beam at the point of bifurcation from the primary equilibrium path. When measured

155 out of the plane of loading, the primary equilibrium path is the trivial path, while the secondary equilibrium path is modelled as a Fourier series of undetermined magnitude.

Figure 2: Buckling model

Figure 2 shows the buckling deformations of one of the girders. These deformations comprise of a lateral deformation ur of the top flange, and of twist rotations tpr and q~B of the top and bottom flanges respectively. By specifying these freedoms, the buckled configuration of the cross-section is defined uniquely if the flanges are assumed not to distort (which is justified as flanges are usually stocky elements), and if the web is assumed to buckle in the plane of its cross-section as a cubic curve. Inherent in this idealisation of the buckled shape is that the bridge deck has both very large translational and lateralrotational stiffnesses. This may be argued on the basis of the shear stiffness of a typical deck, which may have a much lesser flexural stiffness between the adjacent bridge girders and which provides theoretically quantifiable torsional restraint to the girders. If the beam is considered to be simply supported with respect to out-of-plane buckling, the buckling deformations {u} = r are assumed to be given by n

{ul={q}~_sinirc~

(1)

i=l

in which {q} is the vector of Ritz coefficients which represent the maximum magnitudes of the buckling displacements, and where ~ = z/L. Further, if it is assumed that the web buckles as a cubic curve, then u~ = h

a~r/

sin i

(2)

156 where 11= 2y/h. Compatibility of displacements and slopes at the flange-web junction enables the coefficients ai in Eqn. 2 to be written, using Eqn. 1, in the form

[c]{q}

(3)

The strain energies stored during buckling within the flanges, web and elastic restraints can be written respectively as (Bradford 1994) 1

IEIr, CO~ur]" GJsVr f~

U,= 2)ot - ' ~ t , - ~ J Uw

= :

1

"

0

2

+--~LIv-~-J

=

+['~)

-

:]}

(4)

d<

(0

u,,,]

dr/d~

(5)

1

Ua : l L I k , rp~d~

(6)

0

where E = Young's modulus, v = Poisson's ratio, EIyF= minor axis flexural stiffness of flange, GJ = Saint Venant torsional rigidity of flange, k~ = elastic twist rotation stiffness applied at the level of the bottom flange, and D is the web plate rigidity given by Et

(7)

D= i20_v2 )

The variation of bending moments M(~) along the beam induce normal and shear stresses o(~,rl) and ~(~,rl) respectively. These stresses cause a reduction in the potential in the flanges and web given respectively as

Vv=2Lao

k-'~-)

2_ 1 1 11/: Ouw/ v., =2t~h'L~b-20u,,,iO~

k@)

ko~)

OuwlOr

(8)

(9)

where A is the area of the flanges, and VT, B --" X (flT,B

(10)

The moment field is assumed to be specified within the length domain [0,L] which is mapped to [0,1 ] within the ~ domain, and further may be piecewise continuous so as to model concentrated loading. The loose form of this predetermined moment is

157 3

M(r ZtoZb# i=0

with Mo being a reference value, ~, is the buckling load factor, and the coefficients b; (i = 0 .... 3) defining the moment field. The change in the total potential I'I is

ri = u F + u w + u R - g ~ - V w

(12)

which can be written as

I-I=l {q}r[k]{q}

(13)

2 where [k] is the stiffness matrix that depends linearly on 9~. By using the variational form the neutral equilibrium at buckling, that ~srI = 0 for any arbitrary variation of the buckling displacements {5q}, leads to the familiar buckling condition [k(k)]{q} = {0}

(14)

Equation 14 represents a routine linear eigenproblem that may be solved by standard numerical algorithms for the buckling load factor L as well as the buckled shape that is defined by the normalized eigenvector

{q}. GOVERNING PARAMETERS In the generic modelling of this buckling problem, the matrix [k] may be expanded and the relative magnitudes of the terms within it evaluated. In general, the matrix entries depend on the material and geometric properties, and on the moment field specified in Eqn. 11 (Vrcelj & Bradford 2000). Within each entry is the summation of a number of terms whose relative magnitudes for beams of realistic proportions are such that insignificant terms may be omitted. This approach simplifies the formulation greatly. Using this approach, it may be shown (Vrcelj & Bradford 2000) that the buckling load factor is dependent only on the three parameters defined below.

= f(at, K, y)

(15)

In Eqn. 15, the well-known beam parameter is

[ EIW

K=I~ ~

(16)

in which Iw = h2If14 is the warping constant for a doubly symmetric beam, and the dimensionless parameter describing the torsional restraint is

158 A so-called distortional parameter ~, can also be identified in Eqn. 15, which has the form ~'=

DL 2

GJh

(18)

whose alternate form of

,e(D/h) is similar to that ofEqn. 17. An ensemble of buckling solutions has been produced by Vrcelj & Bradford (2000) in which all of the variables that affect the solution of Eqn. 14 have been considered. This study showed that, except for unrealistic values of K, the buckling solutions are embodied in the three parameters in Eqn. 15.

VALIDATION OF SOLUTION The solutions produced by Eqn. 14 were compared with those produced using a line-type finite element (Bradford & Ronagh 1997, Ronagh & Bradford 2001), and the less computationally-efficient software ABAQUS (1998). The results of these comparisons are reported in Vrcelj & Bradford (2000), where close agreement of the solution for specific cases was demonstrated.

NUMERICAL RESULTS The numerical method has been used to study a simply supported I-section beam that would comprise one half of the superstructure for a half-through girder bridge. For consistency with the numerical model, the deck is assumed to restrain fully the bottom (tension) flange against lateral deflection and minor axis rotation, but to restrain partially the bottom flange against twist rotations with dimensionless stiffness a given in Eqn. 17. For particular moment fields defined in Eqn. 11 (uniform bending, a point load at midspan and a point load at a quarter point), the buckling load factor X may be determined as a function of K, a and 7. The load factor is plotted in terms of the dimensionless ratio MolMob, where Mob is the lateral buckling moment for a beam restrained fully against lateral deformation and minor axis rotation at the tension flange level, but which is free to twist during buckling. This buckling mode does not involve cross-sectional distortion. Variations of the dimensionless buckling moments are shown in Figs. 3 - 5 for a beam in uniform bending, a beam with a central point load and for a beam with a point load at its quarter point, respectively. The trends shown in these three figures are very similar, but because of the use of the parameters K and ~, the physical significance of the results may be somewhat obscured in the interpretation. While it might be expected that the effects of distortion would be less for high values of L (or low values of K), this is not evident intrinsically in the figures. This occurs because the high values of ~, for which the effects of distortion are most profound at low values of K are influenced by the high values of L in Eqn. 17.

159

Figure 3: Buckling curves for uniform bending

Figure 4. Buckling curves for central load

Figure 5. Buckling curves for quarter point

160 CONCLUSIONS This paper has presented the results of a Ritz-based method for the rational RDB analysis of simply supported half-through girder bridges. The modelling is generic in nature, and has identified a unique distortional buckling parameter, which, in conjunction with the well-known beam parameter, quantifies the effects of the geometric properties and especially the cross-sectional distortion on the RDB of half-through girder bridges. The presentation of the elastic buckling curves using these parameters reduces the multiplicity of design curves associated with distortional buckling to only a few.

REFERENCES

ABAQUS User's Manual (1998). Pawtucket, Hibbitt, Karlsson & Sorensen Inc., Rhode Island. Bradford M.A. (1992). Lateral-distortional buckling of steel I-section members. Journal of Constructional Steel Research 23:(1-3), 97-116. Bradford M.A. (1994). Buckling of post-tensioned composite beams. Structural Engineering. & Mechanics 2:1, 113-123. Bradford M.A. (1996). Stability of through girder bridges. Proc. Conf. on Structural Steel Developing Afn'ca, Johannesburg, 35-42. Bradford M.A. and Ronagh H.R. (1997). Generalized elastic buckling of restrained I-beams by the FEM. Journal of Structural Engineering, ASCE 123:12, 1631-1637. Lindner J. (1997). Lateral-torsional buckling of composite beams. Journal of Constructional Steel Research 46, 1-3 (Paper 289). Oehlers D.J. and Bradford M.A. (1995). Composite Steel-Concrete Structural Members: Fundamental Behaviour, Pergamon Press, Oxford. Ronagh H.R. and Bradford M.A. (2001). Stability of composite bridge girders. Proc. 1~'tInt. Structural Engineering and Construction Conf., Honolulu, 883-888. Svensson S.E. (1985). Lateral buckling of beams analysed as elastically supported columns subjected to varying axial force. Journal of Constructional Steel Research 5, 179-193. Vrcelj Z. and Bradford M.A. (2000). Design curves for the elastic stability of half-through girder bridges. UNICIVReport, The University of New South Wales, Sydney. Williams F.W. and Jemah A.K (1987). Buckling curves for elastically supported columns with varying axial force, to predict lateral buckling of beams. Journal of Constructional Steel Research 7, 133-147.

Third International Conference on Thin-Walled Structures J. Zarag, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

161

INSTABILITY TESTING OF STEEL PLATE GIRDERS WITH HOLDED WEBS

T. Yabuki 1, y. mrizumi 2, j. M. Aribert 3 and S. Guezouli 4 ~,2Department of Civil and Architectural Engineering, University of Ryukyu, Okinawa, 903-0213, JAPAN 3.4 Department of Civil Engineering & Town Planning, Institute of National Science and Application Rermes, 35043, FRANCE

ABSTRACT The use of folded plates is a possible way of achieving adequate out-of-plane stiffness without needing the use of horizontal stiffeners or thicker plates in plate girders. First, the economical and aesthetic advantages of folded plates in steel girders are underlined. Then this paper compares experimental buckling resistances of I-shaped steel girders having unfolded web plates with those of the folded plates. The tests show that the buckling resistances of the girders with folded webs can be higher than those of conventional I-shaped girders. Finally, the test results are also compared with the buckling resistances calculated from a non-linear finite element approach using shell element and specific algorithms to determine appropriately the lateral-torsional instability.

KEYWORDS Thin-Walled Members, Steel Plate Girders, Folded Plate, Local Buckling, Stability. 1. INTRODUCTION Recent infrastructure development has seen greater demand for structures both economical and aesthetic. Notably with bridges, there is need of new structures that are useful as main members and maintain their aesthetic appeal. Many efforts to create new bridges have resulted in the development of various girders. One example is the hybrid steel girder whose the material of web plate has a lower yield strength than have flange plates. The high shear resistance of the web plate allows for the use of low grade of steel material, making rationalization possible. Other suggestions include the use of corrugated plates as webs, which improves the stiffness of the plates and eliminates the need of stiffeners when using thinner plates. The first purpose of this paper is to propose a new bridge girder model, which appears both economic and aesthetic. This model consists of thin-walled steel plates, in which the webs are plates folded in the spanning direction within the compression zone. This folded line may stiffen the girders laterally and provide structural beauty. Furthermore, this paper presents experimental comparisons of overall, local and interactive buckling resistances of I-shaped steel plate girders having conventionally unfolded web plates with those of the folded plates. The tests show that the buckling resistance of beams having folded web plates can b~ in

162 average, 20% higher than conventionally I-shaped plate girders. Finally, the test results are compared with calculations issued from a non-linear finite element approach using shell element and specific algorithms developed previously by Yabuki, et al. (1994, 1995, 2000) to investigate instability phenomena appropriately. 2. PROPOSAL OF A BEAM WITH FOLDED PLATES IN WEBS Economical designs of plate girders direct to thin webs, normally, as favored by JSSHB (1990) and/or EUROCODE (1992). The conventional welding of stiffeners to allow the use of thin web plates has a disadvantage of high fabrication cost. Instead of stiffener welded to prevent buckling, this study proposes to use webs consisting of plates with folded lines in the direction of the bridge spanning within the compression zone as shown in Figures land 2. The so-folded webs may save time in the shop by eliminating the manual welding required for the horizontal stiffeners. On the other hand, the folded lines on the sides of the girder make the girder height appear lower from the aesthetic viewpoint. Moreover, the light refracting off the folded lines provides outstanding structural beauty. 3. THE EXPERIMENTAL PROGRAMME

Design of Test Beam Specimen The experimental investigation comprised of a set of three ultimate capacity tests performed in folded steel plate girder and that in regular shaped plate girder. The experimental test layout and crosssectional shapes of regular and folded steel plate girders are shown in Figures 1 and 2, respectively. All the test dimensions are presented in Table 1. In the Table, Aflange and Aweb are the flange and web plate slenderness ratios, respectively and Abeam is the relative slenderness for the elastic lateraltorsional buckling of beam. The measured properties of the steel material used in each specimen are given in Table 2. Test specimens A-1, A-2 and A-3 were designed to collapse by overall, interactive and local instabilities, respectively.

Figure 1:General arrangement of test layout

Figure 2: Regular I-shaped and proposed folded cross-sections

163

TABLE 1 DETAILS OF T H E TEST S P E C I M E N S

FLANGE L 0 TEST GIRDERS (cm) ((leg.)

TYPEA

2b (cm)

t/

b/t/ ~.~s,

(cm)

A-1

271.6

0

10.0

0.6

A-2

271.6

0

14.0

0.45

hw' (cm)

WEB . ly Iz J tw hw/tw Zweb (cm4) (cm4) (cm4) (cm)

8 0.540 48.80 , 16 1.060 49.10

0.6 0.6

Cw (cm6)

Abeam

81

0.703

82

0.744

101 13132 4.954 61009 0.879 ..... 207 13653 4.386 126320 0.659 428! 13911 3.991 263264 0.436

....

A-3

271.6

0

20.0

0.32

31 2.010 49.36

0.6

82

0.706

B-1

271.6

5

10.0

0.6

8 0.556 48.80

0.61

81

--

127! 13279 5.005

70838 0.829

,,

TYPE B

B-2

271.6

5

14.0

0.45

16 1.101 49.10

0.6

82.

--.

234 13802 4.438 1369301 0.654

B-3

271.6

5

20.0

0.32

31

0.6

82

--

455 14064 4.044 274408 0.418

1.965 49.36 i,

Note : Iz = Moment Inertia about Major Axis, ly = Moment Inertia about Minor Axis, J = Torsional Constant, Cw = Torsional Warping Constant II

I

I

II

TABLE 2 P R O P E R T I E S OF THE STEEL M A T E R I A L OF T H E TEST S P E C I M E N S

Test Girder

332 338 334 335

448 440 445 444

4.39

382

4.39 4.37 4.38

375 345 367

454 423 410 429

3.15 3.15 3.14 3.15

353 328 341

386 386 382 385

5.85 5.84 5.73 5.81

360 347 348 352

447 448 453 449

392 389 407 396

441 447 419 436

315 321 302 313

428 426 425 426

Thickness t (mm)

Flange(A-I) & Web (A-1.A-2,A-3)

A-A1 A-A2 A-A3 average

5.68 5.71 5.72 5.70

A-B1 Flange(A-2)

A-B2 A-B3 average

i

I :l

l

A-C1 A-C2 A-C3 average

Flange(A-3) ,

TYPE B

Tensile Strength tru(MPa)

Test Coupons

i

TYPE A

Yield Stress trr(MPa)

Plate Components

il

li ii

Flange(B-I) & Web (B-1.B-2,B-3)

B-A1 B-A2 B-A3 average

Flange(B-2)

B-B1 B-B2 B-B3

4.34 4.31 4.39

average

4.35

B-C1 B-C2 B-C3

3.11 3.12 3.12

average

3.12

Flange(B.3)

!

164 An instability classification diagram shown in Figure 3 and proposed by Yabuki, et al. (1995) has predicted the instability mode. In the Figure, Asecao,is a cross-section parameter to describe the influence of cross-sectional properties on the stability classification of the reference girder. Herein, the angle 0 of the fold was equal to 5 degrees and the position of the folded lines followed the Japan Specifications for Standard Highways and Bridges (1990) for application of one layer of horizontal stiffener as shown in Figure 2. Girder test specimens were fabricated by fast manufacturing subassemblies of all components and then welding them together to form the complete girders. Double T-joints accomplished through groove welding were used in holding the flange to the web. This enabled the use of normal welding processes with realistic welding size. Therefore, imperfections of fabrication and residual stresses similar to those found in real bridges would be produced in the test specimens.

Testing Procedure As a whole, a simple beam with stepped cross-sections (termed herein as test setup beam) is assembled including a central part of the test specimen and both side parts of the loading beam units as shown in Figure 1. Lateral movements of the loading beam units are prevented by prevention rigs located near to the loading points as shown in Figures land 4. High-tension bolts through the end plates, as illustrated in Figure 5 connect the test specimen and loading beam units. Accordingly, supports that would stiffen the test specimen to warp and twist with its end diaphragms and the loading beam units and restrict to move laterally with the prevention rigs might be achieved. The simple beam support condition for in-plane bending is provided to the test setup beam by roller bearings at the ends of the beam. The bearings are attached to pedestals (termed herein, support beam unit) that are secured to the test bed floor as shown in Figure 1 and thus provide support on settlement and against out-of-plane rotation. To permit free longitudinal movement during the tests, sliding plates were placed beneath expansion rollers attached to one of the support beam units, respectively. Moreover, supporting conditions that prevent twisting but stiffen to warp with support beam units and their end diaphragms might be achieved to the test setup beam.

Xsection 1.4

i

i / /"

lJocal instability

l.2

9

9

m

Am

9

/

= .s -~0.8

-- - ~ '

~0.6

0.4 ____.1.. 0.20

9

// 9

9,i/i'

I -~ - 4 , I i I

k*,, 9 i

l 9

9

9

9

9149 9149 9149 9 9 9

~- 9

I

I"

--9 /

I /dP ~/ mmm / ~ 9 9 ~ ( ann" 9 9It / l i \ i

1

0

Io,oc.vol 9 I Sm~o, I m / *

1.6

9

9 Boundary for

[

9 * 9 - i ovo~,,-,,,~!,,,y I 9

,,'-'. m . ,. , . , . ,.,~ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Relative Slenderness of Beam

Figure 3: Instability classification of I-shaped steel plate girder under uniform bending moment

165 For the ultimate capacity tests, uniform bending moment was applied to the test specimens by two equally concentrated jack-loads at the loading beam units as shown in Figure 1. Figure 4 illustrates the loading jack device used in the tests. The specimens were tested under stroke control in testing machine. The test-setup-arrangement, in which the test setup beam is loaded only at points of effective lateral restraint, produces an unrestrained length of test specimen subject only to equal end moments. Load cells as shown in Figure 1 are controlled by a microcomputer for measurement of the loads. During testing, the loads are increased by steps of 1.0kN up to the ultimate stage while monitoring the load-deflection relationship displayed on the microcomputer in real time. The test was continued under deflection control, just beyond the maximum test load. Their data are acquired in the computer. 5. DISCUSSION OF RESULTS

Experimental Results The results of tests are summarized in Table 3. The instability aspect for each specimen is mentioned in the following. Figures 6, 7 and 8 collect the load-lateral deflection curves on the middle point of girder depth at the span-center of the test specimens up to collapse as obtained by the tests. Figure 9 illustrates buckling mode for each specimen with the folded steel plates. Figure 6 corresponds to collapse caused by overall instability phenomenon. On the contrary, Figure 8 shows the collapse behavior following local rigidity weakening due to local buckling phenomenon of the steel plates that constitute the tested girder. Figure 7 shows the collapse behavior caused by interactive instability phenomenon of both types. These results show that folding of steel webs appears more efficient against local buckling than lateral-torsional buckling, which is rather logical. From this result, it seems that the test of Type A-3, which is a conventionally I-shaped plate girder designed to collapse by local instability, should be re-examined. Table 3 shows that by using folded plates in the web, the ultimate strength can be increased by 14 to 33%, with an average of 20% or 14 to 16%, with an average of 15% even if the local buckling case is eliminated.

Figure 4: Loading device and side sway protection rigs

Figure 5: Connection of test specimen and loading beam unit

166 M/Mp

~E

0

8

~

~

..,,...,~

"

_~0 6 I--,~'~.y ~

9"-

"

[

~0.4~/ " ~.

~

~

TYPE B-I

0

~M/Mp

r

~

Eo 9 [

............ ~---:r-~--~--i---!

........

i (,,,m~o--o~,l)

L

I/ b/tf=8,hJtw=81

'"

1

~'~ ~0.4 ~;*

~ 0.0025 0.005 0.0075 0.01 0.0125 0.015 Lateral Deflectionat Span-Center Figure 6: Collapse behavior caused by overall buckling ~, M/'Mp 1

............ ~ (Mmax/Mp=O.673,

=~0.6~..f~.~"~~"~:'~'"~~'~: ................... .~............

!

u,~ ~

,

TYPE B-2

~

.-~. ,

h~ [ s!g:'c"~d::;s!0~ ! I' 0

I

l

.............. ..............

:b/t~16, hw/tw=82 I

~

0 0

0.005 0.01 0 . 0 1 5 0 . 0 2 Lateral Deflectionat Span-Center

~o.~ t

I TYPEB-3 I [(Mmax/Mp--0.791) I

!.........-TYi~E~'A:3-.........." !.!....................~.:594!.!

o0

#~

/,

I stoo, Or.,o-SS400 ! t

0.025

Figure 7" Collapse behavior caused by interactive buckling

~0.6 0.4

"::i ....

~'~'"

, J

0.004 0.008 0.012 Lateral Deflectionat Span-Center

UIL 0.016

Figure 8: Collapse behavior caused by local buckling

Figure 9: Different failure modes of the test specimen

167

TABLE 3 TEST RESULTS OF ULTIMATE CARRYING CAPACITIES I-SHAPED STEEL PLATE GIRDERS (TYPE-A) FOLDED STEEL PLATE GIRDERS (TYPE-B)

Type (1)

(2). . . . . 0.741 0.673 0.594

A-1 A-2 A-3

Type (3)

(4)

(4)/(2) (5)

B-1 B-2 B-3

0.845 0.781 0.791

1.140 1.160 1.332

AND FOR

....

TABLE 4 TEST RESULTS AND ANALYSIS OF ULTIMA CARRYING CAPACITIES FOR FOLDED STEEL PLATE GIRDERS

Experimental Resulls

Theoretical Results

Type (1)

Mmax/Mp

Failure Mode

Mmax/Mp

Failure Mode

(2)

(3)

(4)

(5)

(2)/(4) (6)

B- 1

0.845

Overall

0.822

Overall

1.02 8

B-2

0.781

Interactive

0.797

Interactive

0.980

B-3

0.791

Local

0.806

Local

0.981

,

Figure 10: Analysis of the overall buckling case

Figure 11' Analysis of the local buckling case

Figure 12" Analysis of the interactive buckling case

168

Analytical Comparison To analyze the ultimate carrying capacity of folded steel plate girders with initial imperfections due to initial deflections and residual stresses caused by welding, an elasto-plastic finite element analysis was used (Yabuki, et al. 1994, 1995, 2000), based on geometric and material non-linear isoparametric shell elements. It was assumed that material constitutive relationship follows the Von Mises yield criterion and the Plandtl-Reuss plastic flow rule, combined with material and geometric invariance law, strain hardening law and linear unloading law. The non-linear behavior is solved numerically according to both the Newton-Raphson and the Updated Lagrangian formulations where the displacements increase step-by-step. Actual measurements were used for the initial imperfections of residual stress and initial deflection. The values used for material properties and girder dimensions are shown in Tables 1 and 2. Comparisons of the load-lateral deflection curves on the middle point of girder depth at the spancenter of the specimens obtained by the analysis with those by the tests are summarized in Figures 10, 11 and 12, respectively. Table 4 compares the ultimate load carrying capacities obtained through the analysis with the actual test results. These figures indicate that the adopted analysis evaluates well the load-deformation behavior up to the collapse. According to Table 4, the ratio of the test results to analysis results for the ultimate carrying capacity is between 0.98 and 1.028, with an average of 0.996. Thus, it may be concluded that this analysis is quite accurate in evaluating the ultimate load carrying capacity of folded steel plate girders. 6. CONCLUDING REMARKS This paper summarizes experimental and theoretical researches on folded steel plate girders, which are currently in progress at University ofRyukyu, Okinawa in Japan. Test results for the proposed folded steel plate girders subject to a uniform bending moment have been presented and served to prove the reliability of the non-linear finite element method developed for the ultimate capacity (Yabuki, et al. 1994, 2000). Up-to-now, the tests show that the stability strengths of girders having folded web plates can be 14 to 33%, with an average of 20%, higher than those of conventional I-shaped plate girders as reviewed by Beedle et al (1991). However these first results and conclusions need to be confirmed by more tests and the theoretical analyses.

References Yabuki, T. and Arizumi, Y. (1994). Ultimate strength and its practical evaluation of cylindrical steel shell panels under various compressions, Journal of Structural Mechanics and Earthquake Engineering, Japan Society of Civil Engineers, Vol.ll, No.l, pp.37--47. Yabuki, T., Arizumi, Y., Shimozato, T. and Nagamine, Y. (1995). Buckling modes of plate-girders curved in plan, Journal of Structural Mechanics and Earthquake Engineering, Japan Society of Civil Engineers, No.519,1-32, pp.51-~56. Yabuki, T., Arizumi, Y., Aribert, J.M. and Guezouli, S. (2000). An Explicit evaluation of imperfection effects on inelastic lateral-torsional buckling of steel beams, Proceedings of the International Conference on Steel Structures of the 2000 s, lstanbul, pp.209.~214. Committee of Highway Bridges. (1990). Japan Specifications for Standard Highway Bridges and Explanations (I- General, II- Steel bridges), Japan Road Association EUROCODE 3. (1992), Design of Steel Structures Part 1-1, Section 5.5.2, The European Committee for Standardization. Yabuki, T., Arizumi, Y. and Vinnakota, S. (1995). Mutual influence of cross-sectional and member classifications on stability of I-beams, Proceedings of the International Conference on Structural Stability and Design, pp. 125-134. Beedle, L.S. (1991). Editor-in-chief: Stability of Metal Structures - A worldview 11, Structural Stability Research Council.

Section IV COLD-FORMED SECTIONS

This Page Intentionally Left Blank

Third InternationalConferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

171

MODELLING OF THE BEHAVIOUR OF A THIN-WALLED CHANNEL SECTION USING BEAM FINITE ELEMENTS H. Deg6e MSM Department, Institute for Mechanics and Civil Engineering University of Li6ge (Belgium)

ABSTRACT This paper aims at presenting an approach allowing studying the behaviour of channel profiles through the use of beam finite elements. This approach can be considered as an interesting alternative to the use of plate finite elements, for the modelling of structures made of thin-walled members. The main idea consists in superimposing on a classical beam global displacement field, a local field describing the deformation of the profile cross-section, when a local buckling or a distortional buckling occurs. In a first step, the behaviour of the constitutive walls of a profile is studied. These walls are there considered as single plates. Two main classes of walls are studied (internal walls, such as web, and free edge walls, such as flanges). Different kinds of stress state are also considered, in order to be able to model the behaviour under compression and/or bending. In a second step, these isolated wall models are assembled in order to build up a channel section profile. The approach is applied to the computation of critical buckling load under compression and in-plane bending.

KEYWORDS Numerical models, beam finite elements, local buckling, distortion.

INTRODUCTION In the field of civil engineering, a frequent use is made of structural elements for which one dimension (the length) is significantly greater than the two others (the width and the depth). In order to study the behavior of such 'beam' elements, some classical assumptions are made to reduce the behavior of the whole element to the behavior of its axis. Among these assumptions, the conservation of the shape of the cross section can be pointed out. Such an approach leads in most of the cases to an accurate model of the member's real behavior. However, this assumption can be too strong, particularly in the frame on thin-walled structures. Indeed some phenomena that include a deformation of the cross-section can occur in thin-walled members. Such phenomena can be for example local buckling or distortional buckling.

172 If we have a look at the numerical methods available to study such profiles with deformable crosssection, and if we except the finite strip method, very efficient but essentially restricted to the study of isolated prismatic members, we can see a duality between plate finite elements on one hand, and special beam elements on the other hand. Plate models are convenient for almost any geometry, load and support condition, but they can be very heavy for what regards computation time and storage space. At the contrary, beam models are much faster and economic, but their use is more limited. In this paper, we first present concepts allowing generalizing a beam finite element, in order to account for the deformability of its cross-section. Then we applied these concepts to the modeling of a channel section. Here are the criteria that were at the base of the element's development. 9 Conserving the traditional characteristics of beam elements (definition of the element by its axis, concept of generalized internal forces...). 9 Limiting the increase of degrees of freedom compared to a classical element. 9 Representing any shape of cross-section's deformation. 9 Determining the wavelength, the shape and the magnitude of the local phenomenon at the level of the finite element itself, and not through a preliminary computation. 9 Representing different support conditions and connections between structural elements.

BASIC ASSUMPTIONS In this paragraph, we will present the general formulations for strains and stresses. In order to alleviate the presentation, we limit this paper to the study of profiles under in-plane solicitations (compression/traction and/or mono-axial bending). The axes used for the definition of strains and stresses are shown on figure 1. The x-axis is the longitudinal axis of the profile, the y- and z-axis are the global axes of the section, the ~-axis is normal to the wall and the rl-axis is the local in-plane axis.

iI

j v

Figure 1 : Axes used for the definition of strains and stresses Strains

Following the definition of Green, associated with the assumption of moderate rotations, strains are defined as follows (v is the Poisson ratio).

Ex x

dubm( X )

~d2vbm( X )

= ~ - Y

Eyy = -V

dx 2

du bm( x ) d Ev bm( x ) . .dx .. + vy- dx 2 ....

( /

Eoo = - v ~1 Owpl(x,r/

12

OWPt ( x,rl) +

t)x

(I.I)

(1.2)

(1.3)

173

2Ex~ = _~ ~2wPI( X,I~) OXO~7

(1.4)

As it can be seen in equations (1.1), the strains are composed of four parts. The first term is related to the global axial behavior of the profile, while the second one is related to the global flexural behavior of the member. The third term is related to the local flexural behavior of the considered wall, while the fourth describes the non-linear membrane effects due to the out-of-plane displacements of the wall. Stresses

In order to derive the stresses, an in-plane stress state in the plane of each considered wall is used. This leads to the following expressions.

dUbm(x)

S~=EI

d2vbm(x)l y

dx

(2.1)

E

+

[

1-

-r

~2wPI(x,,~7) - v ( 02 wpl ( x ' r l ) +

~X2

01"]2

Syy --0 _

Ow pl ( x , rl)

OX (2.2)

E I__V~O2wPl(x,~7) - ~ a2 wpl ( x'~ )

Srm - -1-' - -VT Sxo = G - (

Ox2 OxO~7

(2.3)

(2.4)

Finite e l e m e n t f o r m u l a t i o n

The finite element formulation is obtained by introducing an appropriate displacement field in expressions (1.1) to (1.4) and (2.1) to (2.4), and then by using the principle of virtual work in order to derive the stiffness matrices. The general shape of the displacement field used for this purpose is given in equations (3.1) to (3.4). ub"( x ) = U l hi( x ) + U 2 h2( x )

(3.1)

vbm( x )= V1h3( x )+Ozl h4( x )+ V2 hs( x )+Oz2 h6( x )

(3.2)

Equation (3.1) describes the global axial behavior of the member, while (3.2) describes the global transverse behavior. These two expressions are those of a classical Bernoulli beam. hi and hz are linear functions, while h3 to h6 are cubic functions.

w PI( X,I~ ) "- f (

x ) w k ( J7 )

f ( x ) = A 1h3( x )+ AOl h4( x )-t- m2 hs( x )+ Ao2 h6( x )

(3.3) (3.4)

Equations (3.3) to (3.4) describe the local behavior of the considered wall of the profile. As shown in (3.3), the displacement of the wall is given by the product of two functions. The first one is the longitudinal modulation of the amplitude of the local phenomenon (3.4). The shape functions hi(x) are cubic and the four parameters A~, Ao~, A2 et Ao2 are the local amplitude and the local slope at each end of the finite element.

174

h

b Figure 2" Shape of the locally buckled cross-section The function wd r/) in (3.4) describes the displacement of the k th wall of the profile in the plane of the cross-section. For what regards the definition of this function, two kinds of walls are considered (see Fig. 2). For the internal wall (web), a superposition of sine functions is used (3.5). For the free-edges walls (flanges), a linear function is used (3.6). 2n+l

izt'r/

Wweb( rl ) = ~.~ Bi cos i=1,3

2n

in'r/

(3.5)

+ ~_~ B i sin b

i=2,4

b (3.6)

W~o~e ( rl ) - 0 r / + ~

Such a formulation allows to use a classical beam finite element discretization, in which the studied structure in defined only by a longitudinal axis, dressed up with the appropriate cross-section.

BEHAVIOR OF ISOLATED WALLS Before studying the behavior of an entire profile, a first part of the study consists in analyzing the behavior of the different walls, considered as isolated plates. As we only consider an in-plane behavior of the profile, three cases have to be studied. If the profile is subjected to pure compression (Fig 3.a), we must consider the web and the flanges under a compressive stress state. In the other hand, if the profile is subjected to bending, the stress state of the web is the same than in the case of global compression, while the stress state of the flanges is of course triangular (Fig 3.b). This part of the study is carried out by using a beam finite element model, in which the cross-section is assumed to be a thin rectangular section. L

t tt

tt

tt

tt

ttttt

Figure 3" Stress states considered - Global compression (a), global bending (b)

Internal walls in compression For what regards the web (or internal wall), two limit cases are considered. The first one is the case of a simply supported plate. This corresponds to a profile with very week flanges. In this case, the transverse shape of the buckling mode is well known. It is a simple sine function. Thus (3.5) can be

175 limited to the first term of the development. Computation results can be found in table 1 for a plate with aspect ratio equal to 4, and different longitudinal mesh densities. The results are in very good agreement with the theoretical values if a minimum of two finite elements are used on each half-wavelength of the buckling mode. A significant reduction of computation time is also observed, in comparison with plate finite element models. The second limit case is a built-in plate. This corresponds to a profile with infinite rigidity of flanges. For this case, it becomes necessary to use more than one harmonic modes in (3.5), because of the more complex shape of the buckling mode. Results are presented in table 2 for the first buckling mode. The deviation from theoretical value is also computed.

Number of half-wavelengths

4

5

i

Plate elements

3

6

7 5.39

ii

Theoretical values

4.00

4.20

4.34

4.70

20 x 10 FE (cpu' 13.1 s)

3.85

4.02

4.22

4.47

I

5.13

40 x 10 FE (cpu" 30.9 s)

3.94

4.13

4.29

4.60

5.27

4 FE (cpu" 1.I s)

4.20

4.60

4.33

5.68

6.26

Beam elements

8 FE (cpu" 1.4 s)

4.01

4.23

4.34

4.77

5.55

16 FE (cpu" 2.1 s)

4.00

4.20

4.34

4.70

5.40

,,

Table 1 9Buckling coefficient ko (=Ocr/OEuler) for a simply supported rectangular panel (b/a = 4)

Theoretical value

40 x 10 plate elements

i

16 beam FE 1 mode

16 beam FE 3 modes

16 beam FE 6 modes

6.97

6.74

9.49

8.27

7.33

0%

3.3 %

36.1%

18.9 %

5.1%

Table 2" Buckling coefficient 1~ (=Oc~/OEu~) for a built-in rectangular panel (b/a = 4)

Free-edge walls in compression In this paragraph, the behavior of a plate simply supported along one side, free along the other side, and subjected to pure compression is studied. This case corresponds to the flange of a channel profile in which the web is supposed to be week. For this case, theoretical values of the buckling coefficient can be found for example in ref [ 1]. Computation results for different aspect ratios are given in table 3 and compared to theoretical values. Aspect ratio

1/2

1

2

4

10

20

100

IFE

5.289

1.641

0.729

0.502

0.438

0.428

0.426

2 FE

4.456

1.433

0.677

0.489

0.436

0.428

0.426

4 FE

4.428

1.426

0.676

0.488

0.436

0.428

0.426

l0 FE

4.426

1.426

0.676

0.488

0.436

0.428

0.426

1.426

0.676

0.488

0.436

0.428

0.426

.

,,

Theory

4.426

Table 3 9Buckling coefficient ka (-'O'c/O'Euler) for a free-simply supported panel in compression

176

Free-edge walls in bending In this paragraph, the same configuration than in the upper point is considered, but the panel is now subjected to pure bending. Results are given in table 4. Aspect ratio

1/2

2

1

4

10

20

100

0.857

0.851

0.856

0.851

i

1 FE

10.578

3.283

1.459

1.003

0.875

2 FE

8.911

2.866

1.355

0.977

0.871

4 FE

8.855

2.852

1.351

0.976

0.871

0.856

0.851

10 FE

8.851

2.851

1.351

0.976

0.871

0.856

0.851

0.856

0.851

....

....

....

Theory

8.851

2.851

1.351

0.976

0.871

,,.

Table 4" Buckling coefficient ka (=ac~/aeu~er) for a free-simply supported panel in bending

BEHAVIOR OF PROFILES

Sectional modes When applied to a whole profile, the description of the web's displacement by a superposition of sine functions (3.5) leads to the notion of sectional modes (Fig. 4). As we only study the bending around the week axis, only the odd modes will be considered for further developments.

Z

Z

Figure 4" Sectional modes for a channel section

Characteristics of the studied profile In order to validate the model, the example of figure 5 is considered. This profile is studied under compression and bending. In both cases, the variant parameter is the depth h of the profile.

h

~t.

tf

b = 90 mm tf = tw = 1.5 mm L=300 h = 20, 30...60 mm E = 21~ N/mm ~ v=0.3

t Figure 5 9Geometrical and mechanical characteristics of the profile

Channel profile in compression Reference results obtained by using a finite strip approach are presented in table 5. In this table, the critical stress and the buckling coefficient of both web and flanges (computed by 4.1 and 4.2) are shown for the different values of the profile's depth.

177

O'cr

(4.1)

ka,we b =

12(1- v 2 ) O'er

(4.2)

k a,1lang e =

1 2 ( 1 - v 2 )~, h ) h (ram)

oct (Mpa)

ka,web

kn,flange

2O

247.9

4.70

0.23

30

232.5

4.41

0.49

40

189.6

3.60

0.71

50

142.3

2.70

0.83

60

102.2

1.94

0.86

Table 5 9Reference values - Finite strip model For a depth of 20 mm, the buckling mode is an instability of the web restrained by the flanges. For a depth of 30 mm, web and flanges are all unstable, while, for depth's value over 40 mm, the buckling mode is an instability of the flanges restrained by the web. Computation results obtained with the modified beam element are presented in table 6. Computation has been carried out with 12 finite elements over the length of the profile, and with a local displacement described respectively by 1, 3 and 6 sectional modes. oc,.3 ~ ,

(Mpa)

h (mm)

ocr.t mode(Mpa)

o=.6,,,,des (Mpa)

Oer.finitestrips (Mpa)

20

258.6

256.3

255.7

247.9

30

232.3

232.1

232.1

232.5

40

189.0

188.0

187.7

189.6

,

50

149.4

146.5

145.7

142.3

60

114.0

109.1

107.8

102.2

Table 6 9Channel profile in compression - Results obtained with a modified beam model The following conclusions can be drawn. 9 For a depth of 30 or 40 mm, a model using a single sectional mode leads to good results. In this case, everything happens indeed as if all the profile's walls were pinned. So the sine function chosen to describe the wall is nearly the exact shape. 9 For a depth of 20 mm, the use of complementary sectional modes becomes necessary. The web is restrained by the flanges, and its exact buckling shape becomes thus more complicated than a simple sine function. 9 For a depth of 60 mm, the buckling mode is initiated by an instability of the flanges, restrained by the web. In this case, the exact deformed shape of the flanges is more complicated than the simple linear function chosen for this study. This explains the error of about 4% obtained on this configuration.

178 The model presented in this paper can then be considered as correct, even if the displacement field chosen for the flanges should be modified, in order to get better results when the flanges' slenderness increases.

Profile in

bending

The same geometrical and mechanical properties than for the case of compression are considered. For each value of the depth, the critical bending moment leading to a distortional buckling of the section is computed. This computation is made by using a plate finite element model and the new modified beam element model, with respectively 1, 3 and 6 sectional modes to describe the local displacement. Results are given in table 7. The following conclusions can be drawn. 9 For the small values of the depth, the deformed shape of the web is very complex. This explains the great differences between the results obtained with 1, 3 and 6 modes for such a geometry. 9 When the depth increases, the deformed shape becomes less complex. The difference between the 1, 3 and 6 modes models becomes then smaller. 9 Even when the displacement of the web is described in a good way, an error of about 30% still remains on the computed critical moment. This is related to the choice of a simple linear function to describe the behavior of the flanges, as it has already been said for the case of compression.

lVL:~.3 modes(kNm)

lVl=.6,,,des(kNrn)

M=.platc(kNm)

20

-1.203

0.869

0.702

0.478

h (mm)

Mer.I

mode ( k N m )

,,

30

3.270

0.744

0.661

0.478 .......

40

1.137

0.679

0.630

0.472

50

0.853

0.629

0.597 . .

60

0.732

0.610

.

.

.

.

0.588

0.459 0.453

Table 7 9Channel profile in bending - Computed critical values of the bending moment

CONCLUSIONS The new model presented in this paper, allowing to study the local behavior of a channel profile by using beam finite elements, leads to good results for the case of compression. For what regards the case of bending, results are qualitatively good, but the linear function chosen to describe the behavior of the flanges appears to be too simple. The displacement field of the flanges should then be completed in order to get more accurate results for this case of bending.

REFERENCES Bulson P. S. (1970). The Stability of Flat Plates. Chatto and Windus, London. Deg6e H. J. (2000). Contribution ~ la prise en compte de la d~formabilit~ de la section droite dans un dl~ment fini de type poutre. Ph.D. Thesis. University of Liege. Belgium

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rightsreserved

179

CODIFICATION OF IMPERFECTIONS FOR ADVANCED FINITE ANALYSIS OF COLD-FORMED STEEL MEMBERS D. Dubina~, V. Ungureanu 2, I. Szabo ! ! Department of Steel Structures and Structural Mechanics, "Politehnica" University of Timisoara, Stadion 1, RO- 1900, Timisoara, Romania 2 Laboratory of Steel Structures, Romanian Academy, Timisoara Branch, M. Viteazul 24, RO-1900, Timisoara, Romania

ABSTRACT The paper analyses the influence of local-sectional geometrical imperfection shapes on the localdistortional buckling modes of stub columns and short beams and their effect on the interactive buckling of cold-formed steel members. The erosion of theoretical interactive buckling load, due both to the imperfections and interactive buckling, is numerically evaluated via ECBL approach.

KEYWORDS Cold-formed sections, compression, bending, imperfection, interactive buckling, erosion.

INTRODUCTION

In numerical non-linear analysis some kind of initial disturbance is usually necessary when the strength of structure is studied. When initial imperfections are used to invoke geometric non-linearity, the shape of imperfections can be determined with eigenbuckling analysis, and they must be affine with the relevant local/overall buckling modes of the member. For these reason, until now, the geometrical imperfections corresponding to local, distortional and flexural buckling modes are taken into account in numerical simulation by means of some equivalent sine shapes, corresponding to the wave-length of each particular instability mode. Rasmussen and Hancock (1988) and, recently, Schafer and Pekoz (1998) proposed numerical techniques, based on Fourier series, to introduce automatically the geometrical imperfections into the non-linear analysis. Based on experimental database and statistical analysis Schafer and Pekoz (1998), established the size of both geometrical and residual stresses and suggested their codification for numerical analysis. Some commercial FEM programs like ABAQUS and ANSYS include facilities to generate sine shapes of local imperfections. The main aim of present paper, which continues a previous one by the authors, is to codify and analyse the influence of geometrical imperfections on the theoretical strength for both compression and bending cold-formed steel members in interactive buckling. On this purpose, advanced FEM simulation and the ECBL (Erosion of Critical Bifurcation Load) approach presented by Dubina et al (1995) are used.

180 INFLUENCE OF SIZE AND SHAPE OF IMPERFECTIONS IN ADVANCED FEM MODELS An advanced FEM sensitivity analysis is used to capture the effect of different imperfection shapes. Tests on compressed cold-formed steel plain and lipped channel members performed by Young and Rasmussen (1995 a,b) at the University of Sydney have been selected for analysis. Measured geometrical dimensions, material characteristics and geometrical imperfections were used in FEM analysis. Two kind of geometric imperfections were taken into account into the FEM model: overall geometric imperfections (with maximum size uo, vo and 0o at the mid-length), and local-sectional imperfections (with amplitude 5). Three different modes were considered for local-sectional imperfections (see Figure 1), i.e. (1) symmetrical, affine with the 1st local eigenbuckling mode, (2) asymmetrical, affine with the 5th mode and (3) directly, by means of shape and size of measured imperfections.

1

I

'

I

5

L_5/

I

3

L_..3 u I

I

I

I

[

I V~

H

'4;,0

Figure 1" Shape of imperfections analysed in this paper The size of imperfections is shown in Table 1. The values of local and overall imperfections in this table are the maximum ones. The membrane and the flexural measured residual stresses change from 15 MPa to 40 MPa. As pointed out for the authors of tests, the residual stresses, compared with the nominal yield strength, which is 450 MPa are negligible, and they were not introduced in the FEM model. TABLE 1 MEASUREDMAXIMUMINITIALLOCALAND GLOBALIMPERFECTIONS

51 (lylm) 52 (mm) Series -1.20 0.24 P36P08151.37 -0.57 P36P13150.71 0.25 !.36P0815+ -1.39 0.38 L36P1315-

53 (mm) 0.13 -0.53 0.25 0.41

54 (mm) 0.17 -0.64 0.33 0.29

55 (mm) 0.67 1.76 -0.94 -0.77

Uo (mm) -0.17 -0.61 -0.46 -0.51

Vo (mm) 0.33 1.14 0.25 0.02

0o (rad) ] -0.00120 [ -0.00378 1 0.00072 ] 0.00292]

The numerical analysis was camed out with ANSYS 5.4, using SHELL 43 elements. This is a 4-node element, allowing for elastic-plastic large strains and deflection analysis. Boundary conditions and loading are set to match those employed in the physical tests. In order to model the pinned support, a supplementary plate was introduced at the ends of profile. The plate mesh includes the node corresponding to the gravity centre of the profile cross-section. The material behaviour model was introduced by means of the multi-linear model (MISO), and the arc-length method was used. Numerical results of this analysis are presented in Table 2.

181 TABLE 2 LIMIT LOADSIN kN Specimen

Tests

P36P0815P36P01315L36P0815+ L36P1315-

40.9 27.0 67.9 41.1

ANsYs with measured imperfections Case 3 Case 2 Case 1 (real (asymmetric sine (symmetric sine imperfections) imperfections) imperfections) 39.72 45.78 36.53 26.93 30.85 24.54 65.33 80.22 64.50 39.39 43.51 38.95

It is easy to observe that, for Case I, the value of ultimate load is 12% lower than the test value, while for Case 2 the ultimate load is 15% higher than the measured ones. This difference, of 27%, rises doubts in regard with the procedure of sine imperfections, affine with the first eigenbuckling mode, generally recommended for non-linear analysis. EROSION OF BUCKLING STRENGTH DUE TO IMPERFECTIONS AND INTERACTIVE BUCKLING

Codification of imperfection modes The results presented by Dubina, Ungureanu and Szabo (2000) shown that both size and shape of initial local imperfections are very important when interaction between local-sectional modes and overall ones occurs in case of thin-walled members. In order to analyse the behaviour of stub columns, the specimens tested in Sydney were considered with their nominal characteristics given in Table 3. For local buckling, the length of stub column was taken equal with three half-waves (one half wave = Bw). In case of distortional buckling, for plain channels, the length of stub columns, Lstub, was establish according with the half-wave length formula proposed by Ungureanu and Dubina (1999), while for lipped channels, the half-wave length formula proposed by AS/NZS 4600:1996 Standard was taken, respectively. These values are Lst~b,L= 290mm for both locally buckled (P) and (L) sections, and Lst~b,D= 420mm and Lst~b,O= 1000mm for distorted (P) and (L) sections respectively. TABLE 3 NOMINAL SPECIMEN DIMENSIONS AND MATERIALPROPERTIES FOR SERIES P36 AND L36

Flanges Br (mm) 36.8 Lips B, (mm)

12.5

Plain channel (P) Thickness Radius t(mm) r~ (mm) 1.47 0.85 Lipped channel (L) Thickness Flanges Web Radius ri (rnrn) t(mm) Bf (mm) Bw (nun) 0.85 1.48 37.0 97.3 Web Bw (mm) 96.9

00.2 Ou (MPa) I (MPa) 550 570 00.2

Ou

(MPa) 500

(MPa) 540

E (GPa) 210 E (GPa) 195

For the case of local-sectional buckling, the size and shape of initial imperfections were taken according to Schafer and Pekoz (1998) proposal e.g. 53 = 0.006B w for web and 51 -~ 1.8mm for flange (Figure 2).

182 u

V<

',z, 0 a) I b) Figure 2" a) Shape of local imperfections; b) positive sign of overall imperfections The different cases of the local-sectional imperfection shape selected for the analysis, are presented in Table 4, together with the corresponding results of ultimate strength, N~.L and M~.L, obtained with ANSYS 5.4 large-deformation elastic-plastic analysis. It has to be emphasised that imperfections like D lcl appear frequently in practice. TABLE 4 TYPE OF LOCAL-SECTIONAL MODE OF IMPERFECTIONS

P L ~ CHANNEL LIPPED CHANNEL . . . . . COMPRESSION BENDING COMPRESSION BENDING Code,,, Shape (kN)~Nu L Code I Shape, (kl~)Mu L 1 Code Shape (k/q)Nu L Code Shape,, (kN)M" L ,~00

1~!

PCDlcl ~

l 12.396 JLCD00 [ ]

74.38 PBDOO

66.73 PBDlcl ~

[~]

95.12 LBD00

4.518

2.332 LCDlcl ~

80.33 LBDlcl ~

4.484

94.81 LBD2cl ~

4.463

..

.

PCD2cl ~

66.43 PBD2cl ~

2.323 LCD2cl ~

PCD3cl ~ . ~

! 41.04 PBD4cl

2.355 LCD3cl ~_,,,_~ 74.52 LBD3cl ~_.x_~ 4.513

.

.

.

.

PCD4ss ~

73.02 PBD3cl ~ , . _ ~ i

2.304 LCD4ss ~

,

PCD5as ~

4.420

61.70 LBD5ss ~ _ . . ~

4.493

9

9

.PBD6as ~

,

PCL2as

.

39.29 PBD5ss~._.~ 2.386 LCD5as ~

i

,

76.63 LBD4cl ~

.

46.15

.

.

.

i

'

.

.

i

i

ILBD6as ~

4.388

'

ii!

..:. LCL2as

I

.

2.383 i .....:' i :

91.81

..

9 :

:'* ........ -:

i

P - plain channel; L - lipped channel; C - compression; B - bending; D i - "i" distortional mode; Li -"i" local mode; 00- without imperfections, c l - imperfection constant over the length; s s- s~,mmetricalsine mode; a s - asymmetrical sine mo.de , In the second step of analysis, the global flexural imperfection of (L/1000) of column length has been introduced in order to analyse the long member behaviour under interactive buckling. The components of global imperfection are Uo, Vo and 0o (see Figure 2), but for the analysis, only the case of-Uo was considered as significant. The local-sectional imperfection shapes for short beams, selected for the analysis, are also presented in Table 4, together with the corresponding results of ultimate bending strength, Mu.L,obtained with ANSYS 5.4 large-deformation elastic-plastic analysis.

183 The short beam lengths, Lshob, for the same sections, were established with formulas given by Ungureanu and Dubina (1999) for plain channels and AS/NZS 4600:1996 Standard for lipped channels in bending. They are Lshob,D= 107mm and Lshob,D= 854mm for (P) and (L) sections, respectively. For these particular sections no local buckling occurs. For the case of local-sectional buckling, the size and shape of initial imperfections were taken according to Schafer and Pekoz (1998) proposal, e.g. 53 ~ 0.006B w for web and 51 ~ 1.5mm for flange (Figure 2). Both deflection and twist imperfections are significant in case of lateral-torsional buckling of thin-walled beams. According to Australian Standard AS4100:1990, the initial deflection, uo, and the initial twist, ~o, were taken as follows: 1000uo/L = 1000~o(Myz/NyL)=-I for LLX 0.6 1000uo/L = 1000~o(Myz/NyL)=-0.0001 for ~.L-r<0.6 where Ny = column elastic buckling load about the minor axis; Myz = elastic flexural-torsional buckling moment; XLT= modified flexural-torsional slenderness. Therefore, these particular imperfections, for the case of lateral-torsional buckling of beams, are specified in that standard only, and they have been also used by Pi, Put and Trahair (1998). For the case of present analysis, only the cases of-uo and ~o overall imperfections were considered as significant.

Members in Compression On the next step of analysis, the erosion of theoretical buckling strength, due both to the imperfections and interaction effect will be examined. In order to do this, also the ECBL approach will be used. Assuming the two theoretical simple instability modes, which are coupling, are the Euler bar theoretical instability mode, N E = 1/~2, and the theoretical local-sectional instability one, N L,th, the erosion coefficient can be computed for different imperfection cases (see Figure 3).

! ! ~L,th

+ect

r I I I I

1

! '

1

1

N (e~.m,~-)

>__

Figure 3" The interactive buckling model based on the ECBL theory The interactive lengths of members, Lint, were established according with ECBL approach in order to match with the local-sectional buckling mode and the overall one on the coupling point. Table 5 shows the results of interactive analysis performed with ANSYS 5.4, for the members having the length equal to interactive length.

184 Four ec,th e ec eL

different types of erosion are considered: = theoretic erosion due to coupling; total erosion due to both coupling and imperfections; = erosion due to coupling; = erosion due to local imperfections.

The following notations were used in Figure 3: N = N / N pl , where N is the ultimate strength of the member; No, represents its corresponding full plastic strength; N L,th = N L,th/Npl, with NL,th the ultimate theoretical short column strength; N L = N L//N pl, NL being the ultimate strength of imperfect stub column; m

= x/N L / Ncr , the reduced slenderness of the member. TABLE 5 RESULTS OF INTERACTIVEANALYSISFOR MEMBERSIN COMPRESSION 'Plain i2hannel PCD00 PCDlcl PCD2cl PCD3cl PCD4ss PCD5as PCL00 PCLlss PCL2as ,.

Lint (mm) 1257 1402 1407 1280 2278 2379 1252 2079 2026 .

.

.

_

.

.

33.15 28.04 27.69 33.55 11.25 10.58 35.86 13.63 14.83 .

Lipped Channel LCD00 LCDlcl LCD2cl LCD3cl LCD4ss LCD5as LCL00 LCLlss LCL2as _ ,

.......

.

.

.

.

.

.

.

.

.

.

.

Lint (mm) 1312 1554 1317 1629 2023 1675 1367 1360 1358 .

.

.

.

.

.

.

.

.

53.22 40.81 53.79 38.18 23.63 36.38 43.46 50.31 50.86 .

In the original ECBL approach N L,th is calculated in terms of the effective widths of the component walls of member cross-section. In this case, the non-linear analysis, without accounting for imperfections, has been used (e.g. D00 and L00 values in Table 4, which lead to NL.th values approximately 10% higher than those based on the effective width). Thus, even if, in both cases, NL.th is not quite the "theoretical local buckling mode", of similar type with the Euler mode of bar, it can be used, for the purpose of this study, as "a measure" of the buckling strength corresponding to theoretical local mode in order to compute the "interactive length" and to evaluate the "erosion". Maximum erosion of theoretical interactive buckling strength, e, occurs in the interaction point, M (~int = 1/~/NL,th ), and is defined as: e = NL,th - N(:L = I/~/-NL )

(:)

The total erosion can be associated with the a imperfection factor used in the Ayrton-Perry formula of European buckling curves, by means of ECBL formula:

et =

e2

(2)

1 - e 1- 0.2~dVNL The N values can be computed for perfect and imperfect cases of both cross-section and member. Therefore, the erosion can be evaluated for different imperfection cases. If no imperfections, the evidence of interactive buckling effect only will be provided.

185 Table 6 contains the values of erosion calculated for different imperfection cases. The ot values, computed in terms of"e", are also shown in Table 6. TABLE 6

Members in Bending The erosion analysis is repeated for the case of members in bending following the same procedure, but with M instead of N, and using the subscript zr to mark the lateral-torsional-buckling. Table 7 shows the results of interactive analysis performed with ANSYS 5.4, for members of interactive length, Lint, established with ECBL approach, too.

,,

TABLE 7 RESULTS OF INTERACTIVEANALYSISFOR MEMBERSIN BENDING Plain Channel Lint (mm) Mu,LT (kN) Lipped Channel Lint (mm) Mu,LT (kN) PBD00 2118 1.006 LBD00 1598 2.752 PBDlcl 2180 0.957 LBDlcl 1611 2.711 PBD2cl 2189 0.990 LBD2cl 1619 2.746 PBD3cl 2157 0.984 LBD3cl 1600 2.779 PBD4cl 2209 0.961 LBD4cl 1635 2.661 PBD5ss 2127 0.977 LBD5ss 1608 2.499 PBD6as 2130 0.954 LBD6as 1647 2.425 ,

The aLr values are computed in terms of"ezr" according to the following formula:

eIw

CtLT = ~ 1- eLT 1- 0.4 M~-L

(3)

186 The resulting values for a~r and erosions are included for the reason of comparison in Table 6, together with compression results.

CONCLUSIONS On the basis of the results of this study the following conclusions can be pointed out: 1. The different shapes of local-sectional imperfections have a different effect on the member buckling strength. It is clear that the sine shape of these imperfections not always represents the critical mode and, in some cases, they lead to unconservative results. 2. The higher sensitivity of the distortional-overall interactive buckling to sectional imperfections is generally confirmed. This can be explained by the lower post-critical strength reserve of the distortional mode, compared with the local one. Therefore, this is understandable, because the local buckling uses the plate buckling model, which provides a stable bifurcation, of higher post-critical reserve than the bar bifurcation, which is used to model the distortion. 3. In case of members in bending the influence of local-sectional imperfection modes is low, but the initial twist unfavourable combined with initial deflection can affect considerably their ultimate strength. 4. Further systematic experimental and numerical studies are necessary in order to identify the critical imperfection modes and to codify these for numerical analysis. 5. The ECBL approach seems to be a suitable tool in order to identify the imperfection critical modes. Therefore, according to the authors knowledge, the results presented in this paper and in the previous one represent a first attempt to evaluate quantitatively, the erosion due to the imperfections in the interactive buckling.

References Australian Standard AS 4100:1990: Steel Structures, Homebush, Australia. Australian/New Zealand Standard AS/NZS 4600:1996: Cold-formed steel structures, Australia. Dubina D. et al. (1995). Interactive buckling of cold-formed thin-walled member, International Conference ICSSD '95-Structural Stability and Design, Sydney, Australia, 30.10-1.11.1995, 49-54. Dubina D. et al. (2000). Influence of Local and Sectional Geometrical Imperfections on the Distortional and Interactive-Overall Buckling Modes of Cold-Formed Members, International Conference on Coupled Instabilities in Metal Structures, Lisbon, Portugal, 14-16 September 2000, 179-188. Pi Y.-L., Put B.M. and Trahair N.S. (1998). Lateral buckling strengths of cold-formed channel section beams. Journal of Structural Engineering, 1t):124, 1182-1191. Rasmussen K.J. and Hancock G.J. (1988). Geometric Imperfections in Plated Structures Subject to Interaction between Buckling Modes. Thin-Walled Structures, 6:1988, 433-452. Schafer B.W. and Pekoz T. (1998), Computational modelling of cold-formed steel characterising geometric imperfections and residual stresses, J. of Constructional Steel Research, 47:1998, 193-210. Ungureanu V. and Dubina D. (1999). Single and interactive buckling modes for unstiffened thin-walled steel sections in compression, hlternational Colloquium on Stability and Ductility of Steel Structures SDSS'99, Timisoara, Romania, 9-11 September 1999, 543-550. Young B. and Rasmussen, K.J. (1995a), Compression Tests of fixed-ended and pin-ended cold-formed plain channels, Research Report R714, University of Sydney, NSW 2006, Australia. Young B. and Rasmussen K.J. (1995b). Compression Tests of fixed-ended and pin-ended cold-formed lipped channels. Research Report R715, University of Sydney, NSW 2006, Australia.

Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

187

INNOVATIVE COLD-FORMED STEEL STRUCTURE FOR RESTRUCTURING OF EXISTING RC OR MASONRY BUILDINGS BY VERTICAL ADDITION OF SUPPLEMENTARY STOREY D. Dubina ~, V. Ungureanu 2, M. Georgescul, L. Fiil6p ~ 1 Department of Steel Structures and Structural Mechanics, "Politehnica" University of Timisoara, Stadion 1, RO- 1900, Timisoara, Romania 2 Laboratory of Steel Structures, Romanian Academy, Timisoara Branch, M. Viteazul 24, RO-1900, Timisoara, Romania

ABSTRACT Two examples of using an innovative cold-formed penthouse steel structure to restructuring an existing single storey industrial building and a three storey high school building are presented. Main framed structure and some basic details are shown in the paper.

KEYWORDS Penthouse, cold-formed steel, built-up sections, bolted connections, sheeting, light, efficient

INTRODUCTION

Restructuring by vertical addition consists of adding one or more stories above the existing structure, resulting in an increase of overall volume of the building. Depending the size and height of the new addition masses, it is necessary to recheck the load-bearing capacity of the original structure in order to decide whether or not to take consolidation measures. In seismic zones this problem can be very serious, and traditional building techniques cannot be always used for this type of restructuring. The necessity to minimise the weight of the new storey structure added above, makes cold-formed steel sections the most suitable solution. The authors imagined a cold-formed steel structure made by built-up members of Johnston (1971) type, by C sections, and special bolted connections to design very efficient penthouses above the existing masonry or RC buildings. The new floor, because the existing terrace (RC slab or timber structure) usually has not enough load-bearing capacity to carry the additional structure, with its dead and life loads, is made by a steel castellated beam grid. The present paper shows two application examples of this constructional steelwork solution focussing mainly the cold-formed steel structure of penthouse.

188 PENTHOUSE STRUCTURE FOR ALCATEL-DATATIM COMPANY IN TIMISOARA General Data

The structure described in the paper represents an extension of the production capacity belonging to ALCATEL-DATATIM Company of Timisoara. The existing constructions belonging to this society are presented in the site plan of Figure la. I

~,~

~

~ ~r

~

r

E

131t

',[+

IcY I I I

_I-~176 ~ GL

' I I

a)

I"'1

b)

I

Stairs

Il

2 x 12.0m

Figure 1: a) Site plan of ALCATEL-DATATIM existing complex; b) Site plan of penthouse superstructure From structural point of view, these buildings are quite different, i.e.: 9 Building "A" (erected in 1971) has a single storey precasted reinforced concrete structure of 2 x 12.0 m span and 4 x 6 m bay, with an eaves height of 5.8 m; 9 Building "B" (erected in 1978) has a single storey precasted reinforced concrete structure of 12.0 m span and 4 x 6 m bay, with an eaves height of 5.3 m; 9 The greenhouse "GH" surrounded by existing buildings, with a surface of 8.0 x 10.5 m 2, covered by a metallic roof sustained by steel tubular columns with a diameter of 130 mm. The roofing is made of single layer glass and the eaves height is of 6.3 m; 9 The social outbuilding "SA" including locker rooms, toilets and rooms for services (in the basement) erected near the greenhouse, was built of masonry and covered with a reinforced concrete deck cast on site. This building has a surface of 5.5 x 9.5 m2; 9 Access gallery "GL" with a width of 3.20 m and an eaves height of 3.8 m, between other existing buildings and buildings "A" and "B", built of a metallic structure; 9 Interior courtyard "CY" located between building "B" and the existing DATATIM building. By project initial data, an extension of the existing production space was required by: a) erection of a ground floor structure over the existing courtyard, called building "G"; b) erection of a penthouse superstructure over the buildings "A" ,"B" and "G" (see Figure lb); c) erection of a staircase in the space of the access gallery, to provide access from downstairs to the first floor level; d) extension of the social outbuilding also by penthouse structure. A reconstruction of the greenhouse was also in view, in order to adapt it to the new assembly resulting from the described extension. The main characteristics of penthouse component buildings are given in Table 1.

189 TABLE 1 MAIN CHARACTERISTICSOF PENTHOUSECOMPONENTBUILDINGS

Name

Destination

Penthouse over Production buildings A+B+G Social Social outbuilding Greenhouse Social Stair case + stairs

Access to first floor

Deck Level [m]

Eaves Height [m]

Built Area

Built Volume [m31

+5,70

4,00

1.066

4.264

+5,00

3,00

85

255

+0,00

8,00

90

720

+0,00

9,00

35,5

320

Some main requirements for the proposed extension have been imposed by the investor and by the architect, i.e.: 9 the live load value inside the new production space equal to 1000 daN/m 2 over the building "G" and equal to 500 daN/m 2 over buildings "A" and "B"; 9 a condition of not interrupting the ongoing production process taking place inside buildings "A" and "B" by the erection of the new structure was imposed by the investor; 9 the proposed solution was not allowed to disturb or interrupt existing services, as electric and water supply, air conditioning system, a. s. o.; 9 the erection solution had to produce the less dust and water infiltration possible inside the existing production space, considering the delicate nature of the inner production technology; considering the difficulties related to the erection process (lack of storage space, lack of operation space for cranes) a superstructure built of light steel elements connected by bolting was required. Due to fact Timisoara is a seismic territory, the penthouse structure had to be light enough in order to avoid a heavy and expensive strengthening of existing RC structure, not properly designed according to seismic resisting criteria. However, the RC structure was supplementary braced. In order to fulfil all upper requirements, a thin-walled cold-formed structure connected exclusively by bolts was proposed by the authors of this paper for the penthouse superstructure. The roofing and cladding of the penthouse were proposed of LINDAB type trapezoidal steel sheet, with suitable thermal insulation. As interface structure, a metallic grid built of castellated steel beams interconnected by HSFG bolts and connected to the concrete structure by cruciform steel elements was provided. The floor of the penthouse is a composite steel-concrete deck. The interface structure is shown in Figure 3, in different phases of erection (see details about in Dubina et al, 2000b).

Figure 2: The interface structure with composite steel-concrete deck and castellated beam grid supported by main RC framed structure of existing building

190

Cold-formed steel structure of penthouse The main structure of penthouse is a cold-formed steel frame with trusses shown in Figure 3.

Figure 3: Main frame structure of penthouse (Section A-A in Figure lb) This structure was made by built-up double C sections, of Johnston type used for both columns and trusses. In order to avoid to charge the existing structure with bending moments from horizontal actions (earthquake and wind), the column supports at both ends are of pinned type and, correspondingly, the structure was properly braced. The diaphragm effect of sheeting in roof structure was also considered. Figure 4 shows the structure during erection and after the roof envelope was completed, while Figure 5 shows the built-up double C members used for columns and truss cross-section.

Figure 4: Cold-formed steel structure during the erection and after the roof envelope was completed

Figure 5: Single built-up column and a view of erected structure made by built-up double C sections with St. Andrew tie bracing system Together with particular connection details used at eaves and ridge of the frame, as well as for base column supports, the use of Johnston type sections represents the main innovative contribution of the

191 authors at the design of this structure. Figures 6 to 8 shows the connection details used for the penthouse structure. 1

1-1

Figure 6: Eaves connection

3-3

L.__3 73 Figure 7: Ridge connection

2-2

+5,30

+_

II, I ,111

Figure 8: Base-column connection The structure was designed on the basis of a 3D static and dynamic analysis. Technical performances proving the efficiency of this structure are shown in Table 2. A very important advantage of this structural solution, accounting for the particular conditions of narrow building site, was the easy erection. Practically, the structure was risen, stick by stick, on the deck of new floor, and there the trusses and frames were assembled and lifted with a light autocrane. Figure 9 shows a view of the erection procedure and the corresponding building facade nearly completed.

192 TABLE 2 TECHNICAL

PERFORMANCES

Characteristic Maximum transversal top drift

Value 19.3 mm

Maximum longitudinal top drift

36.8 mm

Eigenperiod on transversal direction

0.388 sec

Eigenperiod on longitudinal direction Steel weight of the skeleton

0.690 sec 17 kg/m 2

Steel weight of purlins and rails

9 kg/m 2

TOTAL steel weight of structure

26 kg/m 2

Comments

Seismic action

Figure 9: View of the erection procedure and corresponding building facade

PENTHOUSE STRUCTURE ABOVE A THREE STOREY EXISTING MASONRY INFIELD RC FRAMED BUILDING The existing building was an uncompleted three storey masonry infield RC framed construction, built in Tg. Mures, with the initial purpose of some economical activity. That building was bought by "Cantemir" University in the idea to be restructured for teaching activity. An additional storey was estimated to be necessary, but extension with a new masonry storey would be not possible because not enough load-bearing capacity of existing structure was available. Thus, a light gauge steel structure was preferred. Practically, with some small differences, the same type of structure like for ALCATELDATATIM have been used. The main cold-formed steel frames are shown in Figure 10 with the relevant details in Figures 11 to 13. +20.62

. ....... ~:.!~-;'-~. .~.".--.~""~ ~ ............ i

......V.-~-:=..............

,t

"

,

~"T

Currentframe .,.,.~::~,~.~_~.1~,~,_8.3.~

Figure 1O: Main cold-formed steel frames for "Cantemir" penthouse

193

Figure 10: continued 1

1-1

t

Figure 11" Eaves connection

3-3

I.

.J Figure 12: Ridge connection

2-2 i i I I

t~

ri'll.T]

I

Figure 13: Base-column connection

194 Figure 14 shows the penthouse structure with the castellated beam grid interface, after erection, and the whole four storey composite masonry infield RC framed- steel building.

Figure 14: Penthouse structure erected and whole building The weight of main framed structure was 15 kg/m2 only, to which purlins and external walls structure has to be added. Due to the lightness of structure components, no crane was necessary for erection.

CONCLUSIONS Light gauge steel structures are more and more attractive in the recent years. They are characterised by a good price-quality ratio and, very important, they can be built very quickly. Within the actual technological and economical contest, construction industry needs methods to build faster and to obtain better quality "products". Cold-formed steel framed constructions can be a solution to satisfy this need, due to easy prefabrication, reduced erection time, and possibility of high-precision quality control (Dubina et al, 1999 and 2000a). In a seismic territory, like Romania for instance, the use of light gauge steel structures has the advantage of a reduced mass, and they are recommended to be used for restructuring of existing buildings by vertical addition of new storey. Even this structures are considered non-dissipative and have to be considered in the elastic range (q= 1), generally the dominant load combination for design is the fundamental one, except for the case of bracings and colunm-base anchor bolts. The structural system presented in this paper is characterised by the use of cold-formed built-up double C sections, of Johnston type, together with some particular bolted connections details. The technical performances of this system, evaluated by means of two relevant applications, are very good, and it can be recommended for other similar purposes. Therefore, the authors are working now for a two stories vertical addition, based on the solutions presented here. References

Dubina D., Fiilfp L., Ungureanu V, Nagy Zs. (1999), Cold-formed steel structural solutions for residential and non-residential buildings, X V I I Congresso C.T.A., Napoli 3-7 October, vol. 3, 31-46. Dubina D., Ftil6p L., Ungureanu V., Szabo I., Nagy Zs. (2000a), Cold-formed steel structures for residential and non-residential buildings. The 9 th Int. Conference on Metal Structures - ICMS'2000, Timisoara, Romania, October 19-22, 2000, 308-317. Dubina D., Georgescu M., Ungureanu V., Dinu F. (2000b), Innovative cold-formed steel structures for one storey penthouse superstructure of DATATIM-ALCATEL industrial building. The 9 th Int. Conference on Metal Structures - ICMS'2000, Timisoara, Romania, October 19-22, 2000, p. 318-326. Johnston B.G. (1971), Spaced steel columns, Joumal of Structural division, ASCE, Vol. 97, No. ST5, May 1971, 1465-1479.

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

195

EC.3-ANNEX Z BASED METHOD FOR THE CALIBRATION OF (ix) G E N E R A L I Z E D IMPERFECTION FACTOR IN CASE OF THIN-WALLED COLD-FORMED STEEL M E M B E R S IN PURE BENDING M. Georgescu I and D.Dubinal 1 Department of Steel Structures and Structural Mechanics, The "Politehnica" University of Timisoara, Romania, Str. loan Curea nr. 1, RO-1900, Timisoara

ABSTRACT A unified procedure built to calibrate (ct) generalized imperfection factor used in the Ayrton-Perry equation of the European buckling curves is presented by the authors. This procedure is valid for hot rolled, welded or thin-walled cold-formed steel members, either in compression or in pure bending. A review of subsequent design models and present cross section classifications prescribed by EC.3 is thus possible. A practical application of the procedure in connection with the present drafting activity of ECCS-TC.8 Committee aiming to issue the Euronorm EN-1993-1-1 is presented.

KEYWORDS Lateral-torsional buckling, Buckling curves, Imperfections, Coupled instabilities, Generalized imperfection factor, Experimental results, EC.3-Annex Z Procedure

INTRODUCTION

The present model used by EUROCODE 3 to describe the lateral -torsional buckling of steel members in pure bending, represents an extension (somehow artificial) of the existing model used for steel members under axial compression. However, this extension was performed since it provides some practical advantages e.g. a uniform approach using the Ayrton-Perry equation. Thus, buckling curves may be plotted, describing the dependence between the relative reduced moment capacity (MLr) and relative reduced slenderness (ALr) of the member. A major drawback of the present formula in EUROCODE 3 (ENV-1993-1-1) is the discontinuity which appears on the buckling curve for 2LT = 0.4 , since the global imperfection used within the buckling curve equation was taken from the compression members model, proposed by Maquoi & Rondal (1979), i.e.:

196

At present time, the second draft of the European Prestandard pr. EN-1993-1-1 has been issued (as an outcome of the enquiry performed on Eurocode issue ENV-1993-1) where a further discontinuity may appear, by applying distinct values of the partial security factor, i.e. YM0 = 1.0 for short members, treated as a strength ULS case(2Lr _<0.4) and Yu0 = 1.1 for medium and long members treated as a stability ULS case (2Lr > 0.4). Actually, these aspects are common either for hot rolled (HR), for welded or for thin-walled cold formed (TWCF) members. However, owing to their specific features (i.e. characteristic imperfections, local buckling, coupling of instability modes) thin walled cold-formed members require a specific approach, leading to specific buckling curves. The paper is presenting a method of calibration for the generalized imperfection factor "or" (used in the equation of the buckling curves), built on the background provided by EUROCODE.3-Annex Z method. The procedure (valid in case of both hot-rolled or TWCF members) is applied on available sets of test results on the purpose to obtain calibrated realistic values for the "or" factors, based on experimental evidence. A review of the existing classification of steel member cross-sections (on the purpose of buckling analysis) is thus possible, for all types of steel members, either in compression or in bending. The described procedure was applied by the authors on revised Ayrton-Perry models, built to describe in an improved manner the lateral torsional buckling of beams, by using a modified equation for the generalized imperfection, i.e.: r/= a "(2Lr--0.4)

(2)

The specific behaviour of TWCF members was also considered by including the effective cross section factor QLT--13A in the equation of the subsequent buckling curve. Calibration results are presented, leading to subsequent conclusions concerning a reviewed classification of member cross sections.

PRESENTATION OF THE CALIBRATION METHOD The method is based on the ECBL theory developed by Dubina (1993, 2001), where the erosion of the critical bifurcation load of the steel member (owing to the presence of imperfections as well as to the coupling of instability modes) is quantified by means of an "erosion factor" (WLT). The author is proving the existence of the following formulas, linking (WLT) factor with previously defined (CtLT) factor: 2 aLr = 0.6" (~Lr~L r) ~ HR members

(3a,b) ~2r 9 ~ aLr = 1- gLr 1 -- 0.4. ~

~ TWCF members

Thus, by calibrating (lq/LT) factor, the subsequent (s values may be obtained for the studied profile type and the review of the existing cross section classification is possible. In connection with upper topic, a statistical calibration procedure (based on EC.3-Annex Z method) was developed by Georgescu (1998) in his PhD thesis. Furthermore, Georgescu & Dubina (1999) have shown how can this procedure be applied to calibrate "or" values for compression members. In the frame of the present paper, this method is extended to members in pure bending.

197 The procedure is furtheron presented in a "step-by step" manner. For employed notations see EC.3Annex Z text.

Step 1: Definition of a design model using EC.3 formula for the design buckling resistance moment of the beam in lateral-torsional buckling." Mb.Ra = ZLrflwWp"YfY

(4)

YMI

Step 2: Comparison, for each specimen (i), between the design model (lateral torsional buckling strength calculated with the experimentally measured values of the variables) and test results (measured values of ultimate moments). i

Theoretical value: rti "-" M b , R d Test values: re,.

=

MLr. Mpti

"-

M u,exp e

Coefficient of correlation (between rei and rti): /9 > 0.9 Calculation of the correction terms:

(5)

6~ = rei

r~i Mean value correction: n

~

m1. n

Zb i

(6)

1

Error terms: fii = Fti-b N

Error mean value:

6 = --. 1 ~-'6i =1.0 n

Eror standard deviation: Ss=

l

I

1 -~_1.~-~(8/-8~ I

Step 3: Determination of the coefficients of variation (If) and of the weighting factors (at). Coefficient of variation of random error terms (5): V6 ~ S 8 Coefficient of variation of the basic variables (from prior knowledge): VR, = ~V2 (-MLr) + V~ (Mpl) Coefficient of variation of the model: v, -_

+ v:

(7)

Standard deviations in logarithmic form:

QRt =4]n(V t +1)

f

Q: : 4::(V: +1)

(8)

Q = 41n(V +1)

Weighting factors:

QRt aRt

"-

O

and

O

(9)

198

Step 4: Determination of characteristic value 0"~), design value (rd and partial safety factor Fractile factors Uk, and Ukd are extracted from Annex Z tables, according to the number of available test results. Determination of characteristic and design strength using: rk = -b . rm 9exp(-1.64, aR, " Q R ,

- uk, . a~ . Q~

- 0.5- Q 2)

(10)

rd=-b.r m e x p ( - 3 . 0 4 . a . t . Q . t - u e . . a ~ . Q ~ - O . 5 . Q 2) Partial safety factor of the model: YM --rk re

(11)

Step 5: Calibration method and calibration criterion. The calibration method implies increasing constantly the (~tLV)value in a convenient range (identified by a gross search near the expected calibrated value of C~LT).These changes produce (rti) values which are decreasing as (OtLT) is increasing. Thus, the b~ = re--i-ratios will increase, together with the mean rti value correction b. As practically observed by Georgescu (1998), unlike (YM) values, the (bi) values and (b) values are extremely sensitive to (0tLV) change. The importance and sensitivity of (b) values was also observed and mentionned by Byfield and Nethercott (1995). In their paper these researchers emphasize the role of safety factor played by the mean value correction (b). The calibration procedure is run by repeating "m" times the Annex Z procedure, each time using an increased (0teT) value. At each application, the following terms are observed: TABLE 1 - OBSERVED TERMS DURING CALIBRATION PROCEDURE Looking for: Symbol: Name of term: bi > 1,0 bi (i=l ..... n) Correction terms Mean value correction Correlation coefficient Variation coefficient

b p Vr

b > 1.1 P > 0.9 Vr < O.1O0

Safety factor

?g

YM = 1,1

As ((XLT) values increase, more and more (bi) values (initially less than unity) will comply to the relation: b~ = re--L/___1.0

(12)

These values are in fact individual safety factors for each specimen and when ((XLT) reaches such a value that every bi > 1.0 this means: rei > rti,Vi = 1,...,n In other words, for this value of the generalized imperfection factor, the theoretical model is on the safe side compared to the available experimental values. Therefore the calibration criterion proposed by the authors is the following: CR/TERION: The calibrated (aLr) value is corresponding to the obtention of all (bJ values greather than unity and simultaneously of their average (b) greather-equal than 1.1 (this last value being in fact the safety factor prescribed by Annex Z in case of stability phenomena) i.e.: m

199

{~

~>1.0

V i = l ..... n

(13)

>1.1

PROCEDURE APPLICATION ON ' T ' H.R. AND WELDED STEEL MEMBERS The calibration procedure focuses on the model for members in bending adopted in EUROCODE 3ENV-1993-1-1, where the generalized imperfection is taken according to eqn (2) instead of eqn. (1) The experimental results supplied in the frame of EC.3-Background Documentation, Chapter 5 / Document 5.03/ October 1989, (Eds. Sedlacek G. et. al.), have been used to apply the proposed procedure. a) In case of hot-rolled steel profiles a number of 144 test results, from a total of 243 tests (selected by European experts as representative for lateral-torsional buckling of beams) have been available. For what regards the structural shapes used for the tests, the studied profiles are representative for most of the hot-rolled sections used arround the world: I or H sections produced in Western Europe, North America and Japan. It must however be noticed that the depth of the tested beams never exceeded 305 mm so that the representativity of the tested beam population is restricted to this depth range ! Because the 144 tests were carried out by several researchers in different laboratories all over the world, it was accepted that they are well representative of the testing conditions. In order to check for the influence of steel strength on hot rolled profiles classification, the authors have divided the initial 144 hot rolled specimens in two sub-sets i.e. 123 specimens with fyOm < 355 N / m m 2 and 21 specimens with f;om > 355 N / m m 2 The influence of thermal treatment on analyzed profiles framing was also in view, by selecting annealed profiles and calibrating separately on subsequent subset. TABLE 2 - CALIBRATION RESULTS OBTAINED ON H.R. AND WELDED STEEL PROFILES Subset name:

Dibley Suzuki Lindner Trahair LindnerSchmidt U.N. SUB-1 SUB-2 SUB-3 Reunited specimens SUB-4 SUB-5 Reunited specimens

Total nr. of specim.

21 54 11 29 15 90 33 16 144

35 18 53

Spec. within coupling range

Correlation coefficient

Variation coefficient

Safety factor"

(p)

(Vr)

(YM)

Calibrated (GLLT) value

1.2268 1.1640 1.2667 1.1827 1.1579

0.152 0.214 0.181 0.115 0.197

0.096 0.101 0.122 0.098 0.106

1.2505 1.1524 1.2606 1.2110 1.1607

0.181 0.197 0.174 0.044 0.197

WELDED "I" PROFILES 0.962 0.126 0.967 0.095 29 0.991 0.127

1.2461 1.1805 1.2217

0.444 0.232 0.350

HOT ROLLED "I" PROFILES 0.994 0.098 17 0.859 0.094 0.946 0.092 0.926 0.088 19 0.996 0.096

38 13 57

18

0.953 0.993 0.991 0.994 0.994

200 b) In case of welded beams, a number of 71 test results (selected as representative by European experts from a total of 96 tests) have been available. All data concerning the specimens are listed in EC.3-Backgrounds. For every test specimen, all actual properties (mechanical and geometrical) were measured. All the beams were submitted to moment loading. In order to check for the influence of steel strength on welded profiles classification, the initial set of 71 welded specimens was divided in two subsets i.e. 51 specimens with fyOm< 355 N / m m 2 and 20 specimens with f~om > 355 N / mm 2 All specimens having a reduced slenderness less than 2Lr = 0.4 have been eliminated from the application set as irrelevant for the proposed model (out of model definition range). The last proposal of prEN-1993-1-1, to use YM0 = 1.0 on this range of member reduced slenderness (instead of YM0 = 1.1 ) was not yet analyzed by the authors, being part of a future research. A thorough analysis has been performed by subdividing the available test results into various subsets in order to observe the possible differences in terms of (aLT) calibrated values. The results are presented in Table 2. NOTE: The meaning of SUB-1 to SUB-5 denominations are the following: 9 SUB-1 is containing all H.R. specimens with f~o., = 235 N / m m 2 , for which no thermal treatment was applied; 9 SUB-2 is containing all H.R. specimens with f,om = 235 N / m m 2 , annealed; 9 SUB-3 is containing all H.R. specimens with f,o,, = 450 N / m m 2 , for which no thermal treatment was applied. 9 SUB-4 is containing all welded specimens with f;om = 235 N / m m 2 and jyf.... = 314_

N/mm 2

9 SUB-5 is containing all welded specimens with fyom = 450 N / m m 2 and f~om = 690 N / m m 2 OBSERVATIONS ON THE OBTAINED RESULTS (TABLE 1): a) The short member range of ~ ~ [0-0.4] proposed by the model has been confirmed by the excellent corelation values and percent deviations obtained; b) Framing of hot-rolled I members on curve "a" is confirmed; c) Framing of welded I members on curve "c" is confirmed; d) Members made of high strength steel seem to require framing on the previous higher buckling curve ("a0" instead of"a" for hot rolled profiles and "b" instead of"c" for welded members); e) Thermal treatment (annealing) of hot rolled profiles also seems to lead to profile framing on proximum higher buckling curve. According to the second draft of prEN 1993-1-1, a new model was recently proposed by TC.8 Committee of ECCS, i.e. for bending member of constant cross section, the value of XLT for the m

appropriate non-dimensional slenderness ~,LT shall be determined from:

~LT ----

(I)

+

1 4- r ~T -- ~" ~2L'-T but ~ LT < 1,1

(14)

LT

where (I)LT :0,5"[13+(XLT(;~LT--0,2)+~LT ] and ~=0,87. New calibrations are thus required on this model, using upper sets of experimental results and possibly other results, in order to test its validity.

201 PROCEDURE EXTENSION ON T.W.C.F. STEEL MEMBERS The proposed procedure may be also extended on TWCF members in pure bending, starting from eqn. (4). On this base, the non-dimensional form of the resistance moment may be written as:

m MLTMb'Rd _ ZLr 9Weff,y 9fy =~LT" Weff,y -- ZLT" QLT Mp, Wpl,y 9fy Wpl,y

(15)

where Wpl,y denotes the plastic modulus of member gross cross section and Weft the elastic modulus of the effective cross-section In uper relation, the reduction factor (ZLT) is given by the following equation: 1

ZLT --

--2 0.5 < 1.0 (I)LT "}"[(I)2T--~LT]

(16)

where:

(I)LT-- 0.5 " I1 + (XLTCLT -- 0.4)q- ~2LT]

(17)

The relative reduced slenderness of eqn. (17) results from the following equation:

-

/woffly.JMp,

~LT-'VWpI,Y

VM- : Q~-LT''/~p'-v Mr

(18)

! One should observe that for short members (with ~,LT ~ 0,4 ) where ZLT "- 1 and the local instability mode is present only, equation (15) becomes: W~

MLT =

y

' =QLT Wpl,y

(19)

In the frame of the ECBL Theory proposed by Dubina (1993), the following relation exists in the coupling point of local and global instability mode (located on the buckling curve by abscissa -

A,tr = ~

1

)

MLT = (1-- tl/LT)-QLT

(20)

where WLTis called "erosion coefficient" Furthermore, a link relation of the type described in eqn. (3-b) exists in this point, between the generalized imperfection factor "OtLT" and "WET": Thus, by determining the position of the coupling point (i.e. through a calibration of ~dLTusing the procedure proposed by the authors) one can control the buckling curve defined by eqns. (16) to (18) The upper described procedure was applied on the available sets of TWCF profiles in pure bending (lipped channel sections called "C" and channel sections called "U" tested by Lovell in 1985) in order to calibrate realistic "O~LT"values required by test evidence. The obtained results are hereby presented:

202 TABLE 3 - RESULTS OBTAINED ON T.W.C.F. MEMBERS IN PURE BENDING Subset name:

Total hr. of specim.

Spec. within coupling range

Correlation coefficient

Variation coefficient

Safety factor

Calibrated

(p)

fVr)

(~/M)

value

27 9

7 6

0.980 0.956

0.120 0.103

1.2172 1.2119

0.07 11.14

C-Lovell U-Lovell

(~LT)

According to these results, the "U" sections in bending would be framed on buckling curve "a" while "C" sections on buckling curve "a0". This remark must be, of course, limited to the analyzed set of specimens. For a general conclusion, further studies are necessary.

CONCLUDING REMARKS A unified procedure built to calibrate (or) generalized imperfection factor used in the Ayrton-Perry equation of the European buckling curves was presented by the authors. This procedure is valid for hot rolled, welded or thin-walled cold-formed steel members, either in compression or in pure bending. A review of subsequent design models and present cross section classifications prescribed by EC.3 is thus possible. An application of the procedure for members in bending in connection with the present preocupations of ECCS-TC.8 Committee aiming to issue Euronorm EN-1993-1-1 is presented.

REFERENCES Byfield M.P., Nethercot D.A.- "An improved method for calculating partial safety factors" Seventh International Conference on Applications of Statistics and Probability, Paris, 1995 Dubina D. - "Coupled Instabilities in Thin-walled Structures: Erosion Coefficient Approach in Overall-Local Buckling Interaction" - Comission of the European Comunities, cooperation in Science and Technology with Central and Eastern European Countries, Research Report, Ref. ERB 3510PL922443, Liege, October 1993. Georgescu M., Dubina D . - "'ECBL and Eurocode Annex Z based Calibration Proc. for Buckling Curves of Compression Steel Members"- SDSS '99 Proceedings of the International Colloquium On Stability & Ductility of Steel Structures, Eds. Dubina D. & Ivanyi M., Elsevier ,1999, pp 501508 Lovell M . H . - "Lateral buckling of light gauge steel b e a m s " - Research Report, Dept. of Civil engineering, University of salford, 1985 Rondal J. and Maquoi R . - "Formulation d'Ayrton-Perry pour le flambement des barres metalliques" - Construction Metallique, hr. 4, 1979 Dubina D . - "The ECBL Approach for interactive Buckling of Thin-walled Steel Members"- Steel and Composite Structures, vol. 1, Nr. 1,2001-01-05 Georgescu M. - "Coupled Instabilities in case of Thin-walled Cold-formed Members"-Ph.D. Thesis, The "Politehnica" University of Timisoara / Romania, 1999

Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

203

DESIGN ASPECTS OF COLD-FORMED PORTAL FRAMES P. Frti and L. Dunai Department of Structural Engineering, Budapest University of Technology and Economics, Budapest, Hungary

ABSTRACT The paper presents a design method and its background for portal frames built-up from cold-formed profiles. It has a main focus on the most critical point of this type of cold-formed frame that is the comer. To follow the complicate behaviour of the new type semi-rigid connection experimental tests are performed. The results and their application in the design is presented. An other important part of the design is the interactive behaviour modes of the structural members. The interaction of the buckling phenomena is studied by finite strip analysis. Finally, the practical application of the system is discussed.

KEYWORDS cold-formed C-profile, portal frame, semi-rigid connection, test based design, finite strip method

INTRODUCTION General

Thin-walled cold-formed structural elements are widely used as secondary structural systems of steel building structures (e.g. purlins, wall beams). Recently the product range is grown due to the developments in the manufacturing technology. Cold-formed beams and trapezoidal sheeting are used in floors; new kind of light-weight trusses are developed for residential houses and for smaller industrial buildings. The application of portal frames built-up from cold-formed profiles, however is not typical. It is due to the difficulties in the forming of moment resistant joints by the application of uni-symmetrical profiles. In the paper a recently developed portal frame system is introduced, emphasising its typical and most important design aspects.

Structural System The Lindab Small Building System (SBS) is small-span (3-10 meter) complex, light-weight, steel building system, developed in the co-operation of the Budapest University of Technology and Lindab Ltd, Hungary. In the development the most important aspects are the application of cold-formed profiles, the simplicity of the site connections, the simple- do-it-yourself type- erection technology.

204 The skeleton of the structure is shown on Figure 1. The primary load-bearing structure (two hinged portal frame) is built up from single cold-formed C-profiles. The section height is between 150 and 300 mm and the thickness is between 1.2 and 3.0 mm. The moment resistant semi-rigid connections are solved by selfdrilling screws. The screws join the webs of the C-profiles that meet back-to-back. The distance between the frames is 1.0 meter. The secondary load-bearing elements are built-up from cold-formed hat profiles. The distance of the puffins is 400-800 ram, depending on the applied cladding. The wind and side bracings are in the end-bays. It is to be noted, that the sheeting elements of the roof and wall cladding are also considered in carrying the horizontal forces.

Figure 1: Structural system Structural Problems

The above solution of the structural systems arises the following problems: 9 Structural elements: - the single C-profiles connected to each other eccentrically, one of the flanges are usually not braced laterally; distorsional and global buckling modes and also their interaction with local buckling is possible. 9 Structural joints: - structural elements are connected in the joint eccentrically, - forces and moments are transferred only through the webs of the connecting profiles, - the application of self-drilling screws in moment resistant connections brings many uncertainties; yet there is no reliable design method nor practical experience. There is no design standard which can follow directly the above behaviour characteristics. The base of the applied design method is the related part of Eurocode 3 (1996). The complicated joint behaviour is analysed in laboratory and the verification is done by design assisted by testing method. The design values of moment resistance, the stiffness and the ductility of the frame comers are obtained from the tests. The buckling modes and the critical forces of the structural elements are determined with the finite strip method. From the results the slendemess ratios are calculated. The static analysis- using the properties of the structural elements, joints and sheeting plates - is realised with plane- and space flame models, taking into account the stiffening effect of the trapezoidal sheeting. In the following the paper deals with the two most important aspects of the development, namely the test based joint design and the stability design of the members by the finite strip method. -

205 JOINT BEHAVIOUR AND DESIGN

ExperimentalProgram One of the basic requirement of the development that the structure have to be quick an easy to be erected. That is why it is decided to use single C-profiles connected each other back-to-back through their webs, without the application any additional elements (e.g. gusset plates). This simplification in the structural solution, however, results in a very complicated behaviour mode. To create a reliable design model of the joints laboratory test program is performed (Dunai, F6ti, Kaltenbach & K~ill6, 1999-2000) to follow the difficult structural behaviour. During the development, in three steps, a total of 26 full-scale tests are completed. Figure 2 shows a typical test setup with self-drilling screws in the frame comer connection of the specimen.

\ 125125 125 125 125 125,,I

~

,

I-I.5 m

_L~

..'/y/x,'/,"

:: -- ___

Figure 2: Test setup and a typical joint with self-drilling screws To keep the frame in plane hat profile purlins and a C-profile under the joint is used, simulating the lateral supports of a real structure. This longitudinal C-profile is also used to provide aid under the erection of the frames. The concentrated force is applied on the structure with 1.0-1.5 m arm through the beam, adjusting the proper ratio of bending moment and shear force. Table 1 contains the range of varying parameters of the experimental program. It is noted, that beside the self-drilling screws, metric bolts and high tolerance bolts are also used in tests. During the design of the test setup and the loading history the recommendations of Eurocode 3 is followed. TABLE 1. TEST PARAMETERS

Number of experiments 26

Type of fasteners

Number of fasteners 4-22

Thickness of the[ Height of the section ,, I section 1,0-3,0 [mm] I 15.0-300 [mm]

b/t ratio of section 100-200

In the tests the following parameters are measured: concentrated force, displacements of the joints, rotation and stresses (in 2 points on the flanges of the beam, near to the joint). Test Results

The results are evaluated basically by the moment-rotation diagrams. Three different modes of failure are separated: 1) local buckling in the C-profile, 2) pull-out of fasteners and 3) sheafing of fasteners.

206 In the behaviour the interaction of the above characteristics could be observed, as it is detailed in F6ti & Dunai (2000). In most cases, however, one of them is dominant in the failure mode. Due to the type and arrangement of the fasteners local buckling could develop in two main different ways. Figure 3 shows the difference in the two types of failure modes by the moment-rotation diagrams.

~

a"

g

3

~2 oE

,,b"

4

.:,:.3

1

1

0

0

0s

Os

0s

0s

OD8

0.10

0,12

0

0,14

0s

i)s

0s

0s

0s

OD8

I : ~ t i ~ [rad]

Rota-tion Ira d]

Figure 3" Local buckling in a) metric bolted and b) screwed specimen On Figure 3a the behaviour of a classical bolted connection can be seen. There is a large slip about in the beginning of the load application due to the shift between the bolt and the hole diameter. To avoid this slip the gap had to be eliminated. High tolerance bolts or self drilling screws can solve this problem. The previous possibility results in good structural behaviour but requires very accurate fabrication and erection. The usage of self-drilling screws much simpler from practical point of view, and the behaviour is favourable, as it can be seen in Figure 3b. 7

.b"

15

,,

8

~4 Eo3 ~2

~. 10

~ 5

1

0

,

0

0s

0s

0.12

0.18

0

,

ODO

OD1

0s

Rotation [rad]

OD3

0 04

0s

R otation [rad]

Figure 4: Shearing of screws a) one by one b) all at once When the failure mode is shearing, two different types of behaviour are experienced, as it is shown in Figure 4. On Figure 4 a - after the initial nonlinear part- the screw failure happened one by one, while on Figure 4.b all the screws sheared once and caused sudden failure.

~

1,5

~0,5 0 0

0,05

0.1 Rocmion [tad]

0.15

s

Figure 5" Tilting and pull-out of screws

207 Beside the above typical modes of failure a special one is experienced when relatively small number of self-drilling screws are used in thin profiles: the tilting and pull-out of screws. The thin web plate of the profile suffer local bending in the surroundings of the screw instead of shearing the fasteners. The fasteners tilting and as the moment increasing, starting to pull-out. The typical moment-rotation diagram can be seen on Figure 5. Conclusions on the Test Results

After performing 26 tests it is found that for practical design purposes the most suitable connections are the self-drilling screwed joints with spread arrangement. They perfectly eliminate the slip in the initial range. By using suitable number and arrangement of fastener shearing of screw can be avoid and local buckling in the member can be reached instead. They also represent ductile behaviour after the top of the load-bearing capacity is reached. Derivation of Design Values

On the bases of the test results the design values of the joint characteristics are derived. The joint behaviour is characterised by the initial stiffness, moment resistance and ductility. The design values of these parameters are defined as follows: The initial stiffness is defined as the secant stiffness at the design moment level. The characteristic value of the stiffness is taken as a mean value of at least two tests (Eurocode 3 recommendation). The moment resistance is calculated from the maximum moment, measured in the tests. The characteristic moment resistance Mk corresponding to this test is obtained from the test result Madj as it is given in Eqn. 1, according to the recommendation of Eurocode 3. In the equation rlk is taken 0.7. Mk = 0.9"qk"Madj

(1)

When the failure is in the members (local buckling), there is a close relation between the failure moment measured in the tests (Mr) and the cross-sectiorial moment-resistance of the member section (Mc.rd). Due to the test conditions (force transfer through the web, eccentricities) this ratio is about 60%. The ductility is defined as an amount of rotation and it is measured from the zero point until the descending branch reaches the level of the ultimate moment.

ANALYSIS AND DESIGN OF STRUCTURAL ELEMENTS Thin-walled cold-formed profiles under moment and/or axial force lose load-bearing capacity by buckling failure. Due to the geometric properties the following buckling modes must be considered: 9 local buckling, 9 distorsional buckling and 9 global buckling (flexural-torsional or lateral-torsional buckling). It is very important, that the above buckling modes have usually interaction in the behaviour of the structural elements. The interaction phenomena between them have to be taken into account. In this design process the finite strip method is used to obtain the linear critical loads. The structural elements with same cross-sections but different lengths are checked under moment and axial force. It can be handled as one bar buckled in different lengths. The virtual lengths of the elements are equal to buckling half-wavelengths of the examined bar. The results are elastic critical forces for these different half-wavelengths as shown in Figure 6. The diagram is based on a 150 mm high and 1.5 mm thick C-profile subjected to pure bending moment. The curves for local, distorsional and overall buckling modes can be clearly separated.

208

ZE f...

3lJ

0 E m

211

0

9

~,~

uckling

,

,] I1)

I(H}

I~XX)

Half-wavelength [mm] Figure 6: Critical buckling moment for different lengths m

From these critical forces the relative slenderness ()~) for a bar in a given length can be calculated. Below there is an example for the calculation in case of lateral-torsional buckling of members subject to bending by the Eurocode 3 (1996). To calculate the design buckling resistance moment (Mb,Rd)the reduction factor (~x) of Eurocode 3 is used, in the function of )~LTwhich can be calculated as shown in Eqn. 2.

~LT ._ /L "Weft V Mcr

(2)

In the equation fy is the yield strength, Weffis the section modulus of the effective cross-section, and Mcr is the elastic critical moment for lateral-torsional buckling, obtained from the finite strip analysis.

APPLICATION OF THE DESIGN METHOD The above design procedures of joints and structural elements are built into the global structural design method of cold-formed portal frames. On this bases the Lindab SBS system is developed. Its architectural and structural design, fabrication and sale management is supported by an integrated Lindab-SBS SOFT computer program system. The development of the system began in January 1999, the selling of the new product started in the spring of 2000. In the practical application more than 100 Lindab SBS buildings are built by the end of 2000, serving different functions. Among them the smallest is 3x5 meter and the biggest is 10x22 meter floorspace.

Acknowledgement The research work has been conducted under the financial support of OTKA T035147 and Lindab Ltd.

References ENV 1993 Eurocode 3. (1996). Design of steel structures - Part 1.3. General rules - Supplementary rules for cold formed thin gauge members and sheeting Dunai L., F6ti P., Kaltenbach L. and K~l16 M. (1999-2000). Experimental study on frame corner joints built-up from cold-formed C-profiles, Research Reports (1-3) (in Hungarian), Technical University of Budapest, Department of Steel Structures, Hungary P. F6ti and L. Dunai: (2000). Interaction phenomena in the cold-formed frame comer behaviour. The Third Int. Conf. on Coupled Instabilities in Metal Structures (CIMS 2000, Lisbon, Portugal), 459-466.

Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

209

LOCAL BUCKLING AND EFFECTIVE WIDTH OF THIN-WALLED STAINLESS STEEL MEMBERS

H. Kuwamura, Y. Inaba, and A. Isozaki Department of Architecture, School of Engineering, the University of Tokyo, 7-3-1Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

ABSTRACT Local buckling behaviors of thin-walled stainless steel stub-columns were experimentally investigated. Six types of sections, i.e., angle, channel, lipped channel, H-shaped, square box, and circular cylindrical sections were tested. These specimens were formed from two grades of austenitic stainless steels designated SUS304 and SUS301L 3/4H, whose specified yield strengths for design are 235 and 440 N/mm2, respectively. Effective width-to-thickness ratios of unstiffened and stiffened plate elements and limit diameter-to-thickness ratios of circular cylinders were established from the test data. It was found that the effective width-to-thickness ratios must be expressed by different equations for the two strength grades because of the difference in strain hardening.

KEYWORDS stainless steel, local buckling, thin-walled section, effective width, stub-column

INTRODUCTION Stainless steel has a significance in its corrosion resistance, while it has been scarcely used in structural skeletons of buildings due to high price. However, recent change of social mind from mass production and abundant consumption to ecological coexistence with natural environment, sustaining a long life of buildings is of much concern in construction engineering. In that context, stainless steel is expected to be promising material for architectural construction. The research for utilizing stainless steel in building structures has been conformed in the advancement

210

of cold-formed steel, especially in the USA, by G. Winter and his followers, which is resulted in a design manual such as "Design of Cold-Formed Stainless Steel Structural Members -Proposed Allowable Stress Design Specification with Commentary (Lin, Yu, & Galambos 1988)." Japanese history of research on structural stainless steel for building use is back to only a decade. At that period of virtual growth of economy, researchers and engineers intended to use stainless steel in heavy steel construction and then a specification of design and construction was published by SSBA (Stainless Steel Building Association of Japan 1995). The most promising use of stainless steel in buildings is obviously in the form of light-weight members relying on its corrosion resistance, which may compensate for its high cost in fabrication as well as material. Thus, we need a design method of thin-walled sections of stainless steel with a higher strength in order to reduce weight. In this study, SUS301L 3/4H which has about twice strength of SUS304 is studied. The engineering concern in applying such thin-walled members is focused on local buckling of plate elements. In this study, stub-column test is performed to establish effective width-to-thickness ratios on the basis of post-buckling strength.

MATERIAL Stainless steels investigated in this study are SUS304 and SUS301L 3/4H, whose specified yield strengths for design (F-values) are 235 N/mm 2 and 440 N/mm2, respectively. Both stainless steels are austenitic types which are solution heat-treated after cold rolling, but SUS301L 3/4H is further thermal-refining rolled to increase strength. Exceptionally, SUS304 circular hollow sections of 76.3-mm and 48.6-mm diameters are formed from hot-rolled strips. Their chemical compositions and mechanical properties described in mill sheet are summarized in Table 1. It is observed that SUS301L 3/4H strips have higher yield ratios and less ductility than SUS304 strips. TABLE 1 MATERIAL PROPERTIES IN MILL SHEET stainless steel

thickspecification

chemical composition(%)

mechanical properties

ness C

Si

Mn 0.78

P

S

Ni

Cr

{rnrrl) 3.0

0.07

0.52

cold-rolledstrip

1.0

JIS G 3459

1.5

0.05 0.05 0.05 0.04

0.38 1.00 0.26 1.04 0 . 5 3 0.98 0.59 0.96

0.023

0.36

iJIS G 4305 SUS304

~,,

pipe JIS G 4305

SUS301L 3/4H cold-rolled strip ~hermalrefmin$)

3.0 1.5

0.2%-offset tensile yield elongastrength strength ratio tion (~l/mm2~ fN/mm2~ L'%)

0.040! 0.005

8.06

18.30

0.036 0.032 0.034 0.028

0.004: ! 0.007, 0.005 0.006

8.10 8.11 8.33 8.29

18.26 18.45 18.12 18.18

279 .. --312 261

641

0.44

57

633 649 583 612

--0.54 0.43

55 52 53 59

1 . 4 0 0.030 0.005

6.62

17.45

508

829

0.61

43

511

832

0.61

41

stub-column specimen angle, channel, lipped channel,

H-shape,dr square tube lipped channel circular tube(except'bellows) ' circular tube #76. 3 ..... circular tube ~46.8 angle, channel, lipl~d channel, H-shaped, square tube circular tube

Mechanical properties of coupons are investigated also in laboratory as shown in Table 2. It is noted that laboratory test coupons of circular sections are taken from pipes. It is known that stainless steels have following distinguishable mechanical properties in comparison with carbon steels : (1) proportional limit of stainless steel is fairly low and non-linearity appears at a low stress level, (2) strain hardening of stainless steel is considerable and thus yield ratio (yield strength / tensile strength ) is very low, and (3) initial Young's modulus of stainless steel is slightly less than carbon steel ( nominal value of Young's modulus of stainless steel is 193,000 N/mm 2, while that of carbon steel is 205,000 N/mm2).

211 From the first item, yield strength of stainless steel for design is defined as 0.1%-offset yield strength of coupon test (Stainless Steel Building Association of Japan 1995), while 0.2%-offset yield strength is generally used in material specification as in Japanese Industrial Standards. This study follows the def'mition of 0.1%-offset yield strength. TABLE 2 MECHANICAL PROPERTIES OBTAINED FROM LABORATORY COUPON TEST nominal 0.1%uniform ~rupture initail thick- measureJ tensile yield elonga-elonga- Young's thicknes~ offset strength stub-columnspecimen hess (ram) strength (N/ram2) ratio tion tion modulus . . . . (•m) ,, (N/~Eq:~) , , (%) (%~ (]N/mm2) angle, channel, lipped channel, cold-roiled strip 3.0 . . 29.2 . . 249 . . . . . . 203,000 H.shapedr.squaretube 1.0 0.94 257 . . . . 203,000 lippexl' channel SUS304 pipe ~48.6 'i.42 239 600 0.40 57 63 202,000 076.3 1.40 239 694 0.34 69 73 202~000 iPress-formed ~101.6 1.5 1.38. 33 ! 737 0.45 55 60 222~000 circular tube pipe from cold- ~139~8 1.38 318 750 0.42 58 ..... 62 222t000 rolled strip ~165.2 1.36 301 742 0.41 , 57 61 2221000 angle, channel, lipped channel, cold-rolled strip 3.0 3.01 497 845 0.59 40 42 206,000 H.shapedTsquaretube ~48.6 1.51 496..... 932 0.53 45 51 ! 50r01)0 SUS301L 3/4H press-formed d~76.3 1.50 458 928 0.49 46 53 1771000 pipe from cold- d~!01.6 1.5 1.51 451 904 0.50 .45 51 185~000circular tube roiled strip ~!39.8 ! 1.49 420 902 0.47 48 .. 54 1571000 , ~165.2 1.49 420 889 0.47 53 59 188,000 stainless

shape

steel

STUB-COLUMN SPECIMENS

Sections of stub-columns are angle with equal legs, channel, lipped channel, H-shaped, square hollow, and circular hollow as shown in Figure 1. Angle, channel, and lipped channel are cold-press-formed from 3-mm thick cold-rolled stainless steel strips, partly 1-mm thick only for lipped channel. H-shaped sections are built-up from 3-mm thick cold-rolled stainless steel strips by laser beam welding or partly by TIG welding. Square hollow sections are built up from two cold-press-formed channels by laser beam welding. Circular hollow sections are cold-press-formed from 1.5-mm thick cold-rolled stainless steel strips by means of TIG welding except 76.3-mm and 48.6-mm diameters of SUS304 which are formed from hot-rolled strips by automatic arc welding.

r__~BJ ~ L

I

,B - ~ ~ "~rBs

~ B t, - ~

~ D r__~_._d___~ ]j,r

arc weld _

B

~? rL?

p~ess b" .~1Dlt.~es s D['?a~' ~l~li,_t l-/~' bt,press s~

B

b=B-r

~

.

O]? ~dl~ a st__,JF or T I G er laser weld w e l d~ t

t

t

~

!

_

b=B-r d=D-2r

b=B-2r d=D-2r bs=Bs-r

b=(B-t)/2 d=D-2t

d=D-2r

Figure 1 9Shape of stub-column section and method of forming Width in this paper is defined as the width of a flat plate element excluding comer as shown in Figure 2. For example, the width b of an angle is equal to B - r, in which B and rare the whole width and

212

the outer radius of the comer, respectively, and the width d of a channel web is equal to D - 2r, in which D is the whole depth. For an H-shaped section, flange width b and web width d are determined by neglecting weld, because the fillet size by laser or TIG welding is very small. For circular hollow sections, outer diameter D is adopted. Length of each stub-column specimen is three times the whole width of the section. For channel, lipped channel, and H-shaped sections, the length is larger of 3B and 3D. Seventy three specimens of stub-columns are scheduled as listed in Table 3. The section sizes in the table are nominal, while measured sizes, which are much more important for thin plates, are used in analysis. TABLE 3 SCHEDULE OF STUB-COLUMN SPECIMENS section of stub-column angle channel

stainless steel SUS304 SUS301L 3/4H

specified yield stress for design

nominal thickness (mm)

F ~/mm2~ 235 440

SUS304

235

SUS301L 3/4H

440

3

3 SUS304

235

lipped channel

H-shaped

square hollow circular hollow

1 ,, SUS3OIL 3/4H

440

3

SUS304 "' SUS301L 3/4H

235

3

440

SUS304 SUS301L 3/4H

235 440

SUS304 SUS301L 3/4H

235 440

, 3 1.5

nominal width (mm~

B 25"~60 B 25"--~ B 25 ~-50 D 50--- !50 B 25---50 D 50---150 B 50---75, Bs 20---25 D 100"200 B 17"-25, Bs 7--'8 D 33"67 B 50"-75, Bs 2 0 " 2 5 D 100"200 B 50--" ! 50 D 50"--200 B 50"--150 D 50~'200 D 50---200 D 50"-200 D 4 8 . 6 " 165.2 D 48.6 "-~ 165.2

number nominal width-to-thickness of snecimens ratio 6 b/t 6"-- ! 8 .... 6 b/t 6"-- 18 b/t 6--- 15 d/t. 13-,-46 b/t 6--- 15 d/t 13"-,46 ' bit 13"--21, b~t 5",~6 d/t 2 9 " 6 3 b/t 1 3 " 2 1 , bs/t 5"~6 d/t 2 9 " 6 3 b/t 13"-'21, bs/t 5"~'-6 d/t 29"-63 b/t 8--~25 d/t 13"--65 b/t 8"--25 d/t 13"--65 d/t 13---63 d/t 13"-63 D/t 33 "~ 110 D/t 33"-110

6 5 4 4 4

8 8 6 6 5 5

EFFECTIVE WIDTH Application of Karman Equation

Onset of local buckling of a flat plate does not mean the failure, because post-buckling stability with an elevation of strength is usually expected. This post-buckling strength is owing primarily to redistribution of stress in a buckled plate in which higher stress beyond buckling stress is distributed in a section adjacent to its support edge, which is enough to compensate the release of axial stress at the middle of the bent plate. Since the out-of-plane deformation at the maximum strength is not serious, a thin plate can be economically designed on the basis of post-buckling strength in which a concept of effective width is applied. According to Karman, following formula is commonly used in the allowable stress design of light-gage sections of steel, in which C is determined from experiment.

be= C

,

(1)

,N,

The effective width denoted by be or d e, as shown in Figure 2, is a virtual width adjacent to its edge,

213 over which edge stress equal to yield stress equal to the maximum compressive force.

//Ar

O'y is

uniformly distributed and the stress resultant is

e/2~Ar be de/,2~Ar be/~" '-be~2de/2~r bebe

~_~ d~

de/'2~_~XA r

a:zI/ar b:; ~b/2 Figure 2 :Effective width

Effective Width o f Angle Sections

An angle section with equal legs is composed of two unstiffened plate elements each of which is pin-supported at one edge and free at the other. Now, we define an experimental effective width be as follows:

(~,bet+~,Ar)'tYy=emax

(2)

where the first Y. means the summation of effective sectional areas of flat plates and the second is the summation of the comer areas Ar. For an angle section two fiat plates and one corner are involved. O'y is the 0.1%-offset yield stress obtained from tension test, and Pmax is the maximum compressive force of the stub-column. Experimentally determined effective width-to-thickness ratios be /t which are calculated from Eqn. 2 are plotted against actual width-to-thickness ratios b/t in Figure 3. It is noted that be /t-values of angles with small width-to-thickness ratios are greater than b/t-values, because strain hardening beyond full yielding of entire section can be attained. The observed limit value of b / t to assure full yielding is 14.7 for SUS304 and 9.6 for SUS301L 3/4H, beyond which experimental be /t-values tend to keep constant. The yield stress of these plates are 249 and 497 N/mm 2, respectively. Substituting these values into Eqn. 2, the values of C of Eqn. 1 can be obtained with the following results: be be

230 215

for unstiffened plates of SUS304

(3a)

for unstiffened plates of SUS301L 3/4H

(3b)

It is noted that the same number is not assigned to C for the two grades of stainless steels. This indicates that the patterns of stress distribution at the maximum strength are not the same for the two grades of stainless steels. The reason why SUS304 has a higher C-value than SUS301L 3/4H may be attributed to the fact that the former material has larger strain-hardening than the latter.

214 -- .....I 0

20

ii

I

8US304

9

i ..........

I

SUS3OILa/aH

~i

15

................. .. ................. 10

i

I}

0

.

6

5

10

-

14.7 15

20

b/t

Figure 3 9Effective width-to-thickness ratio vs. actual width-to-thickness ratio of angles

Effective Width of Square Hollow Sections A square hollow section is composed of four stiffened plate elements each of which is pin-supported at both edges. As is the case of an angle, Eqn. 2 is applied to square hollow sections with a change of be by d e, from which d e / t vs. d / t is plotted in Figure 4. The observed limit value of d / t to assure full yielding is 39.0 for SUS304 and 23.8 for SUS301L 3/4H which have yield stress of 249 and 497 N/mm 2, respectively. Substituting these values into Eqn. 1, C is calculated and the following equation is established. In this case, too, different C-values are given to SUS304 and SUS301L 3/4H. d e = 615 t ~y de 530

-;-=7,

.-.

>

(4a)

for stiffened plates of SUS301L 3/4H

(4b)

L /- sus ol, ,4, r

40

for stiffened plates of SUS304

...... ["o

sus3o4

"

i ....... .............. 9 '-..............

:

, "........... ".............. ! .............. "......... = ~ .....O ............... : ! ............

i

3o

o

............ i ............ O ...........

i

:

i

i~i

i .............. i .............. i .............. i

i-"i

!o, i

lO

el/i 0

2~.~ i ~-~ 10

20

30

40

i ..................... ~ 50

60

70

d/t

Figure 4 9 Effective width-to-thickness ratio vs. actual width-to-thickness ratio of square tubes

Effective Width of Channel, Lipped-channel, and H-shaped Sections

215 The plate elements constituting channel, lipped channel, and H-shaped sections restrain each other at the state of post-buckling. Here, however, we assume that no interaction acts between the adjacent plates. For example, a channel is assumed to be composed of three independent plates such that a web plate is pin-supported at both edges and flange plates are pin-supported at one edge and free at the other. From this simplification, effective width can be calculated from Eqn. 3 for flange plates and Eqn. 4 for web plates. Effective width-to-thickness ratios predicted by this method are summed up over the entire section, which is denoted by ( ~ be / t)pred. On the other hand, stub-column test gives the experimental effective width as follows:

(~,bet+ZAr)'Cry=mJn{Acry,Pmax}

(5)

From this equation, (~,be/t)exp can be calculated. Both are compared ir~ Figure 5. The predicted width-to-thickness ratios on the assumption of no interaction tend to underestimate the actual ones in experiment. However, the error is less than 20% and conservative, which indicates that effective width of channel, lipped channel, and H-shaped sections can be calculated from Eqns. 3 and 4 on the assumption of no interactions between plate elements.

6oS~ 9 iiiiiiiiii,i!!!!!!!! oo12O ii 84ii 84

lOo12O, ........ .. .............. . . .,......... . . . ............... ............,

~oo

,o

o

~ooi

o

0

20

40

(Zb~ t

60

)pred

(a) channel

80

0

40

80

(Zb~ t )pred

120

(b) lipped channel

..................!

......i.........~........i

o ~ ...... i ........ i ......... i......... i ......... i ........ i 0 40 80 120

(Zbe/t )pred

(c) H-shaped

Figure 5 : Prediction vs. experiment of effective width-to-thickness ratios

Limit Diameter-to-thickness Ratio of Circular Hollow Sections Limit diameter-to-thickness ratio to assure full yielding of a thin circular cylinder for design is generally represented by following equation and the coefficient C2 is determined from experiment.

o) _c: T lim

O'y

(6)

All of the cylinders in this study can sustain full yield strength as shown in Figure 6. Thus, the limit value cannot be obtained. However, from a conservative consideration, C2-values are tentatively assign as follows which are the maximum values verified in the test:

216 C2 = 32,500 for circular cylinder of SUS304 C2 = 46,800 for circular cylinder of SUS301L 3/4H

(7a) (7b)

1.5 ..................................i..................................................... ~I.O 0.5

~

~I

03 ! 32.5xl .................................. i-;.;;............................... - 4 6 . 8 x l

.... ~

0.0 0

;I

03

9 SUS301L a/ H

, , ........... !................. 10 20 30 40 50 ( D / t)cy ( X 103N/rn~)

Figure 6 : Maximum strength of circular cylinders

SUMMARY Thin-walled stub-columns formed from two grades of stainless steels designated by SUS304 and SUS301L 3/4H are tested. Based on post-buckling strength, effective width-to-thickness ratios be /t and d e /t are determined. The be /t of unstiffened plates is derived from angle specimens and is given by Eqn. 3. The d e /t of stiffened plates is derived from square tube specimens and is given by Eqn. 4. The effective width of channel, lipped channel, and H-shaped sections can be calculated on the assumption of no interaction between adjacent plates. The limit diameter-to-thickness ratios of circular cylinders are conservatively given by Eqns. 6 and 7.

REFERENCES Lin S-H., Yu W-W., & Galambos T.V. (1988). Design of Cold-Formed Stainless Steel Structural Members -Proposed Allowable Stress Design Specification with Commentary, A Research Project Sponsored by ASCE, Univ. of Missouri-Rolla, Rolla, Missouri, USA Stainless Steel Building Association of Japan (1995). Design and Construction Standards of Stainless Steel Buildings, SSBA, Tokyo, Japan

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 200I ElsevierScience Ltd. All rights reserved

217

BIFURCATION EXPERIMENTS ON LOCALLY BUCKLED Z-SECTION COLUMNS K.J.R. Rasmussen Department of Civil Engineering, University of Sydney, Australia

ABSTRACT The paper describes a series of tests on thin-walled plain Z-sections compressed between fixed ends. Two tests each were performed on seven lengths covering the range from short stub columns to long columns. Most columns failed by interaction of local and overall flexural buckling. The test results are compared with theoretical solutions for the bifurcation load of a locally buckled point-symmetric column. The theoretical buckling loads require calculation of the instantaneous stiffnesses against minor and major axis flexure. These stiffnesses are obtained using a geometric and material nonlinear finite strip analysis. The theoretical results predict that while overall bifurcation of a non-locally buckled column will occur by pure flexure about the minor axis, the locally buckled column will bifurcate in a flexural mode involving coupling between minor and major axis flexure. This result is verified using a geometric nonlinear finite element analysis.

KEYWORDS Z-sections, Steel structures, Tests, Local buckling, Flexural buckling, Interaction buckling, Finite strip analysis, FEM.

INTRODUCTION

It is well-known that local buckling may influence the overall buckling behaviour of thin-walled sections. The influence depends on the end support conditions (pinned of fixed) and the symmetry characteristics. For a doubly symmetric section, such as an I-section, local buckling reduces the flexural rigidity and precipitates overall buckling at a reduced load but does not induce overall displacements. For a singly symmetric cross-section, such as a channel section, local buckling induces overall bending when the column is compressed between pinned ends but not when compressed between fixed ends (Young and Rasmussen 1997). The present paper focuses on point-symmetric columns, such as Z-section columns. Theoretical results (Rasmussen 2000) have shown that local buckling of point-symmetric columns does not induce overall displacements, as it does in pin-ended singly symmetric columns, but causes a coupling between the minor and major axis buckling displacements. In physical terms, this implies that the direction of overall buckling occurs about an axis rotated from the minor principal axis. According to the theory, torsional and flexural overall buckling are uncoupled. However, the torsional buckling

218 mode may become critical in the case of Z-sections with very slender flanges because local buckling reduces the warping rigidity (El~o) more severely than the minor axis flexural rigidity (E/y). The purpose of this paper is to present tests and finite element analyses of fixed-ended Z-section columns to verify experimentally and numerically the behaviour predicted by the theory (Rasmussen 2000).

TEST P R O G R A M

The test specimens were brake-pressed into section from nominally 1.5 mm thick G500 sheet steel. G500 is an Australian produced steel to AS1397 (1993) with galvanized coating and nominal yield stress of 500 MPa. It has low tensile strength to yield stress ratio and limited ductility of the order of 10-15%. The average measured cross-section dimensions are shown in Table 1 using the nomenclature defined in Fig. la. The coefficients of variation of the measured widths of the flanges (bf) and web (bw) were 0.022 and 0.007 respectively, indicating that a tight tolerance was achieved on the cross-section dimensions. The average measured thickness (T) was 1.58 mm. After removing the galvanizing layer by etching, the base metal thickness (t) was measured as 1.495 mm. The b/t-ratios for the flanges and web were 17.1 and 80.5 respectively. The elastic local buckling stress (trl) and half-wavelength (/) were determined from a finite strip analysis (Hancock 1978), as also shown in Table 1. The local buckling mode is shown in Fig. lb. Local buckling was precipitated by instability of the web. y,v

y

V-

-rwf "Ct

bw

w

.

r_.. ~

.

o

........ :

I bf I (a) Section geometry

(b) Local buckling mode

(c) Overall buckling mode

Fig. 1: Symbol definitions and buckling modes

bf

bw

(n~..4)

(ram) 121.8

t* (mm) 1.495

ro (mm) 3.0

A (ram~) 256.5

ly, (mm) 9573

Ix (mm') 5.01•

a 6.95"

trl (MPa) 137.2

l (mm) 120

* base metal thickness

Table 1: Average Measured Cross-section Dimensions and Section Constants Five tensile coupons were cut from the steel sheets in the same direction as the longitudinal axis of the test specimens. Figures 2a and 2b show a typical stress-strain curve obtained from one of the coupon tests. The average values of initial Young's modulus (E0), yield stress (try) and ultimate tensile strength (o,) are shown in Table 2. The COV of E0, Oy and Ou based on the five tests were 0.039, 0.0039 and 0.003 respectively indicating close resemblance of the mechanical properties. The sheets probably pertained to the same batch. In the absence of a sharp yield point, the yield stress was obtained as the 0.2% proof stress. Significant softening was observed in the vicinity of yield, as shown in Fig. 2a. Based on an average value of

219

0.01% proof stress of ~0.01=383 MPa, as determined from the stress-strain curves, the RambergOsgood n-parameter was calculated as n=9.1.

,~

700

700 '

600

600

500

500

400

~ 400

t~

r~

200

,

,

--

r

200

100

100 00

,

~ 30o

300 r~

,

0.001

0.002 0.003 Strain

0.004

0

0.005

0

I

I

0.02

0.04

0.()6 0.08' 0.10 Strain

0.12

0.14

a) initial curve b) complete curve Fig. 2: Typical Stress-strain Curve E0

O'y

(GPa) 216

....

Cu

(MPa) 533 ,,

(MPa) 576

Table 2" Average Mechanical Properties The test specimens were cut in lengths varying from 250 mm to 1600 mm. Subsequently, the ends were milled fiat to ensure even loading. For each length, two nominally identical specimens were prepared. Local and overall geometric imperfections were measured on all specimens prior to testing. The overall geometric imperfections are shown as u0 in Table 3 corresponding to the measured out-of straightness at midlength in the direction of the major x-axis, as shown in Fig. 1c. The out-of-flatness of the web (Ww0) measured at the centre of the web at mid-length is also shown in Table 3. The averages of the overall imperfection relative to the length (uo/L) and the local imperfection of the web (Wwo) were 1/4050 and 0.19 mm respectively. Specimen zf250a zf250b zf600a zf600b zf600a zf600b zf800a zfS00b zf1000a zfl000b ....zfl200a zfl200b zfl400a zfl400b zfl600a zfl600b Average ,

,

Length (L)

uo

(mm)

(mm)

250 249.5 399.5 399 600.5 60O 799 800.5 1000 1000 1199 1198 1400.5 1400 1597 1599

0.075 0.121 0.121 0.036 0.241 0.188 0.445 0.214 0.401 0.455 0.250 0.314 0.205 0.367 0.359 0.374

uo /L 1/3330 1/2060 1/3300 1/11080 1/2490 1/3190 1/1800 1/3740 1/2500 1/2200 1/4800 1/3810 1/6830 1/3820 1/4450 1/5350 1/4050

Wwo

ME

(mm)

(leq)

(~-,~) 65.3

36.0 35.0 38.0 35.5 36.0 38.0 36.5 33.0 34.0 33.5 32.6

64.5 61.2 61.2 60.2 60.1 57.3 57.2 54.5 51.8 42.5 44.6 35.5 29.5 29.0 29.8

0.28 0.00 0.22 0.'14 0.15 0.14 0.25 0.28 0.30 0.i 1 0.30 0.18 0.23 0.20 0.08 0.18 0.19

35.3

Table 3" Geometric Imperfections and Ultimate Loads

N~

N./N1 1.95 1.83 1.74 1.74 1.71 1.71 1.63 1.63 1.55 1.47 1.21 1.27 1.01

0.824 X:~J

220 The specimens were loaded between fixed ends in a vertical position. The top end platen was rigidly connected to the cross-head, thus preventing flexural and torsional rotations. At the base, a lockable spherical seat was used to ensure full contact between the end platen and the specimen during setup. Once contact was achieved, the seat was locked by tightening a bolt at each comer such that flexural rotations could no longer occur. The specimens were uniformly compressed until failure using a 250 kN capacity MTS Sintec testing machine. The ultimate load (Nu) was recorded, as shown in Table 3, and the test then continued into the post-ultimate range. Readings were taken at regular intervals of local and overall deformations. Local deformations were measured using transducers mounted on an aluminium frame, which was attached to the comers of the specimen at midlength, as shown in Fig. 3. The frame followed the specimen during overall buckling and ensured that the local buckling deformations were measured at the same points in the cross-section throughout loading. Readings were taken at the centre of the web and near the free edge of the flanges. Two transducers were used at each of these location, spaced approximately a quarter-wave longitudinally to ensure non-zero local buckling readings from at least one of the two transducers. In addition, three transducers were used to record overall displacements at midlength. Local buckling of the web and flanges can be clearly seen in Fig. 3. The experimental local buckling load was estimated using the N vs w 2 method, according to which the local buckling load is the intersection of the load axis with the line fitted through the graph of the load (N) versus the square of the plate buckling deformation in the initial post buckling range. The experimental local buckling loads (NlE) are shown in Table 3. They are generally close with an average of 35.3 kN and a COV of 0.053. The average experimental local buckling load (Nm=35.3 kN) was nearly equal to the theoretical value (N1=35.2 kN).

Figure 3: Specimen zf1000a during Test

221

BIFURCATION

ANALYSIS

The general theory (Rasmussen 1997) for calculating overall buckling loads of locally buckled members was applied to fixed-ended point-symmetric columns, such as Z-sections, in Rasmussen (2000). It was shown that the buckling displacements Ub and Vb in the principal x- and y-axis directions respectively are determined from the differential equations, ?t

tr

t

[(Eiy)t u o,t] + [(EIxy)t v ott] + (N cruo ,) = 0 tt tt ? I(Eixy)t UbttI "1-[(EIy)t ]2bt,] + (N crVbt) -" 0

(1) (2)

where Nor is the overall buckling load and (E/x)t, (Ely)t, and (Elxy)t are tangent flexural rigidities calculated at the buckling load, as described in detail in Rasmussen (1997). The tangent rigidities can be found by subjecting a length of section equal to the local buckle half-wavelength to increasing levels of compression and then superimposing small curvatures about the major and minor principal axes at each compression level. The tangent rigidities are the ratios of the resulting moments to the applied curvatures, eg

- A-----~"

(4)

The nonlinear inelastic finite strip local buckling analysis described and applied by Key and Hancock (1993) has been used in this paper to calculate the tangent rigidities. For fixed-ended columns, the governing equations (1,2) and the boundary conditions are satisfied by the displacement field, u b _ v~ = 1 - c o s (.2xz ~)

(5)

L

- c--;

which, upon substitution into Eqns (1,2), leads to the eigenvalue problem, 2

cr (6)

Non-trivial solutions are obtained for,

NCr ~ -

B + ~/B 2 - 4AC

where A=1

(8)

B -- - ( N x + N y )

N x = 4n'2(EI~),

(11)

(7)

2A (9)

N =

L~.

(12)

222 Figure 4 compares buckling curves determined from Eqns (7-12) with the test strengths shown in Table 3. The buckling curves are those corresponding to elastic material behaviour and elastic perfectly-plastic material behaviour with values of Young's modulus (E0) and yield stress (Oy) as shown in Table 2. The Euler curve assuming no local buckling is also included. The base metal thickness and average measured values of flange and web widths were used in the numerical calculations, as shown in Table 1. A local geometric imperfection was incorporated in the nonlinear finite strip analysis, assumed to be in the shape of the critical local buckling mode with a magnitude at the centre of the web of Wwo=0.19mm. This value was the average measured local geometric imperfection, as shown in Table 3, and was chosen according to the recommendation made in Hasham and Rasmussen (2001). Comer radii were ignored in the numerical analyses. One and two harmonics were used to model out-of-plane (flexural) and in-plane (membrane) buckling displacements respectively. The overall buckling loads and test strengths are n0ndimensionalised with respect to the local buckling load (N1=35.2 kN) in Fig. 4. 3.0

,,.

I

~

"

I

I

I

\ . . . . . . :

2.5

Flexural-(u),undistorted ---

Flexural-(uv), distorted (elastic)

99

\ \

,

I

Flexural-(uv), distorted (el-pl)

"

9

Test

\~"..

2.0

1.5 o

2: 1.0

0.5 1

0

500

I

I

I

1000 1500 2000 Column length, L(mm)

/

2500

3000

Figure 4: Test Strengths and Bifurcation Curves It follows from Fig. 4 that the elastic and inelastic bifurcation curves are somewhat higher than the test strengths. It is also noted that the plateau reached by the inelastic bifurcation curve at a length of about 750 mm is a measure of the stub column strength predicted by the finite strip analysis, and that the plateau is 15 % higher than the experimental stub column strength obtained from the tests on 250 mm long columns. This result suggests that the displacement field used in the finite strip analysis may not have been adequate for analysing the localised deformations developing near the ultimate load, as observed in the tests. The discrepancy between the bifurcation curves and the test strengths may also be explained by the presence of overall geometric imperfections and the fact that the material was not elastic perfectlyplastic as assumed in the finite strip analysis. As shown in Table 3, the average overall geometric imperfection in the major principal direction was L/4050, which was not negligible, and likewise, significant softening of the material was observed in the tension coupon tests, as shown in Figure 2a. The twist rotation was calculated from the three overall transducers located at midlength. It was found to be negligible at ultimate for all specimens, thus confirming that local buckling does not produce coupling between torsion and flexure in the bifurcation of point-symmetric sections.

223 DIRECTION OF OVERALL BUCKLING According to the classical theory for non-locally buckled point-symmetric sections; overall flexural buckling occurs about the minor y-axis and is uncoupled from flexural buckling about the major xaxis. However, it follows from Equations (1,2) that for a locally bucked section, flexural buckling about the minor principal axis (Ub) is coupled with flexural buckling about the major principal axis. From a physical viewpoint, this simply means that the direction of overall buckling changes from that of the major x-axis. Elastically, the change occurs because the section loses stiffness near the tips of the flanges which causes the axis of buckling to align itself more with the axis through the web. The change in rotation (Aa) of the axis of buckling can be determined from Eqn. (6),

Aa = tan -1( vb ) = tan-' (Cv)= tan-'

2

9

(13)

Equation (13) is plotted against the nondimensionalised load (N/Nt) in Fig. 5 for both the elastic and inelastic cases. In the inelastic case, the axis of buckling first rotates towards the web, then starts rotating back toward the principal y-axis direction at a load of about N/N~=I.9. The latter change in axis of rotation is a result of the fact that when membrane yield occurs near the comers, the material loses stiffness in this region. 2.5

1 i I l e Abaqus (geometricnonlinear elasticanalysis) Elastic 2.0 bifurcation _

1.5

L= 800mm~ . r~

1.0 -

0.5 -

L=1000mm~

L= 1 2 0 0 m m ~ 0

1

]/'

Inelastic

-

analysis i

i I~ 2 N/N I

I 3

4

Figure 5: Change in direction of overall buckling (Aot), (positive counter-clockwise) It was sought to verify the predicted change in direction of overall buckling using the experimental plots of major (u) and minor (v) axis overall buckling displacements. However, as shown by the inelastic curve in Fig. 5, the change of angle (Aot) was less than about 1~ which could not be detected experimentally because overall geometric imperfections produced displacements in both principal directions from the onset of loading. While overall buckling principally occurred in the major x-axis direction, there was significant random variation in the calculation of Aot--tan-l(v/u). A geometric nonlinear finite element analysis (Hibbert et al. 1995) was therefore performed on three lengths of column, L=800 mm, L=1000 mm and L=1200 mm, using SR4 shell elements for the discretisation of web and flanges. A local geometric imperfection was introduced with the same magnitude (Ww0=0.19mm) as that used in the nonlinear finite strip analysis. A small overall flexural geometric imperfection (u0) of L/10,000 was also introduced to avoid numerical instability and/or possible skipping of the overall bifurcation point. The overall geometric imperfection was purely in

224 the direction of the major x-axis. As for the nonlinear finite strip analysis, the width and thickness of the web and flanges, as well as the Young's modulus, were the average measured values given in Tables 1 and 2. The material was assumed to be linear-elastic. The change in direction of overall buckling from the major principal direction (Aa=tanl(v/u)) was calculated using the values of u and v recorded at the ultimate load, as obtained from the nonlinear analysis. The points are labeled "Abaqus" in Figure 5 and seen to agree with the curve predicted by the bifurcation theory, particularly for ultimate loads less than twice the local buckling load. CONCLUSIONS

Tests have been presented on thin-walled Z-sections uniformly compressed between fixed ends. Fourteen tests were conducted at seven lengths ranging from short to long columns, featuring failure by interaction of local and overall buckling. The test specimens were brake-pressed from high strength steel with a nominal yield stress of 500 MPa. Coupon tests demonstrated significant material softening at stresses well below the yield stress. The tests strengths have been compared with elastic and inelastic buckling loads predicted for a locally buckled section bifurcating in an overall mode. The bifurcation loads were higher than the test strengths with increasing discrepancy in the short to intermediate length range. The discrepancy has been attributed to overall geometric imperfections present in the test specimens, gradual material softening and inability of the nonlinear finite strip analysis to model localised curvature for the number of harmonics chosen. The tests confirmed that local buckling does not produce coupling between torsion and flexure in Z-section columns. According to the bifurcation analysis, the direction of overall buckling changes from the major principal axis in a locally buckled section. This result was verified using a geometric nonlinear finite element analysis. However, the change in direction was less than 1~ for the Z-sections tested and could not be demonstrated experimentally because of random variations in the direction of overall buckling resulting from overall geometric imperfections. ACKNOWLEDGMENTS The tests were conducted by Ms Anh Ton and Ms Ruth Tirtaatmadja as part of their BE honours thesis project. Their substantial contribution to this paper is gratefully acknowledged. REFERENCES AS1397, 1993. Steel Sheet and Strip- Hot-dipped Zinc-coated or Aluminium/zinc-coated, Standards Association of Australia, Sydney. Hancock, GJ, 1978. Local, distortional and lateral buckling of I-beams, Journal of the Structural Division, ASCE, 104(ST11), 1787-1798. Hasham, AS and Rasmussen, KJR, 2001. Nonlinear Analysis of Locally Buckled I-section Steel Beam-columns, Proceedings, 3rd International Conference on Thin-walled Structures, Cracow, Poland. Hibbitt, Karlsson and Sorensen, Inc., 1995, 'ABAQUS Standard, Users Manual', Vols 1 and 2, Ver. 5.5, USA Key, PW and Hancock, GJ, 1993. A Finite Strip Method for the Elastic-plastic Large Displacement Analysis of Thin-Walled and Cold-Formed Steel Sections. Thin-walled Structures 16, 3-29. Rasmussen, KJR, 1997. Bifurcation of Locally Buckled Columns, Thin-walled Structures, 28(2), 117154. Rasmussen, KJR, 2000. Overall Bifurcation of Locally Buckled Point-symmetric Columns, Proceedings, Coupled Instabilities in Metal Structures, C1MS'2000, 163-170. Young, B and Rasmussen, KJR, 1997. Bifurcation of Singly Symmetric Columns, Thin-walled

Structures, 28(2), 155-177.

Third International Conferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

225

BUCKLING LOAD CAPACITY OF STAINLESS STEEL COLUMNS SUBJECT TO CONCENTRIC AND ECCENTRIC LOADING J.Rhodes 1, M.Macdonald 2, M. Kotelko 3 and W.McNiff 1 1.Department of Mechanical Engineering, University of Strathclyde, Glasgow, UK 2.Department of Engineering, Glasgow Caledonian University, Glasgow, UK 3. Department of Strength of Materials and Structures. Technical University ofLodz, Lodz, Poland

ABSTRACT This paper describes the results obtained from a series of compression tests performed on cold formed stainless steel Type 304 columns of lipped channel cross-section. The cross-section dimensions, the column length and the eccentricity of the applied compressive load are varied to examine the effects on the buckling load capacity of the columns. The results obtained from the tests and from a finite element analysis are compared to those obtained from the relevant design specifications in America and in Europe. Conclusions are drawn on the basis of the comparisons.

KEYWORDS Stainless steel, cold formed, columns, lipped channels, eccentric loading.

INTRODUCTION

Design code specifications for cold-formed carbon steel members have been published in many countries, e.g. (1), (2), (3). Stainless steel members have fewer design specifications available, the main design code for stainless steel members being the ASCE code in the USA (Ref. 4). In Europe, Eurocode 3, Part 1.4 (Ref 5) has been introduced recently and is currently under examination in the member countries of the EEC. The Eurocode has taken a substantially different viewpoint on some aspects of design than that adopted by the ASCE, e.g. the evaluation of column capacity. The non-linearity of the stress-strain law is taken into account in the ASCE code whilst the Eurocode uses the initial elastic modulus and assumes a larger imperfection parameter to take care of the degradation of the elastic modulus with increase in stress. In dealing with combined bending and axial loading, the differences between the Eurocode and the ASCE code are compounded by virtue of the different interaction formulae used. In view of these differences it was felt that an examination of the effects of concentric and eccentric compressive loading on the buckling behaviour and load capacity of cold formed stainless steel lipped channel section columns and comparison of the predictions of the ASCE Codes and Eurocode 3, Part 1.4 with experiments would be informative. The interaction formulae incorporate the member moment capacity. Since in a parallel series of tests on the same sections (Ref. 6), the experimental moment section capacity was determined, this is also used in the

226 interaction formulae (for comparison with the theoretical moment capacity) to investigate if any improvements to the code predictions are obtained by using the known true moment capacity.

DESIGN CODE RECOMMENDATIONS

Short to medium length cold formed stainless steel columns of lipped channel cross-section are investigated under two different load conditions. In condition (i) load is compressive and applied through the section centroid (concentric loading), and in condition (ii) the load is applied through a point at a fixed distance from the centroid (eccentric loading). In this paper, design code predictions based on full section properties are presented (reported by Macdonald et al (Ref. 7)). The cross-section dimensions were such that it could be considered to be fully effective, with details of the experimental investigation described later. The design rules given in the ASCE code and in Eurocode 3: Part 1.4, set in a form directly applicable to the particular loading conditions examined experimentally, are given below. In dealing with the Eurocode, Part 1.4 does not directly give details of bending/axial load interaction, but instead refers to Part 1.3 or Part 1.1.

ASCE:

Under concentric loading the design axial strength is given by Pn = 0-85AFn

where A and

= gross cross-sectional area

(N)

(1)

(mm 2)

n2E'

Fn

= flexural buckling stress =

with

K = buckling coefficient = 1 for pinned ends L = column length (ram) r = cross-section radius of gyration (ram)

and

E t = tangent modulus =

(KL/r) 2

(N/mm 2)

EoF.

~/mm2)

F~ + O.OOZnEo(o / Z ) "-~ in which

n Eo

Fv and

t~

= plasticity factor = 6.216 (Ref. 7) = initial elastic modulus (N/mm 2) = virgin or full section 0.2% proof stress = F n (N/ram 2)

(N/mm 2)

In the case of a column subjected to a load of constant eccentricity, the maximum eccentric load P, applied at eccentricity e from the neutral axis, is given by equation (2)

P ~+ P,, where M. = and

9_ Y

Pe

p~

= moment capacity of the cross section

P, = /r 2E,I~,. = Euler buckling capacity L2

(2)

<1

M.(1 - P )

(Nmm)

(N)

Eurocode 3, Part 1.4: Eurocode 1.4 uses the same general rules as for carbon steel members, where the design buckling resistance under concentric loading is given by Eqn.(3)

227

Nh,Ra= X A f y

(N)

(3)

where X is a reduction factor due to overall buckling effects A is the cross-sectional area of the member (mm 2) and fy is the virgin or full section 0.2% proof stress (N/mm 2) The evaluation of the reduction factor is not detailed here, but a Perry-Robertson approach takes account of overall buckling, yielding and imperfections. As mentioned, a higher than normal Perry imperfection factor, corresponding to that of European column curve "c" is used for the stainless steel specimens. For a load applied at constant eccentricity e, and having a fully effective cross section, the relevant interaction formula from which the maximum load P can be obtained is given by equation (4):

P

z .L

+

A

toPe _< 1 M,,/7"M1

(4)

Y MI

where ~/MI is the material factor (equal to 1.1) and the factor 1
~

-

1

P" P

but

~<1.5

z f, A with

p,. = 2,.(2,BMv-4) .

but

,u,, _< 0.9

In the above, 2,. = X/(f,.A) / P, and flM~ is the equivalent moment factor = 1.1 for uniform eccentricity. Note that although they have been given the same symbols, for continuity purposes the moment capacity and axial load capacity given in the ASCE code and Eurocode 3:Part 1.4 are not the same. In the application of the above formulae the values obtained for each individual code have been calculated independently.

EXPERIMENTAL INVESTIGATION In the experimental investigation a total of 77 column tests to failure were made on small lipped channel section stainless steel columns. Figure 1 shows the lipped channel cross-section with dimensions given in Table 1. The specimens were accurately measured and section property calculations were based on midline dimensions. The slenderness ratios tested varied from 38 to 207 for THK sections and from 42 to 234 for THN sections. Forty-tbur tests were carried out with concentric loading and thirty-three tests were carried out with loading applied at 8 mm eccentric to the centroid of the cross-section. Eleven different lengths of column were tested. The end grips were designed such that they would hold the ends of the column, and allow loading to be applied at a given eccentricity through knife edges. The specimens were tested in a Tinius Olsen testing machine, with the vertical displacement and column mid-span horizontal deflection measured during the tests using displacement transducers. Figure 2 shows a schematic diagram of the column test configurations.

228

IIL ~ NeutrA alxis bl "

I. I

I

!

I

!I. i

II

..... .'

Concentric Loading

Figure 1 Geometry of Lipped Channel Sections

EccentrIc

Loadn ig

Figure 2 Column Test Configurations

FINITE ELEMENT ANALYSIS Two types of finite element buckling analyses were used - eigenvalue analysis and non-linear analysis using ANSYS. However, the eigenvalue analysis was used only to verify that the finite element model boundary conditions (i.e.column pin ends and concentric and eccentric loading) were accurate, as this type of analysis takes no account of material non-linearity. A full non-linear analysis was conducted using fournoded shell elements with six degrees of freedom at each node. The non-linear material properties of the stainless steel were defined using the initial elastic modulus and stress-strain data obtained from full-section tensile tests. For a non-linear buckling analysis to be accurate, it is necessary to set an initial imperfection in the structure being modelled. This was achieved by modelling a very large radius of curvature for the lipped channel columns which would approximate the actual imperfections. A parametric model was constructed by defining positions of keypoints to allow for easy alterations to the model for the two different column lipped channel section thicknesses and for the variation in column length. A half-model of a column was modelled using appropriate symmetry commands which helped to reduce the considerable computer processing time. Six column models of varying lengths for each section and loading case were analysed.

RESULTS The results obtained from tensile tests on channel web material showed that the 0.2% proof stress averaged 480 N/mm 2 for THN material and 460 N/mm 2 for THK material. From full section tensile tests, the 0.2% proof stress for THN specimens averaged 520 N/mm 2 and 540 N/mm 2 for THK specimens. Figures 3 and 4 show the graphs of Buckling Load v. Column Length for concentrically and eccentrically loaded THN section columns respectively, showing curves for the test results and, the ASCE, Eurocode and ANSYS predictions.Figures 5 and 6 show the graphs of Buckling Load v. Column Length for

229

70~,

60

9 Experimental I9

- , * - ANSYS

9 \

A50

ASCE (FS)

I/__4

_,,. Eur___o_. (FS)

Z v

t~40

o _1

.E 30 111 20 10

|

I

I

200

400

600

,

l

I

800

I

1000

Column Length (ram)

1400

1200

Figure 3. Concentrically loaded THN specimens 30

II

,

I

25

9 Experimental -

~

=

ANSYS ASCE (FS)

i ~="~ASCE (FS,Mexp) I .,,-., - Euro.l.4 (FS)

A

z

,Euro.l.4 (FS,Mexp)

20

v

"o t~ o _1

15

c

o..,.

r IZl

lO

200

400

600

800

1000

Column Length (mm)

Figure 4. Eccentrically load THN specimens

1200

1400

230 120

O ~.

t

100

9

\

9

~

A

Z

80

"0 r 0 ..I o1 c =m,

60

m

40

'I=-

",

ANSYS

ASCE (FS) !,--- - Euro.l.4 (FS)

~~ 9 I~

Experimental

I t

=,=,,

20

9

200

nun

400

9

600

800

9

lu

1000

1200

1400

Column Length (ram) Figure 5. Concentrically loaded THK specimens 60 9 -

.o

50

Experimental -

ANSYS

' ASCE (FS) "~"ASC

z

E (FS,Mexp)

--9 - Euro.l.4 (FS)

40

"am

'Euro.l.4 (FS.Mexp)

"o o -J9 30 c

.==

:3 m

t=

20

9

200

9 i

400

,

9

600

9

800

i

Ink

1000

Column Length (mm) Figure 6. Eccentrically loaded THK specimens

1200

i

inl

14()o

231 concentrically and eccentrically loaded THK section columns respectively, showing curves for the test results and, the ASCE, Eurocode, modified ASCE, modified Eurocode and ANSYS predictions.

OBSERVATIONS

Figures 3 and 4 show that the design codes agreed reasonably well with the test results, with the ASCE design code based on full section properties being very accurate for the full range of slenderness ratios. The Eurocode predictions, also based on full section properties, are only non-conservative at very low slenderness ratios, with the overall accuracy being very good. The ANSYS predictions however, are very conservative for columns with lower slenderness ratios but very accurate for larger slenderness ratios. For the same columns loaded with a fixed eccentricity of 8mm, the code predictions based on the full section properties, are substantially less than the experimental failure loads, particularly for lower slenderness ratios. It is of interest in this respect that a load of 26.5 kN applied at 8mm eccentricity is sufficient to develop a moment equal to the theoretical moment capacity without taking account of axial load effects. In actuality the moment capacity of the cross section is very much larger than is given by any approach which uses the 0.2% proof stress together with limited inelastic reserve, and this is the cause of the high degree of conservatism. For the cross sections examined the fully plastic bending capacity is approached, and the stresses acting are closer to the ultimate strength than the 0.2% proof strength. In a parallel series of tests (Ref. 6) the bending behaviour of these members was examined and the member bending capacity of both types of specimen was found to be slightly more than twice the theoretical capacities determined on the basis of the codes, due to the combination of very substantial post-elastic capacity and enhancement of the stresses beyond the 0.2% proof point before the bending capacity is attained. Substitution of the actual experimentally obtained moment capacities into the interaction equations (Eqns.2 and 4) results in very good agreement between code predictions and experiment for the eccentric loading case. Figures 5 and 6 show the graphs of Buckling Load v. Column Length for the THK section columns and again the design codes provide rather accurate results for the centroidal loading case, but substantially conservative results, particularly for lower slenderness ratios, for the eccentric loading situation. An improvement in the design code predictions is obtained for the eccentric loading case when the experimentally obtained moment capacity is substituted into the interaction equations (Eqns. 2 and 4). A similar ANSYS prediction was obtained for the THK specimens with the eccentric loading case having the most favourable correlation, but is conservative for low slenderness ratios.

CONCLUSIONS The main conclusions from this investigation is that the new Eurocode for stainless steel members, and the ASCE design specification, give accurate predictions of the buckling load of columns of fully effective lipped channel cross-section under concentric compression by considering the full section material properties. However, the design codes were found to be conservative in the examination of eccentrically compressed columns of the same geometry. A finite element analysis also predicted conservative estimates of buckling load, particularly for columns with low slenderness ratios, with a better correlation obtained for the eccentric loading case. For concentrically loaded columns, the design codes gave accurate predictions of the buckling loads for all but the lowest slenderness ratios examined (<75). In this range, the ASCE was more conservative than the Eurocode. For THN specimens, the ASCE design code showed an excellent correlation to the test results. For THK specimens, the Eurocode showed an excellent correlation to the test results.

232 For eccentrically loaded columns, the design codes were found to be less accurate in their predictions of the buckling loads than for concentrically loaded columns, for the specific members examined, even when full section properties were used. It is known that the bending capacity of these members is substantially greater than the code analysis procedures suggest, and this is the main reason for the over conservatism. Improvements to the design code predictions were gained by using the full moment capacity of the lipped channel sections obtained from tests and substituted into the relevant interaction formulae (Eqns. 2 and 4). It is, however, highly unlikely that the underestimation of bending capacity will be found for members with more slender cross sections, so that this conclusion should not be taken as generally applicable for stainless steel columns. A finite element analysis showed a much improved correlation to the test results when compared to the concentric loading case, with conservatism evident only at low slenderness ratios.

REFERENCES

American Iron and Steel Institute (1996), "Specification for the Design of Cold-Formed Steel Structural Members". BS 5950 (1990), "'Structural Use of Steelwork in Building" - Part 5: "Code of Practice for the Design of Cold Formed Sections". ENV 1993-l-3 ( 1996),Eurocode 3: Design of Steel Structures; Part 1.3: "General Rules Supplementary Rules for Cold Forming Thin Gauge Members and Sheeting". (Edited Draft). ANSI/ASCE-8-90 (1991), "Specification for the Design of Cold-Formed Stainless Steel Structural Members". ENV 1993-1-3 (1996), Eurocode 3: Design of Steel Structures; Part 1.4: "General Rules Supplementary Rules for Stainless Steel". Kennedy, A., (1999) "Beam Deflection Determination of Cold-Formed Stainless Steel Sections". BEng(Hons) Final Year Thesis. University of Strathclyde, Glasgow. Macdonald,M., Rhodes, J. and Taylor, G.T., (2000)"Mechanical Properties of Stainless Steel Lipped Channels". Proc. 15 'h htternational A'pecialty Conference on (?old Formed Steel Structures. St.Louis, Missouri, I /SA.

TABLE 1.

AVEILA(iE DIMENSIONS()F LIPPEDCHANNELCROSS-SECTIONS Radius, 12

Web bl (ram)

Flange b2 (ram)

Lip b3 (mm)

Thickness T

Radius, rl

(mm)

(mmt

(mm)

THN

28.00

14.88

7.45

2.43

1.10

1.10

THK

38.00

17.19

9.99

3.05

0.735

2.255

Section Ref:

(Column Lengths: Varying from 222 mm to 1222 mm in Increments of 100 mm)

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rightsreserved

233

EXPERIMENTAL BEHAVIOR OF PALLET RACKS AND COMPONENTS K.S.(Siva) Sivakumaran Professor, Department of Civil Engineering, McMaster University, Hamilton, Ontario, CANADA L8S 4L7

ABSTRACT

The pallet racks are storage frames consisting of beams and perforated columns connected through semirigid connections along one direction, and braced frames along the perpendicular direction. This paper presents the details of the test setup, and sample experimental results associated with (a) stub column tests, which can account the interplay of the influences due to cold-work of forming, local buckling, and perforations on the axial load carrying capacity of uprights; (b) simply supported pallet beam tests, which can establish the flexural behavior parameters of the beams; (c) pallet beam - to - column connection tests, which can determine the connection flexibility, and the moment capacity of the connections; (d) pallet beam in upright frame assembly tests, which simulate the beam conditions in the actual rack as closely as possible, and determine the allowable loads for the beams; and (e) upright frame tests, which determine the capacity of actual pallet racks as closely as possible. Three identical specimens were used in each one of the tests, and exceptionally consistent results (reflected by the low coefficient of variations) were obtained. Thus, the test results can be treated with high confidence. The experimental procedures described in this paper will be of interest to investigators and manufacturers in the field of cold-formed steel racks.

KEYWORDS experimental, cold-formed steel, pallet racks, semi-rigid connection, stub-column, beam capacity, connection flexibility, upright frame tests

INTRODUCTION Pallet racks are widely used for storage purposes, and are essentially upright frames consisting of braced frames and beams assembled together by easy to assemble beam - to - column connections. Beam ends contain pre-welded heavier gauge(usually) beam-to-column connectors. Regularly spaced holes are prepunched on the uprights for easy assembly of the beams, and to provide flexibility with the beam locations. In a structural sense the pallet racks are frames consisting of beams and perforated columns connected through semi-rigid connectors along one direction, and braced frames along the perpendicular direction. The current cold-formed steel design provisions are not adequate for a successful evaluation/design of such

234 pallet racks, and a theoretical/analytical determination of the stiffness/strength of such pallet racks and their components may be difficult due to the presence of perforations in the upright, connection flexibility, effects of cold working, etc. Tests may be used to evaluate the stiffness, strength, and the overall behaviour of pallet rack components, and of the pallet racks. Various specifications exist (AISI, 1997; CSA,1994; FEM, 1998; RMI, 1997) that provide general details related to the test specimen, test procedure, testing apparatus and fixtures, instrumentation, evaluation of test results, reduction and presentations of test data. In general, the following tests would be needed for such pallet racks; [a] stub column tests, [b] simply supported pallet beam tests, [c] pallet beam - to - column connection tests, [d] pallet beam in upright frame assembly tests, [e] upright frame tests. In addition, tensile coupon tests must be conducted to determine the mechanical characteristics of the materials used in the beams and in the uprights. Recently, these tests were conducted on pallet racks produced by five different manufactures. This paper presents the details of the test setup, [such as test layout, support conditions, loading arrangement, specimen details, instrumentations, and the loading sequence], and sample experimental results, associated with the above tests.

SECTION AND MATERIAL PROPERTIES For each manufacturer, this test program considered a specific beam section, and a specific upright frame, connected together using proprietary beam-to-column connectors. The thicknesses of the beams and uprights ranged 1.77mm - 2.63mm. Due to proprietary nature of the investigation, and in view of the limited space available here, this paper will not include complete quantitative information. The upright columns (overall 83mmx 80mm) are cold-rolled channel sections, with or without lips, with intermediate stiffeners, and with pre-punched holes. The beams (overall 107mmx69mm) may be cold-formed steel members which might have been roll-formed into one piece, or may be cold -formed segments welded together at isolated locations to form a closed section. The dimensions of these components were measured at several cross-sections, during the course of the test program. Due to the manufacturing process, (i.e. cold forming, spot welding, etc.) the cross-sectional dimensions and the thicknesses of these members are not always constant. The dimensional tolerances, manufacturing quality of the uprights and the beams, and the welds were of excellent quality, and in general, the members were found to possess uniformity and were defects free. The beam-to-column connections which were pre-welded to the beam ends contain hookup tabs at the ends, and/or at the sides, and the material thickness of connectors ranged 2.97mm -3.18mm. The origin of the steel, the specified minimum yield stress, mill certificate, and the virgin steel material properties associated with the steel used in these components were unavailable. The mechanical properties of flat elements from the upright, and from the beams were established. Tensile coupons were taken from the flat portions of the uprights, and from the flat portions of the beams at regions of low bending moment and shear. The tensile coupons were cut longitudinally from the flat portions after members have been used in other tests. In general, four tensile coupons each were considered for uprights and beams, respectively, however, two tensile coupons each were taken from two randomly selected upright, and beam members. The material properties of the beam-to-column connectors were not established. Consistent stress-strain relationships were observed. The yield strengths were established using 0.2% offset method. Yield strength ranged 310-351MPa, tensile strength ranged 430-478MPa, while the elongation at ~'racture ranged 30-34.

STUB COLUMN TESTS The manufacturing process associated with upright sections introduces cold-work effects and residual stresses. The relatively thin webs and flanges are liable for local buckling. The uprights also contain perforations. Because of the interplay of these.influences, the axial load carrying capacity of uprights must be established using stub column tests. This test also provides a means to observe, measure, and account for local buckling deformations. The column behavior, including ultimate strength, was established in

235 accordance with the stub column test procedures as described in Part VIII, Cold-Formed Steel Design Manual (AISI, 1997). Accordingly, the length of the stub column must be sufficiently short to eliminate overall column buckling (length_< 20times the minimum radius of gyration), and sufficiently long enough (i) to minimize the end effects, (ii) to represent the existing residual stress patterns, and (iii) so that the center portion of the column would be representative of the repetitive perforation pattern in the full upright. For this study three identical stub columns were cut from representative uprights. Each stub column contained three perforation patterns within its length. The ends were of full sections, and the end planes do not pass through any of the perforations. Efforts were made such that at both ends, the distances from the ends of the sections to the closest perforation were approximately equal. The end planes of the specimens were cut to square and flatness tolerance of+0.001 inch. Figure: 1 shows stub column test setup. Steel end plates were used to transfer the test load uniformly into the stub column. The end plates were milled to have a flatness tolerance of • linch. A thick layer of hydro stone grout was used between the end plates and the machine heads, to facilitate alignment of a specimen. The end plates were leveled, and the verticality of the specimen was checked. The specimen was considered centered when the centroid of the stub column coincided with the principal axes of the test machine. Axial shortening was measured by four Linear Variable Displacement Transducers [LVDT] (NE, NW, SE, & SW LVDT), and two dial gauges ( North & South dial gauges) placed in contact with the top end plate. Three other LVDTs (Top, Middle & Bottom LVDT) were used to detect the local buckling. The stub column was subjected to increasing loads until failure. Free running shortening between the end plates was approximately 0.002inch/min. All LVDTs readings, and the load readings were electronically recorded, whereas the dial gauge reading (division 0.0001inch) and the ultimate load was manually recorded. Figure:2 shows the sample axial load - axial displacement relations for the three identical specimens. For ease of comparison with the other test results, all axial displacements were given corresponding to a column height of 300mm. Study of the load displacement relations from 4 LVDTs and 2 dial gauges indicated that though there were minor differences (perhaps due to mono-symmetric section), overall, the specimens were subjected to uniform loading. Comparison of the average results with that of the dial gauge responses the load - average displacement results can be considered to be the representative Axial load - Axial Shortening relationship for the stub column. In general, no local buckling was observed until the load reached approximately 85% of the ultimate loads. From Figure:2 the three identical specimens produced consistent results. Thus, any one of the load - axial displacement relationship can be used as a representative behavior. Venic~ Load & Load Cell

STUB COLUMN TEST

, m .LL

-----TEST1

---'-TEST2

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Figure: 1 Stub Column Test Setup

gN,_oL3.00mm 0,5

1.0

1.5

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Displacement (mm)

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Figure: 2 Representative Axial Load -Axial Displacement Relations

236 BEAM - TO - COLUMN CONNECTION TESTS The pallet racking frame overall strength and stiffness are greatly influenced by the performance of the beam-to-column connections. These connections are ,in general, semi-rigid connections, and the details vary from manufacturer to manufacturer. It is relatively difficult to establish the connection stiffness and strength through analytical means, therefore, it is necessary to determine these characteristics by tests. The connections were subjected to "The Cantilever Test." Figure:3 shows the schematic view of the test setup, which consists of a short pallet beam(600mm) connected to the center of an upright(910mm) using the beam-to-column connection. Three identical specimens were considered. The upright was fixed at the ends using steel brackets, such that both ends of the upright can neither translate nor rotate. A vertical load was applied to the tip end of the beam, through a ball-and socket arrangement. The vertical deflection of the beam tip, and the lateral deflections of the beam tip were measured using LVDT: 1, and 2, respectively. The upright movements at the mid-height (adjacent to beam connections) were measured using LVDT:3 and 4. LVDT:5 measured the connection opening for increasing loads. Briefly, the test procedure is as follows. The upright was first aligned, vertically leveled, and rigidly fastened to the supports at the ends. Snug tight steel brackets provided the fixed support. The beam was then connected to the upright, and was aligned and horizontally leveled. The load was placed at the tip, and a small pre-load was applied. LVDTs were then attached, and initialized. The loads were then incremented until the joint couldn't sustain any additional loads, or until the beam had undergone significant deflections, and the connectors exhibit significant distortions. In general, the uptight and the beam did not experience any distress. The distress and failure were limited to beam-to-column connection brackets only. At failure the connectors experience considerable distortions, and twists. Some of the hookup tabs experienced considerable distortions and shearing. Non of the connection components fractured or failed. In essence the connectors possessed considerable ductility. Figure:4 shows the moment- tip vertical displacement relations for the three identical connections tested under this test program. The ultimate moment resistance of the connections may be obtained from these graphs. It is clearly evident that the test results are consistent, and any one of the relationships may be used as a representative moment rotation relationship for the beam-to-column connection under consideration. Observations from these tests are (a) the uprights do not experience any deflections, (b) beam experiences lateral deflections. This out-of-plane deflection is inevitable since the beam was supported only at one end, and a single beam-to-column connection is loose against out-of-plane rotations, (c) as shown in Figure:4,connections experience nonlinear load-displacement relations, (d) the connections experience about 8mm of opening of the connections prior to reaching the ultimate capacity. PALLET BEAM-TO-COLUMN CONNECTION TEST

xed End

-'---TEST 1

"-'---TEST 2

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Figure: 3 "The Cantilever Test" Setup

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20

30

40

50

60

70

80

90

100

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Figure: 4

Moment-Rotation Relations

237 SIMPLY SUPPORTED BEAM TESTS The simply supported pallet beam tests are conducted in order to establish the flexural behavior parameters of the beams, such as yield moment, ultimate moment resistance, and the effective flexural rigidity. The beams used were of span length of approximately 2440mm (8'). Three identical specimens were considered. Figure: 5 shows the isometric view of the test layout, support conditions, bracing conditions, and the loading arrangement. The beam specimens were simply supported at ends, and subjected to approximately one third point concentrated loading. Since torsional instability, and lateral instability of the beams were of concern, beams were tested in pairs. One end of the test beam was pin supported, and the other end was roller supported. In order to prevent web crippling of the test beams at the supported ends, the ends of the beams were rested on the rollers, through end plates. These end plates were T- shaped brackets and these brackets were aligned, leveled, and clamped to the beams ends. The beams were spaced approximately 610mm (2') apart, and were braced at four locations. The loads to the two test beams were applied through three transfer beams. Vertically fixed hydraulic cylinder and a 100kN load cell form the primary test load. This central concentrated load was divided equally into two concentrated loads, which were further divided into two equal concentrated loads applied to each of the test beams. The resulting point loads were applied at C=760mm from the end supports. L/C associated with this test setup is about 3, where L is the test span, and C is the shear span. During pilot tests the secondary transfer beams were directly supported on the top flange (compression flange) of the test beams, which led to failure under the transfer beams (perhaps due to combination of flexural - web crippling failure mode). Subsequently, the ends of the secondary transfer beams were supported on two 6.4mm (1/4") plates that have been fastened to the test beams using three bolts. By this load transfer arrangement the loads were applied directly to the webs of the test beams, thereby eliminating any web crippling effects, and bearing effects on the compression flanges of the test beams. In general both beams exhibited failure/distress. All of the beams failed in constant moment regions, indicating satisfactory load application and load sharing between the two beams. At ultimate load the compression flange experienced local buckling accompanied by compression web distress. Vertical Load & Load Cell Test

./

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st Beams

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VDT

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Hinge-Fixed support:

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Figure: 5 Simply Supported Beam Test Setup Vertical displacements were measured at the mid-span. Lateral displacements and rotations of the beams at the mid-span were also measured. Figure:6 shows the moment-displacement relationship for the beams. The measured load has been corrected to account for the dead weight of the transfer beams, braces, etc., thus, the load shown here is the total load carried by two beams. Both beams undergo same vertical deflections, which indicates that both beams were subjected to equal, and symmetric loads. The beams did not experience any lateral deformation, except at ultimate load levels. Three identical specimens exhibited consistent behavior. Thus, any one of the load-displacement relations may be used as a representative behavior. The ultimate moment resistance can be obtained from these relations. The initial portion of moment-displacement relationship is linear from which the linear limit moment may be obtained.

238 SIMPLY SUPPORTED PALLET BEAMS TEST ~TEST

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Moment- Displacement Relations

PALLET BEAMS ASSEMBLY TEST

IN

UPRIGHT

FRAME

The pallet beams in upright assembly test is intended to simulate the beam conditions in the actual rack as closely as possible, and to determine the allowable loads for the beams. Since, the beams are connected to upright frames using the prototype end connections this test is deemed to capture the effects of end restraints on the load carrying capacity of beams. Three identi~zal beam-frame assemblies were considered for each rack types. A uniformly distributed load is simulated in this investigation. The tests were conducted in accordance with the test procedures described in the Specification for the Design, Testing, and Utilization of Industrial Steel Storage Racks(RMI, 1997).

Figure" 7 shows the isometric view of the test layout, support conditions, and loading arrangement. The test assembly consists of two upright frames, not bolted to the floor, and two levels of pallet beams. Since the test is intended to represent the behavior of the rack between inflection points, column baseswere not bolted to the ground, though in actual application the columns may be bolted to the ground. The beams used in the study were of span length of approximately 2440mm (8'). The upright frames used in this investigation were of approximately 915mm (3') wide and 1528mm(5') high. The clear distance between the floor and the test beams (lower level beams) was approximately 675mm, and the clear distance between the test beams and the top level beams was approximately 588mm. All beams were connected to the upright frames using the beam-to-column connections. Safety clips were not used in these connections, though those clips may be used in the actual application of these pallet racks.

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Pallet Beams in Upright Frames Test Setup (Isometric View)

239 The loads to the test beams were applied through 8 loading beams placed at uniform spacing. The cental vertical concentrated load was applied to the primary load transfer beam, which was symmetrically supported by two secondary transfer beams. Due to symmetrical arrangement of the loading system, it can be reasonably assumed that the total load was divided equally to each of the test beam, and was applied as eight closely spaced concentrated loads on each test beam, representing a uniformly distributed loading system. The loading beam flanges rested directly on the top flanges of the test beams. However, the friction between the flanges of the loading beams and that of the test beams was minimized by grinding and smoothing the flanges of the loading beams, and by applying grease to the flanges of the loading beams and to the flanges of the test beams, where ever these flanges come into direct contact with each other. It is important to minimize the friction between the loading beams and the test beams, because otherwise the friction between the loading beam and the test beam may provide considerably more bracing to the test beams, than the bracing that may be provided in the actual use. In reality the pallets and the beams may be worn smooth in use and possibly covered with a film of oil. Vertical displacements were measured at the mid-span. In addition, lateral displacements and rotations of the test beams at the mid-span were measured using 2 LVDTs (2R & 3R) and (2L & 3L) attached to the right test beam, and to the left test beam, respectively. Eight other LVDTs were used to measure the response of the upright, and the response of the beam-to-column connections(shown in Figure:7). In addition to the above measurements, horizontal and the vertical displacements of the top level beams(at mid-span) was monitored using dial gauges. Load was increased until the assembly has reached its ultimate load. The jacking operation continued until the load has dropped below about 90% of the peak load. Due to safety concerns (complete collapse of the test beams may result in transfer beams, loading beams, etc. landing on instrumentation systems) tests were terminated as soon as the beams reached peak load. The measured load has been corrected to account for the dead weight of the loading beams, transfer beams, etc. In calculating the absolute vertical displacements a correction has been applied to reflect the vertical movements of the ends of the beams. In general, both beams undergo same vertical deflections. The beams do not experience any significant lateral deformation, except at ultimate load levels. The upright experiences negligible horizontal displacements. However, beams ends experience relatively larger horizontal displacements indicating the connection opening. At ultimate load levels, the beam-to-column connections experience about 3mm opening. This information may be used, in conjunction with the beamto-column connection test results (Figure: 6) to establish the possible moments in the connections corresponding to beam ultimate load levels. PALLET BEAMS ON UPRIGHT TEST Figure: 8 shows load-displacement relations for the three beams that experienced clear failure. It is evident that the three identical specimens exhibit consistent behavior. Thus, any one of the load-displacement relations may be used as a representative behavior of the pallet beams under consideration. From these graphs the ultimate load resistance and the loads corresponding to beam deflection of L/180(design load) may be obtained. In general both beams exhibited failure/distress. All of the beams failed closer to the mid-span regions indicating satisfactory load application and load sharing between the two test beams. At ultimate loads, in general, the compression flange experienced local buckling accompanied by compression web distress.

70 q -"--TEST1

--TEST 2

--TEST 3 1

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A 45 z40 ~-~a5 3O "J2s 2o is 10 5 o0

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Figure:8 Representative Load- Displacement Relations for Pallet Beams in Upright Frames

240 SUMMARY

The current cold-formed steel design provisions may not be adequate for a successful evaluation/design of pallet racks, and a theoretical/analytical determination of the strength of such pallet racks and components may be difficult due to the presence of perforations in the upright, connection flexibility, effects of cold working, etc. The test program included [a] stub column tests, [b] simply supported pallet beam tests, [c] pallet beam - to - column connection tests, and [d] pallet beams in upright frame assembly tests, [e] Upright Frame tests. In addition, tensile coupon tests were conducted to determine the mechanical characteristics of the of the materials used in the beams and in the uprights. The test program considered pallet racks produced by five different manufactures. Three identical specimens were used in each one of the tests, and exceptionally consistent results (reflected by the low coefficient of variations) were obtained. Thus, the test results can be treated with high confidence. In view ofthe limited available space, this paper does not present the full upright frame tests. Upright flame tests simulate the conditions in the actual rack as closely as possible. The test considered vertical and horizontal loads. Test assembly consisted of three upright frames not bolted to floor, and beams connected at three levels. Two lower level beams are loaded to 1.5 times the design loads, and the top level beams are loaded for increasing vertical loads. Thus, the racks were loaded by increasing vertical and horizontal loads until failure. Racks were subjected to symmetrical and unsymmetrical vertical loading. Horizontal loads were applied along the semi-rigid frame directions, and as well as the along the braced frame directions. Frame stability induced failure along the semi-rigid frame direction was the governing failure mode for these tests.

REFERENCES AISI, (1997), Specifications for the Design of Cold-Formed Steel Structural Members, American Iron and Steel Institute, Washington, DC. U.S.A. AISI, (1997), Cold-Formed Steel Design Manual, 1996 Edition, American Iron and Steel Institute, Washington, DC. U.S.A. ASTM, (1997), A370-96-Standard Test Methods and Definition for Mechanical Testing of Steel Products, American Society for Testing and Materials, Annul Book of ASTM Standards, 1997. CSA, (1994), Cold Formed Steel Structural Members, Structures (Design), S 136-94, Canadian Standards Association, Rexdale, Ontario, Canada. Davis, J.M. and Godley, M.H.R., (1998), A European Design Code for Pallet Racking, Fourteenth International Speciality Conference on Cold-Formed Steel Structures, St.Louis, Missouri, U.S.A. FEM (1998), Recommendations for the Design of Steel Static Pallet Racking and Shelving, Federation Europeenne de la Manutention, Section: X. Pu.Y, Godley, M.H.R., and Beale, R.G. (1998), Experimental Procedures for Stub Column Tests, Fourteenth International Speciality Conference on Cold-Formed Steel Structures, St.Louis, Missouri, U.S.A. RMI, (1997), Specification for the Design, Testing, and Utilization of Industrial Steel Storage Racks, Rack Manufacturers Institute, Charlotte, NC, U.S.A. ACKNOWLEDGMENTS The financial support provided by the pallet rack manufactures is greatly appreciated. Tests were conducted at the laboratory facilities of the Department of Civil Engineering, McMaster University. Special thanks to the civil engineering laboratory technical staff and others who assisted the principal investigator in many different capacities.

Third International Conferenceon Thin-WailedStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

241

STRUCTURAL BEHAVIOR OF COLD-FORMED STEEL HEADER BEAMS S. F. Stephens ~and R. A. LaBoube 2 Department of Civil Engineering, University of Missouri- Rolla, Rolla, MO 65401, USA Department of Civil Engineering, University of Missouri- Rolla, Rolla, MO 65401, USA

ABSTRACT The purpose of this experimental and analytical study was to define the structural behavior of cold-formed steel header beams subject to a combined bending and interior-one-flange loading (IOF) condition as typically occurs in cold-formed steel framed wall construction. A header beam is a structural member that spans window and door openings. The header beam is built-up from cold-formed steel C-sections and cold-formed steel track sections. The C-sections and track are interconnected using self-drilling screws. This study focussed on the IOF loading behavior for both back-to-back (I-beam) and box-beam configurations. Previous research conducted in the area of web crippling strength, bending strength and flange buckling of cold-formed steel members was reviewed and is discussed. The data obtained from the experimental study was analyzed and evaluated to determine the interaction between bending and web crippling for the loading configurations used. The findings of this study were used to establish a design methodology for header beam construction that represents the behavior of these members. New design equations and recommendations for the design of header beams is presented. KEYWORDS Box-beams, built-up headers, cold-formed steel, C-sections, headers, I-beams, IOF loading, web crippling.

INTRODUCTION Today, cold-formed steel is a commonly used building material utilized in a wide variety of applications. Used as studs, joists, beams and trusses, cold-formed steel is making significant advances in the residential building industry. Previously in residential construction wood was used almost exclusively as the prinm'y structural building material. Today, became of its fight-weight, strength, economy and most of all dimensional stability, cold-formed steel is successfully being used in a growing number of residential structures from single-family dwellings to multi-family apartment buildings.

242

Purpose of Investigation This experimental study was initiated as a continuation of a pilot study conducted at the University of Missouri-Rolla (Stephens and LaBoube, 2000). That study was conceived to develop a better understanding of the behavior of built-up header beams as typically used in residential construction in the United States. Residential construction utilizing cold-formed steel building material is often fabricated using the "Prescriptive Method for Residential Cold-Formed Steel Framing," Second Edition (1997) (hence referred to as the Prescriptive Method in this paper). Typical residential construction utilizes header beams fabricated using two C-sections in either an I-beam or box-beam configuration (Figure 2).

Scope of lnvestigation This experimental study consisted of testing header specimens in I-beam (back-to-back) and box-beam configurations. The C-sections used in this study had solid (un-punched) webs, as are typically used in current applications. Sizes of the C-sections and spans chosen for this study were selected to supplement the previous study so that a definitive design procedure for header beams could be developed. Specimen loading was limited to the interior-one-flange (IOF) loading condition which is the typical loading configuration for header beams supporting joist or rafters.

CURRENT DESIGN PROCEDURE The current practice for the design of cold-formed steel header beams in the configurations tested in this study is to treat the load carrying C-sections as two independent flexural elements. Typically, the C-section sizes are based on bending strength, shear strength and serviceability. The header beam is usually a simple span rmmber that is assutned to be loaded uniformly even though the load is nearly always applied through rafters, trusses or joists at some spacing, usually 400mm to 600ram. Composite action or other interaction between the C-sections and the top and bottom tracks is ignored in the design of both the I-beam and boxbeam configurations. The C-sections are therefore assumed to act independently so that bending and shear strengths are determined by using the strength of a single section and then multiplying by the number of sectiom used in the header. In this experimental study, header beams composed of two C-sections were used. Web crippling and web crippling in combination with bending has generally been ignored in the design of header beams.

REVIEW OF PREVIOUS RESEARCH

Prior to 1997, there had been no known experimental work conducted to investigate the structural behavior of cold-formed steel header beams. In 1997, the National Association of Home Builders (NAHB) conducted tests on cold-formed steel back-to-back (I-beam) header beam assemblies (Cold-Formed, 1997). The purpose of the experimental study was to investigate the behavior of built-up I-beam headers as typically used in residential construction. Tests were made on a total of 24 I-beam specimens. Eight different sizes of C-sections, ranging in depth from 102mm to 305ram with varying thickness were used as test specimens. Test specimens were fabricated to correspond to span lengths selected from the header span tables in the Prescriptive Method. According to the researchers, all beams failed by local buckling of the top, compression flange. In a comparison between the calculated bending strength and the tested bending strength, it was found that the ratio of tested to calculated bending strength varied from 0.897 to 1.45. These results seem to indicate that neither web crippling nor the combination of web crippling and bending need be considered in design.

243

The researchers did not propose a new or revised design methodology based on the findings of their study.

EXPERIMENTAL INVESTIGATION The UMR experimental study modeled header beam construction that is typically found in residential construction and as detailed in the Prescriptive Method. A total of 36 new header specimens representing 8 different C-section sizes were tested to supplement the 15 headers tested in the original pilot study (Stephens, 1999). Both I-beams and box-beams (Figure 2) were tested as simple span beams (Figure 1). A description of the experimental investigation including material properties, test specimens, test set-up and test procedures has been summarized by Stephens and LaBoube (2000). The range of C-section test parameters used in the two studies is given in Table 1.

TEST RESULTS Generally, the specimem tested failed in web crippling or a combination of web crippling and bending. The test results that were of primary interest in this study were the ultimate bending strength and the ultimate web crippling strength of each specimen. The ultimate bending strength was calculated using AISI equation C3.1.1-1 (Specification, 1996): Mn -- S~F.),

(1)

The C-sectiom were considered to act independently so that any additional bending strength that may have been provided by the top and bottom tracks was ignored. The ultimate web crippling strength of each specimen was calculated using the following equation:

Pn=Ct2FysinOII-Cl~~]I1 +C,v~]II-Ch~1

(2)

This is a new equation adopted for the next edition of the AISI specification. Eqn. 2 will replace equations C3.4-1 through C3.4-9 in the current AISI specification. The coefficiems C, CR, CN and Ch for built-up I-sections are 20.5, 0.17, 0.11 and 0.001, respectively and 13, 0.23, 0.14 and 0.01 for single web Csections. The results of the tests for both the I-beam and box-beam specimens for the pilot study are summarized by Stephens and LaBoube (2000). The results of this study in combination with the pilot study results are presented as shown in Figures 3, 4 and 5. The calculations for I-beam web crippling strength assumed the header was a built-up section even though it does not meet the specification definition of a built-up section. Web crippling strength was determined using the single web C-section coefficients for the box-beam specimens.

EVALUATION OF RESULTS 1-Beams

The typical failure mode for the I-beam test specimens was by web crippling or a combination of web crippling and bending. Each specimen had a buckle in the web beneath the applied load, which is characteristic of web crippling. Local buckling of the top flange immediately under the bearing plates at the IOF load application point was also present. For specimens tested using the two point load

244 configuration, Figure 1, there was significant compression buckling of the top track between the load points. Because the top track buckled in compression, it was no longer contributing to the bending strength of the specimen when the ultimate strength of the header beam was reached. The results of the pilot study appeared to show that I-beam headers behave similar to built-up sections for web crippling strength. Therefore, for these ~ i m e n s , Pn was calculated using Eqn. 2 and the coefficients for built-up sections. A review of the data for both the pilot study and the current study is shown in Figure 3 and clearly indicates a very good correlation with the interaction equation for bending in combination with web crippling. Box-Beams

The failure mode for the box-beam test specimens was also by web crippling or a combination of web crippling and bending. This type of failure occurred when the webs of both C-sections buckled simultaneously at the load application point. Similar to the I-beam specimens, the top track buckled in compression between the load application points for the two point load configuration. Significant top flange deformation and rotation immediately under the 38ram wide bearing plate was another characteristic of the failure mechanism. Based upon the results of the pilot study, the web crippling strength of the box-beam was actually somewhere between that of a single web C-section and a built-up I section. The results of this study confirmed that observation. Figure 4 shows the box-beam data using Pn calculated for single web Csections in relation to the interaction equation for bending combined with web crippling. Nearly all data points are located on the conservative side of the interaction line indicating the computed value of Pn is conservative. In the pilot study it was shown that if Pn was calculated using the coefficients for built-up I sections, the data points would fall on the un-conservative side of the interaction equation (Stephens and LaBoube, 2000). Figure 5 shows the data points plotted after Pn as calculated for single web C-sections is increased by a factor of 1.4.

CONCLUSIONS The objective of this study was to establish a design methodology for header beams that more closely followed the observed behavior of I-beam and box-beam headers fabricated according to common industry practice in the USA. Based on the results of this study, the following conclusions have been developed: 1. Both I-beam and box-beam header configurations must consider web crippling or web crippling in combination with bending for IOF load points. 2. There is an interaction between the C-sections and the track sections that results in an increased web crippling strength for both the I-beam and box-beam header configurations. 3. I-beam headers can be adequately designed accounting for web crippling at IOF load points when web crippling strength is calculated assuming that built-up I-sections apply. 4. Box-beam headers can be adequately designed accounting for web crippling at IOF load points when the web crippling strength calculated assuming single web C-sections is increased by a factor of 1.4.

245

REFERENCES Cold-Formed Steel Back-to-Back Header Assembly Tests (1997). Publication RG-9719, American Iron and Steel Institute, Washington DC, USA Prescriptive Methodfor Residential Cold-Framed Steel Framing, Second Edition (1997). American Iron and Steel Institute, Washington DC, USA Specification for the Design of Cold-Formed Steel Structural Members (1996). American Iron and Steel Institute, Washington DC, USA Stephens S.F. (1999). "Structural Behavior of Cold-Formed Steel Header Beams for Residential Construction," A thesis presented to the University of Missouri-RoUa for the degree of Master of Science in Civil Engineering, Rolla, Missouri, USA. Stephens S.F. and LaBoube R.A. (2000). "Structural Behavior of Cold-Formed Steel Header Beams for Residential Construction," Fifteenth International Specialty Conference on Cold-Formed Steel Structures, 423-436.

TABLE 1 RANGEOF C-SECTIONTESTPARAMETERS

Parameter

t (mm)

h (mm)

w/t

h/t

Minimum "Maximum

0.879 2.589

152 305

19.62 57.77

83 177

Fy (MPa) ] 242 452

] ]

246

L/2

. . N=38mm

.j ii 1

N=76mm

l

1

>l.5h

1.5h Span Length = L

Single Point Load Configuration

L1

L2

m N=38mm

L3

m N=38mm

N=76mm

R

1.5h

1.5h

> 1.5h

Span Length = L

Two Point Load Configuration

Figure 1" Typical loading configurations.

1.

N=76

247 Top Track Section

FTop Track x / / Section

Screw Locations

Bottom Track ~ . ~ Section

-'-

'-

----_____

Screw Locations

Back-to-Back Channel Sections

1

~-Channel Sections

Bottom Track Section

Figure 2: Typical I-beam and box-beam sections

1.6

......

1.4

1.2

1.0

9

=E ~

9

.~

ff

4'

0.8

9

"~

""

9

, -

0.4

. . .k.P, ,k,

,

,

1.2

1.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.6

1.8

Pt/Pn

Figure 3" I-Beam data with P. for built-up I-sectiom.

2.0

2.2

248 1.8 1.6 1.4 1.2 ee :

9

1.0

~

0.8

0.6

=1

9

] . 0 7 r P' ~ + [ M ' ~ = 1.42 - - - - - - ' ~ 0.4

L,,,)

0.2

.,

iD

i

t

""l :...- ..'" .............

0.0 , ~ 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

P~Pn

Figure 4: Box beam data with P, for single web C-sections.

1.8

1.6 1.4 1.2 0

9

= 1.0 :S

:S

0.8

0.6 0.4 0.2 :: ......

0.0 0.0

0.2

0.4

''

........

0.6

0.8

::::',::::I 1.0

1.2

......... 1.4

1.6

' .... 1.8

' .... 2.0

' ........ 2.2

2.4

P~1.4xPn

Figure 5: Box beam data with P. for single web C-sections.

1, 2.6

Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

249

COMPRESSION TESTS OF THIN-WALLED LIPPED CHANNELS WITH RETURN LIPS J. Yan and B. Young School of Civil & Structural Engineering, Nanyang Technological University, Singapore 639798

ABSTRACT Longitudinal edge stiffeners have been used to enhance the local buckling stress of thin-walled steel channel sections. This in turn leads to the increase of the column strength. Apart from conventional simple lips in channel sections, the use of return lips can further improve the buckling behaviour of the sections and hence improve the column strength. This paper presents the results of an experimental investigation into the strengths and behaviour of cold-formed lipped channels with return lips compressed between fixed ends. A series of tests were performed on channel sections brake-pressed from high strength structural steel sheets. Two different cross-section geometry were tested over a range of lengths which involved local buckling, distortional buckling and flexuraltorsional buckling. The experimental column strengths are compared with the design strengths predicted using the American Specification and the Australian/New Zealand Standard for cold-formed steel structures. Design column curves are also plotted. It is shown that the American Specification generally overestimated the test strengths of the specimens. The Australian/New Zealand Standard is generally conservative, except that it overestimated the test strengths of the specimens having more slender flanges for intermediate and long columns.

KEYWORDS Buckling, Channel columns, Cold-formed steel, Design strengths, Effective length, Experimental investigation, Fixed-ended, Return lips, Structural design, Steel structures, Test strengths.

INTRODUCTION The buckling stress of thin-walled channels can be enhanced by the use of edge stiffeners which provide a continuous support along the longitudinal edge of the flange. Hence, thin-walled channels with edge stiffeners can lead to an economic design. Apart from conventional simple lips used in channel sections, the use of return lips can further improve the column strength of the channels. The current design rules for uniformly compressed elements with an edge stiffener in the American Iron and Steel Institute (AISI, 1996) Specification for the Design of Cold-Formed Steel Structural Members and the Australian/New Zealand Standard (AS/NZS 4600, 1996) for Cold-Formed Steel Structures allow for stiffeners other than simple lips. However, these design provisions were based on

250 the experimental investigations dealing only with simple lip stiffeners on adequately stiffened and partially stiffened elements conducted by Desmond, Pekoz and Winter (1981 ). The extension of these provisions to other types of stiffeners was purely intuitive. Therefore, the design rules for other types of stiffeners such as return lips are not supported by test data. Hence, it is important and necessary to obtain test data for sections with other types of stiffeners. In addition, it is essential to assess the ability of the current design rules for the prediction of column strengths of such sections. This paper presents a series of fixed-ended column tests of thin-walled lipped channels with return lips. The test strengths are compared with the design strengths predicted using the AISI Specification and the AS/NZS Standard for cold-formed steel structures. Design column curves are also plotted. The purpose of this paper is firstly to provide test data for fixed-ended lipped channel columns with return lips for use in international steel design specifications and secondly to assess the ability of the current design rules in predicting the column strengths of lipped channels with return lips.

EXPERIMENTAL INVESTIGATION

Test Specimens Two series of tests were performed on lipped channels with return lips subjected to pure compression between fixed ends. The specimens were brake-pressed from high strength zinc-coated Grade G450 structural steel sheets having a nominal yield stress of 450 MPa and specified according to the Australian Standard AS1397 (1993). The specimens were supplied from the manufacturer in uncut lengths of 4000 mm. Each specimen was cut to a specified length ranging from 500 mm to 3500 ram. Both ends of the specimens were welded to 25 mm thick mild steel end plates to ensure full contact between the specimen and end bearings. The shortest specimen lengths complied with the Structural Stability Research Council (SSRC) guidelines (Galambos, 1988) for stub column lengths. The test specimens had a nominal thickness of 1.5 mm, a nominal lip width of 25 mm, a nominal return lip width of 15 mm and a nominal web width of 150 mm. The measured inside comer radius of the specimens is 2.0 mm. The nominal flange width was either 80 mm or 120 mm and was the only variable in the cross-section geometry. The two series are labeled as T1.5F80 and TI.5F120, where "T" refers to thickness and "F" refers to flange. The numbers following "T" and "F" are the nominal thickness and nominal flange width respectively. The nominal flange width to thickness ratios are 53.3 and 80.0 for Series T1.5F80 and T1.5F120 respectively. Tables 1 and 2 show the measured cross-section dimensions of the test specimens for Series TI.5F80 and T1.5F120 respectively, using the nomenclature defined in Fig. 1. The cross-section dimensions shown in Tables 1 and 2 are the averages of measured values at both ends for each test specimen.

Bl

j

-I

~,-~..... .i",~ T. -, -~-I,,T Yi

ri W

. . . . . .

.._...~

tt

~Drl tl

II ii ii |1 |, ;I |l

i ,r i -. ,~.~. ~_.~.

BW Figure 1" Definition of symbols

251

TABLE 1 MEASURED SPECIMEN DIMENSIONSAND EXPERIMENTALULTIMATE LOADS FOR SERIES T1.5F80 Return Lips Flanges Lips s, sr Brt (mm) (mm) (mm) 17.7 28.2 83.4 T1.SF80L0500 17.6 27.6 83.4 T1.5F80L1000 28.0 83.3 T1.5F80L1500 17.6 T1.5F80L2000 17.3 28.0 83.2 28.1 83.4 T1.5F80L2500 17.6 T~.SF80L3000 17.7 28.3 84.4 'T1.5F80L3500 17.4 28.0 83.3 Mean 17.6 28.0 83.5 0.009 COV 0.008 0.005 Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN * Base metal thickness Specimen

Web

B..

Thickness

t

t"

Radius r,

(ram) (mm) (mm) (mm) 153.4 1.547 152.7 1.564 153.9 1.519 154.3 1.535 153.7 1.532 153.7 1.549 153.9 1.534 153.6 1.540 0.003 0.009 .

1.489 1.506 1.461 1.477 1.474 1.491 1.476 1.482 0.010

2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 0.000

Column Length L

Area

Exp.Ult. Load

A

imm~)

P~e

(mm) 499.5 1000.2 1498.9 1998.2 2498.6 3001.2 3501.2 9

. . . .

,

588 592 578 583 583 593 583 586 "0.010

172.0 166.9 163.4 161.7 158.8 154.8 124.4 ,

TABLE 2 MEASURED SPECIMEN DIMENSIONS AND EXPERIMENTALULTIMATE LOADS FOR SERIES T1.5F120 Specimen

Return Lips

Lips

Flanges

Web

Br!

Bz

Bf

B,,.

Thickness

t

t"

Radius

ri

(mm) (mm) (mm) (mm) (mm) (mm) (mm) T1.5F 120L0500 17.2 27.6 1 2 2 . 9 154.2 T1.5F 120L1000 17.5 28.2 1 2 3 . 7 152.7 T1.5F120L1500 17.5 28.4 1 2 4 . 0 152.7 T1.5FI20L2000 17.2 28.4 124.1 153.5 T1.5F120L3000 17.3 28.3 123.3 153.3 T1.5F 120L3500 17.4 27.9 1 2 3 . 2 153.7 Mean 17.4 28.1 123.5 15314 COV 0.009 0 . 0 1 1 0 . 0 0 4 0 . 0 0 4 ..Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN * Base metal thickness

1.521 1.527 1.524 1.529 1.525 1.530 i.526 0.002

1.473 1.479 1.476 1.481 1.477 1.482 1.478 0.002

2.0 2.0 2.0 2.0 2.0 2.0 2.0 0.000

Column Length

Area

Exp.Ult. Load P~/,

L

A

(mm)

(mm~)

(le,o

497.5 1002.5 1495.6 2001.8 3002.0 3502.2

697 703 703 705 701 703 702 "0.004

166.9 159.3 145.7 139.5 131.3 127.4 , . .

.

The base metal thickness is the effective thickness o f the specimens without zinc coating. In Tables 1 and 2, the base metal thickness (t*) was measured by removing the zinc coating b y acid etching. The thickness of the zinc coating was measured as 29 ~un for Series T1.5F80 and 24 lxrn for Series T1.5F120.

Specimen Labeling The test specimens were labeled such that the series and specimen length could be identified from the label. The specimen label is basically the label o f the test series followed by a letter "L" and the nominal length o f the specimen, where "L" refers to the length o f the specimen. For example, the label "T1.5F80L1500" defines the specimen belonging to Series T1.5F80 having a nominal length of 1500 mm.

Material Properties Tensile coupon tests were performed to determine the material properties o f the test specimens. Longitudinal coupon was taken from the center o f the web plate o f the untested specimen belonging to

252 the same batch of the column test specimens for each series. The coupons were prepared and tested according to the Australian Standard AS1391 (1991), having a gauge length of 50 mm and a width of 12.5 mm. The coupons were tested in a 300 kN capacity INSTRON displacement controlled testing machine using friction grips to apply the loading. The longitudinal strain was measured by using a calibrated extensometer of 50 mm gauge length as well as two strain gauges attached to the coupon at the center of each face. The strain gauges readings were used to determine the Young's modulus. The load, gauge length extensions and strain gauges readings were recorded at regular intervals by using a data acquisition system. The static load was obtained by pausing the applied straining for one and a half minutes near the 0.2% tensile proof stress and the ultimate tensile strength. This allowed the stress relaxation associated with plastic straining to take place. The material properties obtained from the coupon tests are summarized in Table 3. The table contains the nominal and measured static 0.2% tensile proof stress (o0.2), the Young's modulus (E), the static tensile strength (ou) and the elongation after fracture (ca) based on a gauge length of 50 mm. The measured static 0.2% proof stresses were used as the corresponding yield stresses in calculating the design column strengths. TABLE 3 NOMINALANDMEASUREDMATERIALPROPERTIES "i'est Series

! Nominal 00.2 (MPa) T1.5F80 450 T1.5F120 . 450 . . . .

E (GPa) 204 208

Measur'ed 00.2.... ou (MPa) (MPa) 522 554 .,507 550

eu (%) 10 10

_

Test Operation The test setup is shown in Fig. 2. Compressive axial force was applied to the specimen by using a servo-controlled hydraulic machine. The upper end support was moveable to allow tests to be conducted at various specimen lengths. The top end plate of the specimen was bolted to a rigid flat bearing plate connected to the upper end support. The rigid fiat bearing plate was restrained from the minor and major axes rotations as well as twist rotations and warping. The load was then applied at the lower end through a spherical bearing. Initially, the spherical bearing was free to rotate in any directions. The ram of the actuator was moved slowly towards the lower end of the specimen until the spherical bearing was in full contact with the bottom end plate of the specimen with an initial load of approximately 2 kN applied on the specimen. This procedure eliminated any possible gaps between the spherical bearing and the bottom end plate of the specimen. The bottom end plate of the specimen was bolted to the spherical bearing which was then restrained from rotations and twisting by using vertical and horizontal bolts respectively. The vertical and horizontal bolts were used to lock the spherical bearing in position. Hence, the spherical bearing became a fixed-ended bearing which was considered to be restrained against both minor and major axes rotations as well as twist rotations and warping. Three displacement transducers were positioned on the fixed-ended bearing to measure the axial shortening of the specimen. Displacement control was used to drive the hydraulic actuator at a constant speed of 0.2 mm/min during the test. The use of displacement control allowed the tests to be continued in the post-ultimate range. The applied load and displacement transducers readings were recorded at regular intervals by using a data acquisition system. The static load was obtained by pausing the applied straining for one and a half minutes near the ultimate load.

253

Figure 2: Test setup

Geometric Imperfections Initial overall flexural geometric imperfections about the minor axis of the specimens were measured prior to testing. A theodolite was used to obtain readings at mid-length and near both ends of the specimens. The maximum flexural imperfections at mid-length were 1/1540 and 1/1670 of the specimen length for Series T1.5F80 and T1.5F120 respectively.

Test Results The experimental ultimate loads (PExP)obtained from the tests are shown in Tables 1 and 2 for Series T1.5F80 and T1.5F120 respectively. The test results are also plotted against the effective length for minor axis flexural buckling (ley) in Figs. 3 and 4 for the respective series. The failure modes observed at ultimate load of each test specimen are also shown in Figs. 3 and 4, where "L" refers to local buckling, "D" refers to distortional buckling and "FT" refers to flexural-torsional buckling.

254 COMPARISON OF TEST STRENGTHS WITH DESIGN STRENGTHS The fixed-ended column test strengths (PF.xp) are compared in Figs. 3 and 4 with the unfactored design strengths predicted using the American Specification (1996) and the Australian/New Zealand Standard (1996) for cold-formed steel structures. The AS/NZS Standard was adopted from the AISI Specification. The design rules for compression members in the AS/NZS Standard are identical to those in the AISI Specification, except that the AS/NZS Standard has a separate check for distortional buckling of singly-symmetric sections as specified in Clause 3.4.6. The ultimate loads of the test specimens are plotted against effective length in Figs. 3 and 4. The effective lengths for major (lex) and minor (Icy) axes flexural buckling as well as torsional buckling (let) are assumed equal to one-half of the column length for the fixed-ended columns (lex = ley = let = L / 2), where L is the actual column length. This is because the fixed-ended bearings are restrained against the major and minor axes rotations as well as twist rotations and warping. The experimental local buckling loads are also indicated in Figs. 3 and 4. In addition, the theoretical elastic minor axis flexural and flexural-torsional buckling loads of the fixed~ columns are plotted against the effective length for minor axis flexural buckling (Icy) in Figs. 3 and 4. The equations of the theoretical buckling loads are detailed in Young and Rasmussen (1995 and 1998). The design strengths and the theoretical elastic flexural and flexural-torsional buckling loads were calculated using the average measured cross-section dimensions and the measured material properties summarized in Tables 1-3. The base metal thickness was used in the calculation. The fixed-ended columns were designed as a concentrically loaded members, that is, the load is assumed to act at the centroid of the effective cross section as recommended by Young and Rasmussen (1998). For the AS/NZS Standard, the distortional buckling loads were calculated according to Clause 3.4.6 of the Standard. The elastic distortional buckling stresses (foal) were obtained from a rational elastic buckling analysis (Papangelis and Hancock, 1995). For Series T1.5F80, the AISI Specification overestimated the test strengths of the short (ley <- 500 ram) and intermediate (500 mm < ley < 1000 ram) columns, as shown in Fig. 3. However, the AISI Specification underestimated the test strengths of the long columns (ley > 1000 mm), except that the test strength at an effective length of 1750 mm was slightly overestimated. The design strengths predicted by the AS/NZS Standard were conservative for all the columns, except that the test strength at an effective length of 1750 mm was slightly overestimated. The failure modes observed in the tests were combined local and distortional buckling for short and intermediate columns, except that distortional buckling was not observed at an effective length of 250 mm. Combinations of local and flexural-torsional buckling modes were observed for the long columns. The AISI Specification predicted flexural-torsional buckling for all the test specimens, which was in agreement with the failure modes observed in the tests for long columns, but not for short and intermediate columns. The AS/NZS Standard predicted distortional buckling for short and intermediate columns and flexuraltorsional buckling for long columns, which were in agreement with the experimental observations, except for the shortest column. For Series T1.5F120, the AISI Specification overestimated the test strengths of all the test specimens, as shown in Fig. 4. The AS/NZS Standard overestimated the test strengths of the intermediate and long columns, but it underestimated the test strengths of the short columns. The failure modes observed in the tests were combined local and distortional buckling for short and intermediate columns, except that distortional buckling was not observed at the shortest column. Combinations of local and flexural-torsional buckling modes were observed for the long columns. In addition, distortional buckling was observed at an effective length of 1500 ram. The failure modes predicted by the AISI Specification and the AS/NZS Standard follow a similar trend as Series T1.5F80. The AISI Specification predicted flexural-torsional buckling for all the test specimens, which was in agreement with the failure modes observed in the tests for long columns, but not for short and intermediate

255 400

9

Tests AISI - - AS/NZS

350

y 15o

300

Flexural-torsional

a, 250 =

O I,,i t~ =

@

L

200

150

Flexural Buckling

so

9

9 --

=

L,D I

I

,

L,D

.

L,D

~

100 -

L,FT L,FT 9

, ~/L~

L,FT

Buckling

50 I

,

500

I

I

I

I

1000

1500

2000

2500

3000

Effective length, l ~ (mm) Figure 3: Comparison of test strengths with design strengths for Series T1.5F80

400 350

9 Tests . . . . AISI " " AS/NZS

Ix y ~ 1 5 1.~5lZ0 -- ! " |

~

Flexural Buckling

300 250 =

Flexural-torsional Buckling ~

200

t,1

150

L ...~

L,D . .,,

L,D

~

9

..-.

D

,

II

F

T

= 100 @

L,FT

/ / L o c a l Buckling .

.

.

.

.

.

.

.

.

~

.

.

.

.

50 I

500

9

I

1000

.,

I

I

I

1500

2000

2500

Effective length, 1~ (mm) Figure 4: Comparison of test strengths with design strengths for Series T1.5F120

3000

256 columns. The AS/NZS Standard predicted distortional buckling for short and intermediate columns and flexural-torsional buckling for long columns, which were in agreement with the failure modes observed in the tests, except for the shortest column. At an effective length of 1500 mm, distortional buckling was observed in the test but not predicted by the AS/NZS Standard.

CONCLUSIONS This paper has presented the experimental results of fixed-ended column tests on thin-walled lipped channels with return lips. The test strengths were compared with the design strengths obtained using the American Iron and Steel Institute (AISI, 1996) Specification for the Design of Cold-Formed Steel Structural Members and the Australian/New Zealand Standard for Cold-Formed Steel Structures (AS/NZS 4600, 1996). It is concluded that the design strengths predicted by the AISI Specification were generally unconservative for the lipped channels with return lips. The design strengths predicted by the AS/NZS Standard were generally conservative, except for the specimens having more slender flanges. The failure modes predicted by the AISI Specification were generally in agreement with the failure modes observed in the tests for long columns, but not for short and intermediate columns. However, the failure modes predicted by the AS/NZS Standard were generally in agreement with the failure modes observed in the tests for all columns.

ACKNOWLEDGMENTS

The test specimens were provided by BHP Steel Building Products. The authors would like to express their sincere thanks to Miss Siew Ping YEO and Mr. Sun Ping YEN who are undergraduate students in the School of Civil and Structural Engineering of the Nanyang Technological University for their assistance in the experimental program.

REFERENCES

American Iron and Steel Institute, (1996). Specificationfor the Design of Cold-formed Steel Structural Members, AISI, Washington DC. Australian Standard, (1991 ). Methodsfor Tensile Testing of Metals, AS 1391, Standards Association of Australia, Sydney, Australia. Australian Standard, (1993). Steel Sheet and Strip- Hot-dipped Zinc-coated or Aluminium/Zinccoated, AS 1397, Standards Association of Australia, Sydney, Australia. Australian/New Zealand Standard, (1996). Cold Formed Steel Structures, AS/NZS 4600:1996, Standards Australia, Sydney, Australia. Desmond T.P., Pekoz T. and Winter G. (1981). Edge Stiffeners for Thin-walled Members. Journal of Structural Engineering, ASCE, 107:2, 329-353. Galambos T.V. (1988). Ed. Guide to Stability Design Criteriafor Metal Structures, 4th Edition, Wiley Inc., 708-710. Papangelis J.P. and Hancock G.J. (1995). Computer Analysis of Thin-Walled Structural Members. Computers and Structures, 56:1, 157-176. Young B. and Rasmussen K.J.R. (1995). Compression Tests of Fixed-ended and Pin-ended Coldformed Lipped Channels. Research Report R715, School of Civil and Mining Engineering, University of Sydney. Young B. and Rasmussen K.J.R. (1998). Design of Lipped Channel Columns. Journal of Structural Engineering, ASCE, 124:2, 140-148.

Third International Conferenceon Thin-Walled Structures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

257

EXPERIMENTAL INVESTIGATION OF STAINLESS STEEL CIRCULAR HOLLOW SECTION COLUMNS B. Young

and

W. Hartono

School of Civil & Structural Engineering, Nanyang Technological University, Singapore 639798

ABSTRACT The paper describes a test program on cold-formed stainless steel circular hollow section (CHS) columns compressed between fixed ends. A series of tests consisting of three cross-section geometries was performed. The specimens were cold-rolled from stainless steel sheets. The tests were performed over a range of column lengths, which involved local buckling and overall flexural buckling. Measurements of overall geometric imperfections and material properties were conducted. The test strengths are compared with the design strengths predicted using the American, Australian/New Zealand and European specifications for cold-formed stainless steel structures. Furthermore, the test strengths are compared with the column strengths obtained from the design rules proposed by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997). Generally, it is shown that the specifications unconservatively predict the strengths of the tested CHS columns, whereas Rasmussen & Hancock and Rasmussen & Rondal conservatively predict the column strengths.

KEYWORDS Buckling, Cold-formed steel, Design strengths, Experimental investigation, Fixed-ended columns, Stainless steel, Structural design, Test strengths, Tubular members.

INTRODUCTION Cold-formed stainless steel tubular members are used increasingly for structural applications. Members subjected to compression force are one of the major components in a structural design. There are several design specifications available for the design of cold-formed stainless steel tubular columns, such as the American Society of Civil Engineers (ASCE, 1991) Specification for the Design of ColdFormed Stainless Steel Structural Members, the Australian/New Zealand Standard (Aust/NZS, 2000) for Cold-formed Stainless Steel Structures and the European (Euro Inox, 1994) Design Manual for Structural Stainless Steel. In addition, design rules for such columns are also proposed by other researchers, such as Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997). The design rules in the specifications as well as the design rules proposed by the aforementioned researchers are mainly based on the investigations of pin-ended columns. Little test data are available on the strength offtxed-

258

ended cold-formed stainless steel tubular columns. The pin-ended support conditions are rarely realised in practice. In most cases, some degree of rotational restraint is offered at the end supports, and the column is somewhere between fixed and pinned. Therefore, it is also important to obtain test data for fixed-ended columns. The objective of this paper is firstly to present a series of tests of fixed-ended cold-formed stainless steel circular hollow section (CHS) columns, and secondly to compare the test strengths with the design strengths predicted using the American (1991), Australian/New Zealand (2000) and European (1994) specifications for cold-formed stainless steel structures. In addition, the test strengths are compared with the design strengths predicted by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997). Weld

t

ITM

J"l

Figure 1" Definition of symbols TABLE 1 MEASURED SPECIMEN DIMENSIONS FOR SERIES C1

Specimen

Diameter

Thickness

Length L

Area

(mm)

(mm)

(mm)

(ram2)

2.88 2.71 2.83 2.78 2.78 2.68 2.78 0.027

549 1000 1500 2001 2499 3001

781 735 768 753 752 726 752 0.027

89.2 89.0 89.2 89.0 88.9 88.9 89.0 Mean 0.002 COV Note: 1 in. = 25.4 mm. COV = coefficient of variation. C1L0550 C1L1000 CIL1500 C1L2000 C1L2500 C1L3000

A

EXPERIMENTAL INVESTIGATION

Test Specimens The tests were performed on stainless steel circular hollow sections (CHS). The test specimens were cold-rolled from annealed flat strips of type 304 stainless steel. The specimens were supplied from the manufacturer in uncut lengths of 6000 ram. Each specimen was cut to a specified length ranging from 550 mm to 3000 mm, and both ends were welded to stainless steel end plates to ensure full contact between specimen and end bearings. Three series of CHS were tested, having an average measured thickness of 2.78 ram, 3.34 ram, 4.32 mm and the average outer diameter of 89.0 ram, 168.7 ram, 322.8 mm for Series C 1, C2, C3 respectively. The average measured outer diameter to thickness (D/t) ratios are 32.0, 50.5 and 74.7 for Series C1, C2 and C3 respectively. Tables 1-3 show the measured cross-section dimensions of the test specimens using the nomenclature defined in Fig. 1. The cross-

259 section dimensions shown in Tables 1-3 are the averages of measured values at both ends for each test specimen. The specimens were tested between fixed ends at various column lengths. The test specimens were labeled such that the series and specimen length could be identified from the label. For example, the label "C2L0550R" defines the following specimen: 9 The first two letters indicate that the specimen belongs to test Series C2. 9 The third letter "L" indicates the length of the specimen. 9 The last four digits are the nominal length of the specimen in mm (550 mm). 9 If a test was repeated, then the letter "R" indicates the repeated test. TABLE 2 MEASUREDSPECIMENDIMENSIONSFOR SERIES C2 Specimen

Diameter

(mm) C2L0550 168.8 C2L0550R 168.8 C2L1000 168.5 C2L1500 168.8 C2L2000 168.7 Mean 168.7 COV 0.001 Note: 1 in. = 25.4 ram. COV = coefficient of variation. ,,

Thickness

Length L

Area A

(ram)

(mm)

~mm~

3.39 3.36 3.31 3.37 3.26 3.34 0.016

550 548 999 1500 2004

1762 1746 1718 1751 1694 1734 0.016

...

,11

TABLE 3 MEASUREDSPECIMENDIMENSIONSFOR SERIES C3 Specimen

C3LIO00 C3L 1500 C3L2000 C3L2500 C3L3000

Diameter D

Thickness t

Len~h L

(mm)

(mm)

(mm)

(ram~)

4.25 4.25 4.48 4.24 4.38 4.32 0.025

1000 1499 1999 2498 2998

4249 4249 4446 4261 4402 4322 0.022

.

.

.

.

322.5 322.5 320.4 324.1 324.3 322.8 COV 0.005 Note: 1 in. = 25.4 ram. COV = coefficient of variation. M e a n

Area A

.,

Material Properties The material properties of each series of specimens were determined by tensile coupon tests. Six longitudinal coupons (two from each series of specimens) were tested. The coupons were taken from the finished specimens at the location 90 ~ angle from the weld. The coupon dimensions were conformed to the Australian Standard AS 1391 (1991) for the tensile testing of metals using 12.5 mm wide coupons of gauge length 50 mm. The coupons were also tested according to AS 1391 (1991) in a 300 kN capacity Instron UTM displacement controlled testing machine using friction grips. A calibrated extensometer of 50 mm gauge length was used to measure the longitudinal strain. In addition, two linear strain gauges were attached to each coupon at the center of each face. The strain gauges readings were used to determinate the initial Young's modulus. A data acquisition system was used to record the load and the readings of strain at regular intervals during the tests. The static load was obtained by pausing the applied straining for 1.5 minutes near the 0.2% tensile proof stress and the

260 ultimate tensile strength. This allowed the stress relaxation associated with plastic straining to take place. The material properties obtained from the coupon tests are summarized in Table 4, namely the measured static 0.2% (00.2) and 0.5% (o0.s) tensile proof stresses, the static tensile strength (ou), as well as the initial Young's modulus (Eo) and the elongation after fracture (c,,) based on a gauge length of 50 mm. The 0.2% proof stresses were used as the corresponding yield stresses in calculating the design strength of the columns. The measured stress-strain curves were used to determine the parameter n using the Ramberg-Osgood expression (Ramberg and Osgood, 1943), e = m + 0.002

(1)

Eo where e is the strain, o is the stress and n is a parameter that describes the shape of the curve. The parameter n was obtained from the measured 0.01% (o0.01)and 0.2% (00.2)proof stresses using n = In(0.01/0.2) / ln(o0.01/o0.2). This expression provided the values of n = 4, 7 and 5 for Series C1, C2 and C3 respectively, as shown in Table 4. TABLE 4 MEASUREDMATERIALPROPERTIESFROMTENSILECOUPONTESTS Series (GPa) (MPa) CI 191 272 CI 188 268 Mean 190 270 C2 190 291 C2 200 285 Mean 195 288 C3 200 266 C3 203 255 Mean 202 261 Note:l ksi= 6.89MPa.

(MPa) 301 291 296 309 310 310 295 279 287

(MPa) 706 673 690 707 672 690 629 603 616

(%) 62 58 60 61 56 59 68 62 65

7 7 7 5 5 5

Test Rig and Operation The test rig and a test set-up are shown in Fig. 2. A servo-controlled hydraulic testing machine was used to apply compressive axial force to the specimen. Two stainless steel end plates were welded to the ends of the specimen. A moveable upper end support allowed tests to be conducted at various specimen lengths. A rigid flat bearing plate was connected to the upper end support, and the top end plate of the specimen was bolted to the rigid flat bearing plate, which was restrained against the minor and major axes rotations as well as twist rotations and warping. The load was then applied at the lower end through a spherical beating. Initially, the spherical bearing was free to rotate in any directions. The ram of the actuator was moved slowly toward the specimen until the spherical beating was in full contact with the bottom end plate of the specimen having a small initial load of approximately 2 kN. This procedure eliminated any possible gaps between the spherical bearing and the bottom end plate of the specimen. The bottom end plate of the specimen was bolted to the spherical beating. The spherical beating was then retrained from rotations and twisting by using vertical and horizontal bolts respectively. The vertical and horizontal bolts of the spherical bearing were used to lock the bearing in position after full contact was achieved. Hence, the spherical bearing became a fixed-ended bearing. The fixed-ended bearing was considered to restrain both minor and major axes rotations as well as

261 twist rotations and warping. Three displacement transducers were positioned at the loading end to measure the axial shortening of the specimen. Displacement control was used to drive the hydraulic actuator at a constant speed of 0.5 mm/min. The use of displacement control allowed the tests to be continued into the post-ultimate range. A data acquisition system was used to record the applied load and the readings of displacement transducers at regular intervals during the tests. The static load was recorded by pausing the applied straining for 1.5 minutes near the ultimate load.

Figure 2: Test set-up Stub Column Tests

The shortest specimen lengths complied with the Structural Stability Research Council (SSRC) guidelines (Galambos, 1988) for stub column lengths. The measured cross-section dimensions and the measured specimen length of the stub columns are given in Tables 1-3. The stub columns C1L0550, C2L0550, C2L0550R and C3L1000 were tested for Series C1, C2 and C3. Four longitudinal strain gauges were attached at mid-length of the stub columns. The material properties of the complete crosssection in the cold-worked state were obtained for the stub columns. Table 5 shows the measured initial Young's modulus (Eo), the static 0.2% tensile proof stress (or0.2) and the parameter n for the stub columns. TABLE 5 MEASUREDMATERIALPROPERTIESFROMSTUBCOLUMNTESTS

Eo

Series

CI 190 C2 195 C3 202 Note: 1 ksi = 6.89 MPa. i

ao.2 242 247 248

262

Measured Geometric Imperfections Initial overall geometric imperfections o f the specimens were measured prior to testing. The geometric imperfections were measured along the weld o f the specimens. A theodolite was used to obtain readings at mid-length and near both ends of the specimens. The maximum overall flexural imperfections at mid-length were 1/630, 1/2200 and 1/2000 of the specimen length for Series C1, C2 and C3 respectively.

Test Results The experimental ultimate loads (PE~) o f the test specimens are shown in Tables 6, 7 and 8 for Series C 1, C2 and C3 respectively. A test was repeated for Series C2. The test result for the repeated test is very close to the first test value, with a difference o f 0.7%. The small difference between the repeated test demonstrated the reliability o f the test results. Failure modes at ultimate load o f the columns involved local buckling, overall flexural buckling and combined local and overall flexural buckling. TABLE 6 COMPARISON OF TEST STRENGTHSWITH DESIGN STRENGTHS FOR SERIES C 1 Specimen

Test

P~

. _

'"

Pexp PaStE

1.16 C1L0550 235.2 0.98 C1LI000 198.4 0.87 C1L1500 177.4 0.82 C 1L2000 165.1 0.85 C1L2500 151.6 0.84 C1L3000 133.4 0.92 Mean -0.140 COV -Note: 1 kip = 4.45 kN. COV = coefficient of variation. i

Comparison

Paust/m s

PEu,o

PEp PR&H

P~, PR&R

1.1( 0.98 0.87 0.93 0.98 0.96 0.98 0.098

1.16 0.98 0.87 0.81 0.79 0.74 0.89 0.171

1.29 1.09 0.99 0.99 0.96 0.89 1.04 0.135

1.29 1.09 1.02 1.05 1.03 0.95 1.07 0.109

.,

,.

i

TABLE 7 COMPARISON OF TEST STRENGTHSWITH DESIGN STRENGTHSFOR SERIES C2 Corn )arison

Specimen

P~P

PASCE

(k~9 495.6 0.99 492.2 0.99 474.9 0.95 461.0 0.92 431.6 0.86 -0.94 Mean -0.055 COV Note: 1 kip = 4.45 kN. COV = coefficient of variation. C2L0550 C2L0550R C2L1000 C2L 1500 C2L2000

i

....

0.99 0.99 0.95 0.92 0.86 0.94

0.99 0.99 0.95 0.92 0.86 0.94

1.16 1.15 1.11 1.08 1.01 1.10

1.16 1.15 1.11 1.08 1.01 1.10

0.055

0.055

0.055

0.055

263 TABLE 8 COMPARISONOF TESTSTRENGTHSWITHDESIGNSTRENGTHSFOR SERIESC3 Specimen

Test

P~P

Comparison

P~,.

P.+,,

P~,,

P~,,

P~,,

PaSCE

PAust/ ms

PEuro

PRa,H

PR&R

1.00 0.99 0.96 0.93 0.90 0.96 0.046

1.00 0.99 0.96 0.93 0.90 0.96 0.046

1.05 1.04 1.02 0.98 0.94 1.01 0.046

1'05 1.04 1.02 0.98 0.94 1.01 0.046

Oa9 1123.9 1.00 1119.7 0.99 1087.8 0.96 1045.7 0.93 1009.5 0.90 Mean -0.96 COV -0.046 Note: 1 kip = 4.45 kN. COV = coefficient of variation. C3L1000 C3L1500 C3L2000 C3L2500 C3L3000

DESIGN RULES

The design strengths of the columns were predicted using the American (ASCE, 1991), Australian/New Zealand (Aust/NZS, 2000) and European (Euro Inox, 1994) specifications for coldformed stainless steel structures. In addition, design rules proposed by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997) were also used. The design rules to calculate design strengths for cold-formed stainless steel columns in the American, Australian/New Zealand and European specifications as well as those proposed by Rasmussen & Hancock and Rasmussen & Rondal are either based on Euler column strength or Perry curve. Euler column strength requires the calculation of tangent modulus (Et) and the Ramberg-Osgood parameter n to determine the design stress, which involve an iterative design procedure. On the other hand, the design rules based on Perry curve needs only the initial modulus (Eo) and a number of parameters to calculate the design stress. In both design rules, the design stress should be less than or equal to the 0.2% proof stress of the material. Euler column strength is adopted in the ASCE Specification while in the Aust/NZS Standard one can choose either Euler column strength (identical to that of ASCE Specification) or Perry curve. For the purpose of comparison, this paper uses the latter. The European Specification adopts the Perry curve for the column design strength. The design rules proposed by Rasmussen & Hancock adopts the Euler column strength while Rasmussen & Rondal adopts the Perry curve. In calculating the design strengths, effective length (le) was assumed equal to one-half of the column length (L) for the fixed-ended columns (le = L/2). The design strengths were calculated using the average measured cross-section dimensions and the average measured material properties as detailed in Tables 1-5. The material properties obtained from the tensile coupon tests were used to calculate the design strengths for the three specifications. On the other hand, the material properties obtained from the stub column tests were used to calculate the design strengths predicted by Rasmussen & Hancock and Rasmussen & Rondal as required. The three specifications require the determination of effective cross-section area (Ae) of the column, whereas Rasmussen & Hancock and Rasmussen & Rondal use the gross cross-section area (A). In ASCE Specification and A u s t ~ Z S Standard, the effective area was found to be equal to the gross area of cross-section for Series C1, C2 and C3. In European Specification, the effective area was taken as the gross area of the cross-section for all test series. For the ASCE Specification, the tangent modulus (Et) was determined using Equation (B-2) in Appendix B of the Specification. For the Aust/NZS Standard, the values of the required parameters t~, [3 ko and ~,l obtained from Table 3.4.2 of the Standard are shown in Table 9. These parameters depend

264 on the type of stainless steel used in the column. For the European Specification, the values of imperfection factor and limiting slenderness were taken as 0.49 and 0.4 respectively, which were obtained from Table 5.1 of the Specification. These values depend on how the column is made. The test specimens were cold-formed and seam welded. The design rules proposed by Rasmussen & Hancock (1993) are identical to those in the ASCE Specification, except that the tangent modulus and 0.2% proof stress were determined from the stub column tests rather than the tensile coupon tests. As mentioned earlier, the design rules proposed by Rasmussen & Rondal (1997) use the Perry curve as basic strength curve, and specify the imperfection parameter in terms of the non-dimensional proof stress (t~o.2/Eo) and the Ramberg-Osgood parameter n. These values were determined from the stub column tests, as shown in Table 5. Rasmussen & Rondal use the same parameters as in the Aust/NZS Standard, and equations were proposed for determination of these parameters. The values of these parameters are given in Table 9 for Series C 1, C2 and C3. The values of ot and 13are smaller than the values obtained from the Aust~ZS Standard, while the values of ~o and ~,i are close to those of the Standard. TABLE 9 VALUESOF PARAMETERS ct, [3, ~o, ~l Parameter

Rasmussen & Rondal Series C1 SeriesC2 SeriesC3 0.788 0.998 0.792 0.157 0.152 0.161 0.550 0.566 0.545 0.235 0.285 0.229

Aust/NZS All series 1.590 0.280 0.550 0.200

,

_

COMPARISON OF TEST STRENGTHS W I T H DESIGN STRENGTHS The fixed-ended test strengths (PExp) are compared with the unfactored design strengths predicted using the American (PAscE), Australian/New Zealand (PA~tmzs) and European (PEuro) specifications for coldformed stainless steel structures. The test strengths are also compared with the unfactored design strengths predicted by Rasmussen & Hancock (PR~n) and Rasmussen & Rondal (PR,~). Tables 6, 7 and 8 show the comparison of the test strengths with the design strengths for Series C1, C2 and C3 respectively. The design strengths predicted by the three specifications are generally unconservative for Series C1, C2 and C3. However, the design strengths predicted by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997) are generally conservative. For Series C1, the mean values of the test strength to design strength ratios are 0.92, 0.98 and 0.89 with the coefficients of variation (COV) of 0.140, 0.098 and 0.171 for the American, Australian/New Zealand and European specifications respectively. However, the mean values of the test strength to design strength ratios are 1.04 and 1.07 with the COV of 0.135 and 0.109 for the predictions by Rasmussen & Hancock and Rasmussen & Rondal respectively, as shown in Table 6. Similar results were obtained for Series C2 and C3, as shown in Tables 7 and 8 respectively.

CONCLUSIONS An experimental investigation of cold-formed stainless steel circular hollow section (CHS) columns has been described. Three series of CHS having diameter to thickness ratios ranging from 32.0 to 74.7 were tested between fixed ends at various column lengths. The test strengths were compared with the design strengths predicted using the American (1991), Australian/New Zealand (2000) and European (1994) specifications for cold-formed stainless steel structures, and the design strengths were calculated based on the material properties obtained from tensile coupon tests. Furthermore, the test

265 strengths are compared with the design strengths predicted by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997), and the design strengths were calculated based on the material properties obtained from stub column tests. It is shown that the design strengths predicted by the three specifications are generally unconservative for the tested fixed-ended cold-formed stainless steel CHS columns. However, the design strengths predicted by Rasmussen & Hancock and Rasmussen & Rondal are generally conservative. The design strengths were calculated based on an effective length of onehalf of the column length.

ACKNOWLEDGMENTS The authors are grateful to the Ministry of Education of Singapore for the support through an AcRF research grant. The authors are thankful to Mr Choong Kiat TAN and Miss Yah Hwee YEO for their assistance in the experimental program as part of their final year undergraduate research project at the Nanyang Technological University, Singapore.

REFERENCES

ASCE. (1991). Specification for the Design of Cold-formed Stainless Steel Structural Members, American Society of Civil Engineers, ANSI/ASCE-8-90, New York. Aust/NZS. (2000). Cold-formed Stainless Steel Structures, Draft Australian/New Zealand Standard, Document DR 00011, Standards Australia, Sydney, Australia. Australian Standard. (1991 ). Methodsfor Tensile Testing of Metals, AS 1391, Standards Association of Australia, Sydney, Australia. Euro Inox. (1994). Design Manual for Structural Stainless Steel, European Stainless Steel Development & Information Group (Euro Inox), Nickel Development Institute, Toronto, Canada. Galambos T.V. (1988). Ed. Guide to Stability Design Criteria for Metal Structures, 4 th Edition, John Wiley & Sons, Inc., New York, 708-710. Ramberg W. and Osgood W.R. (1943). Description of Stress Strain Curves by Three Parameters. Technical Note No. 902, National Advisory Committee for Aeronautics, Washington, D.C. Rasmussen K.J.R. and Hancock G.J. (1993). Design of Cold-formed Stainless Steel Tubular Members. I: Columns. Journal of Structural Engineering, ASCE, 119:8, 2349-2367. Rasmussen K.J.R. and Rondal J. (1997). Strength Curves for Metal Columns. Journal of Structural

Engineering, ASCE, 123:6, 721-728.

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Section V COMPOSITES

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Third InternationalConferenceon Thin-Walled Structures J. Zara.4,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

269

A MODEL FOR FERROCEMENT THIN WALLED STRUCTURES Donato Abruzzese Department of Civil Engineering, University of Rome "Tor Vergata" Via di Tor Vergata, 00133 Rome, ITALY

ABSTRACT The ferrocement material has been widely adopted in the past in Italy, mostly by famous Nervi, for several kinds of structures. Most of the structures are thin walled, and this kind of structures seem to have a good resistance and as well as a good duration. In the paper is presented a simple and reliable model for thin structural elements made by this composite material, taking into account the information obtained from some experimental results and the construction technology. The model considers the structural element composed by several layers of reinforcing net in a homogeneous matrix of concrete. A large "waved" ttmnel roof designed by Nervi and built in the '50s has been evaluated with the proposed method, and the results were compared with those obtained by Nervi who used less sophisticated mathematical tools.

KEYWORDS Ferrocement, fiber-reinforced concrete, composite material, thin structures, Nervi. THE WIDESPREAD REINFORCEMENT IN THE CONCRETE The limited tensile resistance of the concrete needs to be improved by adding reinforcement, such as steel, fiber glass, carbon fiber or others. The most challenge way to improve in all the directions the resistance of the concrete is to distribute, in a homogeneous way, short fibers in the cement matrix. In a composite material this will result with a behaviour almost equal for tensile and compressive stress. A large literature on the fiberreinforced concrete is available, and also many experimental results can be utilized to investigate better the composite. The ACI Committee guide 549, 1996, states that the ferrocement is a composite material of hydraulic cement and filler reinforced by multiple steel net layers. The characteristic resistance of the matrix should not be less than 35 N/ram 2, while the characteristic of the steel used for the nets should not be lower than the values indicated in the Table 1 (referred to different type of nets, see Fig.l), and the yield stress not greater than 690 N/mm 2.

270

Veld stress fv (N/mmz) E, effective longitudinal

Woven square mesh 450

Welded Square mesh 450

Hexagonal mesh 310

Expanded mesh 310

Wire

138

200

104

138

200

165

200

69

69

(kN/mm~) Er effective transversal

(kN/mm2)

414

Tab. 1 - Minimum recommanded values stress/elasticity modulus of the steel in the net According to the recommendations of the ACI Committee, the structures made by ferrocement should be calculated in the same way as for reinforced concrete. This seems the simplest way to approach the problem, since several things are different between the two composite material. For instance, the way to evaluate the elasticity modulus of the reinforcing wires will differ from type to type of net. Sometime it is required to perform specific test to evaluate the global elasticity modulus for the composite. The ultimate stress analysis of the structure, with this assumptions, will appear as the following:

b----d

__'___ Fig. 1 - Stress distribution for the flexural behaviour of a rectangular beam (limit analysis) THE FERROCEMENT AND PIER LUIGI NERVI

In the 1948, P. L. Nervi patented the ferrocement technique, and started to use several layers of steel net with very thin wires kept together in almost any kind of shape by a mortar composed of cement and sand. Nervi used the ferrocement technique and material for many structures, taking advantage of the moderate thickness, and consequently less weight, relying on the greater ductility of the composite material steel-cement, and assuming the bidimensional behaviour of the new material, which is able to support compressive or tension loads. In fact, the first experimental tensile tests on samples with different number of layers of net, performed in 1949 at the University of Milano by Oberti and Grandori, produced encouraging results which showed well the increasing of the deformation of the material at the first crack. In Tab. 2 the values of the stress (~r and the strain er corresponding to the first crack are presented, versus the number and type of net adopted. The percentage of the steel takes into account only the area of the parallel wires in the longitudinal directions, neglecting the transversal ones. As it is possible to see in Tab. 2, samples with greater steel quantities resulted in greater values of the deformation at the first crack. This behaviour of the ferrocement can be explained considering that the cement, with an high percentage of steel, corresponding to a large frictional surface, can move into an ultra-elastic state, allowing large deformation and ductility.

271 Sample N. layers l 1 2 3 4 5 6 7 8

3 6 4 20 10 10 10 10

P kg/m2 0.4 0.4 0.4 0.4 1 1 1 1

d ,m,m 0.57 0.57 0.57 0.57 0.9 0.9 0.9 0.9

L mm 10 10 10 ..... 10 10 10 10 10

s mm 17 16 17 18 17 17 16 16

g % 0.45 0.95 1.35 2.83 3.74 3.74 3.98 , 3.98

~r E-05 12.5 11.5 11.5 60 66 67 64 64

or orf k g / c m 2 kg/cm2 25 270 20 250 20 250 42 1290 60 1420 60 1440 60 1380 60 1380

Tab. 2 - The experimental results by Oberti & Grandori (1949) The typical tensile response of a ferrocement sample is shown in Fig.2. The stress-strain function can be divided in three phase. In the first phase the composite behaves as linear elastic material, in the second one, starting at the first crack, several cracks appear in the matrix (concrete). This second phase can be considered as the most probably normal life of the structure. In the last phase, starting from the yielding point of the steel, the number of the cracks remains almost the same, but their dimension increases. // /

/

(c~-

/

"'!'-

8

COMI~ITE 9 $TI~IJN

Fig. 2 - Typical tensile behaviour of ferrocement Therefore we can state that the ultimate tensile load of the ferrocement depends neither on the thickness of the concrete nor its quality, since the matrix, once cracked, does not contribute to the overall resistance of the sample, granted only by the reinforcement. In the primary elastic phase the stress-strain relation is defined with the elastic modulus Ec of the composite. The initial value can be calculated: Ec =Em ( 1 - rl Vr) + Er rl Vr

(1)

While the value of the stress in the composite at the first crack is given by the following experimental function (Naaman e Shah): (~cr = (~mu + 25 1"1Sr

1"1Sr< 0.2 mln2/mm 3.

(2)

where Era,= matrix elasticity; ~mu = tensile resistance of the matrix, r I = net effective factor (in the stress direction) The net effective factor indicates the available contribute of the reinforcing net in the stress direction. In Tab. 3 the net effective load factor has been shown depending on the different available net shapes.

272 Woven square mesh 0.50 0.50 0.35

Stress direction . ,

Longitudinal rh Trasversal qt Angle 45 ~ rl4s

Welded Square mesh 0.50 0.50 0.35

. . . .

Hexagonal mesh 0.45 0.30 0.30

Expanded mesh 0.65 0.20 0.30

,=

Tab.3 - Net effective factors for different net shapes As the number of the cracks increases, the contribute of the matrix decreases progressively, and in this phase the value of the elastic modulus Er of the reinforcement can be assumed: Ec= nVrEr

(3)

The maximum dimension of the cracks ~i~max in this phase can be easily calculated with the experimental function proposed by Naaman e Shah: per O'r < 345 11 Sr per Or > 345 1"1Sr

~max = 3500 / Er Wm~ = 20 / Er(175 + 3.69 ( t~r- 345 11 Sr ))

(4) (5)

where ar "- stress in the net when the load is applied Once the steel reaches the yield point, the load remains unchanged and is carried entirely by the net. The collapse load can be calculated: t~cu = fy 1"1Vr (6) THE MODELS FOR THE FERROCEMENT In Fig. 3 is shown the experimental function stress-strain corresponding to a sample of ferrocement of dimensions 25.4 x 9.5 mm and 150 mm long, reinforced with a quarter-inch squared nets. N=

1

5

8

10

14

16

10-

_

-o15

Wov~I Steel Mesh Vr = 0.023 Sr = 0.27 (~ Phase /

t

yeld point

Phase II

10

20

Composite strain, ram*

30

40 I0

50

4

Fig. 3 - Result from tensile test of ferrocement element

Eo'

E1

Fig. 4 - Experimental results of a ferrocement sample

273 The first tensile test required by Nervi in 1949 and performed at the Politecnico of Milano on ferrocement samples produced the graph stress-strain in Fig. 4, in which the strain is limited to the limit elastic deformation e = 0.001 of the steel. The gradient of the function decreases until the value el for the more reinforced samples, while in the range e > el the function is linear, because the contribution of the matrix, already cracked, remain constant until the collapse of the sample. The deformation limit, corresponding to el, does not depend on the reinforcing percentage. In the graph it is possible recognize two different phases. The first one for 0<e<el, in which the concrete is acting the main role, with the secant elastic modulus El, corresponding to the deformation e=Cl; a second one for Cl<e<e r, when the contribute to the resistance is given by the reinforcement, and the elastic modulus can be assumed equal to E'. Let us consider the line from the origin of the coordinates, corresponding to an elastic modulus value: E'o = Eo ( 1 + ~/ n )

(7)

that is corresponding to the starting tangent modulus of the experimental graphs, since the reinforcement steel does not yet reach the yield point. The modulus Eo' can be assumed, instead of the secant modulus, in the first phase. If we write the contribute of the concrete to the resistance of the composite: S t~ic = S - Nd t~c (8) this is the effective contribute, since the load will be distributed considering the reinforcing wires (number N and diameter d). We obtain, for e<el, that the behaviour does not depend on the reinforcement, since we are still in the elastic phase. Also, the modulus in the second phase seems depend on the reinforcement. If we assume eric = el Eo, for e=el, since eric is not depending on the reinforcement, we can then calculate E1 in the following way. Let SN be the thickness corresponding to the nets in the total S thickness, and So the thickness corresponding to the concrete: SN = N d,

So = S - N d

(9) (10)

the elastic modulus corresponding to the thickness of the nets, neglecting the contribute of the concrete in that part, can be calculated: N.n.d 2 EN = E f ~ 4.1.S N

(11)

where Ef = Eo n. Then the value of the effective total elastic modulus E 1 is: EI=E0"S0/S+EN-S~t/S

= E 0 ( 1 . + y . n ) . N 'sd

(12)

The value obtained is very close to that one of the secant modulus of the experimental tests. In the starting elastic phase, the elastic modulus and the crack stress are given by the (1) and (2), after that point we can assume the medium value of the global elastic modulus Ec, given by the (3). The values of the elastic modulus can be always evaluated experimentally. The reinforcing net behaviour is elasto-plastic, with an elastic modulus equal to the effective global elastic modulus of the material matrix-net Er, taking into account for each layer of net the effective resistant area Asi in the stress direction, calculated by means of the global efficiency factor. The

274 behaviour of the concrete can be assumed non linear for compressive stresses, given by the Sargin's formula, and linear elastic for tensile stress, until the first crack. A higher resistance of the concrete can be evaluated considering the amount of reinforcing in the matrix, as calculated in (2). A simplified tensile model of the behaviour of the ferrocement can be obtained by linearizing the described phases and assuming the value for E and o" corresponding at the specified (Fig.5).

~r

--

E2

o.0o4Compression It

I

' i .,

Tension

., 0.9

(~c

Fig.5 - Ferrocement mechanical model

THE PALACE FOR T H E F E R R O C E M E N T ROOF

EXHIBITION

IN TORINO.

THE MODELING

OF THE

The Palace for the exhibition in Torino, Italy, has a rectangular shape with dimension of 96 m x 156 m. The main roof is a cylindrical vault, with the span of 80 m, and internal high 18.10m, with a waved shaped showing a thickness of the cross section ranging from 1 m to 1.60 m. The roof is composed by several precasted elements in ferrocement, with width 2.50 m and length 4.50 m., with some openings for the windows"

Fig.6- The cross section of the waved roof

Fig. 7 - Wires and bars for the reinforcement in the precast ferrocement elements The idea to use precast structural elements in ferrocement allowed to build a very light and not very expensive structure in a short time. Using the traditional concrete would result in a heavy structure, need of casting, and probably the architectural effect will be changed.

275 The variable thickness of the roof is related to the structural assumption of the two hinges arch. If a static calculation of the roof, as it is, is performed, a pressure curve can be observed, not so far from the that one assuming two hinges at the end. Also, the pressure curve due to the dead load only, is very close to the center of the section of the 'beam'. The static calculation has been done considering a distribute load on the roof equal to 150 Kg/m2, an asymmetric load (half roof loaded) and a thermal load of 20~ Our aim has been to evaluate the static behaviour of the ferrocement roof of the Palace of the Exhibition of Turin. The simple constitutive model, obtained by linearizing the experimental curves, offers a reliable instrument to investigate existing structures made by ferrocement. The results of the calculation have been compared with those reported by Nervi in his report. To investigate the flexural behaviour of the ferrocement structural elements the simplified model utilized is based on the concept of homogenization of composite materials. Since the reinforcement is widespread and well distributed, the composite is modeled as an equivalent ideal homogenous material with a different behaviour in compression and in tension. In compression an elastic perfectly - plastic model has been applied, similar to those adopted for the fiberreinforced. The compressive elastic modulus has been calculated by the following homogenization law: Ecr =

Ecru V m +

E,. V r 11

(13)

where Vm, Vr and Ecru, Er are, respectively, the percentage and the Young's modulus of the matrix and the reinforcement. The coefficient rl is the efficient factor of the reinforcement in the stress direction. For the tensile behaviour of the material, considering the high percentage of steel, a simplified model taking into account the different phases of the composite, has been assumed, as described previously. The simple constitutive model, obtained by experimental curves, offers a reliable instrument to investigate existing structures made by ferrocement. The results of the calculation (maximum stress in the concrete ere = 58 Kg/cm2) have been compared with those reported by Nervi in his report (co = 53 Kg/cm2). For each different structural element the ultimate domain M-N has been calculated. The medium safety factor of the roof of the Palace of the Exhibition is slight less then 4. Of course, during the regular activities, when the loads are represented only by the self weight, the safety factor is very large.

ANALYSIS RESULTS AND CONCLUSIONS The aim of the study has been to find a reliable model to evaluate the static behaviour of ferrocement elements, and to assess the safety factor of the structures. As case test, the roof of the Palace of the Exhibition in Torino, designed by P. L. Nervi in 1949, has been analyzed, and the results seem comparable with those calculated by Nervi. The constitutive model assumed seems reliable, and able to describe the behaviour of the bidimensional structural elements in the elastic and ultra-elastic phases. A further experimental investigation on flexural behaviour of ferrocement elements seems useful to validate and refine the parameters of the proposed model.

276 REFERENCES

ACI Committee 549.(1996). Guide for the design, Construction and Repair of Ferrocement. ACI Manual of Concrete Practice 1996. Part 5. ACI Committeee (1996). ACI Manual of concrete practice. Part 5. Balaguru, Naamaan and Shah. (1978) Analysis and behavior of ferroeement in flexure. ASCE Journal of the Structural Division. Vol. 103 Barberio, Goffi and Mattone (1981). Comparison of the flexural behaviour of thin ferrocement and fibre reinforced concrete slabs. International symposium on ferrocement. Rilem. edit by Oberti and Shah Colin, Johnston and Mattar (1976). Ferrocemem behavior in tension and compression. ASCE Journal of the structural division. Vol. 102 Lim, Parasivam, and Lee. (1987). Bending behavior of steel fiber concrete beams. ACI Structural Journal. V01.84. Mansur and Parasivam (1990). Ferroeement short columns under axial and eccentric compression. A CI Structural Journal. Nervi P. L. (1956). I1 ferroeemento: sue r e possibilitY. L'ingegnere. Oberti G. and G. Grandori (1949). Prime esperienze sulla deformabilit~ e resistenza a trazione di provini in ferro eementato. Report of the Laboratory at Politecnico of Milano. Parasivam, Ong and Lee (1988). Ferrocement structures and structural elements. Material Composite. Somayaji and Shah (1981). Prediction of tensile response or ferrocement International symposium on ferrocement. Rilem. edit by Oberti and Shah Wamg Kai Ming (1981). Calculation of strength for ferrocement. International symposium on ferrocement, Rilem, edit by Oberti and Shah

Third Intemational Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

277

EFFECTS OF MANUFACTURING VARIABLES ON THE SERVICE RELIABILITY OF COMPOSITE STRUCTURES A.R.A. Arafath, R. Vaziri, H. Li, R.O. Foschi and A. Poursartip Departments of Civil Engineering and Metals and Materials Engineering The University of British Columbia 2324 Main Mall, Vancouver, B.C., V6T 1ZA, Canada

ABSTRACT A combined deterministic-stochastic simulation approach is presented to demonstrate the manner in which manufacturing-induced variabilities in composite structures can be controlled to achieve targeted reliability levels in structural performance. A finite element code, which deterministically simulates the various physical phenomena during manufacturing of composite structures, is integrated with non-linear structural analysis and reliability analysis to compute the statistics of the parameters that must be controlled at the manufacturing level in order to result in an optimum or reliable structural performance during service. The methodology is demonstrated through a case study that examines the buckling behaviour of a composite plate in the presence of manufacturing-induced imperfections. The reliabilitybased approach adopted here which uses the as-manufactured, rather than the as-designed conditions as the basis for structural analysis can potentially lead to the reduction of uncertainties and costs associated with the use of empirical safety factors currently employed in design of composite structures.

KEYWORDS Fibre-Reinforced Laminated Composite Structures, Manufacturing Imperfections, Process Modelling, Reliability, Buckling, Residual Stresses, Variability.

INTRODUCTION

Background Fibre-reinforced plastic composite materials are replacing traditional ones in many of today's structural engineering applications. When using composites, unlike their metallic counterparts, the complete (largescale) structure is manufactured from the raw materials in one step. Therefore, the manufacturing process and its consequences on the material and geometrical properties of the final structure must be taken into account when considering the response to, and reliability under, the applied loads. Otherwise, the designer

278 of composite structures is forced to use empirical safety factors, which may result in a conservatism that, in many cases, nullifies the benefits that are offered by composite materials. There are many input process parameters that influence, to various degrees, the outcome of the manufacturing, influencing, in turn, the structural performance. Many of these parameters have inherent stochastic variability that result in uncertainty in the final part geometry and material properties. By coupling stochastic and deterministic computational models of the physical events one can study the effect of those variabilities as they propagate through the various stages of the analyses, and eventually influence the response of the composite structure to in-service loading. This would ultimately allow process variables to be controlled or manipulated at the outset (manufacturing level) to result in an optimum or reliable structural performance during service. The objective of this paper is to present a computational framework for analysing the complete sequence of events from the manufacturing stage to the in-service structural performance of a composite structure, while taking into account the stochastic nature of the parameters involved. Specifically, we combine a deterministic model for the manufacturing process and structural analysis with probability-based reliability models to assess the reliability of the composite structure under service loads. To illustrate the approach, we consider the buckling behaviour of a slender laminated composite plate in the presence of imperfections arising from the manufacturing.

Deterministic Process Model Our focus is a high-performance structural component made of advanced thermoset matrix composites typically employed in the aerospace industry. An autoclave process is commonly used to manufacture such structures. Broadly speaking, this process involves stacking of pre-impregnated sheets of unidirectional fibres (commonly called prepreg) at various orientations over a tool of desired shape and then subjecting the whole assembly to a controlled cycle of temperature and pressure inside an autoclave. The process results in the compaction and curing of the composite part. However, because of the residual stresses that build up during the process, the precise shape and dimensions of the final part after tool removal are often difficult to control. In thin-walled composite structures spring-back and warpage are commonly observed process-induced imperfections. A comprehensive, multi-physics, 2D finite element code, COMPRO, has been developed to analyze industrial autoclave processing of composite structures of intermediate size and complexity. The model caters for a number of important processing parameters and the development of residual stress and deformation. This model advances previous work by other researchers and, in addition, accounts for the effects of tool/part interactions, which have been neglected by other investigators. The model assumptions, theoretical background, solution strategies and case studies demonstrating its predictive capabilities have been documented in recent articles by Hubert et al (1999) and Johnston et al (2001). Few investigators have explicitly considered the effect of manufacturing imperfections on the structural performance. Those who have incorporated the as-manufactured conditions in their analyses (e.g. Li et al, 1995) limited them to measurable quantities such as thickness variations. However, using a deterministic numerical process model, such as COMPRO, the final state of a manufactured component with all its spatial variations in geometry, material properties and residual stresses, which do not lend themselves to simple experimental measurements, are readily available as initial conditions for structural analyses.

Probability-Based Modelling Uncertainties in composite structural behaviour arise at different levels (Shiao and Chamis, 1999) such as the material level (uncertainties in the raw material properties), the manufacturing level (e.g. uncertainties

279 in ply orientation, ply thickness, fibre volume fraction, cure cycle) and the structural level (e.g. uncertainties in loading, boundary conditions, geometry). To account for these uncertainties in the analysis and design of composite structures require a stochastic modelling approach. Among several available approaches, sampling techniques (e.g. the Monte Carlo simulation) that choose samples purely randomly from the range of input parameters are theoretically simple. However, since the deterministic models for each input parameter set has to be invoked, the total computational time is a strong function of the number of samples chosen. Typically, a huge amount of computer time and memory is required especially when finite element methods are used for the deterministic computations. An alternative approach is to use analytical-based reliability models, which are theoretically more complex but in return provide powerful and efficient tools for predicting the probability distributions of output parameters of interest. Of particular interest in performance-based design is the inverse reliability problem, whereby design parameters must be found so that different performance criteria are satisfied with target reliabilities. Li and Foschi (1998) have recently developed an inverse reliability software (IRELAN) to study the inverse problem. In this paper, we present how process-induced uncertainties can be included in the analysis of composite structures and how the processing parameters can be controlled to achieve a certain reliability level in structural performance. For illustrative purposes, the buckling behaviour of a composite plate in the presence of process-induced residual stresses and deformations is examined using an integrated deterministic-probabilistic analysis approach. The deterministic process simulation software, COMPRO, the commercial structural analysis software, ABAQUS, and the inverse reliability software, IRELAN, are used here to determine the process parameters that must be controlled to meet structural performance standards with target reliabilities.

M E T H O D O L O G Y AND RESULTS

Statement of the Problem A unidirectional, carbon fibre-reinforced plastic (CFRP) composite laminate with a span of 1200mm, a width of 100mm and a thickness of 1.6mm (consisting of 8 plies of thickness 0.2 mm) is considered for this study. As part of an extensive experimental study (Twigg, 2001), various samples were manufactured on flat aluminum tooling inside an autoclave and the resulting warpages were measured to provide a benchmark for comparison with the process modelling predictions of COMPRO. Initial imperfections resulting from the manufacturing process were computed using COMPRO in terms of three main variables: the fibre orientation angle 0, the fibre volume fraction ratio Vf and the autoclave hold temperature T. The virtually manufactured structures were then considered to be simply-supported and subjected to progressively increasing compressive axial load P as shown in Figure I.

i(w),..7

,1,~,

,,.

"~

-x (u)

t.. IT M

K//

y (v)

Fibre Orientation ~.6mm ~

1200 mm '

t

p

~~,

-5

SIO0

mm

,~1 "7

Figure 1" Schematic of the composite plate loading and boundary conditions. The corresponding non-linear structural analysis was carried out using ABAQUS. The material properties used in these calculations, which are listed in Table 1, were fully-cured properties of the laminate computed using the micro-mechanical models in COMPRO (Johnston et al, 2001). The non-linear relation

280

between the axial load, P and the out-of-plane displacement, w, was obtained by increasing the load P incrementally up to the limiting buckling load Per and computing the corresponding displacement. Using the eigenvalue analysis option in ABAQUS, the critical buckling load for a flat plate, free of imperfections, was found to be Pcr = 29.5 N. Of interest to our subsequent reliability study was the deflection A75~ at a load level of P = 0.75Pcr. TABLE 1 LAMINATEMATERIALPROPERTIES Longitudinal elastic modulus, Eml Transverse elastic modulus, E22 =

126 GPa E33

Major Poisson's ratio in-plane, vl2 Longitudinal shear modulus, GI2 = Transverse shear modulus, G23

10.9 GPa 0.264

Gl3

5.04 GPa 3.28 GPa

Process Modelling The finite element mesh used in COMPRO consisting of 4-noded isoparametric quadrilateral elements under plane strain conditions, is shown in Figure 2. Because of the symmetry, only half of the geometry is modelled. Our experimental and numerical research has shown that the operative mechanism at the interface between the composite part and the tool plays a significant role in the final distortion of parts with various geometrical shapes. At present, our process model in COMPRO assumes a simple, linear elastic shear layer at the tool/part interface as shown in Figure 2. The properties of the shear layer material depend on many parameters, which at present are not fully characterized. These properties are currently adjusted in order to match the experimental measurements of the distorted shape. In our case, using a shear layer modulus of 5.52 kPa resulted in an accurate prediction of the deflection profile as shown in Figure 3.

Figure 2: Finite element mesh used in COMPRO

Figure 3: Comparison of COMPRO predictions of warpage (for half-length) with experiment

281

Structural Modelling Being a 2D model, COMPRO can only provide information on a 2D plane strain section of the composite part. Therefore, to obtain the 3D spatial distribution of residual stresses and deformations necessary for the structural analyses, the output from COMPRO had to be mapped onto the structural elements (plates and shells) in ABAQUS as initial conditions. To accomplish this, all six components of the process-induced residual stresses (3 stresses in x-z plane and 3 stresses in the x-y plane arising from the plane strain constraints) evaluated at the midpoint of the 2D solid elements in COMPRO were initially assigned to corresponding layers of shell elements in ABAQUS (Figure 4a,b). The initial configuration of the composite plate in ABAQUS was taken to be cylindrical with its curved geometry in the x-z plane conforming to the deformed shape predicted by COMPRO (Figure 4b). The shell elements were then allowed to deform freely (i.e. with the plane strain constraints removed) in order to equilibrate under the action of these initial stresses (Figure 4c). Independent, full-blown 3D analyses of a similar problem, a bimetallic strip subjected to thermal loading, were carried out to verify the validity of this approach.

i ~

""~f"-!

L t ' 4-4-d -'

; ~TI"4

i ~.----r-,-t"7-.. , , , t-7---'-

(a)

""

r

'

t /i

(b)

....1 ,",

(c)

Figure 4: Schematic of the procedure used to map the information from COMPRO to ABAQUS

Parametric Study A parametric study was performed to identify the most important process variables that have an effect on the warpage of the plate. The sensitivity factor, which is defined as the ratio of the percentage increase in warpage to the percentage increase in the parameter, was calculated and plotted for each parameter as shown in Figure 5. According to the chart, only 3 parameters (T, Vf, and 0) have much effect on warpage. Hence, only these 3 parameters are considered as random variables in the reliability analysis. 1.6 1.4

T - Temperature Vf - Fibre Volume Fraction 0 - Ply Misalignment HR - Heating Rate p - Pressure HT - Hold Time CR - Coolin~ Rate

t..

o

1.2 1

~ o.8 9- 0.6 9~ 0.4 0.2

T

Vf

0

HR

p

HT

CR

Figure 5: Sensitivity of warpage to changes in selected processing variables

Reliability.Based Determination of Process Control Parameters The distorted shape of the plate after processing, w, is a function of T, Vf, and 8, while the critical load, Pc,

282 is only a function of 0and Ve, which influence the material properties. Since the variables T, Vf, and 19are subject to uncertainty due to possible lack of tight process control, the outcomes w and Per are also random. As a first step, the range in these outcomes is determined by constructing a database for w = A75~ (i.e. displacement at 0.75Per) and Pcr, using COMPRO and ABAQUS in a deterministic manner, allowing T, Vf, and 0 to take values within their respective, likely ranges. These two deterministic databases can be expanded at will by increasing the number of variable combinations studied. For the purposes of this study the performance standards for the plate were defined as follows: 1) The deflection w should be within a range, from 20mm to 40mm, with high confidence (90%). The lower limit will ensure that buckling will not result in a sudden departure from the flat configuration (i.e. there would be some warning before collapse). The upper limit will ensure that the plate will not be excessively deformed. These two conditions can be described by a set of two "performance functions", Gl and G2 as follows

G~ = w(O,V/,T)- 20 G 2 = 4 0 - w(O,V/,T) 2) The applied load P should result in a sufficient reliability against buckling. This condition is expressed as a performance function G3,

o~ =5.,(o,v:)-P Given that the variables are random, the failure or non-performance probabilities correspond to the events G~ < 0,G2 < 0 and G3 < 0. Provided that the statistics of the variables are defined, these probabilities can be readily evaluated by forward reliability procedures. In the software RELAN (Foschi et al., 1999), the iterative evaluation algorithm requires the calculation of the functions G~. G2 and G3 for different combinations of the variables. This calculation is carried out by local interpolation of the deterministic database (Foschi and Li, 2000). The probability of failure in each case is obtained by importance sampling simulation around an anchor point, which is determined by using an approximate response surface and the First Order Reliability Method (FORM). In each case, the probability of failure can be expressed by the associated reliability index ,8.

Example I m

For the inverse problem, let us assume that the task is to find the design parameters V:, T and P , mean values of the corresponding variables, assuming that the remaining relevant statistics are as shown in Table 2. That is, knowing the accuracy with which the angle 0 and the variables Ve and T can be controlled, their mean values are to be found (except for O = 0.0~ Along with these, the mean of the load P that can be applied is to be determined in such a way that the plate response has the following target reliabilities for each of the three performance criteria: 1) For Gt, a failure probability of 0.05, or a target reliability index fl~ = 1.645 2) For G2, a failure probability of 0.05, or a target reliability index ,82 = 1.645 3) For G3, a failure probability of 6.2x10 3, or a target reliability index ~ = 2.500 The first two conditions imply a 90% confidence in the interval 20mm < w <40mm. The inverse reliability computer program, IRELAN, searches for the optimum solution, calculating the reliability indices for different combinations of the design parameters and minimizing the differences between the calculated

283

and target reliabilities, flj and fir (j = 1,2,3), respectively:

j=l

In each case, the reliability indices were calculated by the forward procedure previously described, using the deterministic database for w and Per. Of course, the results are not an exact achievement of the target reliabilities, but rather an optimized solution, and are shown in Table 3. TABLE 2 STATISTICS OF RANDOM VARIABLES Variable 0 Vf T P

Mean 0.00 ~ (To be found) (To be found) (To be f o u n d )

Coefficient of variation 0.50 ~ (Standard Deviation) 0.20 0.10 0.25 .

.

.

.

Distribution Normal Lognormal Lognormal Extreme Type i

TABLE 3 INVERSE RELIABILITY AND OPTIMUM DESIGN PARAMETERS, EXAMPLE 1 Calculated Optimum Design Parameters u

V/ =0.791 T = 356.89 ~ P = 16.39 N .

.

.

.

Performance Requirement G! = w - 20mm G2 = 40mm - w G3 = Pcr- P

Target Reliability Index fiT 1.645. 1.645 2.500

Achieved Reliability Index fl 1.652 1.633 2.497

.

Example 2 In a second design example, we assume that the load P is given (with corresponding statistics), and that the mean of the variables Vf and T are the design parameters that have to be calculated to meet the same performance criteria. That is, process parameters have to be found to satisfy performance under given load demands in the structural application. Let us assume that P has an Extreme Type I distribution, with a mean of 15 N and a coefficient of variation 0.25. Statistics for 0 are as shown in Table 2. The results of this second optimization are shown in Table 4. As shown, the lower load P, with the same coefficient of variation and target reliabilities, allows a slightly lower ~ and mean temperature T. TABLE 4 INVERSE RELIABILITY AND OPTIMUM DESIGN PARAMETERS, EXAMPLE 2. Calculated Optimum Design Parameters

V: =0.786 n

T

=350.50~

Performance Requirement G! = w - 20mm G2 = 40mm - w G3 =Pcr- P

Target Reliability Index/8 T 1.645 1.645 2.500

Achieved Reliability Index fl 1.632 1.646 2.783

284

CONCLUSION The methodology shown here is an effective means of linking manufacturing process control, structural analysis and evaluation of performance reliability for composite structures. Although a flat plate was used here as an application, the method is completely general and could be used with any structural form. The approach adopted here can potentially lead to a new design methodology that uses the as-manufactured rather than as-designed conditions as the basis for structural analysis leading to the reduction or elimination of uncertainties and costs associated with the use of empirical safety factors currently employed in design of composite structures.

Acknowledgements The authors would like to acknowledge the financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC). We would also like to acknowledge the significant interaction and technical support from Dr. G. Fernlund, Mr. R. Courdji, Mr. A. Osooly, Mr. G. Twigg, and Dr. L. Ilcewicz.

References Foschi, R.O. and Li, H. (1999). IRELAN: Inverse Reliability Analysis Software. Department of Civil

Engineering, The University of British Columbia, Vancouver, B.C. Canada. Foschi, R.O. and Li, H. (2000). Reliability and Performance-Based Design in Earthquake Engineering. Proceedings, Int. Conf. for Structural Safety and Reliability, ICOSSAR, Newport Beach, CA. Foschi, R.O., Folz, B., Yao, F., Li, H. and Baldwin, J. (1999). RELAN: Reliability Analysis Software.

Department of Civil Engineering, The University of British Columbia, Vancouver, B.C. Canada. Hubert P., Vaziri, R. and Poursartip A. (1999). A Two-Dimensional Flow Model for the Process Simulation of Complex Shape Composite Laminates. Int. J. Numer. Meth. in Engng, 44(1 ), 1-26. Johnston A., Vaziri R. and Poursartip A. (2001). A Plane Strain Model for Process-induced Deformation of Composites Structures. J. Composite Materials, in press. Li, H. and Foschi, R.O. 1998. An Inverse Reliability Method and Its Applications. Structural Safety, 20, 257-270. Li, Y.W., Elishakoff, I. and Starnes, J.H. (1995). Axial Buckling of Composite Cylindrical Shells with Periodic Thickness Variation. Computers and Structures, 56(1), 65-74. Shiao M. C. and Chamis C.C., (1999). Probabilistic Evaluation of Fuselage-Type Composite Structures. Probabilistic Engineering Mechanics. 14, 179-187. Twigg, G. (2001). Tool-Part Interaction in Composites Processing. M.A.Sc. Thesis, Department of

Metals & Materials Engineering, The University of British Columbia, Vancouver, B.C. Canada.

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

285

POST-FAILURE ANALYSIS OF THIN-WALLED ORTHOTROPIC STRUCTURAL MEMBERS M. Kotetko Department of Strength of Materials and Structures Technical University of L6d~,, Stefanowskiego 1/15, 90-924 Lrd~,, Poland

ABSTRACT The paper deals with the analysis of the load-carrying capacity and post-failure behaviour of thinwalled orthotropic structural members (beams and columns). The problem of post-failure behaviour is solved using the principle of virtual velocities (the kinematic method) based on the assumptions of the rigid-plastic theory modified for orthotropic materials displaying strain-hardening The structural problem is solved in the analytical-numerical way. A generalised load at the global plastic hinge is calculated in terms of generalised displacement. Resuks of numerical calculations carried out for members built mainly from fibrous composites are presented in diagrams.

KEYWORDS Thin-walled orthotropic beams and columns, composite materials, load-capacity, post-failure behaviour

INTRODUCTION With the rapid advancement of technology, the demand for stronger, lighter and tougher structural materials is growing. Among others, there are composites which are able to fulfil requirements of modern and still developing new technologies and they should be treated as materials of future. Nearly all composites display strong orthotropic properties. Also some of sheet metals made from both steel and other cristallic materials indicate certain orthotropic properties after cold-forming or rolling. These properties can be observed not only in the elastic range but in the elasto-plastic range as well. Particularly, in the plastic range sheet metals can be of strong orthotropic properties and those properties depend on the history of plastic deformation. The problem of anisotropic plastic properties of sheet metals caused by complex manufacturing processes like multistage rolling and stretching was comprehensively discussed by Szczepifiski (1993). Composite materials are widely used in thin-walled structures and display very significant orthotropic properties in the entire range of load-strain characteristics. The example of such a material is a fibrous composite with fibres situated perpendicular to each other.

286 Although the post-failure behaviour of thin-walled structures and the related problem of plastic mechanism of failure have been focused researchers attention since early 70-ties of 20th century Horne (Home 1971, Murray & Khoo 1981), the factor of orthotropy was not incorporated into the plastic mechanism analysis until now. The only exceptions are some solutions concerning mainly reinforced concrete structures by Johansen (1962) and Sawczuk & Sokr/-Supel (1993). One of the first proposals of the solution of the plastic mechanism problem when material orthotropy is taken into account was published by Kotetko (1999). The present paper is a summary of recent author's works dealing with the plastic mechanism analysis (post-failure behaviour) in different structural members made from orthotropic materials, mainly fibrous composites.

STRUCTURAL PROBLEM Subjects of investigation were following structural members under bending or eccentric compression: box-section beam, channel-section beam column under bending (both in the flange and web plane) and/or compression as well as channel-section column subject to compression. Typical plastic mechanisms of failure, which are observed in structural members mentioned above, were under investigation (Figure 1). The post-failure analysis was carried out using the basic assumptions of the rigid -plastic theory of thin-walled isotropic structures quoted by Murray & Khoo (1981) and modified by the author for the purposes of orthotropic, strain-hardening material. Thus, it was assumed that kinematically permissible true mechanisms were well developed and plastic zones were concentrated and could be regarded as stationary or travelling yield-lines of the global plastic hinge (monograph edited by Krrlak - 1995, Kotdko 1996). The rigid-perfectly plastic (Figure 2a) or rigid-plastic behaviour displaying a linear strain hardening (Figure 2b) was assumed for an orthotropic material. Material characteristics taken into consideration are simplifications of real material behaviour. Investigations of diphase fibrous composites quoted by Kotetko (2000) indicate their behaviour to be approximated by linear elastic- elasto-plastic characteristics which furthermore are simplified by rigidstrain-hardening or rigid-perfectly plastic characteristics (Figure 2). a)

b)

A

Figure 1 Typical plastic mechanisms of failure: a) - in box-section beam under pure bending, b) three-hinge mechanism in channel-section column, c) - V-shaped mechanism in channel-section beam-column

287 THEORETICAL FOUNDATIONS OF THE ANALYSIS An arbitrary mode of deformation (Szczepifiski 1990) defined by the strain rate ~ which satisfies all kinematic conditions of the problem (i.e. compatibility conditions and the requirement that the velocities on the portion Sv of the surface S should be equal to the given velocities) is called kinematicallypermissible collapse mechanism. The following relation should be fulfilled in that case

s

X i v i d S < ~ (3ijsijdV v

(1)

where Vi* is the velocity vector, c~ij* is the stress tensor corresponding to the assumed strain rate field .* s ij, Xi is the external force vector. Thus, the principle of Virtual velocities derived from (4.2) takes the following form

P'~=~cro.kP (fl, z ) d V

(2)

V

where 6 -generalised displacement at the global plastic hinge, - rate of change of the generalised displacement, P -generalised load, 13- vector of kinematic plastic mechanism parameters, X - vector of geometrical plastic mechanism parameters, ~:ff- strain rate tensor. As a result, relations between generalised load and generalised displacement at the global plastic hinge are obtained. These relations expressed in a graphical form provide us with diagrams called failure curves. An intersection point of a failure curve and a corresponding post-buckling elastic path results in an upper bound estimation of the member's load-carrying capacity. Under assumption that principal stress directions coincide with principle directions of orthotropy the adequate yield criterion is that formulated by Hill (1950). The criterion may be applied not only to sheet metals (like steel after rolling process) but to some diphase composite materials as well, particularly to fibrous composites. Its advantage consists also in limitation of anisotropic parameters to be determined in an experimental way, since Hill criterion introduces only such material parameters which can be determined in simple tensile/compression tests and pure shear test. a)

(~ fO

b)

f

I / (~ 20 I < II II /

(:$1o

(~ 20

t I

,"

///

/

Figure2: Orthotropic material behaviour: a) -rigid - perfectly plastic, b) - rigid - plastic with linear strain-hardening

288 The most convenient form of the Hill criterion is that similar to Mises yield criterion formulated for an isotropic material - 2 2 (3) ~2 = al O'x2 + a2~ + 3a3 r ~ Parameters of orthotropy ffl +~3 are determined by comparing a plastic work performed along an arbitrary direction 3' which specifies a yield-line situation - with the plastic work performed along directions x,y and a third chosen direction 0 = 45 ~ respectively. Magnitudes of these works should be of the same value for all directions. Four material elasto-plastic parameters are here necessary to be determined in the experimental way. Initial parameters of orthotropy alo + a30 can be determined from the yield condition - using four initial yield stresses (Koteqko 1999, 2000).

SOLUTION OF THE PROBLEM

For any yield-line specified by an angle ~' measured from the reference direction x=l, the plastic moment capacity m---pr corresponding to the effective stress ~ris evaluated (Kotetko 2000). A distribution of stresses at the yield-line cross-section is assumed to vary linearly from ~0v (initial yield stress) at the neutral axis to ~r at the boundary layer. The effective stress ~r is evaluated as -:

err

(Oxo +

2

cos4 7' + ~2 sin 4 Y" _ a l 2 sin 2 ?, cos 2 y, + 0.75~- 3 sin 2 (27")

(4)

Initial yield stress av0 is expressed in the similar form like ~ r i n (4) using initial parameters of orthotropy al0 + a30. Generally, the total energy of plastic deformation dissipated at the global plastic hinge takes form P

W(O) = s lj ~ ~ppdfl + ~ F k (mpp, X,,flk. ) + Z f2t (mpp, yF,fl,) j

o

k

(5)

1

where the first component denotes the energy dissipated along stationary yield lines of the length lj, F is the energy absorbed at local travelling hinges whereas f~ - the energy dissipated along nonstationary yield-lines. Details concerning the evaluation of the total energy of plastic deformation (5) and particular components of the energy dissipated at local plastic hinges are given by Kotetko (2000 and 2001). A length of any yield line li, as well as angles 13 of rotation of two adjacent walls of the global plastic hinge along stationary yield lines as well as all components of the vector Z (geometrical parameters of the plastic mechanism) have to be expressed in terms of generalised displacement 6. Thus, the total energy of plastic deformation (5) should be calculated using a numerical procedure only- by means of the incremental method - for subsequent increments of the generalised displacement AS. A generalised load is calculated using a numerical differentiation procedure of the energy with respect to the displacement 8 p= A~(8)

(6)

Two different types of structural members were taken into consideration: box-section or channelsection beams and channel-section columns. In the first case the generalised displacement is an angle

289 of relative rotation of two parts of the beam at the global plastic hinge as it is discussed by Kotetko (1999). The problem becomes more complex when considering columns under compression. Rassmussen and Hancock (1991) elaborated an axial shortening model for the plastic mechanism of channel-section column made from isotropic, rigid-perfectly plastic material. For the purposes of the present solution where the factor of orthotropy itself introduces a high level of complexity of the problem, a simplified model of column shortening has been applied. The model is presented in Figure 3 and consists in the simple trigonometric relation between the shortening 6 and the angle of rotation 0.

I .

.

.

.

p

Figure 3: Axial shortening model It is a far leading approximation, however, in the final phase of failure when a plastic mechanism is entirely developed, it gives satisfactory agreement with results of tests performed by Rasmussen and Hancock (1991). The comparison of numerical results obtained using the present

600000

.............................

400000

' ' ' 1 1 ""7".''",'-~--:--:' . . . . . . "" 7. 'W~"

200000

0.

2

; '; ; shortening

', ; ; ; ; ; ;: 8 mm

Figure 4: Comparison of the numerical and test results: flange depth a= 160, web height b= 360, wall thickness t = 5.1, length 1= 2242 [mm], ay = 375 MPa, 1 - results of the present solution (rigid- perfectly plastic material), 2 - results of test (Rasmussen and Hancock- 1991) solution for the isotropic channel-section column with the corresponding tests results by Rasmussen & Hancock are shown in Figure 4.

EXEMPLARY NUMERICAL RESULTS Selected numerical results concerning load-capacity at collapse of thin-walled beams and columns made from orthotropic materials are shown below. Numerical calculations were carried out for fibrous composites detailed data of which are quoted by Kotetko (2000). The results concerning the boxsection beam under bending obtained before and presented by Kotetko (2000,2001) are confronted with those obtained for channel-section beam-column under bending (Figures 5,6).

290 Figure 5 presents a dependence of the load-capacity at collapse (bending moment at the global plastic hinge) upon the yield stress ratio for principal directions of orthotropy 002 /O0l. The presented diagrams correspond to the box-section beam with and without strain-hardening taken into account, respectively. Analogous diagrams for channel-section beam-column subject to bending in the flange plane are shown in Figure 6. The diagrams represent load-capacity in terms of 002/o0~, for strainhardening and rigid-perfectly plastic material, respectively. The diagrams in Figui'es 5 and 6 were

Mb [kNm]

I

------Mb2

Mbl I

.....

0.25

: . . . . . . .

0.2

.0.1 s.

.

.9

. . . . .

,....~

, 0.1

1

,

~ I

,,

i

!

10

100 0"02/0"01

Figure5 Bending moment at the global plastic hinge of box-section square beam under pure bending: 1 - rigid -strain hardening material, 2 - rigid-perfectly plastic material (0 =5 ~ Flange depth a=100, wall thickness t=1.25, diaphragms distance c=87.5 [mm] Mb/t 2 [kN/mm]

I--" ".2l 14Q 120

.......................

, ..........................

.......................

i ..........................

80

---i . . . . . . . . . . . . . . . . . . . . . . . . . .

60

..i . . . . . . . . . . . . . . . . . . . . . . . . . .

40 ......

o,1

- .

......................

20

i .........................

....... i ......................... : . . . ~ . . : _ ~ _ . ~

I

1o

lOO (~02 /1301

Figure 6: Bending moment at the global plastic hinge of channel-section beam-column under pure bending (three-hinge mechanism) 91 - rigid -strain hardening material, 2 - rigid-perfectly plastic material (8 =5 ~ flange depth a=50, web height b=100, wall thickness t=2 [mm]

plotted for selected values of the orthotropy ratio corresponding to certain composite materials. Thus, the change in those coefficients is not continuous. The diagram in Figure 7 shows an influence of

291 continuously varying orthotropy ratio ExP/Eyp upon the load-capacity at collapse for the channel-section column under compression. Calculations were conducted for a hypothetical set of materials: I

Table 1 E45,p = Exp O45* = O10

yp = 500 Mpa =const. 9 o20 = 40 Mpa = const. ouu,i = 1.25 oi0 (i =1,2)

Gxyp = 0.45 Eyp ZOO2 = 0.6cr10

E p and o01 change continuously

FINAL REMARKS The elaborated analytical-numerical solution of the problem of plastic mechanisms in thin-walled orthotropic structural members enables, in a relatively simple way, to analyse the failure and to estimate a load-carrying capacity of a structure approximately. Results of numerical calculations

P IkN! .........

?r~ . . . . . . . . . . . . . . . . . . . . . . . . . . .

........

?~..

', . . . . . . . .

lr~..

', . . . . . . . .

1o~..

B

',

o.1

so

P

1

lO

lOO

ExP/Eyp Figure 7: Load-capacity at collapse of the channel-section column. Three-hinge mechanismshortening 15= O. lmm. Flange depth a=50, web height b=100, wall thickness t=2 [mm]. provide us with several indications concerning an influence of material orthotropic properties upon the collapse behaviour and load-carrying capacity of thin-walled orthotropic members. The diagrams presented in Figure 5 are nearly symmetrical with respect to the ordinate axis in the certain range of the orthotropy ratio. It means that an orientation of composite's fibres (parallel or perpendicular to the beam axis) does influence the beam load-capacity at collapse. In the case of threehinge mechanism (Figure 6) in the channel-section beam-column we do not observe such a symmetry. In the first case an energy absorbed along yield-lines inclined at certain angles to principal directions of orthotropy plays a predominant part in the whole amount of the total dissipated energy. On the other extreme, the energy dissipated at yield-lines parallel to principal directions of orthotropy is the greatest part of the total energy of deformation when considering the three-hinge mechanism (Figure l b) developed in the channel-section beam-column. Results of calculations give some interesting information about an influence of the strain-hardening phenomenon upon the load-capacity at collapse. Comparing Figures 5 and 6 one can notice that this influence is different for different plastic mechanisms and also for different orientation of fibres with respect to principal stress directions. However it should underlined here that substantially more detailed material data and experimental verification of theoretical results would be of aid in the estimation of this influence.

292 The results indicate a possibility of an optimisation of fibres orientation in composites or, generally, of orthotropic properties in thin-walled structural members. Also here an experimental verification of the obtained results would be desirable.

REFERENCES

Hill R. (1950) Mathematical theory of plasticity. Oxford University Press Home M.R. (1971) Plastic Theory of Structures. Thomas Nelson &Sons Ltd. Johansen K.W. (1962) YieM #ne theory. Cement & Concrete Association, London Kotetko M. (1999) Collapse behaviour of thin-walled orthotropic beams. Light-weight steel and aluminium structures (Proc. of Fourth lnt. Conference on Steel and Aluminium Structures 1CSAS'99), Elsevier, 99-106 Kotelko M. (2000) Plastic mechanisms of failure in thin-walled girders with iso- and orthotropic walls. ( in Polish) Scientific Bull. of L6di Technical University, Nr 844, Ser. Transactions Nr 273, L6d2 Kotdko M. (2001) Ultimate load and post-failure behaviour of thin-walled orthotropic beams, to be published Kotetko M. (1996) Ultimate load and post-failure behaviour of box-section beams under pure bending, Engineering Transactions, 44:2, PWN- Warszawa, 229-251. Murray N.W.,Khoo P.S.(1981) Some basic plastic mechanismsm in the local buckling of thin-walled steel structures, Int. J.Mech. Sci., 23:12, 703-713. Rassmussen K.J.R., Hancocok G.J.(1991) Nonlinear analyses of thin-walled channel-section columns, Thin-Walled Struct., 13:1,2, 145 -176 Sawczuk A., Sok6t-Supel J.(1993) Limit analysis ofplates. PWN Warszawa SzczepifiskiW. (1993) On deformation induced plastic anisotropy of sheet metals. Arch.Mech .45:1, PWN- Warszawa, 3-38 Szczepifiski W., Szlagowski J.(1990) Plastic design of complex shape structures. PWN- Warszawa, Ellis Horword Publ. Ltd

Stability, post-buckling and load-carrying capacity of thin-walled structures with plane orthotropic walls (1995) Ed. by M. Kr61ak,, L6d~. Technical University, ser. Monographs (in Polish)

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

293

MODAL COUPLED INSTABILITIES OF THIN-WALLED COMPOSITE PLATE AND SHELL STRUCTURES M. Krrlak, Z. Kotakowski and M. Kotetko Department of Strength of Materials and Structures, Technical University of L6d~ PL-90-924 L6d~., ul. Stefanowskiego 1/15, Poland

ABSTRACT The problem of buckling and initial post-buckling equilibrium paths of thin-walled structures built of plate and/or shell elements subjected to compression and bending has been solved. Plate and shell elements can be made of multi-layer orthotropic materials. A method of the modal solution to the coupled buckling problem within the first order approximation of Koiter's asymptotic theory, using the transition matrix method, has been presented.

KEYWORDS Composite structures, membrane -flexural coupling, buckling, coupled instabilities.

INTRODUCTION

In the last twenty years numerous studies devoted to stability, post-buckling behaviour and load carrying capacity of thin-walled girders with fiat walls and of shells, including especially the cylindrical ones, have been published. The majority of publications dealing with fiat-walled girders has been discussed or quoted in two collective works by Krolak (1990, 1995). Although girders (beam-columns) built of plates exhibit some advantages, such as, for instance, easiness of construction or an ability of safe operation under the elastic post-buckling after a local loss of stability, their basic disadvantage is a low value of the critical load, particularly for compressed walls of a relatively great width. A value of the critical load in thinwalled beam-columns with fiat walls can be increased through an application of orthotropic materials (including the composite ones) characterised by a high modulus of elasticity along the direction of compression. On the other hand, in many mechanical structures, especially in cars, wagons, ships, aeroplanes or spaceships, thin-walled carrying elements of cylindrical shapes are used. These elements can often be modelled with a system of plates and/or segments of cylindrical shells.

294 FORMULATION OF THE P R O B L E M Let us consider thin-walled girders built of plate and shell elements (in the form of cylindrical shell segments) with closed or open cross-sections. Both the plate and shell elements of the girders under consideration can be multi-layer walls made of orthotropic materials. The materials they are made of are subject to Hooke's law. The classical laminated panel theory (Jones 1975) is used in the theoretical analysis and the effects of shear deformation through the thickness of the laminate are neglected and the results given are those for thin laminated panels. Long prismatic thin-walled structures built of panels connected on longitudinal edges have been considered. In order to account for all modes of global, local and coupled buckling, a plate model of thin-walled structures has been assumed. Equations of stability of thin-walled structures have been derived using a variational method (Dawe &Wang (1994), Kolakowski & Krolak (1995)). The total potential energy variation of a multi-layer sector of the panel H can be written as: b ~I-I = ~ U - ~ W

=

(~x~E; x + l~y~l~y + qTxy~xy )

d~-{ t p~

dy x--0}

(1)

0

where: U - internal elastic strain energy, W - work of external forces, p~ (y). pre-buckling external load in the shell middle surface. The expression f2 = g x b x t = S x t has been employed in the above relation. The above equation says that in the equilibrium state the potential energy FI of the panel has a nonvarying value in the class of permissible displacement variations. Hence, the equation has to be satisfied for all permissible virtual displacements that comply with the imposed constraints. For each panel component, precise geometrical relationships are assumed in order to enable the consideration of both out-of-plane and in-plane bending of the plate: I~1 = U l , 1 + 0 . 5 e 2 = U2, 2 + 0 . 5 I~4 -- --tu3,11

Urn,lUre, 1

1~3-'Ul, 2 +U2,1 +Urn,lUre, 2 + k u 2 u 3 A

Um,2Um, 2 --

ku 3 + 0.5k 2U 2

I~ 5 = - - t u 3 , 2 2

--

+ ku2u3, 2

tku2, 2

I~6 =

(2)

-2tu3a 2 - tku2, i

where: k = 1/R is the curvature of the cylindrical shell segment; ul-u, u2-=v, U3~W, -- the components of the displacement vector in the Xl---x, x2-y, x3-z axis direction, respectively, and e~=ex, e2=ey, e3=2exy=Txy, e4----tKx,e,5----tKy, E;6='tKxy. The summation with respect to the factor m is from 1 to 3 (m = 1, 2,3). Using the classical plate theory (Jones (1975)), the constitutive equation for the laminate is taken as follows:

{N}= [K]{~}

(3)

in which Kij = Kji, where i, j= 1,2,... 6. In the above equations, N~, N2, N3 are the dimensionless sectional forces; N4, Ns, N6 - the dimensionless sectional moments. The reverse relation with respect to (3) can be written as:

{4: [K]-'N= [k]{N} In order to determine the variation of potential energy of a single multi-layer panel, the following identity X 6Y = 6( XV ) - 6X. (5)

295 has been employed. After grouping the components at respective variations, the following system of equations has been obtained: f{[Nl(1 + u u ) + N3ua,2],l +[N=ul, 2 + N3(1 + uu)],2} 6u I dS=0 s

f {[N lu2,1 + N 3(1 + u:,:) - ktN 6 ],1 -t- IN 2 (1 + u2,:) + N 3u2,1 -ktN 5],2 s -k2N2u2-kN2u3,2-kNau3,1]

fs

[(tN4,1 + N~u3,~ + N 3 u 3 , 2 +

(6)

6u 2 d S = 0

kN3u2)~ + (tNs,2 + 2tN6, ] + N2u3, 2 + N3u3,1 + kN2u2),2

+kN2] ~Su3 dS=0 f[l~ i - k i j N j ) ] 6N i d S = 0

(7)

where: i,j=l,2 .... 6 and no summation to 'i'

s

f o r xl =const b

f[N, + NIul,l + N3u,, 2 -tp~

6u 1 dX21xl = 0

0

b

f[N 3 + N,u=, I + N3u2, 2 - ktN6] 5u 2 dx=lxl = 0 o

(8)

b fN4 0

~u3,1 dx21xl = 0

b

(tN4,, + 2tN6,~ + N,u3,1 + N3u3,2 + kNsu2) 8u 3 dx

_l ' = o

o

f o r x2 =const l

J'[N2 + NEU2,: + N3u:, , - ktNs] 6u: dx,[~2 = 0 o

f[N 3 + N2u2,: + N3uu] 5u~ dx~J,,2 = 0 o

(9)

t f N 5 5u3,2 dXl[x2 = 0 0 l

j(tN ,: +

+

+

+

dx,

0

0

for the plate corner, that is to say, for xl=const and x2=const 2tN6lxl Ix2 8u 3 = 0

(10)

Relation (6) is a system of equilibrium equations. Systems (7) are the already employed relations between deformations and external forces, whereas relations (8), (9) and (10) correspond to the boundary conditions at the edge x~=const, Xz=const, respectively, and in the plate corner.

296 An introduction of the full functional into the solution allows for satisfying the subsequent and consistent systems of simplified mathematical models corresponding to the characteristic types of composite panels.

SOLUTION TO THE PROBLEM

The problem has been solved by the variational method using Koiter's asymptotic theory of conservative systems (Koiter 1976). In the solution of the problem and in the computer program developed, the following have been employed Byskov-Hutchinson's asymptotic expansion (Byskov & Hutchinson 1977), the numerical transition matrix method using Godunov's orthogonalization method (Kolakowski & Krolak 1995). The girders under consideration can be loaded with axial compression and bending in the cross-section symmetry plane, that is to say, with the so-called flat bending. The aim of this study is to generate the stability equations and to solve the stability problem (to find the values of the critical load of local and global buckling), as well as to determine the initial equilibrium paths in the elastic post-buckling for the girders described above. As has been mentioned above, after expanding the fields of displacements U k and the fields of sectional forces N k into power series with respect to the buckling mode amplitudes ~ (the amplitude of the n-th buckling mode divided by the thickness t of the wall assumed to be the first one), Koiter's asymptotic theory has been employed: Uk = ~ ( o ) + ~n~(n) + ...

~

= ~o~

+ ~ o ~ +...

(11)

where U~k~ , ~0) are the pre-buckling state fields, and U~kn), Nr n5 -the n-th buckling mode fields. After substitution of expansions (11) into equilibrium equations (6), continuity conditions (9) and boundary conditions (8) (corresponding to the free support at the segment ends), the boundary problem of the zero and first order for the case of uniform compression along the generating lines of the panel has been obtained. In the case of the load that varies along the girder perimeter in the pre-buckling state (e.g. at bending), such a load can be modelled with a step-like varying load (constant in individual strips the girder is divided into). The zero approximation describes the pre-buckling state, whereas the first order approximation, being the linear problem of stability, allows for determination of values of critical loads, buckling modes, and initial post-buckling equilibrium paths. The pre-buckling solution of the k-th orthotropic panel consisting of homogeneous fields is assumed as: .(0~ =0 where Ak is the actual loading. This ulk" (0~ = ( g / 2 - xak)Ak, U2k(0~= X2kAkK12k/K~2 k and U3k loading is specified as the product of a unit loading system and a scalar load factor Ak. Numerical aspects of the problem being solved for the first order fields (like in paper by Kolakowski & Krolak (1995)) resulted in an introduction of the following new orthogonal functions for k-th panel in the sense of the boundary conditions for two longitudinal edges (9):

297 a)q(o)Z(n)/b ~,N~k %,~ /bk

~" n) _. Ulk" (n) _ u(n)

3k,~1

= . (n) U3k •k(n)

n) = U 2 1 k

/b

t kN (n)

(o)w(n)

b ~ = N(n)(1-)~Ai) + ~ , ~ k

~(k n)

- k

=(n)/b k -- ~k,'q

k

(12)

"-- " ' S k (o)~(n)

(n)_

~n) = "~S~,n' =(") /bk + 2t k'~(n)'6~,r k + ~.N~k Ck,r /b k + kk ~.AkN2~ 'lk bkK=,k/K2: k where ~ = x 1 /b and 11= x2 /b. In the case of arbitrary composite structures, the solution of the problem within the first order approximation can be sought in the form of trigonometric series in the light of inconsistency of these functions in the equilibrium equations along the longitudinal direction or in an approximate way. In the majority of studies, only conditions of orthogonality (6) of the assumed class of functions of the problem solution with respect to displacements u--u~, v=u2, w=u3 are fulfilled. In the solution method presented here, additional orthogonality conditions (i.e. (7) and (9)) are satisfied. A separation of variables in the linear boundary problem has been solved by means of Kantorowich's method. The first order solutions may be formulated as follows: ~(~") = X(~n)(rl,) sin m u b ~

~(n) = ~(.) -.~

mub.~

~n) = ~(~)

~(n) _ ~V(kn ) ( I l k ) ~k

COS

(rl~)

mnb.~ e

c o s ~

(~k) sin

mubk~ (13)

E(n) E~)(n ) sin mnbk~"

~(n) = ~k(n)

~k(n) = G(kn)(rlk) sin mubk~

~n) = ~n)(rlk ) sin mltbk~

k =

k

t

*k

(rlk) sin

m~:bk~

e

Ak--(n)' B~"), P(")--'k, ~n), ~n) , V(")" k , G~), H~) (with the m-th harmonic) are initially unknown functions that will be determined by the numerical method of transition matrices. The solution assumed in this way (13) allows one to determine dimensionless sectional forces for the first order approximation in the form: NI~) = ~(") "',k (~k) sin mubk~

N~,) = N("):,k(rlk) sin m~:bk~

N3k( n= ')'~(n) 3 k ('rlk)

mTtbk~ ~?

m~:b N~) = -~(n) , 4~k(rlk) sin ~ k e

~(n) mTtbk~ =-.sk (rlk) sin

~(n) mubk~ N~ ) "-''6k (Ilk) C O S ~

N~)

cos

(14)

The presented way of solution allows for carrying out a modal analysis of buckling of complex composite thin-walled structures. The obtained system of homogeneous ordinary differential equations, with the corresponding conditions of the co-operation of walls, has been solved by the transition matrix method, having integrated numerically the equilibrium equations along the circumferential direction in order to obtain

298 the relationships between the state vectors on two longitudinal edges. During the integration of the equations, Godunov's orthogonalization method is employed. The column global buckling occurs at one sinusoid half-wave on the column length, whereas the local buckling takes place at the number of half-waves m > 1 (with bk<< e ). The developed computer program allows for a division of each plate or/and shell element into several or even more than 30 strips made of different materials and with various wall thickness. The presented solution method enables a multi-modal analysis of buckling. A detailed description of the solution method of the problem under discussion, analogous as in the case of plate structures, has been included in Kefs by Kolakowski and Krolak (1995), Krolak and Kolakowski (1995).

ANALYSIS OF THE CALCULATION RESULTS The developed calculation program has been tested in many aspects, among others in order to determine the range of applicability of the cylindrical shell theory assumed in the considerations. Below, the results of calculations for thin-walled beam-columns with the cross-sections under compression and bending are presented in Figure 1.

f _

----I

----- 7--. --.

1~

I

0

Figure 1: Type of considered cross-section

An influence of the curvature radius R of the flange (or of the cylindrical shell segment rise) on the global and local critical load Nor [N/m] of the column with the length L = 2000 mm, made of a threelayer composite of the following structure of the layers: 9 t~ =0.2 ram, E=200 GPa, v=0.3, Ey=E, G=E/(2(I+v)) 9 t2 =0.6 mm, E =139.3 GPa, v=0.3, Ey =11.1 GPa, G=6 GPa (Shen and Williams (1993)) 9 h =0.2 ram, E=200 GPa, v=0.3, Ey=E, G=E/(2(I+v)) (where t = t I + t2 + t 3 ) has been analysed. The curves in Figure 2 denote, respectively: 1 - global flexural-torsional buckling, 2 - local antisymmetric buckling, 3 - global flexural buckling, 4 - local symmetric buckling.

299 No, [Nm] l

it,

-

A__J,.~.

9 ~iL Imm i - I

Et

. - -

~=

IJl*l[

~, -

-

i

=J r - ~ , A . - - - k i

I

,. i

0,6 0,5 0,4

' -0,5

-0,3

"~""~e"

-0,1

"

" "" "'" 0,1

bl/R 0,3

0,5

Figure 2 Critical load Ner versus b~/R The negative dimensionless curvature b~/R < 0 (bl = 100 mm) corresponds to the flange deflected towards the inside of the column contour, whereas bl/R > 0 - to the flange deflected towards the outside of the column contour. The value of the critical load corresponding to the global flexural-torsional buckling decreases with an increase in the curvature b~/R by nearly 1.5 times. The value of the global flexural critical load undergoes slight changes. A similar situation occurs for the local antisymmetric buckling. The shell element exerts the strongest influence on the local symmetric buckling. Even a slight flange curvature results in a rapid increase in the critical load. The lowest value of the critical load of the beam-column under analysis occurs for: local symmetric buckling mode for -0.5 < b~/R ___0.25; local antisymmetric buckling mode for b~/R = 0.5. The further analysis has concerned the interactive buckling and has been aimed at determination of the sensitivity of the columns under discussion to global and local geometrical imperfections. It has been assumed that the beam-column has the following initial deflections ~g~ = I1.01, ~-10.21. In Table 1 the theoretical load-carrying capacity N, [Nm], the ratios of the theoretical load-carrying capacity, determined within the first order non-linear approximation, to the minimum critical load value, that is to say, NdNm, where Nm = min(Ng, N0, are presented. In each case the sign of the imperfection has been chosen in the most unfavourable value, i.e. so that N, would have its minimum value (for more a detailed analysis see Refs. by Kolakowski and Krolak (1995), Krolak and Kolakowski (1995)). In Table 1 the ratios of the global critical load to the local one, for the same conditions along the symmetry axis of the cross-section, i.e. the ratios of the critical loads: global flexural-torsional (antisymmetric) to local antisymmetric; global flexural (symmetric) to local symmetric, are included as well. TABLE 1 LOAD-CARRYING CAPACITY WITHIN THE FIRST ORDER APPROXIMATION

bl/R -0.5 0.0 0.5

Antisymmetric modes N, [Nm] NdNm Ng/Nl 0.139 0.784 1.949 0.115 0.766 1.626 0.115 0.762 1.344

N, [him],, 0.138 0.0621 0.133

Symmetric modes N~qm 0.858 0.797 0.858

Ng/Nl 4.077 9.085 4.423

300 The interaction of the global flexural buckling mode with the local symmetric one has turned out to be the most dangerous for -0.5 _
CONCLUSIONS The proposed solution method of the problem enables one to analyse the global and local buckling of plate and shell structures, to find out the ranges of geometry variability for the shell element theory assumed and to determine the theoretical limit load carrying capacity within the first order approximation. In the case when the assumed shell theory cannot be used to analyse the plate and shell structure, then the structure buckling can be analysed by replacing shell elements with a great number of plate strips.

REFERENCES

Byskov E. and Hutchinson J.W. (1977). Mode interaction in axially stiffened cylindrical shells. AIAA J., 15: 7, pp. 941-948. Dawe D. J. and Wang S. (1994). Buckling of composite plates and plate structures using the spline finite strip method. Composites Engineering, 4:11, pp. 1099-1117. Jones R. M. (1975). Mechanics of composite materials. International Student Edition, McGraw-Hill Kogakusha, Ltd., Tokyo. Koiter W. T.(1976). General theory of mode interaction in stiffened plate and shell structures. WTHD, Report 590, Delt~. Kotakowski Z. and Kr61ak M. (1995). Interactive elastic buckling of thin-walled closed orthotropic beam-columns. Engineering Transactions, 43: 4, pp.571-590. Krrlak M. (ed.) (1990). Post-buckling behaviour and load carrying capacity of thin-walled plate girders. PWN/Polish Scientific Publishers/in Polish/. Krrlak M. (ed.) (1995). Stabifity, post-critical behaviour and load carrying capacity of thin walled structures withflat orthotropic walls. Technical University of L6d2 Publishers/in Polish/. Kr61ak M. and Kotakowski Z. (1995). Interactive elastic buckling of thin-walled open orthotropic beam-columns. Engineering Transactions, 43: 4, pp.591-602.

Third Intemational Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

301

INDUCED STRAIN ACTUATION AND ITS APPLICATION TO BUCKLING CONTROL IN SMART COMPOSITE STRUCTURES J L o u g h l a n and S P Thompson Cranfield University, College of Aeronautics, Cranfield, Bedfordshire, MK43 OAL, UK

ABSTRACT In this paper experimental tests are described and discussed which illustrate the feasibility of buckling control in composite structural elements using induced strain actuation in a smart technological manner. Compressive tests on simply supported square composite plates which utilise the shape memory effect for buckling control are shown to exhibit substantially reduced post-buckling deflections when under activated control in comparison to those experienced for the uncontrolled case. The alleviation of the post-buckling deflections is shown to result in reduced non-linear stress levels in the post-buckling range and thus it is intimated that the ultimate failure levels of the composite plates can be improved through the application of shape memory control. Particular attention is paid in the paper to tests carried out to ascertain the characteristic behaviour of the Nickel-Titanimn shape memory material employed for actuation purposes. The cyclic recovery force capabilities of the actuator wires utilised in the compressive plate tests is highlighted and a detailed account of the d,,qemfination of the alloys characteristictransformationtemperaturesis given. KEYWORDS

Buckling, Post-buckling, Smart structures, Shape memory alloy, Actuation forces, Nickel-Titanium alloy, Composite plates, Adaptive control

INTRODUCTION

Buckling in thin-walled composite construction can be caused by the de-stabilising loading actions of compression and/or shear. Buckling manifests itself by the structure losing its geometrical shal~ at the critical load levels. In the case of buckling control the aim is to maintain the original structm'al shape in order to avoid buckling and the adverse affects of post-buckling behaviour. In doing this, we are able to improve the load carrying capability of the structure. Buckling control can be implemented through the use of induced strain actuation. The shape memory effect or the piezoelectric effect can be employed for the purpose of buckling control. Investigations into structural control applications utilising induced

302 strain actuation mechanisms have largely focussed on vibration and noise suppression and to a lesser degree on altering structural shapes for improved performance. Less attention has been paid in the literature to the application of smart materials for buckling control although the authors feel there are considerable potential benefits to be gained in this area. Being able to control the influence of imperfections and the adverse effects of post-buckling behaviour would, of course, allow lighter and stronger thin-walled structta~ than would otherwise be feasible. Meressi and Paden [1] employed thin film piezopolymer actuators and strain gauge sensors in their control study of a flexible beam whereby it was shown that the buckling of a simply supported beam could be improved beyond the first critical response level. The work of Thompson and Loughlan [2] illustrates, practically, the feasibility of improving the non-linear behaviour of imperfect composite column strips through the use of piezo~c actuators. The actuators used were encapsulated in E-glass/epoxy to provide electrical insulation and to make the PZT actuator more robust for handling. Doran and Butler [3] have examined the static actuation response of the encapsulated PZT material and Roberts et al [4] have employed such encapsulated PZT ceramics for use in their work on active structural damping. When activated, the actuators impart a localised reactive moment at the column centre. This serves to counteract the influence of the buckling deflections during loading and considerable improvements on load carrying capability are demonstrated through active control. The lateral buckling behaviour of composite beams has been studied by Baz and Chen [5] who utilised SMA wires for the active control of the beams under load. The wires were embedded symmetrically along the mid-plane of the beams and these were then constrained to an external boundary. This approach prevents contraction of the wires upon activation and using this strategy it is shown that buckled beams could be brought back completely to their unbuckled configuration with the appropriate level of SMA activation. The post-buckling response of carbon fibre composite plates employing SMA actuators has been considered by Thompson and Loughlan [6] using a control strategy similar in nature to that of Baz and Chen [5] for composite bemns. Different levels of actuator volume fraction were investigated and the importance of the thermal influence, on structural performance, from ohmic heating is highlighted. The adaptive control of the plates, though not fully complete, is shown to result in considerable reductions in the post-buckling displacements and corresponding stress levels. In this paper the authors discuss this work further and pay particular attention to the operating mechanics of the Nickel-Titanium actuators highlighting the complexities of their Thermomechanical response.

SQUARE COMPOSITE PLATE TESTS

Shape Memory Alloy, SMA, wires can be readily embedded in composite construction for actuation purposes. Nickel-Titanium, Ni-Ti, SMA wires 0.4 mm in diameter have been employed to facilitate the compressive buckling control of the square composite plates during test. In general, SMA's, when heated above a certain temperature, have the unique ability to regain al~arent plastic deformation. If this deformation is prevented then, of course, internal stresses will be generated and these will be equilibrated by the imposed boundary constraints. Actuation control using the shape memory effect is accomplished for the composite plate test ~ i m e n s by pre-straining the embedded Ni-Ti SMA wires and by providing fLxed boundary constraints at the wire ends. Upon actuation, through resistive heating, restoration forces are then set up in the wires and these are then gainfully employed in enhancing the post-buckling performance of the laminated test plates. It is shown that, utilising the considerable control authority generated, even for a small actuator volume fraction, the out-of-plane displacements of the post-buckled laminates can be considerably reduced. The control strategy employed is illustrated schematically in Figure 1, SMA actuators within sleeves, run throughout and along the neutral plane of the laminated plates and perpendicular to the applied compressive loading direction. The strategy necessitates external actuator boundary constraintS, thereby, upon act/vation, preventing free shape memory recovery. Results have been obtained that illustrate the adaptive capability of the control strategy for different alloy compositions and actuator

303

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Figure 1. Schematic Representation of Control Strategy. volume fractions and these have been reported in reference [6]. In the present p ~ details are given which highlight the complex nature of the cyclic response of the Ni-Ti, SMA actuators. This serves to provide a more in-depth understanding of the shape memory phenomenon and of the adaptive control mechanics of the composite laminated plates. The stacking sequences of the laminated plates considered are eight ply systems of [0Y90~]., [0/:1:45/90], & [02/+45],. The dimensions of the plates under investigation are 300 by 310 mm with a 1.0 mm nominal thickness. All specimens are constructed from Fiberdux 913C-TS unidirectional high tensile carbon/epoxy prepreg. For the control strategy employed, the specimens feature SMA actuators, 0.4 mm in diameter, initially pre-strained by 6% and fed through PTFE tubes. Two different alloy compositions were considered for the actuators, these being the near equiatomic compositions of 49.8-50.0 at.% Ni and 50.0-50.2 at.% Ni respectively. Although 0.4 mm diameter actuators were employed for the composite plate tests, the performance of 0.3 mm diameter wires for this application was also investigated. THE SHAPE MEMORY ALLOY TRANSFORMATION PHENOMENON

In general, there are four important characteristic transition temperatures for a SMA. In the stress-free state, the austenite-to-martensite (A-,M) transformation begins at a temperature denoted by M. (martensite start) and ends at the lower temperature Mf (martensite f'mish). The reverse transformation from martensite-to-austenite (M-,A) begins at the temperature A. (austenite start) and ends at a higher temperature Ar (austenite finish). The two phases can coexist at intermediate temperatures.

304 The transformation process is illustrated in Figure 2, [7]. If an annealed material is cooled from At,

phase 1, to Mr, phase 2, while under no applied stress, a self-accommodating twinned martensitic structure develops. The ~ i t e variants are all ct3,st~ographically and energetically equivalent, they differ only in orientation, [8]. Generally, the crystal structure of martensite is relatively less symmetric compared to that of the parent phase, [9].

Phase I -

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Figure 2. Schematic illustration of the phases within a SMA. In the martensitic phase, Ni-Ti is very ductile and capable of sustaining large amounts of deformation. With application of stress, martensite twins become aligned. Such alignment, forming detwinned martensite, phase 3, is attributed to shear-type displacements within the crystal lattice, [10]. It results with the considerable apparent plastic strain in the material. Atoms have shifted without dislocating within the lattice structure. Such dislocations are common to metals in general at high strain levels. When sufficient thermal stimulation is applied to heat the material above the phase transition temperature, and in the absence of an external constraint or an applied load, the lattice shifts to the more rigid, ordered cubic structure, therefore, returning the specimen to its undeformed geometry, phase 1. Providing the alloy has had no prior two-way training, re-cooling reproduces the original twinned martensitic shape, phase 2.

TYPICAL STRESS-STRAIN BEHAVIOUR A typical stress-strain curve, for the 50.0-50.2 at.% Ni alloy, 0.4ram in diameter, is shown in Figure 3. The specimen was loaded at room temperature. The curve iUu.qrat~ is typical for Ni-Ti she~ memory alloys tested at temperatures below A,. It is clearly evident that the curve may be broken down into a series of paths. During the initial loading stage, path 1, the specimen exhibits linear elastic behaviour of the product phase, though a small amount of the parent phase may be present. Such linear elastic behaviour ceases at the first yield point. Note, Ni-Ti, and SMA's of like, exhibit two-stage yielding, [11]. The former yield point depends on the test temperature. The next loading process is accompanied with a reorientation of the martensite variants, path 2. During this phase, stress is nearly constant while the strain increases. Although it appears that there has been permanent plastic deformation, there has been no permanent damage on a microstmctural scale. As long as the specimen is not significantly loaded above the recovery strain limit, which, for this heat-treated alloy composition, is approximately 6%, then, the strain induced is recoverable.

305

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Figure 3. Characteristic stress-strain response for the 50.0-50.2 at.% Ni alloy. Once the martensite has become completely aligned in the preferred orientation, i.e. in the specimen axial direction, the specimen's response to a further increase in load is, once again, elastic, path 3. The strain level at which the material becomes completely aligned is c~dled the domain width and represents the maximum amount of recoverable strain from the material by heating, [ 12]. The continued applic~on of load will result with a load path tending towards the second yield point. It is the second yielding which is associated with the true plastic deformation of the martensite. Continued loading will lead to eventual specimen failure. Upon unloading, path 4, prior to arrival at the second yield point, the residual strain can be recovered by heating the material back to its austenitic phase. As introduced earlier this behaviour is termed the shape memory effect.

NI-TI TRANSFORMATION TEMPERATURES Calculation of the appropriate power level necessary to transform the SMA actuators into complete austenite, therefore, inducing maximum recovery stress, requires the experimental determination of the phase transformation temperatures. The most important transformation temperature being At. Such experimentally determined temperatures can then be inserted into shape memory alloy constitutive equations, along with other experimentally determined material properties, to predict the shape recovery and thus structural response, or simply enable the prediction of an optimal current level necessary for e

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The phase transformation temperatures for both the nickel-titanium alloy compositions considered have been determined. Both the austenite and the martensite transformation temperatures were determined by differential u:mming calorimetry (DSC). This technique measures the rate of heat absorption or emission from a specimen as its temperature is raised or lowered. Any transition will appear as a step

306 change in the heat absorption versus temperature curve. Such a change indicating the difference in the specimens heat capacity. For all the specimens, DSC measurements were performed on a mass of a few milliganm. The DSC curves were recorded in a temperature range from -30~ to 120~ with a heating and cooling rate of 10~ This rate was a common value within published literature and, therefore, was selected for this study. To record the austenite transformation temperature, the sample was first cooled to -30~ within the DSC al~aratus and then heated to 120~ The sample was then cooled back down to -30~ to record the martensite transformation. Transformation temperature measurements were recorded for several samples of each alloy, therefore, providing evidence of consistency in the material behaviour. Typical DSC curves for the 0.4mm diameter samples, with alloy composition 49.8-50.0 aL% Ni, are shown in Figure 4. The upper curve shows the endothermic phase transition to austenite and the lower curve shows the exothermic phase transition to martensite. The endothermic phase transition to austenite is characterised by a single peak, whereas the exothermic phase transition to martensite, upon cooling, is characterised by two peaks, the first being attributed to the formation of R-phase (rhombohedral) martensite, the second peak occurs during the reversion to complete martensite.

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Figure 4. Transformation behaviour of a 49.8-50.0 at.% Ni alloy, 0.4ram in diameter. All peaks initiate at the phase transition start temperature and terminate at the phase transition finish temperature. The formation of the intermediate rhombohedral martensite is attributed to the processing conditions. It has no significant effect on this study and was, therefore, with the exception of cyclic operations, not thoroughly investigated. The phase transitions are characterised as being athermal, the transition proceeds only with a change in temperature. If the temperature is held constant, the transition will not proceed with time. All measured transition temperatures are listed in Table 1. Tylg~cripts M, R and A refer to Martensite, Rhombohedral and Austenite respectively. Subscripts s and f refer to the start and fmish temperatures respectively. As seen in the table, stress free nickel-titanium, with a nominal composition of 49.8-50.0 at.% Ni, requires a temperature in excess of 68~ for complete phase transition.

307 TABLE 1 PHASE TRANSITION TEMPERATURES

It will be of note that, a slight increase in the nickel content substantially lowers the characteristic phase transition temperatures. This is important, as the lower actuation temperatures require a lower power stimulus and this will subject the host laminates to a more acceptable thermal environment. In actual strucUa~ applications the transformation temperatures are assumed to be influenced by stress, o, according to the following linear relationships; M, = M,(o--0) + o'/CM

M f - M,(o--0) + o'/CM

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Af = At(o--0) + o/CA

(1)

The constants CA & C, are stress influence coefficients, [11]. It was seen within the published literature that C^ varies between 4.5-~13.8 MPa/~ and CM varies between 7.0~11.3 MPa/~ It is often assumed, however, that C^ & Cu are equal to one another and have a continuously constant value over a temperature range. As an example, employing a stress influence coefficient of 10 MPa/~ will raise a particular transformation temperature by 1~ for each 10 MPa of axial stress applied to the specimen. The stress free transformation temperatures in equations (1) are clearly those associated with (or--0).

DETERMINATION OF NI-TI SHAPE MEMORY RECOVERY FORCES The ability of the actuating material to operate within the host composite is, amongst others, dependent on the integrity of the SMA. In order to detenmne the true usability of the actuators, the repeatable range of recovery stress that can be generated needs to be assessed. To gain a better understanding of the forces associated with the shape memory effect, tensile recovery stress characteristics of SMA specimens have been determined. Each specimen tested was first uniaxially elongated a predetermined amount, i.e. 2-6%, by loading it in tension, therefore, producing a stress biased martensite microstructure, F igme 3. Testing was conducted in an Instron testing machine. The samples were clamped at both ends, one end bolted to a load cell, the other to a fixed crosshead. This configuration would prevent contraction upon heating the deformed specimens. The wire actuators were heated by a DC power source. Such an electrical power source necessitates suitable insulation from the testing machine. The recovery force, the electrical input and the associated temperature with time were monitored and recorded at a frequency of 10Hz. Each sample was subjected to a number of ~ i n g cycles. A typical operating cycle is defined, via input of a step DC power source, as the heating of the actuator to reach its peak recovery stress. Such a peak recovery stress will be held for a short period of time, and then allowed to cool back to the ambient condition. Cool down is achieved by removal of the DC power source. All tests were conducted at an

308 ambient temperature of approximately 20~ Each cycle was run at intervals of approximately 60 seconds. This activation strategy was employed throughout this progrannne, whereby, we are concerned with a martensite--,complete austenite on-off type response. When embedding in Polymeric-Matrix Composites, PMC, due to the constraints imposed by the surrounding matrix, only partial austenite may be achieved. This is particularly true when embedding the 49.8-50.0 at.% Ni alloy. This is undesirable as we fail to utilise the alloys full capability. The predominant matrix constraint is associated with the maximum imposed power in stimulating the actuators. The drive variable for the SMA material is temperature. Therefore, the temperatme rise, associated with the applied drive current, was monitored by the use of a fine thermocouple located at the surface of the specimen. For validation purposes, the end result, or steady state temperature, was compared to an energy balance solution. In general, the power source applied to the actuators resulted in a SMA temperature greater than the austenite finish temperature. Therefore, any fluctuations in temperature, in response to environmental influences, would have no effect on the measured recovery stress - the predominant parameter under investigation. Experiments were also conducted at lower power levels to assess the knock down effect.

Constrained Recovery Force Results Some of the recorded SMA tensile recovery force curves, generated as a result of the activated wire trying to return to its memorised length, are illustrated in Figures 5 & 6. The horizontal axis exhibits time, the vertical axis exhibits the recovery force. We first concern ourselves with the lower atomic percentage nickel based alloy. Figure 5 shows the cyclic recovery force for the 0.4mm diameter alloy initially subjected to a 6% pre-strain. It is clearly evident that the maximmn recovery force is generated during the first cycle. Subsequent cycles show a decrease in the recorded recovery force. The greatest reduction occurs during the first few cycles. After 20 cycles the recovery force becomes more stable. The behaviour of the actuator during its operation is characterised by an initial sharp rise in recovery force. This being attributed to the step input power source. Lower power/heating rates would result with a lower recovery force gradient. A peak in recovery force will not be r ~ until the material stabilises at a temperature above A , , , or reaches Ate,. An interesting feature occurs during the first cycle. After the force reaches a certain peak value it gradually decreases, even with further increase in temperature. The recovery force then settles down to a constant value, the specimen being maintained at a given constant temperature. Such behaviour was shown to exist in the literature, [ 13], where it is stated that the peak value of the recovery stress, during this cycle, is influenced by the heating rate. The higher the heating rate, the greater the peak recovery stress. After the peak, no matter what the heating rate, the recovery stress would tend to the same value. It would appear, from Figure 5 that this phenomenon exists only for the first cycle. Cyclic behaviour was not discussed in the cited reference, [13]. No defmitive explanation for the aforementioned behaviour is put forward. A plausible conjecture, however, may associate the phenomenon with the materials coefficient of thermal expansion and/or the formation of lattice defects such as dislocations From Figure 5, after 20 cycles, the recovery force level is seen to stabilise at 58N, a drop of 19N compared to cycle 1. Figure 6 depicts the cyclic behaviour of a 0.4ram diameter specimen of the 50.0-50.2 at.% Ni alloy composition, pre-strained by 6%, and activated by a step input of 22.1W/m. For the first actuation cycle, the recovery force reaches a level of 110N which subsequently degrades to a consistent level of 96N over 20 cycles. The rate of degradation is at its highest during the first few cycles, such behaviour being characteristic of nickel-titanium. These recovery force levels are significantly higher, for a comparable power level per unit length, than those previously documented.

309

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310 It is clearly evident that, when deactivating the specimen and allowing it to return to the ambient temperature, the specimen retains a partial force of 23N. This is because the imposed operating temperature cycle remains above the stressed martensitic start temperatme, therefore, preventing the formation of martensite upon cool down. This remaining residual force was typical for all the 0.4ram diameter specimens tested, re.~ardless of the level of pre-stram.

TYPICAL ADAPTIVE PLATE POST-BUCKLING BEHAVIOUR Individual test specimens were located and simply-supported within a loading rig specifically designed for this investigation. The pre-strained SMA wires were subjected to an initial tensioning of 1.5N then clamped within external fLxtures and connected to the power source in parallel. An Instron Universal Testing Machine was employed to apply the compressive loading. The peak out-of-plane displacement was measured at the plate centre using a LVDT. The applied compressive load, LVDT output and thermocouple readings were recorded using PC based data logging software. Figure 7 shows the typical characteristic response of the control strategy. The results shown are for the [02/902], lay-up configuration employing 18 actuators 0.4ram in diameter and of the 50.0-50.2 at.% Ni alloy composition. For illustrative purposes, the applied load and the corresponding peak amplitude central deflection are shown with respect to time. Presenting the results with time allows us to break down the out-of-plane-displacement crave into five distinct phases. During phase 1, a 1.9kN compressive load is applied. The corresponding experimental out-of-plane displacement is seen to reach a peak value of-2.81 mm. At this point, there is reasonable agreement between experiment and FEA. The FEA predicting an out-of-plane-displacement of-2.66 mm. Activation of the 18 SMA actuators connected in parallel, through al~lication of the DC power source, results with SMA phase transition, phase 2. Throughout the tests conducted, the control was a simple on-off type action. The on-action was sent once the loading state reached the prescribed value. The resulting SMA restoration forces act to pull the plate back to the fiat configuration. At the end of phase 2, the energised plates recorded out-of-plane displacement is seen to be -0.8 rnm~ a reduction of 71.5% compared to the uncontrolled state. With the recovery force maintained for the duration shown, phase 3, the out-of-plane displacement is seen to increase to a value of-0.86 mm, this value being 69.4% lower than the uncontrolled state. The slight increase in lateral deformation is attributed to degradation of the matrix dominated material properties, associated with exposure to the elevated temperature, and to a non-uniform thermal stressed state set-up across the plate. At the end of phase 3, the plate deflection tends to its constant value as the temperature profile within the laminate approaches the steady-state. When the actuator power source is removed, the out-of-plane displacement profile is seen to increase rapidly prior to stabilising at -2.02 mm, phase 4. It will be noted that this deflection level is less than the initial uncontrolled post-buckled deflection value of-2.81 mm. This is due, of course, to the presence of residual forces in the actuators when the structural system cools to ambient conditions, as discussed previously and indicated in Figure 6. The actuator residual forces do not influence the steady state condition at the end of phase 3 and thus the 69.4% controlled reduction in post-buckled deflections is repeatable and is maintained with cyclic loading of the structural system. The test ends with the removal of the applied load, phase 5. As the load is removed, the lateral displacement is seen to decay to zero. During phases 1 & 2, good agreement between FEA and experiment exists. The FEA results are not shown during phase 3, as they were not seen to vary by any noticeable amount. Restoration recovery forces not only reduce the peak displacement amplitude, they also alleviate the higher stress levels, at the boundary supports, typical to post-buckled plate configurations as a result of load shedding due to out-of-plane displacements. Figure 8 shows the FEA longitudinal, Ny, membrane

311

Load Control

Exnerimental Behaviour -10/0/90/901s _

_

_

Load

-2.81 mm

~Um~nlml

-2.66 mm

X

,-3

central displacement

Admpg~non.lnoanrFEA

, -2.5

-2.05 mm

,tJ

e

-2.02

2 i -1.5 ~ mm

.-0.80 rnm

mq ~.,

@

,-1

-0.86 rnm

qlm

X

1 ~-o.57 0

, -0.5 mm

100

200

300

400

500

600

Time (s)

Figure 7. Adaptive post-buckling deflection behaviour.

FEA Longitudinal Membrane Force Intensitv Profiles -18

E

! m

.e

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-16

-14 t

......

', .....

[----1.~s

,

,[ ~

~

.......

1.gkN + 8 M A [

,, .......

,.......

,

,

_/ ,

-12 -10

.

t

t

t

e

.

.

.

0

E -4

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

'

o

~

.

. A

/--i

.......

0 0

~

I~ Distance

1~ Along Contrdine

2~

2~

300

(ram)

Figure 8. Adaptive and non-adaptive longitudinal membrane force intensity profiles. force intensity, along the plate's centreline, when subjected to 1.9kN, with and without the adaptive control. At the end of phase 2, utilising adaptive control, it can clearly be seen that, at the plate's knifeedges, the in-plane compressive force intensity is reduced considerably and in fact by the order of

312 62.7%. The tendency of the adaptive plate is to redistribute the loading back towards the plates central region, such that, a more uniform stressed state exists.

CONCLUDING REMARKS

It has been shown that the control of the non-linear post buckling response of composite structural elements is distinctly feasible through the application of induced strain actuation as detmnined fTom the shape memory effect. Utilising embedded wire-form Nickel-Titanium actuators, post-buckling displacement alleviation has been achieved. The control strategy employed utilises SMA actuators, fed through tubing after specimen fabrication, which are constrained to an external boundary. The acumtors are pre-strained and therefore a &twinned martensitic crystal lattice is induced in the shape memory material. The external constraints prevent shape memory recovery, therefore, upon activation; significant stresses accumulate within the actuator and these are then employed to achieve the desired structural response. The research intention was to demonstrate a sufcienfly enhanced adaptive postbuckling structural capability. This intention has been realised and a fairly in-depth understanding of the complex operating mechanics of the adaptive structural system has been presented.

REFERENCES

1. Meressi T and Paden B ' Buckling Control of a Flexible Beam Using Piezoelectric Actuators ', Journal of Guidance, Control and Dynamics, Vol. 16, No. 5, pp 977-980, 1993. 2. Thompson S P and Loughlan J ' The Active Buckling Control of Composite Column Strips Using Piezoelectric Actuators ', Composite Structures, Vol. 32, Nos. 1-4, pp 59-67, 1995. 3. Doran C J and Buffer R J ' Characterisation of Static Actuation Behaviour of Encapsulated PZT ', Second European Conference on Smart Structures and Materials, Glasgow, 12-14 October 1994. 4. Roberts S S J, Butler R J and Davidson R ' Progress Towards a Robust, User Friendly, System for Active Structtwal Damping ', Second European Conference on Smart Structures and Materials, Glasgow, 12-14 October 1994. 5. Baz A and Chen T ' Active Control of the Lateral Buckling of Nitinol Reinforced Composite Beams ', Active Materials and Smart Structures, SPIE Vol. 2427, pp 30-48, 1995. 6. Thompson S P and Loughlan J ' The Control of the Post-Buckling Response in Thin Composite Plates Using Smart Technology ', Thin-Walled StmcUa~, Vol. 36, No. 4, pp 231-263, 2000. 7. C.Boller, et al, 'Some Basic Ideas on the Design of Adaptive Aircraft Structures Using Shape Memory Alloys.', 4th International Conference on Adaptive Structures, Koln, Germany, 1993, pp220-236. 8. L.C.Brinson, et al, 'Deformation of Shape Memory Alloys due to Thermo-Induced Transformation.', Journal of Intelligent Material Systems & Structures, Vol.7, No.l, 1996, pp97107. 9. H.Funakubo, 'Shape Memory Alloys.', Gordon Breach Science Publishers, 1984. 10. J.Perkins, "Shape-Memory-Effect Alloys: Basic Principles.', Engineered Materials Handbook, ASM, 1991, pp4365-4368. 11. C.Lei & M.Wu, 'Thermomechanical Properties of NiTi-based SMA.', AD-Vol.24/AMD-VoI.123, Smart Structures and Materials, ASME, 1991, pp73-77. 12. Y.C.Yiu & M.Regelbrugge, 'Shape Memory Alloy Isolators for Vibration Suppression in Space Applications.', AIAA-95-1120-CP, pp3390-3398. 13. D.Honma, et al, 'Shape Memory Effect in Ti-Ni Alloy During Rapid Heating.', The 25th Japan Congress on Materials Research- Metallic Materials, 1982, ppl-5.

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

313

FAILURE ANALYSIS OF FRP PANELS WITH A CUT-OUT UNDER STATIC AND CYCLIC LOAD A. Muc, P. Kfdziora, P. Krawczyk & M. Sikofi Institute of Mechanics & Machine Design, Cracow University of Technology, Krak6w, Poland ABSTRACT The objective of the work discussed in the present paper is to demonstrate both numerical and experimental investigations of multilayered, laminated, composite plates having circular holes. The analysis deals with static and fatigue type of loading. There are discussed various types of possible numerical modelling of such structures. Then, different failure modes arising in structures subjected to tension or compression are presented with the use of the isotropic finite width correction factors. The influence of geometrical ratios and stacking sequences of laminates on final damage are especially emphasized. The second part of the work is devoted to the presentation of the reflection phoelasticity methods of the analysis and the results for such types of structures. KEYWORDS Composite Plates, Cut-outs, Fatigue, Buckling, Stress Concentration, Photoelasticity, Damage, FE Analysis

INTRODUCTION

The failure analysis of thin plates with a cut-out and made of advanced composite materials are research topics of a great practical importance. The need for a cut-out in subcomponent is typically required by practical concerns so that e.g. bolted joints and bonded joints subjected both to static and fatigue loading conditions are commonly used in different practical engineering constructions. Results of an extensive research in this area are presented and discussed e.g. in Refs [1,2]. They summarize the present state-of-art in the literature and deal with the analysis of both stress concentration effects under tension loads and buckling analysis under compressive loadings. The objective of the present paper is two-fold: to discuss the FE modelling, stress analysis and buckling analysis under static and fatigue loading conditions; the numerical analysis is carried out with the use of the FE package NISA II in order to verify the effectiveness and correctness of the modelling, to present results of nondestructive tests (reflection photoelasticity) in the analysis of damage evolution near opening under static and fatigue loads. The paper is an extension of the methods and ideas discussed by Muc et al. [3-5] where laminate degradation problems due to fatigue phenomena have been demonstrated for composite plated and

314 shell structures. For plates wit a centrally located holes subjected to static and fatigue tension the results of the experimental analysis are presented in Ref. [6]. The terms ,,openings, holes and cut-outs" will be used herein interchangeably

STRESS CONCENTRATION UNDER STATIC TENSION For the stress concentration studies of multilayered, laminated composite structures it is necessary to assure the accuracy of the analysis results. It may be achieved satisfying the following conditions : (i) the mesh should represent the geometry of the computational domain and applied loads accurately, (ii) the mesh should adequately represent the large displacement and/or stress gradients in the solution and (iii) it is necessary to conduct the convergence study. The research in this area shows evidently the advantage of the use of global-local meshing techniques to reduce computational efforts and to obtain an accurate prediction of detailed stress distribution in components. In our approach we propose to introduce a finite number of keypoints located around the hole and then to control the convergence of solutions by a change of they keypoints positions. The convergence is studied by the comparison of the results with theoretical ones, for instance it is known that the stress concentration factor for isotropic plates with openings is equal to 3.0. The example of the used meshes is given in Fig. 1 commonly with the plot of the von Mises (Huber-Mises-Hencky) stress distributions. The geometrical ratios and material properties in the example are following: L---~x= 3 ~

Ly

2' Ly

= 0.02,

E~ = 345[Gea],Ey = 105.2[Gea], Gxy - 40[Gea], V~y = 0.3

Figure 1: Stress distributions around a circular hole In the case of a plate with a hole there are many features of opening and plate geometries that affect the fracture behaviour. In addition, for composite materials due to variety of failure modes there is not completely clear what type of failure criterion can be used. In Ref. [6] the possibility of an application of the isotropic finite width correction (FWC) factor to the analysis of laminated structures has been studied. It has been found that for relatively small holes, d/W _< 0.25 the values of the isotropic and anisotropic FWC are relatively close, d denotes the diameter of the circular openining whereas W (W=Lx) is the width of the plate. It has been confirmed by our studies - see Fig.2. In the computations the same material properties as previously has been taken into account, and fibers have parallel direction to the tension direction (0 ~ fibre orientations). For fatigue loads the discrepancy between the isotropic and anisotropic FWC is even higher since the additional effects of stress ratios have a great

315 influence on the degradation of laminate properties and in this way on the number of load cycles at failure - see Fig.2. In the latter case the corresponding curve have been derived using the computational model proposed by Muc et al. [3,4]. For anisotropic structures there is also observed an effect of the laminate stacking sequence both on mode of fatigue failure and values of the stress concentration factors - see also Muc [7]. .... Isotropic . . . . . . . Laminate - static . . . . . Laminate - fatigue

7

,/...:

:

~.

~

:

i..

:

I

"-'

i

................................... : ....:.=.: :. :.

0,2

0

............ ...........................................

0,4

0,6

Opening/width (d/W)

I

0,8

Figure 2: Comparison of stress concentration factors for plates with circular holes, subjected to static and fatigue tension (S/Smax =0.7, R = 0.1 - see also Fig.4) For both theoretical and experimental analysis (static and fatigue loads) a crack in the direction perpendicular to the axis of tension, starting from the opening, is the observed fracture mode.

PLATES W I T H H O L E S UNDER C O M P R E S S I O N - BUCKLING In the analysis of plates having the centrally located circular opening it is necessary to take into account additional failure m o d e - a global buckling of the structure. In the numerical analysis of that class of problems variety of works exists (see e.g. the review [2]) that demonstrates the effects of: stacking sequences in the laminate, plate geometries and boundary conditions. Figure 3 shows mainly the effects of plate aspect ratios on the values of buckling loads. It is worth to note that for longer, rectangular panels the increase of buckling loads is observed comparing to square plates.

0,297 ~: 0,27-

-,

"~ 0,23

T.

i "~

o,21

--= 9 0,19 "~ 0,17i 0

~._

......

0,2

0,4

0,6

Opening/width (d/W)

Figure 3: Buckling loads for compressed plates with a circular hole

316 In addition, for rectangular panels having small values of FWC (< 0.2) the antisymmetrical buckling modes are dominant modes of failure, and then they switch off to symmetrical ones. Therefore, for long compressed plates with a cut out, even under compressive loads a fracture may preeeed buckling failure mode. Thus, a further analysis is needed to establish the limit load carrying capacity curves for such a structure under compression. The computations have been conducted for bi-directional (textile) composites, having a quadratic symmetry, i.e. Ex=Ey=E = 17.9 [GPa] made of glass fibres with epoxy resin.

EXPERIMENTAL ANALYSIS In the design of composite structures with openings it is necessary to establish failure loads and failure mechanisms and for fatigue loads the fatigue life, i.e. the number of cycles corresponding to the final fatigue damage. However, due to the variability of geometrical and mechanical properties of such structures all theoretical predictions have some margins of error that are commonly introduced in the design by safety factors. The identical situation occurs with all experimental data. Therefore, the question arises how it is possible to monitor continuously the development of failure both for static and fatigue loads conducting various nondestructive tests. Among them one can mention well-known analysis based on: photoelastic and acoustic emission measurements. Now, we focuse our attention on the description of the first type of tests. The verification of the numerical modelling via acoustic emission tests will be discussed in another paper.

Experimentalprocedure Smax

12 -1

~~~ 1 06 ! ~ ~ o 0 0

2

.... S , 4

6

Figure 4: Loading trajectory in fatigue tests All specimens were of a bi-directional, 2-D laminate manufactured out of glass fibres and an epoxy resin. The static Young modulus was equal to 17.9 [GPa]. The specimens had the following dimensions : 250x25x3.5 [mm]. Specimens had a circular cut-out of a diameter 4 [ram] located at their center in order to decrease locally the plate stiffrtess. In addition, all specimens had a steel tabs bonded to their ends with the use of the identical epoxy resin as for manufacturing laminates. A 2 [mm] thick epoxy resin layer with a drilled hole was bonded centrally to one side of the laminate. In order to ensure the comparability of the results for different specimens, the specimens were fixed with the same boundary conditions and were subjected to the identical loading conditions including the velocity of grips. The specimens were clamped at two opposite edges to the Instron hydraulic machine. The photoelastic set-up consists mainly of a non-commercial reflection polariscope and a camera. One lamp was sufficient to achieve a required luminous intensity. A camera captured photoelastic images for selected values of tensile loads (in the static tests) or for selected numbers of cycles (in the fatigue tests). One type of loading was applied to the specimen- a tensile one with a constant velocity. For fatigue tests the loading trajectory is presented in Fig.4. Specimens were initially loaded to the value S, being the average value of the loading cycle and lower than the maximal tensile stress in the static

317 tests (denoted by the symbol Smax). Then the cyclic tension-compression loads were prescribed with the given value of the stress ratio R. Test results

The conducted tests can be divided into three groups" static tests of specimens with no holes, static tests of specimens with a cut-out, fatigue tests of specimens with a circular opening. The results of static tests with no holes are demonstrated in the set of photoelastic images in Fig.5 (originals of all photoelastic images are in colour). Photographs present distribution of matrix cracking damage with the increasing load. The last picture shows the distribution prior to failure that occured at the load equal to 14.5 kN. The photoelastic analysis demonstrates rather qualitatively the evolution of damage and in addition exhibits the nonhomogenity of failure.

Figure 5: Photoelastic images showing damage propagation under static loading The next Figure 6 presents the localized stress concentration around the circular hole under static tension. However, since the matrix failure in the whole structure is also observed (compare with Fig.5) the typical stress analysis with the use of classical photoelastic equations for plane structures is much more complicated than for isotropic plates. Using the obtained images it is possible to survey the gradual damage propagation. The isochromatic fringe patterns should be analysed using softwarebased image processing including the RGB-data for interpreting and processing of colour information. As it may be noticed both stress and strain concentration factors are now strongly affected by the gradual change (degradation) of the laminate properties. Both in Fig. 5 and in Fig.6 there are regions of the stress concentrations caused not only by the localized cut-out but also by the stiffness degradation due to matrix failure. A quantitative approximation of those effects will be presented in the next paper. The next Figures 7 and 8 deals with the photoelastic analysis of specimens with holes subjected to fatigue tension-compression (see Fig.4). Comparing the results plotted in Fig.6 (static) and Fig.7 (fatigue) there is no particular differences. In addition, the images plotted for different number of cycles (Fig.7) do not exhibit a particular differences. On the other hand, the experimental analysis shows evidently the stiffness reduction increasing with the number of loading cycles n. - see e.g. Ref. [6].

318

Figure 6: Photoelastic images for specimens with holes under static tension

Thus, in our opinion the reflection photoelastic methods are not sensitive enough to capture exactly the fact of the stiffness degradation for multilayered, laminated composites. It seems that the photoelastic imges present smeared-out effects (even in the case of the stress concentration) and in this case it is impossible to evaluate and analyse the effects of the stiffness degradation. On the other hand, the stiffness degradation methods are much better visible from the classical 6-e characteristics see Fig.8. As it may be observed the stiffness degradation is directly connected with the change of the 6-e plots. The increasing number of cycles leads to the decrease of the area surrounded by the individual loops, however the change of the slope for the sequence of curves is much smaller and in this sense it may be only calculated from the plots. The measured and then computed stiffness degradation varies for different specimens. It has been found that for the final fatigue damage of structures with openings the reduction of the stiffness belongs to the interval [0.8 , 0.85] E (n=0). However, it should be emphasized that those values are strongly dependent on the type of composite materials and stacking sequences.

Figure 7: Specimens with holes under fatigue tension

(Smax =

12 kN, R = 0.1)

!

....

3:1

|

319

f/:

32

.

!

~.3

1

l

!

3.4

33

32

n = 1 cycle

i

Z

34

Displacement A! [mml

n = 2000 cycles

1

!

i

!

al o

I

32

33

|

34

3.5

Displacement A! [mml

n = 8000 cycles

n = 9500 cycles (prior to failure)

Figure 8" The P -A1 characteristics for different number of cycles - fatigue loads (their parameters are identical to those presented in the caption for Fig.7)

CONCLUDING REMARKS The influence of the circular cut-out on damage loads and damage propagation for composite, multilayered laminated plates subjected to static and fatigue loading conditions have been investigated in the paper. The results shows the effectiveness of the proposed method of numerical modelling on the estimation of failure loads. The experimental studies of stress concentration effects via reflection photoelasticity demonstrate evidently that that method can be used for the qualitative estimations of the damage propagation. Thus, to recapitulate: 9 The insertion of the keypoints in the numerical modelling allows to control the mesh densities at the points of the stress concentrations and to improve the computational effectiveness of the problem. 9 Using numerical models it is possible to evaluate the stress concentration factors for both static and fatigue tension loads. In this way it is possible to estimate and determine the differences of the values for various types of gemetrical and loading parameters describing the structures. 9 Considering buckling mode of failure under compressive loads it should be pointed out that the geometrical ratios as well as stacking sequences have a great influence on buckling loads and buckling modes. It is necessary to analyze also antisymmetrical buckling modes since they may preceed symmetrical modes of buckling failure. 9 In order to obtain stress distributions via photoelasticity methods it is necessary to conduct the computer analysis of images. However, for composite materials other modes of failure (e.g. matrix cracking) change qualitatively the photoelastic images of stress concentration effects.

320 9 There is not observed a visible difference between static and fatigue loading on photoelastic images, although according to other experimental studies the stiffness degradation should result in the significant change in the presented photographs. Such effects can be directly observed by the analysis of the a-e plots for the sequence of number of cycles n.

ACKNOWLEDGEMENT The support from the KBN under the grant PB-847/T07/98/14 is gratefully acknowledged. REFERENCES

[1]

[2]

[3] [4] [5] [6] [7]

Y.S. Choo, Stresses in Composite Panels with Openings, in Numerical Analysis and Modelling of Composite Materials ( J.W. Bull, edt.), Blackie Academic & Professional, Glasgow 1996, pp. 105-127. M.P. Nemeth, Buckling and Postbuckling Behaviour of Laminated Composite Plates with a Cut-out, in Buckling and Postbuckling of Composite Plates, (G.J.Turvey, I.H. Marshall, edts), Chapman & Hall, London 1995, pp. 260-298. Muc A., Krawiec Z., FE Modelling of Laminates Degradation in Thinwalled Structures due to Fatigue Loads, Proceedings IV- WCCM, Buenos Aires ,1998 Muc A.,Krawiec Z., Design of Composite Plates under Cyclic Loading, Composite Structures, 48, 2000, pp.139-44 Muc A., Description of Fatigue Failure Process for Laminated Composite Structures, Proc. ICCM/12, Paryz 1999 Muc A., K~dziora P., Fatigue Strength of Composite Structures with Holes, Proc. X/X Symposium ,,Machine Design ", Swinouj~cie, 1999, Vol. 2, pp. 155-60 (in Polish). Muc A., Design of Composite Structures under Cyclic Loads, Computers & Structures, 2000, 76, pp.211-18

Third International Conferenceon Thin-Walled Structures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

321

SOME NEW APPLICATIONS OF THE THEORY OF THIN-WALLED BARS J.B. ObrCbski Faculty of Civil Engineering, Warsaw University of Technology, Armii Ludowej 16, 00-637 Warsaw, POLAND

ABSTRACT The present author has elaborated and published some papers and books concerning an original and effective complex theory for prismatic bar with any type of cross-section (CS): full, thin-walled (TW) open, closed or open-closed. There were presented some examples of the theory application and its new possibilities. The paper summarize author's observations concerning non-conventional problems analysis and synthesis of structures - specially composite ones. This time the attention is turned on following problems: analysis of simple frames applying also DMEM (Difference-Matrix Equation Method), calculation of the geometrical characteristics for bars with any CSs including typical bridge ones.

KEYWORDS Composite bars, frames, any cross-sections, analysis, dimensioning, calculations exactness INTRODUCTION To the most important - fundamental author's publications belongs (1986,1991,1997a). Moreover, given there theory was developed in the series of the lASS conference papers: Milan (1995), Taiwan (1997,2000d), Moscow (1998a), Madrid (1999a), Istanbul (2000), Chania (2000a), Warsaw LSCE'95, '98,'99 and many others. The theory is very close to the Vlasov's theory (1940/59), which now can be regarded as only its particular case. To the most important effects of the investigations of such theory application, belong almost trivial conclusions: - for the bars with any type cross-sections (CSs) may be calculated its geometrical characteristics taking into consideration its: outlines, masses, strength properties of used different materials, interactions with surrounding media (gas, fluid, soil etc.) - linear and non-linear, 3D etc. - mechanical behaviour of the bars wit any type CSs may be described by the same equilibrium or motion Eqns, Obrgbski (2000), - so, for the bar with any type of CS may be determined: critical loadings of bending-, torsional- or

322 bending-torsional character, for static or for time dependent loadings (Obrebski 1999a, 2000 etc.), the equilibrium or motion Eqns of the bar concern only its small part with the length dx, therefore they may be applied by numerical FDM approach also to non-prismatic bars (Obr~bski 2000a, 2000d), the composite prismatic bar with compact CS may be modelled as set of TW tubes (also composite ones) located one into the other (important for torsion problems (ObrCbski LSCE'95,'97,'99,'2000), the reinforced concrete beams or frames built seemingly of prismatic bars, in the fact often are nonprismatic and characteristics of its CSs are variable on bar length (Obr~bski 2000a, LSCE'2000, 2001), - application of Finite Differences Method (FDM) for more complicated tasks as: for non-prismatic bars with additionally composite CSs, for static, dynamics, dynamical stability; for forced-, free- or/and dumped vibrations, is very efficient (Obrebski LSCE'99, 2000d - for e.g. tall buildings), - the 3D version of FDM introduced and called by the present author as Difference-Matrix Equation Method is possible to be applied to 3D analysis of e.g. tall buildings (paper with R.Szmit), bridges etc. (LSCE'98,'99, Taiwan 2000d), the analysis and dimensioning specially of composite bar structures has in natural way non-linear character, because in most cases we don't know its final bars CSs (Obrebski 2000a). -

-

-

-

The list of similar conclusions given in previous Obr~bski's papers (2000, 2000a, LSCE'2000) is much longer. Many of above problems may be solved applying certain small programs elaborated by present author, or using standard commercial tools as e.g. MathCAD, Obr~bski & Szmit (200(0) etc. Below are given some next important results of comparative calculation of certain series of similar structures, performed in new way. In most cases, they should be regarded as more exact as till now. ON CALCULATION OF GEOMETRICAL CHARACTERISTICS In the Figures 1-5 are given some very important examples of certain series of comparative calculations of geometrical characteristics for similar or identical CSs. They are complementary to presented in previous Obrebski's publications. So, in the Figure 1 are given some full type homogenous CSs modelled as set of tubes in tube (with dimensions of its middle line - bxh) which extend the results presented in papers, Obrebski (1995a,2000a,2000b). In the Tables 1,2 are given results of calculation: 03 values of warping (sectorial) function in its corners A for each tube, I~ - sectorial moments of inertia, Ks - torsional rigidity; all together known up till now only in theory of thin walled bars (rWB). Besides it, are shown: f2 _ double area of closed circuit and L - the length of the circuit. _

a)

b)

c)

d)

e) '

1

?

~[ t._.~_,,

It

,_,j

_

Figure 1: Series of concrete cross-sections and its division on "tubes" Moreover, in the Table 2 are given: A - area of whole CS, I2, I3, Io~ ,Ks - its global moments of inertia and torsional rigidity. Simultaneously, in last row of Table 2 are given results for CS of the Figure 2.

323 In the Figure 3 and Table 3 are given similar data for series of closed type thin-walled homogenous CSs with stiffer comers - with two thicknesses of its walls 6, = 2 c m , 62 = 1, 0.5, 0.1, 0.01, 0.001cm. In the Figure 3d is shown particular case, when $2 =0 - it means the bar with 4 branches working together. In the last case the warping function (see Figure 3e) was calculated as for 4 closed tubes. The obtained results are a little different as for the similar bar when &2 =0.001cm (very thin, Figure 3c). There calculations by FEM or experiment may evaluate these results, only.

I

L~L .

9.5 .

4 2o~"-.

9.s

.

:_,

Figure 2: Steel I cross-section with geometrical characteristics similar to CSs of the Figure 1 TABLE 1 SOME GEOMETRICAL CHARACTERISTICS FOR CROSS-SECTIONS OF THE FIG. 1

Cross-section

bxh (tubes)

f2

L

eli iiii

c~

cm

c~

9x19 7x17 5x15 3x13 lxll 7x19 5x17 3x15 lxll 5x19 3x17 lx15 3x19 lx17 lx19

342 238

56 48 40 32 24 52 4z 3~ 21 48 40 32 44 3~ 4t

-15.267 -12.395' -9.375 -6.093 -2.29] -15.346 -11.591 -7.50t -2.781 -13.854 -8.925 -3.281 -10.363 -3.777 -4.275

iii ii

Fig.la

9

i

Fig. lb

. . . . .

Fig.lc

Fig.ld Fi~.le

!5s 78 22 266 17{ 91 2t 19{ 10; 313 114 34 38

cm 4

i

4351.33 2458.513 1171.87 396.09 42.01 4082.07 1970.45 675.01 72.4", 3071.00 1062.07 114.84 1575.27 171.25 243.67 .

.

.

.

.

i

. . . . .

.

kNcm: 2088.64 i

118.0.08 562.5C 190.17 20.16 1360.6~ 656.81 225.0(] 24.14 752.08 260.113 2.8.12 295.3~ 32.1 36.1{

Similarly in the Figures 4,5 are compared results of calculation for the bridge CS, Figure 4a, modelled as classical TWB with one closed circuit (see Figure 4b,c) or modelled as set of tubes in tube, Figure 5. In the Table 4 are shown calculated values of geometrical characteristics identical as above. There for the second case are observed: discontinuities of warpings (longitudinal displacements & warping normal stresses between tubes 3 & 4) and also flows of shearing stresses di. But in both approaches the resulting flows of shearing stresses from pure torsion di are almost identical (0.0285 & 0.0283 - sum of d~ in last

324 row of Table 4 !!). In the method tubes in tube the disposition of shearing stresses flows is remarkably different as in membrane analogy (see Timoshenko) or in de Saint Venant's theory. Comparing in the same manner (one tube & six tubes) we obtain: for the sectorial moments of inertia (6.678m 6 & 24.754m6); for torsional CS rigidity K, (320 978 511.627kNm 2 & 260 489 113.363kNmZ). TABLE 2 GEOMETRICALCHARACTERISTICSFOR CROSS-SECTIONSFROMTHE FIGS 1,2 bxh

Cross-section

cm

!0x20

Fig.!a Fiz. lb Fig.lc Fig.ld Fig.le. Fi~.2

8x20 6x20 4x20

l

2x2(] 20x20

oi I

~,

, . . .1

i~

A cm2

cm'

cm"

cm'

200 160 120 80 40 59

6666.66 5333.33 4000.00 2666.66 !333.,33. 4666.66

1666.66 853.33 360.0t3 106.66 , !3.33 1333.33

8419.82 ,, 6799.96 4247.92 1746.53 243...6'7 133.000.0

I3

.

"

kNcmZ

i

4041.5_1 ..... 226.6.65 1040.30 327.47 36..!0 160(0)0.

2~_~ --=~

,

[tin] ....

,

I; &.._,.i.

I_'~

.......... '~

-

L 3.s ....-r ~--

~,_I

L.

3.5

~6.BTs

~[-\I ~ - - - ~ ' ~ . . , 20

2L,.99 1

~

~

"

20

~ . . . .

~ =0.001

cm

,

,

,s~2s

......!+\

~

,_

_1

'19.125 15.325

I,._

1.5

.~l .

[<m2]

2~

~ "-.

ts..~s[__+ ]m2s

9

0.006

. . . . .

"~

- ,

15.32519.~25

/

=i ~ -

8.0

o

~2..99

"-'..'

" ~1 g.I~.99

"-517 " ,.L,,."

s.,zs

,6.8,s

!"!

'~6.8~

~

5

~

3

2

5

it

Figure3" Comparative results of warping function for CSs with stiffer comers and with four separate branches There smaller rigidity appears for multi-tube model. Generally we can draw conclusion, that modelling thicker or full CS it is better to apply convex type tubes. It is clear, that modelling such CSs, there may be applied many different divisions of the CS on tubes (closed). But in consequence we should to expect

325 a little different results, too. TABLE 3 WARPINGCOORDINATESFORCROSS-SECTIONSWITHSTIFFERCORNERS,FIG.3

c~ i ] 0.0C 0.0(3 0.04] 0.04] 0.13(]

cm

1.000 0.500 0.100 0.010 0.001

03~

034

cm: -20.013 -22.22 -24.39 -24.93 -24.99

-5.0013 -2.770 -0.609 -0.062 -0.006

032

031

(Sz wall thickness

i ic mi ;

-22.50 -29.115 -35.6"7 -37.31 -37.4~

a)

.-3.oo ~[

0.00

11,60

II

!

[m]

-

,'

O.O,3fi Z.SO

s:io'-

3.~5

c)

c~ 0.00 O.OC , O.OC O.OC

cm ~

i

,,

&5

_

;

_

"i'i " ,' ~.'" Z.50

_q.3.~

3.OO

:I

t 93

z.65

z.65

3.15

I-o.3o9 [m2]~--d+o3o9

Figure 4: Bridge cross-section and its warping function Jo(

Cl

:-:~

,[ ~ _ -

, 3

~:-

,

, ,

,--

b

_

.

: -"11

A., r - - ~

it~2

[-~-dl

"

~

_.

' .i :

- -__., ~lll_.,,, .,.

I

~'~I.

d1

Figure 5" The bridge cross-section modelled as set of six dosed tubes

326 TABLE 4 SOME CHARACTERISTICS OF BRIDGE CROSS-SECTION, FIGS.4,5

Quantity Units

Tube No 1

h/l

,

cba O)b

m2

09c

I1-12

o3~ (be

m2

~s

m 2

....

l,a

m6

Ks di

kNm2 kN/m

i

42.405 30.200 0.00(3 2.68809(] 3.780805 -5.29043~ -0.766598 o.ooo

m~

L

2

,

7.526999 44656888.0 0.004042 ,,

3

39.405 29.800 O.OOC 2.847007 3.507409 -5.468364" -0.839505 o.ooo

4

i

36.445 29.400 0.OO3 3.007655 3.229424 -5.64828(3 -0.911546

7.937119 8.410186 3,9079379.8 , 33883623.C 0.003807 0.003569

ul

6

ii

34.i25 17.000 0.000

32.445 16.603 0.000 .

.

.

30.805 16.200 0.00t3 .

-0.755327 1.252025

-0.733640 1.220877 0.000 0.316259 0.293110 513.,75689.3 47560754.1 0.005779 0.005627 ,,,

.

.

.

-0.711883 1.189659 0.00(3 0.271083 43932778 0.005474

BEAMS AND FRAMES ANALYSIS It is worthy to mention, that Obr~bski (2000a,b) has investigated also application of the method tubes in tube to the series of reinforced CSs as in the Figures 6,7. Generally it should be mentioned, that in all these cases were obtained much higher moments of inertia, remarkably different positions of reduced gravity center and first of all warping functions and sectorial moment of inertia not known in normal analysis of reinforced frames. This all together gives new quality in analysis of many structures. In consequence there we observe much smaller displacements of beams or frames and much more realistic values of normal and shearing stresses. TABLE 5 CRITICAL FORCES FOR ECCENTRICALLY COMPRESSED STRAIGHT BAR

Crosssection Fig. la Fig.lb Fig.lc Fi~.ld Fig.le Fijz.2

P2 kl~ 1480 1184 888 592 296 6483 i

p~ kN 37(] 189 79.94 23.68 2.96 1852I

P,o kN 145539 87969 42974 14180 1610 3390

1~,1 kN 1072495 1278117 4692851 .-369339 -23276 313381

i

.

l~a kN 369 189 79.79 23.64 2.95 1202

M~

M k l

1

kNcrn 47375 253..88 11172 3412 40(3 25276

iii

kNcm -47375 -25388 -11172 -3412 iiii

. . . . . . . .

40(]

-25276

Interesting is possibility to calculate critical forces not only for TWBs but for also for composite frames, too. Such calculations were presented by Obrgbski (2000b,d,2001). There we may observe that critical force is smaller from Eulerian ones. It is shown in Table 5 even for full concrete CSs (99.78% 99.80%). There were considered the eccentrically (6cm) compressed concrete bars with the length 4m and CSs as in the Figure 1. But in the Table 5 is shown also example of the similar eccentrically (10cm) compressed steel beam with the length 4m where critical force has value 64.9% of Eulerian one. Similar comparisons for critical loading of the frames, Figure 7, were given by Obrebski (2000b) There for plane tasks were obtained much higher critical forces for reinforced frames than for concrete only.

327

~

.

IP-'k"

1SOcm b} -

-

~

I

P:lkN

cl

lr~cm

_l_

~

....

Jqj'

i_.

-

-

z'x

H,t

u165

1

I'

-'-_ _ _ _ - 7

--

-- 7

[:36r~1~

k O

i ~ ,

LL . . . . q

I, ~"~ ~t __

Figure 6: Example of reinforced beam with practically non-prismatic cross-section o=1

L

A=I ,

i

#:I

p

zx"'--'~

3-

t=~o~

.p

a:2 A ~ L t/2

Ix

#:0.5 p

zx J_. t/2

P

p

xt~

El =const. L..

~. a)

,,{..,,[

.... b...L}

i

,

1_ lO,dcm

,-

I

....

J-,

,g-a

~

r.,)

toy

d),

,

'Fiil;7~.,~?.i~i'i .........

I IIIIIiiiiiiii

I,l,lu_

d'-d

e)

|-(

Niml:m~llm, ................

, ,

......... I

!

~t.l.h

; ;:~,""'",;,;'; Ig; ~ ; :/"'=-'~""~- :I:~ . . . . . . . . . . . .

!!_~_;

I_ 10,1cm ,-

Ul

',11t3~1~1i

............

!i 111 1I,,,,.-

Ip

IP

w-0 b4=0

-,

' "7- = 2 ~ 'i 'l i ~ '"

ip

!1 0 ,I .3

2t

a:3 A=g .,a:0~ p

L s ' ~ - - z:x z:x z~ ' L~=" _1 ,Lt/3 . . L t / 3 _LI,/3 _}

L_m~~__~

:

L,- 10,,1~

~1 7

Ira'

~ ~ ~

L,- 10~cm

=-

t

'1

i|

1,2_

I

~_d

I_ 10xlcm i-

I

n

_l -1

Figure 7: Examples of structures under critical loadings, Obrebski (2000b) CONCLUSIONS The paper presents examples of possible application of the method of TW bars theory to calculation of bars and frames with full or thick CSs, applying its modelling as tubes in tube. In that case we can obtain much exact internal and critical forces, displacements and stresses. As it follow of calculated results, the moments of inertia may have values up to e.g. 194.87%, critical forces up to 64.9%328.31%, displacements up to 76.49%. We can conclude, that contemporary methods of beams and frames analysis are giving in many cases rather approximate results.

References Obrgbski J.B. (1986) Second order and second approximation theory in static and dynamics of thin-walled straight bars. 1st Int. Conf. on Lightweight Structures in Architecture, Sydney, 815-822 & (1989)

328

Thin-Walled Structures, Appl. Sc. Publ. 8, 81-97. Obrgbski J.B. (1991,1999) Thin-walled elastic straight bars (In Polish). Publishers of Warsaw University of Technology, (lecture notes), Warsaw, pp.452 & Second edition OWPW, Warsaw, Poland, pp.455. Obrebski J.B. (1993) On non-linearities in analysis of thin-walled bar structures, Non-linear Analysis and Design for Shell and Spatial Structures. (Invited lecture), SEIKEN-IASS Sympos.Tokyo, Japan, 197-204. Obrgbski J.B. (1995) On the strength of composite bars. lit. lASS Conf., Milan, SGEditorial, 517-526. Obrebski J.B. (1995a) Some torsion problems in thin-walled and composite bars mechanics. Lightweight Structures in Civil Engineering (LSCE'95), Warsaw, MAGAT, 395-404. Obrebski J.B. (1997,1998) Influence of Local Bar Instability on Strength of Large Double-Curvature Bar Structures. (Invited lecture), lit. lASS Colloquium on Computation of Shell & Spatial Structures, 5-7.11, Taipei, Taiwan, 414-419 & Journal ofthe 1ASS, Vol.39,n.2.(August n. 127),85-90. Obrgbski J.B. (1997a) Strength of materials On Polish) .Lecture notes, Warsaw University of Techn. Faculty of Civil Engin. Micro-Publisher J.B.O. Wydawnictwo Naukowe, Warsaw, Poland, pp.238. Obrebski J.B. (1998a) Mechanics and strength of composite space bar structures, (Invited lecture) Spatial Structures in New Renovation Projects of Buildings and Constructions,Int.lASS Congr.Moscow,Russia,299-306 Obrgbski J.B. (1998b) Numerical examples of the composite bar structures mechanical behaviour, lit. lASS Colloquium on Lightweight Structures in Civil Engineering. Warsaw 30 Nov.-4 Dec., 331-340. Obr~bski J.B.(1999a) Some rules and observations on the composite bar struct, mechanical analysis, Shells & Spatial Struct. from recent past to the next miUenn, lit.lASS 40th Anniv.Cong.Madrid, Spain, B1.37-46 Obrebski J.B.(1999b)Examples of determination of the non-conventional geometrical characteristics for composite bars with any cross-section.LightweightStruct.in CivilEngin. Local Semin.ofIASS Polish Chapter, Warsaw, 3 Dec. 64-77. Obrebski J.B. (2000) Mechanical point of view on modelling of space structures made of composite bars. lit. lASS Symp. Istanbul Turkey, 29.05-2.06, 491-500. Obr~bski J.B.(2000a) Non-linearcharacter of the computationsof compositenot only bar structures. (Keynotelecture) 4th Int. Coll. Computationof Shell & SpatialStruaures, June 5-7, Chania-Crete,Greece, abstr, vol. 558-559& CD-ROM. Obrebski J.B. (2000b) Designing of composite bar structures taking into consideration instability effects, 9th Symposium on Stability of Structures, Zakopane, Poland, 25-29.09, 227-234. Obrcbski J.B. (2000d) Some problems of mechanics and strength of frames made of composite bars. Lightweight Struct. in Civil Eng.Local Semin.of lASS Polish Chapt. Warsaw-Cracow, Poland 1.Dec.43-60 Obr~bski J.B. and Sznfit R. (2000e) Dynamics and dynamical stability of tall buildings, (Invited lecture). 1st lit. Conf.on Structural Stability and Dynamics, Taipei, Taiwan, Dec. 7-9, 85-94. Obr~bski J.B. (2001) On the mechanics and strength analysis of composite structures, (Invited lecture), Int. Conf. SEMC 2001, Cape Town, 2-4.04. Vlasov V.Z. (1940,1959) Thin-walled elastic bars (In Russian), Moscow, Russia.

Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

329

BUCKLING BEHAVIOUR OF THIN-WALLED COMPOSITE COLUMNS USING GENERALISED BEAM THEORY N. Silvestre ~, D. Camotim t, E. Batista 2 and K. Nagahama 2 Department of Civil Engineering, IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL 2 Civil Engineering Program, COPPE, Federal University of Rio de Janeiro C.P. 68506, 21945-970 Rio de Janeiro, BRAZIL

ABSTRACT An extension of the existing Generalised Beam Theory (GBT), intended to make it applicable to orthotropic thin-walled structural members, is presented and discussed. The derived equations are, then, used to analyse the local and global buckling behaviour of FRP pultruded columns and the GBT results are validated through a comparison with numerical ones, obtained by finite strip analyses. Finally, a parametric study is carried out to investigate the variation of the bifurcation stress and buckling mode nature with (i) the column length and cross-section dimensions and (ii) the composite material properties.

KEYWORDS

Fiber reinforced plastics (FRP), Thin-walled columns, Generalised beam theory (GBT), Orthotropic material, Buckling behaviour, Bifurcation stresses, Local plate mode, Distortional mode.

INTRODUCTION

The structural use of composite materials can be traced back to the early 1960s and, since then, it has progressed through several stages. Although such materials have always been extensively employed in the aeronautical industry, civil engineering applications only became meaningful in recent years. Nevertheless, a growing number of companies have already emerged in this particular field and, moreover, they have managed to occupy a non negligible share in a market traditionally dominated by concrete and steel structures. This success is rooted in the fact that composite structural members combine (i) a nearly optimal weight/strength ratio with (ii) increasingly low fabrication costs and (iii) an efficient behaviour under aggressive environmental conditions. This last feature, in particular, makes composite materials ideally suited to be used in off-shore structures, chemical plants, etc. This paper deals with a well defined type of composite structural members, namely thin-walled members with open cross-sections and manufactured, by pultrusion, from a polymer plastic matrix (e.g., epoxy)

330 reinforced with e-glass, s-glass, kevlar or carbon unidirectionally aligned fibers. Such members are usually designated as "FRP" pultruded members and their mechanical behaviour, which depends on the constituent properties and volume fraction, is characterised by (i) a linear elastic stress-strain relations (mostly with relatively low moduli), (ii) the absence of ductility (elastic behaviour up to collapse) and (iii) the presence of a distinct orthotropy. These mechanical properties clearly indicate that the FRP member behaviour is strongly affected by instability phenomena. Therefore, in order to assess the structural efficiency of such members, namely columns, beams and beam-columns, it is essential to investigate their (local and global) buckling behaviour. In particular, this involves the performance of stability analyses, aimed at identifying the relevant buckling modes and determining the corresponding bifurcation stresses. The "Generalised Beam Theory" (GBT) was developed by Schardt (1989) and shown to be a rather powerful analytical tool to study the buckling behaviour of thin-walled (cold-formed) steel members (Davies, 1999). In fact, it provides a general and unified approach to obtain accurate solutions for a wide range of stability problems. By allowing the separation of the fundamental deformation modes, GBT offers possibilities not available through the use of numerical techniques, such as the finite element or finite strip methods (Davies et al, 1994). However, steel being an isotropic material, the existing GBT cannot be readily applied to orthotropic members (e.g., the FRP members under consideration), i.e., members made of materials displaying the stress-strain (constitutive) relations (it is assumed that vvEy= vy~Ex).

to.xt "IEx,.l-.x..>.Ex'.'-.x.x.. olfxxl (:Sss = V,,sEs/(l-VsxV,,s)

l~xs

Es/(1-VsxVxs )

o

0

0

,

1ess~

(1)

Gxs l~,~ J

The objectives of this work are (i) to extend the existing GBT, in order to make it applicable to orthotropic thin-walled structural members and (ii) to use the derived equations to analyse the local and global buckling behaviour of composite FRP columns with commonly used cross-sections. The GBT results are validated through a comparison with numerical ones, yielded by finite strip analyses. Finally, a parametric study is carried out to investigate the variation of the critical bifurcation stress and buckling mode nature with both (i) the column length and cross-section dimensions and (ii) the composite material properties.

GBT

FOR ORTHOTROPIC MATERIALS

The simplifying assumptions adopted in the derivation of GBT for isotropic materials (Schardt, 1989), namely neglecting the membrane shearing strain and transversal extension, with the respect to the longitudinal extension values, remain perfectly valid in the context of the linear stability analysis of orthotropic columns. According with such assumptions, the relevant strain-displacement (kinematic) relations employed are defined by exF• = --ZW,xx

esFs= -- ZW,ss

7xFs= -- 2ZW,xs

M

1

9

Cxx= U.x + 5 (V,'x+

X

W2

,x)

x(u) t

~

ds / l / z(w)

Figure 1: Column geometry, coordinate system and displacement components.

, (2)

331 where x and s are coordinates along the length and mid line of the cross-section wall (see figure 1) and the superscripts ( )" and ( )v stand for the origin of the strain components (membrane orflexural). In order to obtain a displacement representation compatible with the classical beam theory, Schardt (1989) prescribes that each displacement component (u, v or w) at any given point of the mid line of a crosssection comprised of n wall segments (i.e., containing n+l nodes and/or degrees of freedom) must be expressed as a combination of de n+l orthogonal functions (u,), all of which vary linearly between consecutive nodes. One has, therefore, U(X,S) -- UkCk, x

V(X,S) -- Vkr k

W(X,S) -- Wkr k

,

(3)

where the summation convention applies to the subscript k (1 <_k < n+l) and ~x) is a "displacement amplitude function", defined along the column length. On the other hand, as a result of the assumptions underlying the derivation of GBT, all the remaining nodal displacement components (v, and w,) are determined a priori, on the basis of u, values (Schardt, 1989). As a first step towards deriving the GBT fundamental equation for orthotropic materials, let us consider the variation of the column (membrane and flexural) strain energy, which is given by the expression t/2 , fie

(4)

-t/2

where -(2Lis the combined area of the mid planes of the k plates forming the cross-section. Taking into account (i) the displacement approximations (3), (ii) the kinematic relations (2) and (iii) the constitutive relations (1), it is possible to rewrite the stress and strain variation components appearing in (4) as -Ex O~x = 1-Vx~Vsx(ZWkCk'xx + VsxZWk'ssCk)

~SxFx "" -- ZWi~r

(5)

~s

(6)

-Es O.sFs -- ~I__VxsVsx (ZWk,ss(Dk + VxsZWk(l)k,xx)

O•F

2zGxsWk.~bk.x

=-

ZW i,~8r

8~'x~= -- 2ZWi,sSd~i,x

G'xMx"- ExUk~k,xx

8gxMx = UiS~i.xx + (ViVj+WiWj)~j,x~([li, x

(7) ,

(8)

i.e., to express bT.] as a function of the (n+l) nodal displacements u,. Introducing (5)-(8) in (4) and performing the cross-section integration (w.r.t. the coordinates s and z), one is led to the expressions of the four terms comprising ~ , namely the terms due to the internal virtual work done by the (i) longitudinal bending normal stresses, (ii) transversal bending normal stresses, (iii) torsion shear stresses and (iv) membrane normal stresses. They are given, respectively, by ("b.c."- boundary condition terms) t/2

aU:x-f fo:aLdzdn-- f(ExC, ,| ,xx x 03 -t/2

+ vsxExDi~k.xx)5~idx+ b.c.

(9)

L

t/2

BUgs = f f~g~Segsdzdf~ = f(E~Bik*k + vxsEsD~dh,xxlS*idx + b.c. fie

-t/2

(10)

L

t/2

5U~s = f fO~sSg~sdzdf~ =-f(Gx~D~dh,xx)80idx + b.c. f/~ 4/2

L

(11)

332 t/2

+ Xkji(WkdPj,x),x)C$Oidx+ b.c. aU~x = f fo~aex~dzd~ = f(ExCikOk,xxxx | D,. -t/2

,

(12)

k

where the tensors appearing in the r.h.s., stemming from the cross-section integrations of the displacement components and their derivatives, are defined by the relations t3

r

C~ = 12(1-VxsV~x)fwiwkdS

D~ = 12(1-Vx.~Vsx)fw, wk.~ds

Est3 fWi.ssWk.ssds gik= 12(1-VxsVsx) s

D~ = 12(1-VxsVsx)fWkWi.ssdS

S

S

(13)

t3

(14)

S

t3

Di~ = "~"fwi,~Wk,~ds

(15)

S

Ci~k= t fuiukds S

Xkji = t

.] X-'kk S

vyi+wjwi)ds

(16)

Reassembling the terms of ~ and noticing that the 8#j are arbitrary, it becomes possible to write the GBT fundamental (system of) equations for orthotropic materials, which reads ExCik~k,xxxx- GxsDikt~k,xx + Bikt~k+ Xkji(Wkt~j,x),x = 0 9 (17)

Cik = C ~ + C ~

vsxEx | Dik = Di~k- Gxs (Dik+ O~)

This equation can be readily used to perform linear stability analyses of structural members subjected to arbitrary internal force combinations Wk=--ExC~k.x~. In the particular case of uniformly compressed members (columns), IV, corresponds to the axial force (compression) distribution resulting from the applied (axial) loads and includes the load/stress parameter, the critical value of which is sought. When compared with the available GBT fundamental equation, derived in the context of isotropic materials (Davies et al, 1994), equations (17) have the advantage of enabling a more straightforward incorporation of the material properties (constants). Moreover, it is important to remark that: (i) The tensors (13)-(16) differ from their isotropic counterparts (Davies et al, 1994) because of the presence of v~ Vxs (instead of v 2 - C~, D,~, Bik and D ~ and E, (instead of E - Bik). (ii) Equations (17) are slightly different from their isotropic counterparts, as (ii0 the first terms contain the longitudinal Young's modulus Ex and (ii2) the second terms contain both the shear modulus G= and the constant vs~Ex.

LINEAR STABILITY ANALYSIS First of all, attention is called to the fact that, in general, the GBT system of differential equations (17) is coupled. In order to take full advantage of the GBT potential, matrices [C], [B] and [D] (i.e., tensors Ctk, B~kand D~k)must be simultaneously diagonalised, by means of an orthogonal transformation defined by the solution of a standard eingenvalue problem. The corresponding eigenvector components are the nodal values of all the n+l (modal) "warping functions" Uk(4 "rigid-body"and n-3 cross-section deformation functions), each of them related to three section properties (Ca, Bkkand Da - diagonal components of [C],

333 [B] and [D]). All the 2 nd order effects taken into consideration by GBT are included in the last term of (17), which accounts for all the interactions between in-plane stresses and out-plane deformations. In fact, the non-null off diagonal components of matrix [X] (tensor X0k) are responsible for making system (17) coupled, which implies that the column buckling modes are linear combinations of the individual deformation modes. In geometrically linear (1 st order) problems, the removal of this last term from system (17) makes it uncoupled. In order to provide a better grasp of the concepts involved, let us consider the behaviour of a simply supported and uniformly compressed lipped channel column with cross-section dimensions: bweb=20cm, bfla,ge=lO cm, bl,p=4 cm, t=0.6 cm. The column is made of an e-glass/epoxy composite material, manufactured in Brazil and characterised by the elastic constants: Ex=17.236 GPa, Es=6.894 GPa, Gx~=2.895 GPa, Vx~=0.36, V~x=0.144 (Nagahama, 2000). A 9 node discretization of the column crosssection was adopted (3 intermediate nodes) and the simultaneous matrix diagonalisation involved the tensors C~k, B~k and D,k. Figure 2 shows the shapes of the 7 (out of 9) most relevant cross-section deformation modes obtained and table 1 displays the values of (i) the cross-section modal properties (Cu, Bkk e Dkk) and (ii) geometric matrix components (Xljk).

.......... i

!l |

03 I !

Figure 2: Most relevant cross-section deformation mode shapes.

TABLE

1

CROSS-SECTION MODAL PROPERTIES AND GEOMETRIC MATRIX COMPONENTS

Xljk

5 6 7 k ' C'k 0.00286DkkBkk' ...... 1 ' 2 3 ;""4 it i

I

9

28.8

1913.96 475.42 51722.2

'o

0 0 34558

o

0 0 .... 0 0.00237 0.00408 0.00922

o.o1555

0.18101

o

0 0 0 0 0

-01 0 9.258 1

o

0

o

9.258 0 -168.8

o

o

o

0 -0.1149 0 0.90303[ 0 -0.2223 0 [ 0.9910 0

6t,1

0

For the first four modes ( 0 ) - longitudinal extension- C//-area; | and | - major/minor axis bending - C22 and C33- major/minor moments of inertia; | - torsion - C44-warping constant and D44-St. Venant's constant), which are characterised by rigid-body motions, one has B~-O. On the other hand, the remaining three modes (| - symmetric distortional, | - anti-symmetric distortional and ~) - local plate) involve deformations and, therefore, are associated to Bkk~9. It should also be pointed out that no coupling is possible between the symmetric (even-number) and anti-symmetric (odd-number) modes, which stems from the fact that the corresponding Xljk off-diagonal components are null.

334 By incorporating the values presented in table 1 into the GBT equations (17) and assuming a sinusoidal buckling mode shape (with I half-wavelength), one is led to a system of algebraic equations defining an eigenvalue problem, the solutions of which are the roots of the corresponding characteristic equation (determinant). Figure 3(a) displays the variation of the bifurcation stress values orbwith the column length L (logarithmic scale), considering (i) each mode individually (upper, lighter and numbered curves) and (ii) coupling between all the modes (lower and darker curve), and figure 3(b) makes it possible to estimate the "degree of participation" of each individual mode in the column (coupled) buckling mode. Ob (MPa)

,00

\\7

so

i|

6o ................................. i........... i

40

, "

|

'

0%

|

i I

1 1 ! ! [..!i.il !

100%

~

!i

.........

0

\i

i [ii 100 (a)

FD

-~

,

i --i-

....

FTM

F~

i i, ; i i ,~. 1000

L (cm)

1

(b)

Figure 3" Variation, with L, of (a) Orband (b) the modal participation in the column buckling mode. The observation of figures 3(a) and 3(b) leads to the following remarks: (i) For L< 140cm, the "coupled curve" exhibits two local minima, corresponding, respectively, to bifurcation in a local plate mode (LPM) and in a symmetric distortional mode (DM). Depending on the value of the ratio b~t, the local critical stress may correspond to either the LPM or the DM (LPM, in the particular case depicted). Notice also that the critical buckling mode is (i~) a "pure" LPM (| for L<25cm, (i2) an "almost pure" DM (| + a bit of| for 60cm380cm, O'b~O'vr decreases continuously and two (global) buckling modes may occur, namely (iiil) a (major axis) flexural-torsional mode (FTM - |174 for 380cm840 cm. However, it is important to mention that the local minima (bifurcation stress values) do not depend on the half-wavelength number (m) displayed by the column buckling mode. The curves associated to m>l can be obtained by a mere horizontal shift of the m= I curve (the length ranges change accordingly).

335 Next, in order to validate the orthotropic GBT results obtained, they are compared with numerical ones, yielded by orthotropic finite strip analyses (Nagahama, 2000). Two columns are dealt with, namely (i) the lipped channel column analysed previously and (ii) an otherwise identical "Hat-section" (channel with outward lips) column. The GBT and finite strip (FS) results are presented and compared in figures 4(a) (lipped channel) and 4(b) ("Hat-section"). 13"b 90

(~b !

I ~ ~IIi - - GBT

90

I _

80

........

_

70

70

ii Fsl

.................. i............i ............................

6o

60 50

x....i...Ll.........l!! ..

40 30 : l0

(a)

"-' 100

L (cm)

ii\

L (cm)

30

300

1

10

(b)

100

200

Figure 4: Comparison between GBT and finite strip results. (a) Lipped channel (b) "Hat-section" Figures 4(a) and 4(b) show that the GBT and FS results are always practically identical (the difference never exceeds 3%). More specifically, the DM and FTM results are virtually identical and the (small) discrepancies are, basically, all restricted to column lengths associated to the vicinity of either (i) the LPM local minimum (GBT "below" FS) or (ii) the LPM-DM transition (GBT "above" FS). Moreover, it should be noticed that, unlike the lipped channel, the "Hat-section" does not exhibit a DM local minimum. This is due to the fact that the inward-outward lip change significantly decreases the cross-section torsional resistance (warping constant) and, therefore, the FTM bifurcation stress is lowered by a large amount.

PARAMETRIC STUDY

In this section, a limited parametric study is performed, aimed at investigating the variation of the bifurcation stress values and buckling mode nature with the (i) FRP properties and (ii) the cross-section dimensions. Concerning the first aspect, the previous lipped channel column is considered and assumed made of four different fiber reinforced plastics, all with an epoxy matrix and unidirectionally aligned fibers occupying 60% of the volume. TABLE 2 COMPOSITEMATERIALELASTICCONSTANTS ,

,,,

Fibers e-glass (GI) e-glass (G2) ..... kevlar (K) hs carbon (Ci) hm carbon (C2) .

.

.

.

.

.

.

.

Ex (GPa) 40 17.2 75 140 180 ,,

Es (GPa) 8 6.9 6 10 8 ,

,

Gxs (GPa) 4 2.9 2 5 5 ,

Vxs 0.25 0.36 0.34 0.30 0.30 ,

336 These fibers are made of either (i) e-glass (Gi), (ii) kevlar (K), (iii) high strength carbon (Cl) or (iv) high modulus carbon (C2), and their elastic constant values are presented in table 2 (Datoo, 1991). For comparison purposes, the case of the e-glass/epoxy composite (G2) used earlier, with lower elastic constant values (the fiber volume fraction is smaller) is also addressed. ab (MPa)

1

..............................................

i...

!00 1 50

"

0.

:

i

10

L (cm)

,

1O0

1000

Figure 5" Variation of Orbwith L, for five different FRP materials. Figure 5 depicts the variation of Orbwith L, for the five different composite columns. One observes that: (i) The oJ/,~'values are 37MPa (G2), 59MPa (G1 and K) and 108MPa (C! and C2). Notice that, in spite of the distinct material properties, the ~ ' values are identical for (il) G! and K, and (i2) Cl and C2. (ii) The t~ values are 43MPa (G2), 69MPa (G0, 77MPa (K), 138MPa (C!) and 140MPa (C2). Notice that the cr~ values are practically for Cl and C2. (iii) The (global) g/br values essentially increase with Ex (G2---~GI~K~CI---~C2). (iv) Apart from a small horizontal shift, the C! and C2 curves practically coincide, which means that their local and global buckling behaviours are virtually identical. (v) Decreasing the fiber volume fraction (G1--->G2)induces a similar reduction in the oJbe and ~ values LP J LP D (yb.r 7/59"~t~bDt;2/~b.C~l=43/69=O. 625). Next, it is intended to investigate the influence of the lipped channel column cross-section dimensions on the nature of the local critical buckling mode and corresponding orb value. Taking into consideration the cross-section proportions of the commonly used pultruded columns, the following dimension ratios are considered: (i) b,,/t=20 and 40, (ii) b/b,,,=0.25, 0.50 and O.75 and (iii) bl/bw=O.05 to 0. 40. Figure 6 shows the variation of the orb with bdbw, for the LPM (dotted line), DM (solid-dashed lines).

120

Gb (MPa)

40

~b (MPa) beqgw=0.75

b~/bw=0.50

............................................................................

90

bet,w=0.50 b~ow=0.75 bc'bw=0.25

6o

30

bt4bw=0.25

20

30 bl/bw

0, 0

0.1

0.2

(a)

0.3

0.4

0 -! 0

, 0.1

. 0.2

0.3

bl/bw 0.4

(b)

Figure 6: Variation of orb with b/bw and bl/bw. (a) bw/t=20 (b) bwlt=40

337 From this limited parametric study, it is possible to conclude that: (i) As expected, the cr~1' values are practically independent of the lips width (the dotted lines are practically horizontal). They only depend on the web (mainly) and flanges widths. (ii) The r vs. bdbw curves display maximum values for 0il) bdbwzO.16 (b/b,,,=0.25), (ii2) bdb,,,zO.24 (by~b,,,=0.50) and (ii3) bdb,~=0.38 (b/b,,=O. 75). Notice also that, for (unrealistically) large bl/bw and when bfbw < 0.75, cd~is not associated with a local minimum (dashed portions of the curves). (iii) For the lower plate slendemess value (bJt=20), the local critical stress is always associated to the DM (O'crt~OJ~). For the upper plate slenderness value (bw/t=40), on the other hand, it may be associated to either the DM (o'crt~6~) or the LPM (O'cr~(Tlbl'),depending on the b fib,,, value. (iv) The lip width values associated to critical DM-LPM transitions (white circles in figure 6(b) have an obvious practical interest, as they lead to the definition of "optimally efficient" lips (stiffeners). Therefore, design formulae to estimate such width values would be rather useful.

CONCLUDING REMARKS (i) (ii)

(iii)

(iv)

(v)

The existing GBT, developed in the context of isotropic materials, was extended in order to enable its application to orthotropic thin-walled structural members. The derived GBT equations were first validated, by means of a comparison with numerical results obtained through finite strip analyses, and, then, applied to study the local and global buckling behaviour of lipped channel FRP columns. In particular, a mixed flexural-distortional buckling mode was identified, which does not appear in isotropic (e.g., cold-formed steel) columns. Such mode was shown to be critical for intermediate length columns. A limited parametric study was carried out to investigate the influence of (iii0 the composite material properties and (iii2) the column length and cross-section dimensions on the critical bifurcation stress and buckling mode nature. The buckling behaviour of five geometrically identical lipped channel columns made of different fiber reinforced plastics, all with an epoxy matrix and unidirectionally aligned glass, kevlar or carbon fibers, was investigated. The analysis unveiled markedly different behaviours and the results obtained showed the carbon fibers to be the ones providing the highest local and global buckling resistance. Design formulae to evaluate the widths of"optimally efficient" lips (i.e., widths associated to critical DM-LPM transitions) would be very useful in practice and, therefore, should be sought.

References Datoo MH (1991), Mechanics of Fibrous Composites, Elsevier Science, London. Davies JM, (1999), Modeling, Analysis and Design of Thin-Walled Steel Structures, Light-Weight

Steel and Aluminium Structures, Elsevier Science, London, 3-18. Davies JM, Leach P and Heinz D (1994), Second-Order Generalised Beam Theory, Journal of Constructional Steel Research, 31,221-241. Nagahama K, Batista E (2000), Stability Analysis of Glass Fiber Composite Pultmded Members, CDROM Proceedings of the XXIX Jornadas Sudamericanas de Ingenieria Estrutural, Punta Del Este, Uruguay, paper # 2.7.5 (20 pages). (in portuguese) Schardt R (1989), Verallgemeinerte Technische Biegetheorie, SpringerVerlag, Berlin. (in german)

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Third International Conference on Thin-Walled Structures J. Zara~, K. KowaI-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

339

SHEAR CONNECTION BETWEEN CONCRETE AND THIN STEEL PLATES IN DOUBLE SKIN COMPOSITE CONSTRUCTION Howard D Wright 1 Anwar Elbadawy 2 & Roy Cairns 3 1Professor, University of Strathclyde, Glasgow, UK 2 Research student, University of Strathclyde, Glasgow, UK 3 Lecturer, University of Strathclyde, Glasgow, UK

ABSTRACT Double skin composite (DSC) systems are formed from steel-concrete-steel sandwich elements that consist of a layer of un-reinforced concrete, sandwiched between two layers of thin steel plates. These in turn are connected to the concrete by welded shear connectors. Shear studs or bars are used to transfer slip shear between the outer steel skins and concrete core. To investigate the behaviour of the shear connectors when welded to the normally thin steel plates an experimental study, on nine model push-out tests, has been carried out and is presented in this paper. The connectors were 6mm diameter bars welded between each plate providing a connector to plate thickness ratio of 3. Micro-concrete with maximum aggregate size smaller than 2.41 mm was used. The micro-concrete core was pushed through the plates in direct shear. Failure modes are defined for each of three series of tests and more detailed observations regarding the structural action of this form of construction are given. The studies show that failure occurred by yielding for all bar connectors and that the spacing of connectors influenced the extent of cracking in the concrete core.

KEYWORDS Composite construction, Push-Out tests, Shear connectors, Thin-Walled Structures, Micro-Concrete

INTRODUCTION Double skin composite elements (DSC) are basically steel-concrete-steel sandwich elements that consist of a layer of un-reinforced concrete, sandwiched between two layers of thin steel plate. These in turn are connected to the concrete by welded shear connectors.

340 Several experimental and analytical studies have been carried out to understand the behaviour of DSC elements (Oduyemi and Wright 1989, Wright, Oduyemi & Evans 1991, Wright, Oduyemi & Evans 1991, Roberts et al 1996). The main conclusion from these studies was that DSC elements could generally be designed in accordance with normal reinforced concrete practice satisfying the following criteria; (a) Yielding of the tension steel plate. (b) Yielding or buckling of the compression plate. (c) Shear failure of the connectors. (d) Crushing of the concrete in compression. (e) Shear failure of the concrete. (f) Pull out failure of connectors. Of these, (b), (c), (e) and (f) are specific to DSC. Those criteria specific to DSC are influenced by the thickness of the steel plate that is typically only 1/20th of the total element thickness and consequently DSC easily falls within the category of a Thin-Walled Structure. An experimental study on DSC elements will be presented in the paper. The main aim is to investigate the behaviour of bar shear connectors when welded to thin steel plates. The experimental study involves push tests consisting of two thin steel plates connected together through a core of concrete by bar shear connectors. The connector diameter to plate thickness ratio was 3. The concrete core was pushed through the plates in direct shear allowing the behaviour of the bar to plate connection to be observed without the need for full panel bending tests. Current design guidance for concrete slab to steel beam flanges in composite beams merely places a limit on steel plate thickness in relation to stud diameter. The limit is often inappropriate for DSC and the paper will comment on this.

TEST PROGRAM

Nine model push-out tests were fabricated. Micro-concrete was used the mix of which was established by Hossain (1995). Table 1 show the properties of micro-concrete of control mix. TABLE 1 MICRO-CONCRETE PROPERTIES Cube Strength fo~ (N/mm 2) 28.0

Cylinder Strength fo

(N/mm 2) 19.55

Splitting Strength fs (N/ram 2)

Density (Kg/m s)

1.13

2400

Modulus of Rupturefb (N/mm 2)

(KN/mm 2)

4.51

14.55

Ec

,,, _.

The nine models are classified in three series and identified in the text as POT 1 to POT9. A typical pushout test model is shown in figure 1 and full details of each series are given in table 2. In summary: Three test series were carried out as follows: l-Three models with three connectors in one column (1 x3) with spacing 100ram. 2-Three models with six connectors in two-columns (2x3) with spacing 150ram in two directions. 3-Three models with six connectors in two-columns (2x3) with spacing 200mm in two directions. The perimeter of the plates was stiffened with additional steel frame members attached to the plates using a sufficient number of bolts. The models were tested by applying uniformly compressive force over the breadth of the top surface of micro-concrete core to push it through the plates in direct shear.

341 p.. Test

Frame

20 mm DIe. Corner Pin with $poeer~/

Members

?irection

i

o?s o,o. c. . . . . ,,o . , ,

[--,f>'7-

t !tU 6 m m Shear Stud

i

J

I

-]~-4,~ . It. ~ _

11

'.

I ! i

~

I !1

'

I!1

:! -I Te,~ .... ,ember,

I

Ill;

.l.h

!

I i-_-_-_-_-_-_-_-_-~

I

:I ! I

of Loading

I (

'

i

If

I,/"

I ~f:

[

!

I i I :Test

I

=

i

I i I:~

Frame

Members

1" Geometry

'

'

.!

\i

!

\1:1o mm Die. Intermediate

;l

ll

Elevation Figure

m' |-'.

6 ,,n, Shear Stud

..,..I."'.J.,. Connectors

!

I

i-tll

:

;'

I i I : ",.1o mm m I I~-F----//Dio.lntermediofe Coifs !LI

I

:"~ """ ~~.....

Section of

Push-Out

Test Model

TABLE 2 DETAIL MODELS OF PUSH-OUT TEST No

Series

Specimen

1

2

~

POT1 POT2 POT3 POT4 POT5 POT6 ,, POT7 POT8 ......... POT9

Of Studs lx3 lx3 lx3 2x3 2x3 2x3 2x3 i ,, 2x3 2x3

Stud Spacing Horizontal Vertical mm

111171

150 150 150 200 200 200

100 100 100 150 150 150 200 200 200

End spacing mm Top & Left & Bottom Right 100 50 100 50 100 50 75 75 75 75 , 75 75 160 loo 100 100 100 100

Material properties The properties of the steel plates were determined from tensile tests on random samples taken from each batch of steel. A summary of the steel plates tensile test results shown in the table 3. The properties of the bar connectors were determined from tensile tests on three specimens cut at random from the bar material. A summary of the bars' tensile test results shown in the table 4. The micro-concrete consisted of Ordinary Portland Cement, sea-dredged sand of 2.41-mm maximum size. A summary of the results is given in table 1. The properties of micro-concrete in each individual series of models were determined from at lest three tests on 100-mm cubes and three tests of 200 mm long by 100-mm diameter cylinders

342 (for split cylinder tensile and compression cylinder tests). A summary of these test results is shown in the table 5. The models were cast vertically and in stages. Following casting the models were covered with polythene and the micro-concrete was then allowed to cure in air until testing commenced. Figure 2 show the models as cast. TABLE 3 STEEL PLATE PROPERTIES Thickness (mm) 1.93

0.2%Proof stress (N/mm2) . 315

Ultimate stress (N/mm2) 393

E$

(r,~/mm~-) 195

TABLE 4 BAR CONNECTOR PROPERTIES Diameter (mm) 6.22

0.2%Proof stress (N/mm2) , 360

Ultimate stress (N/mm2) 517

ES

(KN/mm2) 196 ,,

TABLE 5 PROPERTIES OF MICRO-CONCRETE IN PUSH-OUT TESTS

Series

Cube Strength (N/ram 2)

Cylinder strength (N/mm 2)

Splitting strength (N/mm 2)

Series 1 Series 2 Series 3

25.33 24.33 29.75

20.31 18.72 22.73

2.53 1.69 1.85

Figure 2: Micro-Concrete casting

Remark

7 days 7 days 12-28 days.

343

Test procedure and instrumentation The compressive force was applied to the top of micro-concrete core of the model by means of a 250 KN actuator using deflection control mode. Model instrumentation is shown in Figure 3. The movement of the micro-concrete core relative to the steel plate was measured by two dial gauges, which were attached to concrete core at 5 cm from the bottom level of concrete core. One was attached on each face. The load slip values were simultaneously recorded and printed.

Figure 3: Model instrumentation

Loading and test observation At the start of each test the initial dial gauge reading was recorded. The compressive force was applied to the model by increasing increment loads gradually until the failure load. The slip between steel plates and micro-concrete core was recorded at each increment load until end of the test. Figure 4 shows the typical load-slip relationship of the push-out tests for series 1, 2 and 3. The observations on each test series is as follows:

Series 1 (Specimen POT1, POT2 and POT3) During the test cracking noises were heard at loads between 10-14kN, 22-28kN and 37-38kN. All specimens cracked vertical in the middle of concrete core along the line of connectors. The specimens failed at a compressive load of 37kN, 40kN and 37kN for specimen POT1, POT2 and POT3 respectively. In specimen POT 1 it was noted that a part of concrete core disturbed one of the dial gauges following lateral movement. In specimen POT3 the micro-concrete core started cracking in the middle of concrete core from the top and separated through the connectors in an inclined crack above the dial gauges. Bar yielding occurred in all the specimens.

344 Series 2 (Specimen POT4, POT5 and POT6) During the test cracking noises were heard at loads between 15-18kN, 44-56kN and 95-107kN. All specimens cracked vertically in two lines through the connectors. The specimens failed at a compressive load of 95kN, 107kN and 101kN for specimen POT4, POT5 and POT6 respectively. Bar yielding failure occurred in all the specimens.

i

./

j

i

i

! !

i ! t .

-1

.....

:

.~

--,r , .

'

,

0

,

.

1

,

.

,

. . . . . . .

2

,. . . .

~. . . . . .

3

4

I,

s.rm3

i ....

I i

w

7

,

5

"

i,

Blip mm

Figure 4: Typical load-slip of the push-out tests

Series 3 (Specimen POT7, POT8 and POT9) During the test low noises were heard throughout and a high cracking noise at failure. On dismantling none of the specimens were found to have a cracked concrete core. The specimens failed at a compressive load of 108kN, 77kN and 86kN for specimen POT7, POT8 and POT9 respectively. Yielding failure of the bars occurred in all the specimens. This led to separation of the connectors from the steel plates.

DESIGN STUDY The behaviour of the DSC systems in general has been reported in many studies. This paper concentrates on the behaviour of the bar shear connectors when welded to thin steel plates. The arrangement and the properties of bar shear connectors could be designed using BS 5950 pt 3 (1990) EC 4 (1994) or studies by Wright et al (1991) Roberts (1994) and Obeid (1986).

Stud~plate ratio BS 5950 limits the ratio between shear stud diameter to steel plate thickness as not greater than 2.5. EC4 uses the same ratio, with an additional criterion that the connector diameter must be less than steel plate thickness. Obeid showed that an appropriate limit to the ratio between stud diameter to steel plate thickness was 3, a figure he derived from a study of 3, 4, 5, 6 and 8 mm shear studs connectors welded to

345 thin steel plates. The tests confirmed that a figure of at least 3 was suitable as there was no indication of plate failure or stud pull-out in any of the tests.

Steel plate buckling and shear connector spacing The ratio of the centre to centre distance between stud shear connectors Sc to plate thickness ts~ is limited in BS 5950. The maximum spacing between studs being 600 mm or 4 times the concrete core thickness and the minimum spacing being not less than the 5 times the stud diameter. The limitation in EC4 is stated as maximum stud spacing to plate thickness ratio of 40. Wright (1993) confirmed that this ratio should not be greater than 40 for stud layouts where a column-buckling mode is likely. Column buckling occurs where the studs are spaced regularly allowing a line buckle to form between rows of connectors. Wright evaluated this ratio when the compression plate was in contact with a rigid medium (as in the case of DSC elements) and found that the limit increased to 52. In this paper this ratio was 50 in series 1, 75 in series 2 and 100 in series 3. During the tests local buckling was not noted however this was due to the fact that these tests are in direct shear with little compression developing in the plates. It should be noted that the direction of the load might not always be perpendicular to the stud layout. In two-way spanning panels the yield lines may be diagonal to the edges. In this case the ratio must be checked with the diagonal spacing between shear connectors. TABLE 6 COMPARISON WITH VARIOUS CODES OF PRACTICE AND RESEARCH STUDIES Data Connector diam/plate ratio Connector spacing

. . . .

Connector spacing/plate thickness Connector capacity

BS 5950 pt.3 <2.5

< 2.5 & tsc< d

>30 mm <200 mm

>11.41 mm <150 mm

Sc>30 Sc<200

<40

Tabulated values not given for 6 mm bar

Conc 4.7kN Bar 9.0kN

EC 4

Previous Studies -- 3(Obeid)

Free 40 (wright) Contact 52 twright) Conc 4.7kN f~~ Bar 6 3kN (~

Push-Out Test

Rema~

100 mm 150 mm 200 mm 50,75,100mm 71,106,141mm

Series 1 Series 2 Series 3 Long Diagonal

6.17-6.59kN 7.92-8.92kN 6.42-8.98kN

Series 1 Series 2 Series 3

Shear stud connector capacity

Previous studies (Oduyemi and Wright 1989, Wright, Oduyemi & Evans 1991, Wright, Oduyemi & Evans 1991, Roberts et al 1996) showed that the strength and stiffness of stud connectors in DSC elements is significantly less than that determined from push-out shear tests. Therefore, the design resistances of studs attached to the compression and tension plates are limited in the some Codes and studies. In BS 5950, the capacities of shear connectors are taken as 80 % and 60 % of the characteristic push test resistance respectively. The values of the characteristic resistance of shear studs ranging in diameter between 13mm to 25mm are tabulated. EC4 limits the characteristic resistance to the lesser of: PRd=0.29ad 2 (f~kE~)~ / Yv or PRd=0.8fu nd2/4~,v. Roberts uses the same equations. Wright (1991) used

346 similar equations but limits the design resistances of shear stud in the tension zone to 50% of the characteristic resistance of shear studs. Obeid showed that the characteristic resistance of 6mm shear studs was 7.27kN in tension and 6.3kN in shear. In the tests reported in this paper the resistance of 6ram bar shear connectors was found to be between 6.2 to 6.6kN in series 1, 7.9 to 8.9kN in series 2 and 6.4 to 9kN in series 3. Table 6 shows comparison between the results of push-out tests with the Codes and other research work.

CONCLUSIONS The aim of this paper has been to investigate the behaviour of the bar shear connectors when welded to thin steel plates. The connector diameter to plate thickness ratio of 3 as found by Obeid has been found to be an appropriate limit based upon the limited information provided by series 2 and 3 tests. Connector spacing to plate thickness ratios greater than those recommended by codes and previous researchers appear to be possible, however the tests did not induce significant compression in the plates and further work is needed to confirm this. The connector capacity determined by both EC4 and Roberts (using concrete failure as the criterion) was less than that found in all of the tests. Series 1 tests had a single row of connectors and may not be representative of real practice. However the pure shear capacity determined using EC4 rules only marginally overestimated the test results for series 2 and 3. This suggests that despite concrete cracking, the full resistance of the bar connector may be used when more than a single row of connectors is provided. Obeid's experimental results (for pure shear on stud connectors) provide the closest match to the minimum test result for series 1. It should be noted that the bar connectors used in the DSC panels reported here are welded to both plates and may therefore provide better resistance than headed studs to tensile forces developed in the connector as it deforms under shear load.

REFERENCES Oduyemi T.O.S. & Wright H.D., An Experimental Investigation into the Behaviour of Double-Skin Sandwich Beams, Journal of Constructional Steel Research, Vol.14, pp. 197-220, (1989) Wright H.D., Oduyemi T.O.S. & Evans H.R., The Experimental Behaviour of Double Skin Composite Elements, Journal of Constructional Steel Research, Vol. 19, pp. 97-110, (1991) Wright H.D., Oduyemi T.O.S. & Evans H.R., The Design of Double Skin Composite Elements, Journal of Constructional Steel Research, Vol. 19, pp. 111-132 (1991) Roberts T.M., et al, Testing and Analysis of Steel-Concrete-Steel Sandwich Beams, Journal of Constructional Steel Research, Vol. 38, pp. 257-279, (1996) Hossain, K.M.A., In-plane shear behaviour of composite walling with profiled steel sheeting, Ph.D., (1995) BS 5950 Part 3.1, The Structural Use of Steelwork in Building, Design in Composite Construction, Code of Practice for Design of Composite Beams, British Standards Institution, London, (1990) European Committee for Standardisation (CEN), Eurocode 4 Part 1.1, Design of Composite Steel and Concrete Structures, General rules and rules for buildings, DD ENV 1-1 (1994) Obeid G. A., Stud welding and its application to ceiling supports, MSc., Cardiff, University of Wales, (1986) Wright H.D., Buckling of plates in contact with a rigid medium, The Structural Engineer, Vol. 71 No. 12, pp. 209-215, (1993)

Section VI DYNAMIC LOADING (CYCLIC, IMPACT AND VIBRATION)

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Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

349

R E G U L A R A N D C H A O T I C B E H A V I O U R OF F L E X I B L E PLATES

J. Awrejcewicz 1, V. A. Krysko 2, A. V. Krysko 2

1Department of Automatics and Biomechanics, Technical University of Lodz, 1/15 Stefanowski St., 90-924 Lodz, Poland, [email protected] 2Department of Mathematics, Saratov State University, B. Sadovaya 96a, 410054 Saratov, Russia, [email protected].

ABSTRACT Forced oscillations of flexible plates with a longitudinal, time dependent load acting on one plate side are investigated. Regular (harmonic, subharmonic and quasi-periodic) and irregular (chaotic) oscillations appear depending on the system parameters as well as initial and boundary conditions. In order to achieve highly reliable results, an effective algorithm has been applied to convert a problem of finding solutions to the hybrid type partial differential equations (the so called yon K~rm~in's form) to that of the ordinary differential equations (ODEs) and algebraic equations (AEs). The obtained equations are solved using finite difference method with the approximations 0(h 4) (in respect to the spatial co-ordinates).

KEYWORDS Chaos, flexible plates, von K~rm~n's equations, finite difference method, partial and ordinary differential equations.

INTRODUCTION Flexible plates periodically excited create a complex dynamical system which can exhibit various dynamical behaviour. A vibrational process can be characterized by complex resonance structures, a collapse of vibration regime leading to a change of a spatial-temporal state, an occurrence of standing or moving waves, a stability loss with respect either to symmetric or non-symmetric modes, and others [Vol'mir A. S. (1956), Kantor B. Ya. (1971), Gould P. L. (1999), Libai A. and

350 Simmonds J. G. (1998), Yu Y.-Y. (1999), Awrejcewicz J. and Krysko V. A. (1999-1), Awrejcewicz J. and Krysko V. A. (1999-2)].

EQUATIONS, INITIAL AND BOUNDARY CONDITIONS The known equations governing dynamics of flexible isotropic plates are taken as the mathematical model. A plate material is elastic and both Hooke's and Kirchhoff's hypotheses are valid. A shell is thin and a hypothesis on an average deflection takes place. The known von K~irm~n equations [Kantor B. Ya. (1971)] satisfy the listed hypotheses, and they have the form _

C~2W

02w Ow 1 V 2 V2w + L(w F) - P~-~x 2 + q, at 2 + r = -12(1 - u 2) V 2 V 2 F = _ l L ( w , w) (1) 2 where w(x, y, t) is a deflection function along z co-ordinate oriented toward the Earth centre, and F( x, y, t) is the Airy's function.

The equations (1) are already transformed to the non-dimensional form. The following relations between dimensional and non-dimensional quantities hold a

= ax,

~ = by,

P~=

~ = 2Hw,

E(2H) 3 b2 Px,

A = -~,

F=E(2H)aF,

t = tot,

q=

~=(2H)r

E(2H) 4 a2b2

The following notation is used: P,(y, t) - longitudinal load along Ox axis; 2H - plate thickness; a, b - plate dimensions; ~ - damping; E - Young modulus; v - Poisson's coefficient (during calculations v - 0.3 has been taken); lower left corner of a plate serves as a origin of a co-ordinate system Oxy. The plate volume is G E {x, yl0 _< x _< 1,0 _< y <_ 1}, - H _< z <_ H. The Airy's function satisfies the following relations: 02 F Txz = - - - Px Oy 2 '

Tyy

02 F = -~x '

02 F Txy = - ~ , cOxOy

(2)

where: T,x, T~y and Txy are the stresses occurring in the middle plate surface, and L(w, F) is known non-linear operator. The appropriate boundary and initial conditions are attached.

ALGORITHM In order to reduce the PDEs (1) with attached boundary and initial conditions to ODEs the finite difference method with the approximations 0(h 4) has been applied. For instance, a difference approximation to equations (1) needs 25 points pattern for 0(h 4) approximation. In spite of that we need to formulate more complex requirements for the off-contour nodes, which permit to take relatively large steps in respect to x and y. The latter leads to essential decrease of an order of difference equations which implies a decrease of a computational time.

351 For the approximation 0(h*) the following mesh is applied 1 (~h = {0 < xi, yj <_ 1,xi = ih, yj = j h , h = - - ~ , i , j = 0, N}, and the PDEs are exchanged by their difference approximations. The following initial conditions are taken wij It=o = A sin 7rxi sin 7ryj,

(3)

wt,ijlt=o = O.

The obtained ODEs are transformed via the relations

dwij dt

(4)

= wij'

to the first ODEs with respect to deflections w 0 and velocities w'ij and to the algebraic equations with respect to the Airy's function Fij: dw'0 -+ ~wij = -{A~(w) + B ( w , F ) } dt t

+ q,

(5) (6)

D k ( F ) = Ek(w).

In order to solve the equations (4)-(6) with the initial conditions (3) and boundary conditions wij = Fij - 0 for xi - yj = 0; xi = yj = 1 the following calculation algorithm is applied. In the first time step the initial conditions (3) are used, and Fij is obtained from the algebraic equation (6). The system of algebraic equations can be solved either using the exact Gauss method or using one of the iterational methods (for instance, the relaxation method). This leads to definition of the right hand side of equation (5). Then, applying the Runge-Kutta fourth order method to (5) and (4), the values of wij are found for the time At, and the calculation procedure repeats. It is possible to apply the Runge-Kutta method with an automatic choice of an integration step. It should be emphasized, that a use of approximation along the spatial co-ordinates 0(h 4) and the Runge-Kutta fourth order method with respect to time leads to identity of the approximations with respect to spatial and time co-ordinates.

NUMERICAL

TESTS

In order to provide numerical tests the multi-dimensional problems of plates theory are reduced to one dimensional (ODEs with respect to time) using finite difference method. Hence, a question concerning a convergence of the methods with respect to spatial and time co-ordinates as well as about optimal choice of a solution to AEs appears. The first part of the mentioned problem is solved using the Runge principle: the solutions with differential integration steps with respect to spatial co-ordinates are compared. As an example square plates (A = a / b w=

1) with the boundary conditions

02w =F= Ox 2

02F =0 Ox 2

for

x=0;1,

352 02w w=

Oy2

02F = F=

Oy 2

=0

y=0;1

for

(7)

and initial conditions (3), and with the initial excitation amplitude A = 1.10 -4 are studied. The longitudinal load is Px = Posinwt (frequency w = 5.72, damping coefficient c = 1). Only first plate quadrant is taken into account due to two axes of symmetry along Ox and Oy, i.e. only symmetric solutions with respect to Ox and Oy are considered. The computational step has been P.=P.sin(ot,=5.72, P.=6, e=l, W.=Asinnx sinny

~.---0.0001 FFT

Poincar6 psudosection 2.0 --I

Power spectrum 1.E-4] t(E-5 -~

1.0-. 0.0-

"~

1.0E,-8-~

0.5--

9 ..o.

1.0E-11nl -2.0

'

I

-2.0

'

1

0.0

-20.0

'

t

-2.0

2.0

0.0

'

0

!"

l"

I

I ' I ' I

2

4

6

8

10

1.0E-12 ~ ' I ' I ' I ' l ' I ' I 0

12

2

4

6

8

10

12

Power spectrum

FFT

0.0-

0.0--

0.0

A=1.25

wt(w)

Poincar6 psudosection

I

2.0

1.0 ~

1.{E--4~ 1.~.8 ! 1.0~ ~

o.s-~ !

1.0E-8 ! 1.0E-9 1.0E-11

-2.O

'

I

'

0.0

-2.O

I

-20.0

'

1

-2.0

ZC

o.o ~!., I"1

0.0

0

2

4

6

I'1'1 8

10

'=-'~i 1.0E-12

12

0

' I'I' 2

4

I'

I'

6

8

I ' I 10

12

A=1.375

w,(w)

Poincar6 psudosection

FFT

Power spectrum 1.0E-4

!

0.0--

:"'"

9 0 e

.o.

'

Figure 1'

!

1.(~-8

0.5 - i

.

1.0~10 1.CE-11

.~,::..;" -2.0

I

1.0E-6

~ ,I ,~ 9

0.0--

-2.0

1.0E~

I 0.0

'

I 2.(

-20.0

i

t -ZO

'

l 0.0

'

I 2.0

0.0 - - ! " 1 " ' 1 0

2

4

1.0E-12

I'1'1 6

8

10

12

'l'

' 2

' 6

' 8

'/ 10

12

Poincar~ sections and pseudosections, wt(w), FFT and power spectra for A =

0.0001; 1.25; 1.375. taken as h - l/S; 1/10; 1/16. The algebraic equations (AEs) has been solved using the Gauss method and the relaxation method. The load amplitude was set to P0 = 6. Analysis of the obtained results leads to conclusion that for further investigations it is sufficient to take h = 1/8 with respect to spatial co-ordinates not only for harmonic and quasi-periodic solutions but also for those of critical and chaotic states. The Gauss method applied to solve linear AEs set leads to decrease of a computational time. Both qualitative and quantitative investigations of complex

353 plate oscillations harmonically and longitudinally excited practically overlap for the considered step h. According to the Runge rule At = 2 . 1 0 -4 has been taken while using the Runge-Kutta method.

INFLUENCE

OF DAMPING

e AND EXCITATION

AMPLITUDE

A.

The influence of the control parameters A and e is investigated for A - 1, v = 0.3, the boundary conditions (7) and the initial conditions (3). The analysis is carried out for A = 1.10-4; 1.25; 1.375 for the fixed e = 1 and P0 = 6. In general, three dimensional phase portraits (wtt, wt, w) and their projections wtt(wt), wit(w), wt(w), the modal portraits w~(w) for the plate point x = y = 0.25, 1

as well as the dependence

1

win = f f w(z, y, t)dxdy have been analysed. In the Figure 1 for the 0

0

given A values the Poincar~ pseudosections, the Poincar~ sections, F F T and the power spectra are shown. Analysis of the computational results shows that oscillations have two frequencies, and their values do not depend on the initial excitation amplitude.

0A

0.4 '

Wav

0.3

0.3

/

0.2

0.2

0.1

0.1 0

Way

9

Ai~

.

0~ 0:~ ~~ ~ ~

~'~' ~' ~

~'

0

-0,

....

lain

-0.1

-0.2

-0.2

-0.3 -0.3

-0.4 -0.4

Fig. a)

Fig. b)

Figure 2" Symmetric (a) and non-symmetric (b) oscillations (P, A sin 7rx sin Try, e = 1).

=

6sin5.72t, W~n =

For each of the mentioned amplitudes an internal synchronization on the frequency w17 occurs. Comparing the computational results for A - 1.25 and A - 1.375 it should be emphasized, that the phase portraits on a plane and in the three dimensional space are rotated for 180 ~ The +

following indicator way - win +2 win with respect to the initial excitation amplitude for symmetric (Fig. 2a) and non-symmetric (Fig. 2b) oscillation modes is constructed. Here the critical values of A appear, for which plate oscillations around a new equilibrium state occur, i.e. a slight change of A leads to a change of the plate state. For non-symmetric oscillations (Fig. 2b) there are two such values of A. This corresponds to a rotation of 180 ~ of the phase portraits, Poincar~ pseudosections and the Poincar~ sections wt(w) in every period of the excited load.

354

Figure 3: Time histories, Poincar~ sections and pseudosections, FFT, power spectra, modal portraits for the indicated parameters: (P0 = 20).

355

P.=P.sin5.72t, P.=26, e = l ,

~t)

~,(~)

Poincar6 pseudosection 1(1o -

(15-

IO.O-

"9 I" o.o-

"'~r

W

i

J

.lo~o

'

o

1 5o

'

I loo

-IO.O o

t e [0,100] axll} -

C

'

~

'

l(x)

I

'

.I(LO

+

i

-20~o

I

o.o

1(1o

[

,

1 o~o

,

,

-lo.o

'

I lo.o

wtt(w)

wtt~vt)

o-

' +:i:ii:i:

i

I o

-20000--t0

'

FFT

F-+i, 0

2

,

,~, 4

B,

, 8

~,: 10

I 0

'

~OE*O" lO

12

12

Wxx(Wx)

loo]

-~o.o ~ -10

, 110

0

,~.~4 -!_

1.0E-7 "I! L

~.os4 5

2

4

I

L

2

4

I

,

1.0E.o -~ 1.o[-t0 --; 1.0E-11 1.0E-12 "i 1.0E-13

1.0E,.10 "i 1.0E-11 --~ 1.0E-12 q 1.0E-13 : . . . . 0

1 t0

Power s ~ectmm

1.0E4 1.0E.6 -~ 1.OE4 -; tOE-7 -; 1.0E-8 -~ 1.0E.e "i

2

o.o

'

toe.+4

~ 6

8

' t 10

0

12

Wxx(W)

6

8

1021

Wx(W)

200.0 --

20.0-

x = v = 0.25

~176~ I ~ 49

2110 --

o

I 0

'

I 2D

-200.0 45

t 0

I 5

-20.0 . . . . .6

Wx(W)

t=22.52

t---46.81

; .....

1 0

t=71.14

:. ",.'~t~.~. -20,0

.....

-so

oo

~o

Figure 4: Time histories, Poincar~ sections and pseudosections, FFT, power spectra, modal portraits for the indicated parameters: (P0 = 26).

The Poincar~ pseudosection is defined as w(t) against deflection shifted in time, i.e. [w(t), w(t+ T)]. A deflection w(t + T) is related to a velocity wt(t) and in result we get the similar properties as in a case of use a full phase plane [w(t), wt(t)]. We have to emphasize, that the above considerations are valid for fixed values of P0 and the applied boundary and initial conditions, as well as for the period of the excited longitudinal load. The non-linear plate dynamics is investigated for the following parameters: fixed values of e = 1 and e = 5, the given boundary and initial conditions, A = 1 . 1 0 -4, P0 E {20; 26}. In the Figures 3 and 4 time histories w(0.5; 0.5; t), phase portraits, FFT for the plate center (x = y = 0.5) as well as the plate surface configurations for different time values are reported. The numerical analysis leads to conclusion that increase of e leads to the essential change of scenario leading to chaos using P0 as the control parameter 9 As an example, for e -- 1 and P0 = 26 we have a chaotic state, whereas for e = 5 the plate undergoes

356 the post critical oscillations.

CONCLUDING REMARKS Flexible plates periodically excited create a complex dynamical system which can exhibit various dynamical behaviour. The vibrational process can be characterized by complex resonance structures, the collapse of vibration regime leading to the change of a spatial-temporal state, the occurrence of standing or moving waves, the stability loss with respect either to symmetric or non-symmetric modes, and others, which have been discussed and illustrated. To conclude we briefly summarize main results of the paper: 1. The new technique with described algorithms and numerical tests have been proposed and applied on a basis of difference approximations to the analysed von Ks163 equations. It possesses many advantages in comparison to potentialy other applicable methods (an error estimation using Runge principle is given, among others). 2. The new non-linear phenomena associated with chaos are reported within one control parameter P. 3. Special tools to analyse systems with infinite dimensions has been applied and their physical meanings have been explained. REFERENCES

Awrejcewicz J. and Krysko V. A. (1999-1). Dynamics of Continuous Systems. WNT, Warsaw, in Polish. Awrejcewicz J. and Krysko V. A. (1999-2). Dynamics and Stability of Shells with Thermal Excitations. WNT, Warsaw, in Polish. Gould P. L. (1999). Analysis of Shells and Plates, Prentice Hall. Kantor B. Ya. (1971). Nonlinear Problems of Non-homogeneous Shallow Shell Theory. Naukova Dumka, Kiev, in Russian. Libai A. and Simmonds J. G. (1998.)The Nonlinear Theory of Elastic Shells. Cambridge University Press, Cambridge. Volmir A. S. (1956). Flexible Plate and Shells. Gostekhizdat, Moskva, in Russian. Yu Y.-Y. (1999). Vibrations of Elastic Plates. Springer Verlag, New York.

Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

357

SEISMIC PERFORMANCE OF ARC-SPOT WELD DECK-TO-FRAME CONNECTIONS W.F. Bond l, C.A. Rogers !and R. Tremblay 2 1 Department of Civil Engineering and Applied Mechanics McGill University, Montreal Quebec, H3A 2K6, Canada 2 Department of Civil, Geological and Mining Engineering l~cole Polytechnique, Montreal Quebec, H3C 3A7, Canada

ABSTRACT A significant number of single-storey steel buildings in North America are located in regions of active and moderate seismicity. These structures typically include a roof deck diaphragm made of steel deck units that are fastened to the supporting steel framing. Arc-spot welds are the most common type of deck-to-frame connection found in Canada and in the United States. This structural system forms a deep horizontal girder capable of transferring lateral loads to the vertical bracing elements. The overall objective of this research was to investigate the possibility of permitting the metal roof deck diaphragm to absorb earthquake induced energy through plastic deformation at the connections. This paper provides details of the inelastic cyclic response for deck-to-frame arc-spot welds with and without weld washers. Test specimens, which included three types of sheet steels and two thickness frame members, were subjected to monotonic, 2 Hz cyclic and simulated earthquake loading. The laboratory results show that the use of weld washers can improve the ductility, strength and energy absorption ability of the connection when weld quality is high. However, caution is warranted because if the weld metal does not fully penetrate into the frame material then premature failure due to shear fracture through the cross-section of the weld may take place. Hence, if adequate weld penetration does not occur, the addition of weld washers may not increase the seismic resistance of the roof diaphragm to expected levels.

KEYWORDS Sheet steel, connections, roof diaphragm, arc-spot welds, washers, seismic, ductility, energy absorption.

358 INTRODUCTION A significant number of single-storey steel buildings; used in North America for light industrial, commercial, and recreational purposes, are located in regions of active and moderate seismicity. To resist lateral loads resulting from seismic ground motion, the structure generally includes a metal roof deck diaphragm and vertical steel bracing. The roof diaphragm is made of steel deck units that are fastened to the supporting roof framing to form a deep horizontal girder capable of transferring lateral loads to the vertical bracing elements. Arc-spot welds are the most common type of deck-to-frame connection used in Canada and in the United States. A hole is bumed in the deck by the arc and then filled with weld metal to fuse the sheet to the framing member. Test results have shown that the common arc-spot weld can carry significant loads but only over a small displacement (approx. 2 mm) (Rogers and Tremblay, 2000a, b). Connections fabricated with weld washers are often recommended for the improvement of weld quality and connectivity when thin deck material (t < 0.71 mm) is to be attached to the frame of the structure (AISI, 1997). The washer dissipates heat induced during the welding process and allows for greater connectivity to occur between the sheet, weldment and frame. The use of weld washers for thicker deck connections is not suggested in design standards because of difficulties in obtaining adequate weld penetration into the frame material. However recently, engineers have begun to specify the use of weld washers for thick deck-to-frame connections to improve the quality and consistency of the welds. In addition to this it may also be possible to rely on the weld washer connections for seismic energy dissipation through local distortion of the sheet around the weld. Tests by Rogers and Tremblay (2000a, b) on arc-spot welds with common circular bolt washers (14.3 mm inner diameter, 26.5 mm outer diameter, and 2.4 mm thickness) revealed that the connections were able to carry loads over large displacements and through numerous cycles, therefore, able to dissipate earthquake energy in the structure. However, the washer that is generally used in practice is a cold formed sheet product (AISI, 1997) that is different from the bolt washers used in these tests. Hence, the weldability and performance of arc-spot connections may change when the standard weld washer for decks is used. This presents two major problems: 1) the use of weld washers for thick decks may not result in adequate weld quality in some cases because of a lack of weld penetration into the frame material, and 2) if an adequate weld is formed, the performance of the connection cannot be predicted because of limited knowledge in the subject area. C O N N E C T I O N TESTS The main objective of this particular investigation was to measure the performance of different arc-spot welded connections with regards to: capacity, ductility and energy dissipation ability. A total of 72 deck-toframe tests (Table 1), in addition to 9 coupon tests, were carried out. The range of tests included three sheet steels, two thickness frame members and the use or non-use of washers. Three displacement protocols were incorporated into the study to impose representative static, cyclic and seismic loading on the connections. The two different thickness frame members were specified to simulated different connection scenarios, where the 3 mm plate represents the leg of a small angle used for the chords of an open web steel joist and the 20 mm plate represents the flange of a large W-section beam. The 0.76 and 0.91 mm sheet steels are the most common thickness roof deck found on the market today, and the 0.76 mm painted sheet steel is often specified for construction projects in the United States.

359 TABLE 1

WELDED DECK-TO-FRAME CONNECTION TEST COMBINATIONS Sheet Steel

3 mm Frame Thickness Arc-spot weld Weld + washer

O. 76 m m Gr 230 0.91 m m Gr 230 O. 76 mm P a i n t e d Total (72)

20 mm Frame Thickness Arc-spot weld Weld + washer

6 8 8

7 -

17 6 6

6 8

22

7

29

14

Test S e t - U p

Connections were composed of a C-shape sheet steel section joined to either a 3 mm or 20 mm plate, which represent either a typical joist chord or beam flange as indicated previously (Figure 1). Welds were formed by a professional welder using a Hobart 6022 electrode for a duration of 3-4 sec at 250 V. The plate end of the connection was attached to a rigid frame that was bolted to the strong floor of the structures laboratory. The sheet steel end was then secured to a 245 kN MTS actuator, aligned, fitted with displacement transducers and finally subjected to loads resulting from the displacement of the actuator ram. An MTS controls system allowed for accurate displacement of the connection through the use of input files detailing the prescribed motion of the actuator. ,70

,-- ,r

Weld

/~:tl0

J- N? 5 " t =~l ~

~/

over 100mmgaugelength

25

diaboltl27mm~" /~'N~9 / // / 3or]--(.:.... - .. // 20 1 ~ -~

Oi,l=mon, moasuro

.5

/ /%@

I~ S h e e t

--/ ent

lat

[ [~'Test Specimen--N ~b / III C-Section ~]l .1.l lll 2-12.7mmdia-N [If"~ ~? p e" c 1i m~eA325b~ n ~

RigidSupport

'\Q\\\\\\\

I

Figure 1: Typical deck-to-structure connection specimen, weld washer details and test set-up

Displacement Protocols Three different protocols were used in the loading of the connection specimens: monotonic, 2.0 Hz and simulated earthquake motion. Monotonic tests were completed for all specimens at a cross-head speed of 0.5 mm / minute. These were followed by cyclic and simulated earthquake tests as illustrated in Figure 2. The cyclic test protocol required displacements that ranged from +0.5 to +15 mm, with 12 increments of 3 cycles at the same amplitude. The amplitude range was determined by estimating the possible displacement demand required through deformation of the steel deck diaphragm for typical single storey buildings, assuming that a significant proportion of the displacement develops inelastically at the fastener location.

360 Once a better understanding of the typical displacement that a connection would undergo during an ,earthquake exists; then more relevant evaluations of the seismic performance of these fasteners can be made at the appropriate intermediate step position. Each simulated earthquake specimen was first subjected to a section of low amplitude cycles (_+0.5 mm), followed by a single 3 mm displacement, additional low amplitude cycles and another 3 mm displacement. This pattern was repeated until a maximum of 15 mm displacement was reached. The earthquake displacement protocol was intended to simulate a worst-case scenario of successive large ground motions on a structure. 16 F

=

L

. . . . . . . .

,,,,,,~,a~l~A~llflllllllllllllllllllllllll

E

~ ......... '"v~vvvvvv~uu/l!llntllllllllllllll _,,

-

o ~ .

.

.

.

.

.

.

.

'12/....... 100

150

.

.

-

.

o i~/_

200

-

...... .........

-2 50

.

.= ,

-20 0

.

-

.

0

.

.

.

.

.

.

.

50

Time (s)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

100

.

.

.

.

.

.

150

Time (s)

2 Hz Cyclic Signal

Simulated Earthquake

Figure 2: Displacement protocol for 2 Hz and simulated earthquake tests

Material Properties Galvanised sheet steels with a specified base metal thickness of 0.76 and 0.91 mm, meeting ASTM A653 (1994) specifications, and painted sheet steels with a specified base metal thickness of 0.76 mm meeting ASTM A366 specifications, were obtained for fabrication of the specimen C-shapes. Material properties of the three different sheet steels were determined from triplicate tests of coupons fabricated and tested according to ASTM A370 (1994) requirements (see Table 2). The 0.76 and 0.91 mm Grade 230 steels meet the requirements for ductility and ultimate strength to yield stress ratio contained in both the CSA S 136 (1994) and American Iron and Steel Institute (AISI) (1997) design standards. The 0.76 mm painted material has a much higher ultimate strength and yield stress, although its ductility is significantly reduced compared with the other mild steel products. The plate sections were composed of standard mild steel cut from Grade 300W CSA G40.21 (1998) bars (fy = 300 MPa, f~ = 450 MPa). TABLE 2 SHEET STEEL MATERIAL PROPERTIES Steel Type Property 0.76 mm 0.91mm 0.76 mm Gr 2301 Gr 230 ~ Painted t base metal (mm) fy (MPa) f~ (MPa) f./fy % elong (50 mm gauge).

0.75 304 375 1.23 22.8

0.89 332 391 1.18 25.3

IASTM A653 (1994) ZASTMA366 (1993) 3fycalculated using 0.2% offset method.

0.71 7773 789 1.02 0.4

361 Connection Test Results and Observations

In total, 90 connection specimens were fabricated, although 18 were discarded for use in testing because of their low weld quality; which was based on an assessment of the percent of the weld perimeter that was connected. However, these discarded tests were used along with the 72 connection specimens to evaluate the average visible perimeter connectivity of a weld, because they reflect the possible range in weld quality for a typical building. The results provided in Table 3 are listed as a function of the sheet steel type and the presence of a weld washer. In general, the addition of a washer significantly improved the visible weld quality of the 0.76 mm sheet steels. The thicker 0.91 mm sheet steel connections without washers exhibited reasonably high quality (94.3%), although the inclusion of a weld washer did bring this value up to 100%. The 0.76 mm painted sheet steels were the most difficult to weld consistently without washers (59.4%). TABLE 3 AVERAGE PERCENTVISIBLE PERIMETERCONNECTIVITY Washer Percent Eft. C.O. K

Steel Type

98.1

O. 76 mm Gr 230 0.91 mm Gr 230 O. 76 mm Painted Overall Average

No Washer Percent E f t C. O. V.

#

0.082

16

100

-

1

100

0.000

9

73.8 94.3 59.4

0.062

26

74.3

98.8

"~ !.

#

0.380 0.112 0.639

30 15 19

0.407

64

. . . . . v, I...................

,^

,

- . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

14~

...............

. z

.......................

w

9

15

"~ -15 .,J

.10

-51~'~

-15

Disp (ram)

~

--

5

10

15

.15

'

15

10

I!

I

Disp (ram)

P20W-09

I0

Disp (ram)

P20W-06

P20-02

.v I

1o

15.

is

~

::zt_

~ -15

10

~ .15

.1

9

-10

-5

.................

5 -,~

Disp (mm)

$20W-03

.is i

isi

Disp (ram)

$3W-04

9

Disp (mini

$3-06

Figure 3" Representative overall load vs. displacement hysteresis Load vs. displacement hysteresis graphs for representative 2 Hz test specimens can be found in Figure 3 (W = weld washer). A dramatic improvement in the load carrying capacity with increasing displacement can be seen for the connections that contained washers. Those that were fabricated without washers ( e . g P2002 & $3-06) revealed an inability to cany loads beyond a displacement of 2.0 mm, as found previously by Rogers and Tremblay (2000a, b). It is important to note that in some cases when weld washers were used and the visible weld perimeter was 100% the connections d i d n o t p e r f o r m at the expected level ( e . g P20W06 & $3W-04). Typically, for connections with weld washers, loads are carried at large displacement by

362 bearing distortion in the sheet steel. The weldment and washer act in a similar fashion to a bolted connection, where the sheet steel piles up in front of the weldment (shank of bolt). If the washer becomes detached from the weldment, or worse if the weldment fractures between the sheet steel and flame, because of lack of penetration into the plate material, the overall energy dissipation ability of the connection will be substantially reduced (Figure 4). For example, specimen P20W09 exhibited extensive bearing distortion of the sheet steel without fracture or loss of the washer, whereas for specimen P20W06 a similar ultimate load was reached, but shearing of the weldment between the sheet and plate at a displacement of approximately 5 mm (Cum. Disp. ~ 160 mm) reduced the overall energy that was dissipated by the connection.

600

--

-~

1200

500 !

| ,,oo

a,~

300

-~'

200 -~

0

P20W-06

,

0

-- ............

,~, 1000 ;

200

P20-02 "

~

400

,

600

C u m u l a t i v e Dis p ( m m )

O. 76 mm Painted 2Hz Test Specimens

,

800

~~ "

600

~

400

.............

-

iii .-.

0

0

200

400

600

800

C u m u l a t i v e Dis p ( m m )

O.76 mm Grade 230 2Hz Test Specimens

Figure 4" Representative energy dissipation graphs An analysis of the test results was carried out for all 72 specimens, including the documentation of ultimate loads, failure modes and ability to dissipate displacement-induced energy. Prior to testing the specimens were categorised based on their visible weld perimeter in order that a comparison of connection performance with weld quality could be reported. Table 4 contains a summary of the results for the 0.76 mm Grade 230 sheet steels, which reflect the overall findings for the other sheet steel types. Connections with a full or near full visible weld perimeter typically outperform connections of lower quality when a comparison of ultimate loads, Pu, is made. However, as indicated previously, visible weld quality is not necessarily directly related to the energy absorption performance of the connection. The lack of weldment penetration into the plate material, which cannot be seen through a topside visual inspection, will decrease the amount of energy absorbed. This issue became known upon premature failure of some of the specimens and subsequent inspection of the failure mode (Bond et aL, 2001). The addition of washers provides more consistent Pu values in comparison with specimens that do not contain washers, and the energy absorption ability of washerless connections is significantly lower. The weld quality of washerless connections also directly affects their ultimate load carrying ability. If the energy absorbing characteristics of an arc-spot weld with washer are to be relied on in the seismic resistance of a roof deck, then steps must be taken to insure that the connection has adequate weldment penetration into the framing member. This may not be possible if one depends on visible inspection methods, hence, a weld protocol including washer size, electrode type and rate of use, etc, must be developed.

363

TABLE 4 CONNECTION TEST RESULT SUMMARY FOR 0.76 mm GRADE 230 SHEET STEEL SPECIMENS

Test Specimen

Deck mm

Frame Displacement mm Protocol

Weld Dia. mm

Visible % Perimeter

PL,

ZEnergy ZEnergy/P.

kN

kN mm

kN mm/kN

100 100 100 100 I00 100

10.4 7.72 11.9 10.6 11.5 9.37

37 22 27 16

3.1 2.1 2.4 1.7

70 70 90 90 70 80

4.66 7.45 9.94 10.9 8.29 6.50

62 74 20 10

6.2 6.8 2.4 1.5

60 40 40 40 50

4.00 2.36 1.64 7.15 1.99

3 35 16 1

1.1 21.1 2.2 0.6

11.6 10.0 12.6 12.4 13.0 10.4

910 1057 95 12

72.4 85.5 7.3 1.2

95 80 95 100 90 100

9.30 4.81 6.95 11.6 10.1 10.9

39 83 35 49

5.6 7.2 3.5 4.5

100 100 100 100 100 100 100

11.4 11.0 11.8 12.3 11.8 12.0 12.0

653 829 1107 113 149

55.6 67.3 93.5 9.4 12.4

Full Visible Weld Perimeter $20-16 $20-20 $20-06 $20-13 $20-01 $20-03

0.76 0.76 0.76 0.76 0.76 0.76

20 20 20 20 20 20

Monotonic Monotonic Cyclic 2Hz Cyclic 2Hz Earthquake Earthquake

$20-17 $20-18 $20-10 $20-11 $20-02 $20-08

0.76 0.76 0.76 0.76 0.76 0.76

20 20 20 20 20 20

Monotonic Monotonic Cyclic 2Hz Cyclic 2Hz Earthquake Earthquake

$20-04 $20-14 $20-15 $20-07 $20-19

0.76 0.76 0.76 0.76 0.76

20 20 20 20 20

Monotonic Cyclic 2Hz Cyclic 2Hz Earthquake Earthquake

$20W-05 t $20W-06 $20W-03 $20W-04 S20W-01 $20W-02

0.76 0.76 0.76 0.76 0.76 0.76

20 20 20 20 20 20

Monotonic Monotonic Cyclic 2Hz Cyclic 2Hz Earthquake Earthquake

$3-02 $3-05 $3-06 $3-10 $3-04 $3-09

0.76 0.76 0.76 0.76 0.76 0.76

3 3 3 3 3 3

Monotonic Monotonic Cyclic 2Hz Cyclic 2Hz Earthquake Earthquake

13.0 12.5 14.9 13.4 14.9 14.5

70-90% Visible Weld Perimeter 12.8 12.2 13.5 14.5 12.0 13.3

40-60% Visible Weld Perimeter 11.9 12.9 13.2 13.8 11.2

Near Full Visible Weld Perimeter 12.3 11.1 11.4 11.8 13.4 13.2

100 70 100 100 100 100

Near Full Visible Weld Perimeter 11.3 12 11.4 13 12.2 12.3

Full Visible Weld Perimeter $3 W-02 0.76 3 Monotonic 10.5 $3W-03 0.76 3 Monotonic 11.4 $3W-04 0.76 3 Cyclic 2Hz 10.3 $3W-05 0.76 3 Cyclic 2Hz 11 $3W-06 0.76 3 Cyclic 2Hz 10.5 $3W-07 0.76 3 Earthquake 11.3 .S 3 w - o 8 o. 76 3 Earthquake 12 ~W in the test specimen name indicates that a weld washer was used.

364 Previous arc-spot weld test specimens by Rogers and Tremblay (2000a, b) did not fail in this weldmentshearing mode mainly because of the different washer type and hole diameter. The typical bolt washer allowed for a more substantial weldment core to be developed, as well as adequate penetration into the framing material. CONCLUSIONS The laboratory results show that the use of weld washers can improve the ductility, strength and energy absorption ability of arc-spot connections when visible weld quality is high and weldment penetration is adequate. However, caution is warranted because if the weld metal does not fully penetrate into the frame material then premature failure due to shear fracture through the cross-section of the weld may take place. Hence, if the seismic resistance of a weld with washers is to be relied on steps must be taken, i.e. weld protocols and washer specifications need to be developed, to insure that the connection will behave as predicted. The thickness of the frame material did not seem to have a significant effect on weld quality in contrast to the thickness of the sheet steel and the type of weld washer. ACKNOWLEDGEMENTS

The authors would like to thank the Natural Sciences and Engineering Research Council of Canada and the Canam Manac Group for their support. The authors would also like to acknowledge the assistance of the laboratory technicians at l~cole Polytechnique; G. Degrange, P. Bdlanger and D. Fortier.

References American Iron and Steel Institute. (1997). "1996 Edition of the Specification for the Design of ColdFormed Steel Structural Members", Washington, DC, USA. Society for Testing and Materials, A366. (1993). "Standard Specification for Steel Sheet, Carbon, Cold-Rolled, Commercial Quality", West Conshohocken, PA, USA. American Society for Testing and Materials, A370. (1994). "Standard Test Methods and Definitions for Mechanical Testing of Steel Products", West Conshohocken, PA, USA. American Society for Testing and Materials, A653. (1994). "Standard Specification for Steel Sheet, Zinc-Coated (Galvanized) or Zinc-Iron Alloy-Coated (Galvannealed) by the Hot-Dip Process", West Conshohocken, PA, USA. Canadian Standards Association, S 136. (1994). "Cold Formed Steel Structural Members", Etobicoke, Ont., Canada. Canadian Standards Association, G40.21. (1998). "Structural Quality Steels", Etobicoke, Ont., Canada. Rogers, C.A., Tremblay, IL (2000a)i "Seismic Loading of Steel Roof Diaphragm Assemblies", STESSA 2000- The 3~ Intemational Conference on the Behaviour of Steel Structures in Seismic Areas, Montreal, Caoad~ 239-246. Rogers, C.A., Tremblay, R., (2000b). "Inelastic Seismic Response of Frame and Side-lap Fasteners for Steel Roof Decks", Research Report No. EPM/CGS-2000-09, Department of Civil, Geological and Mining Engineering, l~cole Polytechnique, Montreal, QC, Canada. Bond, W.F., Rogers, C.A., Tremblay,R., (2001). "Inelastic Seismic Response of Welded Frame Fasteners for Steel Roof Decks", Research Report, Department of Civil, Geological and Mining Engineering, l~cole Polytechnique, Montreal, QC, Canada.

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rightsreserved

365

Dynamic Buckling and Collapse of Rectangular Plates under Intermediate Velocity Impact Shijie Cui, Hong Hao and Hee Kiat Cheong School of Civil and Structural Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798

ABSTRACT This paper investigates numerically the dynamic buckling and collapse mechanism of imperfect plates under intermediate velocity impact. Based on the dynamic response characteristics of plates, dynamic buckling and plastic collapse criteria are proposed to determine the critical loads. The buckling and collapse characteristics are discussed. The effects of damping on the dynamic buckling properties of plates are also investigated. The results indicate that the dynamic buckling and collapse characteristics of the plates under intermediate velocity impact are quite different from those under either high or low velocity impact loads. Furthermore, the effect of damping depends on the impact load duration. For the case of short load duration, the effect of damping is insignificant. Thus, the damping effect can be neglected in the dynamic buckling analysis of structures.

KEYWORDS Dynamic buckling criterion, rectangular plate, intermediate velocity impact, collapse, damping, impact load duration.

INTRODUCTION Since the broad application in many fields of engineering such as the ships engineering, nuclear power plant engineering and civil engineering etc., dynamic buckling of thin-walled structures has been received more and more attention in the last decades. Dynamic buckling of structures under impact loads can be divided into three categories according to the applied dynamic loads and structural responses, viz., high velocity impacted buckling, low velocity impacted buckling and intermediate velocity impacted buckling. For the problems of high and low velocity impacted buckling of structures, there are many publications can be found in the literature (Ari-Gur et al, 1981; Lindberg and Florence, 1987; Simitses, 1987 and Jones, 1989, 1996). For the intermediate velocity impacted buckling of structures, the dynamic load has a moderate amplitude and duration (in an order of milliseconds). The dynamic buckling characteristics of structures may be much different from those of

366 structures under the high and low velocity impacts (Cui et al, 1999). The studies of this problem for structures, especially for plates, however, are limited in literature up to now. Karagiozova and Jones (1992a, b, 1995) investigated the dynamic elastic-plastic buckling of a twodegree-of-freedom model subjected to a rectangular pulse load and two triangular loads. Influences of initial imperfection, dynamic load shape and duration, axial inertia and plastic reloading on the dynamic buckling behaviour of the model were examined in these studies. They concluded that both the initial imperfection and the dynamic load duration strongly affect the dynamic buckling properties of the model, and load duration also has a significant influence on the sensitivity of the initial imperfection. Recently, Cui et al (1999) reported an experimental study of the dynamic buckling of rectangular plates under in-plane intermediate velocity impacts. The intermediate velocity impacts were generated by fluid-solid slamming. The dynamic buckling and dynamic yielding properties were investigated based on the observation of elastic-plastic dynamic response characteristics of the plates. This paper presents a numerical study of dynamic buckling of imperfect rectangular plates subjected to intermediate velocity impact loads by using the computer code ABAQUS (1997). The primary objectives are to investigate the dynamic buckling and collapse mechanism of plates subjected to intermediate velocity impact loads, to define a suitable dynamic buckling criterion and to discuss the effect of damping of the plate material on the dynamic buckling characteristics of the plates.

DYNAMIC BUCKLING AND COLLAPSE CHARACTERISTICS OF PLATES In the present study, compatible mass matrix and Rayleigh type viscous damping are used in finite element modeling. Dynamic load is assumed having a half-sine form as observed in test (Cui et al, 1999). Large deflections are considered by including the second order terms in strain estimation. A bilinear elastic-plastic model is employed to describe the nonlinear property of the material. The material properties of the model are: elastic modulus E=2.1 x 105 MPa, plastic modulus Er=l.519x 104 MPa and the yield stress O'y=289 MPa, respectively. TABLE 1 DIMENSIONS,INITIALIMPERFECTIONANDCRITICALDYNAMICLOADSOF PLATES Plate No. CCP01 CCP02 CSP01 CSP02 CFP01 CFP02

Dimension(mm) L B h 500 325 2.0 500 325 2.0 500 325 2.0 500 325 2.0 500 325 2.0 500 325 2.0

to (s) 0.017 0.017 0.0175 0.0175 0.019 0.019

~ qtc~, qt.c~,E (mm) (N/ram) (N/mm) 0.11 198 192 0.10 202 196 0.11 148 152 0.10 152 158 0.11 78 76 0.10 80 77 mean

error qt~f % (N/mm) 3.13 318 3.06 324 2.70 291 3.95 296 2.63 184 3.90 189 _ 3.23

Six rectangular plates are analyzed in this numerical study. Table 1 lists the dimensions of the plates. In order to calibrate the numerical model, the plates are selected to have the same dimensions as those of the tested plates (Cui et al, 1999). The dynamic load duration to is also taken as those measured from the test (Cui et al, 1999). The initial imperfection of the plates is assumed following the same shape as their first mode of transverse free vibration. Therefore, only the maximum value, ~0ma~,is given in the table. The boundary conditions of the plates are clamped on both top and bottom edges, and the two vertical sides of the plates are clamped (CCP), hinged (CSP) or free (CFP), respectively. In finite

367 element modeling, each plate is divided into 12 segments in each direction in x-y plane as shown in Fig. 1. 8-node isoparametric element is used. Thus, there are a total of 144 elements and 481 nodes for each plate in the numerical calculation.

i i i i ]|| i nnmnn

i ilmmmm i i B B innm!

I'~

q(t)

125 mm ~1.83"3mmj..41.7~_.!83.3 mmA 500 mm

.J

"1 Figure 1: A rectangular plate and its finite element mesh In numerical analysis, a series of dynamic responses of the plates are calculated by loading each plate with half-sine form impact load q(t) with duration given in Table 1 and different amplitudes. Fig. 2 shows the displacement time histories at points A, B, C and D as indicated in Fig. 1 of plate CCP01 under several load amplitudes. Firstly, it can be seen from the figure that the maximum displacement always occurs during the load duration to. The displacement responses of plate decay rapidly after the load application. The displacement responses for other plates, which are not shown here, have similar characteristics as those of CCP01 shown in Fig. 2. This indicates that the plates may buckle during the dynamic load duration. This dynamic response characteristic of the plates subjected to intermediate velocity impacts is different from that of plates under high velocity impacts (solid-solid impacts). For the high velocity impacted buckling, the load duration is usually in an order of microsecond, and the loading will be completed before the deformation of the plate is fully developed. So that dynamic buckling of plate always occurs after load application. As can also be seen from Fig. 2, the maximum displacement response of the plate increases along with the increase of the dynamic load. When the dynamic load is small (e.g., qt<_200N/ram), the maximum displacement of the plate increases gradually. When the dynamic load is large (e.g., qt>200 N/ram), the maximum transverse displacement response of the plate increases rapidly. This is more clearly illustrated in Fig. 3, in which the maximum displacement of the plate is plotted as a function of the dynamic load amplitude. As can be seen from the figure, the curve of the maximum transverse displacement of the plate has a "knee" at Pt near 200 N/ram. When the dynamic load is smaller than this critical load, the slope of the curve is flat, indicating the development of the maximum transverse displacement of the plate is steady as the dynamic load increases. When the dynamic load is larger than the critical load, the slope of the curve becomes very steep, indicating the maximum displacement of the plate increases rapidly for a small increment of the dynamic load. It is obvious that the change of the maximum transverse displacement of the plate from steady increase when the load is small to rapid increase when the load is large is caused by dynamic buckling of the plate. According to this observation, a dynamic buckling criterion of plate is defined as follows: for an impacted plate with a prescribed aspect ratio, the point corresponding to the sudden rapid increase of the maximum

368 transverse displacement is the dynamic buckling critical condition of the plate. The corresponding dynamic load is defined as the dynamic buckling critical load and denoted by qt,crb.

d (ram)

0.08 r

0.4

B

0.06

C

0.04

D

0.02

A

0

d (ram)

C

0.3

B

D

0.2 0.1 ~

2

4

6

8

-0.1 !

-0.02

I

t xl0"2(S)

t xl02 (s)

-o.2 L

(b) qt=200 N/mm

(a) qt=133 N/mm

d (ram)

d (ram)

6

D C

2!

4

-B

B

2

1

,vp -'1 L

4

t xl0"2(s) (c) qt=288 N/mm

6

8

-2

t xl0 2 (s)

(d) qt=332 N/mm

Figure 2: Displacement response time histories of CCP01 under different amplitudes (~c=0.025) Fig. 4 shows the variation of the maximum Mises stresses versus the loading amplitudes. The maximum Mises stress grows with the increase of the dynamic load. Before the dynamic buckling takes place, the dynamic load is small, and the transverse displacement of the plate is insignificant. As a result, the Mises stress of the plate is not large. After the dynamic buckling occurs, the maximum Mises stress of the plate increases sharply because of the rapid increase in the transverse deformation of the plate. As can be found from Fig. 2, the dynamic responses of the plate increase fast after the dynamic buckling took place. When the dynamic load increases to qt=332 N/ram, the residual strains are very outstanding, implying the plate will lose its load bearing capacity. This characteristic can also be found in Fig. 3. It is obvious that, when the dynamic load approaches the "critical load" , the curve of the maximum displacement of the plate shows another turning point, after that, the displacement increases sharply with the increase of dynamic load. This turning point indicates the dynamic response of the plate is dominated by its plastic deformation. Further loading on the plate results in its displacement increasing more rapidly. This indicates that the plate is losing its load bearing capacity at this "critical dynamic load" . To estimate the load carrying capacity of plates, a plastic collapse criterion is defined as follows: for a plate subjected to an intermediate velocity impact, the moment when its maximum

369 displacement response increases sharply with a small increment of dynamic load is defined as the plastic collapse critical condition of the plate. The corresponding dynamic load is the critical collapse load, and denoted as qt,erf. 10 -

I I !

8

~

4 -

,

i 2

0 0

100

200

300

400

q t (N/ram)

Figure 3" The maximum displacement of plate CCP01 vs dynamic load 400 -

300 -

~ 200 -

100

J I

0

I

I

300

4O0

q terb

0

100

200 q t (N/mm)

Figure 4: Variation of the maximum Mises stress of CCPO1 vs dynamic load amplitude

370 DETERMINATION OF DYNAMIC CRITICAL LOADS To estimate the dynamic buckling and collapse critical load, curves of the maximum displacement response of the plates are plotted with respect to the dynamic load. According to the above definitions of dynamic buckling and collapse critical conditions, the corresponding critical dynamic loads can be easily determined. Fig. 3 is an example for plate CCP01. From this figure, it can be obtained that qtcrb=198 N/ram and qtcrr=318 N/mm. The curves for other plates, which are not shown here, are similar. The corresponding critical dynamic loads are listed in Table 1. For comparison, results obtained from the corresponding test by Cui et al (1999), which are denoted as qterbE and qterfZ, are also listed in Table 1. It can be noted from the table that the numerical results of dynamic buckling critical loads of the plates agree well with the experimental results. The errors are about 2.63% to 3.95% for the 6 tested plates with an average error of 3.23%. d (rrrn) a v l-.q t=166N/nma(x3) / ~ 4

i 2 i

2--.q,=200N/rma(x2) / 3"q ,=244N/rr~ /

d (nma) 4

l--q t=100N/rnm , 2---q t=170N/nma 3--q ,=200N/nma / "

'~ 3

4

"\

2 1 ~

,;o-200

500

-1 L x

i

(ram)

x (ram) (b) CSPOI

(a) CCPO1

1--q t=44N/rrrn 2--q t=55N/nma 3--q t=66N/ram 4--q t=78N/ram

d (mm) 8 6 4 2 0 0

100

200

300

400

500

x (mm) (el CFPO1 Figure 5: Distributions of the maximum displacement of plates along the longitudinal direction under different impact loads Fig. 5 shows the distributions of the maximum displacements of plates CCP01, CSP01 and CFP01 along the longitudinal direction and under different load amplitudes. It can be found from this figure that the dynamic buckling modes of the plates are dominated by their fundamental transverse free vibration mode. This is because the initial geometric imperfection of the rectangular plates discussed in this study has the same shape as their fundamental transverse free vibration mode; and the dynamic load duration is in an order of millisecond (from 0.017see to 0.019see), which is closer to the

371 fundamental transverse free vibration periods of the tested plates than to the periods of higher transverse vibration modes. This observation is the same as that obtained from experimental tests of the plates (Cui et al, 1999) but is different from the dynamic buckling mode of plate under high velocity impact. For plates under high velocity impact loads, their buckling modes may be governed by higher transverse vibration modes (Lindberg and Florence, 1987; Karagiozova and Junes, 1996). It can also be found from Fig. 5 that, for the plates with four edges supported (CCP01 and CSP01), after dynamic buckling takes place, the transverse deformation shapes of the plates change to their second transverse vibration modes. While for the plates with two free edges (CFP01), the transverse deformation shapes always follow their fundamental transverse vibration modes. This indicates that, for the plates with four supported edges, their vibration modes in post-buckling process are different from the original buckling modes of the plates. This is because of the effect of different boundary conditions as discussed in the experimental study (Cui et al, 1999). For the plates with two free edges, there is no any lateral support, hence their transverse vibration shape remains in their fundamental transverse vibration mode. In addition, the plates with two free lateral edges under intermediate velocity impact behave as a "wide column" . Therefore, the dynamic buckling of plates with two free lateral edges and subjected to intermediate velocity impact loads can be treated as a onedimensional problem in the analysis. This conclusion was also drawn based on the experimental results reported by Cui et al (1999).

400 300 ! ~ " ' - e " ' - " ~

I

1...t0=O.005s 2---to~).O17s 3...to=O.O50s

100 0l___

,

0

3

_

_

6

_

_

9

~ (%) Figure 6: Effect of damping of plate materials

EFFECT OF DAMPING OF PLATE MATERIALS Fig. 6 shows variations of dynamic buckling critical loads of plates versus different damping of plate materials with three different kinds of load duration. It is obvious that all the relationships between the buckling loads and damping are linear for three different kinds of load duration. The dynamic buckling load increases with the increase of damping. Moreover, the slop of the curve for long load duration is larger than that for short load duration. This indicates that the effect of damping of plate materials is dependent on the impact load duration. For the case of long load duration, the buckling load increases fast with the increase of damping. For the case of short load duration, however, as the damping increases, the dynamic buckling critical load increase slightly. This shows that the effect of damping is less pronounced for plates subjected to intermediate velocity impact loads because their duration is

372 relatively short. This is because, for the plates subjected to dynamic loads, the damping force plays a rule of resisting the flexural deformation of the plate, so that the dynamic buckling load of plate with large damping is higher than that of plate with small damping. However, for the case of intermediate velocity impact, the load duration is in an order of millisecond and the dynamic buckling of plates is always occurs during the load duration as discussed above. Such the duration is not long enough for damping force to be fully developed before the dynamic buckling takes place. Hence, the effect of damping on the dynamic buckling properties is insignificant for the plates under intermediate velocity impacts.

CONCLUSIONS Dynamic buckling of rectangular plates subjected to intermediate velocity impact loads has been numerically investigated. By observing the dynamic response characteristics of the plates, a dynamic buckling critical condition and a plastic collapse critical condition have been defined. The results indicate that, for the present rectangular plates subjected to intermediate velocity impact, dynamic buckling always occurs within the impact load duration. The dynamic buckling mode of the plates is the same as their fundamental transverse vibration mode. However, after the dynamic buckling takes place, the vibration shape of the plates with four supporting edges will change to their second transverse vibration mode because of the effect of lateral supporting boundaries. When two lateral edges of a plate are free, the lateral effect of the plate is insignificant and can be neglected, so that the dynamic buckling analysis can be simplified to a one-dimensional problem. The effect of damping depends on the impact load duration. For the case of short load duration, the effect of damping is insignificant. Thus, the damping effect can be neglected in the dynamic buckling analysis of structures.

REFERENCES Ari-Gur J., Singer J. and Weller T. (1981). Dynamic buckling of plates under longitudinal impact. Israel d Tech 19, 57-64. Cui S. J., Cheong H. K. and Hao H. (1999). Experimental study of dynamic buckling of plates under fluid-solid slamming. Int. J. Impact Eng. 22, 675-691. Hibbitt, Karlsson and Sorensen, Inc. (1997). ABAQUS/Standard User' s Manual; ABAQUS Theory Manual; and ABAQUS/Standard Example Problem Manual. Jones N. (1989). Recent studies on the dynamic plastic behavior of structures. Appl Mech Rev. 42:4, 95-115. Jones N. (1996). Recent studies on the dynamic plastic behavior of structures.... An update. Appl Mech Rev. 49:10, S 112-S 117. Karagiozova D. and Jones N. (1992a). Dynamic buckling of a simple elastic-plastic model under pulse loading. Int. s Non-linear Mech. 27:6, 981-1005. Karagiozova D. and Jones N. (1992b).Dynamic pulse buckling of a simple elastic-plastic model including axial inertia. Int. s of Solid and Structures 29:10, 1255-1272. Karagiozova D. and Jones N. (1995). Some observations on the dynamic elastic-plastic buckling of a structural model. Int. J. Impact Eng. 16:4, 621-635. Karagiozova D. and Jones N. (1996). Multi-degrees of freedom model for dynamic buckling of an elastic-plastic structure. Int. J. of Solid and Structures 33:23, 3377-3398. Lindberg H. E. and Florence A. L. (1987). Dynamic pulse buckling---Theory and experiment, Martinus Nijhoff Publishers, Dordrecht. Simitses G. J. (1987). Instability of dynamically-loaded structures. Appl Mech Rev. 40:10, 1403-1408. Weller T., Abramovich H. and Yaffe R. (1989). Dynamic buckling of beams and plates subjected to axial impact. Computers and Structures 32:3-4, 835-851.

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

373

STRUCTURAL OPTIMIZATION OF THIN SHELLS UNDER DYNAMIC LOADS Susana A. Falco 1, Luiz E.Vaz 2 and Silvana M. B. Afonso a 1-2Department of Civil Engineering Pontificia Universidade Cat61ica do Rio de Janeiro- PUC-Rio Rua Marquis de S~o Vicente 225/30 IL, G~ivea CEP 22453-900, Rio de Janeiro, RJ, Brazil 3 Department of Civil Engineering Universidade Federal de Pernambuco- UFPE Cidade Universitbxia - CEP 50740-030 Recife- Pe, Brazil

ABSTRACT Thin shell structures present immense structural and architectural potential in various fields of civil, mechanical, architectural, aeronautical and naval engineering. This kind of structures are in general designed by the finite element method using the conventional 'trial and error' procedure which can lead, in many cases, to a non-economical design, specially when a dynamic analysis is necessary. Aiming at developing a more efficient methodology for the design of thin shells, this paper develops a structural optimization procedure, using finite element discretisation, to obtain efficient design of thin shell structures under elastic behavior with viscous damping and dynamic loads. The Huang-Hinton finite element, which belongs to the family of degenerated shell elements, is used. The mesh generation on the shell surface is performed by means of a mapping from the parametric plan in the 2D space to the 3D shell surface using Coon patches. The design variables are the key-points coordinates and the Coon patches widths. To solve the nonlinear constrained optimization problem the Sequential Quadratic Programming algorithm is used. Many different optimization problems can be performed by the code. So, one can minimize the volume or the displacement or the acceleration at a point or maximize the first frequency of the structure. On the other hand the volume of the structure can be kept constant while minimizing the displacement at a given point or in all points of the shell. Several examples are considered illustrating the potentiality of this developed tool.

KEYWORDS Thin shells, dynamics, finite elements, structural optimization, Sequential Quadratic Programming.

374 INTRODUCTION The reduction in weight, the limits on displacements and accelerations and the increase in the quality and reliability of structures are considered to be major aspects in the design of structures. The structural shape optimization (SSO) turns up to be an adequate procedure when aiming at designing efficient structural shapes. This work deals with SSO of thin shells under dynamic loads and linear elastic behavior. In this paper an automatic procedure for the optimum design of shells is presented. It includes the geometric modeling and the mesh generation of the structure, the structural analysis by the Finite Element Method, Zienkiewicz & Taylor (1991; 1995), the sensitivity analysis, Dutta & Ramakrishnan (1998), and the structural optimization, Vanderplaats (1999). The geometry of the shell is represented by Coon surfaces which are formed by to series of cubic splines intercepting the key points which lay on the mid surface, Afonso (1995). The sizing and shape variables are the thickness and the lengths of the radii in the key points respectively, which implies a decrease in the number of variables in the project. Once the shell surface is discretized in finite elements, the structural analysis starts. The structural response analysis is performed by means of the Newmark step-by-step method, Newmark (1959). The finite element used is the 9 nodes Huang-Hinton element, which belongs to the family of elements degenerated from 3D elements, Huang (1988). Five DOF are related to each node of this element, corresponding to 3 displacements and 2 rotations. To solve the non-linear constrained optimization problem, the Sequential Quadratic Programming algorithm is used Powel (1978).

OPTIMIZATION PROBLEM

In a design optimization problem, the objective is to find the values of a set ofn given design variables x R" which minimize or maximize a function fix), denoted "objective-function", while satisfying a set of equality (hk (x)=0) as well as inequalities (gr(x)___0 and di (x, u(x, t),u (x, t), ii (x, t),t) ___0) constraints which define the viable region of the solution space. The constraints xlj <__xj ___xuj are called side constraints. Objective-function: f(x) Variable Vector: x ~ Rn Constraints" related to the design variables: hk (x) = 0 v

g (x) _< 0

xlj< xj < Xuj Constraints related to the .dynamic response: di (x, u(x, t),li (x, t), ii (x, t),t) _<0;

k ~ I = [1,..., q] r ~ D = [1,..., m] j : 1..... r for all t ~ [to, tf] i ~ D = [1,..., M]

(1) (2) (3) (4)

Where R is the set of the real numbers. In the above equations, di ___0 represents the set ofM behaviour constraints depending on the design variables x, on the time t and on the system displacements u (x, t), velocities li (x, t) and accelerations ii (x, t). The initial and final instants of time are designated respectively as to e tf. Any point x, which satisfies the constraints is a feasible or viable point. Sequential Quadratic Programming Algorithm

In the previous item the general non-linear constrained optimization problem was presented. In this work, the solution of this problem is performed by Sequential Quadratic Programming algorithm (SQP) Hafika & Gurdal (1992). This method is presently the most popular method for solving non-linear constrained optimization problems and belongs to the family of direct methods. In this method the search direction is obtained by a sequence of approximate quadratic sub-problems defined by Eqn. 5.

375 rain (or max) constraint to

17f d + 1/2 d~ ~ L a G + 17G'd _< 0 H + 171~d = O

(5)

Where the vectors H = (hi ..... hq) t and G = (gl . . . . . gn~ t contain the equality and inequality constraints respectively and VG e 171t are the Jacobian matrices whose lines are the gradient vectors Vg~ e Vh k respectively. The parameters/1, and a are the Lagrangian multipliers, associated with the equality and inequality constraints respectively. The matrix I~L is the Hessian matrix of the Lagrangian function associated to the original problem, Eqn. 6. L(x, 2, a) = f (x)

+ A t H(x)

+

d G(x)

(6)

The calculation of the Hessian matrix is a cumbersome computational task. So, quasi-Newton approximations, such as the Broyden-Fletchen-Goldfarb-Shanno (BFGS) update, are used instead. These approximations are obtained from the gradients of the Lagrangian function I7L calculated in the previous iterations.

DYNAMIC ANALYSIS When the accelerations in the response of the structures for prescribed applied loads are of major importance and activate inertia forces, which must be considered in the equilibrium equations, a dynamic analysis is necessary for the correct evaluation of the structural response. In the dynamic analysis, the structural responses such as the displacements, velocities, accelerations, stresses and internal forces are transient and the applied loads must also be considered as a time dependent function P(t). In this work, linear elastic behavior is considered for the material. The matrix equilibrium equation, which governs the dynamic response of structures under linear elastic behavior and modeled by means of finite elements, is given in Eqn. 7 for a given set of the design variables x, Clough & Penzien (1975). Mii(t) + C d (t) + Ku(t) = P(t)

(7)

Where M, C e K represent respectively the mass, damping and stiffness matrix. The vectors P(t), ii, d and u define the vectors of applied external forces, accelerations, velocities and displacements respectively. The Newmark direct integration method is applied to solve the system of second order partial differential equation defined above. Direct integration methods solve the system of equations step by step, satisfying the equilibrium equations only at discrete time points. The recurrence formula for the algorithm is obtained as follows. At first the equilibrium equations are defined at the instant teAt, Eqn. 8, and then Eqns. 9 and 10 are used to represent the displacements and the velocities at the end of the time interval as a function of the other kinematic variables in the time interval. Mt+Atii +Ct+Atd +Kt+Atu = t+~tP 1

,.~t u=t u+' i=At + ( ~ - a ) ' iiAt 2 +a'+~iiAt 2 t+,xtd='fi + (1 - 6)' iiAt +rt+~'iiAt

(8) (9) (10)

Where ' u is the displacement, ' u the velocity and ' ii the acceleration vectors at the instant t while '+~'u, t+L~t I1 e teat .. u represent the displacement, the velocity and the acceleration vectors at the end of the time interval teAt respectively. In this work, the values of or and 8 in the above equations are o~=]/4 are 5=1/2, which corresponds to the method of constant acceleration. Substituting the equations for t + A t l l e teat 1~!, Eqns. 9 and ] 0, in the equilibrium Eqn. 8 we will have a new equation where the only unknown is the acceleration vector at the end of the time interval, t e a t [i, once the values of the vectors t u, t fi e t ii, at the

376 beginning of the time interval were obtained from the previous time step. Finally the desired recurrence formula, which gives the acceleration vector t+At ii at the end of the time step t+At as follows: ( M .+-ClC -t- c 2K) t+Atii =t+~'R

'§247

.C('u+c;ii).K('u+c'a+c;ii)

(11) (12)

Where c], c2.... , c5 are constants which depend on the values of the Newmark parameters (x and ~i and on the time step At. cl = ~5At ; c2 = o~ At2 ; c~ = (1-~5) At ; c4 = At ; c5 = (1/2-a) At2 (13) Once the acceleration vector at the end of the time interval is obtained in the Newmark method, the displacements and the velocities can be recovered using Eqns. 14 and 15: ' ' u=' u + c,' li + c, ' ii +c, '+~'ii

(14)

'+~ fl = ' fl + c ; ii + c ' / ~' ii

(15)

This procedure is repeated for every new time step until all the time domain of interest is covered. The numerical errors related to direct integration methods are proportional to the step size used.

SENSITIVITY ANALYSIS The aim of the sensitivity analysis i~ to determine the gradients of the objective functions and constraints of the design optimization problem with respect to the design variables. In this work, only the sensitivity analysis of the dynamic response of the structure in terms of displacements, velocity and acceleration will be discussed, once the sensitivity analysis of the static response, of the eigenfrequencies and eigenmodes, of the volume of the structure, etc. have been discussed extensively elsewhere, Hattka & Gurdal (1992). The method used in this work for performing the sensitivity analysis is based on the total differentiation of the discrete dynamic equilibrium equations defined in Eqn. 11. This procedure leads to the Eqn. 16. The derivatives of the stiffness, mass and damping matrices appearing in Eqns. 16 and 17 is then performed by means of the finite difference method. This methodology is known in the literature as the semi-analytical method for sensitivity analysis, Dutta & Ramakrishnan (1998). Thus, the Newmark direct integration algorithm is used for the determination of the dynamic response and of the sensitivities of the dynamic response. These two procedures can be performed simultaneously taking advantage of the same structure of the effective matrix in Eqns. 11 and 16. (M

where

+ c,C + czK)'§

='§

-(M,,

+ c ~ C , , +c2K,,~)'*~'ii

'+~'R -'+~'r -c,~('a + c;iO- x, (' u + c',u + c;ii) "" x=

a,x x - c ( ' , , , x +4//,x ) - X ( ' , , , x +~;a,~ +4/i,~ )

(16) (17)

Eqn. 16 is solved for t+~/i,x In the sequence the sensitivity of the displacements and the velocities are obtained by means ofEqns. 18 and 19:

'+* /g~x= ' U~x +c~u,.+c;/i,. +C2'+~'"/l~x ' +~' ft , x = ' ft ~x+ c .~U~x ' "" +el " ' +~ "U~x "

(18)

(19)

377

EXAMPLES In this example the analysis of a semi-cylindrical shell is performed. The shell is clamped in one of the curved edges and free at the other three edges figure 1(a). It is submitted to an uniformly distributed transient load acting downwards in the direction z with a time variation as depicted in the figure 1(b) Paz (1997). The length 'L' is 100 in, the thickness 'e ~is 1.0 in, the radius 'R' is 10 in and the angle '0' is 90 ~ Clamped ....

Z

~ ~ X

Y

(lb/in2) l

~1I 0

0,25

sec.

Free

(a)

(b)

Figure 1 (a) Semi-cylindrical shell clamped at one curved edges and free at the other three edges. (b) Time variation of the uniformly distributed load in the shell. The material characteristics of the shell are: the elasticity modulus E=3.0xl07 Psi, the Poisson radius v=0.28, the density p=7.24x10 4 lb.sec~/in 4. The displacement and the acceleration time functions are obtained at the point with x=0, from now on denoted point A. The dynamic analyses is performed considering a damping radius of ~=2.5%, figure 2. A timestep of At=0.001 sec. is used and the analysis is carried out until the instant tf = 0.5 seconds. The shell geometry is defined by means of 4x8 Coons regions, 8 in the longitudinal directions (axis y), with a total of 45 key-points and a mesh of 8x20 elements, 20 in the longitudinal direction (axis y). The initial conditions are u(0) = 0 and 6 (0) = 0. 0,5

0,3

ii

0,2 m 0,1

0

0,1

0,2

0,3 Time

(a)

r

.o _o

0,4

0,5

22-iiiiiiiiiiiiiiiii 15000

1o000 0

-2sooo

.

0

.

.

.

.

.

.

.

.

0,1

.

.

.

.

.

.

.

.

.

.

-.. .............................................................................

0,2

0,3

0,4

0,5

Time

(b).,

Figure 2. Time variation at the point A: (a) the vertical displacement and (b) the vertical acceleration. In the case of sizing optimization, the design variables are the thickness of the shell varying only in longitudinal direction. Nine independent design variables are considered at equally distant key-points on the shell surface. In the case of shape optimization, the design variables are the lengths of the curvature radii of the shell and three independent design variables are considered. The localization of these variables at the key-points is depicted in figures 3(a) e 3(b), respectively.

378

2

I

i

2

"~

3

=i

4

.^

6

-L2

,)

7

8

9

'X (a) Thickness along the shell- 9 DV

(b) Length of the radii- 3 DV

Figure 3. Localization of the design variables: (a) Sizing optimization and (b) Shape optimization.

Sizing Optimization: Volume Minimization with constraints in the dynamic response Three cases of volume minimization with constraints in the dynamic response are presented in Table 1. In the first one the vertical displacement and in the second one the vertical acceleration at point A is constrained. In the first problem the bounds for the vertical displacement at point A are + 0.60 in. In the second problem the bounds for the vertical acceleration at point A are + 20000 in/sec 2. In the third case, the displacement + 0.60 in and acceleration + 20000 in/sec 2 are constrained simultaneously. In these cases, the side constraints are also considered for the design variables that are the thickness of the shell, which are considered in 9 key-points as indicated in figure 3 (a). TABLE 1 THREE CASES OF SIZINGOPTIMIZATION Objective-function to minimize Side constraints Constraints in the Dynamic Response (in inches and seconds)

Volume of the shell

Case I

Maximum vertical displacement at point A ut=-0.60 - uz - u.=0.60

V t e [0, 0.5 see.]

e~i, = 0.2 in <_xi _< e~ax= 2.0 in i= 1..... 9 Case II Case HI Maximum vertical Maximum displacement and acceleration at point A acceleration at point A. 0~=-20000 _
L

I I

In Table 2, the optimal distribution of the thickness for the three cases is presented. It is worth noting that the thickness distribution along the longitudinal axis of the shell changes significantly when the constraint is modified from a displacement to a acceleration constraint at point A. In the former case the greater thickness is concentrated close to the clamped edge while in the latter it is concentrated on the free edge. For the third case, the shell widths get thicker at both the free and the clamped edge and decrease gradually at the center, Figure 4. TABLE 2 RESULTS FOR THE OPTIMUMPROBLEMFOR THE VOLUMEMINIMIZATION Optimum values for the .design variables in inches 9 DV

el

e2

e3

e4

es

e6

e7

es

e9

Initial

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

Case l CaseIl Case l l l

0.50 0.50 0.50 0.50 0.50 0.63 0.77 1.09 0.71 2,00 1.26 0.25 0.20 0.29 0.46 0.20 0.20 0.20 1 . 9 0 0.93 0.33 0.33 0.48 0.70 0.70 0.85 0.85

Volume v(x)

3121.15 2010,27 1527.79 2485.20

Disp. l u (x)l 0.46 0.60 2.14 " 0.59

Accel. lii,(x)l 18366 36399 20000 20000 ,

,

379

Figure 4. The thickness distribution for the sizing optimization of the semi-cylindrical shell under dynamic constraints: (a) case I, (b) case II and (c) case III.

Shape and Sizing Optimization: Displacements Minimization with the volume constrained Now the optimization problem is once again to minimize the displacement at point A and, besides the variation of the radius lengths R in 3 key points, the variation of the uniform thickness is also considered. Therefore, four variables are taken in account in this optimization process. Two different cases corresponding to two different side constraints are studied, as it can be seen in Table 3. In the first case, the lower and upper bounds for the shape variables are 5.0 in and 12.0 in respectively. In the second case, aiming at decreasing even more the vertical displacement at point A, the side constraints for the radius lengths are relaxed. So the new upper bound is modified from 12.0 to 20.0 in. TABLE 3 TWO CASES OF SHAPE AND SIZING OPTIMIZATION Displacement at point A

Objective-Function to minimize

Uz ~x, 9 ,

m

Case I

Side

Case H

constraints

Both cases

t ~ [0, t~]

Rmm = 5.0 in _<x <_ Rmax= 12 in Rmm = 5.0 in _<x _< Rmax= 20 in emia = 0.2 in ___x _< emax= 2.0 in constant volume: v(x) = 3121,15 in 3 ,,,

.

.

.

.

.

.

.

.

Equality constraint

The results for the optimization problem are shown in Table 4. The vertical displacement at point A decreases from 0.46 in for the original problem to 0.295 in for the case I and to 0.067 in for the case II, representing an approximate reduction of 3 7 % and 85 %, respectively. The optimum shapes for both cases are depicted in figure 5. TABLE 4 RESULTS FOR THE OPTIMIZATIONPROBLEM Optimumivaluesforthedesign variables~

: Displacement

luz(x)i

Reduction

4 DV

Rl

R2

R3

e

(Inches)

(%)

Case I Case II

9.42 9.91

11.56 11.13

12.0 20.0

0.88 0.67

0.296 0.067

37 85

.

.

.

.

.

380

Optimu~~ 15

A Optimum shape

L rigiml shape """"'"

-10

i R3

-5

0 A~s ,,

5

10

-lo

-5

0

5

10

Axis x

(b) Figure 5. Optimum shape for the semi-cylindrical shell (4 DV): (a) Case I and (b) Case II.

CONCLUSION An automatic procedure for sizing and shape optimization of shell structures under dynamic loads is presented which includes: geometric modeling and mesh generation of the structure, structural analysisvia FEM and Newmark direct integration method, sensitivity analysis via semi-analytical method and a SQP algorithm for solving the optimization problem. A semi-cylindrical shell submitted to a vertical dynamic load is optimized. All the proposed problems were solved adequately. Therefore, the proposed automatic methodology has proven to be a useful and efficient tool for performing sizing and shape optimization of thin shells under dynamic loads.

REFERENCES

Afonso S. M. B. (1995). Shape Optimization of shells Under Static and Free Vibration Conditions, Ph.D. Thesis, University of Wales/Swansea. Clough R.W. and Penzien J. (1975). Dynamics of Structures, New York, Mcgraw-Hill. Dutta A. and Ramakrishnan C. V. (1998). Accurate Computation of Design Sensitivitiesfor Structures under Transient Dynamic Loads using Time Marching Scheme, Int. Journ. for Num. Methods in Engineering, vol. 41, pp 977-999. Falco S.A. (2000). Optimization of Shells under Dynamic Loads, Ph.D. Thesis, Pontificia Universidade Cat61ica of Rio de Janeiro, Brazil. Ha~ka R. T. and Gurdal Z. (1992). Elements of Structural Optimization, Kluwer. Huang H. C. (1988). Static and Dynamic analyses ofplates and Shells: Theory, SoftwareandApplication, Springer- Verlag, London, Berlin, Heidelberg, New York, Paris, Tokyo. Newmark N. (1959). A Method of Computation ofStructuralDynamic, Proc. ASCE 85,EM3, pp. 67-94. Paz M. (1997), Structural Dynamic Theory and Computations, Chapman and Hall, 4th Ed. Powel M. J. D. (1978). Algorithms for Nonlinear Constraints that use Lagrangian Functions, Math. Programming, vol. 14, pp224-248. Vanderplaats G.N. (1999). Numerical Optimization Techniquesfor Engineering Design, 3=r edition, McGraw-Hill Book Company. Zienkiewicz O. C. & Taylor R.L. (1991) and (1995). The Finite Element Method, Vol. 1- 2, McGraw-Hill.

381

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

VIBRATIONS OF COMPRESSED SANDWICH BARS J.Hafirkowiak and F.Romanrw Department of Mechanical Engineering, Technical University of Zielona G6ra, ul.prof.Z.Szafrana 2, PL-65-016 Zielona G6ra, Poland

ABSTRACT The time characteristics and stability regions of simply supported bars, compressed with weak periodic component axial forces, are described by means of perturbation theory. As an example, the Dirac impulse perturbations are considered. The type of initial conditions used is illustrated.

KEYWORDS Periodic axial force, eigenproblem, characteristic exponent, perturbation theory, Ince-Strutt diagram DESCRIPTION AND VISUALIZATION OF ELEMENTS CONSIDERED. Let us consider the displacements of continuous elements with respect to a given axis described by the function u ( t , P ) = T(t) . U ( x , y , z )

(1) where T satisfies the differential equation d2T dt 2

+ [8 + e ~ ] T = 0

(2)

and U is a given function of point P of the element in the unstrained state. In Eq.(2) the two mathematical parameters 6 and e are expressed by physical parameters such as the Young modulus, dimensions of the system, and parameters describing external forces acting upon the system. They also depend upon the discrete variable n labeling possible equilibrium configurations of the bar. Such an interpretation of the function U is possible due to fact that the constant in time function T satisfies (2), for 5 and e equal to zero. This corresponds to the critical values of physical parameters (Euler forces). For transverse vibrations of simply supported bars nrc U - U n(x) = A. sin ~ x,n = 1,2,3.... l

(3)

382 The function W called a perturbation of the system is defined by the external forces acting on the system. In the case of transverse vibrations of a simply supported bar, the perturbation q~ is related to axial force N(t) as follows N(t)=N 0

+

Nlv(t)

(4) and parameters 8 and e occurring in Eq.(2) do not depend on the shape of function q-' and are defined as follows 6 = 6n =co2 (1 - No ) N,r,

(5) where the square of n-th frequency of the free bar 2

4

nrc

co. = ( T )

EJ m

(6) The critical compressed force, for which the n-th equilibrium position of the homogeneous bar is possible, is given by 2

N,,, = ( ~ - )

EJ

(7) Here 1 - denotes a length of the bar, J - an inertial moment of the cross-section of the bar with respect to the neutral axis, m . - a constant mass density. The perturbation parameter 2 Nl C

=o~n-'O)n

~

Nkr,

(8) In the case of transverse vibrations of a simply supported three layer bar, we assume that the structure of the bar leads only to a modification of the critical force (7). We achieve this modification with the help of the hyperbolic displacement distribution hypothesis and with additional assumptions related to sandwich elements, Romanrw (1995). In Fig.1 a simple supported sandwich bar and its cross section is presented. For such a bar, the critical load is Ir2Ej 2 Nkr, = 2b 12 n +

(c+t /2)2 =2E 9b. t. n 2 ( t g h p c + t / 2 ) P n 2E .b . t . c . n 2

lZ(1 +

)

G

l2 U

p~ = n2~ 2 1 - 2 v u l2

J=bt[

2 ( 1 - vu)

1 + ~ (2c + t) 2]

(9) where E represents the Young modulus for facings, G,, vo - the shear modulus and Poisson's ratio for the core material.

383

c o r e e.g. /polyurethane / , , ? f a c i n g s e.g.

,

i ,11,_ I - 7

steel

sheet

/ !

'{ t

.

r

I i i

,

i

(

U[x)

2c

?

_....t

2c§

Figure 1"A simplysupportedsandwichbar In the case of both homogeneous and sandwich bars, there is an infinite number of relations between mathematical parameters 5 and 8 =COn2+ N~ Nl

(10) resulting from their definitions and represented by parallel straight lines in the plane F of parameters 5 and c, see Fig.2.

y Figure 2: ,,Real" values of parameters They provide important information about the element under consideration: their slope informs us about the ratio of the two axial force components and their intersections with 5 axis tell us about its free frequencies. More information can be expressed in Fig.2, if particular values related to definitions (5-8) of parameters 5 and e and expressed by white circles are introduced. To describe various initial conditions, the solutions (1) with different n together with the superposition principle are used. By black circles in Fig.2 we may denote these pairs of parameters 5 and e with n used for the construction of the necessary initial conditions. Taking into account a given class of possible initial conditions, described by the given collection of black circles, we must guarantee that black circles do not enter any instability region described in

384 plane F by Inee-Strutt diagrams, see Fig.4-5. For a given shape of perturbation W, Ince-Strutt diagrams are fixed and do not depend on the two N coefficients in the axial force (4) or on other parameters describing geometrical structure and materials used to make a concrete bar. These quantities influence the slope of parallel lines describing relations (10), and the positions of the black circles on these lines. It is interesting to notice that usually e depends only weakly or not at all on the large values of the Young modulus E. This means that perturbation theory can be used to solve equations like (2) in many cases of vibrating elements. The purpose of the paper is to derive, for the general class of perturbations q~, the time characteristics and stability regions of bars. In our approach, in spite of the explicit definition of parameter 6, we shall assume that this parameter, together with function T, is unknown. This means that Eq.(2) reminds us of an eigeaproblem. According to the Floquet-Bloch theorem for periodic perturbations q~, the different eigenfunctions can be described by means of the characteristic exponent It. They have the following property T ( t ) - T ; (t) = e i~'t ~ , (t)

(11) where arbitrary Ix belongs to the interval <-re/a, n/a> and the function 9 o has the period "a", Davidov (1969). As a consequence of our assumption, the eigenvalue 5 depends on ~t and e

8 = A(#,e) (12) We calculate this dependence of parameter 6 on ~ and e by means of perturbation theory for the above eigenvalue problem. In this way the structure (11) of exact solutions to Eq.(2) is incorporated and an equation for IXcan be derived: COn2+ No e = A(/d,~) Nl (13) To derive (13), the assumption about the analytical dependence of the characteristic exponent Ix was not used. Solutions to Eq.(2) satisfying (11) are called normal solutions, Mierkin (1987). If To(t) is a normal solution then its complex conjugation, the function To*(t) is also a normal solution. The general real solution to Eq.(2) with a given value of the parameter 5 can be constructed by means of the corresponding normal solution (11) by its multiplication with the arbitrary complex number C=]CIe mc after extracting the real or imaginary part.

T ( t ) =1C II~ (t) Isin(/~ 9t + arg ~ (t) + arg C)

(14) Hence, the general solution to Eq.(2) can be interpreted as the sinusoidal vibrations with frequency given by the characteristic exponent kt and with periodically modulated amplitude and phase. The period of modulation is "a". It is a remarkable representation of the general solution to Eq.(2). First of all, it exposes the physical character of the characteristic exponent It. Second, it shows that changes introduced to zero-th order solution (e=0) through slowly varying periodic function 9 o , in the exact normal solution (11), can be reduced to ,,adiabatic" changes in the initial conditions described by constants ICI and arg C.

385 COMPRESSED AXIAL FORCES WITH A SMALL PERIODIC COMPONENT In this case the perturbation parameter e assumes small values and a good approximation to the normal solution (11) can be expressed as follows 1

T(t) =- T~ (t) = - ~ e

ilu.t

+ 0(6)

(15) Parameter L=Na denotes the so called large periodicity introduced here to get a discrete set of values g (Bom-Karman conditions, see Ginter (1979) or Ziman (1977)) numerating a complete, orthonormal set of exponential functions appearing in (15). Complex functions (15) are preferred from reasons of simplicity. According to the standard perturbation approach, e.g., Davidov (1969) or Schiff (1977), improved solutions to the eigenproblem (2) are given by Tu(t) = tu(t) +o~ E tu,(t) + 0 ( ~ u,.u ~t 2-/.t '2

~) (16)

and t

c I~F(t)dt + ~ E Il a(~u,e) = kt 2 +-~ , : u'~u kt - / u

2

+o(~)

(17) The symbols I~> and
=~2rr k - Gk,

k --- +1,+2,...

a

(18) where a is the period of perturbation (t) = E~uk exp(ik 2rr. t) k

a

(19) For other values of index it' in formulas (16-7), the appearing matrix elements L

< gz'l ~lgz >= 1 / L~ e-J;"'~(t)eJ;"dt o

(20) simply disappear. According to the basic postulate of any perturbation theory, the next approximation has to be smaller then the previous one. This means that the denominators in formulas (16-7) should not be small, in contrary to small values of ~ and matrix elements (20). The above postulate is satisfied for (18) with the exception ~ =2

/./,2 (21)

386 or

2 2rr 2 /2 = ( / 2 n) a (22) for n assuming integer numbers. It is seen easily that 7/"

/ 2 = -a- n (23) satisfies these conditions. In other words, for (23), zeros appear in the denominators of formulas (16-7). This is a signal that when It satisfy (23) then for the derivation of formulas (16-7) certain forbidden operations were used. To avoid this a modified perturbation theory has to be applied in the sums which functions with index It' satisfying (21) do not appear [5-6]. We incorporate this function into the zero ~ order term. So, instead of (15), T, (t) = c. t, (t) + d .t_ u (t) + O(~') (24) where t,(t) denotes the exponential function appearing in (15) and which is denoted by lit> in the Dirac notation. ~t in this formula is given by (23). To determine constants c and d, (24) is introduced to Eq.(2) after which Eq.(2) is multiplied by functions t~(t) and t.,(t). The procedure described can be expressed in vector language as follows
§ e . ~ ] } ( c I/2 > §

I-/2 >)-- 0

< -/21{02 + [8 + s . ~v]}(c I2 > +d I-/2 >) = 0

(25) where the second derivative of Eq.(2) is denoted by D z . These homogeneous equations can be solved only when condition 8 = A ( ~ , 6 ) = (~) ~ +El< ~ I~' I- ~ >1-~ < ~ I ~ l a >

(26) exists for any periodic perturbation W and It satisfying (23). Eqs (26) mean that 8 considered as a function of the characteristic exponent It and the perturbation parameter e experiences jumps of magnitude

2 ~ ~ ~b~ ztn

7/'n

(27) a possible mechanism of which is explained in Fig.3 /x

Figure 3" Jumps of functions 8(bt)

387 We interpret Eqs (26) as equations defining stability regions in F. It is interesting that matrix elements appearing in these equations are expressed by the Fourier coefficients of the perturbation W: zcn zcn 1L 2feint) < ~ I ~ I - ~ > - < ~a I~U l- ~a > - - -L! ~ ( t ) e x p ( - ~ a dt=~,

(28) and ~n 1~ iron ~ > = 1 i ~ ( t ) d t =hu0 <~l~l~>--<~ a a L0

(29) see Fig.4

Figure 4" Black color represents instability regions, gray- stability and white- regions requiring the higher order calculations For sinusoidal perturbations, all elements (28-9) are equal to zero with the exception of one element n =1 only. This agrees with canonical results concerning the Mathieu equations. For periodic impulse perturbation

(t)=Z~(t-k.a) k

(30) the matrix elements (28-9) = 1/a (31) In this case Eqs (26) describing stability boundaries of the system under consideration are

fi =(re-n)2 +e/a+~./a+_O(62) a

(32) This leads to the stability regions illustrated in Fig.5.

388

Figure 5: Ince-Strutt diagrams for Dirac impulse perturbation with the same meaning of colors as in Fig.4 They grow with a decrease in the period "a" of perturbation (30). In a similar way one can show for periodic impulse perturbation (t) = Z (-1) k6(t - k . a) k

(33) (compression and tension) that, because in this case matrix elements (28-9) disappear,

8=(re'n) 2 +_O(e2) a

(34) This means that the axis 8 is approached by instability regions on Ince-Strutt diagrams in "needle" fashion. FINAL REMARKS In the paper, for small e, stable vibrations of homogenous and sandwich bars were described: formulas (16-7) and Eq.(13). In addition, equations for stability region boundaries, in space F of parameters 8 and e, were derived, Eqs (26). The results derived depend on the shape of perturbation W. Assuming that the shape of the periodic perturbation W, with fixed period a, is a random quantity, we can obtain Eqs (26) for the averaged boundaries R

8--~ ~ +c.l< ~ I~ I-~ >1-6,< ~ 1 ~ I~ > (35) with the averaged matrix elements

I< ~ I ~ I v >1- I I< ~ I~ I v >1 e [ ~ ] - 8 ~ ' (36) where the averaging is executed with some probability density P(q0, over all possible perturbations qJ, with fixed period a. For a Gaussian P centered on the zero, the averaged matrix elements appearing in the averaged Eq.(35) do not depend on index ~t. In fact, the last ones in (35) disappear and symmetrical Ince-Strutt diagrams emerge. Hence, we come to the

389 conclusion that for a Gaussian distribution of perturbations qJ, all parametrical resonances are equally dangerous (small e). See also Hafirkowiak & Romanrw (2000).

References Davidov A.S. (1953). Quantum Mechanics, MIR Publishers, Moscow (Russian) Dirac P. (1968), Quantum Mechanics, MIR Publishers ,Moscow, (Russian) Ginter J. (1979). An Introduction to Atomic, Molecular and Solid State Physics, PWN Publishers, Warsaw, (Polish) Hafirkowiak J. and Roman6w F. (2000). A Non-singular Description of Parametrical Resonance. J. Tcheor.Appl.Mechanics 38:4, 879-892. Mierkin D.R. (1987). An Introduction to Stability Motion Theory, Science, Moscow, (Russian) Romanrw F. (1995). Strenth of Sandwich Constructions, Technical University of Zielona Grra Publishers, Poland. Schiff L.I. (1977). Quantum Mechanics, PWN Publishers, Warsaw, (Polish) Ziman J.M. (1977). An Introduction to Solid State Physics, PWN Publishers, Warsaw, (Polish)

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Third International Conferenceon Thin-Walled Structures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

391

NUMERICAL SIMULATION OF DAMAGED STEEL PIER USING HYBRID DYNAMIC RESPONSE ANALYSIS T. Ikeuchi j and N. Nishimura 2 IDepartment of Civil Engineering, Tottori University, Tottori, Koyama-minami 4-101, JAPAN 2Department of Civil Engineering, Osaka University Osaka, Suita, Yamadaoka 2-1, JAPAN

ABSTRACT In this study, the damage of pipe piers under Hyogoken-nanbu Earthquake is investigated and the hybrid dynamic response analysis is carried out to clarify the damage process of steel pier under earthquake. The results of analysis show that the main damages are local buckling and concentrate near the cross-section where the plate thickness changes. At first step of the process of elephant-foot buckling, local buckling is observed in one side subjected to bending which is introduced by the ground acceleration. At next step, the second local buckling is observed in opposite side with turning over the ground acceleration. The both local bucklings gradually become large in accordance with the increasing of acceleration amplitude and the buckling mode finally becomes a ring like bulging.

KEYWORDS Steel Piers, Earthquake Damage, Elephant-foot Buckling, Hybrid Dynamic Response Analysis INTRODUCTION Dynamic response analysis is useful method to check the dynamic strength of steel piers under earthquake. However, it takes long calculation time in case when plate or shell elements are employed to consider the local buckling. Several methods were proposed to shorten the calculation time. In one of these methods, hysteretic model is used to consider the effect of local buckling and yielding [Suzuki(1996), Kumar(1997)]. It is difficult to make an accurate hysteretic model. In another method, shell/plate elements are only applied to the part which may have buckling and beam-column elements are applied to other part [Ohta(1997),Nara(1998)]. But this method needs that the part, which has buckling, is known before analysis. Kitada et al proposed the method to combine the response analysis of SDOF (single degree of freedom) system and elasto-plastic finite element analysis, which is called hybrid method [Kitada(1998)]. This

392 method does not require hysteretic model, but can get accurate hysteretic load-deflection relationship by elasto-plastic finite element analysis. In this study, this hybrid response analysis are carried out to investigate the damage of steel pipe pier at Hyogoken-nanbu Earthquake. HYBRID DYNAMIC RESPONSE ANALYSIS Outline o f Analysis Method

In order to reduce computational time, a steel pier is regarded as a SDOF system and that equation of the motion is solved. The read input data hybrid method differs from ordinary response analysis of SDOF in that the horizontal force is given by elasto-plastic finite element analysis, which can consider local buckling and steel yielding. Figure 1 shows the computational procedure of hybrid earthquake response analysis. Impulse acceleration method is 031 set initial conditionI ~0, ~0, ~'0 I used to solve the equation of motion. The details of each procedure 1-4 are shown in Figure 1. 1.For given initial velocity and displacement, initial acceleration [calculate 8tby ] are calculated by the equation of motion. (~) impulse acceleration method ~o = -~ - c ~o - kSo m

(1)

Icalculate Rt by elasto-plastic I (~) finite element method i

2.For last step displacement, velocity and acceleration, next step @ impulse acceleration method displacement are calculated by impulse acceleration method.

I

1

(2)

6 t = 6t_At + 6,_A,At + ~$t_atAt:

~

~

I

no

3.Horizontal force at pier top is given by elasto-plastic finite Figure l:Computationalprocedure element analysis. 4.For horizontal force, next step acceleration and velocity are calculated by next step impulse acceleration method.

cat ~ -~t - c r,-~ - ~ m '-~' ~, = m c

Rt

1 .. m , t~t = 6t-~ + -~(4-~ + 6, )At

(3)

1 +~At

2m

m: mass of pier, k: initial horizontal stiffness of pier, c: damping factor, At: incremental time, ~ :ground acceleration, 5, ~, ~ :displacement, velocity, acceleration at pier top Confirmation of Analysis

In this paper, earthquake response analysis is carried out by using hybrid method as mentioned above. In order to verify the accuracy of the analysis method, the hybrid experiment by Kitada et al is compared with analyzed results. The specimen of hybrid experiment has unstiffened box section. Table 1 shows

393 the property of specimen. Ground acceleration measured at JMA Kobe Observatory (Hyogoken-nanbu Earthquake 1995) was used for response analysis input. Figure 4 and 5 show the time history of horizontal displacement and horizontal load-displacement relationship at the pier top respectively. These figures show that the results of analysis and experiment are in a good agreement. TABLE 1 PROPERTY

Specimen

flange width (r.rn~ 13.5

d-c

OF SPECIMEN

web width

flange thickness

web thickness

height

10.5

4.4

4.4

75

(nm~

(rrrn~

(rrrn~

axial load

(nm~

r ~ i n P/

0.15

4~ 2.0 & ..

o

--:-

oo0iiill o

- 6 y --"--'~

--/~-- ...... -~-

z.

0

_o;;

E

? .......................

-

,~,

--

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^

/~.^

A.

V v v v v v..--~--,,

-2. o-" m o,,

.~_

. o.o

s.o

tOL,,.O ~

,

20.0

IS. O

25.0

timefsec)

9..

time(sec~

(a) time history of displacement

(a) time history of displacement eooo-

6,000

3,000

300o

!, i

I

o

o

N ~

.~

-3,~0

-6.000: -~

0 Horizontal disolacement tern)

5.0

-3ooo-

-eooo-

s. o

0. o Horizontal disolacement

(cm')

(b) horizontal load vs. displacement

(b) horizontal load vs. displacement

Figure 2: Hybrid experiment

Figure 3" Hybrid analysis

NUMERICAL SIMULATION OF STEEL PIERS DAMAGED BY EARTHQUAKE

Outline of pier damages This paper investigates damaged steel pipe piers around the Matsubara intersection of the Hanshin Expressway. As shown in Figure 4, the elevated bridge was separated into eastbound and westbound in this area. The steel pipe piers, which had a height of about 15 meters, were used as single columns. In this paper, these piers are called NP-580 to NP-585 in the eastbound, SP-580 to SP-585 in the westbound. As for these piers, the structure was almost similar. For example, NP-584 was composed of steel pipe segments, which have a diaphragm between welded lines, as shown in Figure 5. By inspecting the damaged locations, all of the damage was caused at the welded line of the piers, especially the parts at which the thickness of the pipe changes. As shown in Figure 6, the damage may be divided into three types: elephant-foot buckling with severe cracks, local buckling on one side and peeling of paint without

394

any residual deformation. The damage details of the each pier are described below. il~ay NP-585 t_.NP-584

~ 0

sP_585

Hanshinexpress highway NP-583

~

~

sP.584

I NP-581

NP-582

~

..~

I

NP-580

0

0

/sP.583

SP-581 I

SP-580

I :pier analyzed in this study _

NP-585

NP-583

NP-584

NP-582

NP-581

NP-580

Figure 4: Grand plan and elevation of piers. 0 t 7, ::~00

Nd=422.

Rt

=39. 3

R t

=57. 9

a =0.176 Nd/I~/=0.101 o --0.130 Nd/N/=0.135

d~

t-4

I~ t a

=57. 9 =0.130

Nd/I~=0.135

inner

1:5

Rf t

=57. 9 =0.130 ,--Nd/I~=0.135

I i

!

~

,-

,

,,~-

t-4

,_

o

r a ~

r

./~-

.

.

l:#t

--44.0 a =0.10 Nd/I~=o. 103

R t

...,

=0.10 ~=0.103

=44.0

r

eq

Y/Y/. Figure 5" Detail of the pier (NP-584)

r

395

Figure 6: Damage state of steel piers NP-584 :Elephant-foot buckling was observed at the middle height of the pier and severe deformation and cracks were observed around the welded line. SP-582 :Local buckling on one side was observed at the middle height of pier and the paint was peeled at the middle height of the pier and near the base. NP-581 :No buckling was observed, but the paint was peeled at three wide areas of the pier.

Numerical Simulation of damaged piers Three piers (NP-584, SP-582, NP-581) which have different type of damage are analyzed. Analysis model is shown in Figure 7. The iso-parametric shell element is employed to express the shape of a pipe suitably. BMC model which can express the stress-strain relationship under cyclic loading [Nishimura(1995)] is used as constitutive equation. Ground acceleration measured at JMA Kobe Observatory(Hyogo-ken Nannbu Earthquake 1995) was used for response analysis input, because these piers were near the JMA Kobe Observatory. The damping factor of the equation of motion is set to 0.02. Figure 8 shows the time history of horizontal displacement at pier top. Horizontal displacement begins to increase from 5.0 see and reaches a peak (P-584: 25.0cm, 5.42sec, P-582:28.0 cm,8.21sec, P-581:27.0cm,9.44sec). After 10.0 sec, the displacement amplitude decreases gradually. Figure 9 shows the relationship between horizontal load and displacement. Peak displacement is almost three times as large as yield displacement. Figure 10 shows deformation and stress distribution of NP-584. The deformation at 30.0 sec shows elephant foot buckling and agrees with the real damage. This figure shows the formation process of elephant foot buckling. At first, the local buckling is observed at one side subjected to bending. Next, the local buckling occurs at another side with turning over the ground acceleration. The local buckling at two sides grows with cyclic loading and becomes a ring like bulging. Figure 11 shows deformation and stress distribution of SP-582. The deformation at 30.0 sec shows local buckling at one side and agrees with real damage. Figure 12 shows deformation and stress distribution of NP-581. The stress distribution when horizontal displacement reaches a peak shows that the yielded zone is founded at wide area and similar with actual damage. These results show that hybrid response analysis can simulate the actual damage and explain the formation process of elephant-foot buckling.

396 422tf

(unit'cm)

tf

585tf

'~---

:~~

~

.,.,J

O~

~176 II:

r r

II

~

.

:

c

~

/

u).l~

N

I

'

: ~

c

o

.

~

c

j

~O'J O .

o

.

r

i!

,q-

'

~ X

NP-584

SP-582

NP-581

Figure 7: Analysis model 30. O-

A

20. 0-!

E

-9

lo. o-! oo!

_ __A --

NP-584 i

A A l i A . I/~/\ I I A A ~ ,,

'~/vvv

AAAA A

~f,,--

"~-10.0-! -20. 0 -i . . . . . . . . 0.0 5.0

15.0

10.0

20.0

25.0

30. 0

time (sec) 30. 0-.

AA

20.0-!

~oo! d

0.0-!

-10.0-!

~ A A A,~A AA/~ A A A A A ~//v~'vv ,-vvv~,vv,,

_..AIII\! '-

rV

.

v

-

.

.... .

.

.

'

-20.0 -i

0.0

5. 0

10. 0

15.0 time (sec)

20. 0

25.0

30.0

30. 0-. 20.0-!

0. 0-!

:~ -10.0-20. 0 i 0.0

AA~./,

A A A r~ A A

~o.o! dIll

--- irV v 5.0

.

10. 0

.

.

.

.

.

15.0

.

.

.

.

.

^ .,.....~

.

20. 0

time(sec) Figure 8: Time history of horizontal displacement

25.0

30.0

397

Figure 9: Horizontal load vs. displacement

Figure 10: Deformation and stress distribution (NP-584)

Figure 11" Deformation and stress distribution (SP-582)

398

Figure 12: Deformation and stress distribution (NP-581) CONCLUSIONS In this paper, the numerical simulation is carried out using hybrid response analysis. These results lead to the conclusions listed below. 1) Earthquake response of damaged steel pipe pier is analyzed and computational time is shortened by using of hybrid method. 2) According to the analyzed results, it becomes clear that local buckling concentrates on the location where the plate thickness changes. 3) It becomes clear that the formation process of elephant-foot buckling dep~ds on the time history of analyzed pier's deformation 4) The analyzedresults show good agreementwith the actual damage. REFERENCES Kitada T., Nakai H., Kano M. and Okada J. (1998). Establishment of a Method for Analyzing Elasto-Plastic and Dynamic Response of Steel Bridge Piers with Single Column by Considering Local Buckling. Proceedings of the Second Symposium on Nonlinear Numerical Analysis and its Application to Seismic Design of Steel Structures. 255-262.(in Japanese) Kumar S., Mizutani S. and Okamoto T. (1997). Circular steel tubes under cyclic loading. Proc. of the 5th international colloquium on Stability and Ductility of Steel Structures. 251-258. Nara S., Murakami S., and Tamari K. (1998). Elasto-Plastic Dynamic Analysis of Steel Bridge Piers Considering Their Local Buckling Behavior under Seismic Load. Proceedings of the Second Symposium on Nonlinear Numerical Analysis and its Application to Seismic Design of Steel Structures. 263-270.(in Japanese) Nishimura N., Ono K. and Ikeuchi T. (1995). A constitutive equation for structural steels based on a monotonic loading curve under cyclic loading. Journal of structural mechanics and earthquake engineering. 513:1-31.27-38. Ohta K., Mizutani S., Nakamura S., Kobayashi Y., Nakagawa T. and Nonaka T. (1997). A Comparison of Commercial Non-linear Structural Analysis Software in Deformation Analysis of Steel Bridge Piers with Circular Section by Elast-plastic Finite Element Method, Proc. of the 5th international colloquium on Stability and Ductility of Steel Structures. 273-280. Suzuki M., Usami T., Terada M., Itoh T. and Saizuka K. (1996). Hysteresis Models for Steel Bridge Piers and their Application to Elasto-Plastic Seismic Response Analysis. Journal of structural mechanics and earthquake engineering. 549:1-37. 191-204. (in Japanese)

399

Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

CYCLIC RESPONSE OF METAL-CLAD WOOD-FRAMED SHEAR WALLS Weili Pan i

and

K.S.(Siva) Sivakumaran 2

1. Former Graduate Student, Currently at Lines Design & Technical Services, Hydro One Networks Inc., 483 Bay St., 7th Floor, Toronto, Ontario, CANADA. M5G 2P5 2. Professor, Department of Civil Engineering, McMaster University, Hamilton, Ontario, CANADA. L8S 4L7

ABSTRACT Shear walls may be used as vertical force transfer system. Shear wall diaphragm design is a widely accepted practice in wood buildings and concrete and masonry shear walls are mainstream of those buildings. In steel buildings, however, the contribution of the cladding is often ignored, and separate bracing system, such as x-bracing, is used. This paper reports a recent experimental investigation conducted in order (1) to determine the in-plane shear strength and shear stiffness of the corrugated cold-formed steel sheet cladding profiles most commonly used in Canadian farm building construction; (2) to exam the performance and behaviour of metal clad - wood framed wall diaphragm under fully reversed cyclic loadings, and compare the performance with that of wood clad- wood framed wall diaphragms; and (3) to study the effect of metal sheet thickness and sheet to purlin fastener spacing on diaphragm shear strength and shear stiffness.

KEYWORDS shear wall, farm buildings, metal cladding, wood frame, cyclic tests, shear strength, shear stiffness.

INTRODUCTION The lateral loads on building structures, such as wind loads, earthquake loads, etc., must be efficiently and economically transferred to the foundations. It requires adequate horizontal distribution systems at each floor and roof levels, and a vertical force transfer system. The floor and roof decks diaphragm design is widely used as the horizontal distribution system. Shear walls may be used as vertical force transfer system. Shear wall diaphragm design is a widely accepted practice in wood buildings, and concrete and masonry shear walls are mainstream of those buildings. In steel buildings, however, although it is generally appreciated that the steel wall cladding improves the strength and stiffness, the contribution of the cladding is often ignored, and separate bracing system, such as x-bracing, is used as vertical load transfer system. Current Cold-Formed Steel (CFS) design standards [AISI, 1996; CSA, 1994] do not include provisions for shear wall design and no comprehensive authoritative design and construction guidelines on this topic exist.

400 Recently, an investigation was conducted on the diaphragm strength, stiffness and the ductility of CFS wall cladding in wood framed shear walls. The aim is to develop design and construction procedures that will facilitate acceptance and use of metal clad shear walls in building and housing construction, particularly in farm buildings. The test program considered three kinds of arrangements, (a) wood-frame with no cladding [to establish baseline shear resistance], (b) wood clad- wood frame panels, and (c) metal clad wood frame panels. Full scale shear walls were subjected to (i) cyclic tests, and (ii)static push over tests. The main objectives of the tests in this study are: (1) to experimentally determine the in-plane shear strength and shear stiffness of the corrugated cold-formed steel sheet cladding profiles most commonly used in Canadian farm building construction; (2) to exam the performance and behaviour of metal clad wood framed wall diaphragm under fully reversed cyclic loadings, and compare the performance with that of wood clad- wood framed wall diaphragms; and (3) to study the effect of metal sheet thickness and sheet to purlin fastener spacing on diaphragm shear strength and shear stiffness. The paper discusses the test arrangements, including loading and instrunaentation related to these tests, and then it presents the comparative results. Complete detail related to this investigation is available in the thesis by Pan (2000).

CYCLIC TESTING OF WALL DIAPHRAGMS The "Construction Guide for Farm Buildings" (Ministry of Housing, 1995) has been liberally used in this study in determining the dimensions for the test specimens, and other test variables. The wall specimens for testing should be representative of those walls used in the actual construction, which includes, the stud spacing, purlin spacing and orientation, cladding type, profile and thickness, fastening type and pattern, etc. This study considered two types[Profile:A & B] of commonly used corrugated cold-formed steel claddings, [panel width of 813 mm (32") and a depth of 13 mm (0.5")], having base steel nominal thicknesses: gauge 28 (0.378 mm) and gauge 30 (0.305 ram), and wood claddings [9.5 mm (3/8") thick plywood ]. Based on tensile tests on steel claddings, the average yield strengthfy was 334 MPa, the average ultimate strength f, was 447 MPa, and the modulus of elasticity E was 192,000 MPa. The height ofthe wall was 3.66m (12'), which was the average height of a stud-framed, single story building, and the width of the test panel was 2.44 m (8'). The material used for framing members was Spruce-Pine-Fir (S-P-F), No.2 or better, dry with a moisture content of below 19%. The studs were of a size of 38 mm x 140 mm (nominal 2"x 6") with a spacing of 610 mm (24"). The top and bottom plates were also of a size of 38 mm x 140 mm (nominal 2" x 6"). Fire breaks were made from 38 mm x 140 mm wood with a spacing of 1220 mm (4'). For shear wall specimens with metal claddings, in addition purlins and shear connectors were provided [bothl 9 m m x 89 mm (nominal l"x 4")]. Ardox spiral nails of 89 mm (3.5") length were used for connecting wood framing members, and 63.5 mm (2.5") long Ardox spiral nails were used for connecting plywood to the frame and connecting double end studs together.63.5mm long Ardox spiral nails were also used for connecting purlins and shear connecters to the studs and plates. 6.35 mm (1/4") Hexagon No. 9 wood gripping screws with neoprene washer were used for connecting sheet metal to purlins and shear connecters. Depending on the configuration of the sheet claddings, 25.4 mm (1 ") long screws and 38 mm (1.5") long screws were used. Figure: 1 shows the test setup. Though, Two-bay Simple-beam Method or The Cantilever Method could be used, since previous tests have shown that both these loading arrangements provide similar results with respect to shear strength and shear stiffness values and load-deflection characteristics, here, The Cantilever Testing Method was used. In this test, the wall assembly was subjected to a fully reversed cyclic racking shear load with the bottom edge anchored to a rigid base and the load applied parallel to the top of the wall with the test assembly being free to displace in its own plane. The test specimen was rigidly supported on the bottom edge using wide flange steel base beams. A wide flange steel load distribution beam was used on the top of the wall specimen. An additional lateral guiding system was provided to prevent the specimen against out-of-plane movement. Some specimens were subjected to vertical loads of 44.4kN, which was applied equally at the both ends of the wall panel. The vertical loads simulate an equivalent load of 2.4 kPa from the roof. These loads were applied on the two ends of the specimen by vertical load jacks reacting

401 through 12.7 mm ( 89 diameter threaded steel rods. Verticality of these loads were ensured through the use of a roller system. The horizontal load was applied through a horizontal load actuator, which was mounted onto the reaction columns, and was connected to one end of the steel load distribution beam using a pin connection. The load actuator had a capacity of 111 kN (25 klbf), and duel action stroke and maximum stroke of 30.48 cm (12"). Diaphragm tests can be conducted either statically or cyclically. However, the fully reversed cyclic load is more representative of the actual load a wall experiences, particularly during an earthquake. The cyclic testing procedures will also be able to examine parameters, such as, strength degradation, stiffness degradation, and energy dissipation capacities of a wall diaphragm that cannot be predicted by the static tests. Though there exist many cyclic test protocols, the New Cyclic Loading Protocol (He et al., 1999) has been used in this study. This loading scheme consists two or three groups of cycles, three identical cycles in each group, and one final unidirectional loading (pushover) until the wall fails. The amplitudes of these cycles groups correspond to the displacement at a certain percentage of the maximum load obtained from the monotonic shear wall tests. As no static monotonic test was conducted under this investigation, the largest test amplitude was selected based on approximate displacement drift of 0.78% of the panel height. For a wall of 3.6 m in height, when the amplitudes are chosen to be 0.25, 0.50, and 1.0 times the largest amplitude, the corresponding three amplitudes are 7 mm, 14 mm, and 28 mm, respectively. After three groups of cycles, the test specimen was pushed over in one direction until failure. For the tests with vertical loading, vertical loads were applied at the both ends of the shear wall before lateral load was exerted. Complete failure of the wall panel was associated with the loading state at which the applied lateral load dropped more than 10% from a maximum, and can not be reached again. The horizontal load and vertical load were recorded through load cells and electronic amplifiers. Lateral and vertical deflections were measured through Linear Variable Differential Transducers (LVDT) placed on the top, and the bottom ends of the specimen. Additional LVDTs were used to measure deflections, and the associated locations of these LVDTs are illustrated in Figure: 1. ~g

,

?/ U/

g~ =

.,

v,. r~

-o

It..

"i:

.m

402 Test Observations

The wood frame only specimens (no cladding) acted like hinges and the frame could hardly resist any lateral load. The wood clad-on-wood frame wall panel with no vertical load exhibited visual damage when the horizontal load reached about 6 kN during the last stage of pushover. The studs were tilted, while the plywood panels tended to act as a whole body and slide relative to each other along the interior sides. As the load was increased, the sheathing nails connecting the plywoods and the studs could not hold these two components together any more. Eventually, the plywoods were pulled out from the framing members accompanied by cracking sounds. Eventually, end studs were pulled out of the bottom plate. The damage patterns of the wood-on-wood frame specimens with vertical loads were similar to the specimen with no vertical load, and these specimens sustained more damage before failure. The studs were tilted and the plywoods slid horizontally under the lateral load. Once again, the plywood of the wall panels acted as a rigid body. Under the horizontal load, the plywood tended to slide relative to each other and pull away from the frame by pulling sheathing nails along with them. The relative slide between the plywoods for specimen with vertical load was significantly larger than the slide for specimen without vertical loads. For these three specimens, the plywoods were detached from the framing members at the top west comer and top east comer of the bottom plywood; the top west comer, bottom west comer, and bottom east comer of the middle plywood. The sheathing nails connecting the comers of the plywood and the wood frame were deformed permanently and pulled-through. Thus, the damage pattern of walls with plywood claddings was the failure of the nail connections of the plywood and the framing members. As for metal clad frames, the resulting damages on the flames with no vertical loads were restricted to the wood frame. With the increase of the lateral load, the studs at the east end of the wall (where horizontal load was applied) were pulled out from the bottom plate of the wood flame. The sheet claddings, however, tended to rotate like a rigid body. The failure of the specimen was the split of the bottom purlin and the pull out of the end studs. There was no damage in the metal sheet or in the screw fastener connections. Five Profile A metal clad specimens were tested with vertical loads. Profile A contains 14mm wide, 13mm deep ribs spaced at 203mm.The behaviour and damage pattern of these five metal clad specimens with vertical loads were, diagonal ripples began to appear between crests (ribs) under the first group of cyclic loads. When the wall panel was pushed, the diagonal ripples between the crests were in one direction, and when the wall panel was pulled, the diagonal ripples appeared in the opposite direction. Between the loading cycles, the rippling completely disappeared. Under the second group of cyclic loads, the behaviour of the sheet panel was the same as that under the first group of cyclic loads except that more and deeper diagonal ripples occurred between the crests. In general, for wall panels with Profile A steel claddings and with vertical loads, the failure began with the local diagonal ripples between crests. With the increase of the lateral load, more and deeper diagonal ripples occurred which finally developed into overall bucking all over the whole panel [Figure: 2 shows the specimen C3V during the final pushover]. Meanwhile, the relative movement of two sheets at the overlap tend to force them to separate, causing the tearing and bearing failure of the metal around the screws at the overlaps. The relative movement of the two steel sheets at the overlap was approximately 15 ram. In some cases, the steel sheet near the overlapping screws was pulled through. No damage was observed in the wood frame, or in the screw fasteners themselves. Profile B contains 50mm wide, 13mm deep ribs spaced at every 163mm.The damage and failure of wall panels with Profile B steel claddings and with vertical loads, were mainly the bearing and tearing failure of the steel sheet in the vicinity of the screws at the overlap of two sheet panels. In some of those places, steel sheet might even been pulled through the screws. The local buckling of the sheet panel (diagonal ripples between crests) or the separation of the two sheet panels at the overlap were not significant, and no overall sheet buckling was observed except where the sheet panel was compressed by the steel beam. There was no damage observed in the wood frame, or in the fasteners themselves. The damage of the gauge 30 (0.305 mm) steel sheets was more significant than that of the gauge 28 (0.378 mm) steel sheets.

403

~J ;> O J= 03

o3 o,,,~ ~J

;>

E

o,~ r

C/3 Cq

o,~

L~

Load Displacement Curve for D3V Q,,

61

~

i

\

I

- 0

20

40

60

80

100

120

!

.

.

.

.

.

1~

1 J

. . . . . . . . . . .

Displacement

(ram)

Figure 3" Sample Lateral Load- Top Lateral Displacement Curve for Specimen D3V [Specimen D 3 V - Metal Clad on Wood Frame, Profile:A, Gauge 30 (0.305mm), with Vertical Loads]

404

Equivalent Static Responses The wall diaphragm design parameters, such as the ultimate shear strength V,,, shear stiffness G', degradation in shear strength and stiffness, energy dissipation, and etc., obtained from this experimental investigation are presented in this section. These parameters were obtained by further processing the test results, such as the sample results shown in Figure:3. Table: l shows the equivalent static responses. It makes comparisons between the wood-on-wood and metal-on-wood wall diaphragms. It also compares the test results of metal-on-wood wall diaphragms with different types of cladding profiles and base metal thicknesses. Obviously, the bare wood frame resistance is insignificant, and thus may be ignored. TABLE 1 EQUIVALENT STATIC RESPONSES Cladding Material

None Plywood Profile A gauge 28 (0.378mm) Profile A gauge 30 (0.305 mm) Profile B gauge 30 (0.305 mm) Profile B gauge 28 (0.378 mm)

Vertical Load None Yes None Yes None

est

Yes Yes Yes

V u.

)eft. at V u.)

kN 0.17 0.28

"nm 12. 73 14.57

7.88

76.57

1 3 . 6 1 99.49 6.01

27.56

6.93/ 79.14 7.58.1 !18.56 10.751 153.51

D Suit p = Deft. at G' (shear c ,hear (Shear (0.4 VuJt) P st Ffness) It("bility) (Ductility Factor) Strength) kN/m

3.23 5.58 2.46 3.82 2.84 3.11 4.41

kq

3.15

5.45

mm

k q/mm

r n/kN

13.54

0.35

2.86

11.65

0.701

1.43

.... 8163

2.32 6.83 8.76

1.801 0.67~ 0.741

0.56 1.51 1.35

33.98~ 17.34~ 17.621

2.40

3.73 2.77 3.10 4.3(3

5.66

The ultimate shear strength V,, (kN) is the maximum load that could be sustained when the diaphragm is loaded in shear up to failure. The corresponding lateral displacement of the diaphragm at V~, is defined as A,~t. Shear strength per unit length S,, (kN/m) may be reported as the ultimate shear strength V,,, divided by the diaphragm width, L. The long-term design shear strength P may be taken as P = 0.4 V,,. Shear stiffness G' is the force per unit lateral in-plane displacement of a diaphragm (kN/mm), and is also an important design parameter of a diaphragm. In some design approaches, shear flexibility c is used which is the reciprocal of shear stiffness G '. The wall ductility factor D is an indication of how much a diaphragm can deflect between the design strength and the ultimate strength. It is determined by dividing the displacement at ultimate load by the displacement at the design load. As indicated in Table: 1, the ultimate shear strength of the wall diaphragms with 9.5 mm (3/8") plywood claddings was higher than any of the metal-on-wood wall diaphragms tested in this study. The ultimate shear strength for a shear wall was significantly higher when vertical loads were applied. Table: 1 shows that the wood-on-wood walls with vertical loads could carry 59% to 86% more lateral load than the wall without vertical load. For metal-on-wood diaphragms the corresponding strength increase was 39% to 70%. For metal-on-wood walls, as expected, the walls with the thicker base metal thickness metal claddings sustained a higher ultimate shear strength. For Profile A wall claddings gauge 28 (0.378 mm) sheet panels was 34.5% higher than that of the walls with gauge 30 (0.305 mm) sheet panels. Profile B gauge 28 (0.378 mm) claddings was 34.1% higher. For the same sheet panel thickness, the ultimate shear strength V,, for the walls with Profile B claddings was about 10% higher than that of the walls with Profile A claddings. The shear stiffness G' of walls with Profile A claddings was much higher than those of the walls with Profile B claddings, a behaviour which was just the opposite to the stiffness of the metal sheet alone. For the walls with Profile A claddings, the stiffness of the specimens with the thinner base metal thickness [gauge 30 (0.305 mm)] sheet panels were again much higher than those of the walls with gauge 28 (0.378 mm) sheet panels. In comparing the ductility factor D, walls with Profile A gauge 30 (0.305 mm) claddings (Specimens D) had the largest ductility factor, followed by walls with Profile B claddings (Specimens F and E). Walls with plywood claddings had the smallest value of ductility factor.

405

Dynamic Responses The cyclic test results provide parameters such as load carrying capacity degradation, stiffness degradation, and energy absorption properties of a wall diaphragm. These data are relevant for earthquake engineering design, since degradation of load or stiffness can cause adverse effects on the performance of the structure. Total Load Degrachltlon(Contraction)

Total Load Degmo~on (Ex~nslon)

e0,

i ao.

~ " " - - - " - "

/// t0,

,

, - ,

,

10

15

k.lnud*

20

25

30

0

k , ~ . d * (mm) t-.--c~v, c=r~-.-mv.~v,mv -*-=,v,,~v -..-,Iv. ~ , , v

(met)

{...?-o~v,c,v...-o..~v.~v, .my .-.-e,v, ~ -..-,,v.,,v,,,v 1

I

Figure 4: Total Percentage of Load Degradation For metal clad - wood framed specimens under consideration, Figure: 4 shows the average percentage of lateral load degradation versus the amplitude for each cyclic load of each specimen under a certain cyclic load. The strength degradation shown is the percentage drop in load after three cycles in the same amplitude group. Figure: 4 shows the total degradations for both extension cycle and for contraction cycles. For metalon-wood wall diaphragms, in general, the percentage of load degradation increases with the increase of the amplitude. Except for the first group of cyclic load, the higher strength degradation always occurred between the first and second cycle, and the strength degradation between the second and third cycle was lesser within the same amplitude group of cycle. Further degradation may occur with continued cycling, but at a much smaller rate. This observation was in agreement with the generally accepted practice and use of having three identical cycles within a group to obtain the lower bound envelope curve (He et al., 1999). For the third group of cyclic loads, the lateral load carrying capacity was reduced by 24% to 49%. These figures indicate that thinner sheathing (Specimens D and E) experiences considerably larger degradation compared to thicker sheathing (Specimens C and F) after three groups of cyclic loads. Also note that Profile A claddings degraded more percentage of load than walls with Profile B claddings. Total Stlffnus Degradation

Total Energy D i n l p d o n

so

,i,-

0

8

9

,

,

10

15

20

0

25

km,t,d, (mm) [-.*-.cavl=~v-.~-mv. ' ~ ew~,v.-.-,,w.,,.~v,~v ' , . mvlmv 1

Figure 5" Effective Stiffness Degradation

30

.

,

6

,,

,.

,

,

~

25

30

!. ' ~ - ~ 9~ -,..,~v. ,~, 9,~, - ~ . m 9m -,,-,,v. n ~ , , ~

Figure 6: Total Percentage of Energy Dissipation

The effective stiffness KEi of a wall panel in a cycle is calculated as follow,

K~,,-(P,+-P,-)ICA~-A~),where

P,§ is the force corresponding to the maximum positive displacement A~+, and P; is the force corresponding to the maximum negative displacement A(.

406 Figure: 5 shows the average total percentage of stiffness degradation (after 3 cycles) versus amplitude for each type of specimen under consideration. Note that in general, the effective stiffness decreased with the increase of amplitude. It was also noted that the stiffness degradation was rapid from the first to second cycle of load and then became more gradual from the second to third cycle within each amplitude. The reduction in stiffness was more pronounced in the metal-on-wood walls than it was in the wood-on-wood walls. It seems that the stiffness of walls with a smaller base metal thickness sheathing (Specimens D and E) degraded slightly faster than the walls with a thicker sheathing of the same profile (Specimens C and F). The stiffness degradation is attributed to the lateral load degradation under cyclic loads at the same amplitudes. Thus, reduction in load resisting capacity is sometimes also referred to as stiffness degradation. Energy dissipation occurs through the friction between fasteners and wall assembly, compression of the metal sheet around the screw fastener, and nonrecoverable deformation in the cladding panel. The value of the energy dissipation is measured by calculating the area of the hysteresis loops from the loaddisplacement curves for each load cycle. Figures 6 shows the average total percentage of energy dissipation versus the amplitude of each kind of specimen for a certain cycle of load The average energy dissipation is used in these figures when more than one specimen of the same type had been tested. It was observed that under this cyclic loading schedule, metal-on-wood walls (Specimens C, D, E, and F) dissipated much more energy than wood-on-wood walls. Metal-on-wood walls dissipated 30.2% to 58.0% of the energy in the third group of cyclic load, whereas wood-on-wood walls only dissipated a maximum of 19.2% under the same amplitude. More energy was dissipated with the increase of the amplitude. Within the same group of cycle, the higher energy dissipation always occurred between the first and second cycle, and the energy dissipation between the second and third cycle was greatly reduced. For example, for the third load group, 25.6% to 52.0% of the energy was dissipated from the first to the second cycle, and only 6.2% to 19.9% from the second to the third cycle. The percentage of change in energy dissipation seemed to be independent of the type of sheathing profile and base metal thickness.

CONCLUDING REMARKS Twelve metal-on-wood, four wood-on-wood 2.44 m x 3.66 m (8'x 12') shear walls and two wood frames have been assembled and tested to study the behaviour of metal-on-wood shear walls under cyclic lateral load. The specimens were tested using the cantilever arrangement subjected to three rounds of cyclic loads followed by a final push over. The damage pattern of walls with plywood claddings was the failure of the nail connections of the plywood and the framing members. In the metal clad - wood frames the eventual failure was due to the bearing and tearing failure of the steel sheet in the vicinity of the screws at the overlap of two sheet panels. The ultimate shear strength of the wall diaphragms with 9.5 mm (3/8") plywood claddings was higher than any of the metal-on-wood wall diaphragms tested in this study. The ultimate shear strength of walls with Profile B claddings was about 10% higher than that of the walls With Profile A claddings. Test results for parameters such as load carrying capacity degradation, stiffness degradation, and energy absorption properties of a wall diaphragm were presented and compared. Further details are expected to be published in scientific journals in the near future. REFERENCES AISI.(1996), Specifications for the Design of Cold-Formed Steel Structural Members, U.S.A. CSA.(1994),Cold Formed Steel Structural Members CAN/CSA-S 136-94, Rexdale, Ont.,Canada. He, M., Magnusson, H., Lam, F. and Prion, H.G.L., (1999), Cyclic Performance of Perforated Wood Shear Walls with Oversized OSB Panels, Journal of Structural Engineering, ASCE, Vol. 125, No.I, pp.10-18. Ministry of Housing, (1995), Construction Guide for Farm Buildings, Ontario, Canada. Pan, W., (2000), Cyclic Behavior of Metal Clad - Wood Framed Diaphragms, M.Eng. Thesis, Department of Civil Engineering, McMaster University, Hamilton, Ontario, Canada, p-251.

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

407

VIBRATION OF IMPERFECT STRUCTURES J. Ravinger and P. Kleiman Department of Structural Mechanics. Faculty of Civil Engineering. Slovak University of Technology. Radlinsk6ho 11, 813 68 Bratislava, Slovakia

ABSTRACT Using the geometric non-linear theory (The Total Lagrange Description) in dynamics we can establish the problem of the vibration of the structure including the effects of the structural and geometrical imperfections. The incremental stiffness matrix can take into account the residual stresses (structural imperfections) and the geometrical initial displacements (geometrical imperfections) as well. The behaviour of columns, frames and thin-walled structures is sensitive to imperfections. This theory and results can be used as a base for the non-destructive method for the evaluation of the level of the load and the imperfections.

KEYWORDS Stability, buckling, post-buckling, dynamic stability, vibration, geometric non-linear theory, initial imperfections, thin-walled structures.

INTRODUCTION The problem of the combination of the linear stability and the vibration of structures was solved a long time ego. By the linear stability we mean the assumption of an ideal structure. Differences, between theoretical results and the reality, forced researchers to search for more accurate models. The slender wen as a main part of the thin-walled structure has significant post-buckling reserves and for a description of them it is necessary to accept a geometric non-linear theory. Burgeen (1951) formulated the problem of vibration of an imperfect column. The problem of the vibration of a slender web as a non-linear system was formulated by Bolotin (1956). The vibration of the rectangular slender web taking into account the geometrical initial imperfections has been investigated by many researchers. (Wedel-Heinen, J. 1991, Hui, D. 1984, Ilanko, S. and Dickinson, S. M. 1991, Yamaki, V. at al. 1983). Author (1994) presented the general theory for the dynamic post-buckling behaviour of a thin-walled panel (a slender web with flanges) taking into account the geometrical initial imperfections and the residual stresses as well. This paper presents a short summary of this theory. The presented results show the peculiarities for the differences of the support conditions for static behaviour and for the vibration of the columns, frames and a slender web as a main constructional element of a thin-walled structure.

408 THEORY

Von K~trrnhn theory has been used for the description of the post-buckling behaviour of a thin-walled panel with geometrical imperfections and residual stresses. By including inertia forces the problem is extended into dynamics. A direct formulation of Hamilton's principle for the dynamic post-buckling behaviour of a slender web leads to a system of conditional equation describing this non-linear dynamic process. In the case of a static problem, linearized assumptions are accepted for the evaluation of the elastic critical load. Analogously, in the case of a dynamic problem, Hamilton's principle is arranged in incremental form for the evaluation of the free vibration. A system of conditional equations in the incremental formulation of the post-buckling behaviour of the slender web or a thin-walled structure can be written as T

KMpA~, P + KmceAa o + giNcwvAaw + Fmr p -FexrP--AFExTP

--0

K iNcw A a w + K INcwpAa e + FiNrW -- FEXTW -- Z~FExTW - - 0

(la) (lb)

where indexes mean M- mass, mc- incremental, #or- internal, Err-external, p - plate, w- web. The incremental stiffness matrix of the web Kmcw is linear and, consequently, we can proceed to the elimination of Eqn. 1b and thus obtain the system of equations K Me A i~ p + ( K iNCe - KINC~ r -, KiNcre )Aap + F m r P - FExrP KiNcw r c~ K m -1~ FINrw + K INC~ r -1c ~ FExrw -- AFExre + K mcre r -' AFExrw = 0 K m K m K 1No,

(2)

Taking out the inertia forces from Eqn. 2 we have conditional equations describing the static postbuckling behaviour of a slender web or large displacements of plates. The Newton-Raphson iteration can be used for the solution of this equations. From a number of rules which are valid in numerical models of static non-linear problems we use one which can help us in establishing the free vibration problem. T

K INC -- J

::Z,

-1

K INC = K INCe -- K INC~ KiNcw K~NC~

(3)

This means that the incremental stiffness matrix is equal to the Jacobian of the Newton-Raphson iteration of the system of non-linear algebraic equations. The problem of the free vibration including the effects of initial imperfections (initial displacement and residual stresses) can be obtained in the following way. We suppose a system Eqn. 2 in equilibrium; then: FINTP - K m T c ~ K INC~ -! T -! FEXTW - I ~ E X T P + KINcwe K ~NCW Z~exrw = 0

(4)

We suppose a zero increments in the external loads : ZkFrxrp = AFrxrw = 0

(5)

The increments of the plate displacements are assumed as : Aap = A~p sin(cot), where co is the circular frequency of the free vibration.

(6)

409 Inserting this into Eqn. 2 we have a problem of eigenvalues and eigenvectors 9

_[K'Nc -

co 2K MP.Idet - -

(7)

0

Eigenvalues are the circular frequencies and eigenvectors represent the modes of the vibrations. The incremental stiffness matrix includes the level of the load and the initial imperfections as well.

EXAMPLES The vibration o f simple supported column

Simple supported column loaded in compression is the most simple example for the explanation of the presented theory. In this ease it is very important how we suppose the edge conditions in the point of the action of the external load. In the case of moving support we have obtained a trivial result. To take in consideration the effects of initial displacements, the support must be fixed during the vibration process. (Fig. 1) w.fa..w,

e,A,I,y J

~ "

~.

F

"~--~---'~---STXnC

.~."

-|

I - ~=o.l 2-~.=0~

.a.VIBRATION

3

l

F/F~.

n~EI I'

~=

n'EI

( ~ m

n r

tar

- ~ t . - o.4

4 - ~t==1.0

W.~ =" 8iIl_~

F-=

r= ~ - ~

1

F/F~

I.O

1.0

3 4

\

9

'I .__,,, . _ _ . . = . .

..

0.5

\

0.5

='=--

,.,-~

/ .~ . . . . - ' " "

-|174 ~.-o ",, \., o

-~ - - - - - . -

10

..

\

wdr 5

~..." ~

0

0.5

1,0

" t,s ~

Figure 1 9The vibration of simple supported column Vibration o f the column with the second mode of the initial displacement

Fig. 2 shows the solution of the vibration of the column where the initial displacement is the combination of the first and the second mode of the buckling. The second mode is dominated. Even the mode of initial displacement is mode 2 (the second mode), the mode of the vibration is the mode 1 (the first mode). We can say, that the initial displacements cause the deviation of the frequency to compare to frequency of the ideal column (column without initial displacements - imperfections) or to the column with moveable support in axis direction. In the case of the second mode of the initial displacement this deviation of the frequency is much smaller. Vibration of the frame

When we have an example where the mode of the vibration is similar to the mode of the buckling and

410

,/

E, A, I, ~/

F

Wo----CXot.Wot + O r ~ . W a

F..=

x~EI is

5 - OL,,= 0.01, ~t,,= 0.I

.zx.

'

6

-ix VIBRATION

7 - ~=o.~,

Wm=Sin~ ~'T-~...~~

- ~,=o.os,

wo = s i n . ~

~=

m,=o.s

~t,,- i.o

r

F/F,,

F/F,.

1.0

1.0

",,,,

0.5

I !~ 0

n'EI ~,AI'

I

--7

,,,

0,5

-7 wCr _...._> 5

I0

(~/~, 0

0.5

1,0

Figure 2 9Vibration of the column with the second mode of the initial displacement if we do not have an additional support, the relationship between the load and the square of the circular frequency is linear and we are not able to take in consideration the effects of initial imperfections. Generally the behavior of the column and the behaviour of the frame is similar. (Fig. 2, alt A movable support). We have arranged the load condition to get the mode of the buckling different to the mode of vibration. (Fig. 3). In that case the relationship between the load and the square of the circular frequency is non-linear. Analogously as in the case of the column we will suppose a different edge condition for static load and for the vibration. If the point of the application of the load is fixed during the vibration process we can take in consideration the effects of initial displacements. Slender web loaded in compression. In the case of the slender web loaded in compression the in-plane edge condition play a crucial role as well. If the mode of the initial displacement is the same as second buckling mode the static postbuckling behaviour could be in this second mode. This initial displacement mode has an influence to the circular frequency, but the mode of vibration is the first buckling mode. (Fig. 4)

E X P E R I M E N T A L INVESTIGATION The experimental investigation of the dynamic post buckling behaviour of the slender web (or a thinwalled panel or a thin-walled structure) needs a high quality equipment and experiences. The presented results has been arranged at The Institute of Construction and Architectures of The Slovak Academy of Sciences.

411 F F= 1.0 w, =

f \

0,8

0.005

__

.

.

_fi0~176

1 .

.

,w,=O.ff~"

,

f,.t"

I' 2500kS/=m

o~.3o4

i "/

0.6

f

STATIC 0,4

i

0,2

VIBRATION -

0

1

2

3

(o2 [s-q ,....._

For = 14.54.106N

5 w~ 1000

4

Figure 3 9Vibration of the frame

P

"b

/I

/

/..

J

P

/

/

.~] .~,.,~.,0ot OF v,.RAr..

T2Eh 2

/

3ll"vzlaZ

VlB'I~ATION

/~i

,r ,,7

-.

~,

-

- - - o q , : 0.01, ~oz:0.15

_

Q5

. T 4 E hz

"

/

"

1.5

2.

~-,,,~-ooooo, 9 ~/1,~,

-'-eco, : 0.15, ec,:O.01

,

3ll-v2l~O er 0

0.5

I.

1~5

-

Fig

i

[

G5

I.

i-.

Figure 4" Influence of the mode of the initial displacement for the vibration of slender web loaded in compression Fig. 5 and 6 show the test of the steel panel loaded in compression. (The square panel with 400mm, flanges 60x8 mm). In this case the connection of the web and flanges has been done by a welding. We used to suppose the distribution of the welding stresses with the positive yield stresses near the edge of the web. In this case the compression loading-unloading process should not produce the redistribution of the residual stresses. To explain the results from Figs. 5,6 needs to accept the "bending" residual stresses into the consideration.

412 P~ h = 3/.9 mm

i. i t t l i t

i Wc

~

9

O= = 14./,5 Nmnri'

K

~0T" 1.22

<= ozs

=

8./.05,

u , - 1166.5 g'

"i l i I t i l i ~ P ,

TE6m~

.P_~. 0

0.5

13

1

1

Figure 5: Comparison of the theoretical and experimental results for the panel with the thickness of the web 3.49 mm.

I I I t I.IiiR

'•wr I

i

pc.

h =2505mm

-c

. .

.

0.,;-0.38

~0= 2.26

tttil tttlp. ,

K=8.86

~ = 7.4t, N m m "~

~"/s

.

~

~

w.= 861.5 ~'

,

/

ca.

z -.,

m . , .

2 i/

/

, \

,

0

OC

1

I

2

_

0

P.

P=~

~" 0 1

2

Figure 6: Comparison of the theoretical and experimental results for the panel with the thickness of the web 2.505 mm. The panel of the thickness of the web h - 1 . 5 3 m m produces a snap-though effect. (Fig. 7). The upper and lower load levels of the jump can be established. The transformation via eigen modes has been used for the theoretical evaluation of this example (Ravinger 1992). Fig. 8 and 9 show the box girder arranged for the experimental investigation. The transverse stiffeners divided the webs into separate parts with the different aspect ratio ( a = a / b - 1.0 - 3.0). Changes of the position of the applied load and the position of the supports can produce wide variety of the combination of the bending and the shear load. The obtained results will be presented.

413 -p=m p

K =9.24 uo=526.Ts-1

liJ~m..Im. P,~ h= 1.53mm !"

o

--

I

p~= 6.97 1 Hill lIlltlll

9

/ /-'~~

THEORY

05

\

~i ~'~,,h 9 1

0

2

wlh

\

.\

O.5

0

15,- P-I.

1.5

2.

P

L

EXPERIMENT

tS!

\ 9

: 1, ~.

1.

\

/

'

/

P I -2

-

o

1.

o

o.s

t

1.:

Figure 7 9Vibration of slender web during the snap-trough

1 12-5 ....,.,1"15:111. . . ,:2.

!

t

=3

w--

4.5,

Figure 8: Geometry of the test girder. CONCLUSION The theory and the experiments have proved the sensitivity of the circular frequency of free vibration to the level of the load and different types of initial imperfections. This knowledge can be used as an inverse idea. Measuring of the natural frequencies can give us a picture about the stresses and imperfections in the structure. This idea represents a base for a non-destructive method for the evaluation of the properties of the frame structure or the thin-walled structure at all. The natural frequency is sensitive even to residual stresses and it is very important detail. Presented results show the peculiarities in the edge conditions. We must distinguish the edge conditions for behaviour of the structure during the application of the load (static or dynamic) and the edge conditions for the free vibration process.

414

Figure 9: General view of the test of the girder REFERENCES

Bolotin, V. V. (1956). Dynamic Stability of Elastic Systems. GITL, Moscow, (in Russian, English translation by Holden-Day, New York, 1964). Volmir, A. C. (1972). Non-Linear Dynamic of Plates and Shells, Nauka, Moscow (in Russian) Burgreen, D. (1951). Free Vibration of Pin-ended Column with Constant Distance between Pinends. J. Appl. Mechan. 18,135-139. Elishakoff, I. - Birman, V. - Singer, J. (1985), Influence of Initial Imperfections on Non-linear Free Vibration of Elastic Bars. Acta Mechanica 55,191-202. Wedel-Heinen, J. (1991). Vibration of Geometrically Imperfect Beam and Shell Structures. Int. J. Solids & Structures 1, 29-47. Rehfield, L. W. (1973). Non-linear Free Vibration of Elastic Structures. Int. J. Solids & Structures, 9, 581-590. Hui, D. (1984). Effects of Geometric Imperfections on Large Amplitude Vibrations of Rectangular Plates with Histeresis Damping. Jr. Appl. Mechan. 55. 216-220. Ilanko, s. - Dickinson, S. M. (1991). On Natural Frequencies of Geometrically Imperfect Simply Supported Plates under Uni-axial Compressive Loading. J. Appl. Mechan. 58, 1082-1084. Yamaki, V.- Otomo, K. - Chiba, M. (1983). Non-linear Vibration of Clamped Rectangular Plate with Initial Deflection end Initial Edge Displacement. Thin-Walled Structures, 1, 101-119. Ravinger, J. (1994). Vibration of an Imperfect Thin-walled Panel. Part 1 : Theory and Illustrative Examples. Part 2: Numerical Results and Experiment. Thin-Walled Structures 19 1-36. Ravinger, J. - ~vol~, J. (1993). Parametric Resonance of Geometrically Imperfect Slender Web. Acta Technica CSAV, 3, 343-356. Ravinger, J. (1992). Dynamic Post-buckling Behaviour of Plate Girders. J. of Constructional Steel Research. 21, 1-3,195-204. ACKNOWLEDGEMENTS

Presented results has been arranged due to the research supported by the Slovak Scientific Grant Agency grant N ~ 45-14-99.

Section VII FINITE E L E M E N T ANALYSIS

This Page Intentionally Left Blank

Third InternationalConferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

417

THE FINITE ELEMENT METHOD FOR THIN-WALLED M E M B E R S - BASIC PRINCIPLES

M.C.M. Bakker 1 T. Pekrz 2 z Assistant Professor, Faculty of Architecture, Building and Planning, Eindhoven University of Technology, The Netherlands 2 Professor, School of Civil & Environmental Engineering, Cornell University, Ithaca, NY 14853, USA

ABSTRACT The application of the finite element method to thin-walled structures often requires non-linear analysis. Whereas in linear finite element analyses errors are easily made, this is even more so in nonlinear analyses. This paper focuses on possible sources of error in linear and non-linear finite element solutions, and gives suggestions how to check and prevent these errors.

KEYWORDS Finite element method, non-linear analysis, errors.

INTRODUCTION The finite element method is a much used tool in current research. Although it is relatively easy to get results, it requires care to guarantee good results. Table 1 gives an overview of possible errors which might occur during (linear and non-linear) finite element analysis. These errors will be elaborated in this paper. This paper is based on experiences in teaching the application of linear and non-linear finite element method to structural engineering students, While the paper focuses on practical aspects which may be simple and self-evident for experienced finite-element users, it is hoped that the information given is useful for inexperienced users. For more extensive guidelines to good finite element practice, see Nafems (1984), Nafems (1986) and Nafems (1992). When users are unfamiliar with (non-linear) finite element analyses, it is best to get acquainted with the program to be used and the interpretation of results by running some simple problems for which exact (analytical) solutions are known. Therefore, the types of errors discussed in this paper will be illustrated with very simple examples, which provide insight in some basic issues encountered in the

418 modeling of thin walled structures. In these examples, only beam and truss elements are used, but similar results can be found when running the examples with shell elements. The finite element method is an approximate method in which equilibrium and/or compatibility conditions may be approximated. This paper will limit the discussions to finite element programs in which only equilibrium is approximated. TABLE 1 OVERVIEW OF POSSIBLE ERRORS

i Realit~ Idealization error ! Mechanics model

- PREPROCESSING I

Input error Discretization error: equilibrium approximated Geometry error: geometry aproximated Shortcomings in element formulation Program bugs [ Finite element model

]

Solution error Convergence error Program bugs ]Nodal displacements

Program bugs

I

SOLVING

[ Derived results according to FE model I Rendering error: postprocessor inter/extrapolates differently than FE model: integration points ~ nodes ~ contour plots

Program bugs I Results according to postprocessor ] ~ Interpretation error: postprocessor shows something else than is expected, for instance averaged instead of unaveraged stresses I Interpretation of resuls

- POSTPROCESSING

IDEALIZATION ERRORS

Choosing a mechanics model Idealization errors are errors caused by a wrong idealization of the real structure, loads and supports to a mechanics model. When a structure is idealized by a mechanics model one has to decide: - the mechanical approach needed to describe the behavior, e.g. beam theory, shell theory or 3D stress theory. This choice implies how the geometry of the structure will be described, namely as a line, a plane or a solid. - assumptions that may be made regarding the magnitude of deflections, strains and rotations. This determines what type of geometrical non-linear analysis should be run. - how the material behavior may be idealized. - whether the influence of initial stresses should be accounted for, and how these residual stresses are distributed through the model.

419 -

-

how accurately the geometry of the structure must be modeled in order to get accurate results. Magnitude and distributions of deviations from the ideal geometry (initial imperfections) must be considered. how loads and supports should be modeled.

When idealizing a structure it is important to know what one wants to learn from the mechanical model. The objective is not to describe reality as accurately as possible, but to find the simplest model resulting in a sufficiently accurate description of reality. Unfortunately, this will often require a lot of insight in the behavior of the structure. In case it is found necessary to build a very sophisticated model, it is best to start the analysis with a simple model, and use a step by step approach to increase the complexity of the model. The choice of the kind of mechanics needed (corresponding to the choice of the type of element) and the appropriate modeling of material properties, initial stresses and initial imperfections are discussed in an accompanying paper by Sarawit, Kim, Bakker and Pekrz. Only the types of nonlinear analysis will be discussed here.

Types of geometrical non-linear analysis With respect to geometrical non-linear behavior a distinction can be made between first order analysis, second order analysis, a large deflection analysis, a large rotation analysis, a large strain analysis and an eigenvalue buckling analysis (see Table 2, and Figure 1). The well known first order analysis is a geometrical linear analysis, in which equilibrium is formulated with respect to the undeformed shape. In the other, geometrical nonlinear types of analysis equilibrium is formulated with respect to the deformed state. A second order analysis accounts for stress stiffening effects (P-5 effects). In a large displacement analysis membrane stresses developing due to out of plane deflections can also be described, but rotations and strains still have to be small. A large strain analysis is the most general geometrical non-linear analysis, in which neither strains, nor rotations or displacements need to be small. A large strain analysis is the only type of analysis in which changes in thickness or area are accounted for. TABLE 2 TYPES OF GEOMETRICAL NONLINEAR ANALYSIS Type of analysis Displacements First order (linear) Very small Second order (stress Small (negligible membrane stresses due stiffening) to out of plane deflections, displacements smaller than the thickness of the beam/plate/shell). Large displacement Large (significant membrane stresses due to out of plane deflections, displacements larger than the thickness of the beam/plate/shell) Large rotation Large Large strain Large

Rotations Very small Small ( < 10~)

Strains Small (< 5%) Small (< 5%)

Small ( < 10~)

Small (< 5%)

Large Large

Small (< 5%) Large

41

i

i

An eigenvalue buckling analysis is a special kind of geometrical non-linear analysis. Usually it is based on a second order formulation. Using the assumption of linear load-deformation behavior up to the

420 attainment of the buckling load the factor with which the loads on the structure should be multiplied in order to get a non-unique solution (the eigenvalue) can be determined. For some structures the load determined by this analysis represents the buckling load of the structure. However, if prebuckling deformations are not small, the load determined does not have much physical relevance (see for instance the eigenvalue calculation in the structure where snap-through occurs). So one should keep in mind that an eigenvalue calculation is a mathematical trick, which can always be performed, regardless of the physical significance. In most cases a linear analysis will not give useful results for the analysis of cold-formed structures, but it is good practice to start every non-linear analysis with a linear analysis. If one wants to model the non-linear load deflection behavior of thin-walled structures, including post-buckling behavior, one needs at least a large deflection analysis or a large rotation analysis, since a second order analysis cannot describe membrane stresses develop due to out-of plane deflections. It is also possible to use a large strain analysis, but in most cases it is much simpler to use a large displacement or large rotation analysis. Note that in defining loads in a large rotation analysis (contrary to first and second order analysis) a distinction is made between follower forces (element loads), where the direction in which the loads act is dependent of deformations of the structure, and nodal loads, where the direction of the loads does not change. It is the responsibility of the user to make sure the right type of analysis is used. A finite element program does not give warnings when displacements, rotations or strains are so large that the assumptions on which the analysis is based are violated, but just gives unreliable results! INPUT E R R O R Input errors are errors made in the preprocessing phase, when building the finite element model. It is best to check input errors by nmning a linear analysis, before starting with the non-linear analysis. Some simple checks on input error are to inspect a plot of the finite element mesh and to run a linear analysis with the model loaded by unit loads. Furthermore the material and geometry properties should be checked carefully, as well as the applied loads and boundary conditions. For more checks see Nafems (1982) and Nafems (1986).

DISCRETIZATION ERRROR In the finite element method the nodal displacements and/or rotations are obtained from nodal equilibrium equations, resulting in the best approximation of the equilibrium conditions throughout the structure for the given finite element mesh. If the mesh is to coarse, the equilibrium conditions are satisfied poorly. This is called a discretization error. A discretization error can be recognized from stress discontinuities between elements. A discretization error may also be recognized at boundaries of the finite element model, where it can be observed how much calculated stresses deviate from known stresses. Furthermore, it is good practice to study the influence of mesh refinement on the results. When one builds a finite element model an important question is how dense the mesh should be for sufficiently accurate results. The required mesh density depends on the stress gradients (larger stress gradients need a finer mesh), and the question whether the mesh should result in adequate stress results, or only in adequate stiffness results. Poorly shaped quadatric elements may loose their accuracy, and thus may increase the discretization error. Poorly shaped linear elements are less shape

421 sensitive, since both ideal shaped and poorly shaped linear element must be able to describe constant stress states. In a physical non-linear analysis (using non-linear material models) with shell elements, the discretization error is very much dependent on the number of integration points used over the area of a shell element. Due to discretization errors, the von Mises stresses in points different from the integration points may far exceed the yield stress. One should be aware, that both an eigenvalue analysis and a physical and/or geometrical nonlinear analysis require a (sometimes much) finer mesh than a linear analysis. Figure 2 illustrates this for a beam element.

GEOMETRY ERROR A geometry error occurs when the geometry of a structure cannot be described exactly by the finite element mesh used. An example of a geometry error is the modeling of a beam with varying height by elements with constant stiffness properties over the length of the elements. Another example is the modeling of a curved shell with fiat shell elements. A geometry error is of interest only when small changes in geometry have a large influence on the results. For the modeling of thin walled structures, the correct modeling of the comer radius might be an issue for some loading conditions (for instance web crippling). In addition, one should be aware of the fact that small changes in the shape of holes may have a large influence on the magnitude of stress concentrations. In general, a geometry error can only be checked by studying results for varying mesh densities. A geometry error does not necessarily result in stress discontinuities. The finite element analysis may result in a solution, which is in perfect equilibrium (no discretization errror), however a geometry error is present due to the use of a wrong geometry!

SHORTCOMINGS IN ELEMENT FORMULATIONS When modeling physical non-linear behavior of thin-walled structures with shell elements, the number of integrations points over the thickness of the shell element is even more important than the number of integration points over the area of a shell element. Whereas a limited number of integration points over the area of a shell element may be compensated by using a fine mesh, this is not the case for a limited number of integration points over the thickness of the shell element. Too few integration points over the thickness will result in an inaccurate prediction of the moment of first yield, the fully plastic bending moment and the interaction between membrane stresses and bending moment. This type of error is called a shortcoming in the element formulation (see Figure 2) Other types of shortcomings in element formulations which might be encountered in shell elements are shear locking and membrane locking. Also one should be aware of the possibility of mesh locking due to large plastic strains (Zienckiewicz and Taylor (1991), Cook, Malkus and Plesha, (1989)). If no results of appropriate benchmark tests (as for instance published by NAFEMS) are available, one should assure oneselve that the elements to be used perform well for the intended application, by running appropriate tests.

SOLUTION AND CONVERGENCE ERROR In literature on the solving of linear and non-linear nodal equilibrium equations in finite element analysis (see for instance Bathe (1982), Cook, Malkus and Plesha (1989) and Nafems (1992)), much

422 attention is given to the discussion of solution errors, that is, errors occurring in the calculated nodal displacements due to ill conditioning of the nodal equilibrium equations and the finite number of digits the computer uses to represent floating-point numbers. In literature on non-linear analysis the emphasis in on convergence errors, that is errors occurring in the calculated nodal displacements in a nonlinear analysis due to the iterative solving of the nonlinear equations, and not much is said about the influence of solution errors. It is probable that solution errors do not directly affect the accuracy of the solution, but may cause convergence problems. Both in linear and nonlinear analysis if the equations are solved exactly, the internal nodal forces calculated from the calculated nodal displacements are in exact equilibrium with the applied external nodal forces. If the equations are not solved exactly, the internal and external residual nodal forces will not be in exact equilibrium and residual nodal forces will occur. Unfommately, both in linear and nonlinear analysis small residual forces do not guarantee accurate displacements (and stresses), and large residuals do not necessarily imply inaccurate displacements (and stresses), see also Table 3. In linear analysis, a good conditioned set of equations and small residuals will guarantee an accurate solution. In nonlinear analysis, a combination of sufficiently fight residual force criteria and displacement based convergence criteria will guarantee an accurate solution. In other cases, engineering judgement will be needed to determine whether the solution is acceptable or not. TABLE 3 INFLUENCE OF RESIDUALS ON ACCURACY OF SOLUTION

Small residuals Good condition/small iterative displacements Small residuals Bad condition/large iterative displacements Large residuals Good condition/small iterative displacements Large residuals Bad condition/large iterative displacements

Nodal equilibrium Accurate

Nodal displacements Accurate

Accurate

Possibly inaccurate

Inaccurate

Possibly accurate

Inaccurate

Probably inaccurate

RENDERING ERROR After the nodal displacements have been calculated, the displacements in the elements, and hence the stresses and strains in the elements can be calculated. One should be aware that the way in which a postprocessor presents this results does not always correspond to the results according to the finite element theory. In a postprocessor the stress distribution over an element will often be determined with a linear interpolation between the stresses in the comer nodes, regardless of the presence of intermediate nodes. Sometimes the stresses in the nodes are assigned the values of the stresses in integration points. In many cases, rendering errors will become apparent only in very coarse meshes, as may be used in simple problems to get accustomed with a program.

423 INTERPRETATION ERROR When looking at stress results it is important to be aware whether on looks at contour plots of element stresses (unaveraged stresses) or nodal stresses (averaged stresses). Plots with averaged stresses may look much nicer but may obscure large discretization errors. Non-linear load-deflection calculations result in dots in a load-displacement diagram. Interpolating a curve through these dots, as many post-processors do, can result in erroneous results. Sharp bends in such curves should be mistrusted, until explained by the model behavior. It is especially important that a sufficiently large number of dots is calculated to determine the ultimate load, otherwise the failure load may be severely underestimated.

P R O G R A M BUGS Program bugs may occur in any fmite element program. It is good practice to record the program version which has been used for an analysis, so that when a program bug is reported in a newsletter, it is possible to check whether the results of a certain analysis might have been affected by this bug. Program bugs are more likely to be encountered in the more exotic options of a program, for instance the application of initial stresses. Therefore when one is using a new option for the first time, it is wise to check this option first on a simple problem for which the exact solution is known.

PREFERABLE ORDER IN CHECKING ERRORS

1. 2. 3. 4.

Check input errors (before starting a nonlinear analysis) Check solution/convergence errors Check discretization errors Check idealization errors as far as possible (check assumptions on magnitudes of deflections, rotations and strains) 5. When one encounters unexplainable results, keep in mind that there might be errors in the element formulation, program bugs, rendering errors and interpretation errors. When comparing finite element results with experimental or analytical results, the error might also be in the experimental or analytical results!

Note that stress discontinuities between elements may indicate either a solution/convergence error, or a discretization error. Therefore it is best to check solution/convergence errors before checking discretization errors.

CONCLUSIONS The avoidance of errors and mistakes in finite element analysis requires a critical attitude towards the obtained results. Much care should be given to checking possible errors as discussed in this paper.

REFERENCES Bathe, K.J. (1982), Finite Element Procedures in Engineering Analysis. Prentice-Hall, Inc., Englewoord Cliffs, New Jersey.

424 Cook, R.D., Malkus, D.S. and Plesha, M.E. (1989), Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York. Nafems. (1984), Guidelines to finite element practice, Nafems, NEL, Glasgow. Nafems. (1986), A Finite Element Primer, Nafems, Department of Trade and Industry, NEL, Glasgow. Nafems.(1992), Introduction to Nonlinear Finite Element Analysis, Nafems, Bimiehill, East Kilbride, Glasgow. Zienckiewicz, O.C. and Taylor, R.L. (1991): The Finite Element Method Volume 2. Solid and Fluid Mechanics. Dynamics and Non-Linearity. McGraw-Hill Book Company Europe, Maidenhead, Berkshire, England.

I

10

.... i

;

:

i

I . . . . . . .

:

i -.... i

spring M = 1000 q,

9 -

H = 0.01 F second order large rotation eigenvalue

_

I ,, -

~ EIA=m , ~ - . / . ~._ .r.

-

e = 1000 mrn11

61 T M 0

i 5

, 10

15

, 25

20

i 30

rotation~ [degrees]

12 I0

i

~

i

lit

/

,P

p

i

~

i

8

6

Z. 4

i'

.._

Z_

.....

......

i

..... !_

o -2

- - first order 1 --- second order ] --- large rotation [ --- eigenvalue

I

EA=I07N ~ m ~

-4

-6 deflection v [ram]

Figure 1" Types of geometrical non-linear analysis

~~" ~

h=lO mm

425

30 .

r

. . . . . .

. . . . .

i. . . . . . i

i

i

I~ 25

. . . . .

i

. . . . . .

t

15

=

i0

t~

5

-7 i

i

-r_~'-

~...-

Results:

,

t

l- . . . . .

............

20

0

i . . . . . i

i

0

i . . . . . .

i

.

i

-~-....-r J

-,-,- -w i

i

" - - ' - ' , " - _ ' - : -~, : - " ' _ ' : ~

~

: ,'___ 1. . . . . . :. . . . . . . . . . . . . . .

beaml

Beam 1" M <Mp for t p > 6 r a d

beam2

due to shortcomings in element formulation

.... theoretical plastic moment

7

Beam 2: M > Mp possible due to

w 0

2

4

6

8

10

discretization error

prescribed node rotation [rad]

M,q) C~,,,

~'~

beam 1

beam 2

integration points ,// "" -""~- ~o

0.5 g

1-~0"2 h

'

.].~0.3 h

, ' "

]_~0.3 h ]_~_0.2 h

-,~

0.5 e

M, qo

g = 3000 mm h = 200mm b=100mm fy = 20 N / mm 2 E = 9600N / mm 2 bilinear elastic-plastic material behavior (no hardening) theoretical plastic moment Mp = ~ fybh 2 = 21-106Nmm

4"

Figure 2: Discretization error and error due to shortcomings in element formulation

This Page Intentionally Left Blank

Third InternationalConferenceon Thin-WalledStructures J. Zara.4,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

427

NONLINEAR ANALYSIS OF LOCALLY BUCKLED I-SECTION STEEL BEAM-COLUMNS A.S. Hasham 1 and K.J.R. Rasmussen 2

1Australian Consulting Engineers 2Department of Civil Engineering, University of Sydney, Australia

ABSTRACT The paper describes the development of an advanced finite element analysis of thin-walled high strength steel I-sections in combined compression and major axis bending. The analysis accounts for geometric and material nonlinearity, and incorporates measured values of geometric imperfections and residual stresses. The failure modes involve local buckling of component plates, in-plane bending and out-of-plane flexural-torsional member buckling. The finite element analysis is shown to be in good agreement with tests. A study is included to devise a simple method for measuring and modeling geometric imperfections in advanced analysis. It is shown that the accuracy achieved by determining the amplitude of the local geometric imperfection from a small number of mid-length measurements is comparable to that obtained by modeling the complete measured geometric imperfection consisting of measurements along the full length of the member at several points in the cross-section. A further study is included to investigate the sensitivity of the finite element analysis to changes in local geometric imperfection.

KEYWORDS FEM, Geometric imperfections, Beam-columns, Local buckling, Flexural-torsional buckling, Steel structures, Ultimate capacity.

INTRODUCTION Great advances have been made in the development of finite element analyses in the past three decades. Commercial computer programs are now readily available for performing advanced (ie geometric and material nonlinear) analyses of deformable structures including steel structures. These analyses have proved capable of simulating accurately the complete loading history of steel structural frameworks collapsing inelastically under large displacements. However, the attention has mainly been paid to frameworks consisting of compact members. The present paper describes the development of an advanced finite element model and assesses the ability of this model to simulate recent tests (Hasham and Rasmussen 1997a, 1998) on thin-walled high strength steel I-section beamcolumns loaded in combined compression and major axis bending.

428 The beam-columns tested in Hasham and Rasmussen (1997a, 1998) were fabricated by welding and included compressive residual stresses of about a quarter of the yield stress. Two types of crosssection were tested, a column-type cross-section with slender flanges and web, and a beam-type crosssection with slender web and stocky flanges. As a result of the slender component plates, the beamcolumns failed interactively by local and overall flexural or flexural-torsional buckling. The measured values of material properties and geometric dimensions are incorporated in the finite element model. Particular attention is paid to the modeling of geometric imperfections. The modeling of geometric imperfections of thin-walled structural members has received much attention for several decades. Founded in Koiter's work, it has become well-known that the greatest imperfection sensitivity in the buckling of elastic structures is found when the geometric imperfection is in the shape of the critical buckling mode. For instance, in analysing a pin-ended Euler column, the greatest reduction in strength can be expected when the imperfection is sinusoidal. In recognition of this result, commercial finite element packages provide methods for superimposing geometric imperfections in the shapes of buckling modes onto a perfect geometry. The procedure requires firstly a buckling analysis of the geometrically perfect member to produce the buckling displacements of selected modes and then a scaling of the buckling displacements before adding these to the perfect geometry to produce the geometry of the imperfect member. In using this procedure, the question arises as to what magnitude of imperfection to use, notably what magnitude of local imperfection of component plates. There is, at present, no generally accepted value to use for the magnitude of local geometric imperfections. Furthermore, a recent study (De Ville de Goyet 1996) on the collapse analysis of a heavy steel cantilever beam challenged the modeling of local geometric imperfections in the shape of elastic buckling modes when applied to inelastic collapse analysis. It was shown that the ultimate load was overestimated by 10% when the local imperfection was in the shape of the elastic buckling mode, compared to using a local imperfection in the shape of the inelastic collapse mode. It was pointed out that the inelastic collapse mode was significantly different from the elastic local buckling mode. Tests on thin-walled steel structural members have included measurements of geometric imperfections to various degree. Test programs undertaken at the University of Sydney on thin-walled columns and beam-columns have generally measured complete profiles of local and overall geometric imperfections. Overall geometric imperfections were deduced from measurements of out-ofstraightness of flange-web junctions whereas local geometric imperfections were obtained from readings taken at the free edge of flange elements or the centre of web elements. Usually readings were taken at regular longitudinal points and reduced to be measured from straight lines fitted through the end point readings. It has been common practice to perform Fourier analyses of such lines of measured imperfections to produce the amplitude of harmonics with one half-wave, two half-waves, etc. In recognition of the fact that buckling modes of simply supported prismatic members vary harmonically in the longitudinal direction, such analyses have been termed spectral analyses (Schafer and Pekoz 1998, Bernard et al. 1999). The idea of expanding measured geometric imperfections in buckling modes was generalised in Rasmussen and Hancock (1988) where both the longitudinal and transverse variations of the imperfections were considered. In expanding the geometric imperfection transversely, imperfection readings at various points in the cross-section were correlated. For example, if imperfections of equal angles were measured by recording the out-of-flatness of the free edge of the flanges, it would be required to consider whether the readings describe twist rotations of the flanges in the same direction or towards each other. In the first case, the imperfection would be a twist of the cross-section, which would be detrimental to the local (or torsional) buckling strength, whereas in the latter it would be a distortion of the cross-section which would have little effect on the local/torsional buckling strength.

429 While sophisticated methods have been developed for expanding measured geometric imperfections in buckling modes, such methods would only find use in a research environment. It is not practical to measure profiles of imperfections along, for instance, the free edge of flanges as it would require a large number of readings. It would be desirable to limit the number of measurements to a small number if possible. Such a procedure is explored in this paper which compares finite element strengths obtained by modeling the complete measured imperfection with those obtained on the basis of a few measurements taken at mid-length.

SUMMARY OF TEST P R O G R A M Four series of tests were performed on welded full-scale slender I-sections fabricated from high strength Grade 350 steel plates. The nominal dimensions of the four series are given in Table 1, using the nomenclature defined in Figs la and lb. The measured dimensions can be found in Hasham and Rasmussen (1997a, 1998). The test specimens have been labelled such that the series, overall length, test type, and ratio of force causing bending to force causing axial compression can be identified from the label. For example, SI1400S0.125 represents a Series I specimen (SI) with a nominal length Lo of 1400 mm. The 'S' denotes a section capacity test (alternatively 'M' denotes a member capacity test), and the '0.125' indicates that the force causing bending was 0.125 times the force causing axial compression. The last number varies between 0.000 (pure compression) and 1.000 (pure bending). Bf I-I 250~ Ll 250I~_, l- -I7- -I tw D J_

_J

L

Lo

|-

I

-|

tweld (a) (b) Fig. 1" Specimen Dimensions

Series

Lo

L1

I II HI IV

1400 1500 2000 4000

740 830.0 1330.0 3330.0

, . ,

B/ [ (nun) 105.0 175.0 175.0 175.0

,,

t/ 8.00 5.00 5.00

[

D

tw

366.0 260.0 260.0

5.00 5.00 5.00

Table 1" Nominal Specimen Dimensions for Series I, II, III, and IV The Series I and II tests (Hasham and Rasmussen 1998) were conducted to determine the interaction curves for the section capacities of a beam-type cross-section (Series I) with a slender web and stocky flanges, and a column-type cross-section (Series U) with slender flanges and web. The Series I and II specimens failed by local buckling of the component plates and in-plane bending. Further tests (Series III and IV) on the column-type cross-section were conducted to determine the interaction curves for intermediate and long members involving overall instability (Hasham and Rasmussen 1997a). The Series III and IV specimens failed by combined local buckling of the flanges, in-plane bending and out-of-plane flexural-torsional buckling. The slenderness ratios Ley/ry were 16.5, 9.8, 26.1, and 52.1

430 for Series I, II, HI, and IV respectively, where Ley is the effective length for flexural buckling about the minor y-axis and ry is the minor axis radius of gyration. The tests were conducted in a dual-actuator rig which applied compressive axial load and major axis bending moments at the ends. In all tests, the end bearings were restrained against twist rotations and minor axis flexural rotations while major axis flexural rotations were unrestrained. Series I and II were braced against out-of-plane buckling at mid-length while Series III and IV were free to buckle out-ofplane. For each series, one specimen was tested in pure compression, one in pure bending, while the remaining specimens were tested under different axial force to end-moment ratios, hence allowing a broad spectrum of axial force to end-moment ratios to be investigated. The moments applied at the ends were equal and opposite in all tests causing in-plane major axis bending in single-curvature. Compression and tension coupon tests were conducted on all stock plates (Hasham and Rasmussen 1997a, 1998). The tension and compression stress-strain curves were similar demonstrating a sharp change from the linear-elastic range to the yield plateau. The average value of Young's modulus was 215 GPa and the average static yield stress was 410 MPa and 360 MPa for the 5 mm and 8 mm stock plates, respectively, based on the compression coupon tests. Residual stresses (arising mainly from welding) were measured using the sectioning technique (Hasham and Rasmussen 1997a, 1998). The average compressive residual stresses in the flanges and the web were 104 MPa and 125 MPa respectively for the Series I specimens. They were 108 MPa and 109 MPa for the Series II-IV specimens respectively. Residual stress models were developed assuming constant values of compressive residual stress in the flanges and web, and tensile stresses at yield in zones around the welds (Hasham and Rasmussen 1997a, 1998). Geometric imperfections were measured on all specimens before testing. The measurement comprised out-of-plane readings at the four free edges of the flanges and the web centre-line for determining local imperfections as well as in-plane readings of the component plates for determining overall imperfections (Hasham and Rasmussen 1997a, 1998). The imperfections were measured along the full length at 25 mm intervals. Subsequently, the overall flexural imperfections for bending about the major and minor axes as well as twist rotations were obtained from the measurements (Hasham and Rasmussen 1997a, 1998). A full description on the procedure for measuring the initial geometric imperfections is presented in Hasham and Rasmussen (1995, 1997a) along with the measurement profiles for all specimens. The averages of the measured local imperfections at the flange tip and web centre-line as well as the measured overall imperfections relative to the length are shown in Table 2 for each series. The values are averages for all specimens of each series based on the measurements taken at mid-length. The imperfections at mid-length are representative values of the local and overall imperfections; they are not necessarily the largest measured imperfections. Series _

Local Web Flanges

ar ',mml

mm',

.... I II III

0.47

0.26

0.18

iv

0.70

0.34 0.39 0.95

0.33

Minor

Overall Major

u

v

(ram) 0.559 0.355 0.260 0.690

(mm) 0.168 0.202 0.449 0.486

Rotation 0 (rad) 0.00132 0.00179 0.00519 0.00799

Table 2: Geometric Imperfections for Series 1-IV at Mid-Span

431 The experimental ultimate loads are summarised in Tables 3a, 3b, 3c, and 3d for Series I, II, III, and IV respectively. They include the ultimate axial force (Nu), ultimate major axis end moment (Me) and ultimate inelastic moment (Mi,), the latter calculated as the sum of the end-moment (Me) applied through the lever arms and the moment due to N-6 effect, where 6 is the measured in-plane deflection at mid-length.

FINITE E L E M E N T NONLINEAR ANALYSIS The finite element non-linear analysis program "ABAQUS" from Hibbert, Karlsson, and Sorenson Inc. (1995), referred as Abaqus to from here on, was used to simulate the experimental behaviour of Isection beam-columns. The Updated Lagrangian approach was used throughout the study which included both geometric and material nonlinearity. Wherever possible, the input data for the analysis included parameters which were measured experimentally, including cross-sectional dimensions, material properties, residual stress distributions, and initial geometric imperfections. In order to simulate the experimental behaviour of the beamcolumns as accurately as possible, all the significant characteristics of the test rig and specimens were incorporated in the finite element model, as described in detail in Hasham and Rasmussen (1997b). Specimen

Experimental

N. I ~six,ooso.ooo

96~ l

SI1400S0.025

.

Me .......I

(kN) ]

Model "M"

M,. IIN., [ M e t

(kN.m)

]1 (kN) i

.

Model "S"

.

.

.

.

N.; ....

I Miut

(kbi.m)

o .......10.90 11990 1 0

i o.lo i 990

['25.1 [ 26.7 [1 905

896 123.5

24.9 11903

sxx400so.050

760 I 50.7

52.4 ~v55 I 50.4 ]52.3 ]744

SI1400S0.125 ' SI1400S0.250 SI1400S0.500 SI1400S0.750 SI1400S1"000 "

624 [ 77.6 464 [ 321 [ 2i0[ 0:.50 I

79.6 i1 598 I 74.3 i ~ II [ '1~ ] 2 [[ [ 12 J 7 I [ 14 ] ~ l[ [ '1{:

Av~ i [ '

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}[ ~ [

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(kN.m)

0

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Model "M" "S"

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N./N.t

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0.9-,~

0.9-,~

25.1 [ 26.9 0.992 0.990 49.6 I 51.6 "1.007 1.022 74.7 / 76.7 1.043 1.038 1( ',.1 ] 104.0 i~ ,.1 I 124.8 lz ;.11146.4 li "4 I 167"4

i

Model "lVI":Results of the non-linear analysis using measured geometric imperfections. Model "S": Results of the non-linear analysis using superimposed geometric imperfections. Table 3a: Experimental and Theoretical Ultimate loads for Series I Specimen

Experimental

Model "M" .

N.

SII1500so.o00 SII1500S0.025 SII1500S0.050 SII1500S0.125 SII1500S0.250

(LN) 852 717 6/)2 4~2 .... 334

M.

i M,.

N.,

(kN.m) (kN) 0 11829 2012 22.5 709 ' 32.7 35.2 598 57.3 59.1 430 75.2 .76. . .7. . 310

.

.

Model "S'"

.

M., 1 M,., (Ida.m)

0 20.0 32.5 53.4 69.7

N.,

(k2q) 0 8i9 21.7 "699 34.8 596 55.5 427 71.6 304

M., I M,.,

(kN.m) 0 0 19.7 21.7 32.4 34.6 53.0 55.1 {58.5 70.3

~todel

......

"M'"

' '"S~;

M/M,

M /M, 1.040

11 [.028 ~ t.011 ] 1.007 ~[ !.074 ]1 1.077....

1.O26 1.OlO

1.082 1.O99

432

SII1500S0.500 SII1500S0.750 SII 1500S 1.000

124 0.10 .

.

91.5 106.7 .

.

.

92.1 106.7 .

.

121 0.0

89.5 101.3

90.4 101.3

118 0

87.2 100.3

88.2 100.3

.

.

1.051 1.064

1.042

1.056

"M" NJNut

"S" NaqV,,

.

"Average ' Coef. of var. .

1.025 1.053

.

Table 3b: Experimental and Theoretical Ultimate loads for Series II

Specimen

Experimental

Model "M"

Me I SIII2000M0.000 SIII2000M0.050 SIII2000M0.125 SIII2000M0.250 SIII2000M0.500 SIII2000M1.000 ,,,

(kN) 799 609 421 292 211 0.10

Model "S"

M,,

(kN.m) (kN) 0 [ 0.10 '~745 25.5 29.1 597 51.9 55.5 399 66.5 68.8 288 80.2 82.2 214 1 0 0 . 0 100.0 0.0

M,, IM, ,

N,,,

(kN.m)

(kN)

(kN.m)

Me/714et M/Met i

0 ' 25.0 49.1 65.7 81.4 98.5

0 28.7 53.1 68.9 84.1 98.5

,,

765 599 397 283 210 0

0 25.1 49.0 64.6 79.7 97.2

0 28.7 53.0 67.8 82.6 97.2

Average Coef. of var.

1.072 1.020 1.055 1.014 0.986 1.015

1.044 1.017 1.060 1.032 1.005 1.029

1.027 0.030

1.031 0.018

Table 3c" Experimental and Theoretical Ultimate loads for Series 11I

Specimen

,,

SIV400OM0.000 SIV4000M0.050 SIV4000M0.125 SIV4000M0.250 SIV4000M0.500 SIV4000M1.000 Average Coef. of var.

Experimental

Model "M"

Nu

Me I M,u

Nut

(kN) 713 582 397 278 187 0.30

(kN.m) 0 1.99 25.5 32.6 49.6 55.6 60.9 67.9 70.6 76.4 92.2 92.2

(kN) 762 583 366 263 173 0.30

i

Met

Model "S"

M,,t

(kN.m) 0 2.48 25.5 33.1 45.7 53.2 57.6 64.3 65.4 70.6 86.2 86.2

(kN) 734 576 365 286 180 0.30

i

(kN.m) 0 7.5 25.3 38.2 45.6 55.8 62.7 74.5' 67.9 74.4 88.5 88.5

Model "M" "S"

N,/Nut NJ'N,,t Me/A'let M/g,t 0.936 0.998 1.085 1.057. . . . 1.081

0.971 1.010 1.088 0.972 1.039

,.070

1.042

1.038 0.061

1.020 0.047

. . . . . . . . .

Table 3d: Experimental and Theoretical Ultimate loads for Series IV The initial local and overall geometric imperfections of each specimen were measured along the full length at 25 m m intervals as mentioned above. Imperfections between the measurement points were obtained by linear interpolation. The measured local and overall imperfections were added to the perfect geometry to produce the final node coordinates. The residual stress models were incorporated in the finite element model using the initial stress facility. The plasticity model consisted of the von Mises yield surface combined with Prandtl-Reuss flow rules. Yielding and strain hardening were incorporated by specifying the true stress versus true plastic strain curve as a multi-linear curve. The 4-node doubly-curved thin or thick shell, reduced integration, hour-glass control, finite membrane strain element (referred to as S4R in Hibbitt et al (1995)) was used. The model used four elements in

433 each flange out-stand and eight elements in the web. The length of the elements was 25 mm along the full length of the member. This produced aspect ratios (length to width) of the elements in the flange outstands of 1:2 and 1:1.17 for the Series I and II-IV specimens respectively. For the web elements, corresponding ratios were 1:1.75 and 1:1.25. A detailed comparison of the finite element results with the test data is made in Hasham and Rasmussen (1997b) including theoretical and experimental load vs shortening graphs, load vs minor and major axis deflection graphs and load vs twist rotation graphs. Only the ultimate loads are compared in this paper, as shown in Tables 3a, 3b, 3c and 3d for Series I, 11,111and IV respectively. The theoretical results are those listed under 'Model M' and include the ultimate axial force (Nut), ultimate end moment (Met), and ultimate inelastic bending moment (Miut). Tables 3a, 3b, 3c, and 3d also show the ratios of the experimental ultimate loads (Nu, Me) to predicted ultimate loads (N~t, Met). The averages of these ratios are 1.0, 1.042, 1.027, and 1.038 for Series I, II, I11, and IV respectively. The coefficients of variation of the same ratios are 0.035, 0.027, 0.030, and 0.061 for Series I, 11, 111, and IV respectively. It follows that the finite element strengths are generally in close agreement with the experimental values, albeit slightly conservative. The maximum discrepancies between the experimental and predicted ultimate loads are 4.5%, 7.7%, 7.2%, and 8.5% for Series I, II, 111, and IV respectively.

THE MODELLING OF GEOMETRIC IMPERFECTIONS Recognising that geometric imperfections in the shapes of critical buckling modes generally have the most detrimental effect on the strength, the most common procedure for modelling imperfections is to superimpose factored local and overall buckling modes onto the perfect geometry. The purpose of this section is to investigate the accuracy of this approach when applied to fabricated thin-walled beamcolumns and to suggest a simple method for determining the amplitude of the local imperfection. The approach taken in this paper was to use the imperfection measurements taken at mid-length, which were representative and not necessarily the largest values of the local geometric imperfection. For the Series I specimens with slender webs, the local imperfection of the web plate was used, while for the Series II-IV specimens with slender flanges, the average local imperfection of the free edges of the flanges was used, both reduced for overall components. This procedure was considered to be so simple that it could be used as a practical method for determining local geometric imperfections in actual thin-walled members. The superimposed overall imperfections about the major x-axis and minor y-axis were in the shapes of the elastic overall buckling modes, viz. sine curves with a single and two half-waves for buckling about the x- and y-axes respectively. For each series, the amplitudes of the major and minor axis imperfections were chosen as the averages of the overall imperfections measured in the y- and x-axis directions at mid-length. The average measured overall imperfections in the x- (minor u) and ydirection (major v) are given in Table 2 for each series. After adding the overall imperfections, an elastic local buckling analysis was performed on the member to determine the critical local buckling mode. While performing this analysis, the members were restrained at the quarter points from buckling in- and out-of-plane in order to ensure that the lowest buckling mode corresponded to the critical local buckling mode. The critical local buckling mode was factored so that the amplitude at mid-length was equal to the average measured local imperfection of the specimens at mid-span, as shown in Table 2. The web imperfection (Sw) was used for the Series I specimens while the flange imperfection (Sf) was used for the Series II-IV specimens. The nodal displacements of the factored critical elastic local buckling mode were then added to the

434 nodal co-ordinates of the model which incorporated overall imperfections, and a geometric and material analysis was performed. The ultimate loads are shown under "Model S" in Tables 3a, 3b, 3c, and 3d for Series I, II, Ill, and IV respectively. It follows that the averages of the ratios of test to finite element strength were 1.005, 1.056, 1.031, and 1.020 for Series I, II, UI, and IV respectively. The coefficients of variation of the same ratios were 0.030, 0.028, 0.018, and 0.047 for Series I, II, III, and IV respectively with maximum discrepancies between experimental and predicted ultimate loads of 3.8%, 9.5%, 6.0%, and 4.2% for Series I, II, 1II, and IV respectively. It follows that the finite element strengths were generally in close agreement with the experimental values. The averages of experimental ultimate load (Nu, Me) to predicted ultimate load (Nut , Met ) show that the 'Model M' strengths (using measured geometric imperfections) were generally closer to the experimental values than the 'Model S' strengths (using superimposed geometric imperfections) for the Series I, II, and III specimens while the reverse was true for the Series IV specimens. On a whole, the 'Model M' strengths are slightly more accurate. However, the coefficients of variation of the ratios of experimental to theoretical strength are smaller for the 'Model S' strengths for all series except for Series II for which they were nearly equal. Hence, the 'Model S' strengths are associated with less variability than the 'Model M' strengths. While this result is surprising, it leads to the conclusion that there is no apparent benefit to be gained by modelling measured profiles of geometric imperfection compared to using the values measured at mid-length. This significant result suggests that an appropriate procedure for modelling geometric imperfections of thin-walled members is to measure local and overall imperfections at mid-length and use these values, or averages thereof, as amplitudes for superimposing local and overall buckling modes onto the perfect geometry. The measurements of local and overall imperfections can be achieved by stretching a string between the ends and measuring the mid-length out-of-flatness at the flange tips and web center-line (local imperfections), as well as the mid-length out-of-straightness at flange-web junctions (overall imperfections).

SENSITIVITY STUDY OF LOCAL G E O M E T R I C IMPERFECTIONS The sensitivity to local geometric imperfections was assessed by varying the amplitude of the superimposed elastic local buckling mode. The method of superimposing overall and local imperfections described above was applied to the Series II specimen SII1500S0.250. The overall geometric imperfections about the major and minor axes were kept constant throughout the analyses, and chosen equal to the average measured overall imperfections about the major and minor axes given in Table 2. The amplitude of local geometric imperfection was increased gradually from a very small value representing almost perfect geometry (Sf=0.0001 mm) to half of the plate thickness. The results of the predicted ultimate loads for increasing values of the amplitude of local geometric imperfection are shown in Fig. 2, where they are non-dimensionalised with respect to the ultimate load of the geometrically nearly perfect member (Pu0---403 kN). It follows from Fig. 2 that the ultimate load dropped by 24.1% when the local imperfection was increased from a negligible value (0.0001 mm) to 2.5% of the plate thickness (0.125 mm). However, by further increasing the amplitude of local imperfection from 2.5% (0.125 mm) to 50% (2.5 mm) of the plate thickness, the ultimate axial load decreased further by only 3%. The measured local imperfection for this specimen was 6.8% of the plate thickness Hasham Rasmussen 1997b), which was in the insensitive local imperfection range. Consequently, the ultimate loads are not likely to have been greatly influenced by inaccuracies in the measured values of local geometric imperfections.

435 1

0.9 0.8 ~"~ 0.7

0.6

0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

5f/tf

Fig. 2: Strength vs Magnitude of Local Geometric Imperfection

CONCLUSIONS The results of a non-linear finite element analysis have been compared with tests on thin-walled high strength steel I-sections in combined compression and major axis bending. The finite element model was found to accurately predict the ultimate loads of short beam-columns with slender beam-type cross-section (stocky flanges and slender web), and short, intermediate, and long beam-columns with slender column-type cross-section (slender flanges and web). The model included measured values of material properties, geometric imperfections and residual stresses. The accuracy of the method of superimposing local geometric imperfections in the shape of the critical elastic buckling mode onto the perfect geometry has been investigated. The results of the non-linear analysis incorporating superimposed geometric imperfections were found to be slightly more conservative than those obtained using measured geometric imperfections. However, the variance of the method was less than that of using measured geometric imperfections. Thus, the method presents an accurate and considerably simpler alternative to using the measured geometric imperfections. The magnitudes of the superimposed local and overall geometric imperfections were determined from readings taken at mid-length. In a practical situation they could be obtained by stretching a string between the ends and measuring the deviations at mid-length. A sensitivity study was conducted to investigate the influence of local geometric imperfection on the beam-column strength of a Series II specimen. It was concluded that the ultimate load was sensitive to very small local geometric imperfections but virtually unchanged when the imperfection exceeded 2.5% of the plate thickness. The fact that the average measured local geometric imperfection (0.18 mm or 3.6% of the plate thickness) of the Series II specimens exceeded this value may partly explain why accurate strengths could be obtained using a superimposed local geometric imperfection whose magnitude was determined from measurements taken at mid-length.

436 REFERENCES Bernard ES, Coleman, R and Bridge, RQ, (1999), 'Measurement and Assessment of Geometric Imperfections in Thin-walled Panels', Thin-walled structures, 33(2), pp. 103-126. De Ville de Goyet, V, (1996), 'Initial Deformed Shape - Essential Data for a Nonlinear Computation'. In: Rondal J, Dubina D & Gioncu V (eds) Proceedings, 2nd International Conference on Coupled Instabilities in Metal Structures, Liege. pp. 61-68. Hasham, AS and Rasmussen, KJR, (1995), 'Section Capacity of Thin-Walled I-Sections in Combined Compression and Major axis Bending', Research Report R716, School of Civil and Mining Engineering, University of Sydney, Australia. Hasham, AS and Rasmussen, KJR, (1997a), 'Member Capacity of Thin-Walled I-Sections in Combined Compression and Major Axis Bending', Research Report R746, Department of Civil Engineering, University of Sydney, Australia. Hasham, AS and Rasmussen, KJR, (1997b), 'Nonlinear Analysis of Locally Buckled I-section Beamcolumns' Research Report R750, Department of Civil Engineering, University of Sydney, Australia. Hasham, AS and Rasmussen, KJR, (1998), 'Section Capacity of Thin-Walled I-Section Beamcolumns', Journal of structural Engineering, American Society of Civil Engineers, 124(4), pp. 351359. Hibbitt, Karlsson and Sorensen, Inc., (1995), 'ABAQUS Standard, Users Manual', Vols 1 and 2, Ver. 5.5, USA. Rasmussen, KJR and Hancock, GJ, (1988), 'Geometric Imperfections in Plated Structures Subject to Interaction between Buckling Modes', Thin-walled structures, 6, pp. 433-452. Schafer, BW and Pekoz, T, (1998), 'Computational Modeling of Cold-formed Steel: Characterizing Geometric Imperfections and Residual Stresses, Journal of Constructional Steel Research, 47, pp. 193-210.

Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

437

THE FINITE ELEMENT METHOD FOR THIN-WALLED MEMBERS - APPLICATIONS A. T. Sarawit 1, y. Kim 2, M. C. M. Bakker 3 and T. Pek6z 4 1,2 Graduate Research Assistant, School of Civil & Environmental Engineering, Cornell University, Ithaca, NY 14853, USA 3 Assistant Professor, Faculty of Architecture, Building and Planning, Eindhoven University of Technology, The Netherlands 4 Professor, School of Civil & Environmental Engineering, Comell University, Ithaca, NY 14853, USA

ABSTRACT Traditionally numerous physical tests have been required to develop and verify newly proposed design procedures. The availability of powerful computers and software make the finite element method an essential tool in such research. Analysis types, material models, elements and initial conditions that are taken into account in the finite element analysis of thin-walled structures are discussed. A list of programs developed in recent studies and example problems of the finite element method application to thin-walled structures are given.

KEYWORDS Cold-formed steel, aluminum, finite element method, buckling analysis, post-buckling analysis, material model, imperfection, residual stress.

INTRODUCTION Current projects at the Thin-Walled Structures Research Group of Comell University mostly involve the study of the behavior of cold-formed steel and aluminum members under various types of loading conditions and the development of design procedures for both the member and frame structure levels. Traditionally numerous physical experiments are required to develop and verify newly proposed design procedures. However, with the availability of powerful computers and software the finite element method has become the main attraction in current research. Computer simulation using the finite element approach provides many advantages over conducting physical experiments, especially in parametric studies. Physical experiments are still conducted but mainly to verify the analytic approach and the assumptions made. ABAQUS, a finite element analysis program developed by Hibbitt, Karlsson and Sorensen, is mainly used in this study. The following is a summary of some of the general issues that are taken into account in the finite element analyses of thin-walled structures.

438 ANALYSIS TYPE

Buckling Analysis The buckling analysis is used to predict the buckling loads and the corresponding buckling shapes. The buckling load is generally used as a parameter in determining the post-buckling strength of members. Elastic buckling solutions for the entire member have also been used in developing alternative design procedures to current AISI (1996) practice such as Schafer and Pek6z's (1998) direct strength approach and Schafer and Pek6z's (1999) method that integrates distortional buckling into the unified effective width approach. The buckling shape is used for the description of the imperfections when the maximum amplitude of the imperfection is known but its distribution is not known. Superposing of multiple buckling shapes may be used as the initial geometric imperfection in post-buckling analysis.

Post-Buckling Analysis In most studies where the problems involve geometric nonlinearity and material nonlinearity prior to failure, a post buckling analysis is needed to investigate the load-deflection behavior. Several approaches are possible depending on the selected algorithm and how the boundary conditions are applied. When the loads can be applied by means of prescribed displacements, and no snapback behavior occurs, a displacement increment method (where proportional displacements are applied) is used. In other cases, the modified Riks method (where proportional loads are applied) is used in order to be able to pass limit points. Both approaches are effective in obtaining nonlinear static equilibrium states during the unstable phase of the response. In both cases initial geometric imperfections must also be introduced to obtain some response in the buckling mode before the critical load is reached.

MATERIAL MODEL In finite element modeling several assumptions and decisions must be made such as elements, boundary conditions, initial conditions, and solution methods. Among all these decisions, the determination of the material model is by far the most difficult and yet ultimately the most important for the determination of the analytic model. The material properties of the cold-formed steel section can sometimes be significantly different from the sheet coil. This is because the cold forming process introduces cold work into the section, especially in the comers. The stress-strain behavior is greatly changed in comer regions where the yield stress is increased but at the same time the ductility is decreased. The changes of the material properties are caused mainly by strain hardening and strain aging. The material at the comers may possibly be anisotropic and, in addition, include residual stresses. An analytical material model that attempts to address these properties can become remarkably difficult especially when there is lack of test data. If test results are available, the input parameters needed to describe the stress-strain behavior are directly obtained from the tensile coupon tests from the different portions of the cross section. If test results are not available, an idealization of the material model that is elastic-plastic with strain hardening is usually made for cold-formed steel, while the stress-strain curve for aluminum alloy is approximated by the Ramberg-Osgood equation.

ELEMENTS

Shell Elements Thin-walled members can usually be idealized as thin shell problems and thus the discrete Kirchhoff thin shell elements can be usually used. Figure 1 (a), (b) show an example of a storage rack post and its base plate modeled with linear quadrilateral (S4R), and linear triangular (S3R) general-purpose shell

439 elements. The triangular mesh is generated by use of the Delaunay triangulation algorithm and the element aspect ratios are kept in the 1/2 to 2 range. Element meshes are usually refined until an acceptable converged solution is obtained. Contact Elements

Many problems involve contact between two or more components. Contact elements used to define the interaction between the different components should be used in these types of problems. For example, connection joint fixity in storage racks which involve interactions between the upright post and its shelf beam or the upright post with base plate and the concrete floor has been studied by using contact elements. Figure 1 (c) shows an example of a storage rack upright post and its shelf beam modeled with contact and shell elements. Contact elements used at the beam end connectors and the upright post flanges define the interaction between the two surfaces. Beam Elements

Recent studies involve modeling the entire rack frame with shell and contact elements as shown in Figure 2. Results agree well with physical tests. However, modeling and computation effort can become very expensive, especially in parametric studies. Alternatively in some cases beam elements may also yield reasonable results. The advantage of beam elements is that they are geometrically simple and have less degrees of freedom than shell and contact elements, but implementation of the beam theory should be considered carefully, especially for thin-walled structures. Neither local buckling of web and flanges, nor distortional buckling of the members can be modeled by beam elements, and thus the failure behavior is not modeled accurately. Available beam elements in ABAQUS that are frequently used are the closed section beam elements and open section beam elements, all of which are Timoshenko beams. Open section beam elements have the warping magnitude as an additional degree of freedom at each node.

Spring Elements Spring elements in conjunction with beam elements are used to model structural frames with semirigid joints. Figure 1 (d) shows the displaced shape of a pallet rack frame. Non-linear torsional spring elements are used at the beam-column joints where the moment rotation relationship used is obtained from a physical connection test. Torsional spring elements are also used at the supports but with base fixity suggested from the design code. Instead of using contact elements, spring elements are used in contact simulation to simplify the problem and obtain first approximation. Figure 1 (b) shows an example of a preliminary study of storage rack base fixity where linear elastic compressive spring elements are used under the base plate to model the concrete slab. Linear elastic compressive springs that may transfer compression but no tension are used to represent the interaction between the concrete footing and base plate. The compressive stiffness of these spring elements is determined as if they were compressing the concrete material.

INITIAL CONDITIONS

Geometric Imperfections When precise data of the distribution of geometric imperfections is not available three approaches have been used in the Comell research. One is to use an imperfection consisting of superposing multiple buckling modes and controlling their magnitudes. The magnitude of imperfection can be controlled by using exiting statistical imperfection data where the maximum values are provided. On the other hand, if imperfection measurements are conducted, the imperfection spectrum generated from the

440 imperfection measurements may be used to approximate the imperfection magnitude corresponding to a particular eigenmode. A statistical summary of imperfection magnitudes and more specific discussion of the application are given by Schafer and Pektiz (1998). Another method is to use a stochastic process to generate signals randomly for the imperfection geometric shape; however, a large number of measurement data is also required to have a reasonable stochastic model. Since the signals are randomly generated, the approach also requires a large number of simulations to obtain statistical results. An initial geometric imperfection shape introduced by superimposing the eigenmodes for the local and distortional buckling is shown in Figure 3 where the magnitude of imperfection is exaggerated for visualization. Residual Stresses

As discussed previously, modeling the material properties can become rather complicated. An appropriate assumption is made difficult by lack of data. Residual stresses are currently modeled as initial stress conditions where the variations of the residual stress through the thickness are given explicitly along the shell sections in the longitudinal direction of the member. However the initial stresses given are not in an equilibrium state. Therefore, an initial analysis is needed to establish equilibrium before the load is applied. The stress state result of this initial analysis is then used as the initial state of the model for following analysis. Further discussion and statistical summary of residual stresses are given by Schafer and Pekrz (1998).

CORNELL UNIVERSITY THIN-WALLED STRUCTURES PROGRAMS Although most of the current research relies on high level numerical analysis which requires commercial programs, recent studies also include developing programs both for research and classroom proposes. Several of the programs developed are listed as follows: CU-FSM The program developed by Schafer (1997) is designed to determine the elastic buckling and corresponding modes for any simply supported thin-walled member subjected to arbitrary stress conditions at its ends. The program uses the finite strip method, which was originally developed by Cheung (1976). Once a cross-section is properly defined, the program proceeds by examining a variety of different lengths for the section. The stress and the shape at which buckling ensues are recorded for each of the lengths. The resulting buckling curve and mode shape may be used to better understand the behavior of the section under consideration. CU-Beam The program developed by Gotluru (1998) is designed for the elastic analysis of continuous beams subjected to bending and torsion. The program performs a geometric nonlinear analysis considering nonlinear effects of transverse load applied away from the shear center. The analysis of the continuous beams is performed by the finite element method. The model consists of beam elements joined at nodes. At each node, seven degrees of freedom are considered; 3-displacements, 3-rotations and a warping degree of freedom. A Newton-Raphson iterative technique is employed to trace the nonlinear load-displacement path. The detailed derivation of the model can be seen in Chen and Atsuta (1977). CU-Plate Buckling & Free-Vibration The program developed by Sarawit is designed to solve three types of problems involving the eigenvalue analysis: plate elastic buckling, free-vibration, and free-vibration with initial in-plane

441 stresses. The program uses the finite element method with thin plate elements. Various boundary conditions and perforations can be applied. The detailed derivation of the model can be seen in Yang (1986).

EXAMPLE PROBLEMS

Simply Supported Plate in Compression In order to study the post-buckling behavior of a rectangular compression plate with idealized simply supported condition for all four edges, a nonlinear FEM analysis is performed using ABAQUS. However, as discussed previously, several approaches are possible depending on the selected algorithm and how the boundary conditions are applied. The combination of different algorithm and boundary conditions are exploited here. The algorithms in study are the modified Riks method and the displacement control method. When the modified Riks method is used, a uniform compressive load converted to consistent nodal force is applied at the traverse ends. The loads are also distributed into the first row of elements to avoid localized failure at the ends. When the displacement control method is used, a uniform longitudinal compressive displacement along both transverse edges is applied. As the plate deforms under the compressive force or displacement, displacement along the longitudinal edges will occur. Two different boundary condition cases along the longitudinal edges are studied. One is to have only roller boundary conditions and let the edges deform freely. This means the edges are stress free and can develop in-plane deflections as the plate buckles. The other is to constrain the edges so that it remains straight, but lateral movement of the whole edge is permitted. To impose this condition, constraint between the degrees of freedom of the nodes along the longitudinal edges is needed such that the translations in the 1-direction (Figure 4(a)) are equal, while the other degrees of freedom at the nodes are independent. Similar to using constraint equations to impose this condition, very stiff displacement spring elements connecting the adjacent nodes along the longitudinal edges are used to constrain their relative displacement in the 1- direction in addition to roller supported boundary conditions. Therefore, with all the nodes along the longitudinal edges having no relative displacement in the 1- direction, their transverse displacement in the 1- direction will be the same, thus the edge will remain straight throughout the analysis. The rectangular plate in this study is 6 in. wide, 24 in. long and 0.1 in. thick. Initial geometric imperfection is introduced by using the local buckling shape shown in Figure 4 (a) from an eigenvalue analysis. Four different imperfection magnitudes are selected, where t denotes the thickness of the plate; 0 (no imperfections), t/100 in., t/10 in. and t in. The material model used is elastic-plastic with Young's modulus 29500 ksi, the yield strength 55 ksi and Poisson's ratio 0.3. Figure 4 shows the yon Misses stress at ultimate load for the imperfection magnitude, t/10 in. for the different cases. Figure 5 compares the results from the eigenvalue buckling analysis, post-buckling analysis, and post-buckling reduction factors given by AISI (1996).

Imperfection Sensitivity Study In order to study the post-buckling behavior of an isolated flange-stiffener model, a nonlinear FEM analysis is performed using ABAQUS. Two types of imperfections, local and distortional modes obtained from the eigenvalue analysis, are superposed to give the initial geometric imperfections as shown in Figure 3. The magnitudes of the imperfection are selected based on the statistical summary provided in Schafer and Pek6z (1998). Ultimate strength of the isolated flange is then found for different magnitudes of imperfection. Two maximum imperfection magnitudes, one at 25% and the

442 other at 75% probability of exceedance, are used. The percentage differences in strength are used to measure the imperfection sensitivity as shown in Eqn. 1.

imperfection sensitivity =

(fu)75%,mp.--(fu)25%imp.

2i((fu )75%,mp."]-(fu)25%,mp.)X l O00~

(1)

Idealization of the boundary conditions at the web/flange junction is made by restraining all degrees of freedom except for the translation along the length. Roller supports are used at both ends. The uniform load has been distributed to the first row of elements to avoid localized failure at the ends. Boundary conditions are shown in Figure 6. The material model used is elastic-plastic with strain hardening and the yield strength 347 MPa. Residual stresses are also included with a 30% yield stress throughout the thickness in the longitudinal direction. The residual stresses are assumed tension on the outside and compression on the inside of the section. The length of the model is selected by using the length that would give the least buckling strength in the distortional mode, which is obtained by using the Finite Strip Method program, CUFSM. Figure 7 shows the results of an imperfection sensitivity study. The results shown are the variation of ultimate stresses as a function of the slendernesses of the plate elements of the idealized flange shown in Figure 6. The error bars in Figure 7 (a) show the range of strengths predicted for imperfections varying over the central 50% portion of expected imperfection magnitudes. The greater the error bars, the greater the imperfection sensitivity. A contour plot of this imperfection sensitivity statistic is shown in Figure 7 (b). Stocky members tending to fail in the distortional mode have the highest sensitivity.

Compact Aluminum Beams in Bending Laterally supported doubly symmetric aluminum I-shaped sections under pure bending are simulated using ABAQUS to compare with the current specification approaches and eventually to formulate improvements to the AA (Aluminum Association) specifications. This study deals with rather compact sections. The study is based on analyzing the parametric study models consisting of various flange and web thicknesses, while the width, depth and length of the models are fixed as shown in Figure 8 (a). After the finite element study results are obtained for the models, linear interpolations (Figure 9) are employed to estimate the capacities of the standard extruded aluminum sections. Simply supported boundary conditions are employed; rollers are attached at both ends of the beam, while one longitudinal degree of freedom is restricted at mid-span to avoid the singularity of the stiffness matrix. In addition, continuous lateral supports are also provided at the web-flange junctions. The load is applied by a concentrated moment at the center of the web at both ends, where the roller supported boundary conditions are located. Rigid beam elements are attached at the ends of the member for the stability of the analyses as shown in Figure 8 (a). The alloy and temper is considered 6061-T6 (extruded) with yield strength 35 ksi, the ultimate strength 38 ksi, Poisson's ratio 0.33 and Young's modulus 10100 ksi. The stress-strain curve is approximated by the Ramberg-Osgood equation. Initial geometric imperfections are generated by elastic eigenvalue analyses as shown in Figure 8 (b). The maximum imperfection magnitude is selected to satisfy the maximum allowable out-of-flatness limitations of Aluminum Standards and Data (1993). In ABAQUS, in case of isotropic hardening, once a stress reaches the ultimate strength, the stress remains constant as the plastic strain exceeds the strain at the ultimate stress, though actual behavior

443 could be a drop as shown in Figure 8 (c). For this reason, if the Von Mises stress of a single point of a member reaches the ultimate strength, it is considered that the whole member reaches the failure. This consideration should be an underestimation of the actual failure mechanism; when a single point of a member reaches the ultimate strength, it would be propagated to the neighboring points of the member before actual fracture occurs at this point. On the other hand, the failure of the member can be initiated by instability due to inelastic buckling of the elements of the member, in which case the ultimate/limit load of the member is obtained before any point of the member reaches the ultimate strength. In this study, these two possibilities of failure are considered at the same time to find a critical load factor. Based on the study, it is found that specification approaches underestimate section capacities compared to the finite element results; a few possible design approaches are suggested.

Preparation of Aluminum Beam Tests In anticipation for the tests to be conducted, a numerical study of a four-point bending test for an AA standard I-beam, I-3xl.64, is carried out. The objective is to develop an accurate and economical approach for a bending test setup using numerical simulations. A successful bending test relies on the location of failure. For a successful bending test, the failure should take place at the inner span where only pure bending moments exist. However, it is possible that the failure might occur near the loading locations because of the combination of moments and shear forces. For this reason, the ratio of the critical distance from a support to the nearest loading location to the total beam length, a/l, is determined so that failure can occur at the inner span, while the total beam length, l, remains unchanged. The same alloy and stress-strain curve is employed as the study of Compact Aluminum Beams in Bending. Two types of nonlinear analyses are executed. One is the analysis of a perfect section without any geometrical imperfection; the other is the analysis of a section with initial geometrical imperfections generated by an eigenvalue analysis. The maximum imperfection magnitude is 0.01 in. Continuous lateral supports are provided at the web-flange junctions. Spreader plates are required at four locations; two at the supports and two at the loading locations. However, since the spreader plate contacts the beam flange without any connections such as welding and bolting, moments should not be transferred to the spreader plate. In addition, when a portion of the spreader plate is not in contact with a flange, no force should be transferred to that area. In other words, only compression should be transferred. The compressive spring elements, which do not transfer tension but compression, are used to model the interface between the spreader plate and beam flange instead of using contact elements. Stiffener plates are required to prevent localized failure at the same four locations. Since stiffener plates are connected to the flanges of a beam usually by clamps and friction, bending moments should not be transferred to the stiffener plates. Thus, linear spring elements are employed to simulate this connection. Based on the parametric study results using the finite element simulations, when the a/l ratio is less than about 0.26, failure occurs at the loading locations (Figure 10b). On the other hand, when the a/l ratio is larger than about 0.26, failure occurs at the inner span (Figure 10a). This is confirmed by the ultimate load factor results as shown in Figure 11; when a/l ratio is less than about 0.26, ultimate load factors are much smaller than they are when the a/l ratio is larger than 0.26.

444 The results are also compared with specification approaches and the finite element simulations using pure bending loading employed in the study of Compact Aluminum Beams in Bending. In addition, the difference between a perfect section and an imperfect section is also observed. Further details regarding the studies of Compact Aluminum Beams in Bending and Preparation of Aluminum Beam Tests can be found in Kim (2000).

CONCLUSIONS Current research relies mainly on the finite element analysis. Physical experiments are still conducted but mainly to verify the analytic approach and the assumptions made. Finite element studies with the modeling assumptions as discussed in this paper show reliable and satisfactory results. However, further improvements in the material model and initial geometric imperfection are possible with better understanding of the physical properties of the member.

ACKNOWLEDGEMENT

The examples given here were obtained in research sponsored by American Iron and Steel Institute, the Aluminum Association and the U. S. Department of Energy. Their sponsorship is gratefully acknowledged.

REFERENCES

The Aluminum Association. (2000). The Aluminum Design Manual. The Aluminum Association. The Aluminum Association. (1993). Aluminum Standards and Data. The Aluminum Association. American Iron and Steel Institute. (1996). AIS1 Specification for the Design of Cold-Formed Steel Structural Members. American Iron and Steel Institute Washington, D.C. Chen, W.F., Atsuta T. (1977). Theory of Beam-Columns I1"ol.2 Space behavior and design. McGrawHill, Inc. Chen, W.F., Han, D.J. (1988). Plasticity for Structural Engineers. Springer-Verlag, New York. Cheung, Y.K. (1976). Finite Strip Method in Structural Analysis. Pergamon Press, New York. European Committee for Standardization. Eurocode 9: Design of Aluminium Structures. Editorial Panel Version 2 (December 1996), European Committee for Standardization. Gotluru, B.P. (1998). Torsion in Thin-Walled Cold-Formed Steel Beams. M.S. Thesis, Comell University, Ithaca, New York. H.K.S. (1998). ABAQUS/Standards User's Manual Version 5.8, Hibbitt, Karlsson & Sorensen, Inc. H.K.S. (1998). ABAQUS/Theory Manual Version 5.8, Hibbitt, Karlsson & Sorensen, Inc. Kim, Y. (2000). Behavior and Design of Laterally Supported Doubly Symmetric 1-Shaped Extruded Aluminum Sections. M.S. Thesis, Cornell University, Ithaca, New York. Sarawit, A.T., Pekrz, T. (2000). A Design Approach for Complex Stiffeners. Fifteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Missouri. Schafer, B.W. (1997). Cold-Formed Steel Behavior and Design: Analytical and Numerical Modeling of Elements and Members with Longitudinal Stiffeners. Ph.D. Dissertation, Cornell Univ., Ithaca, NY. Schafer, B.W., Pekrz, T. (1998). Computational Modeling of Cold-Formed Steel: Characterizing Geometric Imperfections and Residual Stresses. Journal of Constructional Steel Research. Schafer, B.W., Pekrz, T. (1998). Direct Strength Prediction of Cold-Formed Steel Members Using Numerical Elastic Buckling Solutions. Fourteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Missouri. Schafer, B.W., Pekrz, T. (1999). Laterally Braced Cold-Formed Steel Flexural Members with Edge

445 Stiffened Flanges. Journal of Structural Engineering, 125(2). Yang, T.Y. (1986). Finite Element Structure Analysis. Prentice Hall, N.J.

Figure 1: Finite Element Modeling. (a), (b) Pallet Rack Post and its Base Plate Modeled with Shell Elements (von Mises Stress for (b)) (c) Pallet Rack Beam to Column Joint Modeled with Contact and Shell Elements (d) Pallet Rack Frame Modeled with Spring and Beam Elements

Figure 2: (a) Pallet Rack Frame Modeled with Contact and Shell Elements (b) Physical Test

Figure 3: Isolated Flange-Stiffener (a) Local Buckling (b) Distortional Buckling (c) Imperfection

446

Figure 4: (a) Initial Geometric Imperfection (b)-(e) von Mises Stress at Ultimate Load (b) Riks Method (c) Riks Method with Longitudinal Edges Constrained (d) Displacement Method (e) Displacement Method with Longitudinal Edges Constrained 1.2

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448

Figure 9: Interpolated Load Factor Distribution over Slenderness Plane where Mu = ultimate moment, My = yield stress x section modulus, h = clear distance between flanges, b ' - (flange width- web thickness)/2, tw = web thickness and tf = flange thickness

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Third InternationalConferenceon Thin-Walled Structures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

449

FINITE ELEMENT METHODS FOR THE ANALYSIS OF THIN-WALLED TUBULAR SECTIONS UNDERGOING PLASTIC ROTATION Tim Wilkinson 1 and Gregory Hancock I I Department of Civil Engineering, The University of Sydney, Sydney, NSW, 2006, Australia.

ABSTRACT The paper describes the FE analysis used to investigate compact cold-formed tubular members undergoing substantial plastic deformation. The FE program used is ABAQUS and the data is based on actual material tests of cold-formed tubes including nonlinear stress-strain characteristics. The significant variable in the study is the geometric imperfections induced mainly by welding of plates to the tubes. The paper describes how these imperfections are included in the analysis and their effect on the results. A surprising result of the study is the significance of geometric imperfections on the plastic rotation capacity of compact tubular sections.

KEYWORDS Finite element analysis, beams, RHS, cold-formed steel, rotation capacity, local buckling.

INTRODUCTION Considerable effort has been invested in the application of finite element methods to the elastic and inelastic analysis of fairly slender thin-walled tubular sections undergoing local buckling. However, very little research has been performed using FE methods to investigate the plastic rotation capacity of compact tubular sections which only undergo local instability well along the plastic plateau. The reasons for the inelastic local instability have not been clearly understood until now. The bending behaviour of laterally restrained beams is commonly divided into 3 or 4 classes of behaviour as illustrated in Fig. 1. Some steel design specifications, such as Eurocode 3 (ECS, 1992), use 4 classes (Class 1, Class 2, Class 3, and Class 4), while others such as the Australian Standard AS 4100 (SA, 1998) and AISC LRFD (AISC, 1997) have 3 classes (compact, non-compact, and slender). A compact or Class 1 section is suitable for plastic design, and can sustain the plastic moment (Mp) for a sufficiently large rotation capacity (R) to allow for moment redistribution in a statically indeterminate system. Independent flange and web element slenderness limits are prescribed to distinguish between classes. Wilkinson and Hancock (1997,1998) describe tests on cold-formed RHS beams to examine the Class 1 flange and web slenderness limits. The sections tested represented a broad range of web and flange slenderness values, but it would have been desirable to test a much larger selection of specimens. However, a more

450 extensive test program would have been expensive and time consuming. Finite element analysis provides a relatively inexpensive, and time efficient alternative to physical experiments. This paper describes finite element analyses to simulate the bending tests and predict the rotation capacity of Class 1 and 2 cold-formed RHS beams.

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FINITE E L E M E N T MODELLING The program ABAQUS (Version 5.7-1) (HKS, 1997), installed on Digital Alpha WorkStations in the Department of Civil Engineering, The University of Sydney, performed the numerical analysis. In order to model the plastic bending tests, the finite element program should include the effects of material and geometric non-linearity, residual stresses, imperfections, and local buckling.

Physical Modelling and Finite Element Mesh Figure 2 shows the simplified testing arrangement for the RHS beams. The beams were loaded and supported via steel plates welded onto the webs of the RHS. The RHS were supplied by BHP Structural and Pipeline Products (now known as OneSteel Market Mills Pipe and Tube), in either Grade C350 (nominal yield stress fyn = 350 MPa, nominal ultimate stress fu. = 430 MPa) or DuraGal Grade C450 0Cy, = 450 MPa, f~, = 500 MPa). All beams were bent about the major axis, and most reached the plastic moment, Mp, and continued to deform plastically until a local buckle formed adjacent to the loading plate.

Figure 2: Physical model

451 A typical finite element mesh, replicating the test arrangement, is shown in Figure 3. The two relevant symmetry planes, at the mid-length of the beam, and through the minor principal axis of the RHS, have been used to reduce the size of the model.

Element Type The most appropriate element type to model the local buckling of the RHS was the shell element. The $4R5 element, defined as "4-node doubly curved general purpose shell, reduced integration with hourglass control, using five degrees of freedom per node" (HKS, 1997), was used. The loading plates attached to the RHS beam were modelled as 3-dimensional brick elements, type C3D8 (8 node linear brick). The weld between the RHS and the loading plate was element type C3D6 (6 node linear triangular prism). The RHS was joined to the loading plates only by the weld elements. Details of the mesh refinement process have been omitted for brevity.

Material Properties The cold-formed RHS have stress-strain curves that include gradual yielding, no distinct yield plateau, and strain hardening. There is variation of yield stress around the section, due to different amounts of work on the flat faces and comers during the production process, with higher yield stresses in the comers. Details of the material properties can be found in Wilkinson and Hancock (1998). The different material property positions are shown in Fig. 4.

Figure 4: Different material properties around RHS

452 G E O M E T R I C IMPERFECTIONS The initial numerical analyses were performed on geometrically perfect specimens. It is known that imperfections must be included in a finite element model to simulate the true shape of the specimen and introduce some inherent instability into the model, in order to induce buckling.

Bow-out Imperfection Measurement of the imperfections indicated that most RHS had an approximately constant "bow-out" along the length of each beam. For most cases, the web bulged outwards and the flange inwards. The magnitude of the bow was approximately 6 w = d/500 (for the web) where d is the web depth, and 6 f= -b/500 for the flange where b is the flange width. However, the nature of the imperfection immediately adjacent to the loading plate was unknown, as it was not possible to measure the imperfections extremely close to the loading plate. The process of welding a flat plate to a web with a slight bow-out imperfection is certain to induce local imperfections close to the plate.

Figure 5: "Bow-out" imperfection Figure 5 shows a typical mesh with the bow-out imperfection included. Figure 6 shows the moment curvature relationships obtained for a series of analyses on 150 x 50 x 3.0 C450 RHS with bow-out imperfections. The magnitude of the imperfection was either d/500 and -b/500 (approximately the magnitude of the measured imperfections), or d/75 and -b/75 (very much larger than the observed imperfections). Compared to a specimen with no imperfection, the magnitude of the bow-out imperfection had a minor effect on the rotation capacity. In fact, the rotation capacity increased slightly as the imperfection increased. Even when the bow-out imperfections were included, the numerical results exceeded the observed rotation capacity by a significant amount. The conclusion is that the bow-out imperfection was not a suitable type of imperfection to include in the model.

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Sinusoidally Varying Imperfection It is more common to include imperfections that follow the buckled shape of a "perfect" specimen, such as by linear superposition of various eigenmodes. The approach taken was to vary the magnitude of the bow-out imperfection sinusoidally along the length of the specimen. The half-wavelength of the imperfection is defined as Lw. A typical specimen with the sinusoidally varying bow-out along the length is shown in Figure 7.

Figure 7: Mesh with sinusoidal imperfection Figure 8 shows the results of selected analyses for a variety of imperfection half-wavelengths. The code "cont" in the legend to Figure 8 indicates the continuously varying imperfection. The specimen analysed was 150 x 50 • 3 RHS. A half wavelength of approximately d/2 (d is the depth of the RHS web) tended to yield the lowest rotation capacity and most closely matched the experimental behaviour (refer to the specimen with Lw= 70 mm). A half wavelength of d/2 was approximately equal to the half wavelength of the local buckle observed experimentally and in the ABAQUS simulations. In Figure 8, the specimen with Lw = 70 mm experienced a rapid drop in load after buckling, and had a buckled shape as shown in Figure 9 which matched the location of the local buckle in the experiments. A specimen with a slightly different imperfection profile, Lw = 60 mm, had a much flatter post buckling response, and the buckled shape include two local buckles, as shown in Figure 10. Both specimens buckle at approximately the same curvature.

454

Figure 8: Results for sinusoidal imperfection

Figure 9: Specimen with one local buckle

Figure 10: Specimen with two local buckles

To force one local buckle to form, and in the desired location, the imperfections were imposed only near the loading plate, as shown in Figure 11. Figure 8 includes the response of an additional specimen, with Lw = 60 mm, but only the single imperfection. The curvature at which buckling initiated was barely unchanged, but the buckled shape changed, producing the desired shape of one buckle (Figure 9).

Figure 11" Single imperfection

455

Imperfection Size A variety of imperfection magnitudes was considered. The magnitude of the imperfections was varied from 6w = d/2000 to 6 w = d/250, and 5f = -b/2000 to 5f= -b/250. Figure 12 shows the moment curvature graphs for a section with varying magnitudes of imperfection. It can be seen that increasing the imperfection size decreases the rotation capacity. For this example of a 150 x 50 x 3 RHS, applying an imperfection of 1/500 most closely matches the experimental response. 1.4

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P R E D I C T I O N S OF R O T A T I O N CAPACITY A large range of sections was then analysed. The sizes considered were either 150 x 150 (d/b = 1.0), 150 x 90 (d/b = 1.66), 150 • 75 (d/b = 2.0), 150 x 50 (d/b = 3.0), and 150 x 37.5 (d/b = 4.0), with a variety of thicknesses, and different imperfection sizes: d/250, d/500, d/1000, d/1500, or d/2000 (for the web), and b/250, b/500, b/lO00, b/1500, or b/2000 (for the flange). The material properties assumed were those for specimen BS02 (see Wilkinson and Hancock (1997)). Figures 13 to 16 plot the relationship between web slenderness and rotation capacity for each aspect ratio considered and each imperfection size. The results are compared with the tests of Wilkinson and Hancock (1997,1998), Hasan and Hancock (1988) and Zhao and Hancock (1991). It should be reinforced that the ABAQUS analyses were all performed on RHS with web depth d = 150 mm and material properties for specimen BS02 (Grade C450). The experimental results shown in comparison were from a variety of RHS with varying dimensions and material properties. The AS 4100 definition of web slenderness is used, where

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Figure 16: Results for d/b = 3.0

Several observations can be made from the results: Imperfection size had a lesser effect on the rotation capacity of the more slender sections (R < 1), and had a greater effect for stockier sections. For a given aspect ratio, the band of results encompassing the varying imperfection sizes widens as the slenderness decreases. This is an unexpected result of the study. No single curve is a very good match for the experimental results. For example, the ABAQUS results for d/b = 3.0 and imperfection of 1/250 match the experimental results well when Zw = 48, while in the range 58 < ~tw < 65, the results for an imperfection of 1/500 provide a reasonable estimation of the experimental results, and for 75 < ~.w< 85 an imperfection of 1/2000 gives results closest to the experimental values. For d/b = 1.0, the ABAQUS results for an imperfection of 1/250 are close to the experimental results in the range 37 < ~.w< 48, while in the range 25 < ~,w< 35, an imperfection of 1/2000 most accurately simulates the test results. This suggests considerable variability in the imperfections with changing aspect ratios and slenderness, and that as the slenderness increases, larger imperfections are required to simulate the experimental behaviour. There is no reason why the same magnitude of imperfections should be applicable to sections with a range of slenderness values. A possible explanation is that the true imperfections in the specimen were caused by the welding of the loading plate to the RHS. A thinner section was deformed more by a similar heat input, hence larger imperfections were induced. The sinusoidally varying bow-out imperfections simulated the effect of the imperfections caused by the weld, and hence greater imperfections were required as the slenderness increased.

SUMMARY This paper has described the finite element analysis of RHS beams. The finite element program ABAQUS was used for the analysis. The maximum loads predicted were slightly lower than those observed experimentally, since the numerical model assumed the same material properties across the whole flange, web or comer of the RHS In reality, the variation of material properties is gradual, with a smooth increase of yield stress from the centre of a flat face, to a maximum in the comer. A perfect specimen without imperfections achieved rotation capacities much higher than those observed experimentally. Introducing a bow-out imperfection, constant along the length of the beam, as was (approximately) measured experimentally, did not affect the numerical results significantly. In order to simulate the effect of the imperfections induced by welding the loading plates to the beams in the experiments, the amplitude of the bow-out imperfection was varied sinusoidaUy along the length of the beam, and limited to be just near the loading plates. The size of the imperfections had an unexpectedly large influence on the rotation capacity of the specimens.

457 It is likely that the imperfection caused by welding the loading plates to the RHS was a major factor affecting the experientially observed behaviour. The sinusoidally varying imperfections in the ABAQUS model simulated the effects of the localised imperfections in the physical situation. Larger imperfections were required on the more slender sections to simulate the experimental results, since for the same type of welding, larger imperfections are induced in more slender sections.

REFERENCES

AISC, (1997), Specification for Steel Hollow Structural Sections, (AISC LRFD), American Institute of Steel Construction, Chicago, I1, USA. European Committee for Standardisation, (1992). Design of Steel Structures." Part 1.1 - General Rules and Rules for Buildings, (known as "Eurocode 3"), DD ENV. 1993-1-1, Eurocode 3 Editorial Group, Brussels, Belgium. Hasan, S. W., and Hancock, G. J., (1988), "Plastic Bending Tests of Cold-Formed Rectangular Hollow Sections", Research Report, No R586, School of Civil and Mining Engineering, The University of Sydney, Sydney, Australia. (also published in Steel Construction, Journal of the Australian Institute of Steel Construction, Vol 23, No 4, November 1989, pp 2-19.) Hibbit, Karlsson and Sorensen, (1997), "ABAQUS", Version 5.7, Users Manual, Pawtucket, R.I, USA. Moen L. A., De Matteis G., Hopperstad O. S., Langseth M.,Landolfo R., and Mazzolani F. M., (1999), "Rotational Capacity of Aluminum Beams Under Moment Gradient: II: Numerical Simulations", Journal of Structural Engineering, American Society of Civil Engineers, Vol 125, No 8, August 1999, pp 921-929. Standards Australia, (1998), Australian Standard AS 4100 Steel Structures, Standards Australia, Sydney, Australia. Sully, R. M., (1996) "The Behaviour of Cold-Formed RHS and SHS Beam-Columns", PhD Thesis, School of Civil and Mining Engineering, The University of Sydney, Sydney, Australia. Wilkinson, T. and Hancock, G. J., (1997), "Tests for the Compact Web Slenderness of Cold-Formed Rectangular Hollow Sections", Research Report, No R744, Department of Civil Engineering, University of Sydney, Sydney, Australia. Wilkinson T. and Hancock G. J., (1998), "Tests to examine the compact web slenderness of cold-formed RHS", Journal of Structural Engineering, American Society of Civil Engineers, Vol 124, No 10, October 1998, pp 1166-1174. Zhao, X. L. & Hancock, G. J., (1991), "Tests to Determine Plate Slendemess Limits for Cold-Formed Rectangular Hollow Sections of Grade C450", Steel Construction, Journal of Australian Institute of Steel Construction, Vol 25, No 4, November 1991, pp 2-16.

458

Acknowledgements This paper forms part of CIDECT Project 2S (Comit6 International pour le Developpement et l'Etude de la Construction Tubulaire). The first author was funded by an Australian Postgraduate Award from the Commonwealth of Australia Department of Employment, Education, Training and Youth Affairs, supplemented by the Centre for Advanced Structural Engineering and the University of Sydney.

Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

459

ON FINITE ELEMENT MESH FOR BUCKLING ANALYSIS OF STEEL BRIDGE PIER

Eiki Yamaguchi, Yoshikatsu Nanno, Hisataka Nagamatsu and Yoshinobu Kubo Department of Civil Engineering, Kyushu Institute of Technology Tobata, Kitakyushu 804-8550, Japan

ABSTRACT The damage of steel bridge piers in the 1995 Hyogo-ken Nambu Earthquake, Japan, has been found attributable to cyclic horizontal load due to the earthquake coupled with the weight of a superstructure. Therefore, steel bridge pier specimens have been tested under the combination of constant vertical load and cyclic horizontal load in order to study their seismic resistance. Although experiments are undoubtedly important, the number of test specimens has to be limited because of time and/or cost constraints. The numerical simulation of steel bridge piers therefore has been conducted, and good agreement between experimental and numerical results has been reported in the literature. In the present study, however, we show that numerical studies in the literature have often employed the insufficient number of finite elements (rather coarse finite element mesh), so that as we increase the number of elements, the results will change. By analyzing a steel bridge pier with various finite element meshes, we then discuss appropriate finite element mesh to carry out this class of buckling analysis.

KEY WORDS Finite Element Mesh, Finite Element Method, Buckling, Elastic-Plastic Behavior, Steel Bridge Pier, Pipe Section, Cyclic Loading,

460 INTRODUCTION Many steel bridge piers were damaged in the 1995 Hyogo-ken Nambu Earthquake, Japan. The damage has been found attributable to cyclic horizontal load clue to the earthquake coupled with the weight of a superstructure. In an effort to improve behaviors of steel bridge piers during a severe earthquake, therefore, steel bridge pier specimens have been tested under the combination of constant vertical load and cyclic horizontal load. Although experiments are undoubtedly important, the number of test specimens has to be limited because of time and/or cost constraints. The numerical simulation of steel bridge piers therefore needs be conducted, too. In fact, quite a few people have carried out this class of buckling analysis of steel bridge piers with pipe sections. The loading condition is the combination of constant vertical load and cyclic horizontal load. The finite element method is exclusively employed, and the steel bridge pier specimen is modeled by shell and beam elements. Good agreement between experimental and numerical results has been reported in the literature. In the present study, we conduct the buckling analysis of a steel bridge pier with various finite element meshes to examine the influence of finite element mesh on numerical result. Based on this result, we discuss appropriate mesh for this class of buckling analysis.

STEEL BRIDGE PIER

Figure 1 illustrates the pipe-sectional steel bridge pier specimen that we have employed in this numerical study. It is basically a cantilever fixed at the bottom and flee at the top. This is one of the specimens tested at PWRI under the loading condition described later (Public Works Research Institute et al. 1997). In the experiment, the axial compressive force P is 15% of the nominal squash load of the cross section Py, and the horizontal load H is applied to trace the predetermined path of the horizontal displacement 8 at the loading point (Figure 2). 8y is the displacement at the initial yielding due to the horizontal force. The horizontal force at the initial yielding is denoted by Hy. In the present study, we use this steel bridge pier to study the influence of finite element mesh. We consider the same loading condition as that in he PWRI experiment. The material behavior is assumed to be described by the plasticity theory with the yield surface of von Mises type and the kinematichardening model (Chen 1994). This type of constitutive model is available in most of commercial finite element programs. In all the finite element analyses herein, ABAQUS (1997) is used.

461 P t"4 W -

9 f~ t~ fe~

~'//////~

unit" mm

Figure 1" Pipe-sectional steel bridge pier specimen

10

[

5

~o -5 -10 NO. OF CYCLES Figure 2: Predetermined displacement path

FINITE E L E M E N T MESHES

We first conduct the buckling analysis of the bridge pier with the finite element mesh employed by Ohta et al. (1997) (Figure 3(a)). To see the validity of this mesh, we also analyze the bridge pier with four other different meshes. These finite element meshes are shown in Figure 3. The portion modeled by beam elements in FS-I to 4 (Figure 3(b)-(e)) is part of a loading member and is assumed rigid. As clearly observed, FS-4 (Figure 3(e)) is the finest mesh. The numerical results in the form of an envelope of a horizontal load-displacement hysteretic curve at the loading point are presented in Figure 4. This figure clearly shows that 1) the finite element meshes

462

Figure 3: Finite element meshes

2 1.5

.................

i................. i

. . . . . . . . . . . . . . . .

1 0.5

i

~MO

0~'~'-FS-2

2

"'O"FS-3 ----CONV.

6 ~/~Sy

8

10

Figure 4: Horizontal load-displacement curves by meshes in Figure 3 influence the numerical results considerably, especially in the post-peak region, and that 2) the finite element mesh of MO (Figure 3(a)) is inadequate. As the mesh becomes finer, the numerical results tend to converge. The converged load-displacement curve is estimated and plotted as "CONV." in Figure 4. This curve is referred to as the exact load-displacement relationship in what follows. The deformed configuration at 8=8 8y by the mesh of FS-4 (Figure 3(e)) is illustrated in Figure 5. The local buckling is clearly observed. The mesh of FS-4 (Figure 3(e)) gives a pretty good result, but it uses 3560 shell elements and one beam elements, requiring much computational time. The reduction of the number of elements is necessary for practical analysis. To this end, we first evaluate the portion of the pier that can be

463

Figure 5" Deformed configuration (FS-4)

Figure 6: Finite element meshes with different beam-element portions

modeled satisfactorily by beam elements. And we conduct the buckling analysis by the four meshes shown in Figure 6. The results are presented in Figure 7, which indicates that beam elements are applicable down to the portion that may be subject to local buckling. The sufficient number of beam elements is also looked for, and 11 elements turn out to be satisfactory. The numerical results in Figure 7 are not so good in comparison with the exact one. Therefore, we further carry out the analysis using the finite element meshes in Figure 8. As shown in Figure 9, the results converge as the mesh becomes finer. The converged curve is estimated and plotted in Figure 8, which is identical with the exact one. The discretization errors of this set of finite element meshes

464

Figure 7: Horizontal load-displacement curves by meshes in Figure 6

Figure 8" Finite element meshes with different element sizes

(Figure 8) are summarized in Table 1. The errors in the table are evaluated based on the exact one. From this table, one may deduce the number of elements to obtain the results with the accuracy required in analysis.

465

.

.

.

.

.

,-,,,,!

.

.

.

.

.

.

,.,

,

,,.,

1.5 ..............

...............

0.5 0(

2

4

6 ~5/~5y

8

10

Figure 9 Horizontal load-displacement curves by meshes in Figure 8

TABLE 1

DISCRETIZATIONERRORS(%) Mesh 3dy 4dy 58y BS3 2.8 6.9 15.3 (16x15) BS3-2 2.7 4.4 8.3 (24• BS3-3 2.6 3.0 2.2 (32• BS3-4 2.6 2.2 3.5 (40x.38) . ( ) Number of elements in circumferential direction • vertical direction .,

CONCLUDING REMARKS The finite element analysis of a pipe-sectional steel bridge pier under cyclic loading is conducted. The study indicates that finite element mesh used in the open literature is not always satisfactory. It is also found that the portion modeled by beam elements can be large, while the buckling portion needs be modeled by rather fine elements. The table is then provided to give an idea about how many elements are actually needed for required accuracy.

REFERENCES

ABA Q US/Standard User's Manual, Ver.5.7, HKS, 1997.

466 Chen W.F. (1994). Constitutive Equationsfor Engineering Materials, Vol.2, Elsevier, Netherlands. Ohta K., et al. (1997). Applicability of Non-linear Structural Analysis Software to Seismic Design of Steel Bridge Piers. Bridge and Foundation Engineering, 31:8, 33-39. Public Works Research Institute et al. (1997). Technical Report of Joint Research on Seismic Design for Highway Bridge Piers, PWRI, Ministry of Construction, Japan.

Section VIII LAMINATE AND SANDWICH STRUCTURES

This Page Intentionally Left Blank

Third International Conference on Thin-Walled Structures J. Zara~, K. KowaI-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

469

THE ELASTO-PLASTIC POSTBUCKLING BEHAVIOUR OF LAMINATED PLATES SUBJECTED TO COMBINED LOADING R.Grstdzki I and K.Kowal-Michalska 2

~Division of Fundamental Technical Research, Institute of Management 2 Department of Strength of Materials and Structures Technical University of L6d~, 90-924 L6d~., Poland

ABSTRACT This work deals with the analysis of elasto-plastic post-buckling state of rectangular laminated plates subject to combined loads such as: uniform compression, in-plane bending and shear. The plates are built of specially orthotropic symmetrical layers. The analysis is carried out on the basis of nonlinear theory of laminated plates involving plasticity. The iterative approach to the problem is employed using the combination of analytical and numerical solutions based on the Rayleigh-Ritz variational method involving the Prandtl-Reuss equations and Hill's yield criterion. The examples of numerical calculations are shown in diagrams.

KEYWORDS Composite laminated plates, elasto-plastic postbuckling behaviour, analytical-numerical methods

INTRODUCTION The great advantage of composite materials consists in the fact that their material properties can be easily modelled in selected directions or regions. An important type of a plate that appears as a subcomponent of advanced composite structure is the symmetrically laminated plate. The term "symmetrically laminated" refers to plates in which every lamina above the plate midplane has a corresponding lamina located at the same distance below the plate midplane, with the same thickness and material properties. In case when the principal axes of orthotropy in every lamina are parallel to the plate edges, the plates are called "specially orthotropic laminated plates"(Jones, 1975). In the analysis of nonlinear behaviour of thin laminated plates it is usually assumed that the normal to the undeformed midplane remains straight and normal to the midplane after the deformation (classical laminated plate theory).

470 In this work the analytical solution describing the displacement field in the elastic postbuckling state is found out for specially orthotropic laminated plates subjected to simultaneous compression and shear or/and in-plane bending and shear. In the elasto-plastic range the solution of the problem is searched for in the iterative way using an analytical-numerical method involving Hill's yield criterion for orthotropic materials and the relations of flow theory of plasticity. The advantage of analytical-numerical methods, in which the analytical solution of the elastic postbuckling state is used as a space base for the numerical solution in the elasto-plastic range, is very short time of computation and quite good accuracy in comparison with numerical methods.

STRUCTURAL

PROBLEM

~y at;

4

/ --:

i

3 |

i ii i i i i

,.._X

|

r

2

Y

Y

1

2

]

2

Figure 1 Rectangular plates subject to compression, shear and in-plane bending

\y, v

y ~_

2tl=g ~

,ill, tk i

J

Z,W

X,H I

z

Figure 2 Geometry of a plate

471 The rectangular laminated plates subject to uniform compression, in-plane bending and shear, acting simultaneously are considered (Fig.l). The plates are simply supported and initially stress free. It is assumed that the loaded plate edges remain straight during loading. The plates are built of orthotropic layers (lamina) that are symmetrically arranged about the plate middle surface (Fig.2). The layers are called specially orthotropic what means that the principal axes of material orthotropy are parallel to the plate edges. In this case neither shear or twist coupling nor bending-extension coupling exists. The orthotropic material properties in the elastic range of k-th layer are determined by four independent constants: E~,rlk,Gk,Vk, where:

rlk = ~Ey ;

Vk = V~"

V~y = rlkVk,"

The prebuckling displacement and stress fields of a plate are described by a combination of its nodal displacements in the x-direction U~ = U

~

x

~+U

s

y

(1 - - 2X)(l

+U

a

-

-

2 y)

(1)

and additionally: a) for the case of uniform compression and shear acting simultaneously: O

v =0

O

~ x =const.,

O

Oy

O

=const., Xxy=const.

b) for the case of simultaneous shear and in-plane bending: 0 ~-0, 0 vo ~ 0 a ox- linearly distributed, Cry Xxy =const.

CONSIDERATIONS IN THE ELASTIC P O S T - B U C K L I N G STATE According to the non-linear theory of thin plates the relations among strains and displacements are as follows:

~

- iL-~--A

'

,

Ov o%voNv o~ o o%vo Ox + Ox~y ~x Oy

7xYm= ~

F-,xb = --Z

IE ]

lf 01

Exm=~+2 + ~

(w - Wo). O~X2 '

0 ~ (w - Wo).

I~Yb = --Z

cgY2

Wo) cgxc3y

c92 ( w -

,

Yxyb = -2z

(2)

In the postbuckling state the initial geometrical imperfections w0 are taken into account during analysis. The deflection w and initial out-of-flatness w0 are assumed as m

m

w = i=lj=l )-'. ~ fij sin in__xx a sin jny b

w~ = E ~'f~ sin irc---xxsin JrcY i-lj=l a b

(3)

The forms of displacement fields in the elastic postbuckling state are found out in two different ways, by assuming the combination of selected eigen-functions: a) in case of simutaneous compression and shear similarly as in work of Masaoka et al.(1998), substituting the expressions (Eqn.3) in two in-plane equilibrium equations: o~N ONxy + =0, ~x Oy where 9

N x = AllExm + Al2eym,

ON ONy ~Y + -0, ~

Ny = Al2exm +

A221~ym,

(4)

Nxy = A33Yxym. ;

(5)

472 N

and Ars = Z(Qrs)(Zk-Zk-1); k=l

r,s= 1,2,3

k

(6)

E~ . Q22 = rlk E____~. Q12 : rlkVk E______~. Q33 = Gk" Q , 1 - l_rlkVak i_rlkV ~ l_rlkV 2 b) in case of simutaneous bending and shear from compatibility equation in a form: _

04(i)

04(i)

04(i)

( t~2W

2

A22 - - ~ +(-2A12 +A33) &xZ0y2 + A l 1 ~ -~- ~ - ~

-

" /~, 12 . A 1. 2 . . A33. = A22 = AI---L" "~'11 = -A22 -

where:

A00

A00

02(I) --

and N x

02(I) 9

0Y2

A0o

~

Ny

~

1

A33'

W __

&X2 0y2

+

,wo ,wo .

&x--T- 0Y2 ,

(7)

Aoo = AliA22 - A22 (8)

(~2(I)

.

0X2

2

&x0y )

-.-

Nxy

~

~

.

&xo'~y

So, the in-plane displacements u and v can be expressed as mn 2ixx 2in'x 2b~Y} (9) u = u~ + ~ ~ {Bl sin. + B 2 sin ~ c o s i=~j~l a a mn 2jny 2jrcy 2inx b v=v ~ +~E{B3sin +B4sin 9 cos + B s ( y - )} (10) 2 i---1j=l b b a Parameters B~+B5 depend on layers' material constants and amplitudes of deflection functions (Eqs.3) and are determined from Eqs.4 or from Eq.7 with appropriate boundary conditions. The function w describing the post-buckled shape of a plate under combined loads is taken in a form:

e~m m ~ f i ~ .. sin iwx sin JnY w = Cc Em ~ z. e ,ij ss ii nn- -xs ~ i n , Jrc--~Y + Cs E z. ,,j sin in-~xsin jrcy +CbY' i--lj=l a b i=lj=l a b i=lj=l a b

(11)

where 9 b coefficients of Fourier series for a buckled plate (of given ratio a/b and composed of N f~jc ,fijs ,fijlayers) under compression, shear and bending acting separately, Cr C~ ,Cb- free parameters. In order to determine the buckling mode for a considered plate under simple loading the eigen-value problems have to be solved (Lekhnitskij (1947), Jones (1975)) and then the coefficients of eigenfunctions can be determined. The coefficients fij should be chosen in such a way that they provide (with satisfactory accuracy) the smallest values of buckling stresses for calculated values of stiffness matrix Dr~ of a plate and for a given plate aspect a/b, where: k (Zk3-Z3k-I Drs = ~ 1 k~l(Qrs) =

t

r,s = 1,2,3

(12)

MAIN ASSUMPTIONS IN THE ELASTO-PLASTIC RANGE In the elasto-plastic range following assumptions are made: 9 the component materials are orthotropic, elastic-perfectly plastic and obey Hill's yield criterion (Hill (1950)), that can be written in a form: -2 - 2 - 2 - .t2 f =cr =algx +a2~y-al2gxay +3a3 xy (13) where a and x are stress components, a l + a3 are parameters of anisotropy and tr is called the effective stress. The anisotropic parameters in (13) can be determined by four independent yield -2/r wher e ao is the uniaxial yield stress in the tests. For a tensile test in x direction: a~ = tr0

473 reference direction, 6~ois the uniaxial yield stress in the x direction. Taking the x-direction as the reference

direction,

then

al = 1,

and

similarly:

a2 = 6 2 / 6~0,

a3 = 6 2/(3x220),

a2 = 6o2/62~0, a3 = 602/(3x220), a33 = 602/6~o,when 0=45 0 then a12 = al + a 2 +3a3 - 4 a 3 3 . 620,600,x120 denote the initial yield limits, respectively: in the uniaxial tensile tests in the ydirection, in the direction rotated by an angle 0 to the x-direction and in the pure shear test. 9 all assumptions of non-linear plate theory still hold, 9 the forms of displacement functions are the same in the elastic and elasto-plastic range but the amplitudes C can vary arbitrarily, 9 according to the plastic flow theory the increments of plastic strains are described by Prandtl-Reuss equations, where the infinitesimal increments are replaced by finite ones. So the resulting expressions for elasto-plastic stress increments are: - E(1~--xfly 2) [zxex +Vl"iAey-A(Sxx +vrlSyy)], A6x -A6y

.

.

Ey (l_rlV2) [Z~y + VAex -A(Syy + vSxx)] , .

.

(14)

AX~y= G~y(A~/xy- ASxy). where:

--

--

S~x= (2al 6x - a12 6y),

1

-

-

Syy = -~(2a2 6y - a12 6x),

-

Sxy = 2a3 Xxy.

(15)

and the instant value of A has to be determined in each step of calculations.

METHOD OF SOLUTION The iterative approach to the problem is employed using the combination of analytical and numerical solutions based on the Rayleigh-Ritz variational method involving the relations of flow theory of plasticity. This method has been used in many works (e.g. Gradzki & Kowal-Michalska (1999),

(2000)). The analytical considerations are conducted until the expressions (Eqs.2) for strains are determined in terms of eigen-value coefficients f0 and free parameters Co, Cs and Cb. Next the increment of total plate energy has to be found out: AW= !I(O'x + 1 Affx)Z~x + (~y + 1Aay )Z~y +('txy + 1 AXxy)ATx3,~xdydz,

(16)

where: V -denotes the volume of a plate, Ox, oy ,Xx~ denote the stresses before the displacement increment AU is applied and Act, Ae are the increments of stresses and strains produced by AU. The elastic and plastic energy increments are evaluated in a numerical way. In order to accomplish this, the discretisation of a plate is performed. Each layer is divided equally to appropriate cubicoids. The energy values calculated for each cubicoids are summed up for a whole plate. Next the numerical minimisation of AW (Eq. 16) is performed versus independent parameters C. In each step of calculations following aspects are taken into account: occurrence of active, passive and neutral processes, reduction of stresses to the yield surface, strayed from it because finite increments are used. It should be emphasised that during the analytical-numerical solution the response of a plate to the increment of its nodal displacements (Fig. 1) is searched for. The values of the average shear stress and the average compressive stress or the average bending moment corresponding to the nodal

474 displacements of the plate are obtained numerically as a mean value of stresses along the edge y=b, summed up for all layers.

EXAMPLES OF NUMERICAL CALCULATIONS Although the theoretical considerations and elaborated computer programme deal with orthotropic laminated plates, the preliminary numerical calculations have been conducted for square plates (a/b=l) composed from three isotropic layers steel-aluminium-steel. It is assumed that the characteristics of component materials are linear in the elastic range and in the plastic range no hardening occurs. Geometrical and material data are given in Table 1 ( E - Young's modulus, v- Posson's ratio, o0 - yield limit). The calculations have been performed for different ratios of aluminium layer thickness (g) to the plate thickness (h) - Fig.2. TABLE l GEOMETRICAL AND MATERIAL DATA Notation Case I Case II

a/h 100 200

Exterior layers- steel

Middle layer - aluminium EAa=0.7*105 MPa, v=0.3 ao~a=123MPa

Es=2* 105 MPa, v=0.3 ~oS=384 MPa

The results of calculations have been drawn in diagrams showing the dimensionless relations: 9 o*=Oav/OoS versus U : = ( U d a )/(Oos/Es); 9 r,*=xa,c/Xos versus U~*=(Udb )/(XoS/Gs); 9 M*=M,v/(o0bh2/4) versus Ub*=(2Ub/a )/(OoS/Es). In figures 3-4 the curves showing the relations x*=f(Us *) and o*=f(Uc *) are presented for plates denoted as 'T'. For plates subjected to simultaneous compression and shear (AUc*/AUs*=I) the decrease of average stress values relative to the values of these stresses in case of single loads is observed in the elasto-plastic range. It can be noted that in this range (e.g. for Ur the ratio of average compressive stress at combined loading to the average compressive stress at pure compression is almost constant for plates of different ratio g/h. The same remark concerns the ratio of shear stress values. 08, (3"

17

Q7

L I =' ~ L , ~

06

--'0"--2;

~fjJl~r~,ip, v . . ~

(12

v

v

v

v v v v ~

~m U : U~

0

Q5

1

1.5

2

25

Figure 3 Load-displacement curves for three layered plates (g/h=0.4) 1 c - pure compression, 1s - pure shear, 2 c, 2 s - simultaneous compression and shear; AUc*/AUs*=I,

475 o"I:* 0.7 0.6

9

lc

~ls

0.5

---e--2c

0.4

u.

.

.

. nunu =mum ~

. u

0.3

I

2s

-

3c

-n--3s ---0--4c

0.1

---e--4s

' LIe* Us*

0

0.5

1

1.5

2

2.5

Figure 4: Load-displacement curves for plates subject to simultaneous compression and shear 1 - g/h=O, 2 - g/h=O.4, 3 - g/h=0.6, 4 - g/h =1.0 (c - compression, s - shear), AUc*/AUs*=I, The results obtained for plates'IF' subjected to simultaneous in-plane bending and shear are shown in figures 5 and 6.

M*

.r

0.6

o,

,

_~0.3

"

lb

i-"(~ ls i~2b !--O--2s ~-n--3b !--43--3s

- -~'~'~ .D

i i

"

4b

---/~ 4s ~llf

Us*

0 0

0.5

1

1.5

2

2.5

Figure 5: Load - displacement curves for square plates "II" subjected to combined bending and shear b - bending, s - shear, AU,/AUb=I 9g/h=0, 2- g/h=0.4, 3 - g/h=0.6, 4 - g/h=l

476 M* .r

0.6 0.5

,0

1

-Ill

0.4

.4)

l-e-ob'l

0.3 0.2 Ub* U~

0.1 |

0

0.5

1

1.5

2

2.5

Figure 6: Load - displacement curves for square plates "II" subjected to combined bending and shear c - combined, p - pure, b - bending, s - shear, AUdAUb=I, g/h=0.4

FINAL COMMENTS The results of preliminary calculations seem to confirm that the presented method enables to determine the postbuckling behaviour of the multilayered plates in the elastic and elasto-plastic range when subject to the combined loading. It should be remarked that some problems that appear in laminated plates (such as e.g. delamination, cracking) might provoke different behaviour of a plate subject to in-plane loads. It is also known that for many composites the material characteristics obtained in tensile and compressive tests are not identical - in such case the Hill's yield criterion should be replaced by Tsai-Wu criterion.

References Gr~dzki R., Kowal-Michalska K.(1999). Post-buckling analysis of elasto-plastic plates basing on the Tsai-Wu criterion, Journal of Theoretical and AppfiedMechanics, 4, 37, pp. 893-908 Gr~dzki R., Kowal-Michalska K. (2000). Ultimate load of plates under simultaneous bending and shear, Proceedings of Third International Conference CIMS'2000, Imperial College Press, London Hill R.(1950) The mathematical theory of plasticity, Oxford, UK Jones R.M.(1975). Mechanics of composite materials, Int. Student Edition, McGraw-Hill, Tokyo, Japan Lekhnitskij S.G. (1947) Anisotropic plates (in Russian), Moscow, Russia Masaoka K., Okada H., Ueda Y. (1998). ,,A rectangular plate element"for ultimate strength analysis, Thin-Walled Structures, Research and Development, pp.469-476.

Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

477

AXIAL POST-BUCKLING OF THIN ORTHOTROPIC CYLINDRICAL SHELLS WITH FOAM CORE X. Huang and G. Lu School of Engineering and Science Swinburne University of Technology Hawthorn, VIC 3122, AUSTRALIA

ABSTRACT The post-buckling behaviour of orthotropic cylindrical shells subject to axial compression is investigated in the presence of an elastic foam core. The analysis is based on the non-linear straindisplacement relationship for thin shells, and the influence of the foam core is accounted for through a force analysis with the foam being considered as an elastic continuum. The total potential energy of the system including the strain energy, and the work done by axial force, is derived based on an assumed buckled configuration. The equilibrium configuration in terms of non-linear algebraic equations is obtained through minimising the total potential energy. The non-linear equilibrium equations are solved by use of the Newton-Raphson iterative method. The post-buckling loads as well as the classical buckling loads are calculated for the given buckling wave numbers along the axial and circumferental direction. Numerical results show the post-buckling loads have a better agreement with the experimental values than the classical buckling loads. The stabilizing effect of the foam core is also presented in detail.

KEYWORDS Buckling, Post-buckling, Orthotropic Cylindrical Shell, Foam Core, Potential Energy

INTRODUCTION The buckling behaviour of circular cylindrical shells has been studied extensively by many researchers, for example, Timoshenko & Gere (1961). Due to their high specific modulus and strength compared to metallic material, fiber-reinforced composite shells are used in various fields of modern engineering, such as wings and fuselages on commercial aircraft. In these shell structures, loss of stability is of primary concern (Mao & Williams (1998)). An associated problem that has assumed considerable importance is that of a laminated cylindrical shell filled with a material whose Young's modulus is low in comparison to that of the shell material. Recently, Harte et. al. (2000) studied the energy absorption of foam-filled circular tubes with braded composite wall, because this kind of the structure provides high crashworthiness when used as an energy dissipating device, such as the sub-floor assembly of helicopter cabins.

478 There are many studies about the classical buckling, post-buckling theories and experiments of an isotropic shell with a low modulus core (Seide (1962), Almroth & Brush (1963), Goree & Nash (1962), Karam & Gibson (1995)). All of the investigations show the stabilizing effect of the elastic core. But as in the case of a hollow shell, experimental values are much lower than the classical buckling loads since thin-walled cylinders are sensitive to unavoidable initial imperfection. And the post-buckling loads have a better agreement with them. Esslinger et. al. suggested that for design purposes the post-buckling load should be taken as the lower limit of the buckling load. In present paper, post-buckling behaviour of a composite circular cylindrical shell with foam core subjected to axial compression is presented. The equilibrium configuration in terms of non-linear algebraic equations is derived through minimization of the total potential energy. These equations are solved using the Newton-Raphson iterative method. A comparison with available test results indicates that the non-linear post-buckling load gives a better assessment of the strength of the shell with soft foam core than the classical buckling load. The post-buckling loads of the orthotropic composite shell with various foam cores are calculated. It indicates that the stabilizing effect of the foam core is significant. Figure 1 shows a laminated circular cylindrical shell with a constant thickness h, a radius R and axial length of its middle surface L. Displacement components at middle surface are u, v, and w along the x, y and z directions respectively. It is assumed

-~ .~

| L

,// y

that the foam core has the inner radius R c and outer radius R.

\shell

Figure 1" Geometry and the coordinate system of a shell with a foam core INFLUENCE OF T H E FOAM CORE The influence of foam core is accounted for through a force analysis with the foam being considered as an elastic continuum. The pressure p on the outer interface of the cylinder core comes from two sources. Firstly, the foam swells around the circumference caused by the axial pressure, axial strain being the same for foam core and shell in pre-buckle state. Secondly, the outer pressure varies as a result of the buckle of the shell. Assuming a linear, homogeneous, isotropic foam medium and a perfect bonding between the shell and the foam core, and neglecting the influence of the wave numbers, interfacial pressure p can be expressed as l_r/2 P= l+r/2

E/v/

(_ Cro)+

Cl1(1- v/)

2

(1)

1-vf

in which non-dimensional quantity r/ is Rc/R; Ez is Young's modulus of the foam; ClI is Young's modulus of the shell in axial direction; v: is Poisson's ratio for the foam core; (- o"o) is axial stress at the end of the shell and negative sign denotes compression; w is the normal displacement of the core outer interface which has the same form as the shell. It is clear that the first term on the right hand corresponds to the pressure caused by circumferential swelling, the second term denotes the variation of the pressure due to the normal displacement of the shell. Equation (1) can be expressed in a simple form as follows

p=k:(-Oo)+k,w

Here, ky is a coefficient and ke is the "spring modulus" of the foam core.

(2)

479

GOVERNING EQUATIONS From the shell theory, the non-linear strain displacement equations at the shell middle surface are: Ou l(0w) 2 (3a)

=

0v

w

1 0w

+

(3b)

Ou Ov 0%, o%, r,~

Ox

Oy

(3c)

Ox Oy

The changes of curvature of the middle surface are kx

0 2w =- Or.z

ky = 0 2w OyZ

kxy =

02W

OxOy

(4)

Constitutive equations for orthotropic materials can be written in the form

N~y

0

A66

M~y

0

7xy

oiO,, ~ ) D66

k~y

where the definitions of the stress resultants are as usual: h

(Nx, Ny, Nxy )= ~h_.(O'x,Cry,rxy )dz

(6a)

2 h

(Mx,My.M~)= ~h_(crx,o'y,r~y)zdz

(6b)

2

The non-linear compatibility equation is 02j-xx 026' Oy '~ + -Ox --T

027~Y (02w) 2 OxOy

02wOZw

10Zw

COxOy) - Ox~ Oy~

R Ox~

(7)

It is assumed that the normal displacement in the buckled configuration can be approximated by the expression w = f0 + fl sin a mx sin ,B, y + fz sin z a mx (8) where, a m = m x / L and ,B. = n/R. m and n are wave numbers along the axial and circumferential direction, respectively. Introducing the Airy stress function 9 such that 02~

N~ = Oy--2-

02t~

Ny = Oxz

02(~

N~y =- OxOy

(9)

The function 9 can be expressed in terms of the normal displacement w by use of the compatibility equation and constitutive equations, as follow, 1

~ = g l c o s 2 a m X + gacos2fl.y+ gasinamxsinfl.y+ g4sin3a,,,xsinfl,,y--2o'ohy

a

1

+-~Rqx 2

(10)

where, q is the average internal pressure which will be determined by the periodic condition of the shell. The coefficients in Equation (10) are 2 2

Rfl.f, - 4 f 2 gl=

32~'z2Ra ~

(lla)

480 2

amfi2 g2

(llb)

32~-11,fl2

R [A,1,6': + (2A',2 + A66)a2fl: + Az2a: 2

]

2

a=fl" fi f 2

(lid)

The periodic condition is that the tangential displacement must be a periodic function in y, that is

-~dy =0

(12)

After integrating the above equation along the axial direction, the average internal pressure can be expressed as

1 (-~12~roh_ 1 1 2 2 1 ] q= A22R \ "~fo +-~fl,,f, - ' ~ f 2 . .

(13)

The potential energy for the shell subjected to axial compression and internal pressure caused by the core can be written: = ~ ~27rR

FI

l ~[(Nxex + Nyey + Nxy?'xy)+ (Mxkx + Myky + 2Mxyk~y )~lxdy - h 2 (t:)

+ k f (- O'o ) ~ w d x d y + -~1 ke (F)

(- Cro)dY ~L~ - ~ Ox

~wZdxdy

(14)

(F)

Finally, the coefficients f0, fl and f2 would be determined through minimization of the total potential energy. The following non-linear equations are obtained C010"0 + C02fo + C03fl2 + C04f2 =0

(15a)

CllgrO)l + C12]011 + C13]1 + C14713 + C157172 + C16)17 2 -'0

(15b)

C21cro + C220"0f2 + C23]0 +C~f~ 2 +C25f1212 + C26.f2 =0

(15c)

where, C 0 are constants, fo, 3~ and s are non-dimensional parameters defined as

fo = fo/h )1 = fl/h f2= f2/h (16) Numerical solutions of these non-linear equations are obtained by use of the Newton-Raphson iterative method. The classical buckling load can also be obtained from these equations by neglecting non-linear terms. Finally, 4 am + Olla =, + 2(D12 + 066)amP,,22 + O22Pn4+ ke 2m 4 R [Allfln +(2A',2+A'66 )amfln z 2 + A22gtm] , (17)

o" =

hct~ + kzRfl~

For the special case of isotropy, the constitutive relations simplify through the relevant reduction of the material compliances All = 7~22= l/Eh -XI2 = -v/Eh ~ = 2(1 + v)/Eh (18) and stiffness D~1 = D22 = Eh3/12(1 _ v 2)

D12 = Evh3/12(1 - v 2)

D66 = Eh3/12(1 + v)

where E , v are Young's modulus and Poisson's ratio of the shell material respectively.

(19)

481 NUMERICAL RESULTS AND DISCUSSION In order to check the analysis in the previous section, the classical buckling and post-buckling load for the isotropic shell with foam core are calculated and compared with the test result in Goree & Nash (1962). The thin cylinder is characterised by L = 419.1mm, R = 76.2mm and h =0.1524mm. The Young's modulus of the shell is 199.817GPa, and Possion's ratio is 0.30. The shell was filled with the foam with E/=l.378MPa and v / = 0 . 1 . The experimental buckling load is 130.57MPa. According to the previous analysis, the calculated classical buckling load is 242.3MPa, and the postbuckling load is 99.05MPa. It shows the experimental value is between the classical buckling load and the post-buckling load. And the post-buckling load has a better agreement with the test value than classical buckling theory. It can be used for the safe design values of the shell with foam core. The present theory is now used to study post-buckling behaviour of the laminated cylindrical shells with and without foam core. The examples are the [90* /0 ~]2s laminated cylindrical shell with length

L=540mm, radius R=350mm, and total thickness h=l.2mm. Mechanical properties are EI~ = 113,000MPa, E22 = 9,000MPa, G12 = 3,820MPa, and v12 = 0.73. The Young's modulus of the foam is varied from 0 (without foam core) to 5x 10 -s Ell. The Possion's ratio of the foam is 0.1 unchangeably. Figure 2 shows the post-buckling load increases as the stiffness of the foam core increases. The stabilizing effect of the foam core is significant. Comparing the shell with the foam core of E / = 5 x 10 -s EI~ and that without a core, the post-buckling load rises about 48.2%. The numbers in the brackets denote the post-buckling wave numbers along the length and the circumference of the cylinder, respectively. It is interesting to see that both wave numbers increase as the stiffness of the foam increases. The experimental values of Bisagni (1998) are shown in Figure 2 by stars. There is a good agreement between the experimental value and the post-buckling load. In some cases, cylinders are partially filled with a compliant core of foam cylinder with a central bore hole which radius is R c . Figure 3 shows post-buckling loads of the shell with the same parameters as previous example. Only two kinds of foam are selected to plot against the ratio of the inner radius to the outer radius of the foam. The wave numbers along the length and the circumference of the shell are also indicated in the brackets. When r/is less than 0.3, the post-buckling load increases very little.

60 55 ~,

b

o//O

50

~ o ,..2

45

~ .,N

40

o

35

o

30

25

/

~

(6,13)

(5,12)

/e ('~4,If)

/ " (/3,lo) (2,8)

I

0

,

,

10

,

20

,

30

~

40

t

J

50

Er/E l l( • 10"6) Figure 2: Post-buckling load of orthotropic cylindrical shell with various foam cores

482

9......................... .(6, 13) .............

. ...... Q .

....... o(5, 12)

b

b

...

........(4, 11) 0

~.

.,,N

--..--

o

-a (2, 8)

E ~ 1.0 x l O 4 E t l

........... ........Ef--5.0xl04Ell

o 25

J0 0.

'

12 0.

,

I 0.4

.

016

,

018

'

1

i0

'

,7= R./R

Figure 3" Effect of r/on post-buckling load c r

CONCLUSION The post-buckling behaviour of orthotropic shells with foam core is studied. A comparison with available test results indicates that post-buckling loads have better agreement with the tests than classical buckling loads. The stabilizing effect of the foam core is significant and increases with the stiffness of foam core. For cylinders with a compliant cylinder core, when r/ is about less than 0.3 the post-buckling load increases very little.

REFERENCES

Almroth B. O. and Brush D. O. (1963). Postbuckling Behavior of Pressure- or Core-Stabilized Cylinders under Axial Compression. AIAA Journal 1:10, 2338-2341. Bisagni C. (1998). Experimental Buckling of Thin Composite Cylinders in Compression. A/AA Journal 37"2, 276-278. Esslinger M. and Geier B. (1974). Buckling of Structures. Springer-Verlag. Goree S. and Nash A. (1962). Elastic Stability of Circular Cylindrical Shells Stabilized by a Soft Elastic Core. Expermental Mechanics 2, 142-149. Harte AM., Fleck N. A. and Ashby M. F. (2000), Energy Absorption of Foam-Filled Circular Tubes with Braided Composite Walls. European Journal of Mechanics. A/Solids 19 31-50. Karam G. N. and Gibson L. J. (1995), Elastic Buckling of Cylindrical Shells with Elastic Cores II. Experiments. International Journal of Solids Structures 32:8/9, 1285-1306. Mao R. and Williams F. W. (1998). Post-Critical Behaviour of Orthotropic Circular Cylindrical Shells under Time Dependent Axial Compression. Journal of Sound and Vibration 210:3, 307-327. Seide P. (1962). The Stability under Axial Compression and Lateral Pressure of Circular Cylindrical Shell with a Soft Elastic Core. Journal of Aerospace Science 29, 851-862. Timoshenko S. P. and Gere J. M. (1961). Theory of Elastic Stability. New York: McGraw-Hill.

Third International Conferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

483

NONLINEAR STABILITY PROBLEM OF AN ELASTIC-PLASTIC SANDWICH CYLINDRICAL SHELL UNDER COMBINED LOAD L. Jaskuta i and J. Zielnica 2 St. Czarniecki Military College, A1. Wojska Polskiego 84, 60-630 Poznafi, Poland 2 Institute of Applied Mechanics, Poznafi University of Technology, Piotrowo 3, 60-965 Poznafi, Poland

ABSTRACT The paper presents stability analysis and postbuckling equilibrium paths determination for a free-supported sandwich cylindrical shell loaded by longitudinal forces, lateral pressure, and shear. The basic assumptions in the considered problem comprise geometrical and physical nonlinearities, basing on Prandtl-Reuss incremental plastic flow theory. It is also assumed that the shell faces are of different thicknesses and the material properties of the faces are different. The material is compressible and it has the feature of stress hardening. The active processes of loading in the shell material are considered, and the principle of virtual work is applied to derive the stability equations.

KEYWORDS Stability, shell theory, sandwich shells, elastic-plastic shells, theory of plasticity, constitutive relations, strain energy, Ritz method,

INTRODUCTION

Thin-walled multilayered shells are widely used in the manufacturing of modem vehicles, cistems, tanks, and in civil engineering as well. Cylindrical shells are subjected to widely varying combinations of hydrostatic pressure and axial loads. Most commonly used are sandwich shells which consist of two external layers (faces) and one internal layer (core) which joins together the faces. Comparatively thin faces are made of materials of high stiffness (i.e. steel, aluminiurn alloys), and the core can be made of corrugated sheet of metal or plastic foam, porous rubber, cork, and similar. Such shells have good acoustic and thermical isolation, and they represent high stiffness with the possibility to transmit high loadings with small self-weight. Because the external surfaces of the sandwich shells can be smooth and free of local joints, these structures usually represent high aerodynamical properties.

484 The objective of the paper is to derive nonlinear stability equations for the mentioned sandwich cylindrical shell in the elastic-plastic state of stress (see Figure 1), and the solution by Ritz method. A similar elasticplastic approach, but related to conical shell, was presented by Zielnica (1981), and in paper by Zielnica (1987) one can find the linear and nonlinear analyses of elastic-plastic shells. Bushnell in paper (1982) presented a review of the works on numerical methods in the stability of elastic-plastic shells. The subject of the other works written by Zielnica (1997,1998) is focussed on stability of layered shells in the elasticplastic state of stress. In this analysis a sandwich structure is considered which is featured by small weight and high strength and stiffness. The shell consists basically of three principal elements: two faces (external and internal), and the core (so called middle layer). It is assumed that the shell is loaded by longitudinal forces acting along shell's generatrix, by external pressure perpendicular to lateral surface, and by a moment of twist. It is accepted that that faces are of different thicknesses and these are made of isotropic compressible material with stress hardening, and the core resists transverse shear only. We also assume that stress distribution in the faces follows Kirchhoff-Love (K-L) hypotheses according which arbitrary straight line perpendicular to shell middle surface before deformation remains straight and perpendicular to shell middle surface after deformation. Also, active processes of unloading are considered in elastic-plastic state of stress. We accept for the core the assumption which regards the influence of shear in the plane perpendicular to shell middle surface. Taking in mind the relations of nonlinear theory the nonlinear equilibrium paths are analysed, and upper and lower critical loads, and also instability regions are determined for the considered sandwich cylindrical shell under three-parametrical loads.

Figure 1 Structure of considered sandwich cylindrical shell and loadings

CONSTITUTIVE RELATIONS OF THE PLASTIC FLOW THEORY We assume that effective stresses in the shell can exceed the yield limit of shell's material. Thus, the constitutive relations will follow up the Prandtl-Reuss plastic flow theory. This theory assumes that

485 stresses and stress increments are related with strain increments or strain velocities by a physical plasticity law which is associated flow rule and yield condition of the type H-M-H (Huber-Mises-Hencky). These are generalized on stress hardening case. The plastic flow theory allows for more simple and unilateral decription both the plasticity process, and passive processes in particular layers of the cross section of the shell. On the basis of the mentioned flow law, and yield condition, the Prandtl-Reuss equations can be presented in the following form

de~ =~-~ do"0 - $ u 1---~vdcr,, + p

1 O'm = ~ O'U

(1)

1 des

d,~, -

2

0"~

Unknown parameter L in Eqs. 1 we determine using the plastic strain energy ratio ~

-

P = a ~ S z ~ = o'~

&r,

(2)

Thus, we get

(3) Where Et, and Es are tangent and secant moduli of stress hardening, respectively, taken directly from stress-strain curve. We take the forces and moments developed in the shell faces during stability loss in the form: -h

h+h~

-h-h t

h

I

-h

dz h+h~

(4)

~SM~,#= ~ 6a~#zdz+ I&r~#zdz -h-h,

where:h= 89

h

hi = tl, t2

We express stresses and strains from Eqs. 1, using Eqn. 3, by strains and substitute them into Eqn. 4. Then, performing integration we obtain the following governing constitutive relations

8M, = -D,,SX,- D,28Z2 + D138Z,2

(5)

Where Bij, and Dij are coefficients of the local stiffness matrix. The specific expressions for these coefficients can be found in paper written by Zielnica (1981). It is to be note that the coefficients depend not only on material constants and geometrical properties ofthe shell but they depends also on prebuckling stress state in the shell developed by the external load acting the shell.

486 G E O M E T R I C A L RELATIONS Once Kirchhoff-Love hypotheses are applied to the faces and shear stresses are considered in the core, we follow the so called 'broken line' approach (see Figure 2) in the description of the displacements of deformed shell. The relations between displacement vector components u, v, w of arbitrary point in the shell and the displacements of the points situated on middle surfaces of the faces are given by the following expressions when using the 'broken line' hypothesis: W - -+ _

Wi '

u•

u~ + [z_+ 89(c + t,)]w~,,

(6)

v -+= v~ + [z _+ 89(c + tO] We,y

Where the following notation is used: i =1 - upper face, 0,5e < z < 0,5c + t~ i =2 - lower face, - (0,5c + t2) < z < - 0,5c Subscripts ,,+" and ,,-" denote upper and lower face, respectively. Ow

~ = Ox

-z+0.5(c+h)

U~ "2

~:' - J~-" ~-~[z+O.5(e+t,)]

B~_~ .....

I

i

wl

!

.i

Hi

X

'~'~"-.

[0.5(u:uz)-0.5tt,+tg~xl~ . ~ , - u 2 ) - 0 . 5 u2)-0.5 (tl+t2)___...._.....~ (t,+t2)~_}]~~

_~

A'--~,I~

i Uc

0.5(,

" 1:---0 0.50 +tz)

U

A

I! t

z

U1- U2

O~r 2 ~0W

Figure 2 Geometrical configuration of deformed shell

We also introduce the relations between the displacements of the faces which are called the reduced displacements:

487

Ua "- ~ (Ul + U2)

Va -- 89(V1 + V2) Vp = 89(Vl- V2)

Up = 89(Ul- U2)

(7)

If we assume that the core doesn't deform in the z direction and the ratio h/R <<1, then we have the following: Wc = W" = W+ = W

1 1 -t2)w, x + 2Z[ucL p +-~(tl 1 +t2)w, ~] u C =u. +-~(t

=vo

1

2Z[v

1

+Tl '

(8)

]

Nonlinear geometrical relations for particular layers are accepted in the form of

•

•

•

•

W

~'~xy= v'x• + u.y• + w,xw,y

• = w,y + V•.z Yyz

7'~ = w,x + U,~ •

Z~ = w,~

z,•

z;

w +w..

=

l(w ~

~R(U ,

(9)

-

Where zx, ev are the strains along coordinates x, and y, )'~y, yyz, ?'xz are shear strains in the shell, Zx, Zy, Zxy are changes in curvature for the considered shell.

STABILITY EQUATIONS FOR TIlE CYLINDRICAL SHELL In the strain energy approach, which is applied generally in this work, we use the virtual work principle to derive the stability equations for the shell. If the shell is given a small virtual displacement from its equilibrium position we can compare the work done by external forces acting the shell 8L and the change in potential energy of the shell 6W. Thus, we get the following variational equation of equilibrium of the shell in its deformed state. 6Up = 6 ( W - L)=O

(10)

Here W is strain energy in the shell in its deformed state expressed by strain state components, and L is potential of the external loads. It is worth to note that total strain energy of internal forces for the considered shell is equal to the sum of strain energy in particular layers consisting the shell,

W= W+ + W + Wc. The work done by forces and moments developed during stability loss in the faces is given by the following relation: •

•

6 g 7• = 6N~g~ + oW2 g22 +b~•

+ 6M~z~ + 6M;z2 + 26H•

•

In the above relations the bending strain energy is included which is often disregarded in stability

(11)

488 problems of multilayered shells. In order to obtain the final form of the strain energy of the faces it is necessary to substitute Eqs. 5 into Eqn. 11 and perform variation and integration as well. Strain energy of the core, which deforms within elastic limit, is expressed by the formula

l!I 1-v~i Ec / t:=+e~+2vcexctry~+ 2 2 1-vc 2 / + G ~('9"2)1 2 Y ,o,c ~c. + Y ,,:c d V

Wc=-~

(12)

Virtual increment 6L of the work done by external forces is as follows: ab

8 L = r A + ~ ~qfiwdxdy

(13)

00

Here 6_,4denotes virtual work of the extemal forces applied at the contour of the shell, and the second term is the virtual work of the surface pressure q. If we substitute the sum of expressions 11, 12, and 13 into Eqn. 10 we get the final form of strain energy 6Up of the considered shell. We solve the considered problem by Ritz method; using the forementioned equation of the shell strain energy 6Up including geometrical relations 6, 7, and 9. The buckling behaviour of pressure loaded cylinders is strongly dependent on the exact nature of the end support conditions. This has recently been shown to have contributed to what is now considered to be an erroneous view that the buckling behaviour of these shells can be reliably predicted from the classical critical load analysis. But to simplify the present developments a 'classical simple support' condition OU

02 W

Ox

-07

~=v=w=

=0

(14)

(the so-called SS3 condition) will be considered. Although unrepresentative of many shell support conditions, such boundary conditions do provide an approximation of the behaviour of cylinders between adjacent slender ring stiffeners or diaphragms. Other boundary conditions will clearly result in important quantitative difference in the results, but are believed not to change significantly the important qualitative descriptions provided by the parameters emerging from the present analysis. The above given boundary conditions are exactly satisfied by a critical deformation of the form of the displacement vector components w(x, y) = A, sin kx sin(py + ~ )

u~ (x,y) = A2 cos kxsin(py + al)x ) up(x,y)= A 3 coskxsin(py + a2yx)

(15)

v~ (x, y)= A4 sin kx cos(py + a3~) v , ( x , y ) = A 5 sin kxcos(py + a,?x)

Where k-

m]/"

l n

p = -R

O<x
7' - is a parameter equal to tangent of angle of inclination ofbucklewave with respect to shell generatrix. Parameters A 1, A2, A3, A4, A5 in Eqs. 15 are to be denoted in the process of solving. Quantities m, and n are

489 parameters, representing the number of halve bucklewaves in longitudinal and circumferential directions, respectively. These have to be determined from the condition of minimum of critical load. Auxiliary constant multipliers ai in Eqs. 15 are used to control the influence of terms ),x in base displacement functions u~, ul~, v~, v~ on the final results; these take the values 0, or 1, respectively. If we substitute base functions 15 into general strain energy expression Up, and follow Ritz method

OUp - 0

(16)

a4 what denotes the differentiation of the strain energy with respect to parameters Ai (i = 1, 2, 3, 4, 5), we get a nonhomogeneous and nonlinear set of five algebraic equations with unknown parameters Ai, which are the fundamental stability equations of the considered elastic-plastic sandwich cylindrical shell, in the following form:

a)~A, + a,2A 2 + al3A 3 + ai4A 4 + alsA ~ =bllA 3 + bl2A ~ + b13A1A2 + b,4A, A 3 + + bl5 A1A4 + b16A1As + b17

a21A ~ + a22A2 + a23A3 + a24A4 + a25A5 =b21A~ + b22

(17)

a31A ! + a32A2 + a33A3 + a34A4 + a35A5 = b3~A~ + b32 a41A 1 + a42A2 + a43A3 + a44A4 + a45A5 =b4tA~ + b42 asia ~ + a52A2 + as3A3 + as4A 4 + assA 5 = b5tA ~ + b52 Coefficients b17, b22, b32, b42, and b52 include the surface loading q, longitudinal force Nx, and shear force N• The other coefficients in Eqs. 17 depend exclusively on material and geometrical parameters of the shell, and also on parameters m, n, and y.

SOLUTION OF G E O M E T R I C A L L Y NONLINEAR PROBLEM Set of nonlinear equations 17 relates parameters A j, Az A3, A4, As, m, n, and y with external loadings Nx, Nxy. In general solution we shall construct the equilibrium path, i.e. the relation between external loadings, and normal deflection w. The loading is represented by external pressure q, and normal deflection w is represented by coefficient A l (see the form of base functions 15). Longitudinal and shear loadings will be related with q by dimensionless parameters ir and u that will be defined further in the text. So, in the solution of the considered problem we relate load q with parameters Al, m, n, and y, i.e. we represent the load q as the function of deflection w (15)1, number of longitudinal and circumferential buckle half-waves (m, n), and parameter y. Thus, we express parametersA2,A3,A4, andAsin Eqs. 172-175 by AI. Then, we substitute the obtained relations into Eq.17~ to obtain an algebraical equation of the third order with respect to parameter A ~. When we substitute parameters A2, A3, A4, As, which are determined from Eqs. 172-175, into Eq. 17~, and introduce the following dimensionless parameters to, and o:

qR

u-

qR

(18)

then Eq. 171 after rearranging becomes

q(m,n,r) =

e3A1+ e2A~ - e!A~

(19)

490 The coefficients ei(i=l ..... 8) in the above equation depend on geometrical dimensions of the shell, elastic and plastic material constants of the material, and on parameters m, n, ?'. Subsequent step of the solution process is elaboration of special numerical algorithm where the basis is stability equation 19. Also, the objective of the work is the analysis of the influence of geometrical and material parameters on critical loads. The nature of the considered problem points out the following general procedure of the solution aiming the determination of the upper and lower critical loads: i) We accept geometrical and material constants of the shell, ii) We assume values of coefficients ~:, and o , iii) We accept a series of integer values of parameters m, and n, iv) For fixed values of m, and n we accept, starting from zero, a series of consecutive values for parameter A 1, v) Maximum deflection w and loads q, Nx, and Nxy are calculated for each value of parameter A 1, vi) We get a three-parametrical family of curves q (w, m, n, y), Nx (w, m, n, y), or N~y(w, m, n, y) in coordinate systems (q, w), (N~, w), or (N~y, w), vii) From the family of curves we chose the points of minimum values of q, N~, or N~, at the given values of variable w we find a curve that is the solution we search for, viii) Local maximum and minimum of the respective curve show upper and lower, respectively. In order to determine numerical values of the solution the elaborated algorithm was programmed in Fortran95 computer code. The program makes possible to determine the equilibrium paths and critical loads for cylindrical sandwich shells being in elastic, plastic, or elastic-plastic state of stress. Thus, transversal pressure q, longitudinal load Nx, and shear load Nxy can be determined as a function of deflection w of the shell. Solution algorithm, and program of numerical calculation take into consideration a specific feature of elastic-plastic stability of shells. Stability equation 15 is a transcendental function, where the coefficients of local stiffness matrices depend on parameters of external loadings. So, we have to use some iterative techniques to build up equilibrium paths, and to determine critical loads of the considered sandwich cylindrical shell.

REFERENCES Bushnell D. (1982). Plastic buckling of various shells. Transactions ASME, Journ. of Pressure Vessel Technology, 104, 5. Zielnica J. (1981). Elastic plastic buckling of sandwich conical shells under axial compression. Bull. Acad. Pol. Sci., XXIX, 239-251. Zielnica J. (1987). Stateczno~6 powtok stozkowych w zakresie spr~.ysto-plastycznym. Poznan University of Technology, Reports, 182, 1-271. Zielnica J. (1997). Stability of an elastic-plastic sandwich cylindrical shell under combined load. Stability of Structures, VIIISymposium-Proceedings, L6d~-Zakopane 22-26.09.1997, 289-304. Zielnica J. (1997). Inelastic large deformation buckling of a sandwich conical shell - Strain energy formulation and analysis. Proceedings of the XIII Polish Conference "Computer Methods in Mechanics", PCCMM '97, Mar 5-8, 4, 1433-1440. Zielnica J. (1998). Stability analysis of elastic-plastic bilayered conical shells (in Polish). Extended Abstract of papers of the Vf h Conference SSTA ,,Shell Structures- Theory and Applications", Gdahsk, Oct. 10-14, 293-294.

Third International Conference on Thin-Walled Structures J. Zarag, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

491

B U C K L I N G A N A L Y S I S OF M U L T I L A Y E R E D ANGLE-PLY COMPOSITE PLATES H. Matsunaga Department of Architecture, Setsunan University Neyagawa, Osaka 572-8508, JAPAN

ABSTRACT Buckling stresses of angle-ply laminated composite plates are analyzed by taking into account the effects of transverse shear and normal deformations. By using the method of power series expansion of displacement components, a set of fundamental equations of a two-dimensional higher-order theory for rectangular laminates subjected to in-plane stresses is derived through the principle of virtual displacement. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported laminated plate. In order to assure the accuracy of the present theory, convergence properties of the lowest buckling stress are examined in detail. The effects of fiber orientation angles on the critical buckling stress are studied for symmetric and antisymmetric angle-ply laminated composite plates. The present global higher-order approximate theories can predict the buckling stresses of multilayered angle-ply composite laminates accurately within small number of unknowns which is not dependent on the number of layers. KEYWORDS higher-order theory, angle-ply, composite, laminated plate, buckling, orthotropic material INTRODUCTION While a considerable effort has been expended in the analysis of buckling problems of cross-ply laminated composite plates, very few research investigations on angle-ply laminated composite plates can be found in the survey papers (for example, Leissa 1987, Noor and Burton 1989, Reddy 1990). A very flexible design of the structure at the lamina level can be developed in multilayered angle-ply plates by changing the lamination angle. The most appropriate plate stiffness may be designed by selecting suitable values of angles of laminate reinforcements. The mechanical behaviors of angle-ply laminated composite plates are strongly dependent on the degree of orthotropy of individual layers with arbitrary angles of reinforcements, the low ratio of transverse shear modulus to the in-plane modulus and the stacking sequence of laminates. However, in spite of the importance for technical applications, only a few investigations on stability problems can be found for the case of multilayered angle-ply plates. Within the limitation of the classical linear theory, several papers dealing with the buckling problems of angle-ply laminated composite plates has been published.

492 The classical two-dimensional laminated plate theory based on the Kirchhoff hypothesis yields sufficiently accurate results only for thin laminated composite plates. For moderately thick laminated plates with relatively soft transverse shear modulus and for highly anisotropic composites, classical plate theory leads to a significant over-prediction of the buckling stresses. This inaccuracy is due to neglecting the effects of transverse shear strains and transverse normal strain in the laminate. In order to take into account the effects of low ratio of transverse shear modulus to the in-plane modulus, a number of first-order shear deformation theories have been developed. However, the inherent deficiency of Mindlin-type shear deformation plate theory is the presence of a correction coefficient, which is introduced to correct the contradictory shear stress distribution over the thickness of plates and cannnot be found from within the assumptions of the theory itself. The shear correction coefficient should be adjusted when studying higher mode buckling behavior of plates, because the shear strain distribution may differ significantly from the parabolic form of the static shear strain distribution. It has been shown that the classical and Mindlin-type firstorder shear deformation theories are inadequate to predict the accurate solutions of laminated composite plates. In order to obtain the accurate predictions of the gross response characteristics, such as buckling stresses and also the distributions of displacements and stresses at the ply level, a number of contributions based on the three-dimensional elasticity theory have been made to analyze angleply laminated composite plates. However, accurate solutions based on the three-dimensional elasticity theory are often computationally expensive. For one of the best alternatives to the three-dimensional elasticity solutions, the layer-wise theories and individual layer theories have been presented to obtain more accurate information of cross-ply laminated composite plates. These theories require numerous unknowns for multilayered plates and are often computationally expensive to obtain the accurate solutions. The total number of unknowns depends on the number of layers in a laminate and will increase dramatically as the number of layers increases. Without requiring the specification of a shear correction coefficient in the Mindlin-type firstorder shear deformation plate theory, various higher-order plate theories have been developed. A number of single-layer (global) higher-order plate theories that include the effects of transverse shear deformations have been published in the literature. Although various models of higherorder displacement fields have been considered, most of these theories are the third-order theories in which the in-plane displacements are assumed to be a cubic expression of the thickness coordinate and the out-of-plane displacement to be a quadratic expression at most. For thick isotropic plates, a two-dimensional higher-order theory has been developed and has been applied to the statics and dymamics of a very thick plate by Matsunaga (1992, 1997a, 1997b). Natural frequencies and buckling stresses of thick isotropic plates subjected to in-plane stresses have been analyzed by using the approximate two-dimensional higher-order theories. Remarkable effects of transverse shear and normal deformations have been predicted in the results. Recently, Matsunaga (2000) proposed a globa higher-order theory for the vibration and buckling problems of cross-ply laminated composite plates. The modal transverse stresses were obtained by integrating the threedimensional equations of motion in the thickness direction starting from the bottom surface of the laminates. However, general higher-order theories of plates which take into account the complete effects of transverse shear and normal deformations have not been investigated in the stability problems of angle-ply multilayered composite plates. This paper presents a global higher-order theory for analyzing buckling stresses of angle-ply laminated composite plates. The complete effects of transverse shear and normal deformations can be taken into account within the approximate two-dimensional theory. Several sets of the governing

493 equations of truncated approximate theories are applied to the analysis of stability problems of a simply supported multilayered elastic plate subjected to in-plane stresses. Convergence properties of the present numerical solutions are shown to be accurate for the buckling stresses with respect to the order of approximate theories. The present results obtained by various sets of approximate theories are considered to be accurate enough for symmetric and antisymmetric angle-ply laminated composite plates. Two-dimensional global higher-order theory in the present paper can predict the buckling stresses of simply supported multilayered angle-ply plates accurately within small number of unknowns. FUNDAMENTAL

EQUATIONS

Consider an angle-ply laminated composite plate of uniform thickness h, having a rectangular plan a • b. Introducing the Cartesian reference co-ordinate xi(i = 1, 2, 3) on the middle plane of a plate of uniform thickness h, the displacement components in a plate are expressed as v,~ = v,~(x~),

v3 -- v3(xi).

(1)

The displacement components may be expanded into power series of the thickness co-ordinate x3 as follows: oo (~) ~ ~3) vo =

= E

n=0

(2)

n=0

where n = 0, 1, 2 , . . . , oc. Greek lower case subscripts are assumed to range over the integers 1, 2. Based on this expression of the displacement components, a set of the linear fundamental equations of a two-dimensional higher-order plate theory can be summarized in the following.

S t r a i n - d i s p l a c e m e n t Relations Strain components may also be expanded as follows: 7aZ ----

~(~)

n

~/aZ X3,

n=0

~(~)

~/a3 --

n

"ya3X3 ,

733 "-

n=O

~(~)

~

(3)

733 x3.

n=O

Strain-displacement relations can be written as (n)

1 ,(~)

(~) ,

"Y~Z = ~Lv~,z + v,,o),

(~) 1 (~+1) , (~) %3 = ~{(n + 1) v~ -t-v3,~},

(n) (n+l) 3'33 = (n + 1) v3 ,

(4)

where a comma denotes partial differentiation with respect to the co-ordinate subscripts that follow.

Equations o f Equilibrium Introducing stress components s~z, s~3 and 833, the principle of virtual displacement is applied to derive the equations of equilibrium and natural boundary conditions of a plate. In order to treat stability problems of a plate subjected to in-plane stresses s~ which distribute uniformly in xl- and x2-directions, respectively, additional works due to these stresses which are assumed to remain unchanged during buckling are taken into consideration. The boundary conditions are assumed to be traction free on the top and bottom surfaces of the plate. The principle for the present problems may be expressed as follows: (Sa~67a~ + 28aa67a3 ~- 8336733 ~- Sa~(YA,a6VA,fl -4- V3,a

--

(5)

494 where dV denotes the volume element. The in-plane stresses s~ is assumed to be uniform in the thickness direction. By performing the variation as indicated in eqn (5), the equations of equilibrium are obtained as follows: (n)

N,~,~

(n-l)

-

oo ( n + m + l )

n Q~ + ~

o

(m)

s.~

(n)

vz,.~=0,

oo ( n + m + l )

(n-l)

Q.,.-n

o

T +Y~

m=0

s.~

(m) v3,~z=0,

(6)

m-0

where n, m = 0,1, 2 , . . . , cc and (n-t-re+l) 80f~

K

h-+m+a

S0a~k=l E ' ~k+ 172+

:

h~+m+x --

m + 1

'

(7)

where s~ and hk denote the initial in-plane stress and thickness co-ordinate of the lower side of kth layer, respectively, and K denotes the total number of layers in the laminates. The stress resultants are defined as follows: (n)

(n)

(n),

(N~z, Q,~, "1)

:

K E z (k) o(k) (Safe, o a 3 ,

k=l

h n + l __ h ~ + l o(k)'~'~'k+l ~ ]

(s)

n+l

~(k) where oa/~, o(k) s(~ka ) and 033 are stress components of kth layer.

Boundary Conditions The boundary conditions are assumed to be traction free on the top and bottom surfaces of the plate. For equations of boundary conditions along the boundaries on the middle plane, the following quantities: (n)

(n)

r162( n + m + l )

v:,,xl,

v3

(n} u~[Qz+ ~

or

m-O

(n+m+l)

s o~

va,~)j

(9)

m-O

are to be prescribed.

Constitutive Relations For elastic and orthotropic materials, the constitutive relations of the kth layer of angle-ply laminated plates of orthotropic materials can be written in the following expression: o(k)

Oc~~ =

~.(k) n(k) l-)aflAv~Au -{- /_2aB33~/33,

S~3)

~,(k)

-- /-)a3~3~'A3,

o(k) -- /-)33Avery n(k)

~

,.(k)

-~- /-)3333'~33,

(10)

where D~)~s (p, q, r, s = 1, 2, 3) of an individual layer axe the material coefficients transformed by the angle t9 from the Cartesian material co-ordinate (for example, Jones 1975). 0 is the angle between the principal fiber direction and the x~-axis in the individual layers. Stress resultants can be derived from eqns (8) and (10) and eqns (3) and (4) in terms of the expanded displacement components. The equations of equilibrium (6) can also be expressed in terms of the expanded displacement components. M t h Order Approximate Theory Since the fundamental equations mentioned above are complicated, approximate theories of various orders may be considered for the present problem. A set of the following combination of M t h (M > 1) order approximate equations is proposed: 2M-1 m=0

(vm2 ,~

2M-2 m-0

(m) m

(11)

495

where m = 0,1, 2, 3, .... The total number of the unknown displacement components is (6M - 1) which is not dependent on the number of layers in a laminate.

NAVIER

SOLUTION

FOR SIMPLY SUPPORTED

PLATE

In the following analysis, the Cartesian reference co-ordinate x = xl, y = x2 and z = x3 and displacement components u = vl, v = v2 and w = v3 are followed. T h e following combination of the uniaxial (~ = 0) and biaxial (~ # 0) in-plane stresses is taken into consideration: 0 - - ~ S x0x 8yy

and

0 "-- 8y0x 8xy

=0,

(12)

where ~ is ratio of in-plane stresses in y- and x- directions. The Navier approach can be used to find the solution for general angle-ply, simply supported laminated plates. For a simply supported rectangular laminated plate, the boundary conditions on the middle plane (z = 0) can be expressed on the x-constant edges, u=0,

v,x = 0 ,

w=0

(13)

w = 0.

(14)

and on the y-constant edges, u,y = 0,

v = 0,

For unsymmetric response of angle-ply laminated plates, displacements are decomposed into symmetric and antisymmetric components in the thickness direction.

u = ~((2u 9 1)z 2~-1 + i--1

(2u2) z2i-2),

?j--- ~ ( ( 2 v 1)z2i-1- +_ (2/~2) Z2i-2 ) i--1

z 2i-3, (15) i--2 where i - (1), 2, 3 , . . . , M. The expanded displacement components can be expanded in the following double Fourier series: w = ~

z 2 i - 2 nu ~

i-1

--

Urs COS r=0 s=0

=/_.,/_., ~ s sin r=0 s=O

rTr x

sin

(2~-2) ~ ~ (2~-2) rTrx u = ~8 sin~cos r=0 s=0 a

sTry

b"

a cos a

9 b

= A_, A_, wrs sin ~ sin ~ r=0s=0 a b '

'

v

= A., A., v~8 cos - r=O s=O

sTry

b sin - -

a

~D = Z., Z_, ~D~s cos ~ cos a r=0 s=O

(16)

b

b '

where r, s = 0,1, 2, 3 , . . . , oo (r = s :/: 0). The tilde (-) and bar (-) refer to symmetric and antisymmetric displacement components with respect to the middle plane. The equations of equilibrium are rewritten in terms of the generalized displacement components (~) (~) (n) (n) (n) (~) ~rs, ~rs, zDrs of symmetric type and ~ s , ~8, ~ 8 of antisymmetric type. For simply supported

496 angle-ply laminated plates, the total number of the unknown generalized displacement components is (6M - 1) which is not dependent on the number of layers in a laminate. The dimensionless initial in-plane (buckling) stress are defined as follows: A

EIGENVALUE PROBLEM

--

oo /~(1) oxxl~.~2

FOR BUCKLING

(17)

9

PROBLEMS

The equations of equilibrium interms of expanded displacement components can be rewritten by collecting the coefficients for the generalized displacements of any fixed values r and s. The generalized displacement vector {U} for the M t h order approximate theory is expressed as (2M-l) (0)

~,...,

(2M-2)

~

(1)

(0)

; ~,...,

(2M-l) (2M-2)

~

(1)

; ~,...,

(0)

(2M-2) (2M-3).

~

}.

(is)

For stability problems, the stability equation can be expressed as the following eigenvalue problem: ([K] + A[S]){U} = {0},

(19)

where matrix [K] denotes the stiffness matrix and matrix IS], the geometric-stiffness matrix due to the in-plane stresses. NUMERICAL

STUDIES AND RESULTS

Numerical Examples

Buckling stress of angle-ply multilayered composite plates with simply supported edges are analyzed under uniaxial compression (a - 0). The orthotropic material constants of each layer are , (k) , v23 , (k) and,*'12 (k) and other PoisE~k) , E~k) , E(3k) , ~J13f2-(k),~J23f'~(k)and "~12~(k).Poisson's ratio are given by -13 son's ratio can be obtained by the reciprocal theorem. The material properties of the individual layers are taken to be those of typical fibrous composites as follows: El~E2 -- open, E3/E2 = 1, G,2 - G,3 -- 0.6E2, G23 - 0.5E2, v12 -- v13 -- v23 = 0.25.

(20)

The elastic and orthotropic material properties are assumed to be the same in all the layers and the fiber orientations may be different among the layers. The thickness of each layer is identical in the laminates. Both symmetric and antisymmetric laminations with respect to the middle plane are considered. The in-plane stress 8 x0x is identified in each layer. All the numerical results are shown in the dimensionless quantities. The absolute values of buckling stresses are shown in the following results. Convergence of the Lowest Buckling Stress for the Fundamental Mode

Several sets of truncated approximate equations of the present higher-order theory can be derived in accordance to the level of thickness of plates. In order to verify the present solutions the convergence properties of the lowest buckling stress of a square plate are examined in detail for the first displacement mode r = s = 1 in Table 1. It is noticed that the present results for M = 2 - 5

497 are converged accurately enough within the present order of truncated approximate theories. In the following, discussion is made only on the numerical results for M - 5 which are considered to be sufficient with respect to the accuracy of the solutions. It is shown that the present global higher-order theories can provide accurate results for buckling stresses of general angle-ply laminated composite plates. It should be pointed out that the total number of unknowns of the present approximate higher-order theories is (6M - 1) which is not dependent on the number of layers in any multilayered plates. The present theory has the advantage of predicting buckling stresses of angle-ply multilayered composite plates without increasing the unknowns involved as the number of layers increases. TABLE 1 CONVERGENCE OF THE LOWEST BUCKLING STRESS FOR r = s = 1 (h, a/b = 1, a = 0, [ + 8 ~ El~E2 = 40) Number of Layers

Present Solution a/h

0~

M = 1

M = 2

M = 3

M = 4

M = 5

2

15 30 45 15 30 45 15 30 45 15 30 45 15 30 45 15 30 45 15 30 45 15 30 45

0.7652 0.8595 0.8984 0.4172 0.4481 0.4736 0.1728 0.1709 0.1806 0.008950 0.008305 0.008759 0.6252 0.6487 0.6691 0.2935 0.2406 0.2434 0.1148 0.07624 0.07578 0.005743 0.003351 0.003301

0.6729 0.6955 0.7110 0.3807 0.3912 0.4081 0.1637 0.1600 0.1684 0.008730 0.008124 0.008583 0.5396 0.5124 0.5155 0.2609 0.2128 0.2145 0.1075 0.07177 0.07136 0.005628 0.003253 0.003205

0.6566 0.6649 0.6735 0.3717 0.3676 0.3793 0.1620 0.1552 0.1624 0.008725 0.008110 0.008566 0.5361 0.4928 0.4896 0.2597 0.2069 0.2069 0.1073 0.07102 0.07043 0.005627 0.003251 0.003203

0.6564 0.6644 0.6730 0.3714 0.3667 0.3782 0.1619 0.1549 0.1621 0.008725 0.008109 0.008565 0.5324 0.4857 0.4810 0.2588 0.2038 0.2031 0.1072 0.07060 0.06991 0.005627 0.003250 0.003202

0.6560 0.6634 0.6717 0.3713 0.3663 0.3777 0.1618 0.1549 0.1620 0.008724 0.008109 0.008565 0.5312 0.4834 0.4784 0.2584 0.2026 0.2015 0.1071 0.07043 0.06969 0.005626 0.003250 0.003202

5

10

50

2

5

10

50

Variation

of

Critical Buckling

Stress with Respect to Fiber Orientation

Angle

With varying the number of layers and the side-to-thickness ratio, Fig. 1 shows the variation of uniaxial buckling stress with respect to fiber orientation angle 0~ For symmetric and antisymmetric square laminates, the stacking sequence is selected to be [ + 0 ~ 0 ~ 0~ 0~ The critical buckling stress does not always correspond to the fundamental displacement mode r = s - 1. Higher buckling mode number (r = 2 - 5, s = 1) will give the critical buckling stress in consequence of the side-to-thickness ratio, fiber orientation angle and number of layers. It is noted that there are cusps due to the changes of critical buckling mode in Fig. 1.

498 0.6

9

~

~

,

,

.

;

9

,

,

,,,,1

,

"

,~_,...'~...

0.5

|

,

,

0.6

9

9

|

.

;

9

|

,

,

.

;

9

!

,

;

,

,

.

__e._ K = 3

~K=2

--~ r=4

0.5 r ~

0.4

~

0.4

~

o.3

~

0.2

.=_ ,.i

o 0.3 o 0.2

a/h =5

.,,=, i-,

r~

r..)

,/h =10

0.1

A ntisym metric .

0

i

.

1

,

L

.

,

o.1 Symmetric

,

i

,

i

,

,

.

i

,

0 10 20 30 40 50 60 70 80 )0 10 20 30 40 50 60 70 80 90 Fiber Orientation Angle Fiber Orientation Angle (a) Antisymmetric (b) Symmetric Fig. 1: Critical buckling stress AcT vs fiber orientation angle 0~ (a/b = 1, ~ = 0, s = 1, [ + 0 ~ 0~ .], El~E2 = 40)

CONCLUSIONS Buckling stresses of simply supported angle-ply laminated composite plates have been obtained by using a global higher-order plate theory. In order to analyze the complete effects of higherorder deformations on the buckling stresses of angle-ply laminates, various orders of the expanded approximate laminate theories have been presented. It is shown through the numerical examples that the present global higher-order theories can provide accurate results for buckling stresses of symmetric/antisymmetric angle-ply laminated composite plates. The total number of unknowns is not dependent on the number of layers in any multilayered plates. It should be pointed out that the present theory has the advantage of predicting buckling stresses of angle-ply multilayered composite plate without increasing the unknowns involved as the number of layers increases. Re ~ereneea Jones R.M. (1975). Mechanics of Composite Matemal~, McGraw-Hill Kogausha, LTD., Tokyo Leissa A.W. (1987). A review of laminated composite plate buckling. Applied Mechanics Reviews 40:5, 575-591. Matsunaga H. (1992). The application of a two-dimensional higher-order theory for the analysis of a thick elastic plate. Computers 8J Structures 45:4, 633-648. Matsunaga H. (1997a). Free vibration and stability of thick elastic plates subjected to in-plane forces. International Journal of Solids and Structures 31:22, 3113-3124. Matsunaga H. (1997b). Buckling instabilities of thick elastic plates subjected to in-plane stresses. Computers 8J Structures 62:1, 205-214. Matsunaga H. (2000). Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Composite Strucurest 48:4, 231-244. Noor A.K. and Burton W.S. (1989). Assessment of shear deformation theories for multilayered composite plates. Applied Mechanics Reviews 42:1, 1-13. Reddy J.N. (1990). A review of refined theories of laminated composite plates. Shock and Vibration Digest 22:7, 3-17.

Third International Conferenceon Thin-Walled gtructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

499

OPTIMUM DESIGN FOR LAMINATED PANEL WITH CUTOUT: THE GENETIC ALGORITHM APPROACH Zhe Li and P. W. Khong School of Mechanical and Production Engineering, Nanyang Technological University

ABSTRACT Composite laminated structures have become extremely useful because of their high strengthto-weight and stiffness-to-weight ratios. Applications of composite materials are thus increasing in areas where weight saving is most beneficial. For composite laminated with cutout, the classical plate theory suggested that the plate was generally weaken by elliptical hole in comparison to the circular one. In this paper the development and algorithm for the optimum design of the composite laminate panel with cutout are illustrated. The complete optimization procedure involves the integration of the finite element method and the expert system (Genetic Algorithm).

KEYWORDS Design optimization, buckling of composite panel, genetic algorithm, finite element method 1. I N T R O D U C T I O N Structural applications of composite materials are increasing in areas where weight saving is prime important. On the other hand when thin composite material is subjected to in-plane compressive forces, the structural instability become essential consideration in design study. Hence the instability study of composite laminated panel with cutout become an important analysis in the engineering design process. Adopting expert system, Kogiso 7 designed composite laminated plate by the genetic algorithm with memory to optimize the thickness of laminates. It is well known that cutouts or holes cause serious problems of stress concentrations due to the geometry discontinuity. In the literature search, numerous researchers have paid more attentions to the stress concentration of composite laminated structures. Most of the studies in the literature concentrate on the estimation of stresses around the cutouts. The closed form solution to stress concentration around a circular hole in an infinite orthotropic plate was first obtained by Leknitaskii 8 and Savin ~5 using the complex potential method. Gechardt 2 obtained the solution of a finite plane weakened by a circular hole using the hybrid finite element method. The similar problems were studied by

500 Ogonowski ~3 and Lin and Ko 9 using the boundary collocation approach. However these methods suffered some drawbacks, such as large data preparations, high computational time and low accuracy. 18 Recently, Xu et al. 19'2~studied extensively the stress concentration of a finite laminated plate with multiple unloaded elliptical holes using the conformal mapping, the Faber series expansion and the least squares boundary collocation. Numerous methods have been developed and are utilized for design optimization in structural engineering. Although majority of these methods assume that the design variables are continuous, this is not always true in all cases. In most practical engineering design problems, the design variables are discrete in nature. Few methods have been reported for the optimum design of discrete structural systems. TM Using sequential linear programming (SLP) with a branch and bound algorithm for discrete structural optimization has been demonstrated by John. 5 In this research work, an analytical study concerns with the buckling behaviour of composite laminated panel with cutout. The study involves the finite element analysis (FEA) and genetic algorithm (GA) in whole. A commercial FEA tool (ANSYS) is employed as the analytical tool in part of the design procedure. GA is integrated with the finite element analysis tool to form an optimum solution to the design problem.

2. THEORETICAL ANALYSIS The finite element analysis method is adopted as the theoretical analysis of the composite laminate panel with cutout. Generally the finite element analysis has been proved to be a very useful tool, and it has been successfully applied to a wide range of structural problem. 4 The purpose of finite element analysis is to numerically solve complex partial differential equations so as to mathematically describe, or predict, the physical behavior of an actual engineering system under various loading conditions. FEA allows the designer to manipulate and test the effects of all the possible design variables using computer technology rather than by the tedious alternative of actually building and testing prototype designs. For composite panel analysis, the buckling load of panel can be determined by considering the total potential energy that can be written as the sum of the bending strain energy V~ and the membrane strain energy V2, thus V = V1 + V2 (1) These strain energies can be expressed in terms of the out-of-plane deflection function w and the membrane stress resultants Nx, Ny and N~y as follows: V, = -~-,,D{(O~w_~__+._~.i_ja2w~2-2(1 - _

v(a2w~ c3Y2 ~-~,)

dy

(2)

(3) The stress resultant Nx, Ny and N~y can be calculated by the FEA. For simplicity the stress resultants in each element are obtained with respect to a unit applied displacement of the loaded edges. The following shape functions are used in the FEA tool

501

fit8 fu,t Wi

8

rt~ I

"

La3,i

b2"i Oy} b31i

(4)

u=l[u, O-sXl-tX-s-t-1)+uj (l+sXl-tXs-t-1)+uK(l+sXl+tXs+t-1)+u~O-sXl+tX-s+t-1)]

+89 O_s

O_s

)1

(5)

where N,. is shape functions given with u. u, v;, w; are the motion of node i. r is thickness coordinate and t is the thickness at node i. {a} is unit vector in s direction and {b} is unit vector in plane of element and normal. 0x.;, 0y., are the rotation of node i about vector {a}and

{b}. Along equilibrium path the total potential energy is constant and hence its first and second variation are zero. When an expression for the total potential energy is expressed in terms of sum of the loss in the potential energy of the applied loads and the strain energy stored and its first variation is put to zero, the virtual work equation was obtained which it can be used to evaluate the stiffness matrix in the usual way. The second variation of the total potential energy can then be obtained and when put to zero the resulting expression will provide the necessary critical conditions on an equilibrium path and this can be shown as: 1 52 V = ~{ 5q}rlK + BKg[{ 8q} = 0 (6) where V is the total potential energy, 5q is the displacement vector, K is the stiffness matrix, Kg is the geometric matrix and B is the characteristic eigenvalue or critical load factor. In the FEA, the above equation is applied to each element. Generally, the transcendental equation has an infinite number of roots. To represent the stability criterion, the smallest root is the critical load factor, which the structure passes from its stable position to another configuration. 6

3. GENETIC A L G O R I T H M S (GA) Genetic algorithms were originally proposed by John Holland. 3 Numerous publications have established the validity of this technique for function optimization. GA are computationally simple, but powerful in their search for improvement. In addition, it is not limited by restrictive assumptions about search pace, such as continuity or existence of derivatives. Goldberg described the nature of GA of choice by combining a Darwinian survival of the fittest procedure with a structured, but randomized, information exchange to form a canonical search procedure that is capable of addressing a broad spectrum of problems. They combined the concept of artificial survival of the fittest with genetic operators abstracted from nature to form a robust search mechanism. Recently the GA has attracted a great deal of attention. It has been used as optimization tool and has yielded good results in various optimization problems. 1'3'~~ The genetic search begins with the random generation of a population of strings representing design alternatives. Each string (chromosome) represents a candidate solution to the problem. In GA, the chromosome is encoded in a binary string. Each solution is evaluated to yield a performance measure

calledfitness.

502 The population evolves through the application of three major types of genetic operators: selection, crossover, and mutation. A certain probability is applied in the crossover operator, called the c r o s s o v e r rate (pc). It is used to control the number of members that are selected to execute crossover. When a crossover probability of pc is used, only 100pc percent strings in the population are used in the crossover operation and 100(1-pc) percent of the population remains as they are in the current population. The mutation operator is applied to the offspring, and it allows offspring to change further. Like the crossover operator, the mutation operator is also applied with a certain probability, called the m u t a t i o n rate (pro). The need for mutation is to create a point in the neighborhood of the current point, thereby achieving a local search around the current solution. The mutation is also engaged to maintain diversity in the population. These three operators are simple and straightforward. The reproduction operator selects good strings and the crossover operator recombines good substrings from good strings together to hopefully create a better substring. The mutation operator alters a string locally to hopefully create a better string. Even though none of these claims are guaranteed and/or tested while creating a string, it is expected that if bad strings are created they will be eliminated by the reproduction operator in the next generation and if good strings are created, they will be increasingly emphasized. Concerning genetic algorithms and their operators, Goldberg 3 stated the genetic algorithms must has five steps: 1. 2. 3. 4. 5.

Representation: a genetic representation of a solution to the problem. Initialization: a way to create an initial population of solutions. Evaluation: a function that evaluates solutions. Genetic operators: selection, crossover and mutation. Iteration of the genetic algorithm.

Genetic algorithms have also been used to solve constrained optimization problems. In this paper, the following bracket operator penalty term is adopted. = R[g (x)] 2 (7) where g(x) expresses the constrain function. If the constrain function is satisfied, the value of g(x) equals to zero. Otherwise, the value of g(x) equals to the factual value. The penalty term corresponding to the constrained violation is added to the objective function.

4. OPTIMIZATION PROCEDURE Design variables are defined by designer base upon design requirements and specifications. In the present proposed procedure, an overall-controlled command file is created before the entire system can start its operations. In the current case study, this command file consists of ANSYS code. The procedure involves the operations from constructing FEA model, defining physical dimensions and selecting material properties, analyzing theoretical model to finally performing design iterations. The physical dimension, material property and design requirements can be treated as design parameters and variables in the formulation. During iterative process, these design variables will vary and change in magnitude as its requires. Hence the command file is updated automatically every time in the design iterative process. The complete system optimization framework is illustrated in Fig. 1. The role of the finite element analysis is to conduct structural analysis on the model. A command file is written for the FEA tool so that the geometry and dimensions of the structure can be modified and changed automatically during the design iterative process. The FEA tool will perform full

503 analysis for every member of the population. Usually, the output of the analysis is the objective function value or the constraint function value of the given design problem. Thereafter the fitness of a binary string is generated according to the objective function.

Figure 1 The Procedure of Optimization The GA keeps track of the binary string with maximum fitness in the population. Crossover and mutation are then used to produce new offspring. For example, suppose the population size is 100, crossover rate is 0.8 and mutation rate is 0.05. The crossover rate equals to 0.8 expresses that crossover is performed for 80% of the population, i.e. 80 binary strings are changed t~ produce new strings due to crossover. Mutation operator is similar in producing new offsprings. Through the three operators of GA a new population is generated. Once after new population is produced, the members of the population are made as the improved design. A new generation is then started. In every generations the good result is recorded. At the final step, the loop returns to the starting point where the stopping criteria are examined. If they are not met, the new population returns to the step of evaluation. The process repeats until the terminal research has been reached. GA finished the calculations and the best result will retain.

Figure 2 Composite Laminated Panel with Cutout

Figure 3 Finite Element Mesh

5. CASE STUDY The design of composite laminated panel with cutout is often formulated as an optimization problem to deal with the shape of the cutout. Fig. 2 shows the composite laminated panel with cutout subjected to uniaxial compressive forces. The major and minor axial of the elliptical cutout are selected as the design variables of the model. The goal of the design is to determine the best combination of major and minor axial so that the composite panel can achieve the minimum weight.

504 The boundary condition of the composite laminated panel is simply supported at all edges. The finite element model is couple with elastic buckling analysis to form the theoretical model for the composite laminated panel. The model is discretized with all shell elements, and only four plies of laminae are defined. The ply orientation angles are assumed to a set of angles, 0~ 45 ~ -45 ~ 90 ~ A constant thickness of 0.5mm is applied for each ply in the model. The length of the four sides of the panel is 254mm. Being symmetry, only quarter portion of the composite laminated panel is modeled in this study. Thus a quarter FEA model is adopted to represent the entire composite panel in which two symmetry edges of the quarter FEA model are constrained by symmetry constraint operators. A finite element mesh is created with 20 by 20 eight-nodded membrane elements in the radial direction of hole circumference as shown in Fig. 3. The material properties are given that: Marshall ~l has published experimental and theoretical results on the similar study. He related the critical loads with the circular hole's diameters and width of laminate panel. Prior the case study, the authors have verified the current model (by setting major axial equal to minor axial) with Marshall's results. It was found that the FEA's solutions are closed to the published theoretical and experimental results. This suggests that the constructed finite element model is reliable and it is suitable to be adopted in the optimization procedure. From the basic design requirements, the critical buckling strength must be bigger than the allowable critical buckling load. Thus, the optimization problem can be formulated as: minimize f ( x ) = p (102 - ft. xl. x2) (8) subject to gl (x) = acr - a _<0 (9) where x~ and x2 are length of the major and minor axial of the elliptical cutout respectively. Their physical dimensions should be smaller than 63.5mm. 19 is the coefficient of the density of composite laminated panel, g~r is the allowable buckling stress. In collaborating the GA in design optimization, the penalty function methods is employed to solve for the constraint optimization problem. Thus the objective function, f(x) is modified and replaced by the following penalty function: P (x) = f (x) + R [gl (x)] 2 (10) where R (assuming that R=1000 initially) is a penalty parameter for the critical stress constraint. Naturally the GA is a maximization techniques. Since the above formulation is minimization problem, the derived objective function needs to be treated and transformed before the GA can execute. Usually the fitness function can be formed by the following usual transformation: F ( x ) - 1/(1 + P (x)) (11)

6. RESULTS AND CONCLUSION In the application of genetic algorithms, crossover rate and mutation rate is two important factors. These two parameters directly affect search velocity and optimum result. Schaffer et aL ~6 suggested a set of parameters that can produce good GA performance. Hence the GA parameters that used in this case study are 0.95 and 0.005 for crossover rate and mutation rate respectively. In addition, other input parameters for the GA are: population size = 40; generation = 50; string length = 20; substring length for each parameter = 10.

505

Figure 4 The Values of Objective Function in the Initial Population

Figure 5 The Values of Objective Function in the 20th Generation

With the above assumptions, the laminated panel was modeled and simulated numerically. Fig. 4 and Fig. 5 show the values of weight obtained in the initial population and the 20 th generation. It can be observed from these figures that the initial population is fairly spread over the entire search space. After the 20 th generation most members of the population gather in the feasible region, and it is positioned close to the optimum solution. The optimum design solution is obtained only after the 25 th generation with approximately 40• (or 950) new function evaluations, as shown in Fig. 6.

Figure 6 The Values of Objective Function and Buckling Stress in the 25th generation Hence the design solutions for the composite laminated panel with cutout that proposed by GA in conjunction with the FEA method is: Major axial = 2.3456; Minor axial =2.1774; Buckling stress = 38.01; Weight = 10075.59. In this paper, the optimum design procedure is introduced. The procedure utilizes FEA and Genetic Algorithms in assisting engineers in their daily design practices. The FEA enables designers to analysis and improve their designs with ease. The merits of genetic algorithm as a search technique rely on its capability in achieving global optimum solution. The feasibility and accuracy of combining these two methods to perform design optimization are demonstrated through the case study. Generally the proposed procedure yields effective and efficient results to the design optimization.

506 REFERENCES

1. Comba, S. and Davis, L. (1987). Genetic algorithms and communication link design : constraints and operators. InProc. 2na Int. Conf. on Genetic Algorithms, Cambridge, MA, 28-32. 2. Gerhardt. (1984). A hybrid/finite element approach for stress analysis of notched anisotropic materials. ASME Journal of Applied Mechanics 51,805-809. 3. Goldberg, D. (1989). Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA. 4. Jha, N.K. (1990). Computer sided multi-objective optimal design and finite element analysis of cutting tool. Proc Manuf Int 90 Part 4 Adv Mter Autom. Publ by American Soc of Mechanical Engineers(ASME), New York, NY, USA. 59-70. 5. John, K. V., and Ramakrishnan, C. V. (1987). Minimum weight design of trusses using improved move limit method of sequential linear programming. Int. J. Comp. Struct. 27:5, 583-591. 6. Khong, P. W. (1990). The comparison of lower order and higher order finite strip analysis in the stability problems of thin-walled structures. Comp. & Struct. 36:1, 109-118. 7. Kogiso, H., Watson, L.T., Gurdal, Z., Haftka, R.T, and Nagendra, S. (1994). Design of Composite Laminates by A Genetic Algorithm with Memory. Mechanics of Composite Mterials and Structure 1, 95-117. 8. Lekhnitskii, S. G. (1957). Anisotropicplate. 2nd edition. Gostekizdatr, Moscow. 9. Lin, C.C. and Ko, C.C. (1988). Stress and strength analysis of finite composite laminate with elliptical hole. J. Comp. Mater 22:4, 373-385. 10. March, S. and Rho, S. (1995). Allocating data and operations to nodes in distributed database design. IEEE Trans. Knowl. and Data Eng. 7, 305-317. 11. Marshall, H.I. (1986). Buckling of perforated composite plates-an approximate solution, Imech E. 12. Michalewicz, Z. (1992). Genetic Algoritms + Data Structures = Evolution Programs, Springer-Verlag, New York. 13. Ogonowski, J.M. (1980). Analytical study of finite geometry plate with stress concentration. AIAA/ASMW/ASCE/AHS, 21st SDM Conference, 694-698. 14. Rageev, S. and Krishnamoorthy, C. S. (1992) Discrete Optimization of Structures Using Genetic Algorithms. Journal of Structural Engineering 118:5, 1233-1250. 15. Savin, G. N. (1961). Stress distribution around hole, English translation edition, Pergamon Press, Oxford. 16. Schaffer, J. D., Caruana, R. A., Eshelman, L. J. and Das, R. (1989). A study of comrol parameters affecting online performance of genetic algorithms for function optimization. In Proc. 3 rd Int. Conf. on Genetic Algorithms, 51-60. 17. Tam, K.T. (1992). Genetic algorithms, function optimization, and facility layout design. Eur. J. Oper. Res., 63, 322-346. 18. Xu, X.W., Yue, T.M., and Man, H.C. (1999). Stress analysis of finite composite laminate with multiple loaded holes. International journal of solids and structures, 36, 919-931. 19. Xu, X.W. (1992). The strength analysis of mechanically multi-fastened composite laminate joints, Ph.D. dissertation, Nanjing Aeronautical Institute (in Chinese). 20. Xu, X.W. (1995). Stress concentration of finite composite laminates with an elliptical hole. Comp. and Struct. 57:2, 29-34.

Third InternationalConferenceon Thin-WalledStructures J. Zarag,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rightsreserved

507

THE EFFECT OF MEMBRANE-FLEXURAL COUPLING ON THE COMPRESSIVE STABILITY OF ANTI-SYMMETRIC ANGLEPLY LAMINATED PLATES J. Loughlan Cranfield University, College of Aeronautics, Cranfield, Bedfordshire, MK43 OAL, UK

ABSTRACT

The compressive stability of anti-symmetric angle-ply laminated plates with particular reference to the degrading influence of membrane-flexural coupling is reported in this paper. The degree of membraneflexural coupling in the laminated composite plates is varied, essentially, by altering the ply angle and the number of plies in the laminated stack for a given composite material system. The coupled compressive buckling solutions are determined in the paper using the futile strip method of analysis and the buckling displacement fields of the strip formulation are those which are able to provide zero in-plane normal movement at the edge boundaries of the laminated plates. Results are given in the paper for anti-symmetric angle-ply laminated plates subjected to uniaxial compression and these have been obtained from fully converged finite strip structural models. Validation of the finite strip formulation is indicated in the paper through comparisons with exact solutions where appropriate. The natural half-wavelength of the compressive buckling mode of the composite plates is shown in the paper to be significantly influenced by variation in the ply angle. Increasing the number of plies in the laminated system is seen to reduce the degree of coupling and the critical stress levels are noted to tend towards the plate orthotropic solutions. The ply angle corresponding to the optimised buckling stress for any particular laminate is noted in the paper to be influenced by the support boundary conditions at the plates unloaded edges. KEYWORDS

Membrane-flexural coupling, Compressive stability, Antisymmetric, Angle-ply, Laminated plates, Critical stress, Finite strip method, Natural half-wavelength

INTRODUCTION AND DISCUSSION OF RELATED RESEARCH

Many interesting results [1-6] pertaining to the structural response of anti-synnnetric laminated plates have been presented over the years. The non-linear behaviour of anti-symmetric cross-ply plates resulting from bend-stretch coupling is detailed in the work of Prabhakara [1]. The post-buckling analysis of this work highlights the presence of pre-buckling rotations for the thin plates considered

508 and thus for the simply supported boundary condition, lateral bending deflections are noted to be in evidence throughout the compressional loading history of the plates. Compressive buckling solutions have been presented by Sharma et el [2] for anti-symmetric cross and angle-ply laminated plates. The free edge boundary condition is considered in this work and plates with a free edge are considered to behave in a linear manner. Strictly speaking the free edge boundary condition is unable to react the twisting moments set up in the angle-ply laminates as a result of stretch-twist coupling or the bending moments set up in the cross-ply laminates due to bend-stretch coupling and as such the compressive loading of these anti-symmetric laminated plates with a free edge will have non-linear equih'brium characteristics throughout the loading process. In the work of Sharma et al [2] the non-linear aspects associated with the free edge condition have not been considered. In the case of anti-symmetric cross-ply plates coupling between membrane shearing action and out-ofplane bending or twisting action does not exist and thus it is possible, in this case, to apply shear loading and the plate will remain perfectly flat until the onset of buckling. This problem has been considered by Hui [3] who obtained the shear buckling response of simply supported rectangular plates using the Galerkin p r ~ e end by assuming a summation of the double sine series to represent the out-of-plane buckled state of the plates. To eliminate the confusion in the research community at the time, with regard to whether an anti-symmetric laminated plate would respond in a linear or non-linear manner when subjected to in-plane compressive or shear loading, the work of Leissa [4] set out in some detail the conditions for such plates to remain flat and thus to yield linear buckling behaviour. As an example it is demonstrated by Leissa [4] that for an anti-symmetric angle-ply plate subjected to uniform or linearly varying in-plane normal stresses it is necessary to provide twisting moments but not bending moments at the plate boundaries in order for the plate to remain in the flat configuration. It is also noted that the necessary twisting moments can be supplied straightforwardly by either clamped or simply supported conditions at the boundaries. The validity of the simplified reduced bending stiffness approach in its application to anti-symmetric laminated plate construction has been investigated, in some detail, by Ewing et al [5] for the case of compressive loading and corresponding to simply supported edge conditions. It is shown in this work that the reduced bending stiffness method can give large errors in the buckling loads for angle-ply plates when constraints are imposed on the membrane displacement conditions at the plate boundaries. It is noted that the reduced bending stiffness method gives no consideration to the degree of membrane constraint existing at the plate edges. It is also of note that Ewing et al [5] treat the compressive loading of a simply supported anti-symmetric cross-ply plate as a l i n ~ buckling problem which is, of course, in contrast to the non-linear study made by Prabhakara [1] which indicates bending of the plate from the onset of membrane compressive loading owing to the existence of bend-stretch coupling. The buckling and postbuckling study of Jensen and Lagace [6] also indicates, both theoretically and experimentally, the non-linear response of the simply supported anti-symmetric cross-ply configuration. The postbuclding response of composite laminated plates has been investigated by Dawe and Wang [7] who developed the spline finite strip method to determine the geometrically non-linear behaviour of the laminates. The method developed is versatile in that it is able to deal with a range of end conditions and can accommodate laminates, which possess a high degree of anisotropy. Of particular interest in this work are the results presented for anti-sym_metric cross-ply laminates which illustrate a distinct non-linear response of the plates from the onset of end compressional displacement. This is especially true for the thinner laminates since these are associated with higher levels of material coupling interaction. The general characteristics of the compressional postbuckling curves presented by Wang and Dawe [7] for anti-symmetric cross-ply laminates have been noted to closely support the findings of the earlier compressive postbuckling studies by Prabhakara [1] and by JerlSen and Lagace [6]. Wang and Dawe [8] have extended their earlier work [7] to incorporate laminate through-thickness shearing effects. This was achieved through the development of a spline finite strip approach based in the context of first-order shear deformation plate theory. It is shown in this work that for moderately thick plates the

509 postbuckling curves based on shear deformation theory differ significantly to those determined on the basis of classical plate theory. Loughlan [9,10] has examined the effect of bend-twist coupling on the shear buckling performance of thin laminated composite plates and stiffened panels. The finite strip method of analysis was employed in this work. The support condition considered at the plate and panel boundaries was taken as that normally associated with the attachment to a rigid shear diaphragm and both balanced and unbalanced symmetric laminates have been studied. It is shown in this work that for laminated plates which have a fixed number of constant thickness plies of a given degree of material anisotropy, the stacking sequence of the pries significantly alters the degree of bend-twist coupling in the laminates. Stiffened panel structures which have exactly the same geometry and which are made from the same material are thus shown to possess quite different levels of shear buckling capability. It has been pointed out [9] that this is due solely to lay-up configuration and thus to the degrading influence of bend-twist coupling. The shear buckling response of long plates has been shown [10] to be associated with a considerable degree of amplitude modulation in the buckled mode shape. The finite strip formulation employed is shown to have sufficient flexibility to deal with the complex distortions of the shearbuckling mode. In this paper the compressive buckling behaviour of thin anti-symmetric angle-ply laminated composite plates is studied through the use of the fmite strip method of analysis. The effect of membrane-flexural coupling on the compressive buckling response of the thin anti-symmetric laminates is investigated.

ANALYSIS APPROACH

The perturbation or buckling displacements chosen for the finite strip formulation employed in this work are those which satisfy exactly the compatibility and equilibrium conditions at the strip simply supported loaded ends. In the present case these conditions are, of course, set-up appropriately for an antisymmetric angle-ply laminated material. The simply supported conditions satisfied at the strip loaded ends are the $3 edge boundary conditions. These result in zero lateral and in-plane normal displacements at the boundary as well as zero moment and in-plane shear. The boundary conditions satisfied at the unloaded edges of the uniformly compressed plates are those of compatibility and these are imposed in the f'mite strip s t r u ~ models. Equih~orium conditions are satisfied approximately at the unloaded edge boundaries through the utilisation of suitably refined models and these have been shown to be able to yield fairly acctmee converged solutions for the compressive critical stress levels. The reader is referred to references [9,10] for detailed information regarding the analysis approach. In these works the shear buckling performance of symmetric laminates have been considered with due consideration being given to the effects of bend-twist coupling and also the $2 boundary condition is satisfied exactly at the strip ends. In the present work for antisymmetric laminates the [B] matrix is now non-zero and in order to satisfy the $3 boundary condition the in-plane perturbation displacements u and v must now vary along the length of the strip corresponding to the sine and cosine functions respectively. SOME TYPICAL RESULTS The results given in the paper are those pertaining to composite plates manufactured from high strength carbon-epoxy pre-impregnated uni-directional plies with a ply thickness of 0.125 mm and with the following ply material properties; E, ffi 140 kN/mm2,

E2 = 10 kN/mm2,

O~2 - 5 kN/mm2,

vl2 = 0.3

Antisymmetric angle-ply plates have in-plane orthotropic [A] stiffness properties and also the bending [D] stiffness matrix for this lay-up configuration is orthotropic in nature. Such laminates have a non-

510 zero coupling [13] stiffness matrix, which determines the degree of interaction associated with the membrane-flexural response of the plates to any specified loading. The non-zero [13] matrix links inplane shearing action to out-of-plane bending action and also in-plane extensional or normal loading to twisting of the laminate. It is clear therefore that for the case of in-plane shear loading the simply supported plate would not be able to react the bending at its boundaries and hence the simply supported in-plane shear problem would be non-linear in nature from the onset of loading. For the case of in-plane compressive loading however the simply supported boundaries would be able to react the twisting tendency of the plate [4] and thus the plate would remain flat until the onset of buckling and the critical stress levels can thus be determined through normal eigenvalue prcmedures.

30

i

=

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E--~-=14.i G--ZZ'10.!, Vn=~3 /~

,,, ^

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2

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Ptm AspectRatio(an))

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C o m p m u M I Bucking Loads for Simply Supported Rectangular AnU-Symmetdc Angk)-P~ Plates (e 1 45)

Figure 1 shows the characteristic compressive buckling curves for simply supported rectangular antisymmetric engle-ply plates [0/-0]. for n values of 1-4 and for the ply angle of 45 ~ The curves are depicted in a non-dimensional manner such that all of the laminates have the same non-dimensional orthotropic solution. The orthotropic solutions have been determined simply by setting the coupling [B] coefficients to zero in the finite strip analysis procedure. It was found that modelling the plates with 100 longitudinal strips was more than sufficient to provide extremely accurate converged compressive buckling solutions. The exact solution procedure for the simply supported $3 boundary

511 condition is reported [11] and comparisons of the finite strip solutions with those determined using the exact procedure have been found to be indistinguishable. It is of note that for the 45 ~ ply-angle the natural half-wavelength of the compressive buckling mode is that which is associated with square buckles. The characteristic garlanded nature of the curves with the garlands flattening out with increase in the plate aspect ratio and levelling off at the minimum critical buckling stress is evident in Figure 1. It is also clear that the highest degree of membrane-flexural coupling is associated with the thinner laminates and that as the number of plies in the laminated stack increases the buckling solutions tend increasingly more towards the plate orthotropic solution which is, of course, devoid of the influence of coupling.

Figure 2. Influence of Ply-Angle on Critical Buckling Slmss and Natural Half-Wavelength of the Buckling Mode

The influence of changing the ply angle is shown in Figure 2 for the lay-up configuration of [O/-O]n with n = 3. The compressive critical stress levels of this six-ply laminated system for the ply angles of 0 = 15, 45 and 60 ~ axe depicted and these are seen to be significantly different. The minimum critical buckling stress for the laminate with 0 = 15~ is seen to be of the order of 50% lower than that of the 45 ~ plate whilst the 0 = 60 ~ laminate can be seen to have a minimum critical stress level which is about 18% lower than the 45 ~ plate. The ply angle of 45 ~ was found to be the optimum with regard to the compressive buckling performance of square plates for this six-ply system and for the case of the $3 simply supported boundary condition. All other ply angles in the range 0-90 ~ were found to give lower critical stress levels for square plates and this is reflected in Figure 2 for the two ply angles of 15 and 60~ respectively. It will be of note that the ply angle bears a significant influence on the natural halfwavelength of the compressive buckling mode. For long simply supported plates, square buckles exist

512 only for the 45 ~ ply-angle since this results in equal in-plane and equal flexural stiffnesses in the two mutually orthogonal longitudinal and transverse directions of the laminated plate respectively. For ply angles greater than 45 ~ the natural half.wavelength, k, of the buckled mode shape will be less than the plate width b. For ply angles less than 45~ it is found that ~, is greater than b. In Figure 2 it is shown that g is of the order of O.Tb for the ply-angle of 60~ and of the order of l.Sb for the ply-angle of 15~ It is also of note in Figure 2 that the square plate with 60~ ply-angle is associated with two longitudinal buckles and that the plate with 15~ ply-angle can realise the highest critical stress level of the three plates considered for some plate aspect ratios less than unity.

Figure 3. UniaxlalCompressive Buckling Loadsfor Simply Supported Square A n t i - S ~ Angle-Ply Plates

The effect of ply-angle on the buckling performance of square simply supported plates having different numbers of plies in the laminated stack is shown in Figure 3. The ply-angle is varied from 0-90~ and as mentioned previously it is clear that for the thicker laminates the optimum ply-angle with regard to buckling performance is 45 ~ The full lines in Figure 3 are the exact solutions for the square plate buckling problem according to the procedure outlined by Jones [11]. The finite strip solutions, which have been determined using 100 longitudinal strips across the plate, are noted to be identical with the exact solutions. The buckling displacement fields employed in the finite strip formulation are those which satisfy, automatically, the simply supported compatibility and equilibrium $3 edge boundary

513 conditions at the strip loaded ends. The $3 compatibility conditions are then imposed at the unloaded edge boundaries in the finite strip structural model and the remaining equilibrium conditions are then satisfied approximately through a sufficiently refined structural model. The employment of 100 strips in the plate models is noted in Figure 3 to give indistinguishable results in comparison with the exact solutions. For ply angles greater than 60~ it should be noted that the buckling stress levels shown in Figure 3 are those associated with the development of two buckles in the loading direction.

Figure 4. Effect of Boundary Conditions on the Buckling Performance of Square Plates.

A comparison of the buckling performance of square plates with different support conditions on the unloaded edges is shown in Figure 4. The plates are simply supported at their loaded ends and have a six-ply stacking arrangement as indicated. The influence of the unloaded edge support condition on the critical compressive stress level is clearly evident and as expected. We note that the clamped-clamped, CC, condition and the simply supported-simply supported, SS, condition give the highest and lowest critical stresses respectively with the clamped-simply supported, CS, condition giving stress levels lying between these two bounds. It is also of note that the optimum ply-angle with regard to buckling performance is influenced by the unloaded edge support condition. For the SS condition the optimum ply-angle is 45 ~ For the CC condition this reduces to about 42.5 ~ whereas for the CS condition the optimum ply-angle is noted from Figure 4 to be about 48.75 ~

CONCLUDING REMARKS The effect of membrane-flexural coupling on the compressive buckling performance of thin antisymmetric angle-ply laminated composite plates has been examined in this paper. The fmite strip method has been used to determine the buckling solutions and the multi-term nature of the finite strip

514 buckling displacement fields in conjunction with the appropriate level of structural modelling has been able to readily account for the compressive coupling response of the plates. The degree of membraneflexural coupling in the laminated plates has been varied by altering the ply angle and by varying the number of plies in the laminated stack. For laminated plates which have a fixed number of constant thickness plies it is shown in the paper that for a given degree of material anisotropy the ply angle significantly influences the natmal halfwavelength of the compressive buckling mode. The ply angle of 45* is associated with square buckles for the case of simply supported long plates. Ply angles less than 45* result in longer natural halfwavelengths whereas ply angles greater than 45 ~ produce shorter naUual half-wavelengths. For the degree of material anisotropy considered in this paper, which is typically that corresponding to high strength carbon-epoxy pre-impregnated plies, the compressive buckling behaviour of square plates has been shown to be associated with two longitudinal buckles for ply angles greater than 60~. For any particular laminate the ply angle corresponding to the optimised compressive buckling stress for square plates is noted in the paper to be influenced by the support boundary conditions at the plates unloaded edges. For simply supported square plates the optimised angle for laminates with more than 4 plies is 45 ~. For square plates with clamped conditions on the unloaded edges the comparable angle is about 42.5 ~ and for square plates with one unloaded edge clamped and the other simply supported the ply angle for optimised compressive buckling is about 48.75~ It has been shown that thinner laminates are more prone to the effects of membrane-flexural coupling and that as the laminate thickness gradually increases, by increasing the number of plies in the laminated stack, the compressive buckling response tends increasingly more closely towards that of the orthotropic plate solution. REFERENCES

1. Prabhakara, M.K., Post-Buckling Bvhaviour of Simply Supported Cross-Ply Rectangular Plates. Aeronautical Quarterly, 27 (1976) 309-316. 2. Sharma, S., lyengar, N.G.R. & Murthy, P.N., Buckling of Anti-symmetric Cross- and Angle-Ply Laminated Plates. Int. Jou. Mech. Sci., Vol. 22, 1980, pp 607-620. 3. Hui, D., Shear Buckling of Anti-symmetric Cross-Ply Rectangular Plates. Fibre Science and Technology, 21 (1984) 327-340. 4. Leissa, A.W., Conditions for Laminated Plates to Remain Flat Under Inplane Loading. Composite Structures, 6 (1986) 261-270. 5. Ewing, M.S., Hinger, R.J. and Leissa, A.W., On the Validity of the Reduced Bending Stiffness Method for Laminated Composite Plate Analysis. Composite Structures, 9 (1988) 301-317. 6. Jensen, D.W. and Lagace, P.A., Influence of Mechanical Couplings on the Buckling and Postbuclding of Anisotropic Plates. AIAA Journal, Vol. 26, No. 10, 1988. 7. Dawe, D.J. and Wang, S., Postbuclding Analysis of Thin Rectangular Laminated Plates by Spline FSM. Thin-Walled Structures, 30 (1998) 159-179. 8. Wang, S. and Dawe, D.J., Spline FSM Postbuclding Analysis of Shear-Deformable Rectangular Laminates. Thin-Walled Structures, 34 (1999) 163-178. 9. Loughlan, J., The Influence of Bend-Twist Coupling on the Shear Buckling Response of Thin Laminated Composite Plates. Thin-Walled Structures, 34 (1999) 97-114. 10. Loughlan, J., The Shear Buckling Bchaviour of Thin Composite Plates with Particular Reference to the Effects of Bend-Twist Coupling. International Journal of Mechanical Sciences, 43 (2001) 771-792. 11. Jones, R.M., Mechanics of Composite Materials., McC~aw-Hfll Kogakusha LtcL, 1975.

Third InternationalConferenceon Thin-WalledStructures J. Zarafi,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

515

HOMOGENEOUS AND SANDWICH ELASTIC AND VISCOELASTIC ANNULAR PLATES UNDER LATERAL VARIABLE LOADS D.Pawhs Department of Mechanical Engineering Principles, Bielsko-Biala Branch of Technical University ofL6d~, Willowa 2, 43-309 Bielsko-Biala, POLAND

ABSTRACT

This paper presents the results and observations connected with the time histories of deflections of thin, annular plates. Homogeneous and three-layered plates are the objects of the analysis. The plates are loaded in their surface by compressive lateral stresses. Rheological models of linear viscoelastic medium describe the material of homogeneous plate and the material of the layers composing the sandwich plate. During the analysis of dynamic behaviour of three-layered plate with viscoelastic core the observations earlier obtained for viscoelastic, homogeneous plate were used. Results show essential influence of values of material parameters on viscoelastic behaviour of plates. Geometrically non-linear and dynamic problem has been solved using Finite Element Method in ABAQUS system.

KEYWORDS

annular,homogeneous,sandwich,plate, viscoelasticity, buckling, dynamic stability,Finite Element Method

INTRODUCTION

Relatively small weight and good strength qualities of sandwich comtructiom decide about their wide application. These qualities are obtainable by suitable division of physical characteristics of layer materials. ORen these constructions are three-layered, built of rigid, thin outer layers and relatively thick, ,,soft" core. Especially, material and structure of the core are responsible for the proper stiffness and lightness and for the other important qualities of such constructions, like: capacity for vibration damping under dynamic loading. Materials showing the characteristic behaviour of viscoelastic body fulfil above mentioned qualities. Detailed observations of homogeneous, viscoelastic plates, Pawhs (1996),(1998),(2000), in comparison with well known behaviour of elastic plates, Trombski (1972), Wojciech (1979), indicate the possibility to use viscoelastic material for sandwich plate layers. The analysis of dynamic behaviour of viscoelastic

516 homogeneous and sandwich plate is the subject of this paper. In three-layered structure of plates the middle layer (the core) has the physical properties of rheological material of plate analysed earlier as homogeneous plate.

FORMULATION OF THE PROBLEM

Thin, isotropic, annular plate is the subject of the analysis. Plate with clamped edges has been loaded in its surface by compressive stress, which is evenly distributed on plate outer perimeter - Figure 1. Radial forces are constant in time or rapidly increasing proportionally to time according to the formula: p=st

(l)

where: p - compressive stress; s- rate of plate loading growth; t -time.

P~

-t------~\\\\\\\\'~ ~

Figure 1: The scheme of annular plate Non-linear relations describe the geometry of homogeneous or sandwich plate. Plate has preliminary deflection. Physical relations of plate material are defined by relations of linear elastic or linear viscoelastic constitutive equations for these bodies. Viscoelastic qualities of material of plate layers have been described by three elementary rheological models: Maxwell, Kelvin-Voigt and standard. The behaviour of analysed plates has been observed considering dynamic stability of plate; evaluating the values of critical parameters, ie: time, deflection, stress and analysing the damping vibration initiated in overcritical region of plate loading. As a criterion of loss of plate stability the criterion presented in Volmir's work was adopted, Volmir (1972). According to this criterion the loss of plate stability is in the moment of time when the velocity of the point of maximum deflection reaches the first maximum value.

NUMERICAL CALCULATIONS The calculations were made using Finite Element Method in ABAQUS system version 5.5,5.8 ; in the Academic Computer Center CYFRONET-CRACOW (KBN~C3840\CD\034\1996). The annular sector of the plate (1/8 part) has been a subject of the consideration. The geometric nonlinear and dynamic analysis of the problem was led for the system built of 4-node 3D shell elements. Viscoelastic behaviour of layer material has been approximated with single term of Prony series for the shear relaxation modulus, Hibbitt,Karlsson&Sorensen (1996) :

o. o- Oo(.-,:(.-

(2)

517 where: q P1 ' x~ - material constants; E Go = _______L__0.instantaneous shear; values Eo and Vo have been described in the *Elastic* option. 2(1 + Vo) The sandwich structure of the plate has been defined using the COMPOSITE parameter to specify a shell cross-section. The essential calculations were carried out changing of the damping rl coefficient values, which describe the viscoelastic properties of plate layers and changing the thickness of the core layer in sandwich plate,too. Viscosity constant 11 determines the value of x~ parameter (2); x~ = f(xR), where xR = 11 _ relaxation k2 time in extensional relaxation function E R( t ) = k~ + k s e -'''R , Hibbitt,Karlsson&Sorensen (1996).

ANALYSIS OF R E S U L T S Results presented in this paper are for homogeneous and sandwich plates. Linear elastic qualities of plate material are described b y E0 = 1.67-105 MPa, vo = 0.25; dimensions of plates a r e rz = 0.5 m , rw = 0.2 m ; the system of layers in sandwich plate is showed in Table 1. TABLE 1 MATERIALAND GEOMETRICDATAFOR THREE-LAYEREDPLATES Figure Figure 3

Layer i outer layers core

Thickness [m] 0.0005 0.001

Material linear elastic: E0 = 1.67-105 MPa ,Vo = 0.25 viscoelstic described by Kelvin-Voigt model: qP = 0.99 , 't,c = 7 3 - 1 0 4 s

.....Figure 4

outer layers core

0.0005 0.001

linear elastic: Eo = 'i.67.10 ~ MPa ,vo 0.25 viscoelstic described by Kelvin-Voigt model: q = 0 . 9 9 , Xl~ = 73-10-~ s

Figure 7

outer layers

0.001

viscoelstic described by standard model: q~ = 0.55, T~ = 40.8-10-4 s

core

0.004

viscoelstic described by Maxwell model: q~ = 1.0, x~ = 7 4 - 1 0 -s s

outer layers

0.001

viscoelstic described by standard model: q~ = 0.55, x~ = 40.8.10-' s

core

0.004

viscoelstic described by Maxwell model: q ~ = l . 0 , x 12I = 7 4 - 1 0 -5 s

outer layers

0.001

viseoelstie described by standard model q~ = 0.55, x~ = 40.8.10-' s

0.002

viscoelstic described by Maxwell model: q~ = 1.0, x~ = 7 4 . 1 0 -2 s

i

Figure 8

Figure 9

core

ii

518 In the case of Kelvin-Voigt model, which doesn't describe the relaxation phenomenon, using the *VISCOELASTIC* option the values q~P and x~ (2) have been calculated for standard model-Figure 2, which for E~--,oo approaches Kelvin-Voigt model.

Figure 2: Standard model Numerical calculations were realized for the increased value of Young's modulus E~ = 1.67.107 MPa. In the case of standard model the calculations were led for equal values of Young's moduli 9El = E. The linesl,2 of the diagrams determine: linel - time histories of plate maximum deflection; line 2 - time histories of velocity of plate maximum deflection and value of the first maximum velocity, which marks the loss of plate stability. Analysed plates have been compressed by time-variable loading, according to the Eqn. (1). Results showed in Figures 3,4 are for axially symmetrical form of loss of plate stability, (designated by re=O). Outer edge is compressed by increasing stress with rate of plate loading growth equal: s = 1700.5 MPegs. Figures 7-9 are for the case when the loss of plate stability is with four transverse waves in the direction of the circumference of plate, m=4. In this example the parameter s is equal: s = 9473 MPa/s. Comparing the time histories showed in Figures 3,4 , one can observe evident influence of only changing value coefficient-viscosity constant q - of viscoelastic material of plate core.

I

i

vs x,~ v$ xul

14 S4

I

I

2

|1 _

x

g" Y -102 2

TIME [s]

10"*

Figure 3" Time histories of sandwich plate with Kelvin-Voigt core; z~ = 73.10 4 s

519

15 L~ul

1 z

yIIXJOLI

le$ ~e)ll 14 vs ~enx 14

,~a~z

('10"'1)

TJLoTe~

'4.1. eel+e4 9L . e e | 4 4 2

$

~1o

;% x E

]# T [

~5 L

I T Y

0

0

2

4 TINE

I

6

I

8

[s]

('10"*-2)

Figure 4: Time histories of sandwich plate with Kelvin-Voigt core; z~

= 73-10 4 s

With the increase of value of viscosity constant ( increase of TI G ) oscillating vibrations initiated in overcritical region of plate loading disappear; critical time to loss of plate stability extends and maximum deflections decrease. This behaviour of sandwich plate is ,,determined" mainly by the core.

2 klmm I Z

va3~u~ y $ N e ) B 14 V~ J N ) n 14

elM.|

I

('10"'2)

l

,siTes

41.00B4-04 §

1

T

|-z

-2

0

2

I

4 TIIE

i [el

6

I

8 ('10.*-2)

Figure 5: Time histories of homogenous, Kelvin-Voigt plate; z~ = 73.104s

520

,,~,

v$ M 0 1 9 14 v :) ~lolg 14

Z

,AeT6~

!

('10"'1)

1

I

41. en444 9t .en+el

5 h O lr H E

[

Ts v E L 0 C I T Y 2

0

4

6

TI~

8 ('10"**-2)

i,]

Figure 6 " Time histories of homogenous, Kelvin-Voigt plate; x c, = 7 3 . 1 0 -~ s Using this kind of core material to analysis the behaviour of homogeneous, axially symmetrical plate, it has been found that with the increase of viscosity constant rl - plate becomes more stiff- Figures 5,6. The thickness of the homogeneous plate is equal: h = 0.001 m" rate s is equal: s=1700.5 MPa/s.

=,,

1.t 2

y,~,n~,

I1 Mill 14 V~ MID" t4

,,=, TA~T0~

1

28

!

('10"'1)

+1.111§ +t.,n.l.0a

i

!

/

9_4 -

F

,I

s 20 P

k IN T

[

V

12

8

E o

c4 Y

2 0 0

I

I

1

2

TI~

3

[s]

4

(-10.*-2)

Figure 7: Time histories of sandwich plate with Maxwell core; x c = 7 4 . 1 0

-5 S

521

16 s I~11.| ('10"'1)

1

lr~ ~4,I,B

2

v $ iqel]t

14 14

I

1

I

7AITO~ +1.0%Z~4 § .eez~e2

~12 S P c E X

Z8 N T

[

T ~4

T 0

0

1

2

TIME

[s]

3

4

$

('10"*-2)

Figure 8: Time histories of sandwich plate with Maxwell core (0.004 m); z G,= 74.10 -2 s 4

i. l~l VAILIS)).]I Y~ )'JeJi 1~ v~ NeSa t4

T,~.~n. |Tel +1.00|+O4

J

u

u

(,10--2)

9i. e em.l.4~

L

1

3

fO

ii

i

1

~

-

u

-2

0

1

2

TIME

I

3

[s]

I

4

~

('10-*-2)

Figure 9: Time histories of sandwich plate with Maxwell core (0.002 m); z,~ = 74.10 -2 s Opposite behaviour has been observed analysing the homogeneous plate, but made of material described by Maxwell model. With the increase of value of viscosity constant behaviour of plate approaches the behaviour of elastic one, Pawlus (1996), (1998), (2000).

522 Use of this kind of material for the core of sandwich plates is presented in Figures 7-9. The material of external layers is described by standard solid. With the decrease of values of viscosity constant and with the increase of core thickness the vibrations in overcritical area of plate loading disappear and are damped. Increase of values of viscosity constant extend the critical time to loss of plate stability, too.

CONCLUSIONS Results presented in this paper show the essential connection between physical properties of plate material (particularly the core material) and the time histories of deflections and sensitivity of dynamic stability of homogeneous and sandwich plates. One can observe that the choice of rheological model and its elastic and viscosity parameters are important in description of approximate qualities of real plate materials. During the analysis, characteristic viscoelastic behaviour of elementary models presented in work Findlay&Lai&Onaran (1976) has been noticed in behaviour of the considered viscoelastic plates.

References

Findlay W.N.,Lai J.S.,Onaran K. (1976). Creep and Relaxation of Non-finear Viscoelastic Materials with an Introduction to Linear Viscoelasticity. North-Holland Publishing Company-Amsterdam, New York, Oxford. Hibbitt,Karlsson&Sorensen, Inc. (1996). ABA QUS/Standard. User's Manual, version 5. 6. Hibbitt,Karlsson&Sorensen. (1996). ABA QUS/Standard. Example Problems Manual version 5. 6. Pawlus D. (1996). The Behaviour of Viscoelastic Ring-Shaped Plate under Variable Load (in Polish). Doctoral Thesis, L6d~ Technical University Branch in Bielsko-Biata. Pawlus D. (1998). Dynamic Stability of Viscoelastic Annular Plate (in Polish). Scientific Bulletin of the Technical University of L6~ Branch in Bielsko-Biata 32, 123-143. Pawlus D. (1998). The Behaviour of Ring-Shaped Viscoelastic Plate. Proceedings of the Int. Colloquium LSCE '98, Warszawa, 152-157. Pawlus D. (2000). Dynamic Stability of Annular Viscoelastic Plate-Calculations in ABAQUS System (in Polish). Proceedings of the IXth Symposium ,,Stability of Structures", Zakopane, September 25-29, 243-247. Trombski M. (1972). The Problem of Ring-Shaped, Orthotropic Plate in Non-linear Formulation (in Polish). ScientO~c Bulletin of L 6 ~ Technical University.32. Wojciech S. (1979). Numerical Solution of the Problem of Dynamic Stability of Annular Plates (in Polish). Journal of Theoretical and Applied Atechanics 2,(17). Volmir A.C. (1972). Non-linear dynamic ofplates and shells (in Russian). Science, Moskwa..

Third International Conference on Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

523

LOCAL BUCKLING BEHAVIOUR OF SANDWICH PANELS Narayan Pokharel ! and Mahen Mahendran 2 1PhD Research Scholar, 2Associate Professor Physical Infrastructure Centre, School of Civil Engineering, Queensland University of Technology, Brisbane, Q 4000 Australia

ABSTRACT The use of sandwich panels in Australia has increased significantly in recent years due to their widespread structural applications in building systems. However, the lack of design rules and standards is of concern to the designers of sandwich panels. The European standard for the design of sandwich panels is for metal faces and polyurethane or polyisocyanurate cores whereas sandwich panels generally used in Australia comprise of polystyrene foam and thinner and high strength steel faces. When subject to axial bending and/or compression, the steel plate elements of profiled sandwich panels are susceptible to local buckling failures. A research project was therefore undertaken using experimental and finite element analyses to investigate the local buckling behaviour of Australian sandwich panels and to develop appropriate design formulae. This paper presents the details of experimental studies on the steel plates supported by polystyrene foam core, the results and their comparison with available design formulae.

KEYWORDS Sandwich panels, steel plates, polystyrene foam core, local buckling, effective width. 1. INTRODUCTION Sandwich panels are composite structural elements, consisting of two thin, stiff, strong faces separated by relatively thick layer of low-density and stiff material. The Faces are commonly made of steel, aluminium, hardboard or gypsum and the core material may be polyurethane, polyisocyanurate, expanded polystyrene, extruded polystyrene, phenolic resin, or mineral wool. In Australia, sandwich panels are commonly made of expanded polystyrene foam cores and thinner (0.42 mm) and high strength (minimum yield stress of 550 MPa and reduced ductility) steel faces which are bonded together using separate adhesives. Since before the Second World War, sandwich construction has been widely used in aircraft and many structural applications. In recent years, sandwich panels are increasingly used in building structures particularly as roof and wall cladding systems. They are also being used as internal walls and ceilings. Because of their good thermal properties, they have been

524 used in cold-storage buildings. The structural analysis of sandwich panels with thin flat faces has been investigated as early as 1940's, particularly for aeronautical applications (Allen, 1969). However, research and development of sandwich panels with foamed facings began only in late 1960s, pioneered by Chong and his associates (Chong and Hartsock, 1993). Due to the increasing interest in the use of structural sandwich panels, a good deal of research has continued in recent years (Davies, 1993). In the building industry, the steel faces of sandwich panels are generally used in three forms: flat, lightly profiled, and profiled, as shown in Figure 1. The faces of sandwich panels serve various purposes. They provide architectural appearance, structural stiffness, and protect the relatively vulnerable core material against damage or weathering. Tensile and compressive forces are supported almost entirely by faces. Flat and lightly profiled faces can carry only axial forces, as their bending stiffness is negligible, whereas profiled faces can carry both axial forces and bending moments. Similarly, core of the sandwich panel has many functions. Basically, it keeps the faces apart and stabilizes them against local failures, and provides shear connection between faces. Hence, sandwich panels represent an excellent example of the optimum use of dissimilar materials.

Figure 1: Sandwich Panels Design of sandwich panels becomes complicated due to the presence of two dissimilar materials (i.e. core and faces) acting together. A large number of researches have been undertaken in sandwich construction to investigate their buckling behaviour and thus obtain rational design procedures. At present, there is only one design document called "European Recommendations for Sandwich Panels (ECCS, 2000)" for the design of sandwich panels. This document is mainly based on the metal faces and polyurethane or polyisocyanurate cores. Hence this document can not be adopted in its original form to design the Australian sandwich panels, which are the combination of high strength steel faces and polystyrene foam cores. i, b ~.1 ~' "

Figure 2: Local Buckling of Sandwich Panels (Jeevaharan, 1996)

b

Figure 3: Critical b/t Ratios of Profiled Sandwich Panels for Local Buckling

Under the action of different loading such as gravity, wind, snow, temperature gradient, and others, the profiled faces of sandwich panels are susceptible to elastic local buckling (Figure 2) due to axial compression and/or bending. The conventional design treatment for the local buckling phenomenon of sandwich panels utilizes the concept of effective width. But this is applicable only for a relatively low b/t ratio (see Figure 3), generally less than about 200 and can not be extended to thinner plates. Davies and Hakmi (1991) proposed an enhanced effective width formula based on limited number of tests. These tests did not cover a wider range of plate width/thickness (b/t) ratios. To cope with the inadequacy of the design rules, a detailed investigation of local buckling behaviour of profiled

525 sandwich panels was conducted using extensive series of laboratory experiments. An investigation using finite element analysis is currently under way and the results are not yet available. Therefore, this paper presents the background theory of local buckling, details of experimental studies, local buckling results and their comparison with available design formulae.

2. RESEARCH INTO LOCAL BUCKLING BEHAVIOUR

2.1 Theory Thin steel faces supported by a thick foam core can be considered as a plate on elastic foundation. Mathematically the problem can be modelled as shown in Figure 4. A simply supported rectangular plate is subject to an applied stress p along the two transverse edges. The longitudinal edges of the plate are assumed to be simply supported. The length of the plate in x-direction is large compared with the width. The critical buckling stress O'crof this plate is given by (Davies and Hakrni, 1990):

Figure 4: Steel Plate in Compression with Core Support

~, =K

(1)

12(1-v .r2)

where K is the buckling coefficient and given by K

=

1 + n 2r

+ Re[1 + n2r

24(1-v2)(1-v~)Er and

R = zc3(1 + v c)(3 - 4v c)Et

,=

Ibl '

(2)

(actual)

.simpli. e .

(3)

Jr 3 Es The critical buckling stress itself does not provide any satisfactory basis for design, but it can be used as a useful design parameter. It is well known that in cold-formed steel design, the width to thickness ratios (b/t) are usually large, hence local buckling becomes a major design criterion for the compression members. Buckling of these elements may occur at a stress level lower than the yield stress of steel. However, the elastic local buckling does not necessarily represent the collapse of the members. Failure will occur at a load higher than the elastic-buckling load. Thus, post-buckling behaviour is important for the optimum design of cold-formed steel members and this raises a significant analytical problem. For the cold-formed steel members without any foam support, such local buckling problems are treated for design purposes by utilising the concept of effective width. A widely used effective width formula in many national and international standards including the European recommendations is the "Winter" formula. The principle associated with this is: the width b of the compressed element is replaced by the reduced value of the width, be#, when calculating the section properties for use in the design calculations. The "Winter" formula takes the form:

526

b~.tr=pb p = ~ [ 1 - - - 0 ~ 2 1 for 2>0.673 p = 1.0

for 2 < 0.673

(5)

E,K where be#--- effective width, fy - yield stress of the steel, Es = Young's modulus of steel, t - thickness of the steel plate, K - buckling coefficient (- 4.0 for simply supported plate without foam core). This effective width approach can be extended to the profiled faces of sandwich panels by modifying the buckling coefficient K to take into account of the core support. As seen from Eqn. 3 and 4, the influence of the composite action between faces and core is modelled by the dimensionless stiffness parameter R. The critical buckling stress o'er can be found by minimising the buckling coefficient K with respect to the wavelength parameter ~. Hence the condition OK / 0r = 0 from Eqn. 2 gives: 2

2n4r

R(2n2r z + 1)(n202 + 1)_!2 = 0

(6)

/

If the elastic support given by the core is ignored, the buckling coefficient, K in Eqn. 1 has the value K = 4. 0. If the elastic support provided by the core is utilised, Eqn. 6 can be solved for r using a suitable numerical method and K can be evaluated. For design purposes, a number of explicit mathematical formulae have been proposed for the solution of Eqn. 6 to determine buckling coefficient K for sandwich panels with profiled faces. These mathematical formulae are given below: 1. By Hassinen (1991) based on the half-space assumption

K-4-O.415R+O.703R 2

with

F1

R= b Ec

2. By Hassinen (1991) based on the simplified foundation model

K-4-O.474R+O.985R z

7L~-:j

with R -

.EcGc

E/2

(7)

(8)

3. By Davies and Hakrni (1990) based on the simplified foundation model

with R = 12(1-v2)x/EcGclb[

K = [16+ll.8R+0.055R2] '/z

7/'3El

(9)

4. By Davies and Hakmi (1990) based on the simplified foundation model by replacing R with 0.6R K =[16+ 7R + 0.02R2] ~/2 (10) 5. By Mahendran and Jeevaharan (1999) based on the simplified foundation model to include a greater range of R from 0 to 600 K = [16 + 4.76R!29] i/2

with R=

(11) ~3E:

In the current European Recommendations for Sandwich Panels, Part I: Design (ECCS, 2000), the following formulae have been recommended for predicting the value of K. These expressions are applicable for 0 _
K=[16+7R+O.OO2R2] 1'2

with

R=O.35~/EcQ[b] 3 E:

(12)

In order to determine the ultimate strength, K derived from one of the equations from 7 to 12 can be used in the effective width method by substitution into the general eqn. 5 for plates. However, none of the explicit formulae explained above can predict suitable values for the wider range of b/t ratios. For

527 the low range (b/t less than 200), their predictions may be good. But for thinner plates, these formulae either overestimate or underestimate the strength. Figure 5 demonstrates the inadequacy of the design method when Davies and Hakmi's test results are compared with predictions from the design method using Eqn. 9 for K and Eqn. 5 for beth. 1

,

0.8

~

~ 0.6 ~9 0.4 0.2

9

Winter Formula[ Results

Test

o 0

0.5

,, ,,, 1.5

1

2

2.5

2

Figure 5: Effective Width (bert/b) vs Modified Plate Slenderness 2

2.2 Experimental Investigation 2.2.1 Test Specimens In order to investigate experimentally the local buckling behaviour of sandwich panels, a series of laboratory tests was conducted on plate elements supported by polystyrene foam as used in the profiled sandwich panels. The experiments were essentially compression tests of fiat steel plate elements with varying b/t ratios. To cover a large range of b/t ratios (between 50 to 500), both the thickness and width of the plates were varied. Also to observe the effects in different grades of steel, the experiments were conducted in two grades, one mild steel with a minimum yield stress of 250 MPa and the other high strength steel with a minimum yield stress of 550 MPa. For each grade of steel, different nominal thicknesses were chosen. The widths (b) of the plates chosen were 50, 80, 100, 120, 150, 180, and 200 mm. The lengths of the plates were chosen as three times the width (b) plus 10 mm for clamping. As the foam thickness has negligible effect on the buckling strengths (Jeevaharan, 1996; McAndrew, 1999), a constant thickness of 100 mm was used throughout the test. The steel faces and foam were glued to each other by using a suitable adhesive. The specimens were tested at least after 48 hours of attachment to ensure the adhesive was set and steel face and foam were joined properly. Details of experimental program and test specimens are given in Table 1.

2.2.2 Test Set-Up and Procedure A specially constructed test rig was used to hold the test specimen for the compression test. Test rig consisted of a base plate and two vertical supports. Two vertical clamps used to hold the steel plates were attached to vertical supports. The vertical supports were adjustable in both horizontal and vertical directions to accommodate the required plate width and length, respectively. Plate lengths up to 600 mm can be held between these vertical support edges. The boundary conditions along the longitudinal edges of the plate were designed to simulate the real condition present on the plates of the profiled face supported by adjoining plates. The vertical clamps allow the vertical displacement and free rotation about the edges, hence well representing the simply supported condition of longitudinal edges. The test specimens were placed in the test rig between two loading blocks. A schematic diagram of the test rig is given in Figure 6. The compression tests of the steel plate elements were carded out using a Tinius Olsen Testing Machine. Two linear variable displacement transducers were used to measure out-of plane deflections. The axial displacement was recorded by the Tinius Olsen Machine. A compression load was applied at a constant rate of 0.5 mm/min until the failure of the specimen. The buckling and ultimate loads of each test specimen were recorded. The buckling load was noted by visual observation of plate buckling. The ultimate load was the maximum load carded by the specimen, and was directly taken from the machine reading. Hence the buckling load was approximate, but the ultimate load could be considered exact.

528

TABLE 1 TEST PROGRAM TO INVESTIGATE LOCAL BUCKLING BEHAVIOUR Plate Test

Width

Series

b

1

|

2

|

(mm) 50 50

3 4 5

E (GPa) 226 230

Ratio Spec. 52.63 1.00 62.50 ~ 0.80

bmt , (MPa). (GPa) 0.93 326 216 0.73 , 345 , 217

Ratio 53.76 68.49

, 0.60 0.42 , 0.95 . 0.80 . 0.60 0.42

682 726 637 656 682 726

235 239 226 230 235 239

83.33 i 0.60 119.05.0.40 84.21 I 1.00 100.00 0.80 133.33 0.60 190.48 0.40

0.54, 0.39, 0.93 , 0.73 . 0.54. 0.39

, 218 220 , 216 . 217 . 218 220 |

92.59 128.21 86.02 109.59 148.15 205.13

0.95

637

226

105.26

1.00

0.93 . 326

. 216

107.53

100

0.80 , 0.80

656

230

125.00, 0.80

0.73 . 345

, 217

100 100

0.60 , 0.60 0.42 , 0.42 0.95 0.95 0.80 0.80

682 726 637 656

235 239 226 230

166.67/ 0.60 238.101 0.40 126.32, 1.00 150.00 0.80

0.54 , 0.39 , 0.93, 0.73 i

, 218 , 220 , 216 217

136.99 185.19 256.41

0.60 i 0.95 0.80 0.60 , 0.42, 0.60 , 0.42 I 0.95 , 0.80 ,

682 637 656 682 726 682 726 637 656

235 226 230 235 239 235 239 226 230

200.00, 157.89, 187.50 , 250.00, 357.14, 300.00, 428.57, 210.53, 250.00,

|

9

. 100

11

12

|

80 80 80

13 , 120 14 i 120 15 1 1 2 0 16,: 150 17 1 5 0 ,

18

, 150

19

, 150

20

, 180

21

, 180

2 2 , 200 23 , 200

S p e c . . bmt 0.95 0.95 0.80 , 0.80 0.60 0.42 0.95 0.80 0.60 0.42

.

0.95.,

,

0.60 0.95 0.80 0.60 0.42 0.60 0.42 0.95 0,80

b/t

.

(mm)

. fy

|

360 368, 326 345 360 368

360 368 326 345

E

b/t

fy

6 7 8 10

G250 Steel Plates Thickness ... Measured

(MPa) 637 656

50 50 80

|

G550 Steel Plates Thickness Measured

|

0.60 0.54 360 , 218 1.00 0.93 , 326 216 0.80 0.73 345 217 0.60 0.54 , 360 , 218 0.4,0 0 . 3 9 1 3 6 8 220 0.60 0.54 , 360 218 0.40 0 . 3 9 , 368 1 2 2 0 1.00 0.93 , 326 , 216 0.80 0.73 , 345 , 217

:

0.60, 0.6,0 682 235 333.33, 0.60 0 . 5 4 , 3 6 0 , 218 24 , 200 0 . 4 2 . 0.42 726 239 476.19. 0.40 0 . 3 9 . 3 6 8 . 220 I 25 . 200 Note: fv - measured yield stress of steel, E - measured Younlz's modulus b/t ratio - plate width b/bmt, Spec. - specified thickness bmt - estimated base metal thickness based on measured total coated thickness

F i g u r e 6" S c h e m a t i c D i a g r a m o f T e s t R i g

129.03 164.38 222.22 16!.29 205.48 277.78 384.62 333.33 461.54 215.05 273.97 370.37 512.82

529 2.2.3 Results and Discussion

In this section, test results obtained from the compression tests of foam supported steel plates and their comparison with available design formulae are presented. Altogether 25 test results for G550 and G250 steel plates each are reported here. 180

140 .... DaviesFormula (Eqn. 9) ....... Mahen&an's Formula (Eqn. I I)

120 100

_ .:

..

160

/

__

140 9 120 -

..

80

j Davies Formula (Eqn. 9) ' J . . . . . . . Mahen&an's Formula (Eqn. I 1) i

! .....

100 9

K 60 40

.

..,.o'" ~

9

609

40-

~~~~..

20

~ "

.-

.,-!"

80-

f

+.. . ....-"f 9

/

i /

~.ccs2ooo(r.q~. 12)

9 9

I

9 9 ,,,.S~'" .~-""

200

0 0

100

200

300

400

500

0

100

200

b/t

300

400

500

600

b~

(a) G550 Steel (b) G250 Steel Figure 7: Buckling Coefficient K vs b/t Ratio Buckling coefficients K evaluated from different design formulae together with the test results are plotted against b/t ratios in Figure 7 for both G550 and G250 steel plates. The experimental K was calculated by using Eqn. 1. The non-dimensional parameter R was calculated by using simplified foundation model using Eqn. 4, and then the predicted K values were determined using Eqns. 9, 11 and 12 as proposed by different researchers. In calculating R and K, the experimental values of material properties of polystyrene foam and steel faces were used. These values were taken from Jeevaharan's (1996) experimental results. The experimental values for foam are Ec=3.8 MPa, Gc=1.76 MPa, v~= 0.08. The experimental values of Young's modulus and yield stress for both G550 and G250 grades of steel with different thicknesses are given in Table 1. Poisson's ratio for steel is v = 0.3. As seen in Figure 7, for low b/t ratios, the experimental K values are close to those predicted by all three Equations. But, for higher b/t ratios, Eqn. 9 and Eqn. 11 overestimate the K values whereas Eqn. 12 underestimates it. However, for higher b/t ratios, K predicted by Eqn. 11 (proposed by Mahendran and Jeevaharan, 1999), are nearer to the experimental values in comparison to Eqn. 12 (proposed by Davies and Hakmi, 1990). This shows that Eqn. 11 is more suitable for high b/t ratios than Eqn. 9, however, it is still inadequate for slender plates with very high b/t ratios. i_ Davies Formula(Eqn. 9) . . . . . . . Mahendran'sFormula(Eqn. !!) . . . . . ECCS2000 (F.qn. 12) , TestResults

\ 0.6

~0

"~ 0.4

_ 9 ~1~

100

200

[ ] I i

0.8

Results

0.2

400

i

0.4

. .. . . . . . . . . .

9................

300

' ~ T e s t

0.6

500

0

*

0

! 00

200

300

**

400

b:'t

(a) G550 Steel

(b) G250 Steel

Figure 8: Effective Width of the Steel Elements Stiffened by Foam

t

9

500

600

530 As explained earlier, the purpose of evaluating the local buckling stress and buckling coefficient is to determine the effective width of the steel plates. In designing the cold-form steel structures, effective width is an important design parameter. Hence the available formulae should be able to predict realistic values of effective width. Figure 8 shows the comparison of experimental values of effective width of steel plates with the predictions made by different formulae (Eqns. 9, 11, 12). It can be observed that the predicted value of effective width is always higher then the experimental value. For lower b/t ratios, the difference is reasonably low, but for higher b/t ratios the formulae predict very high effective width values compared with experimental results. So for slender plates, none of the formulae could estimate reasonable values of buckling coefficient K and effective width beff. With this extensive series of test results, it can be concluded that the current design formulae are not applicable for slender plates. Further understanding of this problem and improvement to design formulae can be made after the completion of finite element analysis, which is currently under way.

3. CONCLUSIONS In recent times, the use of sandwich panels in the Australian building industries has increased considerably. However, lack of design standards in Australia has been a problem for designers. Therefore, research into the behaviour of structural sandwich panel is essential. Local buckling behaviour of profiled sandwich panels was investigated using an extensive series of laboratory experiments. The experimental results were compared with currently available design formulae. The results indicate that these design formulae can not be used for slender plates. Improved buckling and strength formulae have to be developed for these plates based on experimental and finite element analyses.

4. REFERENCES

1. Allen, H.G. (1969). Analysis and Design of Structural Sandwich Panels, Pergamon Press, New York, U.S.A. 2. Davies, J.M. (1987). Design Criteria for Structural Sandwich Panels. Journal of Structural Engineering 65A:12, 435-441. 3. Davies, J.M. (1993). Sandwich Panels. Journal of Thin-Walled Structures 16, 179-198. 4. Davies, J.M. and Hakmi, M.R. (1990). Local Buckling of Profiled Sandwich Plates. Proc. IABSE Symposium, Mixed Structures including New Materials, Brussels, September, 533-538. 5. Davies, J.M. and Hakmi, M.R. (1992). Postbuckling Behaviour of Foam-Filled Thin-Walled Steel Beams. Journal of Construction Steel Research. 20, 75-83. 6. Davies, J.M., Hakmi, M.R. and Hassinen, P. (1991). Face Buckling Stress in Sandwich Panels. Nordic Conference Steel Colloquium, 99-110. 7. Davies, J.M. and Heselius, L. (1993). Design Recommendations for Sandwich Panels. Journal of Building Research and Information, 21:3, 157-161. 8. ECCS, (1991). Preliminary European Recommendations for Sandwich Panels, Part 1, Design. European Convention for Constructional Steelwork (ECCS), No. 66. 9. Hassinen, P., (1995). Compression Failure Modes of Thin Profiled Metal Sheet Faces of sandwich Panels, Sandwich Construction 3-Proceedings of the Third International Conference, Southampton, 205-214. 10. Mahendran, M. and Jeevaharan, M. (1999). Local Buckling Behaviour of Steel Plate Elements Supported by a Plastic Foam Material. Structural Engineering and Mechanics, 7:5, 433-445. 11. McAndrew, D. and Mahendran, M. (1999). Flexural Wrinkling Failure of Sandwich Panels with Foam Joints. Proc. 4th lnt. Conf. Steel and Aluminium Structures, Helsinki, Finland, 301-308. 12. Pokharel N. and Mahendran, M. (2000). Buckling Behaviour and Design of Sandwich Panels. Proc. of the Queensland Civil Engineering Postgraduate Conference, December 2000, 27-38.

Section IX O P T I M I Z A T I O N AND SENSITIVITY ANALYSIS

This Page Intentionally Left Blank

Third International Conference on Thin-Walled Structures J. Zarag, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

533

OPTIMAL DESIGN OF STEEL TELECOMMUNICATION TOWERS BY INTERIOR POINT ALGORITHMS FOR NON-LINEAR PROGRAMMING N. A. Cerqueira l, G. S. A. Falcon 1, J. G. S. da Silva 2 and F. J. da C. P. Soeiro 2 1North Fluminense State University, UENF Civil Engineering Department, Campos, Rio de Janeiro, Brazil (*) 2State University of Rio de Janeiro, UERJ Mechanical Engineering Department, Rio de Janeiro, Brazil (*)

ABSTRACT This paper proposes an analysis methodology for optimal design of steel telecommunication towers using non-linear programming techniques and including safety against global instability. The methodology seeks to minimise the weight of the steel structure and to study the critical load associated to global stability of the tower, satisfying mechanical and geometric constraints. Usual constraints, such as displacements at the top section of the structure, stresses in the main legs of the steel telecommunication tower and member buckling are also included. In the optimisation process, the structural behaviour is verified through a mixed model of truss and spatial beam. The optimisation algorithm used in this work is the interior point technique for non-linear programming. This algorithm requires an initial feasible configuration and it obtains a sequence of feasible points with decreasing cost. The fact of only dealing with feasible points makes this technique particularly suitable to structural optimisation. The structural models, in a first order elastic analysis, are investigated in terms of their qualitative and quantitative behaviour focusing on the actual structural safety provisions. KEYWORDS Structural Optimisation, Structural Analysis, Steel Structures, Steel Telecommunication Towers INTRODUCTION Optimal design means the obtainment of the best solution for a certain structural model, normally associated to its cost. In this work the design of a steel telecommunication towers without affecting integrity and durability is aimed. In this context this paper presents an automatic procedure for the optimum design of steel telecommunication towers using non-linear programming techniques and including safety against global instability.

534 The instability problem is originated by the interaction of buckling modes associated to critical loads and it would make unfeasible the practical use of the obtained structures. Thus, in the present work, we include design constraints that verify a lower limit for critical loads and the interaction among global buckling modes. The analysis methodology seeks to minimise the weight of the steel structure and to study the critical load associated to global stability of the tower, satisfying mechanical and geometric constraints. Usual constraints, such as displacements at the top section of the structure, stresses in the main legs of the steel telecommunication tower and member buckling are also included. In this approach, the problem consists on size and shape optimisation. The size optimisation problem consists of the determination of the optimal cross section of the bars, while shape optimisation determines the optimal nodal location. Thus, the design variables are the cross sections dimensions and a set of coordinates of some nodal points. Besides the usual constraints on element stresses and nodal displacements, the model defines a set of constraints on the nodal coordinates. These constraints correspond to symmetry conditions or to the alignment of some set of nodes on the same bar. Beginning from a feasible initial configuration this technique defines an iterative sequence that converges to the optimal solution of the problem. It is requested to calculate, at each iteration and for the current design variables, the objective function, the constraint values and the corresponding gradients. The fact of only dealing with feasible points makes this technique particularly suitable to structural optimisation. In the optimisation process, the structural behaviour is verified through a mixed model of truss and spatial beam. The optimisation algorithm used in this work is the interior points technique for nonlinear programming discussed in Herskovits & Falcon (1997) for the solution of Karush-KunhTucker first order optimality conditions. A computational environment was developed for the optimum design of steel telecommunication towers. Independent computational modules for geometric modelling, structural analysis, sensibility analysis and the optimisation algorithm compose this computational environment. The finite element method is used for the structural analysis, Estrella Jr (1998), allowing for futures applications of the analysis methodology to other types of structures. For sensibility analysis we use the finite difference techniques while semi analytic techniques are under development. Structural models, in a first order elastic analysis, are investigated in terms of their qualitative and quantitative behaviour focusing on the actual structural safety provisions.

OPTIMAL DESIGN Optimal design consists in the numerical minimisation of a function submitted to a set of nonlinear smooth equality and inequality constraints. We want to find the best design that minimises a given cost and satisfies all the requirements of feasibility. Considering X the design variables, f(X) the objective function, gi(X); i=1,2 ..... m, the inequality constraint and hi(X); i=1,2 ..... p, the equality constraints, the optimisation problem can be written in all generality, Eqn. 1, where g and h are functions in 9t" and m, p and n are the dimension of inequality constraints, equality constraints and project design, respectively, as shown in Eqn. 1.

535 Minimise f(X) Subject to g(X) <_0 h(X) = 0

(1)

The objective function represents the design quality on a quantitative basis. For this purpose, in this work the steel telecommunication tower own weight is defined as a function of the design variables which are the tower members cross sectional dimensions as shown in Eqn. 2, where ai represent the cross section area of the i-th bar, ), is the specific weight of the structural members material and li is the length of the i-th tower member. nd

f ( X ) = E a i )' li

(2)

i=l

The main objective of the developed strategy is to minimise the weight of the steel structure and to include the critical load associated to global stability of the tower, satisfying mechanical and geometric constraints. The mechanical constraints are the displacements at the top section of the structure, stresses in the main legs of the steel telecommunication tower. Members buckling are also included. Stress constraints are given by Eqn. 3, where oi is the i-th bar stress and O~dm is the corresponding maximum allowable stress.

g~(X)= IIlOil 1

(3)

aadm Nodal displacements constraints are shown in Eqn. 4, where uj is the displacement of the j-th degree of freedom and U~dmis the correspondent maximum allowable displacement.

d

gj(X) =

lujl

------

1

Uadm

(4)

One of the shortcomings of an optimum design is that structural systems sensitive to initial imperfections can be generated, mostly when the structures have coincident critical loads for different buckling modes, due to interactions among these modes. This fact reduces the structural failure load. Thus, besides the usual constraints, associated to allowable stresses and nodal displacements, this analysis methodology adds one more constraint related to the tower global structural stability. Despite the importance for practical applications, this constraint has only been considered by a few authors. So, to guarantee global structural stability safety it is necessary to include the constraint expressed in Eqn. 5, where 2Ler represents the first critical load of the structural o system and ~. is its minimum allowable value.

gk~(X)

=

1

~'~ -

(5)

536 Upper and lower bounds, Xvub and Xvlb, respectively, are also considered for the design variables Xi, which are cross-section dimensions and nodal coordinates. The correspondent constraints gwb and gv~bare presented in Eqn. 6 and 7. X =~-1 grub(X) Xvub

gvlb(X) = 1 - - ~

X

Xvlb

(6)

(7)

THE FEASIBLE INTERIOR POINT METHOD

Feasible direction algorithms are an important class of methods for solving constrained optimisation problems. At each iteration, the search direction is a feasible direction of the inequality constraints, and, at the same time, a descent direction of the objective. A constrained line search is then performed to obtain a satisfactory reduction of the function, without loosing the feasibility. The fact of giving feasible points makes feasible direction algorithms very efficient in engineering design, where function evaluation is in general very expensive. Since any intermediate design can -be employed, the iterations can be stopped when the cost reduction per iteration becomes small. The stop criteria adopted in the present work is a tolerance for the absolute value of objective function in three consecutive iterations and also a small value for the search direction modulus. In the present research we use the Quasi-Newton feasible direction algorithms, which uses fixedpoint iterations to solve the non-linear equalities, included in the Karush-Kulm-Tucker optimality conditions. With the objective of ensuring convergence to admissible points, the system is solved in such a way as to have the inequalities in Karush-Kuhn-Tueker optimality conditions satisfied at each iteration. The algorithm used in this work is presented and the computer implementation is discussed in Herskovits & Falcon (1997).

EXAMPLES In this work a computational environment was implemented for steel telecommunication towers optimisation. The main objective is to obtain a minimum weight structure satisfying mechanical and geometric constraints. Independent computational modules for geometric modelling, structural analysis, sensibility analysis and the optimiser algorithm compose this computational environment. Initially some examples associated to structures with known behaviour are tested to validate the developed methodology and also the use of the critical load constraint is verified from the standpoint of size optimisation. In sequence, a study related to the structure of an actual steel telecommunication tower is presented. Table 1 shows various optimisation cases developed in this effort where X represents a vector with design variables, [a] are dimensions and [~] nodal coordinates.

537

TABLE 1 OPTIMISATIONCASES Case/Objective. A - Size Definition of optimal bars cross sections Case 1" With constraint load factor Case 2: Without constraint load factor ,,

Design.Variables

constraints

X=[a]

g~,gd,glf,gvlb,grub g~,ga, gv)b,grub ,,

B- Shape Definition of optimal coordinates of free nodal points Case 1" With constraint load factor Case factor . . 2". Without . . constraint . . . load .

,

gS, gd,glf gvlb,grub g~, gO gvlb,grub

x:[x]

Ten-Member Cantilever Truss

The plane truss of ten bars and six nodes is shown in Figure 1. The structure is subjected to a load, F, equal to 444.80kN at nodes 2 and 4 in y-axis direction, Figure 1. The material and geometrical properties are listed in Table 2. In sequence, the plane truss was optimised for the following cases: Case A - The design variables are the cross sections of all bars: X = [al,a2, ,a~0]; Case B - The design variables are the coordinates of nodes 1 and 3: X = [Z~x,Z~y,Z3x,Z3y].

~

5

TABLE 2

3

TEN-MEMBER CANTILEVER TRUSS PROPERTIES Zl

....

Y F

X

Elasticity Modulus Specific Weight Stress Limit Displacement Limit Load Factor Limit Initial Cross Section Minimal Cross Section

Figure 1: Ten-member cantilever truss

E(MPa) ~(Kg/mm3) ' t~adm(MPa) Uad~(mm)

~,adm 'ao(mm z) amin(mm ~)

6.90X 104 2.80x 10-6 175.0 20.0 1.0 24193.50 64.50

Table 3 presents a comparison of the optimum solutions obtained using the proposed analysis methodology for the ten-member cantilever truss. TABLE 3 OPTIMISATION RESULTS. TEN-MEMBER CANTILEVER TRUSS

Structure

Case

. . . . Ten Bar Truss

A-i A'2 B-1 B-2

Design Variables Number 10 " 10 .... 4 4

Total Constraints Number 19 18 19 18

Initial weight

(kS) 71.44 71.44 71.44 71.44

Finai weight

(kN) 42.41 23.30 64.19 53.59

In case A-2 (Table 3) where only size optimisation was considered without the critical load constraint, three constraints became active in the optimum. These constraints are the vertical displacements in nodes 1 and 2, and traction in bar 5 which links nodes 3 and 4 (Figure 1). Buckling critical factor was 0.25, less than unity, which is the minimum safe value to avoid

538 problems of global instability. In case A-l, size optimisation was considered with the critical load constraint. In this case the critical load constraint was activated, giving a more conservative solution with buckling critical factor equal to unity. In case B-2 (Table 3), the shape optimisation problem without the critical load constraint was solved. The vertical displacement of node 2 was the active constraint, reaching the upper bound. In this ease the calculated buckling critical factor was 1.26. In case B-I, the critical load constraint was considered and activated very quickly in the problem solution with a buckling critical factor equal to unity. The results related to size optimisation stress the importance of the use of the critical load constraint to guarantee the structure stability. In the case of the shape optimisation problem a better understanding of the sensibility analysis needs to be achieved. Steel Telecommunication Tower

The steel telecommunication tower investigated here is a truss type geometry configuration. It should be pointed out that the present work is based on a real structure, designed, fabricated and erected by a state-owned company, EMBRATEL (1998). Actual member properties were used in this analysis. The structure possessed a square cross section divided into two segments: the lower part was pyramidal while the upper part was trapezoidal. Rolled angle sections connected by bolts composed the main structure. The geometry configuration of the simply supported tower structural system, depicted in Figure 2, presented a total height of 40.0m.

TABLE 4 STEEL TRANSMISSION TOWER PROPERTIES

Elasticity Modulus Specific Weight Stress Limit Displacement Limit Load Factor Limit Initial Cross Section Minimal Cross Section

E(MPa) 7(Kg/mm 3)

2. lxl 0 ~ 7.90x10 "6 t~adm(1V[Pa) 250.0 Uadm(mm) 360.0 ~adm 1.0 a0(mm2t 2840.0 amin(mm") 239.0

Figure 2: Tower geometry layout Based on an extensive parametric investigation developed by Policani et al (2000), a modelling strategy combining three-dimensional beam and truss finite elements was proposed. In this methodology the main structure uses beam elements while the bracing system utilises truss elements.

539 The used beam finite elements presented seven degrees of freedom per node, six associated with translation and rotation displacements in space and the last representing the warping degree of freedom. The acting load considered in the analysis where self-weight and two wind load cases (perpendicular and diagonal to the tower face). The material and geometrical properties are listed in Table 4. Initially the steel telecommunication tower was optimised for the following cases: Case A - The design variables are the cross sections of all bars: X=[a~,a2, ,an] Case A~ - The vector with variables is X=[al,a2], where a~ represents the cross section of the tower main structure members, and a2 the cross section of the bracing system bars; Case A2 - The vector with variables is X=[al,a2, a3,a4], where al e a2 represent the cross section of the tower main structure members, and a3 and a4 the cross section of the bracing system bars. Table 5 presents a comparison of the optimum solutions obtained using the proposed analysis methodology for the steel telecommunication tower. TABLE 5 OPTIMISATIONRESULTS. STEELTELECOMMUNICATIONTOWER

Structure

Al-1

Steel A1-2 T e Iecomm uni carl on A2-1 Tower A2-2

Total Constraints Number 302

Design Variables Number 2

Case

.

.

.

.

.

Initial weight (kN)

Final weight (kN)

.

247.60

104.62

2

301

247.60

95.65

4 4

302 301

247.60 247.60

79.88 69.43

.

.

.

In all the studied cases the initial design was the same. A structural shape with 2840.0 mm 2 of initial cross sectional area was used in all members of the tower. This resulted in a structure with an initial weight higher than the weight of the actual tower. This structure did not violate any of the design constraints. In case A~-I, a size optimisation problem, the critical load was active. The final buckling critical factor was the unity. In case A~-2 a lighter structure is obtained but with a critical factor of 0.34, which is about one third of the recommended value to avoid tower global instability. In this situation the active constraint was in stresses. In case A2-1, also a size optimisation problem, the active constraint was again the critical load. The buckling critical factor reached unity. In case A2-2, again, as expected, buckling critical factor was equal to 0.20, five times lower than the recommended value. In this case, again the stress constraint was activated. One can verify that as the number of sets of structural shapes increases the tendency is to obtain a lower final weight for the telecommunication tower structure and, thus, closer to the weight of the original structure. The authors intend to continue this work using an increasing number of sets of structural shapes to obtain a better tower configuration. It is convenient at this point to emphasise again the idea that it is very important to use the critical load constraint in order to obtain a structure that is safe against global instability. The solution will be closer to reality.

540 FINAL REMARKS This paper proposes an automatic procedure for the optimum design of steel telecommunication towers using techniques of non-linear programming and including safety against global instability. The analysis methodology seeks to minimise the weight of the steel structure and to study the critical load associated to global stability of the tower. The most common constraints, such as displacements at the top section of the structure, stresses in the main legs of the steel telecommunication tower and member buckling are also included. The proposed analysis methodology, substantiated by the finite element method, has shown coherent results when the global structural behaviour was considered. Despite these results, it becomes clear to the authors that the continuation of this research has to deal with case studies and comparisons with other steel towers found in practice. The inclusion of the critical load constraint has shown to be very important. Without that consideration the optimum solutions present extremely low critical loads, which is in disagreement with current design practice. One can note clearly that the inclusion of the critical load constraint is very severe. Nonetheless it is difficult to reduce the objective function keeping critical load factor above unity. The sensibility analysis is very important for this methodology and needs to be better evaluated. In this work usual finite difference techniques were used and the results have significant variations depending of the adopted step when perturbing the several variables of the problem. Moreover, this technique produces a higher computational cost.

REFERENCES

Empresa Brasileira de Telecomumcaq~es, EMBRATEL (1998). "'Mem6rias de C/dculo e Plantas do Projeto Executivo de uma Torre Padr~o de Telecomunica~fies". Correspond~ncia Privada. Estrella Jr. L.F.E. (1998). "FINLOC Program: Structural Analysis Program for Trussed or Framed 3D Steel Structures". North Fluminense State University, UENF. Herskovits J. and Falcon G.S. (1997). "On the Computer Implementation of Feasible Direction Interior Point Algorithms for Non-linear Optimisation". Structural Optimisation Journal. Vol. 14, pp. 165-172. Policani M.N., Silva J.G.S. da, Estrela JOnior L.F., Vellasco P.C.G.S. and AndradeS.A.L. (2000). "Structural Assessment of Steel Telecommunication Towers", International Conference on Steel Structures of the 2000's, Steel Structures 2000, Istanbul, Turkey, September 2000.

Third InternationalConferenceon Thin-Walled Structures J. Zarag,K. KowaI-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

541

DESIGN OPTIMISATION OF SHELL STRUCTURES WITH DIMENSIONAL ANALYSIS RESOURCES

Maria del Pilar A. M. R. C. Gomes ~ Department of Structures of Architecture Faculty, Federal University of Rio de Janeiro, Cidade Universithria, CEP: 21945-970 Rio de Janeiro, Brasil

ABSTRACT The objective of this work is the determination of the limit load of shell structures. The limit load is determined by a computer program for geometrically non linear analysis based on Finite Element Method. The computer program has been developed in the Civil Engineering Department of COPPE/UFRJ and it was included in the D. Sc. author's thesis. This software uses degenerate quadratic elements for shells such as Serendipity, Heterosis or Lagrangiam. Shell structures divided into finite elements with different thickness values are presented to a computer program. The results of this program analysis and the use of the n-numbers described through the dimensional analysis allow the development of graphics. With these graphics, it is possible to determinate with great precision the limit load of shells with similar shape and also to modify the shell thickness if necessary, which makes it possible to choose the best solution for a particular shell with similar shape. The limit load obtained by the graphics allows the choice of the ideal thickness for a specific shell type. According to shell dimensions and material, it leads to the optimisation of its design.

KEYWORDS Shell Design Optimisation, Reversal Shell Structures Analysis, Dimensional Analysis, Reversal Cover Shell Structures, Reversal Thin-Walled Shell Structures, Shell Design with Graphics.

INTRODUCTION The thin-walled structures can be optimised using dimensional analysis resources. With the aid of a shell computer program it is possible to analyse any shell structure. The computer program allows physically or geometrically non linear analysis. An iterative incremental analysis was used and the integration may be selective or reduced.

542 The structure was divided into finite elements, Serendipity elements with eight nodes were used. The elements were divided in layers. By increasing the load, it is possible to see where the damage occurs in Gauss points into the layers. With the computer results, it is possible to prepare a curve using the load and corresponding deflection. The principal parameters, enough for the analysis, were related using dimensional analysis. The load can be horizontal, vertical or to have any other direction. Graphics for reversal shell structures with conoidal shape were prepared according to the analysis of a uniformly distributed load acting in normal direction in relation to the shell surface. The graphics were prepared with the computer results and dimensional analysis concepts. Structures to cover walk ways, with conoidal shell shape and structures to protect areas against wind acting directly were idealised in this way. METHODS OF ANALYSIS The structure was divided into finite elements, six Serendipity elements with eight nodes were used, four of them in the corners, and the remaining four in the middle of the edges. A quarter of the structure was analysed using the symmetry.

~y \ /

/\ ~26

22; 17~

18

14 i 9'

!.0

12

27 28

29

t23

24

19 .20

21

Y

I H 15

16

ll |2

13

34

]

5//~]g

Figure 1" Conoidal shell finite elements

543 The elements were divided in six layers, numbered from the base to the top. Increasing the load, it is possible to see where the damage occurs in four Gauss points inside the layers. With the computer results, it is possible to prepare a curve using the load and corresponding deflection. Using dimensional analysis the principal parameters, enough for the analysis, were related. The parameters corresponds to the structure shape, the load and the material resistance. The shell can be idealised to cover an area, to contain a fluid or in many other uses. Structures to cover walk ways, with conoidal shell shape and structures to protect areas against wind acting directly were idealised using this technique. The load can be horizontal, vertical or have any other direction. Graphics for reversal shell structures with conoidal shape were plotted analysing a uniform distributed load acting in normal direction in relation to the surface. The graphics were prepared using the computer results and dimensional analysis concepts. DIMENSIONAL ANALYSIS The parameters correspondent to the shape of the shell structure, the load and the material resistance were related using dimensional analysis. The parameters of the shape were related to the length of the shell. For the loads, distributed load was used, normally applied against the shell surface. A matrix was prepared with these parameters, and three n-numbers were determined by solving the two equations. The related parameters are:

where:

H 1 0

(~0

~1

~2

q -2 1

-2 1 ~3

~4

~5

h = shell thickness % = limit tension of the material L= length of conoidal shell projection 6 = vertical displacement q = distributed load

The equation system is: O~l- 20~2+ 0~3 + o~4-2 % = 0

(1)

(Z2 + 0~5 = 0

Solving this equation system, the n-numbers are: n l -- q/(~0

(2)

n2 = 5/h n3 = L/h The first n-number corresponds to the relationship between the uniform distributed load and the material resistance, the second one presents the relationship among the displacement and the thickness, and the third one, among the length and the thickness.

544 The graphics are used to verify if it is possible to design a similar shell with different parameters, for instance by choosing the length of the shell, the thickness and the allowable displacement. With the relationship among the length and the thickness, i.e. rt3 = L/h, and the relationship between the displacement and the thickness, i.e. rt2 = 6/h, it is possible to determinate n~ = q/co directly from the graphics, and then, to determine the limit load for the shell structure using 00, for any material resistance. By appropriately choosing the data corresponding to the shape, it is possible to analyse a thin-walled shell, and determine the limit load for wind pressure with the aid of the graphics. Also, by changing the resistance of the material or changing the thickness, one can verify how the limit load varies. It is also possible to analyse a shell that covers a run way, determining the limit load for a set of shape data, using the graphics and choosing the material resistance. It is possible to choose the ideal thickness for a specific material, and optimise the shell with the aid of the graphics. C O M P U T E R RESULTS AND GRAPHICS The computer results were obtained for different thickness values. An iterative incremental analysis was used. Increasing the load, it is possible to prepare a curve with the load and the corresponding deflection. Using rt-numbers, the load is divided by the material resistance, and the deflection is divided by the thickness. These results correspond to the curve that related the length to the thickness. Other L/h curves are determined by changing the thickness for the same shell, and repeating the process. Five thickness values were used in the following graphics.

350 lUh!= 300

.....

(-9 250 UJ C3

x 200 .

e,.o

ii.i

too

....

.

...........

.

..............

i

...............

.

i ..............

~.-.i.q. 0. ..........

c: 4~

F-- 150

i

-o t0 o .J

"-" 100

i

i

i

i

L h .300

50 0

0

5

10 15 20 25 (Deflection I Thickness) x 10 E-2

Figure 2' Conoidal shell graphics

30

35

545

EXAMPLES OF GRAPHICS USE Thin-walled structure analysis F o r a thin-walled structure, verify if it is possible to use the following data in a limit wind-pressure analysis: L = 30.00m h = 30.00 cm 6 = 1.0cm 7z3=L/h = 3000 / 30 = 100

groph,c,, > ~Zl = f = 350 X 10 .6

7t2=6/h = 1 / 30 = 3.3 x l 0 2

Using Cro = 20.0 M N / m 2 for material resistance, determine the limit load o f the wall structure. q = f . o0 = 350 x 10.6 x 20 = 7000 x 10.6 = 7.00 x 10 .3 M N / m 2 = 7 k N / m 2 In the same thin-wall structure, reducing the thickness to h = 20 cm 9 7t3=L/h = 3000 / 20 = 150 r~2=5/h = 1 / 20 = 5.0 x l 0 "2

graphics

> rt~ = f = 190 x 10.6

Increasing the material resistance to o0 - 30.0 M N / m 2, the limit load for the same thin-wall structure is: q = f . a0 = 190 x 10.6 x 30 = 5700 x 10 .6= 5.7 x 103 M N / m 2 = 5.7 k N / m 2 R e d u c i n g the thickness, increasing the material resistance, and c o m p a r i n g the limit load, it is possible to c h o o s e the best solution for the thin-walled shell in wind pressure problems.

R u n way shell cover analysis D e t e r m i n a t e the limit load for a reinforced c o n c r e t e shell to cover a run way, using the following data: L = 12.00m 8 = 2.00 cm h = 8.00 cm rt3=L/h = 1200 / 8 = 150

graphics

rt2=6/h = 2 / 8 = 2.5 x l 0 .2

>rtl =f= 325 x 10 .6

I f c0 = 20.0 M N / m 2 is the material resistance, then the limit load o f the structure is: q = f . g0 = 325 x 10.6 x 20 = 6500 x 10 .6 = 6.50 x 10 .3 M N / m 2 = 6.5 k N / m 2 g = 0.08 x 25 = 2.0 k N / m 2 p = q - g = (6.5 - 2.0) = 4.5 k N / m 2 I f the thickness is r e d u c e d to h = 6 cm and the material resistance is increased to Oo = 30.0 M N / m 2, then the limit l o a d is: 7t3=L/h = 1200 / 6 = 200 rtz=6/h = 2 / 6 = 3.3 x l 0 "2

graphics ) ~ 1 ~

=

225 x 10.6

q = f . a0 = 225 x 10.6 x 30 = 6750 x 10.6 = 6.75 x 10"3 M N / m 2 =

546 6.75 kN/m 2 g = 0 . 0 6 x 2 5 = 1.5kN/m z p = q - g = (6.75 - 1.5) = 5.25 kN/m 2 By reducing the thickness and increasing the reinforced concrete resistance, it is possible to control the allowable overload on the shell. CONCLUSIONS With the aid of dimensional analysis resources, it is possible to analyse any type of structure, including shells with complex shapes. For shells, the relationship between the thickness and other parameters, for example the length of the shell projection, the deflection and the limit load, is very important. The graphics were prepared for a particular shell shape, but the same process can be generalised for any shell shape. The objective, when preparing these graphics, is to solve the shell thickness problem for a particular material quickly. This idea is very important, because shell structures without suitable thickness might not supports their own dead load. ACKNOWLEDGEMENTS The author wishes to thank Prof. Fernando Luiz Lobo Barboza Carneiro, who has taught her how to use dimensional analysis, a very powerful tool for solving many shell structure problems, including the application presented in this work.

References Carneiro F.L.L.B. (1993). Dimensional Analysis, UFRJ, Rio de Janeiro, Brazil Gomes M.P.R.C. (1993). Elasto-plastic and Geometrically Non-linear Analysis of Shells by Finite Element Method, Coppe/UFRJ, Rio de Janeiro, Brazil Gomes M.P.RC. (1996), Analysis of Shell Structures with Similar Shapes, In Proc. of Joint

Conference of Italian Group of ComputationalMechanics and Ibero-Latin American Association of Computational Methods in Engineering, XVII CILAMCE, Padova, Italy Gomes M.P.R.C. (1996), Dimensional Analysis for Hyperbolical-Paraboloid Shells, In Proc. of Joint Conference of Italian Group of ComputationalMechanics and lbero-Latm American Association of Computational Methods in Engineering, XVII CILAMCE, Padova, Italy Gomes M.P.RC. (1996), Stability Verification of Prestressed Concrete Shell Structures, In Proc. of Joint Conference of ltalian Group of Computational Mechanics and Ibero-Latin American Association of ComputationalMethods in Engineering, XVII CILAMCE, Padova, Italy Gomes M.P.R.C. (1997), Limit Load of Structural Shells with Similar Shapes, In Proc. oflCCBE-VII, Seoul, Korea

547 Gomes M.P.R.C. (1997), Abacos para determinagao da Carga Limite de Cascas Semelhantes, In Proc. of XXVIll Jornadas Sul-Americanas de Engenharia Estrutural, S~o Paulo, Brazil Gomes M.P.R.C. (1997), Carga Limite de Cascas Estruturais com Formas Reversas, In Proc. of XENIEF, Bariloche, Argentina Gomes M.P.R.C. (1998), Design Optimization of Structural Shells with Reversal Shapes, In Proc. of Joint Conference of IV WCCM and XIX CILAMCE, Buenos Aires, Argentina Gomes M.P.R.C. (1998), Otimizagao do Projeto de Cascas Estruturais com Formas Reversas, In Proc. oflII SIMMEC, Ouro Preto, Minas Gerais, Brasil Gomes M.P.R.C. (1999), Optimizaci6n del disefio de bovedas y estructuras similares, In Proc. of IV COMNI, SeviUa, Spain Gomes M.P.R.C. (1999), Abacos para projeto de cascas estruturais, In Proc. of XX CILAMCE, Sao Paulo, Brazil Gomes M.P.R.C. (2000), Optimizag~o do Projecto de Cascas Estruturais, In Proc. of V1 CNMAC, Aveiro, Portugal Gomes M.P.R.C. (2000), Graphics to determinate the limit load of shells with similar shapes, In Proc. of Fourth International Colloquium on Computation of Shell & Spatial Structures, IASS-IACM-2000, Chania, Greece Gomes M.P.R.C. (2000), Design Optimization of Conoidal Shell Structures, In Proc. of XXI CILAMCE, Rio de Janeiro, Brazil

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Third International Conferenceon Thin-Wailed Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

549

VECTOR OPTIMIZATION OF STIFFENED PLATES SUBJECTED TO AXIAL COMPRESSION LOAD USING THE CANADIAN NORM Susana Ang61ica Falco and Khosrow Ghavami Department of Civil Engineering Pontificia Universidade Cat61ica do Rio de Janeiro - PUC-Rio Rua Marquis de S~o Vicente 225/301L, G/tvea CEP 22453-900, Rio de Janeiro, RJ, Brazil

ABSTRACT This paper is concerned with the vector optimization problem of longitudinally stiffened plates subjected to axial compression load. This leads to a non-linear vector optimization problem with a finite number of sideconditions. As an application it is considered the optimal design of such plates where the ultimate buckling load should be maximal and the weight minimal. Among the different formulae and recommendations for the prediction of the ultimate buckling load, the Canadian norm CAN-S 136-M89 (1989) was chosen for its simplicity and for producing one of the best results. The chosen design variables are the number, the thickness and the height of the stiffeners. Several constraints are considered, Brosowski and Conci (1983), and constrains avoiding the failure due to the lateral buckling for torsion of the longitudinal stiffeners are introduced considering different norms Falco (1996). These side-conditions created a bounded feasible set of points. The compromise solution of this multi-objective optimization problem is established by an interactive procedure using the Efficient Points Method, developed by Brosowski (1985), which leads the problem, with n objective-functions, to a problem with (n- 1) ones. In the particular case of the optimization of stiffened plates, it is used a simplification procedure ofscalarization, which consists of the minimization of only one objective-function with only one variable within an established interval. An example is considered illustrating the potentiality of this method for practical cases. KEYWORDS Vector optimization, Stiffened plates, Ultimate loading, Buckling, Efficient points, Scalarization, Simplified method, Non-linear problems.

550 INTRODUCTION Stiffened plates are very efficient structures, as a large increment of the strength created by a small addition of weight used as stiffeners. The collapse mechanism of stiffened plates subjected to axial compression load is complex because it depends on several geometricvariables beside the mechanical properties of the materials used and buckling failure modes. Predicting the buckling behavior is possible both analytically and through numerical methods, such as finite element method, which need to use a large number of elements and is not considered in this work. A semi-analytical method based on the Canadian norm CAN-S 136-M89 (1989) is considered in this paper for the prediction of the ultimate buckling load. This method has been chosen based on an extensive experimental investigation realized in the Civil Engineering Department ofPontificia Universidade Catolica of Rio de Janeiro, Ghavami (1994). In this program a total of seventeen tests on simply supported steel plates were executed. The main variables considered were the type of stiffeners cross-section and the spacing between them. Plates with three types of longitudinal stiffeners i.e. L, T and rectangular (R) crosssections with and without transversal stiffeners o f t cross-section have been studied. The comparison of the calculated loads showed that the Canadian norm, the Murray's method, Murray (1975), and the AISC, American Institute of Steel Construction (1978), gave the response very close to the experimental ones, Falco (1996) and Brosowski, B. and Ghavami k. (1997). Based on these results, the Canadian norm was chosen for the simplicity of its formulation comparing with others methods and it is given in detail. In this paper, the design problem is carried out for an isotropic plate stiffened by rectangular stiffeners parallel in equal distance along the axis of the load application. The geometrical characteristics of this stiffened plate are shown in Figure 1, where L, B and t are the length, width and thickness of the plate, respectively, and tl, d and b are the thickness, height and distance between the stiffeners, respectively. The axial load is denominated as Px.

Figure 1. Geometrical characteristics of the stiffened plate under axial load. Otten the design of such structure has several requirements such the case of minimization of the weight and maximization of the buckling collapsing load. Thus, the designer of this structure is confronted with the problem of satisfying two conflicting objectives; such problems are called multi-objective or vector optimization problems. In general, the objective-functions do not attain their optimum in a common point of the set of feasible points. Intuitively, it is clear that one can expect only compromises in the sense that an improvement of any of the objective functions may presumably lead to a worsening of the value of some of the other objectives. Compared with ordinary optimization one has to develop a new concept of a solution, which is called an efficient compromise. In this work, this compromise solution of this multi-objective optimization problem is established by an interactive procedure using the Efficient Points Method, developed by Brosowski (1985).

551 ULTIMATE BUCKLING LOAD USING THE CANADIAN STANDARD The Canadian Standards is based on the concept of the effective width, including the effects of local buckling. The project stress ap, or reduced stress of buckling is calculated multiplying the stress of the global elastic buckling by the reduction coefficient 0.833, which takes into account the initial imperfections of the plate and the welding residual stress, Eqn. 1. The effects of the inelastic buckling are considered from the equations of the curve of CRC (Column Research Council), Figure 2. Therefore, after calculating the project stress, the obtained stress is adjusted using the curve of CRC. Thus, the critical stress of buckling a~ is obtained comparing the project stress with the proportionality stress, based on the limit of the proportionality ~y/2, Table 1. Op= 0.833 OE = 0.833 rc2E

O'a :

/~ ~y

~y

(1)

2 t~ Y

4~p

~ TABLE 1 CRITICALSTRESSOF BUCKLING

OE =~2E/(kL/r)2

2

For % > o r / 2 9 o~ = O'y/2

Oy-4--~n I

For~p
IINELASTIC BUCKLING

"

aa=Op

t ELASTIC BUCKLING

= kL/r Figure 2. Buckling CRC Curves.

Thus, with the critical buckling stress o,, the failure compression load Pf is calculated through Eqn. 2, where ~a is the reduction coefficient and A~ is the effective area calculated using the effective width (be).

(2)

P f = ~a Ar O'a

Therefore, for the calculation of the ultimate buckling load just a #action of the width, called effective width is considered. The width of the plate is considered totally effective (b=be) when the width is smaller than the value limits given in Eqn. 3. When the width of the plate is larger than the width limits, Eqn 4 is used for the calculation of the effective width being considered O~x= o,. The coefficient k is 4 for stiffened elements in compression and has the value of 0.5 for non-stiffened elements in compression Roger (1988). bc > biim= 0,644 t

,[ V

(~ max [ 1 - 0 , 2 0 8

( 3) max

~ max-

]

(4)

552 EFFICIENT POINTS METHOD The vector optimization problem, defined in Eqn. 5, is solved using the Efficient Points Method. In this problem two or more functions denominated Objective-functions are either maximized or minimized. This should be accomplished by determining the values of certain project variables, which are the parameters chosen to describe the system. In most of the problems we will find restrictions so that the project is feasible or viable. Objective-functions Variables Vector Constraints

ps (x) = (pl(x), p2(x),..., pt(x))T

(5)

x ~ R"

gi (x) _<0

i ~ D=[ 1,..., m]

Thus, p,(x) is the vector oft Objective-functions that define the solution x, in the n dimension space, gi (x) are the i inequality restrictions that limit the space of the solution. Any point x that satisfies the restrictions is a viable point and the set of all these points is the viable region Z. The main characteristic of the vector otimization problems is the appearance of the conflict of objectives. In the present analysis the scalarization process proposed by Brosowski is introduced, Brosowski and Conci (1983). The method transforms a problem with t objective-functions in a problem with k_<(t-1) objective-functions. The substitute objective-function is given by Eqn. 6, where y is an scalar and Z` ~ R t is chosen arbitrarily. The group of parameters Z`is given by A={ Z`~Rt I z`t= 0 }. For each Z` ~ A the Parametric Otimization Problem is defined: POP(Z`). Pip(x)]

= y = ps(x)-

z`s

s=l .... ,

t

(6)

To minimize the substitute objective-function means to find the value x* (weak efficient point), so that: P[p(x*)] = min y = min[ps(X ) - ks] l_<s_<_t

(7)

Defining the vector otimization problem with t objective-function: ps(x)=pl(x),..,pt(x), subject to gi(x)_<0, i = l , . . . , m. For each Z` ~ A, it is considered: POP(z`):Minimize y

Subject to:

(x,y) ~ Z(Z`)

Z(z`)is the set of feasible points for the POP(Z,) and it is given by the Eqn. 8. Z(Z`)={(x,y)~RN+I I V gi(x)<0 A V p,(x)-L~< y} comM=l .... , m & s ~ T = l ieM seT

..... t

(8)

P(Z`) is denominated the set ofaU points of minimum of the POP(Z`). Each element of P(z`) contains the pair (x*, y*) ~ Z x R, for which an active restriction exists.

OPTIMAL DESIGN PROBLEM FOR STIFFENED PLATES In the project of optimization of stiffened plates studied in this work two objective functions are considered: the transversal section of the plate pl(x) and the collapse load p2(x). It is assumed that the plate has given a width B and a length L. The sought variables are the number N, the height d and thickness t~ of the stiffeners. Thus the cross section of the plate is given by Eqn. 9, where the coefficient tx represents the specific weight of the material. As the coefficient ~t and the length L are known, the first objective function is formulated in relation to the weight of the plate divided by IXand L. pl (N, t~, d) = w (N, tl, d) / IX L = Bt + dt~ N

(9)

553 The collapse load, given by Eqn. 10, is calculated using the Canadian norm. The failure load (Pf) is calculated multiplying the N stiffeners for the failure stress of the plate (r and for the effective area (A0. P2 (N, h, d)= - Pf = - N c~a/L,

(10)

It is introduced the following dimensionless quantities dividing all given values by the length of the plate: z = t/L, xl = tl/L, 8 = d/L, 13= B/L, F=r / L and a variable oq = z~ 6 Therefore, the objective functions are transformed in the following equations: ql (N, a~8) = p~ (N, r

) / L 2 = j3x + Otl N

(11)

q2 (N, a l , 8 ) = p2 (N, or,,8 )/L 2

(12)

The feasible set Z is defined by nine restrictions detailed in Brosowski and Conci (1983) and Falco (1996). These restrictions are listed below: g,(N, e t , , 8 ) = ~o~- ~E-< 0 g,(N, it,, 8) = 30 ~ ( N + I ) - 13-<0 gs(N, oq, 8 ) = 4x2(N +1) 2 _ (0,456 + 82)ot? < 0 132 84

(13) (14)

(15)

ga(N, o q S ) = -N + 1 < 0 gs(N, oq, 8) = - oq < 0

(16)

g6('N, CXl,8) = - 8 < 0

(18) (19)

(17)

gT(N, or1,8) = Noq- 138 < 0 g8a(N, 0l,l . ~ ) = 8 2 / O t l _ 10 3qt~/~x/~y _< 0

(BS 5400)

gsb(N, r

(API B U L 2U)

= 8~/etl - 5 < 0

g9(N, cz18)= - z f / o t 1 + 1/2 < 0

(20) (21) (22)

In Eqn. 13, ~E is the Euler stress and ac, is the critical buckling stress of a simply supported plate on all sides under in-plane loading. In Eqn. 14, the ratio b/t>30 is considered, that limits the number of stiffeners of the plate. In Eqn. 15, the restriction is to avoid the buckling in the stiffeners that happens as sudden collapse. In Eqn. 16, it is assumed that at least one stiffener is used. In Eqns. 17, 18 and 19 it is assumed that the amounts of the thickness and height of the stiffeners are positive and that the thickness of the stiffeners is limited by the distance between them. To avoid the failure due to the lateral buckling for torsion of the longitudinal stiffeners, the ratio between the height and the thickness of the stiffeners, d/t~, should be limited. For this, two different norms are studied: BS 5400, British Standard Institution (1983), Eqn. 20, and API BUL 2U, American Petroleum Institute (1987), Eqn. 21. The last geometric restriction is the relationship between the thickness of the plate and of the stiffeners, Eqn. 22. Thus, the set of feasible points of the VOP is defined in the following way: 9

Z = ( - ~ {(N,c~l, 8) ~ N x R 21 gi(N, txl, 8) _< 0}

(23)

i-I

For each N given and considering the restrictions g5 <- 0 and g6 -< 0, it is written: 9

Z N = ( " I {(Oq, 8) ~ R 2 [ giN*(oq,8) < 0} i=l i;e5.6

(24)

Considering the restrictions g2 <--0 and g4 < 0, lateral restrictions are introduced to calculate the number of stiffeners. Therefore, a value of maximum N given by Eqn 25 exists. Thus, the formulation for the feasible set is given in Eqn. 26.

554

Nmax=INT

13

-l=max{N~N[

(25)

N<(-~-x "

N

z= N#'({N }xz.) =I

(26)

Considering the set ZN for fixed and feasible values of N and 0[.1 ~ 0, the restrictions of the Eqn. 14 to 22 are transformed in function of the variable ot~. With the restriction gl (N, ot~, 6)_<0, it is obtained the lower limit 8~ (ot~) for variable 15and for the restriction gT(N,Otl, 8)<0, another lower limit is given, Eqn. 27.

8 _> 81N(or,)

8 _>8~(otI)= N.ot 1

(27)

r -

With the restriction g3(N, oq, 8) <_0 is obtained the upper restriction shown in Eqn. 28. For gs(N, oh, 8)<_0, another upper limit is given, Eqn. 29. 8 < 83N tot1)=

being 15>0 and

20)

6 _<# N = 13,73 a x / - ~ - t / ~

(BS 5400)

or

8 <__84N = ~

m=

4"1:2(N + 1)2

132

(API BLR~ 2U)

(28) (29)

For g9(N, (I,1.8) --< 0, with fixed N, a lower limit for 6 is given in function of oq, Eqn. 30. >_ ~ N ((I, 1) = (I'-'~1

2x

(30)

Therefore, it is defined the boundary feasible set as shown in Eqns. 31, 32 and 33. A detailed discussion of the functions 80N, 8~ and ~ is given in Falco (1996).

(~N(0tl) = m,x[8~ (Otl),~ N (Oil) ] ~oN(Of.l) ---- max[(~iN(otl),~6N (Oil)] ~(Otl) = mint53N(Otl),SN(0tl)]

(31)

(32) (33)

Thus, for each N, being l 0, the element/5 belongs to a compact interval. ZN = {((~I, 8) E R 2 [ 6N(o~I) _< 5(0~1) _< 67N(0[,1) }

(34)

EXAMPLE A square plate is analyzed with the following dimensions: L=B=650 mm, t=4.80 mm, the elastic modulus E = 195000 MPa, the yield strength Oy =224 MPa and the Poisson's ratio v=0.3. The maximum number of stiffeners calculated by Eqn. 25 is equal to 3. The Figure 3 presents the set of feasible points ZN, for N =1, 2 and 3. The bounded feasible space is defined by the restrictions of the Eqns. 31, 32 and 33. The stop criterion is given for j,~x = 150, thus one hundred points are calculated, reaching the maximum value of interruption ofp1=5686.56 mm 2. The step size for Pl is (Aotl)L 2= 10 mm 2. The initial value for the searching of (Ot~mmN) is 0. lxl 0 "4. The relative accuracy for oq~a, is 0. lxl 0 "3. The global minimum for the first objective-function is searched moving through the different stiffeners calculated for this plate (N=l,2 ..... Nm.,x). The value ofoq in the intersection of 80N(Oil) and 87" (oq) is searched. Thus, each local minimum is obtained for each stiffened plate, as shown in Table 2. The efficient global minimum calculated for the first objective-function corresponds to the plate with one stiffener.

555

Figure 3. Feasible space Z1; Z2 and Z3

556 TABLE 2. MINIMUMEFFICIENTPOINTS FOR Pl

N

A (mm2)

d (mm)

tl(mm)

pl (mm2)

p: (MN)

pz/pl(MPa)

pl/p2(mm2/MN)

1 2 3

166.56 455.42 885.25

28.86 33.74 38.41

5.77 6.74 7.68

3286.54 3575.39 4005.18

0.24 0.58 0.78

71.5 162.7 193.7

13981.9 6145.1 5161.3 .

.

.

.

The second objective-function is the minimization for each N stiffener, in an interval varying with d. This function is analyzed to equidistant intervals, being chosen an interval Ad =0,1 mm. Therefore, for the same cross section of the stiffeners, a second objective-function is calculated in different intervals to which the value Ad is added. The smallest of the obtained local minimal is calculated. Then, a new search of the minimum point of the second objective-function is initiated, in the interval [80(N, oq), 87(N, Ctl)], for a next value of the first objective-function. In Figure 4, it is observed that the minimum points drop in different plans (px, p2). In all the intervals of Ad and for all the admissible values of N, the global minimum of the second objective-function is always found at the end of the interval, in 87(N, ot~J). As it can be observed in the Figure 5, the second objective-function presents a slight decreasing inclination.

Figure 4. Minimization of p2, for a given N in different intervals of p~. In Figure 6, the transformed feasible set for each stiffened plate in superposition is shown. For each number of stiffeners, all the minimum efficient points are calculated. Increasing values of pl imply an increase in the cross section of the plate, it is possible to pass for different numbers of stiffeners. When passing a value of p~ for a new number of stiffener, a jump may be observed in the value of the second objective function. The set p(ZN) is separated. Thus, as a small decrease of the area reached through the passage of a number of admissible stiffeners N for (N-l) a considerable decrease of the collapse load is produced. These curves can be efficiently used by engineers and planners. For example, to calculate a plate with a maximal axial load of project of 0,7 MN a cross section of 3800 mm 2 is obtained. The measurements ofthe different parts of the plate are found in the final data of the program. Otherwise, to obtain a plate with one stiffened end, the compression load will be 0.41 MN.

557

p2(MN)-0.30 N=I

-0.40

-0.50

-0.60

-

-

-

-

L_ p1=3387 m m 2

N=2

--

p l =3687 m m 2 -0.70 - -

-0.80

N:3

--

-0.90

~

I 0.040

~

0.045

I 0.050

'

1 0. 055

\

'

,

I 0.060

pl =4187 mm 2

'

I 0.065

'

1 0.070

' 0.075

Figure 5. Variation of" P2with delta, for a certain value of p~ r==d

Figure

6. Transformed

feasible

space.

558 CONCLUSION In this paper an optimal design problem of longitudinally stiffed plate has been modeled considering the case of minimization of the weight and maximization of the buckling collapsing load. In this particular case, the compromise solution of the non-convex designed problem was solved using a simplification procedure of scalarization through the Efficient Points Method. The simplicity of the numeric formulation of this process justifies its application, which consists of the minimization of a function with only one variable, within a certain interval. Among the different methods for the prediction of the ultimate buckling load, the Canadian norm was chosen for its simplicity and for producing one of the best results comparing to the experimental ones. This method is based on the concept of the effective width as for AISC and Murray's methods but its application is relative simpler. Through an example it was shown the potentiality of this method for optimum design of the stiffened plates in practice.

REFERENCES AISC-American Institute of Steel Construction (1978). Specification for Design, Fabrication and Erection of Structural Steel for Buildings, New York. API BUL 2U American Petroleum Institute (1987). Bulletin on Stability Design of Cylindrical Shells, 1st edition. Brosowski B. and Conci A. (1983). On Vector Optimization and Parametric Programming. Segundas Jornadas Latino Americanas de Matematica Aplicada, 2, 483-495. Brosowski, B. (1985). A Criterion for Efficiency and some Applications, Preprint Univ. Frankfurt. Brosowski, B. (1989). A Recursive Procedure for the Solution of Linear and Nonlinear Vector Optimization Problems, Proceedings of a GAMM-Seminar, Siegen, Springer Verlag, Berlin. Brosowski, B. and Ghavami k. (1997). Multicritera Optimal Designed Plated. Part II. Mathematical modeling of the optimal design of longitudinally stiffened plates. J. of Thin-Walled Structures, 28 (2), Elsevier Science, pp. 179-198. BS 5400, British Standard Institution (1983). Steel, Concrete and Composite Bridges, Part 3, Code of Practice for Design of Steel Bridges, BSI, London. CAN-S 136-M89, Canadian Standards Associations (1989). Cold Formed Steel Structural Members. Falco S. A. (1996). Otimizar Vetorial de Placas Enrijecidas sob Compress~o Axial. MSc. Thesis, Pontificia Universidade Cat61ica, Rio de Janeiro, Brazil. Ghavami K. (1994). Experimental Study of Stiffened Plates in Compression up to Collapse. J. Construct, Steel Research N ~ 28, pp. 197-221 .Manchester University Simon Eng. Lab. Report. Murray, N.W. (1975) Analysis and design of stiffened plates for collapse load. The structural Engineer. 53, pp.153-158 Roger C. (1988). Cold-Formed Steel Structures, University of Alberta, Canada, 3rd-Colloquium on Steel Structures. Wieland, U. (1992) Anwendung der Vektoroptimierung auf den Entwurfversteit~er Platten. Diplomarbeit, Universit~,t Frank~rt, 175 p.

Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

559

BICRITERIA OPTIMIZATION OF SANDWICH CYLINDRICAL PANELS AIDED BY EXPERT SYSTEM R. Kasperska 1 and M. Ostwald 2 1 Institute of Technology, Pedagogical University of Zielona Grra, Zielona Grra, al. Wojska Polskiego 69, Poland 2 Institute of Applied Mechanics, Poznafi University of Technology Poznan, ul. Piotrowo 3, Poland

ABSTRACT The aim of this paper is to present the results of bicriteria optimization of the sandwich, thin-walled, cylindrical panels under combined loads. The objective functions are weight and flexibility. The design variables are the thicknesses of the layers. The set of constraints includes the stability condition, stress conditions, the importance of theoretical models, and finally, technological and constructional requirements. The optimization problem was solved with the help of the Pareto concept of optimality and by means of a computer program, which includes elements of the expert system. Results of numerical calculations are presented in the form of tables and diagrams.

KEYWORDS multicriteria optimization, sandwich shell, Pareto-optimum, expert system

INTRODUCTION

The main advantage of sandwich structures (plates, panels, shells) is a very beneficial relation between the weight and the carrying loads. Therefore these structures are widely applied in different branches of industry and in civil engineering. The range of application may be considerably extended by a suitable determination of the shell parameters and core construction. The most important fault is connected with the structure's susceptibility to global and local buckling. Other faults are connected with the manufacturing and assembling processes, and with adoption of adequate mathematical models to the structural analyses. In the literature, the available theoretical models and expressions describing the state of stresses and the panel deformations are unreliable. According to Adeli (1988) and Eschenauer (1990), thin-walled sandwich-type structures are very interesting objects for multicriteria optimum design and knowledge engineering. Multicriteria optimization (or poly- or vector optimization) requires the formulation of the optimization criterion set, design variables and the set of constraints.

560 BICRITERIA OPTIMIZATION MODEL OF SANDWICH CYLINDRICAL PANELS In the paper, the results ofbicriteria optimization of the sandwich thin-walled cylindrical panels under axial compression and external pressure are presented. The model of an open cylindrical shell (presented in Figure 1) is composed of three layers: two thin carrying layers, called the faces, and a core placed between them. q

\-.. ',, ,

/

./

"" ""E2, v2, 72, [a2] ,"'E3 v3, ~'3, [~3] El, vl, ~'1, [~1]

//

\9

\ .,\

/

\, '\/" Figure 1 Model of sandwich cylindrical panel The bicriteria optimization problem, which used the concept of Pareto optimum, was formulated as follows: Q(~)=[QI(~), Q 2 ( ~ ) ] = ( 1 - W A G A ) . Q I ( ~ ) + W A G A . Q 2 ( ~ ) ~ minimum (1) where: ~ = [hl,h2,h3] - vector of the design variables (the thicknesses of the panel layers), Q) (x), Q z ( x ) - optimization criteria, WAGA - criteria weight factor, 0 <_WAGA <__1,0. The fundamental condition in the optimal design of structures is to formulate the optimization criterion, design variables and the set of constraints. The first criterion of the minimum structure weight is of the following form: Q~(~)=a.b(h171 +h272 +h373) [kg] ~ minimum, (2) where 3'i- material density of i-layer. The second criterion is the requirement of maximum structure stiffness in the form of minimal deflections of the shell. This paper adopted the simplified model of procedure because the equations describing the shell deflections are not sufficiently reliable. Ostwald (1993) formulated the condition of minimal structure flexibility. Flexibility is defined as the converse of the panel bending rigidity and it represents some qualitative measure of the panel deformability. For the cylindrical panel this condition is defined as follows: Q2(ff.) : D--~

1

[1/MNm]

~

minimum,

where: D(K) - E. h, h 2(h I +2 2h _3 + 2h)2 )2 [MNm] - bending shell stiffness. E. = E1 = E2. Young's modulus 4(h, +h )(1 v n and Vn= V, = V2 - Poisson's constants for the faces. Similarly, Ostwald (1993 & 1997) established the optimization criteria for closed cylindrical shells. The stability condition is the main constraint. In this constraint, the initial geometric imperfections connected with manufacturing and assembling defects are taken into account. The set of constraints is defined as follows:

561 1) Permissible critical load must be greater than the axial compressive force Px: p lin P e r per "cr > Px, 0L.n

where the upper critical load piinor[MN/m] is calculated as the smallest positive root of an algebraic equation: A, .Px4 +A 2 -Px3 +A3 9Px2 +A4-P x +A 5 : 0 , [Px]: 1,

p,in = B-(h, +h 2 +h3):--px, B=ZBi=~'=3 ,=3 Ei h~ , cr a-b ,:1 i=1 1 - v? A1, A2, A3, A4, A5 are factors that depend on physical and geometrical parameters of the panel, see Sekulski (1984). According to Bushnell (1987), the factor ot= 1,4 takes into consideration the influence of initial deflections (geometrical imperfections) on the value of the critical force, and n=1,25 is a safety factor. 2) Reduced stresses in the faces cannot exceed the permissible stresses: ar~di = 3/(ax~(Px, q))2 + (ayi (Px, q))2 _ (axi (Px q))(~3yi(Px, q)) < [a, ]. 3) Deflections of the panel center cannot exceed the values, which are determined by a constructor: 5 q-a 4 w= ---C< 384 D(~) - Wp,r, ( 48k)_ 1536 ( x 9th(rd2X)'~ El,2 x2 .hlh2h3 where: C= 1+57t2 j 51t5 ~c-~-(Tt/2L) l + k + 4~ ), k = 0.,v~.2)~a2i~ I + h ~ ) 4) The importance of critical load equations leads to the assumption, see Sekulski (1984) R

>30. h~ + h 2 + h 3 5) When the technological and constructional requirements are considered, the following conditions are assumed: 0,1mm _< hi,h2_< 3,0mm and lmm _< h 3 < 50mm In this problem, 11 conditions are taken into account.

EXPERT SYSTEM AIDED MULTICRITERIA OPTIMIZATION The optimization problem is solved with the help of the OPTIKON program, which has been written in Delphi 4.0 language for Windows 95/98/NT. Some Dynamic Link Libraries (DLLs) of the expert system shell, taken from the PC-Shell (ver. 2.3), have been used in program realization. The diagram of computer program, which includes elements of expert systems, is presented in Figure 2. The OPTIKON program uses a procedural-declarative knowledge representation. Two approaches to declarative representation of knowledge are available: object-attribute-value (OAV) triplets and production rules in the form of IF-THEN rules. The knowledge base consists of numeric information, the sets of facts and rules and the mathematical models of thin-walled structures (plates, panels and shells). There are two kinds of models in the knowledge base: structure model and optimization one. Knowledge about structure model includes information about: kind of structure (a plate, a panel, a shell), the shape, thickness::::~: and dimensions of structures, - core properties and construction, the number of structure layers, - structural materials applied to every layer, structural parameters, -

-

-

-

562 kind of loads (axial compression, external pressure), mathematical models of structures. Knowledge about optimization model enables a user: to select design variables and the ranges of their values, to specify the number and kind of optimization criteria, to define the set of constrains, to specify the way of normalization of objective functions, to choose optimization methods, to select control parameters for optimization methods, to choose preference functions (the min-max method, the TOPSIS method, the global criterion methods). -

-

-

-

EXPERT Expert or I ~ engineer /

knowledge

m C~

Input 1

usER

~..CO~R

KNOWLEDGE

.~t- 9Knowledge

acquisition module

~ w Z

. . . .

l Fact base

1

Inference mechanism

w

D

............

OPTIMUM

SOLUTION

---I

Rule base

tn [ Criteria 7 -7 ~l I Constraints~ .Q -

BASE

Structural] model

base

.........

~-----P --r.n l Explanation! facility

-~ Requirements

SYSTEM

-7

[Normalization[

I

o WPreferences] I ~ l Control -I < parameters j

; ................

/

,

~ ~

! ..... ~

----7

Libraryof ; '= ' ~ i optimization I ~T .. ~ methods LI OPTIMIZATION !

..................

Solution evaluation

et of compromise~. .... solutions f

Figure 2 Diagram of program for multicriteria optimization of thin-walled structures The expert module enables the choice of different optimization procedures from the library of optimization methods. This library includes deterministic and stochastic algorithms, methods of discrete and continuous optimization (for instance, the systematic search method, Hooke-Jeeves algorithm, simulated annealing, genetic algorithm). The best optimum solutions are generated in discrete and continuous sets of design variables (results with practical and theoretical meaning). The OPTIKON program has the ability to provide explanations that clarify its functioning and recommendations. The explanation facility provides three types of knowledge corresponding to the following explanation labels: How, Why and What Is. The inference mechanism is backward chaining, which takes the form of hypothesis verification.

N U M E R I C A L

C A L C U L A T I O N S

A N D

C O N C L U S I O N S

In the paper, some examples of using of the OPTIKON program in multicriteria optimization of structures are presented. The following data were assumed in the numerical calculations: the faces are made of aluminum alloy PA6, E~,2 = 7.06.104 MPa, v~,2 = 0.3, ~/1,2= 2780 kg/m3, [ai] = 0.75 R~= 195 MPa,

-

563

-

-

the core is made of foam plastic, E3 = 53 MPa, v1,2 = 0 , ~/],2 panel middle surface curving radius R = 2000 mm, panel length and width a = b = 1.0 m, axial compressive force Px = 0.1, 0.2, 0.3, 0.4 MN/m, external pressure q = 0, 0.001, ..., 0.007 MPa, permissible deflection Wp= = 30 mm.

--

210 kg/m 3,

Figure 3 shows the objective space for the sandwich cylindrical panel under the compressive force Px=0.4 MN/m and external pressure q=0.007 MPa. This space includes 25707 points for all vectors of the design variables, which satisfy constraints. The results of numerical calculations are presented in Figure 4 and Table 1.

222.4138

200

-ff Z

..::.?... ,~.~.-....:'..

160

~

,

~%',:':..

~"". ' t?~"~" ~ . ; t ~..."

~ 120-

"6 117.511~

-

i!

i

80-

~ LL

::. ..

9 ":'i:.

40--

t 3.0591

' !

0

I

8 9~1240

1

I

16

12

20

W e i g h t of p a n e l

....

I

24

27.1800 i

28

[kg]

Figure 3 The objective space for the panel under loads Px=0.4 MN/m and q=0.007 MPa

i ~A=~ I'"'117.5117 -

I

~

I

1

.=~.1 m , . ~ . ,

-1

P-o., M ~ m . , - O . O O , . P .

MESP SOLUTIONS

iI

-~

I .Ao,."I i

100 --

i

"6 "~

[

I

I

:

I

I

50 - -

E 5

I[ SOLU'IIOI ioEAL 3.0591 0-8 9.1240

~ " ~ 12

Preferred solution

.

' 16

10 20

Weight of panel [ k g ]

!

[ WAGA-1 1 :

24

2 7 . 1 8 0 0 28

Figure 4" The set of Pareto-optimal solutions in the objective space

564

TABLE 1 OPTIMAL THICKNESSES OF PANEL LAYERS ACCORDING TO THE DIFFERENT OPTIMIZATION PROCEDURES FOR Px=0.4 MN/M, Q=0.007 M P a

No ] WA GA 1.

0

2.

0.1

,.,

Method i

MESP 0.9 SW 0.9 AG 1.1 HJ 1.1255 MESP .... 1.'1 SW 1.1 AG 1.1 HJ 1.1078 MESP 1.1 , SW ,i. 1.1 AG 1.1 HJ 1.0864 MESP [ 1.1 SW 1.1 AG ! 1.1 HJ 1.0742 ! MESP 1.1 SW 1.1 AG 1.1 HJ 1.0653 MESP 1.1 SW : 1.1 AG: 1.1 HJ 1.0578 MESP 1.0 SW 1.0 AG 1.0 HJ 1.0508 MESP 1.0 SW 1.0 AG 1.0 HJ 1.0437 MESP 1.1 ~ SW 1.1 I AG 1.1 ,' HJ , 1.0435 [ MESP 1.6 SW 1.6 AG 1.6 HJ 1.5641 MESP 3.0 SW 3.0 AG 3.0 HJ 2.9999

I

,

0.2

:

-.-,

,.

4.

0.3

5.

0.4

i

9

6. 0 , 5 1 :

,

i 7. 0 . 6 ,

~

8.

0.7 i

I 9. ! 0 . 8 i

I 10.

11.

Optimal thickness [ram] . . . . .h. .,. . . . . I h= i h.,

0.9

1.0

,

,,,,

,,

~,

i

ii

'

,_'! ....

,

,

~

i .[1/MNm] .....

..

,

,,

.,, :

,,, ~

,,

,,

,

:

,_,.

,_

,

i

, ~;.

,

,.

ii

,,

Qt

[kg]

1.4 ..... 13.0 9.1240 1.4 13.0 9.1240 ! 1.2 13.0 9.1240 1.1255 12.2567 8.8316 1'.1 17.0 "~).6860 1.1 17.0 9.6860 1.1 17.0 9.6860 1.1078 14.8454 9.2771 1.1 20.0 10.3160 1.1 20.0 10.3160 1.1 20.0 10.3160 1.'0864 ~ 19.5782 10.1517 '1.1 24.0 11.1560 1.1 24.0 11.1560 1.1 24.0 11.1560 1.0742 23.5672 10.9217 1.1 28.0 11.9960 1.1 28.0 11.9960 1.1 28.0 11.9960 1.0653 27.4431 11.6859 1.1 32:0 12.8360 1:1 32.0 12.8360 1.1 3Z0 12.8360 1,0578 31.5488 12.5064 1.1 39.0 14.0280 1.1 39.0 14.0280 1.1 39.0 14.0280 1.0508 36.2865 13.4625 1.1 43.0 / 14.8680 1.1 44.0 15.0780 I 1.1 43.0 i 14.8680 1.0437 42.2402 14.6732 1.1 50.0 u 16.6160 1.1 49.0 16.4060 1.1 50.0 16.6160 , 1.0435 i 4 9 . 9 9 9 9 . 1 6 . 3 0 2 0 1.6 ' 50.0 19.3960 1.7 50.0 '19.6740 1.6 50.0 19.3960 1.5641 ]49.9999 19.1967 3.0 50.0 27.1800 3.0 50.0 27.1800 3.0 50.0 27.1800 ] 2.9999 49.9999 27.1799

,.

_,!

JJ

II

,

,

i

ii

_.

::

;

,

i

i

'_

......

!

,

,,

,

,

i

,.'

,.

]1

i

'2

,

'.

[

Q:

[I

Aci Active Preferred const constrains . . . . solutions ,

117.5117 1 117.5117 i 112.1703 ] 127.9018 1 S, o'r,a ffrea~, O'reag_. 71.5347 71.5347 71.5347 91.4304 o'r,al, 52.6392 '1 52.6392 52.6392 55.5687 Oredl, O'red2 ,, . . . . 37.1986 37.1986 37.1986 39.5230 O'redl, O'red2 27.6750 27.6750 27.6750 TS 29.7761 O'redl, O'red2 n 21.3904 MM,P1.P2 MM,P1,P2 21.3904 21.3904 M/VI,PI,P2 22.9229 err,at, O'red2 .. MM..,P1,P2 15.3412 TS TS 15.3412 15.3412 [ TS 17.5983 I o',.,a~, O'red2 12.6815 i 12.1248 12.6815 13.1842 Clrredl, O'red2 8.9750 9.3368 8.9750 9.4815 , h3 6.0513 5.8619 6.0513 6.1986 h3 3.0591 3.0591 3.0591 3.0591 i h,. hi, th2, hz ,

.

.

.

,

,_

,

,

.

ii

[

9

,,

,.

[

,

,

~ ,

T h e p a p e r presents the solutions for the p a r a m e t e r W A G A ~ [ 0 , 1], o b t a i n e d by the m e a n s o f M E S P p r o c e d u r e (systematic search method), S A (simulated annealing) and G A (genetic algorithm) based on the discrete set o f design variables .... these solutions have practical meaning. T h e lower line presents the results o f H o o k e - J e e v e s p r o c e d u r e (HJ) with an interior penalty function, based on the c o n t i n u o u s set o f design

565 variables they show which of the constraints is active. The condition of structure stability (S) is of decisive importance to the task of scalar optimization for WAGA=0 (Weight as the optimization criterion). Geometrical constraints, which determinate permissible thickness of panel layers, are of crucial importance for WAGA=I (flexibility as the optimization criterion). For WAGA 0+0.7 the strength conditions are active constraints too. The last column presents the preferred solutions. The abbreviation M]VI means that the preferred optimal solution was obtained with the help of the min-max method, P 1 - with the help of the global criterion method with the norm p=l, P2 - with the norm p=2 and TS - with the help of the TOPSIS method (Technique for Order Preference by Similarity to Ideal Solution). In most cases optimal parameters of the panel in the vector optimization are obtained for WAGA=0.5. It corresponds to equal weight of the two optimization criteria. Figures 5, 6, 7 and 8 present the optimal values of design variables h~, h2 and h3 depending on external pressure q for the panel under different load Px. ....

,1

I

i, ~:,~].o,r,.,z,,,o,

1.2 -{~

;

,,

,w,o,.o, I .

I

9 ~,o, .....

,.,

-- -- -- Thicknessof h=

~3---

]

,'~

[;.

E•---

}

~'~

[. . . . . . .

~---

L.

El-

~J

''

-{ P~=O,; MN/m

0.8

. . . .

....

"

"-..-

*

.~ ._.9.

e-

T'7 ;->"

.,..a

0,31MN/m

-'~

"'vr

/ -

m

==

,

.z

Px=0, 2 MNIm .~I

u.

0.4 --

F

O.4

T

i

T

I

VECTOR OPTIMIZATION ! {Preferred solutione ) }

l

0--

t

i

[

0--

0.005 External pressure q [MPa]

0 External pressure q [MPa]

Figure 5 The change in the face thickness (scalar optimization, WAGA=0)

,,

i T

T

J-

32 --[

Px=O 3 MNIm

1

0.007

Figure 6 The change in the face thickness (vector optimization) t

I

I

,,1

I

I VECTOR OPTIMIZATIONI(Pre, . . . . d s o l u t i o n . ) l

_:.

0.006

r

....

l

,/

)

E

r.

==

10

Px=0, 2 MNIm

28--

'~

._o -=_

z~/

"~' Px=0,2 MNIm

0 24

Px=0,1 MNIm

8

I 0

0.001

0.002 0.003 0.004 0.005 External pressure q 0VlPa]

0.006

PX=0,1 MNIm

J

2O 0.007

Figure 7 The change in the core thickness (scalar optimization, WAGA=0)

0

0.001

0.002 0.003 0.004 0.005 External pressure q [MPa]

0.006

0.007

Figure 8 The change in the core thickness (vector optimization)

This Page Intentionally Left Blank

Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

567

SHAPING OF OPEN CROSS SECTION OF THE THIN-WALLED BEAM WITH FLAT WEB AND MULTIPLATE FLANGE E. Magnucka-Blandzil, R. Krupa 2 and K. Magnucki 3 1Institute of Mathematics, Poznafi University of Technology ul. Piotrowo 3, 60-965 Poznafi, Poland 2Metalplast Invest plac Wolno~ci 6, 61-738 Poznafi, Poland 3 Institute of Technology, University of Zielona G6ra al. Wojska Polskiego 69, 65-625 Zielona Grra, Poland

ABSTRACT The subject of the analysis is a thin-walled prismatic beam with flat web and multiplate flange. The beam is loaded at both ends and submitted to pure bending. Open cross section of the beam is described by five dimensionless parameters controlling its shape. Basic values of cross section area resulting from classical theory of beam bending, warping function and warping moment of inertia resulting from general theory of warping torsion have been determined. The parametric shaping of the cross section is reduced into the problem of maximization of bending moment for fixed cross section area. Constraints of the solution result from geometrical, strength, and stability conditions. Numerical analysis has been performed.

KEYWORDS Thin-walled beam, monosymmetrical open cross section, warping function, lateral buckling.

INTRODUCTION Thin-walled structures are endangered with destruction mainly due to insufficient strength of material or loss of stability. The strength and stability of thin-walled beams, particularly coldformed, are the subject of present-day research. JOnson (1999) developed a classical theory of thin-walled beams, taking into account the change in their cross-section shape. Kotakowski, Kowal-Michalska and K~dziora (1997) numerically analyzed stability of thin-walled beams of open and closed cross-section profiles. Kubo and Kitahori (1998) performed experimental investigations of the stability of a monosymmetrical I-bar of various flange widths. Pi, Put and Trahair (1999) discussed the results of stability analysis of thin-walled channel and Z-bars. Optimization of the thin-walled structure is their further improvement. Zyczkowski (1990) made a basic contribution into development of shell optimization taking into consideration their

568 stability. He discussed in his monograph the results of research carried out in the domain. Bochenek and Zyczkowski (1990) have optimized an arc of I-section under stability constraints. Magnucki and Magnucka-Blandzi (1999) have determined an optimal shape of thin-walled beam section of variable thickness. Magnucki and Monczak (2000) have optimized a cross-section profile of a beam subject to bending. Manevich and Raksha (2000) have optimized a channel cross section of a thin-walled beam. Rao (1995) discussed mathematical essentials of optimization applied to technical problems. The topic of the work is a thin-walled beam subject to pure bending. The optimization is performed with a view to defining the cross-section profile of the beam of maximal safe bending moment.

ANALYTICAL DESCRIPTION OF THE OPEN CROSS SECTION The curvilinear shape of the cross-section for many purposes is unpractical, therefore it is often replaced by a piecewise straight line (Fig. 1). Total depth of the beam is H = 2a + t. The web of the beam is located in the middle of the cross-section width.

b&Y

0

H

z ~''

Figure 1: The open cross-section of the thin-walled beam The cross-section is characterized by the following dimensionless parameters b x1 = - , a

c x2 = - , a

d x 3 = --, a

and the angle 13.

(1)

The beam cross section area (2)

A l = 2 a t . fi

where 1 + 2cosl3- sin 13 A -- A ( X l ' X 2 ' X 3 ' ~ ) -- 1 +

2 cos 13

X1 + X2X3 .

The cross-section geometrical stiffness of torsion 2 Jt = -~tS a" f l 9

(3)

569 The coordinate of the centroid of cross-section

1 f~ z0--~a~,

(4)

where

f2 "- f2(Xl)X2)X3~)--Xl

I

X1 4" X2 -- X2X3 / ' 4cosl3

The moment of inertia of the cross-section area with respect to the y and z axes 1

dy =-~a3t" f3,

(5)

dz =2a3t" f4,

where 6cosl3 x~ + x 2 + X2X 3

A

x, ( 1 + ~1- t g 2 ) f4 = f4(xl,x2,x3,~5)= 2cos ~ fl~ +~ fll tgl3xl 13x~ +2cosl3 + 1 .-(I,~§ 3

3

)3

)'

tg~ +x2]-

fll = f l l ( X l , X 2 , ~ ) = l - - f l x l

The sectorial coordinate with respect of the point B shown in Fig. 2.

I

..... ~

3

i

2

i

-~ ff--'~ i

z-"

Figure 2: Plot of the sectorial coordinate-point B. Values of these coordinates are as follows:

toB~ = 0 '

1

, coB2 =--a2x~f~l 2

co~ - ~1 a ~x, (2+x2 - fll + x ~ ) .

1

c~ =-~a2x~(x2 -f~l),

f.OB4 --- -~1 a 2xi (2 + x 2 -f~,),

570 The distance from the shear center A of the cross-section to point B (Fig. 2)

zA-zs=-~1 AI03sydA=laf' 6 74,

(6)

where

f5 -- f5(Xl,X2,X3,~)- ~3Xl f 03B3 "+03B4 -'1-[2(1 -- X2).-F fll X2X 3

+~

a

] 03Bl }+-~[(3--X2)f-OB3+(3--2X2)O3B2 6cosl3

{[2(1- X2X3 )+ l]f.0B5 + (3 -- X2X3 )03B4 }"

The main warping function (the main sectorial coordinate) 03 for the open cross-section is shown in Fig.3.

i 3 5 2

60" 1

B

A

Z

Figure 3" Plot of the main sectorial coordinate-point A. Values of these coordinates are as follows: o3i = a 2 "~i

for

i = 1,2 ..... 5,

where ~.

1

f5

031=--6f11~4 ' 1 c~ =-2 xl

...

1 (x2 _ 1)~_ "

('t)2 ='6

x,

,14 --"~-fll,

...

1

lf5 (x z - fl, ) - - - 6f4

f5 (2+X2-fl, ) 6l f4 '

.~ =1-~ x, ( 2 + x 2 - f ` , + x~x~ )- ~. ~ ( l 1_ x z x 3 ) o~ The warping moment of inertia of the cross-section (the sectorial moment of inertia) 2

J,o =-~aSt"f6, where

(7)

571

"~ +2cOs~ X1 f6 = f6(Xl'X2'X3'~) = IfllO~

(~ + ~ + ~1~2)+ X2(~ + ~ + ~2~3)+

+Xl(~ +~42 +~3~4)+ X2X3(~2+~2 +~4~5)]. Parameters x 1,x2,x3,[3 determine the above geometric quantities of the open cross-section and they are parameters in the optimization problem.

FORMULATION OF PARAMETRIC OPTIMIZATION P R O B L E M A thin-walled beam with open cross section is in pure bending state - the bending moment M is constant. The optimization problem is defined as follows: the area A1 of the cross section of the beam is constant. For this value the parameters xt, x 2, x 3 , fS, the thickness t and the depth H are sought for which bending moment M will be will maximum. The strength and stability conditions are constraints of the solution. The depth of the beam (Fig. 1) n = 2a + t,

from which

a = 0.5(H - t).

(8)

Substituting the size a into the cross section area (2) one determines the parameter x 1 controlling the width of the beam x 1 = 2 (H

-

A-----!---~ t)t 0 +x2x3 -

)I

cos13 1 - sin [3+ 2 cos [3

(9)

The strength condition of the beam M H <

jT--f

- (~allow,

from which

M < 2 d~ -

n

Gallow 9

Substituting the inertia moment Jz (5)2 and the size a (8)2 into this condition gives (10)

M < M 1, where M, = (H -t)3 tCrat,ow f4 (x i, [3). 2H The critical moment for the beam under pure bending state McR =-'~-- J r 2 ( 1 + v ) + 7

J~

"

Substituting expressions (3), (5)1, (7) and (8)2 one can write the global stability condition for the beam (11)

M<M 2,

where M2 = 16)~2nb x 2E Ht 22 (H_t)3 I H - t 3t

f3(xi'~

I

2 H2A1 § X28)L 0+V) (H-t) 4

t

f6(x,,~

,

572

= __L. the relative length, H

n b - the safety factor.

E - Young's modulus, v - Poisson ratio. The critical stresses for the flange - the rectangular plate 7t2E ) ( t ] 2 OCR=g(l_v2 ~ ,

or

4x2E)( t ] 2 1 CrcR:3(1-v2 tI-t 7x "

The local stability condition for the flange

M a < crcR, Jz

M < ~Jz cr cR.

from which

(12)

a

Substituting expressions (5)2, (8) and (8)2 one can write the local stability condition for the flange M < M 3,

(13)

where

2nEEt 3 f4(xi,[3 )

The objective function - the maximization problem (14)

M = max {M l ~ M 2 ~ M 3 } x i .[3,t,H

where

x i (i = 1,2,3)and 13 - dimensionless parameters - continuous variables, t - the thickness of the beam w a l l - t h e discrete variable (At = 0.5 mm), H - the depth of the beam - the discrete variable ( A H = 20ram), - the symbol of conjunction.

N U M E R I C A L ANAl

~Y i i

I l.~j.~q.~ ~ .1~~./.i..~.~ m-~m..Pl~Pl~l' [ ~ ~'~1

i Z

Fig.4. The optimal cross section of the beam

573 Numerical

analysis

has

been

performed

for

steel beams:

the

Young's

modulus

E = 2.05.105 MPa, the Poisson ratio v = 0.3 and the allowable stress tr~ttow= 205MPa. The cross section area A~ = 2340mm 2, the relative length of the beam ~. = 12.5 and the factor of safety n b = 1.6 in stability conditions. Results of the numerical analysis are following: x I =1.54, x 2 =0.69, x 3 =0.50, 13=0.38rad, t =3.5mm, H = 200mm and the maximal bending moment Mmax = 24.8 kNm. The optimal cross section of the beam is shown in Fig.4.

CONCLUSIONS Optimal cross section of the beam under pure bending has no flat web provided by the calculation model. The web of the beam has been located in the middle of the cross section width. The optimum solution has been determined by the strength condition and the conditions of global stability of the beam and local stability of the flanges.

References Bochenek, B., Zyczkowski, M. (1990). Optimal I-section of an elastic arch under stability constraints. Engineering Optimization 16, 137-148. J6nson J. (1999). Distortional warping functions and shear distributions in thin-walled beams. Thin-Walled Structures 33, 245-268. Kotakowski Z., Kowal-Michalska K., K~dziora S. (1997). Determination of inelastic stability of thin walled isotropic columns using elastic orthotropic plates equations. Mechanics and Mechanical Engineering 1:1,79-100. Kubo M., Kitahori H. (1998). Lateral buckling capacities of thin-walled monosymmetric Ibeams. Second Int. Conf. On Thin-Walled Structures. Research and Development Ed. Shanmugam N. E. and others, Elsevier, Amsterdam-Oxford-Singapore-Tokyo, 705-712. Magnucki K., Magnucka-Blandzi E. (1999). Variational design of open cross section thinwalled beam under stability constraints. Thin-Walled Structures, 35, 185-191. Magnucki K., Monczak T. (2000). Optimum shape of open cross section of a thin-walled beam. Engineering Optimization 32, 335-351 Manevich A. I., Raksha S. V. (2000). The optimum design of compressed thin-walled column of open cross-section. 9-th Symposium on Stability of Structures, Zakopane/Poland, 189-196. Pi Y.L., Put B.M., Trahair N. S. (1999). Lateral buckling strengths of cold-formed Z-section beams. Thin-Walled Structures 34, 65-93. Rao S.S. (1995). Engineering Optimization. A Wiley-Interscience Publication. New York, Toronto, Singapore. Zyczkowski, M. (ed.) (1990). Structural optimization under stability and vibration constraints. CISM Udine, Springer-Verlag, Wien- New York.

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Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

575

TWO-CRITERIA OPTIMIZATION OF THIN-WALLED BEAMS-COLUMNS UNDER COMPRESSION AND BENDING A. I. Manevich', S. V. Raksha z

Department of Theoretical Mechanics and Strength of Materials, Ukrainian University of Chemical Engineering, Dniepropetrovsk, 49005, Ukraine (2~Department of Lift-Transport Machines, Dniepropetrovsk State Railway Technical University, Dniepropetrovsk, 49010, Ukraine

ABSTRACT A two-criteria optimization problem for thin-walled members of open cross-section with one axis of symmetry (of the channel type) subjected to compression or/and bending in the symmetry plane under stability constraints is solved. The objective function vector components are the axial compressive force and the bending moment. Optimal configurations of thin-walled members for marginal cases of loading - axial compression or pure bending - are obtained, Pareto-optima are constructed and <>optimal projects are derived as a function of a single leading nondimensional parameter of weight. The obtained optimal projects are universal, in distinction from the optimal projects for special loading cases, and applicable under arbitrary combination of compression and bending. Comparison of standard profiles (specified by the design codes) with the optimal projects enables us to indicate <>standard profiles which are far from optimal ones under any loads.

KEYWORDS Structural Optimization, Multi-criteria minimization, Thin-walled Members, Pareto-optima

INTRODUCTION

The minimum weight design problems for thin-walled bars - beams under compression or bending with respect to their stability were investigated in a few works already in 60-70-s on the base of various theoretical models (see review by Zyczkowski & Gajewski (1983)) and were refined later (Yoshida & Maegawa (1979), Manevich & Raksha (2000) and others). Optimal profiles of compressed or bent thin-walled members which are found for the separate loads are essentially different As in exploitation they undergo, as a rule, the action of various loads, separately or in a combination, there arises the question: what configuration of the profile may be considered as optimal with account of such multifarious character of the loading? This problem of the choice of a <>optimal project may be formulated and solved in the framework of the theory of multi-criteria optimization of structures (vector-valued optimization) which was intensively elaborated in last decades.

576 In the paper the two-criterion optimization problem for thin-walled members of open crosssection with one axis of symmetry (of channel type) subjected to compression and bending (in various combination), under stability constraints is solved. The objective function vector components are axial compressive force and bending moment. An efficient method of nonlinear programming is used for solving the optimization problem. At the first stage optimal configurations of thin-walled members are derived for the marginal cases of loading - axial compression or pure bending (with using exact linear solutions for local buckling). Optimal dimensionless parameters are presented as a function of a single leading parameters - a nondimensional weight parameter. At the second stage the Pareto-optima are constructed in the plane (P- Mz) where P is the compressive force and Mz is the bending moment. Finally, at the third stage <> optimal projects are derived with using a global criterion, comprising P and Mz. The obtained optimal members are universal, in distinction from the optimal projects for special loading cases, and applicable at arbitrary combination of compression and bending. Comparison of standard profiles (specified by the design codes) with the obtained optimal projects enables us to indicate <>standard profiles which can not be optimal at any loads considered.

STATEMENT OF THE PROBLEM AND THE METHOD OF SOLUTION

Let us consider a thin-walled member of open cross-section shown in Fig.l, simply supported on its edges and subjected to compression and bending (in the symmetry plane) in various (a priori unknown) combinations. The length L , force P, the material properties (elasticity modulus E, Poisson's ratio v) are considered given. The profile thickness t, widths of the web and flange b~, b2 are design variables (thickness of all elements is considered to be constant implying cold-formed members). Y

9I

_ .

t:~b~.

I

I ~

I ~,

I

I

I I . . . . .

Figurel: Cross-sections of a bar and design variables

_1

I.. . . . . . .

Figure 2: Buckling modes of a channel

The objective function depends upon formulation of the optimization problem We consider sequentially following optimization problems: 1. The optimization with a single criterion - maximum of the critical compressive force P or maximum of the bending moment Mz for given total cross-section area. 2. Constructing Pareto-optimal cross-sections in the framework of the two-criteria optimization problem. 3. Optimization with using a global criterion which comprises P and Mz. The optimization problems are stated as nonlinear programming ones. In the single-criterion optimization for a compressed column the objective function is P = max min (Pi., Pc Pt ), where Ps, Pc el are critical forces for the flexural (euler) mode in the symmetry plane, for

577 torsional-flexural mode and local mode respectively (see. Fig.2). For a beam under pure bending the objective function is Mz = max rain (Mzr Mz~), rae Mzto Mz~ are critical bending moments for torsional-flexural mode (out-of-plane buckling) and for local buckling. The overall buckling stresses for the bar under compression or bending were calculated according to the theory of thin-walled bars (f.e., Vlasov, 1959). The local buckling stresses were calculated according to (Manevich & Raksha (1996)). All the plates that constitute the thin-walled bar-beam are divided on several strips for which the longitudinal stress may be considered as a constant, and the exact solution of the buckling problem is constructed by conjugation of the solutions for all the strips, with exact conjugation conditions at the contact lines (in distinction from the finite strips method which uses power approximations for displacements). For each the strip with local coordinates (x, y) b the solution of the differential equations of stability was assumed in the form w = w(r/) sin mrc~:, r / = y / L, ~: = x / L (x is the longitudinal coordinate). Boundary conditions on free edges and conjugation conditions on the contact lines result in a characteristic equation which determines the local critical stresses (after minimization in m). All details of the calculations are dropped here. Because of closeness of the critical stresses for a cluster of short-wave local modes it is necessary to take into account many local modes. Note that dependencies of the local buckling stresses on the half-wave number m can have two local minima, one of them being determined by the web slenderness, another - by the flange slenderness. In view of these features the local buckling stresses were calculated for a wide range of half-waves numbers, i.e. instead of one constraint we consider a set of constraints (as a rule, for values m=2-25). Strength constraints, as a rule, were not imposed, i.e. the assumption was assumed that the material is elastic and the yield limit is sufficiently high. As it will be seen from the obtained solution this assumption is valid in a range of relatively low values of the weight parameter G* or the load parameter P* (see below), for usually employed steels and alloys. For other G* values the presented solutions determine idealized optimal configurations. The nonlinear programming (NP) problem was solved by the linearized method of reduced gradient (Manevich (1979)).. This method realizing the idea of changing an independent variables set in a vicinity of the admissible domain boundary by means of the linear operations with the sensitivity matrices, effectively overcomes familiar difficulties arising at solving the NP-problem (in particular, connected with zigzag-type motion in a boundary vicinity). Our experience of many years employment of this algorithm has shown its high efficiency and reliability. As a rule, 10 - 20 iterations were required to achieve the optimum with relative error of order 10.3 . For generality of analysis all the objective functions and constraints were formulated in dimensionless parameters of weight, load and stress:

A G* =---=103 L~

P* = ~ P1 '

L 2 .E

M* =

06 '

M L 3 .E

10 s or* = - cr - 10 3 '

E

(3)

where A is the cross-section area (the scaling multipliers are introduced in order to deal with parameters of order of unity). The cross-section also was characterized by nondimensional geometric parameters b2 / bl, t //91 .

RESULTS OF THE SOLUTION All optimal nondimensional parameters are determined by specifying the single weight parameter G* (all dimensional parameters are determined when additionally the length L is given). We considered the range of parameter G* (0; 0.6) in which the assumption about elastic

578 deformations of the material may be justified according to results of given solution (for usual materials). In this range several values of G* have been chosen for which the optimization problem was solved. Optimization with a single criterion. Columns under compression or bending.

Solution of the optimization problem for centrally compressed columns of channel crosssection based on the exact solution of the local buckling problem for the column as a plates assemblage was obtained in Manevich & Raksha (2000). Here we present only some results of the solution. For bars under pure bending earlier only approximate solutions have been obtained based on simplified expressions for the local buckling stresses. Here an exact solution for this case is presented. The optimal columns under compression have equal critical stresses for the torsionalflexural mode and one of the local modes. For overall bending mode (in the symmetry plane, Fig.l) critical stresses are found to be higher by 15-25 %. The optimal beams under pure bending also have equal critical moments for the torsional- flexural mode (out-of-plane buckling) and one of the local modes. The calculations show that the optimal profiles under compression as well as under bending have nearly constant values of flange width to web width ratio in entire range of G* considered. For compressed channels b2 //91=0.42 -0.43; .in the case of pure bending b2 / hi= 0.530.56. The thickness parameter t/b~ depends upon G* as follows.

TABLE 1 PARAMETER l/b 1 FOR SINGLE-CRITERIONOPTIMA Parameter G* t/b~ compression bending

0.1

0.2

0.3

0.4

0.5

0.0139 0.0223

0.0174 0.0283

0.0200 0.0323

0.0219 0.0357

0.0238 0.0389

Parameters of cross-sections, optimal for compression (max P*) and pure bending (max Mz*) are presented in Figure 3 versus the cross-section total area parameter G*. Optimal crosssections turn out to be rather different for compression and for bending.

579

b21bl

t,/bl max

0,55

max

Mz

Mz

0,03

0,5

,-

, , . . _ . - - . ,

.....---

.-,.

I

0,02 i.

0,45

0,4 0,35

~ m a x

P

max

c omp r omi s e project(P,Mz)

M, ~t y

0,1

I'

c omp r omis e project(P,Mz)

••blY]

- ~~]b2 "

P

0,01

]b2

i

i

i

0,2

i

0,3

i

i

0,4 G*

I

.

0,1

i

0,2

I

ii

i

i

0,3

I

0,4 G*

a b Figure 3: Nondimensional geometrical parameters of optimal bars in single-criterion optimization and <>projects. The load parameters P* and Mz" versus the weight parameter G* for both single-criterion optima are shown in Fig.4. The calculations show that for the single-criterion optima dependencies of the critical force P* and critical moments in two planes Mz~ and My" upon G* can be approximated with high accuracy (the error less than 1%) by the power functions. These approximations are presented in Table 2. Pr

:

:

:

i

!

0,6 . . . . . . i. . . . . . . . I

0,4 . . . . . . . . . .

0

0

0,1

f ..... ' .... +----~ ....

I

I

0,2

0,3

,,!

0,4

0,5 G*

Figure 4: Load parameters P* and M~z versus the weight parameter G* for the single-criterion optima. TABLE 2 APPROXIMATIONS OF CRITICAL FORCES AND MOMENTS VIA THE WEIGHT PARAMETER G * FOR SINGLE-CRITERION OPTIMA

Optimal bar under compression

Optimal bar under pure bending

P* = 2.794 G* 1.668

P* = 2.453 G ' 1 6 7 6

M z = 3.805 G* 1.978

M z = 6.527 G* 1.995

M y = 16.334G* 2.0025

My = 14.184 G* 1.956

580

We see that the optimal channel under condition max P* can carry out the moment Mz which approximately by 40 % less than the optimal channel for max Mz*, and, correspondingly, the latter bar can carry out the compressive force by 12 % lesser than the optimal bar for max P*.

Two-criteria optimization

Pareto-optima There were constructed the Pareto-optima for two criteria max P*, max M,* for a set of values G*. The Pareto-optimal projects were obtained by means of minimization in P" with constraints on M~* which gradually increased Mz* from the value obtained for the optimal bar under compression up to the value obtained for the optimal bar under bending. In Figure 5, a the Pareto-optimal solutions are presented in the plane P'-Mz*for G*=0.4. In order to obtain the Pareto-optima for different values of G* in a compact form, they were recalculated in parameters P*/G *1"668, M*/G *1"995, with account of the power approximations which are presented in Table 2. Results are presented in Figure 5, b. The curves for various G* almost coincide in these variables, and therefore these curves may be used approximately for any G* values. In Figure 6 the dashed lines present optimal projects obtained in the single-criterion optimization, and the solid lines show Pareto-optimal solutions for several G* values (each point on this lines presents limit values of P" and M~* for a certain optimal project obtained for a given G*).

--"~i Pareto-c'urve '

1,o --rX:

, G*=o,4

comproml s e ~ ' ,

project

~

0,8 ............. M . , y 0,53

i

0,55

I i 0,57 0,59 1:'*

M Z/GIBil

6,0 ....

:

G* -o,1...o, 4 &-1,995

5,0 . . . . . .

fl

M~_~y

;-~---{"

4,0

2,4

1

2,5

i

"%

.... !' I I 2,6 2,7 I:,'~/G/~'

b

Figure 5" Pareto-optimal projects for two criteria - P , M~*

581

Mz

0,8

0,6

...

Y

\G'=o,3

i

"f

0,4 0,2

13

0,1

0,2

0,3

0,4

0,5

0,6

P*

Figure 6: The single-criterion optimal projects (dashed lines ), compromised projects (solid line) and the Pareto-optimal solutions for several G* values. uCompromise>> projects

With the aim of finding a <) project the problem of optimization was solved with the global criterion in the form:

(5)

F = P /P m=+M~ /M ,.~x

where P'max and M*, m = are the values of P* and M*,, obtained in single-criterion optimizations in P* and M * . The projects obtained are presented in Figure 3 (dashed lines) and indicated by the cross in Figure 5. In Figure 7 cross-sections of the optimal bars for max P*, max M*~ and the <) optimum are compared for given G*=0.4 and length L=I m . We see that the compromised optima are close to the single-criterion optima obtained for max M*. They can carry out almost the same limit moment, but the limit force is less about 10% in comparison with the project obtained under condition of max P* (for the same G*). It is worthwhile to compare the optimal values of nondimensional parameters with those for standard profiles . In Table 3 there are presented dimensions of channel cross-sections according to Russian standards GOST 8278-83 and corresponding values of nondimensional parameters b2/ b,, t~ b,. !

8,

....

|

42,8

lll

2,60

1.2,17

,

z3,

,i

76,9 ~. 2,62 . . .

|

!.. 39, ,i

p. 3 ,0 .I!

b C Figure 7: Optimal profiles obtained in single-criterion optimization (a, b), and <
582 TABLE 3 DIMENSIONS OF BENT STEELCHANNELPROFILESACCORDINGTO GOST 8278-83

b2, m m 32 50 '50 60 "60 ' 60 80 80 80 80 80 125 100

bl~ mill

60 80 100 120 120 140 160 160 180 200 200 250 300

.

.

.

t, m m 3 4 3 " 4 5 4 4 5 5 4 5 6 8

.

.

.

.

.

.

....

.

bzl b, 0.53 0.625 0.5 0.5 0.5 0.43 0.5 0.5 0.44 0.4 0.4 0.5 0.33 .

.

.

.

.

t I bl 0.05 0.05 0.03 0.033 0.042 0.029 0.025 0.031 0.028 0.02 0.025 0.024 0.027

In Figure 8 domains of optimal nondimensional parameters in plane t~ b,, b21 b, for singlecriterion and two-criteria optima are shown. The dark points present values of these parameters for the standard profiles. |

b21bt ( 9 - standard - optimal

0,6 0,55

,

!

. . . .

profiles parameters

,

,

i

,

i

_~|

.I., 1 : ~.! ! ! i ! i .~ ' i i i i i. - ~ i ! ! i.i : ~ [ .

,max M= I

max

0,4

. . . .

,

9

0,35

. . . .

9 '!

.

I

01

,

i

i

,

9

r

,

',

S

I1 I

~ . . . . ~. . . . . I I -I . . . . . I

'

. . . .

0,03

,

dI

9

. . . . . . . . .

|

~- . . . .

0,02

,

I

I

-

|,

,

~ - - 4 - - , t

-

i

............... %: . . . .

_.

,. -

;

0,5 0,45

' zone I

I

'

"~ . . . . .

',

0,04

,

,

"~ . . . . .

1

0,05

,

t/t

Figure 8: Domains of optimal nondimensional parameters and the standard profiles. We see that the majority of the profiles fall in (or close to) the optimal zone for compromised projects, but some profiles are rather far from this zone. So we can indicate those profiles which are not optimal at any load considered (any combination of axial force and bending moment in the symmetry plane).

REFERENCES Manevich A.I. (1979). Stability and Optimal Design of Stiffened Shells. Kiev-Donetsk, 152 p.(in Russian).

583 Manevich A.I., Raksha S.V. (1996). Local and Coupled Buckling of Thin-Walled Bars under Compression and Bending. In: Theoretical Foundations Of Civil Engineering, 4:1, part 2 (Proc. of the Polish-Ukrainian seminar, Warsaw, July 1996), Dnepropetrovsk, 270-275 (in Russian). Manevich A.I., Raksha S.V. (2000). Optimal Centrally Compressed Bars of Open CrossSection. In: Theoretical Foundations of Civil Engineering..Proc. of the Polish-Ukrainian seminar, Warsaw, 484-489 (in Russian). Manevich A.I., Raksha S.V.(2000). The Optimum Design of Compressed Thin-walled Columns of Open Cross-section. In: Stability of Structures. IX Symposium (Zakopane, IX 2000), 189-196. Vlasov V.Z. (1959). Thin-Walled Elastic Bars. 2-nd edition. Moscow, GIFML, 568 p. (in Russian). Yoshida H., Maegawa K. (1979). The Optimum Cross Section of Channel Columns. Int. J. Mech. Sci., 21: 3, 149-160. Zyczkowski M., Gajewski A. (1983). Optimal Structural Design With Stability Constraints. In: COLLAPSE. The buckling of structures in theory and practice. Ed. J.M.T. Thompson, G.W. Hunt. Cambridge Univ. Press.

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Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

585

OPTIMIZATION OF VOLUME FOR COMPOSITE PLATED AND SHELL STRUCTURES A. Muc and W. Gurba Institute of Mechanics & Machine Design, Cracow University of Technology, Krakbw, Poland

ABSTRACT The aim of the paper is to discuss the possibility of using genetic algorithms in the volume optimization of composite laminated plates and shells. A particular attention is focused on the conjunction of genetic algorithm procedures with FE codes enabling us to evaluate easily all possible types of required objective functionals. The presented coding procedures of design variables are also especially directed on the use of FE packages. Two particular cases of volume optimization problems are formulated and solved: (1) thickness optimization (a constant area), (2) optimization of the area occupied by a structure (a constant thickness). The volume optimization problems are always subjected to additional constraints that are introduced by an augmented objective functional. Two problems dealing with the stress distribution analysis for plates and buckling of cylindrical panels illustrate the effectiveness of the proposed method of optimization.

KEYWORDS Optimization of Volume, Genetic Algorithms, Plates, Shells, Composites, FE Analysis.

INTRODUCTION Nowadays structural optimization techniques based on finite element (FE) analysis and applications of approximation algorithms have reached a high level of reliability and efficiency but up to now mainly in the area of isotropic structures. A number of studies have been published on the optimization of composite structures (see e.g. WCSMMO Proc. [1-3]) dealing mainly with particular problems and using numerical and optimization algorithms directed only for solving the selected problems. The existence of many local optima and the lack of generally acceptable definitions of design variables are the essential complications connected with the optimization analysis of composite structures. To avoid these difficulties and to prevent divergency in searching for the global optimum it is necessary to propose the conjuction of the FE techniques used in computations of an objective functional with the efficient optimization algorithms. This stabilises the optimization process and reduces significantly the computational effort.

586 The present paper focuses the attention on the volume optimization for composite plated and shell structures. For this purpose, the technique and methods proposed in Refs [4,5] are adopted. However, in contrast to the methods and results discussed in [4,5] the optimization of volume is considered herein. This problem is understood and formulated in rather general sense; i.e. it means that not only the thickness variations are studied but also the possible eliminations of various inactive (according to the prescribed criterion) elements of structures can be taken into account.

CODING OF DESIGN VARIABLES AND OPTIMIZATION PROBLEMS Using the FE approximation to evaluate the objective functional it is possible to change (redefine) various groups of data corresponding to: (i) the existence (or not) of a finite element (or groups of them) in the initial mesh - the reduction of the volume, (ii) the variations of thicknesses at nodes of a finite element e t c . - it may also change the total volume (an increase or a decrease of it). Thus, having a 2-D structure representig plate/shell each finite element is described as the term of the 2-D matrix: a(i,j)

(1)

where the coefficients i,j characterize uniquely the position of the finite element in the given mesh. The operation of coding is directly connected with the description of the above-mentioned finite element properties. The existence of the finite element at the i-th row and the j-th column can be determined in the following way:

10a(i, j) = 80. =

the existence of the ij- th FE in the mesh the lack of the ij - th FE in the mesh

(2)

whereas the thickness of the individual FE is characterized by the natural number k multiplied by a real constant, i.e.: a(i,j) = 0.1 * k(i,j), k(ij)e{ 1,2 .... K}

(3)

Both the value of the real multiplier (now it is taken to be equal to 0.1) and the total number of k values in the sequence is completely free. There is also an equivalent representation of the thickness distributions in the given mesh by real numbers. It is assumed in advance that the thickness of the structure can belong to the following interval:

t ~ [a,b],

a = {0,0,0,0,0,0}, b =/1,1,1,1,1,1}

(4)

The lower and upper bounds of the interval are expressed by the strings of the 0 and 1 numbers presented in Eqn (4). Using the binary representation the interval [a,b] may be divided into 26 - 1 subintervals. Increasing the number of positions in the string one can divide the initial interval into subintervals with the required density and in this way express the various thicknesses in the mesh. The thicknesses are piecewise constant on the FE used in the analysis. In this way one can define the total volume of the meshed 2-D structure as follows: I J

V=tEESiy i=1 j--I

(5)

587 or I

J

V=O.1A~'~k(i,j)

(6)

i=1 j = l

where A denotes the area of each individual FE (identical for each i,j), and t is a plate or shell thickness. Now, the optimization problem can be written in the following explicit form:

Min V[a(i,j)] a(i,j)

(7)

where the pair of coefficients (i,j) characterizes the properties of the ij-th element. In addition, the problem (7) is subjected to different equality (or inequality) constraint conditions. They form is prescribed for the particular optimization problems. Design variables have to be selected among a(ij) given, discrete variables. In fact, the relation (7) determines two different optimization problems since two different coding methods (5) and (6) have been introduced. They may be joined and treated as a single optimization problem but such a problem will not be analyzed here. It should be emphasized that the formulated optimization problem (7) has few characteristic features given below: it is a discrete optimization problem, it is strongly dependent on the type of the FE discretization and the particular classes of 2-D FE are (according to our experience) the most convenient in our analysis - they are presented in Fig. 1, m since the optimization problem (7) is a discrete problem it should satisfy the general rules and theorems of the combinatorial analysis.

?:---.~-,. a

a

a

2a

Figure 1" Types of FE used in the optimization problems GENETIC ALGORITHMS The attributes that define a particular genetic algorithm include: coding, crossover, mutation and selection. Each individual in a population is called as a chromosome that carries out the basic information about the problem. In our case it deals with the form of the mesh. The chromosome is defined as follows:

chromosome =

a(1, j ), ~ a( 2, j) ..... ~" a( I, j) \j=l

j=l

(8)

j=l

and in general it consists of I*J numbers equal to the number of FE in the mesh. In the case of coding given by Eqn (2) some of the positions in the chromosome (8) have to be always occupied by FE

588 (preserved) and they form so called schemata. Those places are strictly determined by the form of boundary/symmetry conditions, positions where the external loading is acting on the structure etc. They are further called as the base points in the algorithm. The individuals in the initial populations (chromosomes) are generated in a random way. As the thickness variation is considered a random generator produces series of natuarl numbers in the given range (see Eqn (3)) for each FE in the mesh with the exception (or not) of the base points. In the second case of coding defined by Eqn (2) the random generator determines whether the ij-th FE exists in the mesh or not (coded by 1 or 0, respectively). However, in such a case it is necessary to restrict a complete freedom in the choice of the mesh since the mesh should consists of the series of joined FE (at least at one edge) and in addition, series of FE should joined the introduced base points. The strategy and results of such a pseudo-random choice of the single chromosome are schematically presented in Fig.2 for two types of FE (see Fig. 1). The crossover operations used to generate from a pair of parental chromosomes a pair of children are limited herein to a single-point crossover. The experience with different types of the crossover operations shows evidently (see Muc,Saj [6]) that the single-point crossover gives the best results for preserving schemata (the base points). For generation of a new mesh the crossover operation is defined as the multiplication operation of two matrices (1) representing different chromosomes. If the result of the multiplication is equal to 1 that FE is selected as the point of the crossover. Of course, the base points are not taken into account. For the thickness optimization problem the crossover operation is defined in the classical manner.

Figure2: Forms of randomly generated chromosomes for different types of FE (three base points) Mutation introduces local random changes into the design and allow us to create new features. It is especially important in the case the random choice of the FE mesh. Mutation operation is understood in the sense of adding (or subtracting) to the existing mesh of a new FE. For the thickness distribution optimization mutation corresponds to the change of the FE property (thickness). A new value of the thickness is evaluated in a random way from the admissible range of the natural k number variations. The selection process is the most important from the point of view of the reproduction of a new, better than previous population of chromosomes. Different performances of the genetic algorithm have been studied (see Ref [6]) including: (i) the classical roulette wheel method, (ii) the stochastic tournament procedure, (iii) the tournament with random pairs and (iv) the ranking selection scheme to observe their influence on the convergence and effectiveness of the optimization process. Good results have been obtained for the tomament with random pairs conjugated with the eUitist method (the choice of the best individuals- the best fitness value).

589 An augmented objective functional (instead of V - see Eqn (7)), used to rank the designs with constraints, is defined as follows: V* = V + p l a -

(9)

P2fl

where t~ means the violation of the most critical constraint, whereas 13 is a margin of the same constraint; they are both positive. Pl is a penalty parameter, and p2 is a bonus parameter for constraint margins. From the computational effectiveness point of view it is not worth to start the optimization algorithm with a very dense initial mesh. Commonly, at the beginning the rectangular FE (see Fig. 1) are used to discretize the problem and to find the first approximation of the optimal solutions. Then, more dense meshes are introduced to the first approximation of solutions (triangular ones) in order to tune (improve) previous solutions and to obtain the better approximations. This procedure resembles evolutionary strategy. Both crossover and mutation operations do not allow us to loose other quasioptimal solutions. However, it should be emphasized that the use of genetic algorithms does not allow us to find (to determine) one global optimum but few quasi-optimal solutions having the similar (best) values of the fitness functions. It is also connected with the existence of a variety of local optima for composite structures.

NUMERICAL RESULTS

Rectangular panel Let us consider the simply supported, rectangular panel having the following, geometrical, dimensionless parameters Lx/Ly = 2, t/Lx = 0.01 and loaded by a concentrated force at its c e n t e r Fig.3. The optimization problem is subjected two two additional inequality constraints: Uma x <__Uadmi s

and

Uy --
(10)

where the first condition in Eqn (10) describes the constraints for the Mises equivalent stresses, and the second for the y translation at the point where the load is applied. Due to the symmetry of the problem a quarter of a plate is modelled with the use of the plane stress triangular finite elements (NKTP=2). At the beginninig of the optimization process each eight of triangles is grouped to form a rectangular. Three base points are introduced herein.The panel is optimized for minimum material volume with the constraints (10) taken into account in the augmented form (9). The optimization results for different material combinations are plotted in Fig.4. ~"

Zr

w

Lye> I p

Figure3: The initial sizes of the structure

590

Figure 4: Optimal shapes (presented fight half only) As it may be seen a large amount of material savings can be obtained with the use of GA procedures. However, the optimal results are similarly as previously highly affected by the material parameters. The rotation of fibre orientations results simply in 20 % savings in volume (mass).

Buckling of cylindrical panel

Figure 5: Optimal thickness distribution for a quarter of the cylindrical panel (the projecetion on the xy plane - the concentrated force is located at the left, lower comer of the panel) The next example deals with the optomization problem of the thickness ditribution for the cylindrical panel subjected to the action of a normal concentrated force P located at its center. Therefore the optimization problem (7) is subjected to the following constraints: 2 cr = "~"admis

(11)

t < taxis

(12)

and

where Zcr denotes the buckling load parameter. ~,admis means the initial buckling load parameter computed for the constant thickness distribution over the area of the structure. This value of the constant thickness is denoted as taxis, and t is a thickness distribution over the shell structure. In the numerical example that value have been evaluated for following geometrical parameters of the panel: tadmis=0.5, R/tadmis = 160, L/R = 2, f/R = 0.5, where R - the radius, L - the total length and f - the shallowness parameter. Assuming that the Young modulus is equal to 30 [GPa] and the value of the conentrated force P = 10 [N] the buckling load parameter ~,admis is equal to 3.519. Using the

591 augmented optimization problem (9) and introducing one base point o n l y - at the place where the external force is located, the optimal thickness distributions have been obtained. The optimization results are plotted in Fig.5 and the final volume is equal to 40% of the initial one. Probably it is possible to obtain further reduction of the volume (thickness) by the increase of the thickness at the base point over the admissible value t = 0.5. It should be mentioned that at the beginning of the optimization process the shell has been divided into four rectangular parts with two admissible thicknesses only, i.e. K - 2 in Eqn (3) and the parameter 0.1 has been replaced by the number 0.25. Finally, a quarter of the structure have been divided into 512 triangular FE, K was equal to 10 and the parameter 0.1 in Eqn (3) was replaced by the number 0.05.

CONCLUSIONS An application of genetic algorithms to the volume optimization of plated and shell structures has been demonstrated. The modelling of optimization problems is strongly connected by the form of coding design variables. Different coding methods based on the integer or real representations of design variables may be easily implemented to the design codes. The problem of coding is associated with the description of the FE properties; FE codes are used in the evaluation of the objective functionals. Using the augmented form of the objective functionals various additional constraints can be simply inserted into the optimization analysis. The results shows the applicability of the proposed method of the volume optimization to a vast class of optimization problems including stress, displacement or buckling constraints. Numerical results demonstrate also the influence of fibre orientations on the final (optimal) value of the volume. However, with the use of genetic algorithms we are not able to prove that the optimal values are global ones. They should be rather treated as one of possible quasi-optimal solutions. It should be also mentioned that the presented version of the volume optimization by the elimination of some FE from the initial mesh can be also understood as a variant of material optimization. In the material optimization problem two or more different materials may exist and the objective is to find the optimal distribution of two materials having different mechanical properties. The current method may be easily adopted to the analysis of such class of optimization problems because the eliminated from the initial mesh groups of FE may have an attribute of different mechanical (material) property. The analysis in this area is now carried out and the results will be reported in the next paper. REFERENCES [ 1] [2] [3] [4]

[5] [6]

G. Rozvany, N. Olchoff (edts), Proceedings 1st WCSSMO Congress. Pergamon, 1996 R. Gutkowski, Z. Mr6z (edts), Proceedings 2"d WCSSMO Congress. Warszawa, 1997 C. Bloebaum (edt), Proceedings 3 rd WCSSMO Congress. Buffalo (1999) Muc A., Gurba W., Application of Genetic Algorithms to Shape Optimization of Composite Structures with Holes. Proceedings International Conference ICCST/3. Durban, 2000, pp. 147 -151. Muc A., Gurba W., Genetic algorithms and finite element analysis in optimization of composite structure, Composite Structures,2001, (accepted for publication ) A.Muc, P. Saj,Transverse shear effects in discrete optimization of laminated compressed plates under buckling and FPF constraints, Proc.ICCM/11,1997, Woodhead Publ., Vol.l,pp.331-341

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Third Intemational Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

593

SENSITIVITY ANALISYS OF STRUCTURES MADE OF THIN-WALLED ~-PROFILES K. Rzeszut, W. Kaekol and A. Garstecki Institute of Structural Engineering, Poznafi University of Technology, Poznafi, Poland

ABSTRACT The paper presents a study of the stability of columns and beams made of single Y~cross-section and double E close-open cross-section. Linear stability and nonlinear static analyses were performed. In the nonlinear case initial imperfections were accounted for. The sensitivity analysis with respect to cross-sectional parameters and to initial imperfections is presented. Numerical examples were solved using Finite Strip Method and Finite Element Method, both of them employing shell elements. In this way global and local instabilities were studied. Pure global instability was also studied using FEM with Vlasov beam elements.

KEYWORDS thin-walled structures, stability, sensitivity, imperfections. INTRODUCTION New technologies of cold rolling, and of corrosion and a fire protection gave rise to a wide application of cold-rolled thin-walled elements in structural engineering. The cold-rolled elements are produced from very thin steel sheets what is advantageous from economical reasons. However, these structures show great tendency towards local instability and are sensitive to initial geometric imperfections. Therefore, the application of advanced analysis methods of this type of structures is well motivated. Design engineers are well aware that the analysis of thin walled cross sections has to account for initial imperfections and that the structural response can take form of the local instability. Shell theory must be used for the analysis of these effects. However, application of FEM with shell elements leads to very large dimensions of matrices and is often unacceptable by designers. On the other hand, application of beam elements, i.e. Vlasov type elements, limits the analysis to global effects and moreover the prediction of the errors following from the Vlasov's assumption of undeformed crosssection is difficult. Therefore, it is reasonable to apply Finite Strip Method (FSM) to this class of problems [10]. It has been well proved that FSM is very efficient in static and stability analyses of plates and plate assemblies. The theoretical background of the method was presented by Cheung [1, 2]. The application of FSM to geometrically nonlinear problems has been demonstrated in [4], the interaction of local and global buckling problems was solved in [8], [9] and [3]. The problem of

594 optimal design of E cross-sections was solved with the use of FEM [10], too. However, due to the simplicity of beam theories, it is still worthy the further improvement of these theories and study of the limits of their application. This paper presents the sensitivity analysis with respect to cross-sectional parameters. The sensitivity operators were derived using Direct Differentiation Method [7]. The sensitivity with respect to initial imperfections was studied too, by the perturbation of the parameters which controlled the intensity of the imperfection. The numerical examples were solved using the FSM code [6] and the FEM code ABAQUS [5]. Beam finite elements were used for the comparison, too.

PROBLEM FORMULATION Let us limit the considerations to single structural element (column or beam) made of single E crosssection or double E close-open cross-section. Assume that the material is linear elastic, strains are small and the load is conservative. The geometrical non-linearity is allowed for. For simplicity we use the discrete model based on FSM or FEM approximation.

Linear buckling analysis Finite Strip Method formalism The buckling loads are calculated by solving the eigenvalue problem which can be written in the matrix form as: (K~

+ J~KmGn)Um = 0,

(1)

where K~ denotes the linear elastic stiffness matrix, KGmn is the initial stress (geometric) matrix (being a linear function of a base stress state S~j), A, is the load multiplier, Um is an eigenvector which describes a shape of buckling mode and m and n denote the consecutive number of harmonics in displacement shape functions [1, 2]. The application of the FSM to buckling problem in the form of (1) for simply supported columns provides uncoupling of (1) and as a result one obtains a small number (m) of eigenvalue problems to be solved. It is a great advantage of the FSM over the general FEM formulation. The critical buckling loads are computed as ),.i'P, where P is the vector of reference loads (the base state). Usually the lowest value of a load multiplier (~.min)and its associated buckling mode are of designer's interest.

Finite Element Method formalism The eigenvalue problem for linear buckling can be written in similar form as (1). However, this set of equation does not uncouple in any way and for fine meshes it usually requires large computing efforts. Advantageous is that we can easily introduce various mixed boundary conditions, intermediate constraints (supports), stiffening elements (diaphragms), etc. We can also use different ways of improving the accuracy in regions of stress or strain concentrations by applying p and/or h type adaptive methods.

Nonlinear analysis Using a standard finite element approach the system of nonlinear equilibrium equations can be written symbolically in the form of:

F(U) = 0,

(2)

where F is the global generalized force vector and U denotes for displacement vector. Applying the Newton method for solving (1) we arrive with the following matrix equation:

595 (K ~ + K G + K o ) U =P,

(3)

KU = P,

(4)

or:

where K ~ is the small-displacement stiffness, K c is the initial stress and K U is the load stiffness matrices respectively. The Riks method is used to solve (3), for both stable and unstable postbuckling behaviour. The postbuckling analysis is carried out by introducing a geometric imperfection pattern in the "perfect" geometry so the response in the buckling mode before critical load is reached. Imperfections are introduced by perturbations in the geometry as a linear superposition of buckling eigenmodes, from the displacements of an eigenvalue problem analysis (1). The lowest buckling modes are assumed to provide the most critical imperfections, so these are scaled and added to the perfect geometry to create the perturbed mesh. The imperfection thus has the form: M

~ ; = X w~;,

(5)

i=1

where ~bi is the i th buckling mode and wi is the associated scale factor, M is the total number of eigenmodes extracted in the buckling analysis.

Sensitivity analysis Linear bucklingproblem For the sensitivity analysis the preferable form of the buckling equation follows from the virtual work theorem: OU T (K ~ + 2 K G (S))U

= 0,

AOTI,

(6)

Setting b'U = U we obtain: U T (K ~ + ~,K G (S))U

= 0.

(7)

Variation of (7) with respect to design parameter s leads to: /

2O"UT (K ~ + 2 K G (S))U +

uTiOK~ Os 8s + 8 2 K G (S)U

+ A r K G (S) ] U = 0.

s

(8)

/

Hence OK ~

- Ur ~ S s U -

Os

/~Ur r K a (g)U

uTKC(~)U Using the normalization of U, namely u T K G u j ~, = _U T

OK ~ 8s

or:

(9)

= ~/j, equation (9) takes the final form:

8sU - ~u r6K c (g)u,

(10)

596 aK ~ 82 = - U r ~ S s U as

aK G 8sU-~,ur

- 2U r as

aK c ^ ~u. aS

(11)

where S denotes the matrix of stresses. Nonlinear static problem Rewrite the eq. (4) in the form: K(s, U ) U = P ,

(12)

where s denotes the vector of design parameters. Assume that we have solved the eq. (12) at each incremental step. Let the response of the structure is expressed by a functional G being a differentiable function of design variable vector s and nodal displacement vector U: G = G(s,U).

(13)

The structural sensitivity operator with respect to design parameters is defined as the derivative of (13) dG aG aG dU . . . . I-~ ~ . (14) ds c3s aU ds The derivatives dU/ds are implicit functions and have to be computed from (12). Differentiation of (12) leads to" IOK OK d U ] dU ~ + ~ - U + K(s,U) =0 c3s aU ds ~ '

(15)

which can be rewritten briefly as: KdU-p, ds

(16)

where: ~

K = K(s, U) +

aK

" aK and P = 9U . aU ~s

(17)

The matrix K is the tangent stiffness matrix, which is computed in element by element way using the semi-analytical method. The solution of (14) takes now the form: dG aG c3G ~ - 1 " ~ ....... + K P. (18) ds as aU We compute the final sensitivity operator dG/ds by summation the sensitivity at each iteration step [7]. Note that for the geometrically nonlinear statical problem the sensitivity operator (14) can be computed using the Adjoint Variable Method, where we need only the information referring to the final state of the primary (real) structure and tangent stiffness matrix of the adjoint structure. However, we are using the Direct Differentiation Method and the incremental approach because we are going to allow for the plasticity in the future. NUMERICAL EXAMPLES Consider a simply supported column under uniform static and conservative load. The sensitivity analysis in stability problem is carried out for open and close-open cross-section (Fig.l) with respect to variation of thickness and width of the section assembly. The stability analysis of local and global buckling (Fig. 2), employing FEM and FSM precedes the sensitivity analysis.

597

16 i 17 15_[_

I

16 Ils

17

151

14

its

14 1

4

lli~t

5~ 3 4

3

6

2

7 I

, ~

2

11

19

5

8 II

lo

12I

Figure 1" Mesh of)-', and 2x~ cross-section

O'cr. 10 3

I

- -0--

[FSM] [Vlasov el]

9 [Shell el]

r 0

tb 50

100

150

200

Figure 2: Buckling stresses Ocr plot for open cross-section Z profile Note that for a range of l/b < 40, where I denotes the length and b the width of flange of the column, the interaction of local and global buckling appears. The results for FSM and FEM are in very good agreement with each other. For l/b=25 the sensitivity derivatives computed with the use of formula (11) are presented in graphical way in Fig.3 and Fig.4.

Figure 3: Sensitivity derivatives of the lowest buckling load for ~ profile

598

Figure 4: Sensitivity derivatives of the lowest buckling load for 2x~ profile For the column, for which the global buckling load is close to local o n e (Pglobal < Plocal) the nonlinear analysis was performed to study sensitivity to local type imperfections. Postbuckling analysis was performed using the Riks method by introducing a geometric imperfection in the "perfect" geometry as a linear superposition of buckling eigenmodes. The buckling mode which was scaled and added to perfect geometry to create perturbed mesh is shown in Fig.5.

Figure 5: First buckling mode The unstable postbuckling paths for a node corresponding to a central buckle (Fig.5) with the scaling factors w i = a < b < c are shown in Fig.6. It is seen that the structure is sensitive to local imperfections. The nonlinear response depends strongly on the mode of imperfections as well as on theirs magnitude. The results of nonlinear sensitivity analysis of considered structures will be presented during the conference.

599 .

.

.

.

.

.

.

.

.

.

.

.

.

V

._....____..__.._

i+

.....i

I

. . . . . . . . . . .

I

I

~:~

Figure 6: Load-displacement paths for the unstable response

CONCLUDING REMARKS In the paper the sensitivity analysis with respect to cross-sectional parameters and to initial imperfections of columns made of single Y. cross-section and double ~ close-open cross-section was presented. Numerical examples were performed using Finite Strip Method and Finite Element Method, employing in both shell elements. The applied methods for analysis as well as for design sensitivity analysis have proved to be very efficient. The application of FSM to design sensitivity analysis allows in a simply way to introduce analytical derivatives of stiffness matrices. It gives possibility to answer very quickly what changes are required to obtain higher buckling loads. The presented examples demonstrate that structures under consideration are sensitive to imperfections and it can result in unstable behaviour when two buckling modes are close to each other. It is believed that presented analysis may contribute in improving design of structures considered.

References [1] M.S. Cheung, Y.K. Cheung (1972). Static and dynamic behaviour of rectangular plates using higher order finite strips, Build. Sci., 7, 151-158. [2] Y.K. Cheung (1976). Finite strip method in structural analysis, Pergamon Press, Oxford. [3] A. Garstecki, W. K~kol, K. Rzeszut (2000). Analysis of thin-walled bars with open and closedopen cross-section, Engineering Transactions, 4(in print). [4] J.T. Gierlinski, T.R. Graves-Smith (1984). The geometric non-linear analysis of thin-walled structures by finite strips, Thin-Walled Structures, 2, 27-35. [5] Hibbitt, Karlsson & Sorensen (2000). ABAQUS/Standard Manuals. [6] W. K ~ o l (1990). Stability analysis of stiffened plates by finite strips, Thin-Walled Structures, 10, 277-297.

600 [7] Kleiber, M., H. Antfinez, T. D. Hien, and P. Kowalczyk, Parameter Sensitivity in Nonlinear Mechanics, John Wiley & Sons, 1997. [8] S. Sridharan (1983). Doubly symmetric interactive buckling of plate structures, lnt. J. Solids & Structures, 19, 7, 625-641. [9] S. Sridharan, Ali H. Ashraf (1986). An improved interactive buckling analysis of thin-walled columns having doubly symmetric sections, lnt. J. Solids & Structures, 22, 429-434, [10]G. Thierauf (1990). Thin-walled structures and related optimization problems, Thin-Walled Structures, 9, 241-256. Acknowledgment Financial support by Poznafi University of Technology grant- DS-11-0 is kindly acknowledged.

Section X PLATE STRUCTURES

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Third International Conference on Thin-Walled Structures J. Zarag, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

603

BUCKLING LOADS OF VARIABLE THICKNESS PLATES Alexander Alexandrov and Moshe Eisenberger Faculty of Civil Engineering, Technion- Israel Institute of Technology Haifa 32000, Israel

ABSTRACT This work gives exact solutions for the buckling loads of a rectangular thin plate tapered in thickness in the directions parallel to the two sides, and compressed in these directions, with various combinations of boundary conditions. The calculation of critical values of the inplane forces applied on the edges of a plate at the initiation of buckling was done using the extended Kantorovich method. In the solution, an exact method for the stability analysis of compressed members with variable flexural rigidity is used (Eisenberger 1989,1991). The buckling load is found as the inplane load that makes the determinant of the stiffness matrix equal zero. New exact results are given for many cases.

KEYWORDS Plates, variable thickness, tapered, buckling, exact stiffness matrix, extended Kantorovich method. BASIC FORMULATION The functional of the total potential energy of a rectangular thin plate of dimension Lx x Ly in elastic stability is

7c=-~

~x z +-2

+ ( Oy2 ) + 2 v D ~

ax ~

Nxx

+ Nyy

oy---T-+2D(1-

+ 2Nxy ~

"t, axOy )

dxay

dy ,

(1)

Eh3/120 - v2)is the bending rigidity, v is the Poisson's ratio, h(x, y) is the thickness of the NI, Ny are inplane forces in x- and y-directions, taken positive in tension, and Nxy is inplane shear

where D = plate,

force. In the classical Kantorovich method the solution for two-dimensional problem is taken in the form

w(x, y) = ~ X,(x) * Y,(y) . i=1

After substituting the assumed solution (2) in the functional (1) it take the form

IIID(O2X)= n" = -~ ~x 2 Y z + DX =(O=Y']= [ oy =) + 2 vD(a=X']yO=Y2(l_v)D(~)=(.O~.)=]dxdy X ox , ) Oy---W +

(2)

604

[

--21 II ux"

2+ 2U~y X OXj

2y2 + NvX 2 - -

~

~OY dy,

(3)

integrated allover the plate of size Lx x Ly. The minus sign before the second term of the functional is due to fact that the forces we apply are compressive. The flexural rigidity of the plate can be represented by multiplication of two functions: the first is a function of rigidity in x-direction and the second function of rigidity in y-direction as D = D, Dy, in which Ehx3n

Dr

12{1 v2 )

'

3

Dy=hy

(4)

where h~,hy are functions of the variation of the plate thickness. If we specify one of the functions a

priori, X for example, we can perform the integration over X, and obtain a problem of minimization of a function of one variable only. After integration the functional take the form lli[ (02Y~ 2 02Y I_~.) 2] S1DvY2 + S2D~ + 2S3DyY + S4Dv dy 2 " -~,ay2J -j -

rc = -

--

SsY 2 + S6

2

+ 2S7Y

dy

(5)

where the coefficients S~ to S 7are L,~

Lx

Lr

Lx

$2 = oi ,, . ~~ ' $3 = 0i~,, :~ ~dXox2 ' $7= oi ~ . x ~dXox ~, = oI *,., ~&---gx ~Ox---Tdx' ~ ~ ,: ~:~ Lx

Lr

Lx

x dx , S 5 = IN= "-&xOX dx , S 6 : I N,XXdx S 4 = 2 f (.ID~,I- v ) OX-&--&OX 0

0

(6)

0

The function that make the functional stationary (bring the energy of the plate to its minimum value), must satisfy the differential equation that results from the first variation of the functional. Taking the variation of Eqn. 5, integrating by parts and equating each term to zero, we obtain /-).,,]-~+ 2S2

-~--3-+ 32 ~'2

+S3DyqS3Dy-S4Dy3t-S6

[3D~aD~

02D,] 2S3---~--$4---~+$7-$7]3Y ---~+ES, Dy +S 30Y 2 -S,

"~+

Y=O

(7)

The remaining terms form the boundary conditions that the function Yhas to satisfy. For y=Ly:

~ -

s2 ~

7

g

- s, - ~ -

s7 ~ = o,

02y S2Dy --~ + S3DyY = 0

(8) (9)

For y=O: 9

E

S2

--~

40v s30vly E .

32y

]

.---~'at" S3 - ~ ' - -- S 7 Y : O ,

- S2Oy - - - ~ - S 3 D y Y = 0

(10) (11)

We assume that the shape functions for the strip element are polynomials and we have to find the appropriate coefficients. We take the flexural rigidity as the following polynomial variation along the strip

605

I

k

i=o

i=0

D~=ED~,x' , Dy=ED~,Yi

(12)

We introduce two new local variables ~' = x/L~ and q = y/Ly. Now Eqn. 7 can be rewritten as

4 + I ot4[T2De(~7)]~ + otnI Rz2 aDe(u)q ~a3Y(U) ot4T2c32De -~5_(r]) + a2T3Oq (r]) oq

J

aq

+~r.D,(~)-a~r.D,(~)+a'r6]

aq 3

+ a ~2T3 aD~(rl)oq

~

OY(q) [T~D,7(q)+a2T302D,7(q)

+~176

Or/ +

k

k

i

Or/5~-T'

a2T4~or/

]Y(r/)

0 = '

(13)

Lx

where D~(q)= ~Dy,I21~rl'= ~"D~,q , at=--.r The coefficients T~ to T7 in Eqn. 13 are i=0 i=0 a.,y T,-

1

1~2X(4) 0 2 / ( 4 )

o

O~ 2

,

er

O~ ~

=

o

o

2~!4) d4

r, = INr162 aX(r162162 o o4 04

o4

T6 = IN,,TX(r162162

r 7 = INr162

0 I

~

o

)OX(r162

T4 = 2 I D r 1 6 2

'

(14)

de,

0

I

2 ZD~is162 = ZDr and Nr162 N~L2~, N,m = NyyL~ and Nr = N~yL2x . The solution i=o i=o Y(q) is assumed as the infinite power series

where Dr

Y(r/) = ~7~Yf/ (15) i=0 Calculating all the derivatives and substituting them back into Eqn. 13 we obtain the following equation, for the term Y~§ in Eqn. 15 as a function of the first four terms in the series

' Y~+4= T2(i + 1Xi + 2Xi 1+ 3Xi + 4)D, oa4 I -a4T2~"~(i-k,=,

+ 1Xi-k + 2Xi-k + 3)

(i-k + 4)D,n,Y~_k+4 - cr42T2 ~ (k + 1Xi-k + 1Xi-k + 2Xi-k + 3)D,,+,Y~_,+3 -

o~4T2~

k=0 (k + 1Xk + 2Xi - k + 1Xi - k + 2)D,k+2Y~_,+2 -

,--o Or2(r3 + T3 - r 4 ) ~ ( i - k + lXi-k + 2)D,7,Yi_k+2-0t2T6(i + lXi + 2)Yi+2i k=0 cr2 (2T3 -T4)~"(k + 1Xi-k + 1)D,Tk+,Y,_k+,-a(T7 -TTXi + 1)Ym k=0

T~ D,TkY~_, +ot2T3~'(k + IXk + 2)D,7,+2Y~_k+ T,Y~

(16)

k--O k=O The first four terms should be found using the boundary conditions. The terms in the stiffness matrix are defined as the holding actions at both ends of the strip, due to unit translation or rotation, at each of the four degrees of freedom, one at a time. Thus, there are four solutions for Y(q). Then the stiffness matrix can be formed using Eqs. 8-11 L~ D,~' (0)r,.~ S 0 , i ) = 6T:D,~ (0) ~_ Y~3 + 2T: -73y

'

Ly

606

[T4D,(0)- T6 -- T3Dq(0)]

Y/,! + 7'3 ~

D'o(0)-

,,o

(17)

D (0) Yi.o

S(2,i)=-2T2D,~(O)

Y~.2- r3 -~x

(18)

v

S(3,i)=-T:D,~(1)

~]k(k-lXk-2)Y~j , - T 2 --5-D,7 .

k=3

+[T4D,(1)-T6- T~D,(1)] ~kE~,__.-

k=2

T3 L-~LLLL~D;(1)-

Y~,,

(19)

~0o n,l(1) ~. ] Z k ( k - 1)Y~,, + r 3 Y~ (20) y ,=2 Lx k--0 The buckling loads for variable thickness strip of plate can be found as the inplane forces, that cause the determinant of the corresponding stiffness matrix to become zero. The accuracy of the function Y, which was computed in the analytical method, is higher than that in the "approximate" x-direction (the function X that was specified a priory). To improve the obtained solution, Kerr (1969) proposed an extended Kantorovich method, in which, after a Kantorovich solution in one direction is obtained, the "approximate" and "analytical" directions are interchanged, using the obtained function Y as the specified a priory function, and compute function X, using the above procedure. The iterations are repeated until the result converges up to the desired accuracy. This method allows for the assumed initial solution X not to satisfy the boundary conditions, and then it will take only one more iteration to converge to the exact solution.

SO, i)= T:D,7(1)

NUMERICAL EXAMPLES

A wide range of problems can be computed using the proposed method. Some of the numerical results summarized below. Unlike the plate of uniform thickness, there are no discontinuous mode changes. The plate with very small taper undergoes smooth, although very rapid, mode changes as the side ratio increases through that corresponding to a discontinuous mode change of the uniform plate.

Example 1: Rectangular plate with linearly variable rigidity loaded in the x-direction The values of the dimensionless buckling load P = ex,c,,b2/z2D o for the SSSS, CSSS (built in at thick end, and simply supported at thin end), CFSS (clamped at thick edge), SFSS (free at thin edge) plates with linearly variable thickness in x-direction hx(~) = h0(1 + fl~:), hy = 1.0 are given in Tables 1 to 2 for values of fl = 0.125, 0.25, 0.5, 0.75, 1.0, respectively. Throughout the computation the value of Poisson's ratio was taken as 0.25. Diagrams of buckling load variarion and modes are shown in Figs. 1-4.

Example 2: Rectangular plate with linear variable rigidity in y-direction loaded in the x-direction In this example a SSSS plate with linearly variable thickness in y-direction hy (r/): h o(1 + fir/), h x = 1.0, was considered. The results are summarized in Table 3 and Fig. 5.

Example 3: Rectangular plate with linear variable rigidity loaded in two directions The values of the dimensionless buckling loads Px = PX,crb2 / z2Do, Pr = YPx for the square SSSS plate ( a = 1.0) with linearly variable thickness in x-direction are given in Table 4. In Fig. 6 buckling modes are given for two particular cases.

607

TABLE 1 VALUES OF P FOR SSSS AND CSSS PLATES WITH LINEARLY VARIABLE THICKNESS Ssss

]

csss

# 0.5

-0.1250.25 7.4678 8.7769

0.5 11.6656

0.75 14.9102

1.0 18.5055

0.125 12.2619

0.25 14.2500

0.5 18.5552

0.75 23.2870

1.0 28.4332

(1,1)

(1,1)

0.7

5.4101

6.3476

8.3918

10.6538

13.1265

7.6897

8.8630

11.3755

14.1050

17.0466

0.8

5.0143 (1,1) 4.7582 (l,1) 4.8748 (1,1) 4.9735 (2,1) 4.6295 (2,1) 4.5022 (3,1) 4.4133 (4,1) 4.3538 (5,1)

5.8721 (1,1) 5.5306 (1,1) 5.5634 (1,1) 5.4991 (2,1) 5.1138 (2,1) 4.8367 (3,1) 4.6782 (4,1) 4.5774 (5,1)

7.7184 (l,1) 7.1183 (1,1) 6.8960 (1,1) 6.5268 (2,1) 5.9560 (2,1) 5.4163 (3,1) 5.1342 (3,1) 4.9584 (4,1)

9.7311 (1,1) 8.7749 (l,l) 8.2480 (1,1) 7.5735 (1,1) 6.7578 (2,1) 5.9592 (3,1) 5.5538 (3,1) 5.3046 (4,1)

l 1.9041 (1,1) 10.5124 (l,l) 9.6503 (1,1) 8.6539 (1,1) 7.5599 (2,1) 6.4917 (2,1) 5.9592 (3,1.) 5.6357 (4,1)

6.6527 (1,1) 5.5828 (l,l) 5.1741 (1,1) 5.0037 ....(2,1) 4.7267 (2,1) 4.5133 (3,1) 4.4145 (4,1) 4.3539 (5,1)

7.6320 (1,1) 6.3388 (l,l) 5.8044 (1,1) 5.5082 (2,1) 5.1629 (2,1) 4.8389 (3,1) 4.6783 (4,1) 4.5774 (4,1)

9.7141 (1,1) 7.9178 (l,1) 7.0966 (1,1) 6.5272 (2,1) 5.9750 (2,1) 5.4165 (3,1) 5.1342 (3,1) 4.9584 (4,1)

I 1.9591 (1,1) 9.5912 (l,l) 8.4439 (1,1) 7.5741 (1,1) 6.7672 (2,1) 5.9592 (3,1) 5.5538 (3,1) 5.3046 (4,1)

14.3650 (1,1) 11.3622 (1,1) 9.8534 (1.,!) 8.6569 (1,1) 7.5651 (2,1) 6.4917 (2,1) 5.9592 (3,1) 5.6357 (4,1)

(1,1)

1.0 1.2 1.5

2.0 3.0 4.0 5.0

(1,1)

1

20

(1,1)

I

I

(1,1)

I

I

,,1I'..... t\ ' ' ' % _ _ ~ _ j _ _ . _ _ t _ _ _ , _ _ _ J _ _ • h '\\ ~ ' ' -

-%

121-~1~2--'1..---51~

-

J

-

-

J- -

-

I

.-~ -4 --

--

4- --

--

[.- --

b

'r .... ~ .1~.~_,. . /

.

~

q - -

I

~

' '

--I--

J

--

-I

' '

(1,1)

(1,1)

I

I

_

_

.t. _

--

--

4.- --

' '

_

--

I,.- --

.

T - -

- -

.

.

--I / 'l

~

' '

--I--

--

7 - -

i

; - - ' - - ' ,

(1,1)

oI

-

~\

....

(1,1)

(1,1)

i

'

I

I

I

I

I

I

I

I

l

t

I

I

I

i

I

r

,-

r

I

-

-.-

-

7

I

I

-

I

I

- r

I

.I . . . . . . I. . . . . . . !. . . . . . .I .

' ~L - - , - -. ' ,, _~ ~~ , '

.

.

I

.

.

'

-

7

'

'

= 1.5 in E x a m p l e 1

1

I -

-

r

'

I

I

I

I

I

1

'_

I

'

'

'

.

___'

'

C S S S Plate

t~x/tox

-r -

'' .' . ' . . ' . ' r--,--~--r--~--r

a D i a g r a m for E x a m p l e 1

F i g u r e 2: B u c k l i n g m o d e for

(1,1)

I

,~\\\_, I

(1,1)

(1,1)

~\\\ "" .. .. .. .. .. .. .. . . . .

I

S S S S Plate F i g u r e 1" P -

(1,1)

-,,~.,~~1 '' '' '' '1\ W~.,';ql-r--'---1

-4 -,]

9- - ~ - - , - - ~ -1 _,_ _ 9

(1,1)

(1,1)

,lta.~

. . . . .. . . . .

F - - ' - - - ' - -

(1,1)

I

L

' '

(1,1)

.. .. .. .. .. .. . . . . . .

, iL _ _ , _._ ~ ' ~. _ ~.- - ~ -.~ _ _. ~ . , - - ' - -

(1,1)

'

608

TABLE 2 VALUES OF P FOR CFSS AND SFSS PLATES WITH LINEARLY VARIABLE THICKNESS CFSS

SFSS

P , 0.125 3.1774

0.5 ,

0.7 , 0.8 , 1.0 ,

2.0 3.0 ,

0.5 4.6506

0.75 5.8303

1.0 7.1707

0.125 2.4751

(l,l)

(1,1)

Q,l)

(1,1)

(1,1)

2.8199

3.1047

3.7352

4.4475

5.2423

(1,1)

(1,1)

(1,1)

(!,1)

2.7758

3.0176

3.5473

4.1394

,

(1,1), 4.7948

0.25 2.8083

0.5 3.5760

0.75 4.4858

1.0 5.5447

(l,l)

(1,1)

(1,I)

(1,1)

(l,1)

2.7514

3.0163

3.6055

4.2751

5.0268

(1,1)

(1,1)

(1,1)

(!,1)

(1,1)

2.7722

3.0085

3.5239

4.0997

4.7377

(1,1)

(1,1)

(1,1)

(1,1)

(1,1)

(1,1)

(I,1)

(1,1)

(1,1)

(1,1)

2.7226

2.9153

3.3252

3.7717

4.2569

2.7006

2.9022

3.3211

3.7710

4.2568

(2,1)

(2,1)

(2,1)

(2,1)

2.6629

2.8299

3.1741

3.5383

(2,1)

(2,1)

(2,1)

(2,1)

2.5936 (2,1) 2.5459 (2,1) 2.5099

2.7335 (2,1) 2.6520 (2,1) 2.5812

3.0142 (2,1) 2.8633 (2,1) 2.7221

3.3029 (2,1) 3.0769 (2,1) 2.8628

3.6038 (2,1), 3.2957 (2,1), 3.0047

2.5775 (2,1) 2.5455 (2,1) 2.5099

(2,1),

(3,1)

1 2 , (2,1) 15

0.25 3.6294

(3,1)

2.4920 4.0 (4,1) t2.4812 5.0 (5,1)

(3,1)

(3,1)

(3,1)

2.5457 (4,1) 2.5243 (5,1)

2.6518 (3,1) 2.6095 (4,1)

2.7572 (3,1) 2.6939 (4,1)

(1,1), 3.9264

(2,1)

(2,1)

(2,1)

(2,1)

(1,1)

2.6285

2.8046

3.1594

3.5295

3.9213

i (2,1)

2.86282.4920 (3,1) , (4,1) 2.7783 2.4812 (4,1) , (4,1)

,(2,1)

(2,1)

(2,1)

(2,1)

2.7205 (2,1) 2.6515 .(2,1) 2.5812

3.0050 (2,1) 2.8626 (2,1) 2.7221

3.2961 (2,1) 3.0762 (2,1) 2.8628

3.5986 (2,1) 3.2951 (2,1)... 3.0047

(3,1)

(3,1)

(3,1)

(2,1),

2.5457 (4,1) 2.5243 (4,1)

2.6518 (3,1) 2.6095 (4,1)

2.7572 (3,1) 2.6940 (4,I)

2.8628 (3,1) 2.7783 (3,1)

Figure 3: P - a Diagram for Example 1

Figure 4: Buckling mode for

tlx/tox

= 1.5 in E x a m p l e 1

609 TABLE 3 VALUES OF P SSSSPLATE WITHLINEARLYVARIABLETHICKNESSIN Y-DIRECTION ,-,.,

SSSS

0.5 0.7 0.8 1.0 1.2 115 2.0 3.0 4.0 5.0 .....

..

.

0.125 7.4645 (1,1) 5.4199 (1,1) 5.0291 (1,1) 4.7887 (1,1) 4.9502 (1,1) " 5.1931 (2,1) 4.7887 (2,1) 4.7887 (3,1) 4.7887 (4,1) 4.7887 (5,1) .

0.25 8.7633 ( 1 , 1 ) 6.3891 (1,1) 5.9345 (1,1) 5.6565 (1,1) 5.8490 (1,1) 6.1252 (2,1) 5.6565 (2,1) 5.6565 (3,1) 5.6565 (4,1) 5.656 (5,1) .

. . . . . . . . . . . . . .

...

.

.

Figure 5: P - a

.

.

.

.

.

.

.

.

....

.

P 0.5 11.6112 ( 1 , 1 ) 8.5741 ( 1 , 1 ) 7.9891 ( 1 , 1 ) 7.6394 ( 1 , 1 ) 7.9067 ( 1 , 1 ) 8.2344 ( 2 , 1 ) 7.6394 ( 2 , 1 ) 7.6394 ( 3 , 1 ) 7.6394 ( 4 , 1 ) 7.639 (5,1) .

.

.

.

0.75 14.7942 ( 1 , 1 ) 11.0979 ( 1 , 1 ) 10.3820 ( 1 , 1 ) 9.9678 (1,1) 10.3289 ( 1 , 1 ) 10.6817 ( 2 , 1 ) 9.9678 (2,1) 9.9678 (3,1) 9.9678 (4,1) 9.9678 (5,1) .

.

.

.

.

.

1.0 18.3175 (1,1) 13.9730 (1,1) 13.1280 (1,1) 12.6602 (1,1) 13.1362 (1,1) 13.4811 (2,1) 12.6602 (2,1) 12.6602 (3,1) 12.6602 (4,1) 12.6602 (5,1) .

Diagram for SSSS Plate and first buckling mode for t~y/toy = 1.5

SUMMARY

The extended Kantorovich Method (Kerr, 1969) in conjunction with the Exact Element Method (Eisenberger, 1989,1991) were used to obtain the buckling loads and modes of variable thickness plates. Many new values are given for comparisson by other numerical methods. REFERENCES Eisenberger M. (1989). Exact static and dynamic stiffness matrices for variable cross section members. In 30 th Structures, Dynamics, and Materials Conf., Mobile, Alabama. AIAA, U.S.A., 852-858. Eisenberger M. (1991). Buckling loads for variable cross-section members with variable axial forces. Int. J. Solids Structures 27:2, 135-143. Kerr, Arnold D (1969). An extended Kantorovich method for the solution of eigenvalue problems, lnt. J. Solids Structures 5, 559-572.

610 TABLE 4 VALUES OF ex AND er FOR SSSS PLATE WITH LINEARLYVARIABLETHICKNESS IN X-DIRECTION SSSS (linear variation in thickness)

-20 2O 10 7 5 3

0.7 0.5 0.2 0.0 -0.2 -0.5 -1.0

0.125 0.25 -0.2520,5.0415 -0.2978,5.9572 (1,1) . (1,1) 0.2280,4.5601 0.2692,5.3845 (1,1) (1,1) 0.4352,4.3522 0.5137,5.1374 (1,1) (1,1) 0.5984,4.1886 0.7061,4.9428 i (1,1) (1,1) 0.7977,3.9886 0.9410,4.7051 (1,1) (1,1) 1.1962,3.5886 i 1.4100,4.2301 (1,1) (1,1) 1.5944,3.1888 1.8778,3.7556 (l,l) (1,1) 2.3897,2.3897 2.8087,2.8087 (1,1) (1,1) 2.8100,1.9670 3.2983,2.3088 (l,l) (1,1) 3.1830,1.5915 3.7311,1.8656 (1,1) (1,1) 3.9732,0.7946 4.6408,0.9282 (1,1) (1,1) 4.7582,0.0 5.5306,0.0 (1,1) (1,1) 5.9161, -1.1832 6.7966, -1.3593 (1,1) (l,l) 8.2546, -4.I273 9.1866, -4.5933 (2,1) (2,1) , 9.8476-9.8476 ~I 11.2651,-11.2651 (2,1) l (2,1)

1.0 0.75 -0.5263,10.5263 -0.6697,13.393 l (1,1) (1,1) ., 0.5999,11.9999 0.4732,9.4638 (I,I) (1,1) 1.1402, I 1.402l 0.9007,9.0069 (1,1) (1,1) 1.5619, 10.9332 1.2354,8.6480 (1,1) (1,1) 2.0725,10.3624 1.6421,8.2105 (1,1) (1,1) 3.0765,9.2295 2.4466,7.3397 (1,1) , (1,1) 4.0550,9.1099 3.2377,6.4754 (1,1) (l,l) (1,1) 5.9208,5.9208 3.7332,3.7332 4.7713,4.7713 (1,1) , (1,1) (1,1) , 6.8508,4.7956 4.3660,3.0562 5.5514,3.8859 (1,1) (1,1) (1,1) 7.6387,3.8194 I 4.9186,2.4593 6.2226,3.1113 (1,1) ..... (1,1) i (1,1) 9.1771,1.8354 6.0537,1.2107 7.5662,1.5132 (1,1) . (1,1) (1,1) , 10.5125,0.0 i 7.1183,0.0 8.7749,0.0 (1,1), (1,1) (1,1) 8.5259, -1.7052 i 10.2894, -2.0579 12.1241,-2.4248 (1,1) (l,l) (1,1) , 11.0093, -5.5047 12.8858, -6.4429 14.843o, -7.4215 (1,1) (1,1) (1,1) 13.8888,-13.8888 16.3688,-16.3688 18.8071, -18.8071 (2,1) (2,1) (2,1) 0.5 -0.4028,8.0550 (1,1) 0.3632,7.2639 (l,1) 0.6923,6.9229 (1,1) 0.9507,6.6549 (1,1) 1.2655,6.3276 .. (1,1) 1.8916,5.6748 (1:1) 2.5122,5.0244

Figure 6: Buckling mode for SSSS plate of Example 3 with

t~x/tox = 1.5

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michaiskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

611

BUCKLING AND POST-BUCKLING BEHAVIOR OF PLATES ON A TENSIONLESS ELASTIC FOUNDATION A. S. Holanda and P. B. Gon~alves Department of Civil Engineering Catholic University, PUC-Rio 22453-900, Rio de Janeiro, RJ, Brazil [email protected]

ABSTRACT This work presents a geometrically non-linear analysis of plates resting on a tensionless foundation with emphasis on the non-linear behavior and stability of the structural system. For this, a non-linear finite element formulation based on von Khrm~n non-linear plate theory, modified by Mindlin's hypothesis is developed. The foundation is described by a third degree polynomial function of the plate transversal displacement. To obtain the non-linear equilibrium paths, the Newton-Raphson algorithm is used together with path-following strategies. To solve the unilateral contact problem, a linear complementarity problem is derived and solved by Lernke's algorithm. Numerical results considering linear and non-linear foundation models show that the tensionless non-linear foundation has a remarkable influence on the buckling and post-buckling behavior of the plate.

KEYWORDS Plates, finite element method, unilateral contact, tensionless foundation, plate instability, post-buckling of plates, linear complementarity problem. INTRODUCTION The stability analysis of plates is an important topic in structural engineering and has been studied by several authors in the past (Chia, 1980). The influence of a linear elastic foundation on the stability and non-linear behavior of plates is also well studied in the literature (Naidu et al., 1990, Raju and Rao, 1988). The stability of plates on a non-linear elastic foundation has also received some attention in recent years. Shen (1995) presented an analytical study of the post-buckling behavior of orthotropic rectangular plates on non-linear elastic foundations with cubic non-linearities and, recently, the authors presented a formulation for geometrically non-linear analyses of orthotropic plates resting on nonlinear elastic foundations with quadratic and cubic non-linearities (Holanda and Gon~alves, 1999). However, in several engineering applications, the foundation can't provide tensile reaction and, under certain conditions, some portions of the plate may lift-off. In these circumstances, if the unilateral character of the foundation is not taken into account, the engineer may incur considerable error, as shown in this paper. Although the equilibrium analysis of plates on a tensionless foundation has been

612 the subject of some papers in recent years (Khathlan, 1994; Akbarov and Kocattirk, 1997; Hong et al., 1999), no study on the stability analysis of plates resting on a tensionless foundation has been found. In the present paper, a geometrically non-linear analysis is presented for slender plates resting on a non-linear tensionless elastic foundation with emphasis on the stability and imperfection sensitivity of the structural system. The plate theory is based on von K~xrn~-a non-linear plate theory modified by Mindlin's hypothesis and the non-linear foundation used in this paper is modeled by a third degree polynomial equation. In order to obtain the non-linear equilibrium path, the Newton-Raphson method is used together with path-following strategies. The non-linear model for plates resting on a non-linear foundation is implemented in the Femoop program, which is an analysis program, based on a finite element formulation. The Finite Element Method is used to discretize the plate and the foundation. To overcome the difficulties in solving the plate-foundation equilibrium equations together with the inequality constraints due to the fi'ictionless unilateral contact condition, a variational formulation equivalent to these equations is obtained from which a linear complementary problem (LCP) is derived and solved by Lemke's complementary pivoting algorithm (Silva et al., 2001; Holanda, 2000).

GEOMETRICALLY NON-LINEAR ANALYSIS OF PLATES In this paper von K~trm~n non-linear plate theory, modified by Mindlin's assumption, will be used to analyze the behavior of plates resting on an elastic tensionless foundation (Crisfield, 1991). Based on these assumptions, the displacement components at any point in the plate (~, ~, ~ ) may be expressed in terms of the corresponding middle-surface quantities by the relations:

(x,y,z) = u(x,y) + zOy (x,y) (1)

(x,y,z) = v(x,y) -zO,, (x,y) ~ ( x , y , z ) = w(x,y)

where z is the normal coordinate, u and v are the in-plane displacements of the middle surface in the x and y direction, respectively, w is the transversal displacement and 0,, and 0y are the rotations of the normal relative to the x and y directions. The components of the strain-displacement relations for the plate are given by:

& 2L&)

8 p = 6y

= O~t~

,(el = 2~,+)

(2)

-g+ ~y

~b

=Z

ky

kxy

& OOx

= z

& OOy

OOx

I~s

=

=

Y~

tT~

~- -0~

(3)

613 where the superscript p denotes the extensional and shearing components at points in the plate middle plane (membrane action), b is related to the curvature components (bending action) and s denotes the vertical shear strains. Considering a linear elastic material, the constitutive matrix is composed of three diagonal submatrices. For an orthotropic material, this matrix is written as

D=

Dp

0

0

0

Db

0

0

0

Ds

I 1

(4)

The plane stress component is given by

-

v

y

Ey

1 - Vxy Vyx

0

0

~ 1

(5)

(1- V~y vyx)Gxy

where Gxy =

Ey(1 + Vyx)+ Ex(1 + Vxy)

(6)

t is the plate thickness, E is the elastic modulus and v is the Poisson ratio. On the other hand, the bending component is written as D b = t2---~DP 12

(7)

and, finally, the shear component is given by the following expression

D,=kt

IG~ 0

0 I Gy, '

5 k=-6

(8)

where Gxz and Gy~ are analogous to (6).

ELASTIC FOUNDATION MODEL In this paper, a new non-linear elastic foundation model is employed. This model includes the wellknown linear Winkler model and the non-linear elastic foundation model with cubic non-linearity used by Shen (1995). The reaction of this foundation is given by:. P = K ~ w + K 2w 2 + K 3w 3

(9)

where K1 is the Winkler foundation stiffiaess and K2 and K3 are the non-linear elastic foundation parameters.

614 CONTACT PROBLEM A critical step in the analysis of contact problems is the selection of a numerical methodology to deal with unilateral contact constraints. In this work mathematical programming methods are used to solve the contact problem. This enables one to solve the contact problem by directly minimizing the potential energy containing explicitly moving boundaries and the associated inequality constraints and thus maintaining the original mathematical characteristics of the problem (Ascione and Grimaldi, 1984; Silva et al., 2001). In the present formulation the plate and foundation are treated as separate bodies in frictionless unilateral contact at the interface. Based on these assumptions, a linear complementarity problem (LCP) is formulated and solved by Lemke's pivoting algorithm (Silva et al. 2001; Holanda, 2000).

FE FORMULATION The present formulation for non-linear analysis of plates resting on a non-linear foundation was implemented in the Femoop program (Martha et al., 1996). This is a structural analysis program based on a FE formulation. It can be used to perform static elastic, materially and geometrically non-linear FE analyses. The program is written in C++ language and employs the concepts of objected-oriented programming (OOP) to build an easily extendable framework for computational mechanics (Holanda and Gon~alves, 1999; Holanda, 2000).

NUMERICAL RESULTS A rectangular simply supported isotropic plate resting on a linear elastic foundation subjected to an inplane compressive load q is considered. The plate has length b = 3.0, width a = 1.0 and thickness t = 0.002. The material properties are: E = 100000 and v = 0.25. In the analysis the following foundation parameter is used: k~ = K~ a 4 ~.4D

(10)

where k~ is the non-dimensional Winkler foundation stiffness and D is the flexural rigidity of the plate. The response of the plate for increasing values of the foundation stiffness parameter kl is obtained (kl =0, 1, 10 and 100). To model the structural system, 48 isoparametric finite elements (Q9) were used for the plate and 48 for the foundation, as shown in Figure 1.

I~>

:--

Figure 1 - FE mesh.

615 The results are shown in Figure 2, where the plate load parameter is plotted as a function of the central deflection. The plate without foundation (kl = 0) exhibits a stable symmetric post-buckling response, as expected (Chia, 1980). For a plate resting on a tensionless foundation, two different branches, emerging from two distinct bifurcation points are observed. When the plate buckles, the number and size of the contact area depend on the buckling direction, as illustrated in Figure 3, where the plate deflection associated with the two different branches of the post-buckling response is shown for a load level of 4.0 and kl = 100. When the central deflection is negative (Figure 3a), there is only one small contact region. When the central deflection is positive (Figure 3b), there are two contact regions. In both cases the displacements in the non-contact region are much larger than the displacements in the contact area. These differences in the deformation patterns explain the differences in the critical load and in the post-buckling effective stiffness. 7.00

i 6.00-~ i

!

kl = 100

kl=1

5.00

k l = 100 \ \

=

4.00

3.00 -kl = 0

2.00 -0.04

-0.02

0.00 central deflection

0.02

0.04

Figure 2 - Equilibrium paths. Load parameter versus central deflection. The difference of the two bifurcation loads increases as k~ increases, but in all cases a stable postbuckling response is observed. A detailed analysis of this problem considering perfect and imperfect plates, as well as linear and non-linear foundation models can be found in Holanda (2000).

616 0.012

.

.

.

.

O.OOe

O.(XN

0.000

I

'

o.~

I

'

~

I o,~

'

i o.~

'

I

r

J

o.~

t.~

o.~

o.~

'

) o.~

'

1 o.~

'

t

'

o.~

1.~

(a) load factor = 4.00. (w-) (b) load factor = 4.00 (w+). Figure 3 - Plate deformation (kl = 100). The influence of the non-linearity of the elastic foundation on the post-buckling response is illustrated in Figure 4, where the load factor is a function of the central deflection for a foundation with ky=l 0 and different values of k3 (k2=0). As one can observe, the cubic non-linearity of the foundation has a remarkable influence on the post-buckling behavior of the plate. While a hardening foundation increases the load carrying capacity of the plate in the post-buckling region, a softening foundation can not only decrease the plate effective stiffness but also change the post-buckling response from stable to unstable thus increasing the plate imperfection sensitivity. 5.00 =5 k3=

4.00

k3=5

I

=u

3.00

2.00

k3=-3 -

-

k3=-5

1.00 -0.04

! -0.02

0.00

0.02

0.04

W

Figure 4 - Post-buckling behavior of a plate on a non-linear tensionless foundation.

617 CONCLUSIONS A non-linear FE formulation was used together with incremental-iterative strategies to study the buckling and post-buckling behavior of plates resting on a non-linear tensionless elastic foundation. To solve the contact problem, a linear complementarity problem is derived and solved by Lemke's pivoting algorithm. The results show that the critical load and the characteristics of the post-buckling response are influenced by the foundation parameters and by the buckling direction. This is a nonclassical stability problem where the buckling and post-buckling characteristics depend on the number and size of the contact and non-contact regions, which are not known a priori and have to be obtained during the solution of the non-linear equilibrium equations. For a linear Winkler type foundation the post-buckling response is always stable but the effective stiffness and the buckling load depends on the buckling direction and foundation stiffness. On the other hand, when non-linear term are considered in the foundation model the post-buckling response may be stable or unstable depending on the relative value of the non-linear coefficients. For some softening foundations, unstable post-buckling responses with a high degree of imperfection sensitivity were obtained. Finally the results show that the response of the plate with unilateral characteristics is rather different from the behavior of the plate without foundation and also of the plate resting on a foundation that reacts both under tension and compression. So, in practical situations where the foundation cannot provide tensile reaction the engineer may incur in considerable error if the unilateral character of the foundation is not taken into account, specially when the foundation exhibit a non-linear behavior.

ACKNOWLEDGMENTS The authors acknowledge the financial support provided by the Brazilian Research Agency CNPq.

REFERENCES Akbarov, SD and KocatOrk, T (1997) On the bending problem of anisotropic (orthotropic) plates resting on elastic foundations that react in compression only. International Journal of Solids and Structures 34(28), 3673-3689 Ascione, L. and Grimaldi, A. (1984) Unilateral contact between a plate and an elastic foundation. Meccanica 19, 223-233 Chia C. (1980) Nonlinear Analysis of Plates. McGraw-Hill, New York, USA. Crisfield, MA. (1991) Non-linear Finite Element Analysis of Solids and Structures. Vol. 1. John Wiley and Sons, New York, USA Holanda AS and Gongalves PB. (1999) Buckling and post-buckling behavior of orthotropic plates on non-linear elastic foundations. In: Structural Engineering and Mechanic. Vol. 1. Eds. C. Choi & W. C. Schnobrich, Techno-Press. Seoul, Korea, pp. 457-462. Holanda AS. (2000) Analysis of the equilibrium and stability of plates with contact constraints. Ph. D. Thesis, Civil Engineering Department, Catholic University, PUC-Rio, Rio de Janeiro (in Portuguese). Hong T, Teng JG and Luo YF. (1999) Axisymmetric shells and plates on tensionless elastic foundations. International Journal of Solids and Structures 36:5277-5300

618 Khathlan AA. (1994) Large-deformation analysis of plates on unilateral elastic foundation.

Journal of Engineering Mechanics, ASCE; 120(8): 1820-1827. Martha LF, Menezes IFM, Lages EN, Parente Jr. E and Pitangueira, RL. (1996) An OOP Class Organization for Materially Nonlinear FE Analysis. In: Proceedings of the CILAMCE 96. Veneza, 229-232. Naidu NR, Raju KK and Rao GV. (1990) Post-buckling of a square plate resting on an elastic foundation under biaxial compression. Computers and Structures; 37(3):343-345 Raju KK and Rao GV. (1988) Thermal post-buckling of a square plate resting on an elastic foundation by finite element method. Computers and Structures; 28(2): 195-199 Shen H. (1995) Postbuckling analysis of orthotropic rectangular plates on nonlinear elastic foundations. Engineering Structures; 17(6): 407-412. Silva ARD, Silveira RAM and Gongalves PB. (2001) Numerical methods for analysis of plates on tensionless elastic foundations. International Journal of Solids and Structures, 38(10-13): 20832100

619

DEGREE OF WALL JOINT WORK TOGETHER WITH STIFFENING RIB IN STEEL BUNKER M. I. Kazakevitch and D.O. Bannikov Department of Bridges, Dnepropetrovsk State Technical University of Railway Transport, Dnepropetrovsk, Ukraine

ABSTRACT During the design of different kinds of thin-walled structures, the dilemma 'how many thicknesses of wall is it necessary to include in the cross section of the stiffening rib' often arises, and for many common constructions it often equals 30. In the process of finite element simulation of ribbed walls for steel bunkers some interesting conclusions were made. The most significant were that the real degree of wall joint work is higher then accepted, it isn't a constant value and it depends on the load and stiffness of the wall and the rib. Numerical computing was done by means of SCAD software and a good correlation with experimental data is reported.

KEYWORDS Ribbed wall, degree of joint work, FEM, bunker. INTRODUCTION Today in structural engineering it is very often required to design different thin-walled structures. In practice the approximate calculation method based on the division of a structure into simple single elements is widespread. One such element for shell and plate structures is a stiffening rib within the wall and usually these schemes are designed and calculated as beams fully or partly fixed at their ends. The cross section of them represents the cross section of the rib plus some additional part of the joined wall. In other structural areas such as the aircraft, car, railway and shipbuilding industries, the value of the joint wall part is different and changes over a wide range- [Ref. Filin & Sokolova (1957) or Strigunov (1984)]. Figure 1 shows the cross section of a rib with the part of a wall (k) which often equals 30 for structural constructions- [Ref., for example, Rucovodstvo (1983)].

620 t

k-t

1

Figure 1" Cross Section of Stiffening Rib with Part of a Wall. FINITE ELEMENT INVESTIGATION For the purpose of this investigation, a steel prismatic pyramid shaped bunker was chosen which is a very complicated construction. Design of bins covers the analysis of thin shells and stiffened plate structures with uncertain load distributions, including a range of different important factors which are not well understood to date. The method of designing such bunkers is exactly the same as discussed above - [Ref. Rucovodstvo (1983) and ESDEP (1994)]. Due to the complicated nature of this bunker structure, it was decided to perform a finite element analysis (FEA) to simulate a wall which represents a part of a bin flat wall with stiffening ribs. The FEA model is shown in Figure 2.

Figure 2: FEA Model of Investigated Ribbed Wall of Bunker. Only half of the wall was modelled as it is symmetrical, and all of the ends were simply supported with the exception of the symmetrical axis where the boundary conditions of deformation symmetry were established. The load represented in the model was a normal pressure uniformly distributed over all of the wall surface. The geometric non-linearity was also taken into account during modelling. During the process of the investigation, factors such as the rib step and thickness of the wall (and, as a result, general wall flexibility) were varied.

621 In all of the cases considered, the ribs were taken as the same where they had the following nominal dimensions: 25x0.9cm for wall and 20x 1.5cm for shell. The applied pressure was taken as a constant value at 0.1 kg/cm 2. The joint number of wall thicknesses (k) that work together with the rib was determined by the difference in stresses on the inner surface of the wall and the external part of the rib, taking into account the existence of deformations in both directions. MAIN RESULTS AND CONCLUSIONS The calculation of the parameter k was made only for the middle rib of the modelled wall to exclude any boundary influence. The main results obtained from the research are summarised in Table 1. TABLE 1 VARIATION OF THE NUMBER OF WALL THICKNESSES IN JOINT WORK WITH THE STIFFENING RIB

Table 1 shows that the part of a wall that is included in the joint work together with a cross section of a stiffening rib, is not a constant value. Including is active enough and higher than usual, but it cannot be concluded if it is a unexpected result. For example, Hvalla analytically investigated the value of k for the case of a flat rib and concluded that a considerable part of the wall strengthened it [Ref. Blahe (1959)]. In addition, from the physical experiments that were conducted in 1971 by the organisation Dnerpproektstalkonstruktsija to obtain the parameter k under the circumstances given in Table 1, the value 103-119 was obtained. Table 1 also shows that within the wall thickness range 1.2-1.6cm the included part of wall according to this investigation is more then the rib step. In Table 1 it is marked in italics and physically it means that all the wall draws in joint work. The final conclusion is that the parameter k depends on wall and rib stiffness, and at the same time, does not possess linear characteristics. From this investigation, it can be concluded that the results obtained could be useful not only for the bunkers studied, but also in every branch where ribbed walls are necessary. REFERENCES 1. Blahe F. (1959). Ustojchivost metallicheskih konstruktsi, Strojizdat, Moskva, Russia (in Russian). 2. Rucovodstvo po raschetu i proektirovaniju gelezobetonnih, stalnih i kombinirovannih bunkerov. (1983). Strojizdat, Moscva, Russia (in Russian). 3. Structural Systems (1994). Bins. ESDEP WG. Lecture 15C.2, 1-31. 4. Filin A.P. and Sokolova A.S. (1957). Stroitelnaja mehanika korablja, Rechnoj transport, Leningrad, Russia (in Russian). 5. Strigunov V.M. (1984). Raschet samoleta na prochnost, Mashinostroenie, Moskva, Russia (in Russian).

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623

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

PURE DISTORTIONAL BUCI~ING OF CLOSED CROSS-SECTION COLUMNS Kunihiro Takahashi~, Hitoshi Nakamura1 and Kazuhiko Imamura1 1 Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522 Japan

ABSTRACT The existence of the pure and uncoupled distortional buckling of closed cross-section columns under axial compression is investigated. This type of global buckling mode has thus far been unrevealed, obscured by numerous local buckling modes. A new concept of shear centers of distortion of closed cross-sections is introduced for the analysis of buckling. The elastic buckling load with distortional mode is obtained analytically and expressed in a simple formula. It is important to note that the pure distortional buckling modes often become predominant, showing the lowest critical loads.

KEYWORDS Thin-Walled Column, Buckling, Distortion, Closed Cross-Section, Box Cross-Section, Shear Center

INTRODUCTION Thin-walled columns are known to exhibit flexural buckling and torsional-flexural buckling under axial compression. In addition to these buckling modes, a third mode; a distortional buckling mode of open cross-section columns (Takahashi (1988)), has been studied. However, the pure and uncoupled distortional buckling of closed cross-section columns does not appear to have been widely investigated,

I

O Figure 1: Pure distortional buckling of closed cross-section columns

624 especially from the perspective of one-dimensional beam theory. As it is common for this buckling mode to remain unrevealed due to the lower critical loads of local buckling (Krolak, Kolakowski & Kowal-Michalska (2000)), we have had few opportunities to observe this elegant mode of distortion, which is shown schematically in Figure 1. In this study, we examine the existence of this overall buckling mode of closed cross-section columns analytically within the framework of one-dimensional beam theory. Although it is possible to obtain buckling loads by numerical or semi-analytical calculation (Schardt (1989)), it is important to verify the existence of this peculiar distortional buckling mode analytically and to express the critical load in a simple formula. We initially discuss the theory of closed cross-section beams along a theme consistent with the onedimensional theory of open cross-section members. A new concept of shear centers of distortion of closed cross-sections is introduced, extending the classical beam theory. Based on these fundamental discussions, the principle of virtual work is descr~ed for the incremental components of displacement. Using the method previously introduced by the authors in the analysis of open cross-section columns, differential equations are derived for elastic bifurcational buckling of the closed cross-section columns with pure distortional deformation under axial compression. Buckling loads of this type can be calculated by a simple formula. From the theoretical analysis, we discuss how an overall buckling mode with pure uncoupled distortional deformation can be seen in certain doubly symmetric closed cross-section columns. For a column with a box cross-section, local buckling is more likely. However, by using longitudinal stiffeners, the local buckling mode is constrained and the global distortional mode becomes the mode that gives the lowest buckling load. On the other hand, for octagonal columns, it can be seen that very often the pure distortional buckling modes become predominant, showing the lowest critical loads. It is also notable that elastic bucking loads are often higher than the plastic buckling loads, however, we confine our attention to the elastic phenomenon of instability. Numerical firtite element analyses for the instabilities of columns are also carried out in order to verify these analytical results.

THEORY OF DISTORTION To develop a foundation for theoretical distortional instability for thin-wailed members, it is desirable to investigate the theory of distortion within the framework of beam theory. Distortion is cross-sectional deformation or flattening. As shown in Figure 2, classical beam theory relates to flexural buckling, the theory of warping torsion relates to torsional-flexural buckling, and the theory of distortion addresses distortional buckling.

Figure 2: Beam theory and theory of buckling

625 Table 1 shows the relationship between two methods of analysis for thin-walled members. As shown in this Table, the present study is based on the precise analysis of distortional deformation along a theme consistent with the warping theory of open cross-section members. In this analysis of deformation, a concept of shear centers of distortion must be introduced, which will be discussed in a later section. TABLE 1 THEORY OF THIN-WALLED MEMBERS

Open section member

Closed section member

Method of analysis Deformation of cross-section Series expansion Theory of warping torsion Generalisd Beam Theory (Vlasov (1961)) (Schardt (1989)) Theory of distortion (Takahashi & Mizuno (1978)) Theory of warping torsion Present analysis and distortion (Vlasov (1961))

LOCAL AND GLOBAL BUCKLING In descrying deformation, it is important to distinguish between the in-plane bending and the out-ofplane bending of each constituent plate of a thin-walled member (Takahashi &Mizuno (1978)). Table 2 shows the types of distortional buckling. Members with the displacement of nodal points of the cross-sections are subjected to the in-plane bending of their constituent plates. The in-plane bending of a plate is inevitably accompanied by outof-plane bending. We have called this type of distortion Type I distortion as shown in Table 2. The distortional deformation without the displacement of nodal points of a cross-section is subjected to only the out-of-plane bending of constituent plates. We have called this type of distortion Type II distortion. Distortional buckling with Type I distortion is not local, rather, global buckling of a member. In contrast, distortional buckling with Type II distortion is primarily local buckling. In developing a theory of distortional buckling, it is crucial to distinguish between Type I and Type II distortion. In this study, we limit the discussion to buckling with Type I distortion. TABLE 2 TYPES OF DISTORTION

SHEAR CENTERS OF DISTORTION FOR CLOSED CROSS-SECTIONS As we have defined above, Type I distortion is deformation with the displacement of nodal points of the cross-section. In a previous study, a concept of degrees-of-freedom of distortion is introduced in the theory of distortion for open section members (Takahashi & Mizuno (1978)). For closed section

626 members, it is also poss~le to define the degree-of-freedom of distortion in the same manner. Here, we focus on doubly symmetric cross-sections, because we confine our attention to the pure or uncoupled distortional buckling. The degree-of-freedom in this case is expressed as re=n~4(n: number of nodal points) excluding bending and torsional deformation. We can tentatively locate the rotation center of each constituent plate arbitrarily within the condition of compat~ility at a node, as shown in Figure 3(a). The tentative warping function ~ is determined from

aq,# = - m h# ax,

(t)

for the distortion of the f-th degree-of-freedom ( f = l-m), where/art is a tentative rotation ratio with respect to the first constituent plate, h# is the distance between the tentative rotation center and the plate, and xl is the coordinate along the middle line of a plate. Coordinate systems are shown in Figure 3(b). Equation (1) arises by virtue of the description of deformation contained in the next section.

Figure 3: Rotation centers and coordinate systems From the condition of an orthogonal system, the following relations are required

where F denotes the area of a cross-section. From the above conditions, orthogonalized warping functions q): are obtained. The warping function satisfies the next equation

dq,/ = - ~ : h~ ,tx,

(3)

and from this equation, the value of ~h! can be obtained. Using the compat~ility condition at a node, the distance hI and the positions of the rotation centers are determined. An example of the shear centers is shown in Figure 4(a) for a box section. From the reciprocal theorem, transverse forces acting on these shear centers cannot cause the distortion of the corresponding f-th degree-of-freedom. For this reason, the rotation centers after orthogonalization can be defined as shear centers. Deformations of a member under transverse forces are demonstrated in Figure 4(b).

Figure 4: Shear centers of distortion

627 THEORY OF DISTORTIONAL BUCKLING FOR CLOSED CROSS-SECTION COLUMNS We can use the same method introduced in a previous paper (Takahashi (1988)). The principle of virtual work for the finite deformation theory of a column is described as:

~ (s. + as., )o(aE.)av -j"

=.. F.8(~u.)ax.+I... __o' r.~(~u.)ax.-- o

(4)

where zlEu : the Green strain tensor ( A E u = (1/2)(Aut, a + d u j , I +Auk, iAuk, J) ); Aut : increment of deformations from before-buckling to after-buckling in the Xt directions (I=1,2,3); XI : coordinate in the reference configuration; dui 9deformations from before-buckling to after-buckling in the xi directions (/=1,2,3); x~: coordinate in the current configuration; St~ : second Piola-Kirchhoff stress tensor in the reference configuration (superscript (0) means a quantity in the reference configuration in this paper) ; zlSz~: increment of the second Piola-Kirchhoff stress tensor; V: volume of a column in the reference configuration; and F3 is the force per unit length of X1 as it relates to the first Piola-Kirchhoff stress tensor and the normal of the cross-section in the reference configuration. We take the before-buckling configuration as the reference configuration, and the after-buckling state as the current configuration. The summation convention is used for tensor indexes. Applying Gauss' divergence theorem and substituting the balance equation of force, we have

2s.~

+ ~ . _ , . ( t s . ~ - F~ )8(zl.~)axr~c~.:o ' ( t s . ~ - F. ~(,~u.),iX.= 0 (5)

where t is the thickness of a plate in the reference configuration. The second and third integrations of the above equation should vanish because of the boundary conditions. Holding the second-order terms with respect to d u in the first integral, we can rewrite the above equation as an integral over the current volume v 9 1

(6)

where Ac~i is the increment of the Cauchy stress tensor and Asi/is the increment of the infinitesimal strain tensor (,4e# = (1/2)(dui, ~ + duj, i)) . Components of deformation in the x~ directions are expressed as follows: m

/lu~ = A~cosa + z l ~ i n a + ~., AO:,~: /'--1 m

zlu 2 = -A~sina + A ricosa + ~ AO:v r

(7)

f=l m

/'=1

/l: = # : ( h : - x 2 + u ~ r ) , v : = # r j : +u~ y ,CO: = -

g:hrdx ~ o/

(8)

where A ~ , d r I 9rigid translations in the x, y directions; A 0 / ( jr = l-m): rotational angle of the first constituent plate (at an origin of xl) in distortion of the f-th degree of freedom;/4 " rotation ratio with respect to the first plate; j, , h, 9distances in the Xl, x2 directions from shear centers; So," coordinate xl where q~ =0; co," warping function; ui'" additional deformation due to accompanying out-of-plane bending in the xi direction; a 9the angle of a plate; and prime (') expresses the derivative with respect to X3.

628 Assuming the shear strain ~31(=(1/2)(Au3,1+ Aura)) is zero in the middle plane of a plate, and using Eqns. (7) and (8), we obtain

d fo: = - g: ht dxl

(9)

The linear relationship between the Cauchy stress and the infinitesimal strain is shown as follows:

zla~ = E#~azlc~a Enn = Emz = E333s = E ,

(10) Em~ = Era1 = E~n = E3~ = E3m = Em~ = 2G

where E and G are the Young's modulus and shear modulus, respectively. The first term of Eqn. (6) is rewritten as a ((M:,,AO:AO, + C,,::,,~O':AO',)+ (E1y,~r"~ + EZ,~,7 *~ + Er.,,:,aO*:~))dx, =

(11)

=

(12) s t 3

=

~M:Mgdx 1 ,

where s 9length of the contour line of a cross-section, and Mr is the bending moment of the contour line of a cross-section with the unit width in the x3 direction. The second term of Eqn. (6) becomes

f=t o=i

p~162 + AO;C:~Ar +(,t,7'+ AO)S:r

+ (zg'C: + zl,fS: + zlO;R:~ )e~o;)ax,

(13)

C/=ffl ~r' (2:cosa - v ssina)dF, S: = -~1 ~F(J,fsina + vfcosa)d.F , Rso 2 = f1f ~r"(2I"2o + v : v o ) d F

(14) where P o (= o.33OF) is a buckling load. Substituting (11) and (13) into Eqn. (6), we finally obtain the next balance equations: ~r

po,ar _ po

c:~o} = o =

JFJ.A~

ww

- P 0 zlrl

w

m

IEI~,:AO

#w

k

_ ~.,(p

- P

0

~S/AO: = 0

(15)

fl 0

R:~2 +C_Mro)AO~m + ~.,M:oZlOo -

o=1

P~162

+ S:Ar/')

= O

o=~

For a column with a square box cross-section, the cross-sectional constants C/and S: become zero, and from Eqn. (15) it is obvious that the bending and distortion modes are uncoupled. We can also see here the existence of the pure distortional buckling. The balance equation in this case is EI A r 1 7 6 1 6 2

I

=0

EI xA r1 9, - P 0 zl rl 9 =0

LF_~I~plLlO:n-... (P~ 1 I~ = -4~aSt,

"J-

(16)

GJn)AO: + MnAO ~ = 0

4 Jn = ~ at3,

Rll 2 =

where a is the length of a side of the square box cross-section.

-la23'

Et 3 Mn = 4 ~ a

(17)

629 DISTORTIONAL BUCKLING LOAD UNDER AXIAL COMPRESSION Assuming solutions of equilibrium equations for a simply supported column as A~: = ~ Ar .=1

nrc x3 l

ArI = '

A,/'sin nrc x3 .=l

l

ZOf = '

Ao,,si n nn" x 3 .=1

(18)

l

and substituting these forms into Eqn. (16), we can obtain simple formulae for critical buckling loads of a column with a square box cross-section as follows: Px=EL<

'

P~:P~

P'=

(=, rn='

"t, l ) +M,,

}

+C~,, / R , /

(19)

where P=, Py or P~, is a compression force ( - P 0). The first two equations express the Euler buckling load, and the last equation denotes the distortional buckling load of a column with a square box crosssection. As the value of the term GJn is usually very small, we find that the last equation has the same form as the previously obtained formula for an open cross-section column (Takahashi (1988)). The same procedure for obtaining critical buckling loads is applicable to columns having morecomplicated cross-sections.

RESULTS OF CALCULATION Figures 5(a) and 5(b) show the buckling loads Pc, (Ix, Pr or P~) calculated from the above equations for a box section and an octagonal section. Results obtained from the finite element method using NASTRAN are also plotted (dots: local buckling modes; squares: global buckling modes). As shown in Figure 5(a), the local (Type II) buckling loads by NASTRAN are much lower than the global (Type I) distortional buckling loads. Small dots plotted in this figure at the length I/a--6.67

Figure 5: Results of calculation

630 correspond to critical buckling loads by NASTRAN. The number of dots indicates that the global distortional buckling mode at l/a=6.67 is the 95th critical load, and that is the reason why this elegant buckling mode has been unrevealed and hidden, until now, behind these numerous local buckling modes. However, by using longitudinal indentations in the plates, the local buckling mode is constrained and the global distortional mode becomes predominant, showing the lowest critical loads. We can show examples of modified box cross-sections having this type of bucking load. On the other hand, as shown in Figure 5(b), for a column with an octagonal cross-section, the distortional buckling loads are the first mode and are lower than any local buckling loads.

CONCLUSIONS The existence of the pure and uncoupled distortional buckling of closed cross-section columns is confirmed theoretically and numerically. It is important to note that the distortional buckling loads of columns with certain closed cross-sections often become critical. The authors would like to express their gratitude to Mr. T. Taguchi and T. Mizoguchi. References Krolak M., Kolakowski Z. and Kowal-Michalska K. (2000). Global and Local Instabilities of Tubular

Pole Structures with Intermediate Stiffners Subject to Combined Loads. in Proc. Third Int. Conf. CIMS, eds. Camotim D., Dubina D. and Rondal J., Imperial College Press, 171-178. Schardt R. (1989). Verallgemeinerte Technishe Biegetheorie, Springer-Verlag Takahashi K. and Mizuno M. (1978). Distortion of Thin-Walled Open-Cross-Section Members (OneDegree-of-Freedom and Singly Symmetrical Cross-Sections). Bull. JSME 21:160, 1448-54. Takahashi IC (1988). A New Buckling Mode of Thin-Walled Columns with Cross-Sectional Distortions. inApplied Solid Mechanics-2, eds. Tooth A.S., Spence J., Elsevier, 553-573. Vlasov V.Z. (1961). Thinwalled Elastic Beams, Israel Program for Scientific Translations for the N.S.F. and the Dept. of Commerce, Washington, D.C., U.S.A.

Third InternationalConferenceon Thin-WalledStructures J. ZaraL K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rightsreserved

631

ELASTO-PLASTIC LARGE DEFLECTION OF UNIFORMLY LOADED SECTOR PLATES G.J. Turvey 1 and M. Salehi 2 Engineering Department, Lancaster University, Lancaster, LA1 4YR, UK 2 Mechanical Engineering Department, Amirkabir University of Technology, Tehran, IRAN

ABSTRACT The governing equations for the elasto-plastic large deflection response of pressure loaded fiat sector plates are presented in outline. In contrast to earlier studies by the authors, based on the Ilyushin fullsection yield criterion, the present formulation is based on the Von Mises layered yield criterion and the Prandtl-Reuss flow theory of plasticity. It is shown that the first yield pressures predicted with the layered yield model are only about two-thirds of those predicted with the Ilyushin criterion. Detailed response plots for the centre deflection and the centre radial stress resultant and couple are presented for 600 slender and stocky plates with four edge condition combinations ranging from simply supported to clamped and in-plane fixed to in-plane free.

KEYWORDS Dynamic relaxation, elasticity, finite differences, large deflection, plasticity, plate, pressure, sector

INTRODUCTION The steel plating between the radial stiffeners of flat circular end closures to cylindrical pressure vessels has a sector planform and in service is subjected to uniform pressure loading. Hence, isotropic sector plate analysis is relevant to the design of this type of structural fabrication. A number of elastic small and large deflection analyses of sector plates have been carried out during the course of the last century. Several significant small deflection analyses were carried out before the advent of the digital computer and are described by Timoshenko and Woinowsky-Krieger (1959). These approximate closed-form solutions are limited in so far as few boundary condition combinations have been considered, most probably because of the difficulties associated with the singularity at the apex. Solutions for the elastic large deflection response of sector plates are limited and, in general, have been obtained using approximate numerical techniques. Several years ago the authors presented an elastic large deflection analysis for uniform pressure loaded sector plates with simply supported and

632 clamped edges, Turvey & Salehi (1990). They obtained numerical results using a finite difference formulation of the DR algorithm, Otter, Cassell & Hobbs (1966). As far as the authors are aware few studies of sector plates have been undertaken, which account for both geometric and material nonlinearity. Recognition of this situation prompted the authors to investigate their elasto-plastic large deflection response. The first results of their investigations were published recently, Turvey & Salehi (1997 & 1999). These results were computed using the Ilyushin full-section yield model and the flow theory of plasticity. Although this yield model is computationally efficient, it tends to be over-stiff. Thus, the yield pressure is over-estimated and modelling of the postyield response is less accurate. A more accurate approach is to employ the Von Mises layered yield model in conjunction with the flow theory of plasticity. In this model the thickness of the plate is subdivided into a number of layers, so that the local plastification of the whole plate thickness (implicit in the Ilyushin model) at first yield is avoided and growth of the plastic zones from the upper and lower plate surfaces is modelled realistically. Therefore, this paper focuses on the elasto-plastic large deflection analysis of uniformly loaded sector plates using the Von Mises layered yield model. New results are presented for deflections and stress resultants and couples at critical locations in slender and stocky, simply supported and clamped, uniformly loaded sector plates.

PLATE GEOMETRY Figure 1 shows a sector plate, subtending an angle ~ at its vertex, and the positive co-ordinate directions, r, 0 and z. For uniform pressure loading there is symmetry about the radial centreline; therefore, only one half of the plate is modelled in the analysis. c

Z

A

Figure 1: Sector plate geometry, transverse pressure loading and positive polar co-ordinate directions The geometric and material properties of the sector plate may be expressed non-dimensionally as:-

P- o

(1)

In Eqn. 1 ro and ho are the radius and thickness, Oo and Eo are the yield stress and Young's modulus. 13, is-referred to as the plate slenderness, because its value is dominated more by changes in the radius to thickness ratio. For mild steel plates 13varies from about 0.4 (stocky plates) to 3 (slender plates). This small range is beneficial, because computations need only be carried out for a few 13values to include all practical geometries. Here they have been carried out only for the two extreme values.

633 GOVERNING PLATE EQUATIONS The governing system of equations includes four subsets, viz. equilibrium, strain/curvaturedisplacement, constitutive and boundary condition equations. These equations are introduced without derivation and with minimal comment.

Equilibrium Equations ONr

Or

l (M'rV" N o ,~+ l ONre +. . . . r r O0

&2 + - ~ + ~ ~ + r OrO0

2

r ~ 002

N " o~r 2+ N

O,

+

Or

9

Or

r O0

N o ---~+ + r -;T005)

+ Nr~ + Or z

Or

+-;-Z-~)

e

1 ON e 2 ++ - Nre = 0 ; r O0 r

ONre

+ 2Nro

w r OrO0

)

10w r2 - ~

+q=0

(2)

In Eqns. 2 Nr, No, N~o are the stress resultants, Mr, M0, M~o are the stress couples, w is the deflection and q is the transverse pressure. Also, the first two of Eqns. 2 define equilibrium in the radial and circumferential directions and the third defines it in the z-direction. The underlined terms in the latter equation account for m e m b r a n e action, i.e. stretching of the plate middle surface within the large deflection regime.

Incremental Strain~Displacement Equations In an elasto-plastic analysis, the strains depend on the final stress state and the loading history. Consequently, an incremental formulation has to be adopted. The incremental strain and curvature relations are as follows:~6 ~ = ~+~

A e Oa

1 Ou "-

Ax ~

o "Ae~ -c~,~ ,

-- V

1 0 2w r 2

002

=- 1 u+

Ov l bw ~ dr. Or r Or O0

.]. ~

10w r Or

o . Atr tc~

+ ~1

r

o . -e~,~

2r 2

o

.

ErO, b

,

o AK

r

2 0 2w

=

w Or 2

__ ~

2 0w

'

_

o

.

X r,O

'

o

- r- OrO0 ~ + -;T-ff~ - XrO.O

(3)

In Eqns. 3 er, co, ~o are the direct and shear strains, K:r,~ , K:~oare the curvatures and twist and u, v are the radial and circumferential displacements. The symbol, A, denotes an incremental quantity, the superscript, 0, denotes a quantity associated with the plate mid-plane and the subscript, b, refers to the cumulative value of the quantity before the application of the pressure increment. The total strains and curvatures are determined, by adding the cumulative incremental strains and curvatures to the incremental strains and curvatures arising from the current pressure increment.

634

Incremental

Constitutive

Equations

The incremental stress resultants and couples are computed from the incremental strains and curvatures and the elasto-plastic tangential rigidities as follows:-

{AN}=I~~[AP~z)~6~176

;{AM}=I~~[AP~dzI{A6~176

(4,

In Eqns. 4 {AN}, {AM} are vectors of incremental stress resultants and couples, {As~ {A~:~ are vectors of the incremental strains and curvatures, and [A p] is the matrix of elasto-plastic stiffnesses. The latter are integrated through the plate thickness, which is sub-divided into seven layers, to give the tangential rigidities. The tangential stiffness matrix, [AP], in Eqns. 4 is obtained from the Von Mises yield criterion and the Prandtl-Reuss associated flow rule as:-

c3F

c3F r

(5)

1 inEqn'5[A]=oE~176

v 01

0 O - v0)

;F =

2 + cr~ _ o.ro.~ +

;

denotes the vector

2

of stress components and v is Poisson's ratio. As before, total stress resultants and couples are determined by updating the total values at the end of the previous pressure increment with the incremental changes due to the current increment.

Boundary Conditions The sector plates were assumed to have all edges either simply supported or clamped. In addition, the in-plane condition along all edges was assumed to be either fixed or free. Thus, the following four edge conditions were used in the numerical computations:Along the curved edge (r = r0) U=V= W=M r

=0

Ow

u=v=w=m=0 Or Along the radial edge (0 = 0~ u=v=w=M

0=0

[SS-FI] ; v = w = N r = M r = 0 [SS-FR] ; Ow

[CL-FI];v=w=~=N,=0 Or

[CL-FR]

[SS-FI];u=w=N e=M e=0

[SS-FR];

Ow 0w u=v=w=--=0 [CL-FI];u=w=--=N 00 a0 Along the radial line of svmmetry_ (0 = d~/2) Ow v =~ = N r e = M ro c~O

=0

o=O

[CL-FR]

(6a)

(6b)

(6c)

635 NUMERICAL SOLUTION OF THE GOVERNING EQUATIONS The DR iterative procedure was used to solve the finite difference approximations to the governing system of equations. To apply the DR algorithm, the boundary value problem, defined by Eqns. 2 - 6, first has to be transformed to initial value format by adding damping and inertia terms to the right hand sides of Eqns. 2 as follows:02gt 00~ [LHSof Eqn.2]=p~ - ~ +k~ ~Ot (7) In Eqns. 7 p= is the mass density, 1~ is the damping factor, t is the time and ot = u, v, w. Introducing approximations to the temporal derivatives in Eqns. 7 allows them to be re-arranged in terms of the velocities before and after the time increment, i.e. in initial value format. Thus, the velocity equations are:-

(c30t~" 1 1 (1-k~* ~Oa~bAt{LHSofEqn.2} 1 Ot) = O+k:) ? at,) Pa

(8)

In Eqns. 8 the modified damping factor, k~*, is a function of At and p,~ and the superscripts, a and b, denote the values of the velocities at the start and finish of the current time step, At. The displacements are obtained by integrating the velocities as follows:-

a" =a b+ \Or)

(9)

Eqns. 2 - 9 are the complete set of equations for the application of the DR iterative procedure. However, before they may be used in numerical calculations, they have to be discretised. In the present case, interlacing finite differences were used. The DR iterative procedure is straightforward. All displacements, velocities, stress resultants and couples are set to zero and a pressure increment is applied to the plate. The resulting velocities are then calculated from Eqns. 8. Integrating the velocities, using Eqns. 9, gives the displacements. The displacement boundary conditions are then applied using Eqns. 6. The incremental total strains (elastic and/or plastic) are determined from Eqns. 3. These are then used in conjunction with Eqns. 4 and 5 to calculate the incremental stress resultants and couples so that the total stress resultants and couples may be determined. Thereafter, the stress resultant and couple boundary conditions are applied (see Eqns. 6). This overall procedure is repeated until convergence is achieved for the current pressure increment. Then a further pressure increment is applied and the whole process is repeated. The process stops when the maximum prescribed pressure is reached. Convergence of the iterations for each pressure increment is assured by an appropriate choice of damping factor. Further details of the finite difference approximations to the governing equations and the DR iterative procedure may be found in Turvey (1978) and Turvey & Salehi (1990).

NUMERICAL RESULTS AND DISCUSSION The first set of non-dimensional results, presented in Table 1 for stocky ([3 = 0.4) and slender ([3 = 3) 60 ~ sector plates, were computed using an 8.5 x 8 interlacing finite difference mesh. They show a comparison of the Von Mises and Ilyushin results at first yield. It is evident that the Von Mises first

636 yield pressures are only about two-thirds of the Ilyushin values, confirming that the latter yield model is over-stiff. Because modelling of the elasto-plastic large deflection response is computationally intensive, pressure versus centre deflection etc responses are only presented for stocky and slender 60 ~ sector plates for each of the four boundary condition combinations. These results are presented in Figures 2 -4. Figure 2 shows the pressure versus centre deflection response. It is clear from Figure 2a that the in-plane edge restraint only affects the response at the highest pressures in stocky sector plates, whereas in slender sector plates (see Figure 2b) the centre deflection is considerably larger at high pressures in the absence of any in-plane edge restraint. The pressure versus central radial stress resultant responses are shown in Figure 3. It is evident that the stress resultants in stocky clamped and simply supported plates become very large at pressures about two and three times the first yield pressure respectively (see Figure 3a). Moreover, the stress resultant appears to increase more rapidly in the presence of in-plane edge restraint. This is particularly so for simply supported plates. However, once there has been substantial yielding of the plate section, the stress resultant magnitude appears unaffected by the degree of in-plane edge restraint. In the case of slender sector plates, restraining the in-plane displacements at the supports enables the stress resultant to grow more rapidly, as is clearly visible in Figure 3b.

TABLE 1 VON MISES (VM) AND ILYUSHIN (IL) FIRST YIELD PRESSURES AND ASSOCIATED CENTRE DEFLECTIONS ETC FOR STOCKY AND SLENDER 60 ~ SECTOR PLATES Iv --- 0.3] 32

0.4

3.0

qro4/(Eoho4) [at yield]

w/ho [at yield]

Nrro'/(Eohfl) [at yield]

MrroZ/(Eohoz) [at yield]

Boundary Conditions [All Edges] i

4.13 (vM) 2.7 (IL).... 4.145 2.75 4.55 3.03 4.55 3.03 33.6 20.3 31.4 21.45 34'.1 22.6 34.1 22.7

0.04549 (VM) 0.02979 (IL) 0.04576 0.03038 0.01645 0.01095 0.01645 0.01095 0.3218 0.2105 0.3388 0.2346 0.1225 0.08147 0.1230 0.08199

0.01152 i~/M) 0.004952 (IL) 0.004316 0.001901 0.0013'66 0.000604 0.00074'23 0.0003422 0.5780 0.2475 0.2168 0.1005 0.07565 0.03346 0.04036 0.01807

0.09577 (VM) 0.1066 (IL) 0.09654 0.1101 0.09634 0.1022 0.09658 0.1030 0.6671 0.6283 0.7078 0.6836 0.6977 0.6737 0.7133 0.6997

SS-FI SS-FR CL-FI CL-FR SS-FI SS-FR

CL-FI CL-FR

Figure 4 shows the central radial stress couple versus pressure responses for stocky and slender sector plates. For the stocky plates (see Figure 4a) there is a noticeable reduction in slope at about 1.5 and 2 times the yield pressure for simply supported and clamped edge conditions respectively. Under simply supported edge conditions the central stress couple reaches its peak value at roughly 2.5 times the first yield pressure, whereas under clamped edge conditions the stress couple continues to increase even when the pressure reaches 3 times the yield value. Moreover, the in-plane edge conditions do not

637 appear to influence the central stress couple. However, for slender sector plates (see Figure 4b), the inplane edge condition does affect the peak values of the stress Couple - higher values being reached in the absence of any in-plane restraint, especially in conjunction with simply supported edges. It also appears that changing the flexural edge condition from simply supported to clamped produces a small increase in the peak value of the stress couple when the edges are fixed in-plane, but not when they are free in-plane.

10.0

,.z'-

8.0

6.0

/

4

E~1764.0

I~

~

s~s"'la'%~'7

I? J2.011 f

--

o c,-,.,>~.,,.8.,,,=

0.0 - P ~ - ~ , q:3.o! I 0.00 0.04 0.08 0.12 0.16 0.20 0.24 W ho

(a)

100.0

/

7""

f""

80.0 4 60.0 q ro

,0/,, . ,,(__

4

o

o ' y

E~17640.0

20.0 0

(b)

0.0

.

0 0.3

~ 0.6

0.9

1.2

W ho

Figure 2: Pressure versus centre deflection - (a) 13= 0.4 and (b) 13= 3.

638

10.0

8.0 m.,==:=:==~ 9

6.0

4

qro

Eoh~

4.0

P ---0.3 9 SS-F!I q.--'2.7 E] SS-FR,q=2.75 CL-FI, q =3.03 0 GL-FI,qy:=~.95.8.5x12 A CL-FR~qv=3.03

I /

I 2.0 --if 9

I

0.0 - 9 0.00

I

i

I

0.02

0.04 2 Nr

I

0.08

0.06

r o

3

Eoho

(a)

100.0 .

.

.

.

,,,;,li,

80.0

4

qro

le"

tO'

lO/~ m

i

m" 9

60.0

.I 9

9

m"

Eoho 40.0_t7;~ ~ . l ' 9 /~-'~,ff.~~ , . .m ,~j~

2O.O -ifvI.4/"

u=o.~

9 $S-FI, q=20.3 12 $ S - ~ , (~ =20.3

"

, CL-n., ~ ,

511~"/"/ ii

0.0 "! 0.0

CL-FR,qT=22-7

0

i

0.4

i

0.8

i

2

1.2

i

1.6

2.0

Nr r o

3

Co)

Eoho

Figure 3: Pressure versus central radial stress resultant - (a) [3 = 0.4 and (b) [3 = 3.

639 0.12 0.10 -

/,i

0.08 Mr

2

r o 4

Eoho

0.06

-

0.04 -

0.02 -

,,/"

/ / ps

o

,E~

0.00 -i 0.0

CL_.~_nL~,--~_~03 ....

A CL-FR, q =3.03

'I

2.0

' 'I' "

I

4.0

6.0

qro

'i

8.0

10.0

4 4

Eoho

(a)

0.8 0 (20o

0.62 Mrro 4

0.4-

Eoho

0.2-

//

C S$-FR. qT=2q. CL-Flt.g, ==22"6

/9

0.04 0.0

Ill

II

20.0

J

40.0

J

60.0

nl II liB ill ii

80.0

i

i

100.0

4

qro

4

(b)

Eoho

Figure 4: Central stress couple versus pressure - (a) 13 = 0.4 and (b) [3 = 3.

640 CONCLUDING REMARKS The governing system of equations for the nonlinear response of pressure loaded sector plates, based on the Von Mises layered yield plasticity model, have been outlined. In comparison to the Ilyushin full-section yield model, it has been shown that first yield pressures predicted with the present elastoplastic constitutive model are about one-third lower. The centre deflection, stress resultant and stress couple responses with increasing pressure have also been presented for stocky and slender 60~ sector plates with simply supported and clamped edges. It is shown that the in-plane edge restraint has a significant effect on the response of simply supported slender plates, especially as the elasto-plastic state in the plate becomes well-developed. The in-plane edge restraint is shown to have little influence on the response of stocky plates, except for the radial stress resultant and then only over part of its response range, when the plate edges are simply supported. The results presented quantify extremes of response for 60~ sector plates and provide benchmark data for checking similar analyses, based on alternative numerical techniques. ACKNOWLEDGEMENTS The second author wishes to acknowledge financial support for this research provided by the Iranian Ministry of Higher Education. Both authors wish to record their appreciation to colleagues in the Engineering Department for encouragement and support. REFERENCES

Otter J.R.H., Cassell A.C. and Hobbs R.E. (1966). Dynamic Relaxation. Proceedings of the Institution of Civil Engineers (Research and Theory) 3:2, 633-656. Timoshenko S.P. and Woinowsky-Krieger S. (1959). Theory of Plates and Shells, McGraw-Hill, New York, USA. Turvey G. J. (1978) Large Deflection of Tapered Annular Plates by Dynamic Relaxation. Journal of

the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers 104:EM2, 351-366. Turvey G.J. and Salehi M. (1990). DR Large Deflection Analysis of Sector Plates. Computers and Structures 34:1, 101-112. Turvey G.J. and Salehi M. (1997). Full-Section Yield Analysis of Uniformly Loaded Sector Plates. In Trends in Structural Mechanics: Theory, Practice & Education, Roorda J. and Srivastava N.K. (eds.) Kluwer Academic Publishers, Dordrecht, Holland, 289-298. Turvey G.J. and Salehi M. (1999). Elasto-Plastic Response of Uniformly Loaded Sector Plates: FullSection Yield Model Predictions and Spread of Plasticity. In Proceedings of CIVIL-COMP 99 F, 157169.

Section XI SHELL STRUCTURES

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Third Intemational Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

643

VALIDATION OF ANALYTICAL LOWER BOUNDS FOR THE IMPERFECTION SENSITIVE BUCKLING OF AXIALLY LOADED ROTATIONALLY SYMMETRIC SHELLS G.D.Gavrylenko 1 and J.G.A.Croll 2 1Professor, Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev, Ukraine 2 Professor, Head of Department, Department of Civil Engineering, University College London, London WC1E 6BT, UK

ABSTRACT Doubly curved shells of revolution, with cylindrical and near cylindrical form continue to provide < solutions for important classes of structural problem. Although considerable progress has been made towards a full understanding of the highly non-linear, imperfection sensitive, buckling that takes place under axial loading, the insights provided by the widely varying approaches to the problem have not always been fully realized or reconciled with each other. The work described in this paper brings together analytical approaches that individually have played an important role in helping our understanding of the problem, but together provide even greater insights into this complex form of shell buckling behaviour. The use of computing power to solve the non-linear equations governing shell buckling is now well entrenched in the shell buckling community. At the Institute of Mechanics, Kiev, new algorithms have been developed based upon a finite difference descretisation of the non-linear differential equations. However, without an appropriate conceptually simple interpretation it is often difficult from such non-linear solutions to draw any generalizations of behaviour that could help influence conceptual design. A conceptually simple interpretative framework is in this joint programme provided by the so-called <
644 NOTATION Aij

element of eigenvalue matrix equation (20)

AIj, Aii D E Ex, E0, Ex0 i

constants defined in equation (21)

iem

J K 1

mx, mo,mxo Mx,Mo,Mxo n

nx,no,nxo Nx,No,Nxo r, t RSM U, V, W

U,V,W V x,O Xx,Xo,Xxo (X ~x, E;0, E;x0

bending stiffness (-Et3/12(1- kt2)) modulus of elasticity total membrane strains number of circumferential waves in critical mode i for minimum critical load number of axial half-waves in critical mode membrane stiffness (- Et/(1- ~2)) length of shell incremental moment resultants total moment resultants inward normal coordinate incremental membrane stress resultants total membrane stress resultants radius and thickness of shell reduced stiffness method incremental displacements in (x, 0, n) directions total displacements in (x, 0, n) directions total potential energy axial and circumferential coordinates total bending curvature changes non-dimensional coefficient (-=12R 2) incremental membrane strain

~em

non-dimensional axial wavelength parameter (~-j~/L) non-dimensional load parameter (---o/E) classical critical load [(- R'l(3(1- p2))-la] critical load minimum critical load Poisson's ratio radius of meridional curvature uniform axial compressive stress loading classical critical axial stress critical axial stress minimum critical axial stress

Xx,XO,%xO

incremental bending curvature changes.

A Acl Ar Aem I.t P (5 ~el Oc

Superscripts

(..) (...)" (...)" (...)~

stresses and strains linearly dependent upon incremental displacements stresses and strains quadratically dependent upon incremental displacements belonging to reduced energy model belonging to fundamental path

645 INTRODUCTION Lower bound estimates of the buckling loads for axially loaded unstiffened shells near to cylindrical form are suggested and realized. The authors used an approach which has been early created and published in articles [1-3]. The full review of articles describing of reduced stiffness method are contained in our earlier articles [4-5]. The procedure of nonlinear numerical experiments and approach for defining of lower bounds for the buckling are given in articles [6-9]. 1. CLASSICAL BUCKLING ANALYSIS The total potential energy of an axially loaded shell near to cylindrical form, shown in Fig. 1 to be of length, /, radius, r, thickness, t, and meridional radius of curvature, P, undergoing a kinematically admissible total displacement (U, V, W) about an unloaded and underformed state may be written

V=

12n

l

xXx + MeX e + 2MxeX~e]rdOdx + 2g

Ix=Z

[NxE x + NeE 0 + 2NxoEx0lrd0dx- ~otU rdO. 0 Ix=0

2

(1)

+

mx0 n"+'~"~ m[~ m no

n0x

0x

/i

nx0 Fig. 1. Notation and convention adopted for geometry and internal stress and moment resultants. The fundamental path may be approximated by the axisymmetric membrane solution (N xF, N Fo,N~e) = F = [-(1 + ~ r ) o ~ , (~ + r ) E ' 0] -_ (-at, _r ct, 0) with corresponding strains ( E ~, EoF ,Exo) P t9 P '

(2)

646 where o is the magnitude of the externally applied end stress; E is the modulus of elasticity; la the Poisson's ratio. Equilibrium states other than the fundamental may then be examined in terms of the incremental displacements (u, v, w) about this fundamental state, where U -- Ur(o)+u; V - vr(o)+v;

W - WF(o)+w.

(3)

The values of full forces, moments and strain may be represented in forms N, - NxF(O)+ n, ; E x = E~((~)+~, ; E o - E~ ( o ) + Go 9

N O = N~(o) + n o 9

F

Nxo -= NFo(o) + n~o = nxo ;

(4)

E xO-- E xO(~ + e xO= e xo F

Mx = M~(o) + mx = mx"

Xx - Xx(O)+Xx = x ~ x0 - XF(C) + Xo = Xo"

M o = MoF ( a ) + m o = m o; F

M ~o -= M xO(~ + m xO= m xO

X xO- X xO(~ + XxO= X~o,

(5)

where nx, n o, nxo, e x, e o, exo ... are incremental stresses and strain about the fundamental state. The equivalence of the total bending stresses and strains and the incremental bending stresses and strain in equation (5) is a consequence of the assumed membrane fundamental state. For present purposes it will be convenient to break the incremental membrane strains into their components that are linear (~'x, G'o, G'xo) and quadratic (e"x, %, e~o)in the incremental displacements, and to define the associated membrane stress resultants (n'x, n o, n'xo) and (n x, n o, n~o) as respectively those derived from these linear and quadratic strains. The equation (1) may be replaced by the following form V = V o +V: + V 2 + . . . .

(6)

where the contributions to the total potential energy that are independent, Vo , linear, V~, quadratic, u in the incremental displacements would for the present membrane fundamental state then be given as Vo

x~Ex F +NgEg +2NxoExo r d 0 d x 0 0

Vl

=

1

N xEx F'

"2"o

1

2n ]1

]

~

+

0

2

(7)

o'tu o

1'~![ m

TM

!!t,

1 12,~

rd0 0

n'ExF)+(No~e'o+n'oE F) +2(NxFoe'x9+n~oExo)lrd0dx

V2 =~-! ![nxex + n;eo + 2n:oexo]rd0 dx +~-!

+_

I trtU F

rd0

(8)

o

xXx + moxo + 2mxoXxo]rd0dx +

]

-

NFex +nxExF)+(NoFeo +noEo) r d 0 d x = U u +UB+~'~ +~'~ +V~ +~M~

(9)

where U M and U B are the membrane and bending strain energies based upon the linear stress and strain, while the contributions to the quadratic strain energy arising from the interactions between the fundamental state and the non-linear stresses and strains have the meaning -

I

F

V~ = ~ - I J N x e x

"

rd0dx,

~x 9v~, =~

I

F

IIn"xEx rd0dx"

V ~ = ~ - I J "Noe F o r d 0 d x ; V~ "o = ~1 I I n ; E o ~ rdOdx.

(10)

647 The quadratic components, V2, of the total potential energy control the stability of equilibrium of the fundamental path and from them the eigenvalue problem for the critically stable states is derived. Using the notation and convention for incremental quantities indicated in Fig.l, the linear strain displacement relations in equations (8 - 9) are 0~

W

&

p

e,0=1

c32w 10u pOx

1

0v

l c3u

"

1 .02w

"

,

0V_w)

r(N

Ov

X'o -

X'xo :-

'

1.02w Ov 1 c~ +--+r('0xc~0 0x O N )'

(11)

while the relevant components of a small strain but finite rotation theory will be taken as

ex =~'L(~-) +(-~--+ )5 ;

e;=--2r2 ( )2+(.~_+v)2 .

(12)

The linear stress and moment resultants are related to the linear components of strain through the constitutive relations t

t

t

t

n x =K(e x +rte o); f

r

t

r

t

n o =K(e o +l.te x)"

t

t

mx = D(Xx +laXo );

r

t

nxo =K(1-g)exO;

r

r

t

(13)

mo = D(X0 +l.tXx ); mxo = D(1-~t)Xxo 9

The relevant quadratic stress components are given by re

tt

tt

tt

n x =K(e x +ge o ) ;

tt

tt

(14)

n o = K ( % +ge x ) ,

where K = Et/(1- rt:), D = Et3/12(1- ~t2), and the nonlinear strain are defined in equations (12). The expression for the energy (9), taking into account expressions (11- 14), becomes

V2=~!

~---+OX p .r.c. ~.

Di2~[(0Zw

r

-

20-.)

-

Ov

w)

r00

! ~___.'~2 1 02w + -] -~ + -2(]P OX rE 00)

+ 2(1- ~t)( 02w + -Ov - + -1- - r 2 (.0x60 0x P

1

. ( a:u

+

.L.~_(~___.+ -l Ou]21 r dO dx + r00

1~)(] a2w l&)

I.t)L~-T + p &

7--~-Y'+7~"

+

-x = --o ~-o rdOdx+V~, +V~, +V~ +V~,,

--x~ + V-'e where V ~ + V-xM + V-o~ = - ~ (oK 1-1a

!L L~

2)

+ Pr

(15)

rd0dx.

(16)

0

For a shell with classical simply supported ends satisfied by a mode u = uij cos i0cos jrcx/1, v = vii sin i0sin jrcx/l, w = wi~ cos i0sin jwx/l, in which there are i circumferential waves and j axial half-waves, the equation (15) becomes

(17)

648

V~ = V: ~r4r2 -1 +lot

E

r

- ;~uij + ivij -(1 + p)Wij

12

r

-

+2(1-gt) (~uij + 9 wij) (ivij -wij) +

-(i2 + ~'2)wiJ - 9 ~uiJ +ivij

+2(1-gt) (~2w,j +p;~uij

ij

(~v,j - iu,j) 2 +

ij

r )21} + ~rl~l + ~/~I + ~'0M+ V~M +(-~,iwij + ~,vij - ~-iuij where

(18)

= N_.2 r VI~1 + VI~I + ~'1~I+ ~,o=_ g ,- , )(~.~ +-i2)w~ " ~ = 12R 2 = 12(r/t) 2 X = jTt/l. P ,--,

-

o

(19)

Stationarity of the total potential energy (18), with respect to arbitrary kinematically admissible displacements (u0, vij, w~j ), allows formulation of the eigen-equations in the form 0, A31 A32 A33

(20)

w

where A,, = ~2 + 1-1a i2 + (r/p): [)2 + 2(1_ gt)iz]; 2 ot

A, z = _ 1 +gti~ - r/___99(2_ gt)iZ. 2 ot

A, 3 = L(r+~t) + ~ L [;~2 +(2_~t)i2]; 9 9 = r) i A2, - i ( l + l t - - - [ i 2 +(2-g)L2]; 19 ot

A22

-

-

1-~t~2+i 2+ 1 [i ~+2(1 2 (z ~

~

~

~t);~2]

9

A33--A33 - AA33 ; A 0 = Aji (i ~ j) ;

A'3a = 1 + (r)2 + 21.t_r+ _ ( i 1 2 +;~2 )2 . (A~3)~ l = ( 1 - l.t2)(~, 2 -Li~). p p ~x 9 The resulting solution of the eigenvalue problem may be written

(21)

I-

o.._~s_= 1 |A,33 - F 2AnA'3A23 - A"A~32 - A22A~3 1 " Al1A22 - A n E (A"33)c1 L l

(22)

Fig.2 gives dependence of the lowest critical eigenvalue Crc,o~/col, where a~ = 0,605Et/r is the lowest critical load from classical theory for the cylinder.

l-

l/r=3 I

-0,04

I

0,25 t I

-0,02

I

[

C~ ,,z

0

I

0,02

I

i

r/p

Fig. 2. Dependence ao / Ool from r/p for shell near to cylindrical form (fit = 100) for various 1/r.

649 2. L O W E R B O U N D B U C K L I N G

ANALYSIS BY REDUCED STIFFNESS METHOD.

The non-linear circumferential membrane strain energy provides a crucial stabilising contribution to the axially loaded cylinder. This leads to the simple idea that a lower limit to the shells post-critical loss of stiffness, or incremental energy, would be provided by the reduced incremental quadratic total potential energy 1

,

v; = 700

,

1

%x + mo~o + 2mxoXxo r dO dx +

+1-

e~ + nxE x +N~e

,

,

xe: + n'0e'o + 2n;oe;0 r dO dx +

;] rd0dx.

(23)

2o0 The expression for V~ will have the components ~'~ + ~'~ + ~'~ as in equation (18). Their sum will be equal "x + V~ "~ + V~ =x = - 2--~ IO' t 2[z( l2~V~

+ t.tr)x,2 + (It + 2B2 -r- r ) i 2 ] 2w,j . P P P

(24)

(A;3)RSM =~-I I (2-- g 2 +t.tr)~, 2 +(It + 2bt2 -r- - r ) i 2 ] . P P P It is possible to write (23) in the form given by

(25)

Introducing the notation

V~x +V~ +V~ -

2r 2 E IAa/

:

(26)

From equations (20) bat making use of (25)-(26) the eigenvalue may now be written

[

RsM = A33 +

1

~llTA= - A~2

(A;3)RSM '

(27)

where the value o~, is the reduced stiffness method (RSM) critical stress. The minimal value A~m = o~,/E is realized for case having a single axial halfwave j=l and connected critical mode lcm. Fig.3 represents curves O~m/ Ool=fir/t), defined by RSM. 0>6 0,5 0>4. 0>3 0>2 0>1

=

'

0

0

500

I

1000

I

I

I

1500

2000

2500

I

r/t

Fig.3. Dependence O~m/ Ocl upon r/t, for shells having 1/r= 1 and near to cylindrical form with various r/p>0.

650

9

Analysis of Fig.3 allows the following conclusions to be drawn: Non-dimensional minimum lower bound critical stress, decreases with growth of r/t;

9 the growth of r/p leads to growth of O'cm/t~cl and especially in comparison with a smooth cylindrical shell (r/p=0); 9 when r/t trends to 2000 and more values acrn /C~elstabilize and almost do not exchange. METHODS ESTIMATION OF CARRYING CAPACITY OF THIN-WALLED RIBBED STRUCTURES.

3.

Two methods of defining carrying capacities for thin-walled ribbed structures, having initial imperfections of geometry, have been examined and compared: 9 numerical method based on using of nonlinear theory of shell together with method of finite differences, [8,9] created by Prof. G. Gavrylenko (NAS Ukraine, Kiev); 9 method of an analytical calculation suggested by Prof. J. Croll (UCL, London). Numerical method [6-9] uses the procedure of defining of minimal critical loads in compressive ribbed cylindrical shells having given initial imperfections of form with take in account an eccentricity of ribs and their discrete placing. As local so regular axisymmetrical and nonaxisymmetrical imperfections are investigated. The parameters of minimal critical load Pl = Per / P~ is determined (p~,~-criticalload of nonideal shell, pcl=0,605EtF/r, where F- area of cross section of skin ). The theoretical basis of analytical estimation of carrying capacity is expounded in articles [ 1-3] and named by authors reduced stiffness method - RSM. In Fig. 4 a representative shell [ 10] is examined; this has the following characteristics: 1/r=l; r/t=400; ds/ts=3; ts/t=3; k=40, where d s and t s are height and width of longitudinal stiffening elements, k- number of stringers. Plmin is found for indicated shell having axisymmetrical imperfections w 0 = B ~ sin mwx/1. Others type of imperfections give larger values pl 9In Fig.4 are represented: 9 two curve (m = 3 & m = 5, using a finite difference mesh having IxJ = 61 x31, continuous lines - solution of nonlinear problem (O - m=3, 13 - m=5) and dash lines- solution of linear problem ( O - m=3, II-m=5); 9 straight line- result received by RSM. Values p t m founded by numerical and analytical methods draw together when 1,2 < Bin, / t < 2,5. P1 1,5 1,25

J

. . ~

9

..

,Qo

o

"'t) ....

0...

0,75 0,6

i

0

~

1

i

i

2

i

i

lOmnl/t

Fig. 4 Comparison of minimum lower bound non-dimensional critical stress pl founded by RSM and numerical method.

651 4. CONCLUSIONS. The reduced stiffness method [ 1- 3] is extended to shells near to cylindrical form ( r/o ~ 0 ). Suggested and realized is an approach which allows definition of an analytical solution for lower bounds for the buckling of these shells. In result of calculations has been found: 9the values of classical critical stress of shells with positive Gaussian curvature are close to equal that for cylindrical shells; 9in contrast shells with negative Gaussian curvature have classical critical stresses that tend to zero with growth Ir / p J.

Despite of above mentioned deduction the following main result was received: o non-dimensional minimum lower bound critical stress found by RSM predict lower knockdown with growth of r / p for shells with positive Gaussian curvature;

9when r / t ratio tends to 2000 - 3000, values of the minimum lower bound stabilize at a constant asymptote;

9 for a longitudinally stiffened cylinders, lower bounds

of non-linear imperfection sensitivity studies are shown to be accurately bounded by the analytical predictions from the reduced stiffness method.

REFERENCES 1. Batista, R.C., (1979)Lower bound estimates for cylindrical shell buckling. P h . D . thesis, University College, London. 2. Croll J.G.A., and Batista R.C., (1981) Explicit lower bounds for the buckling of axially loaded cylinders. Int. J. Mech. Sci., 23, 331-343.. 3.Croll J.G.A., (1995) Towards a Rationally Based Elastic-Plastic Shell Buckling Design Methodology. Thin-Walled Structures, 23, 67-84. 4. Croll J.G.A., Gavrylenko G.D. (1998) Substantiation of Reduced Stiffness Method//International Journal ~Strength of Materials)), N5. 39-58. 5. Croll J.G.A., and Gavrylenko G.D., (1999) Reduced Stiffness Method in the Theory of Smooth Shells and the Classical Analysis of Stability (review). Strength of Materials, 31, N2, 138-154. 6. G.D. Gavrylenko, (1995) Stability of Smooth and Ribbed Shells of Revolution in a Nonuniform Stress-Strain State, Int. Appl. Mech. 31, N7, 501-520. 7. G.D. Gavrylenko, (1983) Basic Nonlinear and Linearized Equations of Imperfect Ribbed Shells of Revolution. Int. Appl. Mech. 19, N7, 610-614. 8. Gavrylenko G.D. Stability Ribbed Cylindrical Shells with Nonuniform Stress-Strain State. Kiev, Nauk. Dumka. (1989), 176. 9. Gavrylenko G.D. (1999) Stability of Imperfect Ribbed Shells, Kiev, Printed in Institute of Mathematics of NAS Ukraine.. 190. 10. Ellinas C.P., Croll J.G.A., & Batista R.C. (1981) Overall buckling of stringer stiffened cylinders //Proc. Inst. Civ. Engrs., Part II,, 71, June, 479-512.

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Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

653

ON THE ANALYSIS OF CYLINDRICAL TUBES UNDER FLEXURE F. Guarracino and M. Fraldi Dipartimento di Scienza delle Costruzioni, Universit~ di Napoli "Federico II", via Claudio, 21 - 80125 Napoli, ITALY

ABSTRACT In the present paper the collapse behaviour of infinitely long, cylindrical, elastic tubes under pure bending is investigated. This behaviour is characterised by a smooth global maximum on the load-deflection curve which is due to the well-known yon K~rm~in effect, that is to the progressive flattening of the cross section of the tube under bending moment. Even if this phenomenon has been extensively investigated in many classical works, nevertheless the comparison of the predictions from these approaches with experimental data shows that there are still some discrepancies which seem to be worth of further considerations. To this purpose a straightforward solution is presented which takes fully into account some effects in the deformation of the cross section which appear to have been neglected in the past and which, on the contrary, can give reason for several experimental findings.

KEYWORDS Cylindrical tubes, ovalization, bending strength, limit moment. INTRODUCTION The present work is concerned with the analysis of the bending strength and with the determination of detbrmation, stiffness and stresses of originally straight tubes. In fact, it is known that the cross section of thin walled beams deform during bending and that this deformation affects the bending strength giving occurrence to a flattening instability phenomenon. This phenomenon was firstly described and modelled by von Kmanfin (von K~rm~n, 1911) in the case of curved tubes and, successively, by Brazier (Brazier, 1927) and Chwalla (Chwalla, 1933) in the case of originally straight tubes. A large number of works have been published on this subject, both from a theoretical (e.g. Ades, 1957, Reissner, 1961, Fabian, 1977, Gellin, 1980) and an experimental point of view (e.g. Ellinas and Walker, 1985, Corona and Kyriakides, 1988) and several quantitative results for the non-linear behaviour of stresses and deformations in the tube have been made available, both in the purely elastic and in the elastic-plastic range. However, some of these results happen to differ quite significantly from each other (as it is the case, for example, of the limit moment derived from Brazier's paper and that derived from Chwalla's one) and many of them are not able to predict or give full reason of some experimental findings, so that, in the opinion of the authors of the present work, it is still worth to attempt to formulate a more precise formulation of this classical problem. Therefore, in this paper a straightforward modelling of the phenomenon is presented, and the relative solution is

654

obtained in a closed form. The analysis is carried out in the purely elastic range, but the findings can be usefully extended to the elastic-plastic range and this extension will be the object of a forthcoming work. The developed formulation leads to a clear explanation of the asymmetry of the deformed shape of the crosssection of the tube with respect to the neutral axis of flexure, which has been pointed out in some experimental works (Ellinas and Walker, 1985, Corona and Kyriakides, 1988). This asymmetric behaviour seems to have been overlooked in practically all the previous solutions of the problem, with the noticeable exception of the Brazier's work in which, however, the asymmetry deriving from the contribution of the St. Venant's solution of the flexure problem is first taken into account in the formulation of the variational problem and then discarded in the final solution. In the present paper it is shown that there are also other terms, apart from the St. Venant's ones, that contribute to the asymmetry of the problem and that the characteristics of inertia of the deformed cross-section are affected by this deformation mode. Finally, two numeric examples, one of which relative to a tube extensively tested at the University of Surrey (Ellinas and Walker, 1985), are presented. FORMULATION

OF THE PROBLEM AND RELATIVE SOLUTION

Let us start by taking into consideration the initial shape of the circular cross-section of a cylindrical tube, as shown in Figure 1. r is the mean radius and t denotes the thickness of the wall. /

y

M

/ M

/

M

Figure 1 Under the Bernouilli's hypothesis that the cross-sections of the tube remain plane on bending, we assume that the displacement field of the points of the cross section in its own plane results (see Figure 2) u=~'+~=

VCr 2

sin2~:+t7

2

v= V+ ~=-

(1)

VCr 2

2

cos 2~: +

where ~" and V are the linear components of displacement, yielded by the St. Venant's solution to the flexure problem, and t~ e F represent the non-linear part of the displacement field, due to the yon K ~ m ~ effect, that is to the ovalising pressure produced by the longitudinal stresses which give origin to a resultant directed towards the centre of curvature of the deformed element of tube, as shown in Figure 1. The usual linear bending moment-curvature relationship can be assumed to hold true

1 R

dip ds

c. . . . .

M El(c)

(2)

where R is the radius of curvature of the initially straight longitudinal axis of the tube, dip is the slope between

655

two the

sections

at

distance

t

ds

after

deformation

has occurred,

Y

M

is

the

bending

moment,

E

is

s

~

u

,

, "; 8

,

'

udr ~ "' --~fl

Figure 2 Young's modulus and I is the second moment of area of the section about its neutral axis. On account of the cross-section deformation in its own plane, I results a function of the curvature, that is I= I(c). The ovalising pressure p acting on the unit of area of the tube wall in the normal direction to the neutral axis, can be expressed as

p . dR.tc . . cr~ctrd~ . . . c2tEy c2trEsin~

ds

rd~

(3)

where dR.z is the force acting on the element of circumference ds = r sin~:, and o2 is the stress normal to the shell thickness, which can be evaluated by the classic Navier's formula. To the purpose of determining the expression of the non-linear displacement field, t7 e ~, we take into consideration the initially circular cross-section subject to the ovalising pressure p. As it is usual in problems of this type, we make the hypothesis that the system a e ~ is inextensional and that the shear deformation through the shell thickness is negligible. With reference to Figure 2, let us impose the equilibrium of a generic element of the shell in the deformed configuration, and write the compatibility relationships between the components of displacement and the slope fl(~) in the plane of the cross section, as well as between the slope fl(~) and the curvature Z(~:). We are thus led to the following set of differential equations

a=a,(g) rot(g) r ~ ag = + ' ~ ' ( g ) = - E J

ar (4) + :,, (r - - r fl(r

rz(4)- ~fl(4)

656 where tT,. and 12. are, respectively, the components of the non-linear displacement field in the normal and tangential direction to the undeformed cross-section, fiB(C:) is the bending moment acting through the shell thickness and J is the flexural stiffness of the shell, that is tadl

J =

We now define the load Q(r

(5)

12(1-v 2)

see Figure 3, as (6)

Q ( ~ ) = ds~c~ p " r " d v

Y P

Q(O

I

,@x

x

Q Figure 3 Quite straightforwardly, the expression for the shell bending moment, 9Yt(~:), results OB(~)

= Q.[r.(1-cos~)]-a(~).[r.(1

-

cos~:)-d]+X

(7)

~r

where X is an unknown parameter, Q = ds~j p- r - d ~ is the resultant of the vertical pressures, which equates the vertical reaction of the support, and d is given by ~r

d s . ~ j p . [r . ( 1 - c o s ~ ) ] d= ~ Q(r

. r . d~

= r'sin 2 2

(8)

The unknown parameter X can be obtained by means of the virtual work theorem, that is I ~ 9"A ( ~ ) . r " d~=O .IO

EJ

~

X

--- ~ ! c 2 t

. E . r3ds

(9)

4

Thus, the expression (7) of the shell bending moment, 9B(~), takes the form 93~(~) . ds ~: cos ~:) + sin2 "~] . .c2t. E. r 3 [ - ~1 + c~ ~" (c~ --2

(10)

We are now able to pursue the solution of the problem represented by the set of differential equations (4). From the first of the equations of the system (4), provided we take into consideration the following additional conditions,

657

\9, r

....

0

(symmetry of the displacements with respect to the y-axis)

(11)

(the displacement field is inextensional) we obtain

c2 r 5 _ V 2 t2r = - - - 7 ( 1 )- coseC:

(12)

and by integrating the second equation of the system (4), under the condition fi, (~: =0) = 0 , we have c2r 5 fi.~(~) = ,10 t~, (~)d~: = -2.-7(1- 1,'2 ) sin 2~

(13)

Given that we can express the displacement field in the reference frame x-y as fi = -fi, cos ~: - fi.~sin ~: = -ti r sin ~ + iT, cos ~

(14)

on account of the formulae (12) and (13), the equations (1), which fully describe the deformation of the genetic cross section in its own plane, take the form

u =

VCF2

c2r5

2 sin2~'--7(1-v~)/-[c~176

2

sin2~']sin~:}

Vcr2 t-c2r5 1 v = - - - - - - - c o s 2~:----7-(1-v2){-[cos2~:]sin~:-[ sin 2~:]cos~:} 2

(15)

Figure 4 Figure 4 shows the deformed shape of the cross section in the x-y plane. It is worth underlining that it results by no means symmetrical, contrary to what happens to practically all the solutions proposed in literature. As the value of the external bending moment grows, the square terms in the longitudinal axis curvature, c, tend to play a more significant role in the deformation of the cross section. Moreover, as the curvature c increases, the thickness t of the tube wall and the length ds of a generic element of its circumference become t ' = x][t +(G' -u~)] 2 + [u.~'-u~] 2 (16) ds '= x/[d(r cos ~:+ u)] 2 + [d(r sin ~ + v)] 2

658

where u~' and u~ are, respectively, the displacement components in the direction normal to the mid-surface of the shell of two points which, along the same normal, represent the intersections with the external and the internal boundary of the shell, u]' and u~ are the displacement components of the same points in the direction tangent to the mid-surface of the shell. After some algebraic manipulation, we can write

t'= t41 + c2v2r 2 + 2 c r v s i n ~ = V(c,~)-t ds '= ds~/1 + c2v 2r 2 + 2 c r v s i n ~ = V(c,r

(17)

ds

so that the shell circumferential strain, e,, results e.~ = - - - - ds

=v-i

(18)

This strain, according to the hypothesis made that the system fir and tT. is inextensional, is due to the linear part of the displacement field only. As a direct consequence of the asymmetry of the deformed shape of the cross section with respect to the x-axis, which can be described by means of the strain factor ~(c,~:), the position of centre of the mass, G, of the deformed section results displaced with respect to the origin of the axes x-y. The actual y-coordinate of G can be easily obtained by means of the formulae expressing the actual area of the cross section, A ', and of its first moment, S'. We have

A'=~ t'ds'=fci'~r.t.~2(~).d~ (19)

S '= ~st'. (y + v)ds '= ~ci'r.t. ~r2(~). [r sin r + v(r so that it results

, y~ . S A'

ft21r r . t . ~ 2 ( ~ ) . [ r s i n ~ + v ( ~ ) ] . d ~

. ~

.

. . ~,i=r.t.~2(r

.

= ( c r 2 v)

[1- 3 " c Z ' r 4 " ( 1 - v z ) ] 4t2 [l+(c.r.v)2l

(20)

The actual value of the second moment of the area, l(c), which appears in the moment-curvature relationship (2), is l(c) = I,, - A ' y 6 =

r.t.

(~).[rsin~+

-

Y6

9

,

r t

(~:) -d~:

(21)

that is, in a closed form,

l(c) =

[I,, + ~ f i ( r , t , v ) . c 2k ] ~=' [1 + c 2- v 2- r 2]

(22)

where

f ~ ( r , t , v ) = ~~. .{/5. 5. t 2 . v 2 - 6

9r 2 -(1- V 2 )}

f 2 ( r , t , v ) = 8.---~3.{20 ~'r7 . t 4. v. 4 . r 2 (1-v2)-[8 - t 2 91,,2 - 5 - ( 1 - v 2)]} (23)

L - (r,t,v) = x" 8./3 r 9 . V.~ 2 "{2"t4

.V 4

_ r z . ( 1 - v 2 ) . [ 2 0 . t 2 .v 2 - ( 1 - v2)] }

/~'. r 15 . V 4

f , ( r , t , v ) = t----.--.--y~- 5 - (1 - v2) 8. Finally, equation (2) and equation (22) allow us to write the relationship between the external bending moment, M, and the longitudinal axis curvature, c, in the following form

659

[l,,+~fi(r,t,v).c

zk ]

k=i [1 + C2- V2- r 2]

M(c)=E.c.I(c)=E.c

(24)

This is a compact and quite handy expression, formally similar to that proposed by one of the present authors under the assumption that the deformed shape of the cross-section could be approximated to an ellipse (Guarracino and Minutolo, 1996). It allows the direct evaluation of the limit bending strength of the tube as a function of the curvature of its longitudinal axis, as well as of its actual stiffness. Its simple expression can turn useful in the analysis of more complicated problems as, for example, pipelaying in deep waters (Guarracino and Mallardo, 1999). NUMERICAL EXAMPLES AND CONCLUSIONS We consider the cases of two different tubes, with D / t ratios equal to 500 and 14, respectively. Both tubes are high grade steel made, with E=2.07 • d a N / c m ~"and v=0.3. The first case is that of a very thin tube, characterised by an overall diameter D= 100.2 c m and a wall thickness t=0.2 c m , whose collapse can be obtained experimentally for suitable lengths in the purely elastic range. The bending moments vs. curvatures plots, see Figure 5, show the results from the present formulation as a continuous line and the results from the Brazier's solution as a dashed line. The attainment of the limit moment results evident in both contexts, with a difference of about 5% in the value of the maximum bending moment (Mlim=4.60425X106 d a N c m according to the present formulation and MHm=4.3854x106 d a N c m according to the Brazier's solution) and about 9% in the value of the critical curvature (c=4.39443x10 -~ c m -~ according to the present formulation and c=4.03871 x l 0 -5 c m 1 according to the Brazier's solution). M

5xlO ~

l

4xlO ~ 3•

I

/

\ '\\

6

j

\

'\, \

2x10 6

"'\

\

I

\\\

\

IxlO ~

! \ \

/ ,

. . . .

L

0.~2

,

,

.

|

0.~4

.

,

,

~

0.00006

,

,

.

/ ~

0.00008

J

,

.

i

C

0.0001

Figure 5 Similar findings were obtained in the second case, which is the case of a pipeline extensively tested at the University of Surrey-UK (Ellinas and Walker, 1985), characterised by an overall diameter D= 16.2 c m and a wall thickness t= 1.2 cm. It must be noticed that, as it is the case for the vast majority of tubes of practical interest, the actual collapse of this tube cannot be obtained in the purely elastic range. Nevertheless, in order to perform a comparison in the elastic range, we plotted the results from the present formulation as a continuous line and the results from the Brazier's solution as a dashed line, see Figure 6. After the proposed solution, the limit moment results M~im=2.60323• d a n c m in correspondence of a critical curvature c=1.070205x10 2 c m -I. After the Brazier's solution, we have Mlim=2.47863• d a N c m and c=0.983094x10 2 c m ~ It can be concluded that, according to the present formulation, the values of the limit bending moments, as well as those of the critical curvatures, do not differ significantly from those deriving from the Brazier's theory in the elastic range. This is meaningful, because Brazier's results have received a significant experimental validation in the elastic range in the past seventy years. However, it is noticeable that the shape of the deformed cross section deriving from the present analysis differ

660 significantly, to the best of the author's knowledge, from those proposed in other works. The asymmetry of the deformed shape gives origin to a difference of about 25% between the maximum value of the longitudinal compression and the maximum value of the longitudinal tension in the vicinity of the limit bending load in the first of the cases considered above (tf=-3305 daN/cm 2, ot=2648 daN/cm2). This fact, apart from being in accordance with the experimental findings, gives also reason for the different locations of the plastic zones under bending, as well as for the onset of the bifurcation buckling of the ovalised tube (Fabian, 1977).

I

4xlO 7

f / I

3xi0 7

2xlO

I

7

I/

\\\\\

IxlO ~

O.005

O.01

O.015

I O.02

O.025

Figure 6

References Ades, C.S. (1957). Bending strength of tubing in the plastic range. J. Aero. Sci., 24, 605-610. Brazier L.G. (1927). On the flexure of thin cylindrical shells and other thin sections. Proc. Roy. Soc. A, 116, 104-114. Chwalla, E. (1933). Reine Biegung schlanker, diinnwandinger Rohre mit gerader Achse. ZAMM, 13, 48-53. Corona, E. and Kyriakides, S. (1988). On the collapse of inelastic tubes under combined bending and pressure. Int. J. Solids Structures, 24, 505-535. Ellinas, C.P., Walker, A.C. et al. (1985). A development in the reeling method for laying subsea pipelines. Proc. 1'~tPetr. Tech .Austr. Conf., Perth, Australia, 26-29 November. Fabian, O. (1977). Collapse of cylindrical elastic tubes under combined bending, pressure and axial loads. Int. J. Solids Structures, 13, 1257-1270. Gellin, S. (1980). The plastic buckling of long cylindrical shells under pure bending. Int. J. Solids Structures, 16, 397-407. Guarracino, F and Minutolo, V. (1996). Analisi della ovalizzazione di condotte circolari in regime di spostamenti finiti. Scritti in onore di M.Ippolito, Ass. Idrot. It., Napoli (in italian). Guarracino, F. and Mallardo, V. (1999). A refined analysis of submerged pipelines in seabed laying. Applied Ocean Research, 21,281-293. Reissner, E. (1961). On finite pure bending of cylindrical tubes. Osterr. Ing. Arch., 15, 165-172. yon K~rm~in, Th. (1911). Ueber die Form~inderung dtinnwandinger Rohre, insbesondere federnder Ausgleichrohre. Zeitschrifi des Vereines deutscher Ingenieur,. 45, 1889-1895.

Third InternationalConferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

661

BUCKLING OF ABOVEGROUND STORAGE TANKS WITH CONICAL ROOF Luis A. Godoy and Julio C. Mendez-Degr6 Department of Civil Engineering, University of Puerto Rico, Mayagiiez, PR 00681-9041, Puerto Rico

ABSTRACT The buckling of aboveground circular steel tanks with conical roof is considered in this paper. The specific source of loads investigated is wind action during hurricane storms in the Caribbean islands. The structure is modeled using a finite element discretization with the computer package ALGOR. Bifurcation buckling of the shell is computed for a given static wind pressure distribution. Then the bifurcation loads and buckling modes are compared with the evidence of real tanks that failed during hurricane Georges in Puerto Rico in 1998. Several pressure distributions are assumed for the roof of the tank, and it is shown that the results are highly sensitive to the choice of pressures.

KEYWORDS ALGOR, Bifurcation analysis, buckling, finite elements, hurricane winds, metal tanks, mode shape, shells, wind pressures. INTRODUCTION The failure of tanks employed to store water and oil in the Caribbean Islands has been studied in recent years by the first author. Buckling of aboveground circular steel tanks with and without a roof was observed in St. Croix in 1990 (hurricane Hugo), St. Thomas in 1995 (hurricane Marilyn), and in Puerto Rico in 1998 (hurricane Georges). For tanks without a roof, or for those that lost the roof before the cylindrical part buckled, it was possible to reproduce the expected behavior using computer modeling. For example, a tank without a roof that failed in St. Thomas was modeled using standard wind pressured distributions around the circumference (Flores & Godoy 1997). For such a pressure, the computer model displays buckling for the wind speeds usually found during a hurricane. However, a far more difficult job is faced in an attempt to model the failure of the cylindrical shell in a tank with a conical roof or a shallow dome. The main questions regarding the pressure distributions on the roof are not answered within the current state of the art. In this paper we employ computer modeling to identify adequate pressure distributions that are compatible with the structural evidence showing buckling due to hurricane winds.

662 The search for adequate modeling of loads due to natural hazards is of great importance for the prediction of the safety of structures. There are various ways in which this is done at present. On the experimental side one can perform a full-scale test on a real structure or instrument it until an event occurs. The use of small-scale models is another possibility. Computer modeling of the environmental action on the structure is now possible, specially thanks to advances in Computational Fluid Dynamics. And there is the possibility of linking the failure of uninstnmaented structures to the loads that led to their failure. This work explores the last possibility within the context of tanks exposed to hurricanes winds. For tanks with a roof (either conical, spherical, or flat), the literature on wind load buckling reduces to a few contributions. Some extensive books on the design of tanks (Myers 1997, Ghali 1979) do not consider buckling under wind load. Early studies in the 1960s on the wind pressures on cylinders with shallow cap roofs were reported by Maher (1966), while an extension for fiat roof was published by Purdy, Maher & Frederick (1967). The literature on silos may also be relevant for short tanks; however the aspect ratio of tanks (height to diameter) of interest in this work is of the order of 1/5, while for silos this is higher than 1. Esslinger, Ahmed & Schroeder (1971) investigated silos with a dome roof under wind. This work is also discussed by Greiner (1998). Other wind tunnel tests were performed by Resinger & Greiner (1982), but there is no information about pressure distributions on the roof. The influence of group effects in silos under wind was reported by Rotter, Trahair & Ansourian (1980) from buckling experiments in a wind tunnel. The silos tested were aligned in a single row perpendicular to the direction of the wind, and closely spaced. Esslinger, Ahmed & Schroeder (1971) tested two cylinders with spherical cap roof separated by a distance of the order of a diameter, and for various directions of wind incidence. For two tanks aligned in the direction of the wind, the first tank shelters the second one, and develops pressures on the windward side, suction close to 90 degrees from the wind direction, and small pressures on the leeward side. The roof has suction on a small part of the windward side, and pressure on the leeward side. Such non-uniform pressure distribution on the roof is also found for other orientations of wind. The influence of an internal operating vacuum may further modify the wind pressures. Flores and Godoy (1997) studied the nonlinear dynamic response of short tanks and found that inertia effects were not significant in this class of shells, so that static analysis could well be carried out to estimate instability under wind load. In the following sections we describe the computational finite element model employed and consider a specific structure to investigate its buckling failure. Several pressure distributions are assumed for the roof and the results of buckling pressure and mode shape are compared with the evidence from real cases. COMPUTATIONAL MODEL A metal circular tank with a conical roof under a static pressure distribution (Figure 1) is modeled in this paper in order to evaluate bifurcation buckling loads and modes. Because there is evidence that the real tank failed during hurricane Georges in Puerto Rico, then we have an upper limit to the wind velocity at the time of buckling. The buckling mode observed in the real tank includes plasticity effects associated to an advanced post bucking behavior; however, the localization of the mode at the windward meridian and the mode shape are considered similar to the initiating elastic buckling mode. Preliminary studies using geometrically nonlinear analysis indicate that instability in this case has small displacements in the pre buckling equilibrium states and that the maximum load and mode attained by the tank are well represented by a bifurcation study. This is by no means a general conclusion and applies only for the geometry of the tanks considered, which is rather short. At least for open tanks, it has been shown (Godoy & Flores 2000) that the aspect ratio and thickness slenderness are crucial to determine the type of instability that may be found in the structure.

663 The general purpose finite element package ALGOR (1999) was employed to build the computational model using 2250 shell elements, including 1250 quadrilateral elements for the cylinder and 1000 elements (either quadrilateral or triangular) for the roof. The boundary conditions at the bottom of the shell are assumed as clamped. Only one-half of the tanks is modeled, with symmetry in the plane of incidence of wind. The mesh of elements is shown in Figure 2. A series of bifurcation buckling analysis were made in order to identify adequate pressure distributions on the roof, which are compatible with the evidence observed in real structures. This was accomplished by calculating the critical bifurcation buckling mode and pressure and comparing them with the maximum load expected to occur during a hurricane. CASE STUDIED The theme structure in this paper is a tank with variable thickness, as shown in Figure 1. The tank has 30.5 m of diameter and a height of 12.2 m. It is made of steel with the assumed properties listed in Table 1. The tank is located in Pefiuelas, an industrial area in the south of Puerto Rico, where a large number of tanks were build in order to store petrochemical products for the many industrial companies that developed several years ago. TABLE I DATA ASSUMEDIN THE ANALYSIS

Figure 1. Dimensions of the tank.

Figure 2. Finite element mesh.

This tank was damaged by winds during hurricane Georges in 1998, as shown in Figure 3. The tank was empty at the time when the hurricane hit the area; it belongs to a group of tanks that are separated by a distance of approximately 50 m, and its location is close to the coast (about 300 m from the coast).

664

Figure 3. Tank investigated that buckled during hurricane Georges in 1998. ASSUMED PRESSURE DISTRIBUTIONS

It was assumed that the pressure distribution around the circumference has positive values on the windward meridian, and negative pressure (suction) on the rest of the cylinder, and was modeled as a constant unit pressure in the vertical direction. This pressure pattern was used by many authors before (ACI-ASCE 1991, Flores & Godoy 1998, 1999). In order to propose a pressure distribution for the roof of the tank we used the photographs as a guide to how the tank buckled under wind load. For this purpose we used an inverse technique of cause and effect, in which the photographs showed the effect of the hurricane winds. Once the buckling mode shape was available, we attempted to find the pressures that caused this failure. Several bifurcation buckling analyses were made using ALGOR for different pressure distributions on the roof. Then the buckling load was calculated for each case. Every load assumption used caused a different buckling load and mode shape in the structure. Finally, we searched for a buckling load similar to the load expected during a hurricane that caused a buckling shape similar to the one photographed just after the hurricane occurred. RESULTS FOR DIFFERENT PRESSURE DISTRIBUTIONS ON THE ROOF

In Load Case 1 the tank was modeled by taking the influence of the roof with upper boundary conditions instead of the roof itself. This class of models is attractive because one does not model the roof with finite elements, but it neglects the pressures that may act on the roof surface. First, the cylindrical tank clamped at the top was considered. This lead to a bucklin~ mode consistent with what was observed in the structure, but for a high load factor o f k c = 3.35 kN/m" (or wind speed of v = 66.9 m/s). For a simply supported condition on top, the values changed to ~c = 3.33 kN/m 2. This model shows a buckling mode shape similar to the real mode. For a free condition at the top it is not expected that the model simulates a roof, and is only considered here as a reference case. The load factor changed to ~c = 1.3605 kN/m 2, but the mode shape from such computations was very different to the one expected. Load Case 2: In order to include the influence of the roof into the model we assumed different patterns of wind pressures acting on the roof. As a first option, we included a downward constant pressure as a percentage of the maximum pressure (1 kN/m 2 ) applied on the walls. The distribution of pressure (ACI-ASCE, 1991) applied around the circumference has a maximum value of 1 kN/m 2. For this load

665 case we obtained the results plotted in Figure 4. The buckling mode of each of these cases was very similar to the real one, as shown in Figure 5.

Figure 4. Load Case 2

Figure 5. Load Case 2.

Load Case 3 was a variable pressure load acting on the roof. It was assumed that the meridian of the shell has pressures with the same sign, and following a circumferential distribution similar to a tank without a roof. This pressure load on the roof has the same orientation as the pressures acting on the walls, and we call this a "variable pressure" because it varies in the circumferential direction. The magnitude of the pressures varies according to the direction with respect to the incidence of the winds. The positive pressures produce a downward pressure on the roof, while negative magnitudes of pressure produce an upward pressure (suction). The scope of this analysis was the same as in Load Case 2, varying the percentage of the pressure load to study the variation of the critical load, Figure 6. In this case the critical buckling loads were very high, exceeding the expected range.

Figure 6. Load Case 3.

Figure 7. Load Case 4.

Load Case 4: Consultations with wind-experts led to the recommendation that an upward pressure should be present, following the experience with pressures on buildings with rectangular plan. For this case we applied a constant upward (negative) pressure to the roof. In this model only suction occurs on the roof. With full maximum constant pressure (lkN/m 2) acting upward on the roof the buckling load was -2.98 kN/m 2. With 90% of the maximum constant pressure acting the buckling load was-1.83 kN/m 2. Negative values of critical buckling load mean that the pressure distribution should be applied in the opposite direction. The buckling mode for this case was very interesting: The roof suffered large buckling deflections instead of the walls, and the buckling mode shows a totally different shape, Figure 7. This shape is far from what is observed in the real situation.

666 Load Case 5: Wind-tunnel experiments in Germany (Esslinger, Ahmed & Schroeder 1971) have shown that for silos, the pressure on a conical roof is negative on the windward part of the roof and positive on the leeward part. At the center of the conical roof the pressures are zero, and they take non-zero values on a ring which spans half way between the center and the edge of cone. Our cases 5 and 6 take that into account. For Load Case 5 we applied a variable pressure but with the "inverse orientation" of the pressure acting on the walls (Greiner, 1998). That means that the roof is modeled with positive pressure in the places where the wind is negative on the walls, and negative (suction) pressure in the places where the walls have positive pressures. The critical buckling loads are similar to the expected load corresponding to the wind speed measured for the area in which the tanks are built. The variations of the critical load due to the increase of variable pressure load are plotted in Figure 8. Notice that this behavior is similar to what was obtained for Load Case 2. Furthermore, the modes of buckling computed were also similar to those expected in comparison with the photographs (Figure 9).

Figure 10. Load Case 6. Load Case 6: We assumed that the pressure acting on the wall only affect some part of the roof. In this case the pressure was applied over the circumference of the roof to a distance of one fourth of the diameter of the roof measured from the comer (the junction between the roof and the wall). This distribution left the middle section of the roof without any pressure. The orientation of the load was the same assumed for the Load Case 5. The critical load computed by ALGOR was 3.03 kN/m 2. The buckling mode shape for this case is also similar to those obtained from Load Cases 2 and 5. The buckling mode shape corresponding to this distribution is shown in Figure 10.

667 DISCUSSION For the first load case (Load Case 1) the tank was modeled using three different boundary conditions instead of the roof. For the tank clamped at the top the buckling load was larger than the expected range of values (wind speed larger than that registered in the area in which the tank is built), although the buckling mode shape was similar to the one expected. The same situation occurred with the tank modeled with a simply supported condition at the top. For a free condition at the top the buckling load was smaller than the other two cases but the buckling mode shape was totally different of the one expected. In Load Case 2, a downward constant pressure was applied on the roof. The results obtained vary depending on the magnitude of the pressure used. As discussed before, the magnitudes considered vary in percentages of the maximum constant pressure (lkN/m2). For 50% up to 120% of the maximum constant pressure the critical buckling load resulted in values very close to the one expected. These values decrease as the percent of full pressure acting on the roof increases. The estimated range of the velocity pressure for an open area is between 2.40 kN/m 2 and 3.00 kN/m 2, corresponding to wind velocities from 56 m/s up to 67 m/s (125 mph up to 150 mph). The values between 10% and 40% lead to high wind speeds. In terms of the buckling mode shape, the results obtained for the different magnitudes of pressure were similar to the shape expected according to the photographic evidence. For Load Case 3, in which the applied load was a variable pressure acting on the roof, the results were not satisfactory in terms of the buckling load. That means that if we applied on the roof a distribution that varies according to the direction of winds (positive pressures on the cylinder produce a downward pressure on the roof, while negative magnitudes of pressure on the cylinder produce an upward pressure on the roof) we obtain very large values of the critical buckling load. These values also changed the sign if we increase the pressure to more than 50% of the value of the pressure on the cylinder. Such pressure distribution is not recommended in this context. For Load Case 4 (constant upward pressure on the roof) we obtained negative values of buckling load, which means that the distribution used in this case was not satisfactory. This case considers also the distribution used on the walls, and for this reason this is not a good distribution to model the wind. The buckling mode shape obtained was very different from the one expected, i.e. the tank model failed in the roof instead of the walls. In Load Case 5 we applied a variable pressure distribution but with the inverse orientation of the pressure acting on the walls (Esslinger, Ahmed & Schroeder 1971, Greiner 1998). The critical buckling load computed in this case was similar to that computed in Load Case 2 and the results are also satisfactory. The modes of buckling were also similar to those obtained for Load Case 2. For Load Case 6, which is a variation of Load Case 5, we obtained a critical buckling load just larger than the one obtained for the 100% of the load in Load Case 5. The value was over the expected range, but very close to the maximum load expected. The buckling mode shape was similar to those modes obtained for load cases 2 and 5. CONCLUSIONS The pressures distributions obtained for silo structures with shallow cap roof by Esslinger and coworkers (1971) seem to be an adequate representation for short tanks with conical roof. In this case the pressure distribution on the roof is not uniform, but has suction close to the meridian of incidence of wind, and positive pressure on the leeward side of the shell, while the central part of the roof does not have pressures. For such pressure distributions the buckling pressure (and associated wind velocities)

668 and the buckling mode shape were consistent with those found in the field following hurricane Georges in 1998. This preliminary study illustrates the sensitivity of the buckling response with the pressure distribution assumed on the roof. More detailed experimental evidence of pressure distributions for short tanks is required, and this could be obtained from wind-tunnel experiments. Such studies are now being performed at the University of Puerto Rico at Mayagtiez. ACKNOWLEDGEMENTS

This work was sponsored by the US National Science Foundation (NSF) under grant CMS-9907440, and by the US Federal Emergency Management Administration (FEMA) under grant PR-0060-A. The support of both institutions is greatly appreciated. REFERENCES

ACI-ASCE Committee 334 (1991) Reinforced concrete cooling tower shells: Practice and Commentaries. American Concrete Institute, New York. ALGOR (1999) Linear and nonlinear static and dynamic finite element stress analysis. ALGOR lnc., Pittsburgh, PA, USA. Esslinger M., Ahmed S. & Schroeder H. (1971). Stationary wind loads of open-topped and roof-topped cylindrical silos (in German). Der Stalbau, 1-8. Flores F. G. & Godoy L. A. (1998) Buckling of short tanks due to hurricanes, Engineering Structures, 20(8), 752-760. Flores F. G. & Godoy L. A. (1999) Forced vibrations of silos leading to buckling, Journal of Sound and Vibration, 224(3), 431-454. Ghali A. (1979), Circular Storage Tanks and Silos, EFN Spon, London. Godoy L. A. & Flores F. G. (2000). Imperfection-sensitivity of wind-loaded tanks. Submitted for possible publication. Greiner R. (1998). Cylindrical shells: wind loading. Chapter 17 in: Silos (Ed. C. J. Brown & L. Nilssen), EFN Spon, London, 378-399. Maher F. J. (1966) Wind loads on dome-cylinder and dome-cone shapes. ASCE Journal of the Structural Division, 92, 79-96. Myers P. E. (1997), Aboveground Storage Tanks, McGraw-Hill, New York. Purdy D. M., Maher P. E. & Frederick D. (1967). Model studies of wind loads on fiat-top cylinders. ASCE Journal of the Structural Division, 93,379-395. Resinger F. & Greiner R. (1982). Buckling of wind loaded cylindrical shells: Application to unstiffened and ring-stiffened tanks. In Buckling of Shells, E. Ramm (Ed.), Springer-Verlag, Berlin, 305331. Rotter J. M., Trahair N. S. & Ansourian P. (1980). Stability of plate structures, Proceedings, Symposium on Steel Bins for Bulk Solids, Australian Institute of Steel Construction / Australian Welding Research Association, Sydney, 36-42.

Third International Conference on Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

669

ON THE COLLAPSE OF A REINFORCED CONCRETE DIGESTER TANK Luis A. Godoy and Sandra Lopez-Bobonis Department of Civil Engineering, University of Puerto Rico, Mayagtiez, PR 00681-9041, Puerto Rico

ABSTRACT The investigation following the collapse of a large reinforced concrete dome, which was part of a digester tank, is presented. The shell was constructed in 1987, and had construction errors related to the location of the single layer of reinforcement, which were discovered as a consequence of the collapse. The shell collapsed without the occurrence of any natural hazard. It is believed that a high internal pressure developed on the day of the collapse because of a problem with a valve, which allowed the discharge of large quantities of sewage inside the tank and filled the structure completely. A finite element analysis of the structure shows the stress levels in the structure, and support the hypothesis of a failure mechanism which coupled the construction errors with the internal pressure. Finally, the strengthening of the shell with externally bonded fiber composite sheets is described as a possibility to improve the safety of other tanks in similar situations.

KEYWORDS Collapse, composite materials, construction errors, digester tanks, reinforced concrete, shells. INTRODUCTION The failure of reinforced concrete shells has been reported in a number of cases, as mentioned in the texts by Billington (1990), Godoy (1996), Gould (1999), and others. For example, Ballesteros (1978) reported the collapse of an elliptical paraboloidal shell during the process of removing the formwork; the structure showed clear imperfections in the geometry and had construction defects. Many cases of reinforced concrete tanks (or similarly shaped structures) that fail due to structural or construction problems are not reported in the open literature; however, this lack of publication does not help other researchers to learn lessons from failures. This paper reports on the collapse of a large reinforced concrete shell dome, which was part of a digester tank, under an internal pressure produced by the accidental filling of the tank. The change in the operational conditions resulted in a limit state for which the tank was not adequately reinforced. Possible ways to repair existing tanks in the same plant with composite sheets are also considered.

670 THE SHELL STRUCTURE INVESTIGATED The digester studied here is sufficiently far from other digester tanks in the same facility so that it can be considered as an isolated structure. The structure is a closed reinforced concrete containment structure formed by a cylindrical shell, a sector of a spherical shell roof, and a conical shell at the bottom, as shown in Figure 1. Part of the structure was built underground. The horizontal radius of the cylinder is r0 = 14.5m, the height is 10m, and the wall thickness is 0.5m. According to the plans of the design, the cylinder was reinforced with two layers of steel (#6 bars at 250mm in the vertical direction and #9 bars at 38mm in the horizontal direction).

Figure 1. A typical digester tank considered in this study.

CL

11 0.228m #,1@20

0.30rnT ~ .1" .

#4Q20cm

0.50m

3 m

!

11.5

,

Figure 2. Reinforcement of the dome of the digester tank. The dome is a spherical cap (Figure 2) with radius of curvature R = 30.8m and thickness t = 228mm, so that R / t = 135. This is considered a thin shell for reinforced concrete, and should have been designed according to the ACI provisions already published at the time of design (ACI Committee 334, 1986). As a reference value, large reinforced concrete cooling towers have R/t of the order of 150. The dome is connected to a cylinder by means of a ring. The maximum elevation of the dome with respect to its

671 supports on the ring is approximately 2.90m. A PVC liner was attached to the bottom surface of the dome. The steel section in both directions was found to be #4 at 200mm, leading to As = 645mm2/m. For the area of concrete Ac, the ratio As / Ac -- 0.28% is a low amount of reinforcement for this class of shells. The structure was constructed in 1987. The dome should have been designed to resist primary compressive forces due to self-weight and accidental loads, but sufficient bending capacity should have been provided for situations other than gravity load. For this shell, which is exposed to an aggressive environment, it would be expected to have a 38mm concrete cover.

Figure 3. Partial view of the dome after the collapse.

Figure 5. Details of damage of the shell.

Figure 4. Location of the reinforcement.

Figure 6. Details of cracks in the shell.

CONDITIONS OF THE STRUCTURE PRIOR TO THE COLLAPSE Figure 3 shows the structure following the collapse of a large part of the dome. Because part of the structure did not collapse, it was possible to observe some details of the construction: (a) The single layer of the reinforcement was not placed at the center of the thickness, as indicated in the drawings, but it was displaced to the bottom surface of the shell. This has important consequences for the membrane and bending resistance of the shell.

672 (b) The concrete cover on the bottom surface of the shell was not sufficient. For this shell, which was constructed to operate in an aggressive environment, the concrete cover was found to be only 12.7mm and perhaps less in some zones. This is illustrated in the photograph of Figure 4. (c) Furthermore, the concrete cover in the zone close to the supports of the dome on the top surface of the dome was not adequate: in some parts of the shell it was possible to see the steel bars in the meridional direction. (d) Corrosion of the steel bars had occurred for some time. This could be seen at many places on the external surface of the dome. Corrosion has the consequence of reducing the effective diameter of a steel bar in a localized way. (e) Meridional cracks are clearly visible on the external surface of the dome (Figures 5 and 6), even in parts of the shell sufficiently far from the area that collapsed. Those are not new cracks formed as a consequence of the collapse, but were formed some time ago. Cracks larger than 3mm in the meridional direction occur at a spacing of about 2m. This reduces the bending capacity of the shell for negative hoop moments. (f) Circumferential cracks on the external surface are visible. Again, this reduces the bending capacity of the shell for negative meridional moments. Both meridional and circumferential cracks reduce the tensile capacity of the shell. (g) There are some fiat parts of the shell between meridional cracks, with the consequence that the shell had rotated taking the cracks as hinges. SEQUENCE OF EVENTS LEADING TO THE COLLAPSE OF THE DOME There was no report of high winds, earthquake, or small amplitude ground motion on the day of the collapse. However, a problem was reported on a valve, which caused the filling of the digester up to the top, with large internal pressures acting on the dome. A worker observed that material stored in the digester was being discharged at the top of the roof. A large crack (l.5m) formed on the external surface in the meridional direction, and large quantities of sewage material started flowing through the crack, coming from inside the digester. Next, a bulge formed close to the crack, and extended for at least 2m in the circumferential direction. The amplitude of the bulge (the elevation with respect to the external surface of the shell) was estimated to be at least 0.10m. The 3m part of the shell that had double reinforcement remained in the structure and the central part of the dome collapsed towards the inside of the digester. No explosion was reported. All debris were found inside the digester. STRUCTURAL CONSIDERATIONS The main loading condition of the dome shell is self-weight, leading to compression in two directions. The self-weight of the structure produces in-plane stresses of the order of 363KPa. Such stresses are small compared to the compressive strength of the concrete, f'c = 20.7MPa. Even if the weight is increased by accidental loads, the stresses are still small. Failure of concrete under compression is ruled out as a main cause of the collapse. A buckling load of the dome was estimated (Billington 1990, pp. 320) using E = 21.2GPa, leading to a critical pressure pC = 360KPa. This pressure is much higher than any pressure associated to gravity action. Thus, it is not likely that buckling under the main compressive (membrane) action led to a limit state in the shell. It has been shown that for thin-walled shells geometric distortions produced by various causes may induce bending stresses of the same order as the primary stresses. The collapse of several large reinforced concrete shells has been attributed to this effect. A review of several cases and their causes is reported in Godoy (1996). This dome clearly had significant geometric distortions, as reported by engineers previous to the collapse. Since no measurements of the actual shape of the dome were performed, it is difficult to assess the amplitude and extent of such distortions. At the time the shell

673 showed signs of a critical condition, an engineer reported a bulge with amplitude of about 0.10m and extending with a diameter of at least 2m. This may be a significant source of stress concentrations in the shell. But for the reinforcement present in the shell, an internal pressure may be the triggering cause of the collapse. Notice that a reinforced concrete spherical cap is extremely efficient to resist self-weight because it can develop compression; however, if the load is reversed the dome becomes a most inefficient structure under tension. Had the reinforcement been placed on a single layer at the center of the thickness, a tensile force T could develop due to the contribution of the steel section As. The concrete could have taken a compressive zone at the lower part of the thickness, to produce the required bending and thus equilibrate the internal pressure p'. Cracks were clearly present in the dome, in both directions. Such cracks were not new, and may have been produced by a variety of reasons, including the early life of concrete, thermal action, and others. A factor that must have played a role in the crack formation is the position of the reinforcement, which was displaced towards the bottom of the cross section for some unknown reason. In the real situation, the as-built shell has the steel reinforcement on the inner side. Under bending produced by the reversed load, the tensile force T had to be very large since the distance between the tensile and the compressive forces is small. Furthermore, the compressive force C developed by the concrete small section (the concrete cover) must be extremely high. For a steel with yield stress a y = 450MPa, a limit state could be reached with an internal pressure higher than the self-weight of the dome. FINITE ELEMENT STRESS ANALYSIS A shell with dimensions similar to the central part of the dome, for which a single layer of reinforcement was present, has been studied using a finite element model. The structure was considered as an axisymmetrie solid with quadrilateral elements, as illustrated in Figure 7. The data assumed is shown in Table I.

6 ,L2:_:: ~.,-._3__

~ ...........

1

Figure 7. Dimensions of the dome investigated in the analysis. The stresses and displacements of the shell have been computed under an internal pressure to simulate the influence of the sewage at the time a valve permitted the filling of the tank under pressure. It is not known the value of the pressure that was induced by the sewage, so that a reference value of 0.30 times the self-weight is adopted for the computations. Since this is an elastic analysis, the pressure should be scaled to evaluate a limit state.

674

TABLE I PROPERTIES FOR CONCRETE AND STEEL

Properties

Concrete

Steel

E, Elasticity Modulus

20.68 GPa

200 GPa

Mass density

2402.7 kg/m3

7861.4 kg/m3

v, Poisson ratio

0.15

0.29

Thermal Coefficient

.0000108

.0000117

Shear modulus

9.0 GPa

77.2 GPa

v. ";5

,,,

!

[

i ,

0.1

~zr

0-

- oooo

!

.

i

]

15~00

200 ~ 0

!

!1 oloo '

]

i

I

!

:

2 5 1 ~ ~

350'000

; -0.05 ~ !

-v.'

!

1 i

: i - vr ~. 4|~,

t

'

,

t

Stresses (N/m2) r

E=Econcrete/10--e--E=Econcrete/3

-"

E=Econcrete(2/3)

•

E=Econcrete

Figure 8. Stresses in the dome with and without deterioration o f concrete. o.0008

i Econcrete*l/10

0.0007 ~....,,~.~-~.~

g

le

~. 0.0005

Ec~

E

,

i f

" ~ ~ ~ , ~§ Econaete'2/3

Econ~me 2.07E§

a

!

I

i

'

0.0000 2.07E*08

5.21E*09

I 1.02E+10

!

I 1.52E+10

, 2.02.E+10

2.52E+10

Elasticity M o d u l u s (NIm'L2)

Figure 9. Displacements in the shell for various conditions o f the concrete.

675 The stresses in the meridional direction are shown in Figure 8 for E = Ec. This is a linear response including membrane and bending action. For the perfect shell to crack under internal pressure p', the value of pi should be 3.65 times the self-weight of the shell. However, there are clear signs that the shell had serious deterioration and cracking prior to the occurrence of internal pressure. To investigate the elastic stresses in the shell including damage of the concrete, parametric studies are shown in Figure 8 for several values of the modulus of elasticity. The actual area affected by damage was assumed over the central part of the dome, extending one-third of the total arc in the meridional direction and on the top half of the thickness. The results depart from classical shell theory assumptions, i.e. a linear distribution of stresses through the thickness is lost, with a significant reduction of stresses on the top part. To compensate for that, the stresses in the lower part of the thickness are largely increased, in order to maintain the net tensile force on the overall section (membrane contribution). In the deteriorated section with E = 0.1 Ec the tensile stresses increase by 40%. The bending moment, however, now acts in the opposite direction. The influence of the deterioration of concrete on the maximum displacements is shown in Figure 9, leading to a 100% increase for the case with E = 0.1 Ec. STRENGTHENING THE DOMES In view of the catastrophic consequences of the construction errors for some loading conditions, the safety of other digester tanks in the same plant and with similar characteristics to the tank that collapsed had to be evaluated. Possible ways to strengthen a dome were considered, and the most convenient way evaluated was the use of externally bonded fiber composite sheets.

V

-

_

,o

T

/

,

/

00 Stresses

ro00-0o (N/m21

-~-One layer of CFRP -e-Without strengthening Figure 10. Stress redistributions in the shell with an externally bonded carbon fiber sheet. Composite sheets have been employed to strengthen reinforced concrete bridges, columns, and beams, and this is an area of great interest in terms of research and civil engineering practice. A carbon fiber composite (CFRP) could be laid on the external surface of the thickness, to restore the tensile capacity of the dome in case of a reversed bending situation. Due to limitations of space it is not possible to describe in full the behavior of the dome with such reinforcement, and only one plot of the stress redistribution in the meridional direction is shown in Figure 10.

676 The main problems to be considered in the design of this reinforcement are the shear transfer between the composite and the concrete, and how the behavior of the shell is modified under normal (selfweight) conditions. CONCLUSIONS The preliminary conclusions of this work may be summarized as follows: (a) The shell could resist normal operation under self-weight and accidental loads due to gravity. This was done even taking cracking and the real location of the reinforcement into account. (b) Effects due to buckling of the shell have not been a crucial factor that could explain the collapse. Seismic or wind load were not identified on the day of the collapse so as to study a dynamic action as a possible explanation. (c) Because the digester was accidentally pressure-filled on the day of the collapse, it is expected that high internal pressures developed on the dome. If the shell had been constructed as designed (with a central layer of reinforcement in two directions) then it is expected that it would have resisted an internal pressure. But for the actual shell with the reinforcement at the bottom of the thickness, with cracking on the top part of the concrete, and with geometric distortions, it seems that the structure would not be able to take the internal pressure. (d) Strengthening the dome on the top side with carbon fiber composite sheets may be a convenient way to improve the safety of a structure in such conditions. The results show that the stress levels are reduced in the concrete, and that the bending capacity is restored thanks to the tensile contribution o f the composite. REFERENCES ACI Committee 334 (1986), Concrete Shell Structures: Practice and Commentaries, American Concrete Institute, pp. 14. Ballesteros P. (1978). Nonlinear dynamic and creep buckling of elliptical paraboloidal shell. Bulletin of the Int. Association for Shell and Spatial Structures, 66, pp. 39-60. Billington D. P. (1990), Thin Shell Concrete Structures, 2 nd Ed., McGraw-Hill, New York. Godoy L. A. (1996), Thin-Walled Structures with Structural Imperfections: Analysis and Behavior, Pergamon Press, Oxford, UK. Gould P. L. (1999), Analysis of Plates and Shells, Prentice Hall, New Jersey.

Third Intemational Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

677

CLOSED CYLINDRICAL SHELL UNDER LONGITUDINAL SELF-BALANCED LOADING V. L. Krasovsky and G. V. Morozov Prydnieprovsk State Academy of Civil Engineering and Architecture, Chernyshevsky Street, Dnepropetrovsk, 49600, Ukraine

ABSTRACT One of factors, influencing stress-strain state (SSS) of large-sized vertical cylindrical tanks for oil storage, is an subsidence of foundations. The magnitudes of this subsidence are defined as vertical displacements of the lower edge of the tank shell. The constituents of these displacements that correspond to the second and subsequent harmonics of trigonometric series may lead to considerable stresses in the tank shell. The problem formulated in this paper may be reduced to solving the SSS problem for a closed cylindrical shell at non-uniform kinematic loading applied to one of the edges. The problem is solved using a specially developed automatic calculation package in finite differences. The investigation of the problem has shown a number of effects in the shell behavior at specified kind of loading. Particularly, it was found that in case of free edge at unloaded edge of the shell considerable radial displacements occur in its vicinity that lead to increase in the bending stresses. It is also found that when the circular displacements at this edge are restricted (v-0), the SSS in the shell is close to the momentless one.

KEYWORDS Cylindrical shell, kinematic loading, storage tank, foundation subsidence, finite differences method.

PROBLEM F O R M U L A T I O N The problem of SSS definition in a closed cylindrical shell (see Figure 1) is considered. The shell is loaded at one edge by the longitudinal kinematic load: *

ny

u = Asin--k-- ,

(1)

where A is the amplitude of displacement, n is the number of harmonics that defines the alternation of longitudinal displacements of the shell edge in the direction of circular co-ordinate y.

678

S=

i

....

d

N

M21

l~

Figure 1: Cylindrical shell

MI

MI2

& ~ N!

d Q2

M1

Figure 2. Internal forces and displacements in shell

The solution is built on the base of equations of the linear theory of cylindrical shells written in displacements, Vlasov (1949): # OW

02U 1 - # 02U ~ + ~ ~ + Or,2 2 Oy2

l+bt 02V

1+#

1-,u 02v 10w 3-,u

02u

20xOy

+

+--~-c~ OxOy R Ox

2

02v

+ ~ ~ + -~ 20x 2

--m-C # OU (03U 1-# R Ox

Ox3

2

(03W

1-#

&3

2

~ c R Ok, 2

03w 1-#2 Ox20y

+

3+ 1 OV --C3-# . 03V

03U

Ox@2

R Oy

03W ) 1 - # 2 )+ X =0 OxOy2 Et '

2

Et

Y =0,

(2)

(3)

+

Ox20y

(4) + c R V2V2w + ------:---+ R

+

R2

Et

Z = 0.

where x, y are the unknown variables (longitudinal and circular, respectively); u, v, w are the displacements in longitudinal x, circular y and radial z directions (figure 2); R, t are the radius and thickness of the shell, E, # are Young's modulus and Poisson's ratio of the material; X, Y, Z are load components in the directions x, y, z; ~ . _ .

t2 12R

V2

9

__

02 02 ~ +~

_

Laplace's operator.

The expressions for components of the internal forces (see Figure 2) that correspond to the equilibrium equations (2-4) can be expressed as follows: N1

1_#2

= ~

+#

-c

+---C

,

s~ 20+#) Oy Ox

MI:D

R --~+#

OxOy)

-

-'~+#7

, N2= 1- #2

S2 =

w ,

+~-+,u-~- +c OY2 ),

+--+c~

2(1+#) Oy ox

OxOy

,

(5)

(6)

(7)

679

M12 = -D (1-/.t) /

R 0x +OxOy)'

Q1 =D

M21 = - O

0x2

2

2 . -~-

-

0y2 + 2R 0x0y

+2-~j

0x

+

,

(8)

w ,

(9)

Q2:D

(1-P) R Ox2

R 2 Oy

03, ~-T +

w .

(10) where D is the cylindrical stiffness. The expressions for the generalized shear ( S 1 ) and transverse (Q1) forces, according to Kolkunov (1987), are as follows: S1 =S 1+M12 R

D(l-p)

6 ~ 6 1 ~ ~ + +

0v 0y

3 02w 2R 0xOy'

(ll) Q1 = Q1 +-

=D

0y

0x2

-

~

2

0y2 + - 2

3 - ,u O2v

03w

0x0y - ( 2 - / t ) 0X0y2

03w

0x 3 .

(12) Boundary conditions on the lower edge of the cylinder were taken in the form: u = u*, v = 0, w = 0, M~=0. Different boundary conditions were considered on the upper edge. The solution and the analysis of presented problem have been performed within the framework of an automated calculation package using the finite differences method (FDM). This package was developed for solving various shells problems, connected with calculation of vertical cylindrical tanks. All calculations have been performed on base of FDM models for cylindrical shells. The difference templates of extra-high accuracy have been used for presentation of initial differential equations.

RESULTS AND ANALYSIS In the case of flee support of shell on the upper edge (NI "-" 0, S l * - 0, Q1*= o, MI = 0) it was found that even slight longitudinal displacement of the unloaded edge may cause considerable radial displacements on the upper edge. The nature of these displacements corresponds to the longitudinal kinematic loading of the lower edge. In this case the bending stresses or(M2) prevail but the membrane stresses are negligible. The deformed shell at n = 4 is shown in Figure 3 (a, b). In Figure 3 (c) the maximum radial displacements w are shown with solid line (marked "o") along the length of the shell with parameters R = H = 10 m, R/t = 1000, A = t, n = 2. It has been found that the increase of amplitude of displacements A of the lower edge of the shell leads to proportional increase of the radial displacements and stresses on the opposite free edge. The proportional increase of deformations and stresses can be also seen when the shell height H increased. The increase of number of harmonics n for given longitudinal displacements at the constant values of these parameters results in an increase of radial displacements proportional to n 2.

680 To check the possibility of such an effect in real shells the experiments have been performed on samples made of solid paper. One of models is presented in figure 4. The shell 1 is installed on a cross-piece 3. The dead load 4 is applied to the lower edge. Such loading causes the state close to that described by (1) with n= 4. The radial displacements of the shell on the lower edge are restricted with a rigid plate 2 installed inside the shell. The results of the experiments confirm the qualitative peculiarities of the shell deformation obtained in the calculations. As it is seen on the photograph, the shape of sample is similar

to the configuration obtained in calculations. To examine the behavior of the shell at different boundary conditions and to reveal the conditions that considerably influence the SSS at given kind of loading the following test has been performed. The boundary conditions were varied at constant level of load for the shell (R = H = 10 m, R/t = 1000, A = t , n = 2) on the upper (loaded) and lower (unloaded) edges. The results are presented in Table 1. TABLE 1 DISPLACEMENTS, FORCES AND STRESSES IN THE SHELL WITH DIFFERENT BOUNDARY CONDITIONS (R = H = 10 M, R/T = 1000, .4 - T, N = 2) Displacements, forces and stresses Boundary conditions (upper/lower)

w/t

N~ (kN) 0.77 0 (1)

N2 (kN)

S~ (kN)

M~ (kNm)

M2 (kNm)

(MPa)

2.92 (u)

0.435 (1)

6.41e-3 (u)

21.3e-3 (u)

1.53 (u)

O'IV

N! = 0, $1 = 0, Q1 = 0, M1 = 0; u=u*, v=O, w=O,M~=O

4.0 (u)

N1 = 0, $1 = 0, Ql = 0, M1 = 0; u= u, v= 0, w= 0, w ' = 0

3.99 (u)

3.30 (1)

58.9 (u)

5.13 (1)

0.614 (1)

0.184 (1)

32.9 (u)

u - 0 , S~=0, Q I = 0 , M 1 = 0 ; u=u', v=O, w=O,M~=O

1.70 (u)

2060 (1,u)

615 (1)

26.8 (1)

1.27 (1)

0.383 (1)

250 (l)

N l = 0 , v=0, Q 1 = 0 , M I = 0 ; u=u*,v=O,w=O,M~=O

0.234 (1)

2090 (1)

625 (1)

1180 (1)

0.556 (1)

0.168 (1)

225 (l)

N~ = 0, v= 0, w= O, w'= 0; u=u , v=O, w=O, w ' = 0

0.209 (1)

2090 (l)

625 (1)

1050 (1)

1.67 (1)

0.501 (1)

274 (l)

Momentless solution (upper edge: N1 = O, v = O)

0.305 (1)

2090 (1)

,

.,

,

,,.

1050 (1)

182 (1)

681 Here the maximum values of the corresponding characteristics are given that occurred in the area of either unloaded lower (1) or unloaded upper (u) edge of the shell, in this table $1 is the shear force. The maximum equivalent stresses o~v are defined according to the energy theory of strength:

O'IV =

~/ 2

2 2 O"x + Cry + 3"r x - O ' x C r y

,

N1 6M1 where crx = ~' + ---~--,

N2

S1

6M2

O'y = ~ t + / 2

'

Z"X - - ~

t

As it is seen from the Table 1 the rigid fixation (in comparison with hinge) does not considerable influence the behavior and the general SSS. At the same time the rigid fixation of the upper edge does not change the SSS too much. This SSS passes on from the bending (moment) one in the case of free edge to the state close to the momentless one. It becomes apparent from the considerable reduction in the radial displacements and bending stresses and in the increase (by one or two orders) in the membrane stresses. It is found that the most important factor among boundary conditions that defines the SSS is the restriction of the circular displacements of the unloaded edge (v=0). To show this effect in Figure 3(c) together with maximum radial displacements of the shell with free upper edge (curve "o"), the displacements obtained for fixed upper edge are presented in circular (v=0; curve ~v))) and longitudinal directions (u=0; curve (~u))). The solution of the momentless SSS problem (v=0) has been found also on the base of the momentless theory of shells [6]. The results of this solution (line "Momentless solution" in Table 1) show practically complete coincidence of the basic forces N~ and S~ obtained in the momentless and moment solutions at the corresponding boundary conditions. In practice the canonical boundary conditions (free edge, rigid fixation and others) are realized rarely. Usually the edges of shell are elastically supported. The walls of large-sized tanks are supported at the upper edge with a stiffness ring. To study the influence of reinforcement of unloaded edges on the shell behavior the problem of thinwalled cylinder SSS has been solved. The cylinder is stiffened with a ring of rectangular cross-section which is fastened to the butt-end symmetrically with respect to the medial surface. The equilibrium equations for the ring are accepted as Karmishin et al (1975). At given geometry of the cylinder and loading (R = H = 10 m, R/t = 1000, A = t, n = 2) the dimensions of transversal cross-section of the ring were varied (a is the ring width, b is the ring height ). The results of calculations are presented in Table 2. As in Table 1, here the maximum values of obtained displacements and forces are shown and the points where they were defined are marked: (1) - at the lower edge, (u) - upper edge of shell. The values Jr and .L are the moments of inertia of the ring in its plane and out-of-plane, respectively. The first line of the table contains the results of calculations when the ring is absent (free edge). It is seen from the table that the increase of the ring rigidity leads to the decrease of maximum radial deflections in the area of junction of the ring and the shell. At the same time it makes the normal and shear stresses rise. When the rigidity increases, the maximum values of these forces move from the point of junction with the shell toward the lower edge to which the self-balanced kinematic load is applied. Thus, when the ring rigidity increases the smooth transformation occurs from the bending stress state to the momentless one. It was also found that in real storage tanks it is almost impossible to restrict the radial displacements in the area of coupling of the shell with a ring. The stiffness of the ring has to be so high that it is hard to make such ring with accepted shape of cross-section. This can be explained by high level of shear forces emerging in the shell that are transferred to the ring and causes the bending. To stiffen the free edge of the cylindrical shell at the self-balanced longitudinal loading it is recommended to use roofs, rigid membrane coverings. If it is impossible, and if one has to use the rings of ri-

682 gidity (as for example in large-sized vertical tanks with a floating roof) the emergence of self-balanced loads on a shell should be restricted. Speaking about restrictions it should be pointed that these effects of non-homogeneity of radial displacements in the radial direction have been observed in storage tanks of large volume, reinforced with a ring of rigidity, in the process of filling up, i.e. when the foundations settle. TABLE 2 INFLUENCEOF THE RING STIFFNESSON DISPLACEMENTSAND FORCESIN THE SHELL Displacements and forces

Geometry of ring b

a

Jr" 102

N1

N2

S12

ml

M2 (~-M)

(u)

0.770 (1)

2.92 (u)

0.435 (1)

0.00641 (u)

0.0213

Jx" 104

~) 4.00

(u)

'0.1

0.5

10.4

0.417

39.3 (u)

29.8 (1)

46.6 (u)

17.2 (1)

0.603 (u)

0.201 (u)

0.2

1.0

1.67

6.670

3.2]. (u)

395 (1)

119 (1)

206 (1)

0.483 (u)

0.161 (u)

'0.3

1.5

8.47

33.80

1.81

1120 (1)

336 (1)

565 (1)

0.611 (1)

0.183

0.4

2.0

(u)

1630 (l)

488 (1)

816 (l)

1.164 (l)

0.344

0.40 (u)

1890 (l)

565 (l)

985 (u)

1.440 (1)

0.433

0.5

(u)

2.5

26.0 65.1

107.0 260.0

0.87

(1) (1) (1)

References Krasovsky V.L., Morozov G.V.(1997) The Stress-Strain State of Closed Cylindrical Shell Under Longitudinal Self-Balanced Loading of its Edge. Theoretical Foundations of Civil Engineering, Warsaw, 133-138 (in Russian). Krasovsky V.L., Morozov G.V. (1998) Stresses in a Wall of Steel Vertical Cylindrical Tanks under Foundation Subsidence. Lightweight Structures in Civil Engineering. Proceedings of International Colloquium, Warsaw, 131-136. Vlasov V.Z. (1949) General shell theory and it application in technique, Moscow (in Russian). Kolkunov N.V. (1987) Bases of elastic shell calculation, Moscow (in Russian). Karmishin A.V., Laskovets V.A., Machenkov V.I., Frolov A.N. (1975) Static and Dynamic of Thin Walled Shell, Structures, Moscow (in Russian).

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

683

COUPLED INSTABILITY OF CYLINDRICAL SHELLS STIFFENED WITH THIN RIBS (theoretical models and experimental results)

A. I. Manevich Department of Theoretical Mechanics and Strength of Materials, Ukrainian State University of Chemical Engineering, Dniepropetrovsk, 49005, Ukraine

ABSTRACT Coupled instability of longitudinally stiffened cylindrical shells under axial compression is considered. The attention is focused on features of theoretical models for shells with thin ribs of large bending stiffness for which the buckling and postbuckling behaviour can be strongly affected by local displacements of stiffeners. The interaction of overall modes with two types of local modes (<<skin buckling>> and <>) is taken into account. The exact analytical solution of the linear buckling problem with a plate model for stiffeners, the Koiter's asymptotic method and the amplitude modulation approach (in a specific form) are employed in the analysis of the coupled buckling. Comparison of the theoretical solution with the experimental data is carried out, and limitations on the use of the asymptotic theory for prediction of the failure loads are discussed. KEYWORDS Stiffened cylindrical shells, buckling, coupled instability, experimental investigations.

INTRODUCTION The first theoretical and experimental investigations of coupled instability of stiffened shells have been carried out in 70-s (Koiter (1976), Byskov .& Hutchinson (1977), Manevich A.I. et al. (1971), Manevich (1977, 1979) and were refined in 80-s (Byskov & Hansen (1980), . Hue et al (1981), Manevich (1983, 1985). In this paper we focus our attention on cylindrical shells stiffened with relatively small number of thin-walled stringers of large bending stiffness (welded, riveted), behavior of which may essentially differ from that of shells stiffened with a large number of weak fibs. Due to the ribs large stiffness such shells can carry out considerable load after the skin local buckling. But for these shells it is principally important to account for effects of the ribs local buckling.

684 In the paper are briefly summarized results of the theoretical investigations of stability of cylindrical shells stiffened with longitudinal thin ribs (Figure 1) which were carried out by the author in the framework of the Koiter's theory of initial postbuckling behavior. The exact analytical solution for the linear local buckling problem allowing for the local deformations of stringers is used. The theoretical conclusions are compared with qualitative and quantitative results of experiments which had been carried out by the author and his collaborators. The experiments confirm the main conclusions of the coupling buckling theory, but the asymptotic approach is found to be inadequate for prediction of ultimate loads of shells with preliminary local ~skin buckling)) since the theory of initial postbuckling behavior does not account for some important features of buckling process in these shells.

Tu7

Figure 1: Stiffened cylindrical shell and cross-sections of ribs THE L I N E A R P R O B L E M

In the first investigations of the coupled instability in stiffened shells rather simplified theoretical models for analysis of buckling modes, especially, local modes were employed; later they were gradually refined. Byskov & Hutchinson (1977) used the sinusoidal approximations of local modes, i.e. the stiffeners only determined the periodicity of the local mode in the circumferential direction, their stiffness was not taken into account. Byskov & Hansen (1980) took into account the ribs torsional stiffness. Hue et al (1981) accounted for the axial rigidity and the eccentricity of the fibs, in order to describe the redistribution of stresses between the skin and the fibs. But the ribs were considered as bars which do not lose their stability (parametrical terms in the equations of equilibrium were neglected). An exact solution of the linear local buckling problem for a stiffened shell as a skin-fib system allowing for the local buckling of the ribs themselves has been obtained in papers (Manevich (1983), Manevich (1985)). Side by side with the bar model for ribs (with account for the parametrical terms in stringers equilibrium equations) the plate model for stringers of rectangularand T- cross section was employed. For shells with thin ribs one should discern two types of local modes (Figure 2): 1) ~skin local buckling, modes initiated by instability of the skin between stiffeners (with wavelength of order of distance between stiffeners), 2) ~rib local buckling)) modes initiated by buckling of the stiffeners walls (wavelength of order of the rib height).

685 The bar-beam model of ribs is adequate for the local modes of type 1 (critical stresses and modes obtained with using the plate model and bar model differ slightly, see upper drawing at Fig. 2). But for the local modes of type 2 only the plate model is applicable. The lower drawing in Fig. 2 illustrates large differences in the buckling modes obtained with the bar model (displacements of the skin between the ribs are large) and the plate model ( displacements of the skin are negligible). The bar-beam model can not describe real modes and therefore considerably overestimates the local critical stresses crx. In Fig. 3 the dimensionless local critical stresses cr~ = O-xl0 3 / E are presented for the shell R/L=0.5, R/h=300, ks=60, co=td/(ha)=0.2 (R, L, h = radius, length and thickness of the shell, t, d = rib thickness and height, ks=the number of fibs, a = distance between the fibs) via the number of longitudinal half-waves m for various values of ribs ((slenderness parameter)) t / d = 0.10; 0.08; 0.06; 0.05; dark points (solid lines) correspond to the plate model, light points (dashed lines) to the bar model (with account of the parametrical terms). G

J

E 10

. . . . beam model plate model

4.s

~

./0.08

\

,,/'/,-" ' ',

~

1-"skin local buckling"

,

3.s

d ~

~

010

0.06

,~" " ",'

~

,,,

..y: 2 -"rib local buckling" ~, ~ ~, ~,r ..P ~- '" ''d ~" ~,-,b ~ ~ .~ ~ ~

2.6 3

Figure 2: Two types of local modes - - - - - the plate model, ..............the . bar-beam model.

5

7

O

ill

Figure.3: Comparison of local critical stresses for beam model and plate model of ribs. L/R=0.5, R/h=300, ks=60, co=tdl(ha)=0.2, t / d = 0.10; 0.08; 0.06; 0.05.

When ribs are relatively thick ( t / d > 0.07) both the models result in nearly coincident results; for thinner ribs the bar model overestimates cr~ and this overestimation increases with increasing m. THE NON-LINEAR COUPLED BUCKLING P R O B L E M The Koiter's theory of initial postbuckling behavior results in the following expression for the potential energy of a shell in the case of two interactive modes - overall (i=l) and local (i=2): 1 22 1 ~2_) 1 2 P=~ao + ~ a 1 ~ ' 2 ( 1 - 2 1 +sazr"~(1-~-2)+a122G,~'22 + 1 + 4 allll

1

, 2

~': + 4 a 2 2 2 2 ~"~ - a l ~"1 ffl ~

, 2

(1)

+ 0 2 ~'2 ~'2 "~2

where ~ is the amplitude of ith mode (normalized, in given case, by the condition of equality of the maximal deflection to the shell thickness h), ~i is the amplitude of initial imperfection in ith mode, 2

686 is the load factor, ,;1,i is the critical value of 2 for ith mode (the periodicity and symmetry conditions are accounted for in Eqn.1). Coefficients a~,,, a2222 govern the postbuckling behavior at one-modal overall and local buckling and are determined by solving the nonlinear problems for one-modal buckling, coefficient a m (or all22, if am=0) governs the coupled buckling. Equilibrium paths are determined by the equations OP/cT~i =0. These equations together with the condition of vanishing jacobian I=0 determine the limit load ';[,(~'1,~'2 )as a function of amplitudes of the initial imperfections. Here we consider in detail only the coupling buckling problem. There exist two principal approaches to the analysis of the interaction between the overall and local modes in stiffened shells, both based on W. Koiter's works: 1) the asymptotic approach, 2) the method of modulation of the local mode amplitude. When the asymptotic method is employed (Byskov & Hutchinson (1977)) the cubic terms in the potential energy vanish due to conditions of periodicity and symmetry ( a~::=0), and the mode interaction can be accounted for only in the second asymptotic approach (by coefficient az~:2) through solving a boundary problem for the mixed second order mode. But the presence of a cluster of short-wave local modes with close critical stresses makes this boundary problem badly conditioned (Koiter (1976)). The amplitude modulation method takes into account that overall deflections either promote or suppress the local buckling depending on the direction of the overall deflection. This method deals with the amplitude of the local mode as a slowly varying function, thereby allowing for the mode interaction already in the first nonlinear approach ( a~22s0). It can be expected (and this is confirmed by experiments (Manevich A.I. et al. (1971), Manevich (1979)), that in the case of very short wavelengths (in particular, for the ~ribs local buckling>> modes) the tendency to the coupled buckling exhibits itself only on portions of a shell with certain direction of the overall deflection. When the overall buckling is oscillatory then zones on the shell surface with different character of buckling - coupled and uncoupled - alternate. So the following simple approach can be proposed (Manevich (1983)) which may be considered as a limit variant of the amplitude modulation method. We assume that the coupled buckling occurs only on portions of shells with certain direction of deflection in the overall mode, i.e. on the half of the shell surface (Fig.4). We remain the procedure of the asymptotic method in calculating the coefficient a m , only the domain of integration in corresponding integrals is to be changed. If the sign of a122is found to be negative then the mode interaction occurs namely in the chosen zones; but when a~22 is positive we have to change the domain of integration (to take other half of the shell surface) that leads only to the change of the sign of a~22(in this case the coupled buckling takes place on the shell portions with the opposite direction of the overall deflection). Numerical analysis of the solution (Manevich (1983)) shows that in the case of external stringers the unstable postbuckling behavior is pertinent to zones with radial overall deflections directed inward of the shell; in the case of internal stringers it remains valid for relatively thick fibs, but for thin ribs the coupled instability is pertinent to zones where the overall deflection is directed along the external normal. In intermediate t/d range (including the domain of optimal parameters) values of a~22for internally stiffened shells are small. So the conclusion can be drawn about much more pronounced mode interaction and imperfection sensitivity of shells with externally spaced stringers. This effect is similar to the effect which has been revealed by Hutchinson & Amazigo (1967), but it has another nature (in the later paper the <> scheme was employed and the mode interaction was not considered).

687 The second important feature of the nonlinear behavior of shells with thin ribs is the presence of two local minima in dependence of the limit load on m (if to account for the interaction between the overall mode and each local mode separately). The first minimum is determined by the minimum of the linear critical stresses cr~ and usually corresponds to a <<skin local buckling>> mode, the second by the maximum of the coefficient iau~t and corresponds to a <> mode. The lam! values turn out to be much greater for <> modes with wave-lengths of order of stringer height than for those with wave-lengths of order of the distance between stringers. At small t/d the second minimum is found to be global, even in the cases of higher o-~ values. In Fig. 5 the dimensionless limit stresses of coupled buckling versus < t/d are presented for the shells with the same parameters as in Fig. 3, for internal (a) and external (b) stringers at various combinations of initial imperfections amplitudes, solid lines - for the plate model of rib, dashed lines - the bar model (the m values were chosen which correspond to the minimal limit load). Curves for the overall and local buckling linear critical stresses are also presented. At imperfections amplitudes which do not exceed 0.5 h the limit stresses are lower than the linear critical stresses in the range of optimal values t/d approximately by 40% at extemal stringers and by 10-15 % at internal stringers. Note also that the optimal value t/d approximately corresponds to a value below which errors of the rib model become considerable; at intemal stringers this value almost coincides with the minimum point of iam[ in t/d. The interaction between overall buckling and the <> modes, as a rule, is found to be more pronounced than that of <<skin local buckling>> modes. Thus the local modes with prevailing displacements of ribs can govern the carrying capacity even in the cases when the linear critical stresses for them are higher than for <<skinlocal buckling>>.

I 3,fi

3,6

3

3

2

2

0,04 0,06 0,08 0,1 t/d a

.-"

0,04 0,06 0,08 0,1 t/d b

Figure 4: Alternation of zones with <
Figure 5:

the case of <> modes external

k,=60, co=td/(ha)=0.2 in the case of intemal (a) and

Limit stresses of coupled buckling versus

slendemess parameteD> t/d for the shell L/R =0.5,

(6) stringers at various combinations of imperfections amplitudes: (0.5, 0), (0.1, 0.1), (0.25, 0.25)-curves 1, , 3,.

688 ONE-MODAL OVERALL AND LOCAL BUCKLING Here we restrict ourselves only with some brief remarks. The nonlinear buckling problem for onemodal overall buckling in stiffened cylindrical shells (coefficient a,,l~in (1) and imperfection sensitivity) was investigated in 70-s (Hutchinson & Amazigo (1967), Brush (1968)) on the base of the "distributed stiffness" scheme. Results of the analysis turn out to be very sensitive to features of the theoretical model. But independently of these distinctions the solutions show that the imperfection sensitivity at the one-modal buckling decreases with increasing ribs stiffness, and for shells of large bending stiffness under consideration the coefficient a,~l, turns out to be positive or very small negative quantity. Thus coefficient all,, has no crucial effect on shells of this class, and thinness of stringers and their large bending stiffness do not introduce principal changes in the theoretical model. As for the uncoupled local buckling (coefficient a2m), it is evident that due to periodicity of the local modes in the circumferential direction the second order problem for local modes is reduced to that of a cylindrical panel with ribs on the longitudinal edges (and with corresponding periodicity conditions). W.T. Koiter has shown (Koiter (1956)) that the initial postcritical behavior of a cylindrical panel simply supported on its edges is governed by the parameter (b= width of the panel):

0 = ~/12(1-

b ~V2)2--~ -~1-~

(2)

The postbuckling behavior is stable at o < Oo= 0.64 and unstable at 0 > 0,. Later the value 0, for a cylindrical shell was refined with account for the stringer torsional stiffness (Stephens (1971)) and finite longitudinal stiffness and eccentricity of the ribs (Hue et al (1981)). The upper bound of the range of stable postbuckling is found to be higher (in the examples considered - up to 0.9). For the shells with stiff thin-walled stringers at construction of the second order local modes the possibility of"rib local buckling" is to be accounted for. A solution of the problem with account of the parametrical terms in the equations of equilibrium of stringers was obtained in (Manevich A.I. (1989)). In the examples considered the 0o value was found to be slightly lesser in comparison with results (Hue et al (1981)), but quantitatively the effect is small (0.=0.85-0.89). Shells stiffened with heavy fibs, as a rule, have a small number of them, and the Koiter's parameter usually is large: 0. >1. So for these shells coefficient a2222<0, and their local postbuckling behavior is unstable. From expression (1) one may easily conclude that in this case the limit load can not exceed local critical load ;% (at arbitrary other coefficients a~1~!, am). However this conclusion contradicts to experimental data (see below). SOME RESULTS OF THE EXPERIMENTAL INVESTIGATIONS Let us now compare the theoretical results with data of the experimental investigations of stability of compressed cylindrical shells stiffened with thin stringers of moderate and large bending stiffness which were performed in the Dniepropetrovsk State University and were partially presented earlier in (Manevich et al. (1971), Manevich (1979), Kostyrko et al (1983). There have been tested several hundred small-scale shells made from high-strength steel (o'0.2=780 MPa ) stiffened with angular cross-section stringers made from the same steel (spaced on the inward or outward of the shells). Parameters of the shells were varied in rather wide ranges: R= 70 - 150 mm, L= 80380mm, h= 0.19 - 0.28 mm, the number of stringers k, =12 - 48. The boundary conditions were close to the simple support. The amplitudes of initial imperfections, as a rule, did not exceed 0.5 h.

689 Here we outline only some results of these experiments in order to examine adequacy of the initial postbuckling theory for predicting the limit loads. At first consider experimental observations and qualitative results. The interaction between the overall and local modes explicitly manifested itself in the character of prebuckling deformations and in the buckling patterns. In all the tests when large discrepancies between the linear overall buckling loads and the experimental failure loads were noted the loss of carrying capacity occurred with simultaneous appearance of overall buckles and local folds of ribs walls in spite of that the linear buckling stresses for <) modes were higher than the experimental limit stresses. So the experiments have shown the crucial role namely of the <> modes. The experiments have confirmed also the theoretical conclusion about weaker mode interaction and imperfection sensitivity of shells with internal stringers in comparison with externally stiffened shells. First, for shells with internal ribs there were noted, as a rule, relatively small discrepancies between the experimental failure loads and the predictions of the linear theory. Second, this conclusion is confirmed by the following experimental observations: 1) failures of the shells with external stringers occurred sharply, with a snap, while the internally stiffened shells lost their carrying capacity smoothly; 2) many shells with internal stringers after the appearance of overall buckles could carry an increasing load, and the failure load was by 10-15% above the overall buckling load. Let us consider now some quantitative results. At comparison of the experimental and theoretical limit loads it is convenient to separate all the shells on following two groups: A - the shells with close values of linear critical stresses for the overall (Co) and local (O'L) modes (Cro z crL) or with higher local critical stress (~L > Cro); B - the shells with much lower local critical stress: crL << cro. Typical results for the shells of group A are presented in Fig. 6 where the experimental and theoretical values of dimensionless critical and limit stresses cr 103 / E are given versus the length to radius ratio for one series of the shells with internal stringers. There are presented the curves of the linear local critical stresses, linear overall critical stresses, uncoupled overall buckling limit stresses with overall imperfection ~* =0.5 and interactive buckling limit stresses for the same amplitude of overall imperfection and two values of amplitude of the local imperfections (~*=0 and (e*0.1). The experimental limit loads were sufficiently close to the theoretical values for the interactive buckling and on the whole rather far from the linear critical loads for overall and local buckling.

e~ _ . 1 0 ~ l. 2

1

. . . . . . . . .

t overallmode

. . / ' . ' , ,local mode '~ ~-',-~ -~--

~,~'~lznear

i,

I; : I', ',1 i

~

theory

i

l .

i

overall

0

'

I

'

'

,

2

,

|

,

t

3

,

,

,

,L 4R

Figure 6: Theoretical curves or- L/R and experimental data for the shells R=86 ram, h=0.19 mm, 24 stringers4x4.5x0.34mm

690 But for the shells of group B (with preliminary local <<skinbuckling>>) the discrepancies between the experimental and theoretical limit loads remain considerable. The values of Koiter's parameter /9 for these shells lay in the range /9=0.87 - 1.59. As we noted above, this values correspond to negative values of the postbuckling coefficient a2222,i.e. to the unstable local postbuckling behavior. So the theoretical limit load 2, accordingly to the Koiter's theory is lower than the local critical load 22. However for many shells the experimental limit stresses were higher than the linear local critical stresses crL. As a rule, they lay between o"L and linear overall critical stresses Cro: crL < O'lim < o"o . The buckling process in all these shells was rather complicated. In Fig. 7 typical diagrams of deforming are presented for the shells with internal (a) and external (b) stringers. At first the local buckling mode appeared (the <<skinbuckling>> mode - the stringers were only twisted). After a small decrease the load increased again until the overall buckling occurred. In the process of increasing load the local mode usually underwent transformations, and new local modes could appear, with small falls and subsequent rises of the load. Moreover, after the appearance of the overall mode the load can again rise ( at internal stringers) by 10-20%, in spite of a partial loss of rigidity, until the failure load was reached. The theory of initial postcritical behavior can not describe such a buckling process. It can only pretend to predict the first critical point ( local skin buckling) in dependence on imperfections and the initial slope of the postbuckling path. If one would constrain himself with this two items then the Koiter's theory turns out to be adequate. Local buckling always happened with a snap and a fall of the load, in concordance with the theory prediction. However after reaching certain local deflection this mode became stable. The theory of initial postbuckling in the first and second approximations is not able to predict such changes in character of the postbuckling. It follows that this theory, basically, can not predict the limit loads of shells, if the first critical point does not coincide with the failure load or is not a stable point. Thus the Koiter's asymptotic theory which correctly describes the initial postbuckling, can not be used for prediction of failure loads of rather wide class of stiffened cylindrical shells in which the buckling process includes several stages (transformations of an initial mode, the appearance of new modes with unstable initial and stable following behavior and so on).

L

kN

7,

kN

o retail buckl~ng~,,~

/buckhng ! r 0

10

~

~ ~

overall

~ckl~ng

\

2R=143ram, h=O.19 ram, L=130 ram, 24 internal stringers 2.3x3.0x0.234 0,1

0.2

0.3

0.4

0.5

EL, mm

0

0,1

0.2

0.3

0.4

0.5

,A ./.,~ mm

b Figure 7: Typical diagrams of deforming stringer-shells with internal (a) and external (b) stringers

691

References Brush D.O. (1968). Imperfection Sensitivity of Stringer Stiffened Cylinders. AIAA 9'. 6:12,. 24452447. Byskov E. and Hutchinson J.W. (1977). Mode Interaction in Axially Stiffened Cylindrical Shells. A/AA J. 15: 7, 941-948. Byskov E. and Hansen J.C. (1980). Postbuckling and Imperfection Sensitivity Analysis of Axially Stiffened Cylindrical Shells with Mode Interaction. J. Struct. Mech. 8:2, 205-224. Hue D., Tennyson R.C. and Hansen J.C. (1981). Mode Interaction of Axially Stiffened Cylindrical Shells: Effects of Stringer Axial Stiffness, Torsional Rigidity and Eccentricity. 3'. Appl. Mechs, Trans. of ASME, Ser. E 48:4,. 915-922. Hutchinson J.W.and Amazigo J.C. (1967). Imperfection sensitivity of Eccentrically Stiffened Cylindrical Shells. AIAA 3". 5, 3,. 392-401. Koiter W.T. (1956). Buckling and Postbuckling Behaviour of a Cylindrical Panel under Axial Compression. NLR Report N 476. Amsterdam. Koiter W.T. (1976). WTHD Report N 590 (Delft Univ. of Technology, 41 p. Kostyrko V.V., Krasovsky V.L. and Manevich A.I. (1983). About the Effect of Stiffeners Parameters on the Carrying Capacity of Compressed Shells. Civil Engineering and Design of Buildings 3, 5255. Moscow (in Russian). Manevich A.I. (1977). Coupled Loss of Stability of a Longitudinally Stiffened Cylindrical Shell. Hydroaeromechanics and Theory of Elasticity 22, Dniepropetrovsk, DGU, 104-114 (in Russian). Manevich A.I. (1979). Stability and Optimal Design of Stiffened Shells. Kiev-Donetsk, Vyshcha shkola, 152 p.(in Russian). Manevich A.I. (1983). Loss of Stability of Compressed Longitudinally Stiffened Cylindrical Shells at Finite Displacements with Account for Local Buckling of Ribs - Plates. Mechanics of Solids, Izv. of USSR AS 2, 136-145. Moscow (in Russian). Manevich A.I. (1985). Stability of Shells and Plates with T-cross-section Ribs. Civil Engineering and Design of Buildings 2, 34-38. Moscow (in Russian). Manevich A.I. (1989). Nonlinear Theory of Stability of Stiffened Plates and Shells with Accounting for Mode Interaction. Thesis on Dr. Sci. degree. Dnepropetrovsk (in Russian). Manevich A.I. et al. (1971). An Experimental Investigation of Stability of Longitudinally Stiffened Cylindrical Shells under Axial Compression. In: Design of Spatial Structures 14, Moscow, Stroyizdat, 87-102 (in Russian). Stephens W.B. (1971). Imperfection Sensitivity of Axially Compressed Stringer Reinforced Cylindrical Panels under Internal Pressure. AIAA J.. 9: 9, 1713-1719.

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Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

693

INSTABILITY MODES OF STIFFENED CYLINDRICAL SHELLS

J. Murzewski Faculty of Civil Engineering, Politechnika Krakowska, 31-155 Krakrw, Poland

ABSTRACT

Circular cylindrical shell with circumferemial stiffening frames (tings) is subject to axial force Ns. Two-dimensional state of stress occurs in the shell because of stiffening action of the frames. Thanks to this fact a small enhancement ~g of resistance of the elastic cylinder may be taken into account; however, much stronger reduction 7~of the resistance may happen if circumferential plastic hinges arise along the stiffening rings and in the middle of the cylindrical segment (Figure 1). Quite different equations have been proposed for local elastic instability of thin walled cylindrical shells. Coupled plastic and elastic instability modes are taken into consideration. Probability principles are applied to assess interaction of plastic and elastic instability limits and a coupled reduction factor cp is derived as a function of shell slenderness r/t and stiffener spacing a/t (Figure 2). Severe reductions of resistance of transversally stiffened shells with not too thick walls indicate that design and construction of large diameter cylindrical shafts of towers, guyed masts etc. without longitudinal stiffeners rather shall be avoided..

KEYWORDS

critical strength, cylindrical shell, elastic-plastic buckling, partial safety factor, plastic hinges.

694 INTRODUCTION

Certain of the background material relating to behaviour of stiffened cylinders under axial compression and/or bending have been published more than 40 years ago (Becker, 1958). Perfect elastic behaviour of the stiffened circular cylinder is a classical problem presented by Timoshenko (1962), Flugge (1967) etc. In the 1960-ies some novel concepts about plastic instability were derived by the author from theoretical considerations and experimental tests as results of research projects on steel shell galleries (Murzewski, 1964, 1967) and tubular guyed masts (Murzewski, 1968,1969). Coupled instability ideas came before the proper time to attract interest and discussions of the specialists. Code-writers and designers were not disposed to study non-classical plastic problems and the new rules has not been implemented. There is some progress in recent publications and standards on structural design, but a new Polish draft on design of steel towers and masts (prPN-B-03204) is still conservative. No advantage of plastic behaviour is allowed; however, reduction of buckling resistance with respect to both the ideal Euler's critical stress and the yield strength has been always taken into account. Now, computer facilities and a better knowledge of coupled instabilities may help to reassess instability modes of cylindrical shells. A very simple problem is taken into consideration in this paper: a transversally stiffened cylindrical shell (tube) subject to axial compression (Figure 1).

Figure 1: Stiffened cylindrical shell before (a) and after plastic hinge formation (b)

695 STABLE ELASTIC BEHAVIOUR

First, two-dimensional state of stress is assumed in a perfectly elastic shell under axial compression. Normal stresses ox, fly of the cylindrical shell in axial (x) and circumferential (y) directions (Figure 1) are balanced by axial tension

fff

of the stiffening frame. The stresses are derived from:

a) the Hooke's law for the shell and the frame

gx =(t~x -Vt~y)/E,

E:y =(Oy-VOx)/E , 8f --fff/E;

(1)

b) compatibility condition between the shell and the frame

9

gv = g f ,

o

(2)

c) integral equilibrium equations in a cross-section and an axial section of the cylinder atOy +Aft~f =0 ;

Ao x =N s,

where E = 205 GPa,

(3)

v=0,3 - elastic constants for structural steel,

Af, a - cross-section area of stiffening frames (rings) and their spacing (length of the segment), A = r~ r t, r, t - cross-section area, middle surface diameter and thickness of the cylinder, Ns [kN] - the compressive axial force, Ns > 0.

Solution of the set of linear equations (1), (2), (3) is as follows

~

-

N 2rtrt

-o

o

'

o

o o l+ot(1-v

gx = where

E

l+ct

Y

va l+a

= - ~ o

2)

o o '

ct = Af/a t - non-dimensional frame spacing,

v 9 l+ot ~176

o,

o ~ = ~

(4)

v

~ Y = ~ ~E = ~l+0t f

;

(5)

o0 = Ns/A >0 - the compressive stress.

Length of the cylinder decreases and radius of the shell increases Aa = - a z ~ ,

Ar = ~ r

(6)

696 The effective stress, so called according to the Huber-Mises yield criterion, is reduced thanks to the two-dimensional state of stress VOt,

VO~

oefr = o o 1 - 1 - ~ a +

=WOo
(7)

The ultimate load NR hardly can be enhanced in proportion ~ because plastic instability may happen earlier, even for shells of moderate slenderness r/t < 50 for which local shell instability may be neglected according to the Polish design standard PN-90/B-03200.

PLASTIC

SHELL INSTABILITY

Kinematic approach of plasticity is applied. Circumferential plastic hinges are supposed around stiffening frames and also in a distance Xo from them at a critical level of axial stress oo (Figure 1). The shell still remains in the elastic state of deformation (4), (5), (6). The axial stress Oo does not change along the section of the cylindrical shell. Virtual displacement 6 is supposed in the middle section of the shell, Xo<x
(8)

(Be 1 -8~;2)r =8 and the integral equilibrium equation must be satisfied O'y I ~ + G y 2

(l-O

+~Of

(9)

-- 0

The solution is as follows 1-~ 8 1 - ~ ~ E " 8Sxl = v ~ - t~Yl =--I-I-(X r ' l+ot r '

(~y2 --

~ + ot 8 E, l+ot r

&:x2 "- V

~+~ 8 l+ot r

1-~ 8 6Sy I . . . . l+ct r ~y2

~+a 8 l+ot r

for x = 0

(10)

for Xo< x
(11)

Non-dimensional distance ~ = xo/a is defined as a free parameter. The frame spacing a+Aa decreases and the radius r+Ar of the middle section of the segment increases because of virtual displacement 8.

v ot ( l - O a 8 8a=-l+ct r

8wl

l+t~

8

and 8w 2

et8 i~ ++ct

forx=0

(12)

697 Circumferential plastic moments mred [kNm/m] are reduced because of the axial stress Oo,

m r e d - 4V

L 1-

(

\ fd J

(13)

[kNm/m]

with fd = Rm/1,33 - design strength of steel relative to central value Rm (median) of the yield strength.

The virtual work of external force, 8Ws = Ns 8a, and virtual work of internal bending moments, ~SWR = 4 m~d 2rtr 8/xo, shall be equal at limit state of equilibrium. After eliminating A.~5 from both sides of virtual work equation we get vot ( 1 - ~ ) a 4 t fd . . . . Oo= 9 9 1l+ot r ~a 4 ~

~0o

(14)

fd )

The second degree equation (14) with unknown reduction factor g=~oo/fa and constant c = ~

vtx

has

a positive solution Z = I 1 + (c~(1a ~ .~) t / ~- 2

c~(1-~)

-

2

a2 --7)

(15)

r.t-

The kinematic approach, with a free parameter ~, gives an upper bound of plastic capacity. The best estimate will be if the modal value ~=0,5 is selected, i.e the plastic hinge will happen in the middle of the length a of the cylinder segment ....

Z=

I 1+ (c--ff. a.~_.t) 2 c a2 . . . 8 ~ -. )

(16)

The factor Z can drastically reduce stability of the cylinder shell.

P.ex.:

r = 500 m m ,

t = 10 m m ,

0,0005 ot = ~ = 0,05 1.0.0,01

u

0,705 ~ 0,993

0,710

Ar = 5 cm 2

0,3-0,05 c=~ = 0,0143 1 + 0,05

V : ~/1-0,0143 + 0,01432- : 0,993, X

a = 1000 m m ,

-

Z=

v = 0,3 9

1,02 0,0143.~ = 2,857, 0,5-0.01

1+\

8 7 ----~

= 0,705,

reduction factor of strength fd with respect to plastic instability.

698 ELASTIC-PLASTIC INSTABILITY

Experimental tests do not agree with theoretical predictions of local elastic instability of thin-walled cylinders. Therefore an empirical factor C used to be introduced to the classical equation for axially compressed cylindrical shells

oct

3"C t ~ . E~/1 - v 2 r

[MN/m 2]

(17)

The equation (17) has been modified by Z.Mendera (1990) who had assumed that geometrical imperfections will reduce the elastic shell stability limit Ne the more the shell slenderness ratio r/t is higher. His equation for design value of stability limit of thin-walled cylinders of a cross section area A is

t ) 1,5 N e =2.EA.

[MN]

(18)

The Eqn.(18) has been modified again for low and moderate slendemess, applying probabilistic formula of interaction of elastic and plastic instability modes

Nu -I/u = Ne -i/u +Nd -l/V with

(19)

o - the Weibull coefficient of variation.

Design value of plastic compression resistance Nd of the cylinder was defined by Z.Mendera (1990) in the same way as it would be for tension resistance Nd = A fd

(20)

however, taking into account plastic instability of stiffened cylinders it should be NRd = A fd Z/~

(21)

Local instability curves (PZM= Nu/Nd for definitions (17), (20) and (PJM = N for definitions (17), (21) are represented in Figure 2 as functions of shell slenderness r/t and non-dimensional frame spacing a/t. The geometrical data are the same as they are in the numerical example in the preceding chapter. Imperfection parameter 0=0,625 corresponds to instability curve e from the standard PN-90/B-03200.

699

xx

--.

i"'-

---......

---....-...

"'r--;.:

"--" --"- . . . . . . . . . .

..... i

t

~

Z::

7__-_ ! .....

i ii~l-i

Figure 2: Resistance of the elastic-plastic local instability of cylindrical shell

Eqn (19) has been derived from assumption that elastic instability and plastic failure are independent random events and the Weibull distribution functions characterise random strengths NE, NR. Multiplica-tion principle of probability functions 1-F(N) was used in earlier derivations [Murzewski, 1969, 1974]. A simpler derivation may be given if summation principle of hazard functions of independent events is applied. The hazard function h(N) is defined the logarithmic derivative probability 1-F(N) of down-crossing a given level N. The Polish buckling curves q~(~.) have the same probabilistic background

-''~

(22)

where n =l/t~ - imperfection parameter calibrated for steel members depending on residual stress; = 1,15~/N Rd/N cr

--

~/

~'p

"

relative slenderness ratio of a member subject to compression;

NRd = NRm/Tm - design value of resistance,

NRm - central value (median) of the plastic

resistance; ~m = 1,152 = 1,33 - partial safety factor for steel members according to Polish design standards.

The Eurocode (ENV 1993-1-1, 1992) defines the design strength in reference to nominal values of yield strength fy with a partial safety factor yM0=l,1 and the same value yMl=l,1 for members subject to elastic buckling. This gives unsafe, --20% higher design values for resistance of slender members.

700 CONCLUSIONS Coupled elastic-plastic instability of transversally stiffened cylindrical shells should be checked with account that plastic hinges may occur around the stiffeners before the full plastic strength cannot be reached even in perfectly axial compression. Design specifications, which allow to neglect local instability, are not safe enough. Second-order radial forces compress the circumferential stiffener after plastic hinges have been formed (Figure l c) and it may be not able to resist. Its buckling constitutes so called inward-bulge, i.e. general instability of the stiffening frame and adjacent parts of the shell. Evaluation of the second-order effect may give a basis to dimensioning sufficiently strong frames. A practical conclusion is that cylindrical shells of large diameter, stiffened by frames only, should not be constructed in civil engineering unless drastic reductions of their capacity under axial load were allowed. This rule is well known in aeronautical engineering [Becker, 1958] where systems of frames and longitudinal stiffeners have been applied as a rule.

REFERENCES

Becker H. (1958). Handbook of Structural Stability. Part VI- Strength of stiffened elastic plates and

shells, NACA TC 3786, Washington Flugge W. (1972). Powloki. Obliczenia statyczne. Arkady, Warszawa Mendera Z. (1986). Interaction of Elastic and Plastic Instability in Cylindrical Shells with Imperfections.

Proc. 1ASS Syrup. on Shells, Membrane & Space Structures, Osaka 1986, Vol. 1, Elsevier, Amsterdam, 17-24 Murzewski J. (1964). Post-buckling behaviour and panel instability of stiffened cylindrical shells,

Proc. 1ASS Symp. : Non-classical Shell Problems, Warsaw 1963, North Holland, Amsterdam, 749-767 Murzewski J.(1967). Tests on models of cylindrical stiffened shells. Archiwum In~/nierii Lqdowej, 13:3, 375-397 Murzewski J. (1968). Instabilitatsfragen der dunnwandigen und ausgesteiften Kreiszylinderschalen.

Bauplanung-Bautechnik, 22:4, 173-174 Murzewski J. (1969). Wytrzyma|o~6 spawanych maszt6w powIokowych.

Archiwum Inzynierii

Lctdowej, 15:3, 571-583 Timoshenko S., Woinowski-Krieger S. (1962). Teoria plyt i powlok, Arkady, Warszawa

Third InternationalConference on Thin-Walled Structures J. ZaraL K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

701

REFINED STRAIN ENERGY OF THE SHELL Ryszard A. Walentyfiski Department of Building Structures' Theory, Silesian University of Technology, Gliwice, PIA4-101, Poland

ABSTRACT The paper provides information on evaluation of the formula for shell strain energy with a computer algebra system. There is proposed a formula for strain energy compatible with a definition of stress tensor resultant. The integral has been evaluated with the MATHEMATICA"system. It was shown that the strain energy can be presented as a linear combination of stress resultant components multiplied by respective strain tensors. The presented numerical example shows the difference of deformation, stress resultants and energy for the katenoidal shell subjected to thermal influences. There was found that strain energy do not derive the formulas for tensors of stress resultants.

KEYWORDS Nonlinear, elastic, theory of shells, constitutive relations, strain energy, computer algebra, MATHEMATICAL MathTensor, tensor analysis.

1

INTRODUCTION

Most of shell theories postulate the Strain density functional in a form proposed by Novozhilov (1959), which is compatible with his approximated constitutive relations derived for orthogonal system of coordinates. Further theories, for example Kraus (1967), only adopted this approach to tensor notation. Mazurkiewicz (1995) stated that the applied the strain energy density for the parallel layer does not have to be expressed with the Novozhilov approach. This paper proposed another functional (20) that can be used to evaluate shell energy density. It is compatible with exact definition of resultant stress components (1) and (2). The symbolic and numerical computations were made with the MAT~MAT/CA" system of computer algebra, Wolfram (1996), and its external package for tensor analysis Math Tensor, Parker & Christensen (1994). The graphics is also produced with the system.

702 The considered shell is thin so Kirchhoff assumptions hold. The functional can be, of course, completed with components responsible for transverse forces influence. The most of notations in the paper are compatible with Bielak (1993).

2

RESULTANT STRESS TENSORS

The tensors stress resultants, stretching forces and moments, should be computed from the following integrals, Bielak (1993)

N ij-- f

~l ~

( ~ r J - Z b / ) ri r dz

(1)

"~

(~/-- Z b j)

(2)

-h

M ij =

Z dz

-h

where strain tensor in 3D space %j and stress tensor zsj are computed from (3)

~.i = Yq - 2 z pi j + Z20ij . r ij = ,,1. g'J.. gPq + 2 g g'P. gqJ )

,~/ij

(4)

Contravariant componets of metric tensor can be derived from . .

a ij (1 - z2) - 2 (H aO - biJ) (1 - n z) z

g'J =

Z2

(5)

Present scalars are defined with

Z= ~=

(6)

1-2Hz+K~

~t -

vE

(7)

l_v 2 P

E 2(l+v)

=

(8)

where E is a Young modulus and v is a Poisson ratio. H and K denote average and Gaussian curvatures, respectively. Tensors aiy and bq form coefficients of the first and the second differential form of the reference surface. Invariants g and a are determinants of the metric tensor gij and aq, correspondingly. The strains in shell are denoted with three symmetrical tensors ~j, PO and Oq. They are measure of the change of the three differential forms of the reference surface, respectively. For example

1

Yij = ~ (aij - &ij)

(9)

It has been found, WalentyIiski (1999), that stretching forces tensor (1) can be evaluated as a sum of three tensors. Ni j = ~[ij + ~[ij + ~ J + 0

(h5)

(10)

703 Each of the tensors in (10) depend on one of reference surface strain tensors. /gij = 2 E h (1 - h 2 K) (v a ij Tpp + (1 - v) T/j) + l_v 2 (11)

4 E h 3 9,,pq (v b,,q (H a ij + b iy) + (1 - v) b pi (bqJ + H 6qJ)) +

^.. 4 E h 3 (2HtSqJ-bq j) (vaqipp p + ( 1 - v ) pqi) 8 E h 3 (vaiJbpqppq + ( 1 - v ) bp'p pj) N 'J = 3 (1 -

3 (1 -

[V.ij

§

(12)

2 E h 3 (v a ij Oapp + (1 - v) 0ij) =

(13)

If we compute the (1) with precision to the third power of the shell thickness we receive that it depends on two strain tensors MiJ = ~,liJ + n i J + O (h5) (14) where ff/1/ij ---

4 E h 3 (v a ij bpq Tm

+ (1

- v) bp i Tpj)

_ 2 E h 3 ( 2 H ~qJ -

3 ( 1 - v 21

bqj) (v a qi Tpp + 3 ( 1 - v 2)

(1 - v) Tqi)

(15)

and 1~ ij =

4 E h 3 (v a ij PPP + (1 -- v) pij) 3(1_112 )

(16)

The refined formulas (10) and (14) satisfy the last equation of equilibrium (17) Epq (N m - brP M "q) = 0

(17)

Many theories apply the simplified formulas where stretching forces tensor is computed from (18) and moment tensor from (16), only. NiJ * ~,

3

STRAIN

2 E h (v a ij T~ + (1 - v) T/j) 1 -v 2

(18)

ENERGY

Strain energy in volume V in orthogonal Cartesian system of coordinates is computed from the integral.

E=f2rPq~'pqdV

(19)

(v)

By analogy to the definitions (1) and (2) the strain energy will be postulated here as a following integral

E --"

U dA : (A)

5 (A) -h

( 6 / -- Z brq ) T pr 'ypq dz dA

(20)

704

The appropriate notation in the MATHEMATICA'/MathTensorof the inner integral in this definition, which will be called further shell energy density, is Energy[k_]

:=

Tsimplify [

AbsorbKdelta [

integ/@Expand [ Normal [

Series [ 1 - Z (Kdelta[13, u5] - zb[13, u5]) 2 tau[u4, u3] gammastar[14, 15], {z, 0, k}] ] ] ] ]

The result of the computation with precision to the third power of the shell thickness is Energy [2 ] E h ( - 3 + 5 h 2 K ) ( a pt) (a qs) (Tpq)(~8/st) 3 (l+v)

Eh (-3+5h2K) 2Eh3H

v (apq) (a st) (ypq) ('/st)

3 (-i + v 2)

(apr) (aqs) (br t) (Ypq)('s

+

3 (l+v) 2 E h 3 H v (apq) (ars) (br t) (~'pq)('~'st) 4 E h 3 H v (ast) (bpq) (Ypq) (Yat) + 3-3v 2 3-3v 2 2Eh3 (aqs) (brt) (bpr) (u (Tst) 2 E h 3 v (ars) (br t) (bpq) ('s 3 (l+v) -3+3v 2 4 E h 3 H (aqs) (bpt) (Vpq)(Yst) 2Eh3 (apt) (brt) (bqs) ('~pq)(Yst) -+ 3 (i + v) 3 (i + v) 4Eh3 (bpt) (bqs) (Ypq) (Yst) 2 E h 3 v (aPq) (brt) (brs) (Ypq) ("(st) 3 (i +v) -3+3v 2 4 E h 3 v (bpq) (bst) (Ypq)(Yat) 8 E h 3 H (apt ) (aqa) (%'st) (Dpq) + + 3-3v 2 3 (l+v) 8 E h 3 H v (apq) (ast) (Yst) (Ppq) 2 E h 3 (apt) (aqs) (brt) (Yst) (Dpq) + + 3-3v 2 3 (l+v) 2 E h 3 v (apq) (ars) (brt) ('/st)(Ppq) 8 E h 3 v (ast) (bpq) (]fat) (Ppq) 3 - 3 v2 + -3 + 3 v 2 16Eh3 (aqs) (bpt) (Tst) (Ppq) 8 E h 3 v (apq) (bat) (Yst)(Ppq) + + 3 (l+v) -3+3v 2 +

2 E h 3 (apr) (aqs) (br t) (Ypq)(Dst) 3 (l+v)

.... +

+

2Eh3v

-

4 E h 3 (apt) (a qs) (Dpq) (Pst) 4 E h 3 v + 3 (l+v) 2Eh3 (apt) (aqs ) (Yat) (Opq) 2 E h 3 v + 3 (l+v)

(apq) (ars) (brt) (Ypq)(Pst) 3-3v 2

(apq) (ast) (Ppq) (Psi) + 3-3v 2

(apq) (ast) ('/st)(Opq) 3-3v 2

The received result is a symmetrical, positive definite form, which is equal to zero if the shell is subjected to the rigid motion (in constant temperature). In this case strain tensors are equal to zero, so differential forms of the reference surface do not change, compare eqn. (9). According to the well-known theorem of differential geometry differential forms defines the surface with a precision to the position in space.

705 The received formula can be used directly in the further symbolic and numerical computations but can be simplified with MathTensor. After some simplifications of the formula - like absorption of the metric tensor, canonicalization, application of geometrical rules and simplification - we receive e h ( 1 - hEK) ( ( 1 - v ) Tpq ),qp + v ),p~' 7,qq) l _ v2 +

U= + + + +

((I-

2 E h3H

v)

bpq~'qrrl "at"vbq," rp q ~rr)

3 ( 1 - v 2)

2 E h3 bqP br s ((1 - v) ~pr,yq _~.V,ypq ~sr)

3 ( 1 - v 2) 8 E h3 n ((1 - v) yqP ppq + v ypP p q )

3 ( 1 - v 2) 4 E h 3 ((1 - v) ppq pqP + Vpp p pqq) 3 ( 1 - v 2)

~-

~-

~-

(21)

+

2Eh3 (2 ( 1 - v ) bpq),rPPq~ + Vbq p (~rrppq +~pqprr))

l_v2 +

2 E h 3 ((1 - v) yqP Opq + v ypP Oqq)

Now we will show that the shell energy density can be presented as a linear combination of the resultant stress tensors multiplied by the appropriate strain tensors. To do this we will introduce five scalars, which express the work of internal forces. The first three represent the work of stretching forces tensors ~/t,q (11), ~/m (12) and ~'Pq (13) on the first strain tensor y,.j

U.=~N Tpq= 1

-

pq

_ E h ( 1 - hZK)((1--V)~pq~q p + V~pP~qq)+

1 - v2 2 E h 3 H ((l - v) bp q .yqr),rp + V bqp "ypq"yrr) + + 3(1--V 2)

(22)

+ 2 E h 3 bqP brs ((1 - v) ~pr ~sq + V "~q "~sr)

4 E h 3H ((1 - v) yqP ppq + v ypP pqq) -

_

3

+

(23)

2Eh3 (3 ( 1 - v)bpqyrPpqr + VbqP (2%rppq + ypqprr))

U~ = ~Pq ~,pq =

2E h 3 ((1 - v) ypq Oqp + vypP Oqq) 3 ( 1 - v 2)

(24)

706 The last two represent the work of moment tensor components 11740(16) and Mij (15) on the second strain tensor Pij 4 E h 3 ((1 - v) ppq pqP + vppP pq) Up,,, = -MPq ppq = 3 (1 - v2) (25)

U,/p = - ~ I pq p pq =

_ 4 E h 3 n ((1 -v)TqPpp q + V~pPpqq) b 3 ( 1 - v 2)

(26)

2Eh3 (3 ( 1 - v)bpq%/'pq" + Vbqt'

(~/rrppq + 2~lpqprr))

3 ( 1 - v z) It can be easily verified that the strain energy density (21) can be expressed with

U = Urr + Um + U~r + Upp + Um, + O (h5 )

(27)

It proves that stretching forces components work on the first strain tensor and moments work on the second strain tensor. Below we present an extract from the Mathematica notebook which verifies it. S i m p l i f y / @ T s i m p l i f y [AbsorbKdelta [Expand[El - E n g - F A r - E a r - E m g - Eat ] ] ] 0

Some shell theories apply the approximate formula (28) where

E h (( 1 - v) "yq TqP + v T / yqq) Ur/*=

1_v2

This definition is compatible with simplified constitutive relations.

4

NUMERICAL EXAMPLE

Figure 1: Deformed reference surface. Simplified approach in the background.

(29)

707

Figure 2: Physical displacement vector w 1 (meridian direction) and w3 (normal direction) components and rotation vector d~ (meridian direction) component. [w i] = I.tm and [d l] = p m / m .

Figure 3: Physical stretching force Nil (meridian direction), N22 (parallel direction). [N/y] = N.

Figure 4: Physical bending moment M12 (meridian direction), M21 (meridian direction) and transverse force Q1 component. [Mij] = N m and [Q1] = N.

Figure 5: Energy components distribution for refined approach. There is considered a katenoidal shell which meridian is defined with a function f ( x ) = aocosh (a~)

(30)

The shell subjected to thermal influence consisting in difference of temperature on both limit surfaces. The following numerical data were applied: shell thickness 2 h = 0.2 m, radius in the neck a o = 5.0 m,

708 shell height I = 10.0 m, difference of temperature At = 20 ~ Young modulus E = 324000 kPa, Poisson ratio v = 1/6, thermal expansion coefficient a, = 10-5. The shell boundaries are fixed. The calculations were made with application of both refined and simplified constitutive relations. The results were presented in figures. Fig. 1 presents deformed reference surface. As we can see reference surface deforms when the shell is solved with application of refined constitutive relations. The deformation of the reference surface is small, Fig. 2, but enough to produce stretching forces, Fig. 3, which are absent in simplified approach and remarkable difference in moments and transverse forces, Fig. 4. The strain energy density is also remarkably different for both approaches, Fig. 5. In this case the energy density component U ~ is crucial and other components are mutually comparative.

5

CONCLUSIONS

Formula (27) shows that moments' tensor can be derived from the density of strain energy (21) as a derivative with respect to the second strain tensor. The stretching forces' tensor can be derived similarly by differentiation of strain energy density with respect to the first strain tensor, but only if we neglect influence of the third strain tensor. Presented numerical example shows that nevertheless the strain energy density usually depends mostly on components (22) and (25) but stress resultant tensors cannot be computed from the simplified formulas (15) and (18) that can be derived from (28). It may result in significant errors in computations. The received formulas can be applied in nonlinear theory of shells and can be easily expanded to multilayered shells and then applied for large strain theories. It will be shown in further contributions.

REFERENCES

Bielak S. (1993). Theory of Shells, Silesian University of Technology, Gliwice Kraus H. (1967). Thin Elastic Shells, an Introduction to the Foundations and the Analysis of Their Static and Dynamic Behavior. John Wiley and Sons, Inc., New York-London-Sydney Mazurkiewicz Z.E. (1995). Thin Elastic Shells, Linear Theory. Warsaw University of Technology, Warsaw Novozhilov V.V. (1959). The Theory of Thin Shells. P. NoordhoffLtd., Groningen, The Netherlands Parker L. and Christensen S.M. (1994). Math Tensor: A System for Doing TensorAnalysis by Computer. Addison-Wesley, Reading, MA, USA Walentyfiski R.A. (1999). Refined constitutive shell equations with MathTensor. Proceedings of the 3rd International Mathematica Symposium IMS'99. RISC, Hagenberg-Linz, http ://south. rotol, ramk. fi/-keranen/IMS99/ims99papers/ims99papers.html

Wolfram S. (1996). The Mathematica book. Wolfram Media/Cambridge University Press, Champaign, IL, New York, NY, USA

Third InternationalConferenceon Thin-Walled Structures J. ZaraL K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

709

Numerical and Experimental Studies on Generalised Elliptical Barrelled Shells Subjected to Hydrostatic Pressure P. Wang* Department of Mechanical Engineering, The University of Liverpool, Liverpool, L69 3BX, UK

ABSTRACT Numerical and experimental results on mild steel cylindrical and generalised elliptical barrelshaped shells under external hydrostatic pressure are discussed in this paper. Based on the cylinder geometry, the mass-equivalent elliptical barrel-shaped shells are analysed and the optimal barrel geometry is derived. BOSOR5 and ABAQUS software are employed to perform the numerical calculations. Results show that the load carrying capacity of elliptical barrelled shells can be increased up to 1.46 times above the load capacity of mass equivalent cylinder. Four experimental specimens were manufactured and loaded up until they failed and the discrepancy between experimental results and numerical predictions varied from -1.1% to 6.9%.

KEYWORDS

cylinder, elliptical barrel-shaped shell, bifurcation, collapse, FEA

1 INTRODUCTION The stability characteristics of cylindrical shells subjected to axial compression, external pressure or combined loading have attracted much attention in the past because of their technical importance. Recently, studies have shown that the buckling performance of cylindrical shells under axial compression or combined loading can, contrary to the usual belief, be improved by bowing out. These barrel-shaped shells which with positive Gaussian curvature perform well under external pressure and their stability domain are expected to outperform that of the plain and equivalent cylindrical geometry. Kruzelecki [1] considered optimisation of both wall-thickness redistribution and barrelled curve profile determination. Blachut [2,3] analysed both elastic and elastic-plastic buckling of barrelled shells. Other relevant papers can be found in Blachut and Wang [4], Zintilis and Croll [5], Lukasiewicz and Wawrzyniak [6]. Very few experimental results on barrel-shaped shells can be found in Odland [7] and Schmidt and Krysik [8]. Blachut and Wang [9] also considered the performance of composite barrelled shells and indicted that the structural response function is discontinuous and non-smooth due to different types of failure modes. * Currently at EngineeringDepartment, Lancaster University,Lancaster, LA1 4YR, UK.

710 It should be pointed out that most previous researches concentrated on barrelled shells with constant positive Gaussian meridional curvature. To the authors' best knowledge, other possible meridional forms have not been examined, e.g., elliptical meridional curvature, neither analytically nor experimentally. This paper performs a parametrical analysis of elastic-plastic elliptical barrelled shells which are subjected to external hydrostatic pressure only. Four experimental specimens were manufactured according to numerical results and tested up to failure.

2 P R O B L E M STATEMENT AND ANALYTICAL T E C H N I Q U E Consider a cylindrical shell with radius R0, length Lo and wall thickness to, which is subjected to external hydrostatic pressure P (Fig. l a). For a given material volume (weight) Vcylinder, the meridional profile of the mass equivalent elliptical-shaped shell (Fig. l b) can be derived under such restrains: the material volume of the cylinder and elliptical-shaped shells are the same; the axial length of the two structures are the same; the truncation radius Ro is kept unchanged; and, the meridional profiles of the elliptical shells are determined by a generalised ellipse equation which involve four variables nl, n2, n3 and b, (x/n3) nl + (z/b) n2 = 1.0

(1)

The material volume of the cylinder and the elliptical shell can be expressed as follows:

Wcylinder -- 2~;RoLoto

(2)

2[/~ Vellipse = J0 2n (n3 q l . O - (z / b)n2) tdz

(3)

Vcylinder -- Vellipse

(4)

Since

The wall thickness t of the mass-equivalent elliptical shell can be determined by following equation using direct numerical integration, 2rcRoLoto =

2f/~ ,,o 2~: (n3 ~1.0 - (z / b)n2 ) tdz

(5)

The mathematical solution procedure for wall-thickness t of elliptical shells suggests that variable n3 is dependent on values of other three variable nl, n2, and b. In other words, the elliptical shell geometry can be determined once variables nl, n2, and b are given. Typical profiles of the elliptical shells for cylindrical shell of ~ = 2.0 are shown in Fig. 2 with assumptions of that, n~ = n2 and b/Lo = 1.0. It is clear to see that the physical meanings of variables n3 and b are actually the ellipse radius on R-axis and Z-axis respectively. Fig. 2 shows that the meridional profile of elliptical shell of n~ = n2 = 1.0 will result in a triangle form surface geometry which is also the limit position attainable. For nl(n2) > 1.0, the meridional profile is actually part of the generalised ellipse determined by Eqn. (1). When variables nl or n2 approach relatively large values, for example, nl = n2= 5.0, the meridional curvature will approach to a form of straight line and thus the geometry of the elliptical shell will approach to the geometry of cylindrical shell. Since elliptical shells possess same truncation radius R0 and axial length Lo as that of the cylinder, the wall-thickness t of the elliptical shells must be thinner than cylinder wallthickness to due to the surface' s bowing-out. For example, for cylinder of IxJR0 = 1.0, R0/t0 = 33.33

711 and b/Lo = 1.0, the derived wall-thickness ratio t/t0 is 0.67 for (nl,n2) = (1.0,2.0) and 0.98 for (nl,n2) = (10.0,2.0). .-200 .

~

i

i ,

~

,

J.

,.

_,.

-

~

!

at'ha 9 at 9 nt - I.S

-lS0

"

n, 9nt 92.0 nt 9

-t00

i I

-50 0

It~

R

1110 150

Figure 1: Geometry of cylindrical (Fig. 1a) and elliptical barrelled (Fig. l b) shells.

Figure 2: Different meridional profiles of elliptical shells (Lo/Ro =2.0, b/Lo = 1.0, nl = n2)

In all pertaining analyses, the boundary conditions (BC) are applied to the top and bottom openings of the shell structure and the only non-zero BC is the axial movement of the shell's top and bottom edges. All shell structures are subjected to external hydrostatic pressure and therefor, the loading type is actually combination load of pure external pressure P plus axial compressive load with edge loading density of RoP/2. Mild steel material properties are assumed and elastic-perfectly plastic material with modules of elasticity E = 207 KN/mm 2, yield point Ovv = 300 N/mm 2 and Poisson's ratio v = 0.3 are employed throughout numerical calculations.

3 NUMERICAL RESULTS BOSOR5 [10] software has been adopted as the main numerical tool in this paper to perform collapse and bifurcation analyses. ABAQUS [11] software is also used for some selected points. Since collapse and non-axisymmetric bifurcation are the two possible failure modes of cylindrical and elliptical shells, both non-linear axisymmetric stress (collapse) analysis and bifurcation analysis from both software were performed to predict the failure modes and loads. The meridional curve was modelled by 91 small-segments in BOSOR5 and both 3-node axisymmetric line element SAX2 and doubly-curved 8-node shell element $8R5 were used for FE modelling in ABAQUS.

3.1 Buckling Pressure of Cylinder (Lo/Ro =1.0 and Ro/to --33.33) Numerical results of cylindrical shell from both analytical codes are presented in Table I. It is seen that the two methods predicted almost same collapse pressures. As for bifurcation failure, ABAQUS can not predict bifurcation failure mode while BOSOR5 predicted that the Lo/Ro =1.0 and R0/to =33.33 cylinder will lose its load carrying capacity by bifurcation with 7 circumferential waves in eigenmode. TABLE 1 PREDICTIONSOF BIFURCATION[ COLLAPSEPRESSURES Pbif (MPa)

Peon (MPa)

HEImH ABAQUS

i0.52

10.38

* Value in bracks denotes the circumferentialwave numbers in eigenmode

712

3.2 Buckling Pressure of Elliptical Barrelled Shells Due to the fact that the geometry profile of the elliptical barrel-shaped shells could only be determined once all three variables n~, n2 and b have been decided, enormous calculation efforts are required to draw the whole picture of the performance of the elliptical shells. Subjected to the limitation of the calculation resources, it is unpractical to fulfil this target. Thus, additional restrains were introduced to transform the 3-variable-involved problem to 1-variable-involved (by setting variable b unchanged and nl = n2) and 2-variable-involved (by setting variable b unchanged) problem.

3.2.1 One-Dimensional Analysis of Elliptical Shells (b/Lo = 1.0 and nl

=

n2)

For cylindrical shell of Lo/Ro =1.0 and R0/to =33.33, by setting pararheter b unchanged, the only variable nl(= n2) is needed to determine the geometry of the elliptical barrel-shaped shells in onedimensional analysis. Numerical results obtained from BOSOR5 are shown in Fig. 3 and the point of wall-thickness t = 3 mm corresponds to the cylindrical shell structure. It is seen from Fig. 3 that, without adding additional material and without losing the cylinder's axial length, the load carrying capacity of the cylindrical shell could be improved simply by re-arranging the material distribution. The peak point (nl = n2 = 2.1) fails at pressure 14.30 MPa which is 41% higher than the ultimate failure pressure of mass-equivalent cylinder. It is worth to point out that, with the continuous changing of variable ni = n2, the failure modes of the elliptical barrelled shells also change. Cylindrical shell and elliptical shells of n~ = n2 > 2.5 fail by bifurcation; elliptical barrels of nl = n2 < 2.5 fail by collapse, including the optimal (peak) point. 15.0

4.0.'

~.)

-

x

~s

10.0

3.0

i j

~ m

112 2.5

5.0

2.0 Collapse . . . . . .

0.0

i

2.0

2.2

i

B ifurcation

i

2.4 2.6 Wall-thickness t (ram)

1.5

, i

2.8

3.0

Figure 3: Variation of bifurcation/collapse load for elliptical shells of b/L0 = 1.0, n~ = n2

1.6 i

1.0

i 1.5

i 2.0

!

2.5

i 3.0

! 3.5

4.0

Ill

Figure 4: Contour plot of elliptical barrelled shell's performance (b/Lo = 1.0)

3.2.2 Two-Dimensional Analysis of Elliptical Shells (b/Lo = 1.0) In this section, the imposed restrain in Section 3.2.1of setting n~ = n2 is freed and the aim is to find the optimal structural geometry in the whole searching domain of (nl, n2). The optimal point is defined as such that the load carrying capacity at this point is higher than all its surrounding points. The searching process was performed like this: after a coarse study at some even-spaced points in (nl, n2) domain, finer computations will only concentrate on the region where optimal point could possibly occur based on the coarse study results. In this paper, four values of variable b were evaluated, i.e., b/Lo= 0.75, 1.0, 1.25 and 1.5. Only results of b/Lo= 1.0 are discussed in detail. 2-D contour plot was achieved to reveal the elliptical shells' performance of b/L0 = 1.0, shown in Fig. 4. The optimal point happens at (nt, n2) = (2.2, 2.0) and the magnitude of failure pressure is 14.71 MPa, which is 46% higher than that of the cylinder. Also, the failure mode at this optimal point is collapse. Fig. 5 illustrates the squashing process of the optimal barrel by external

713 hydrostatic pressure. The horizontal axis represents the axial displacement of the top (bottom) edge and the vertical axis represents the applied external pressure. The yield pressure of the optimal elliptical barrel at which the structural stress level at some parts of the structure exceeds the yield strength of the material is Py = 9.17 MPa. Deformed profiles of the meridional surface also show the spread of bending in the process of increasingly applied external pressure. Three other values of variable b were checked to find their respective optimal elliptical barrelshaped shells. The results are listed in Table 2 with information as where these optimal failure pressures were obtained and Fig. 6 illustrates how optimal pressures various versus variable b. TABLE 2 FAILURE PRESSURES OF OPTIMALBARREL-SHAPEDSHELLS b/Lo 0.75 1.00 1.25 1.50 .

.

.

.

.

.

.

.

.

.

.

.

.

.

....

Popt (MPa) 14.53 14.71 14.68 14.33 .

.

.

(nbn2) (4. 6 , 1.9) (2.2, 2.0) (1.29, 2.09) (1.005, 2.05)

.

.,,

..

n3 (mm) 114.5 115.0 113.3 111.7

1.48 16.0 It

1.46 ~, 12.0

== o.

Q. 9~

8.0

1.44

n

ir m

X 1.42'

4.0

0.0

0

0.01

0.02 0.03 0.04 Axial Disp. (ram)

0.05

0.06

Figure 5: Pressure versus end-edge axial displacement of optimal elliptical shell (b/L0 = 1.0). Magnitudes of yield and collapse pressures are marked and meridional deformed profiles are given

1.40 0.70

0.90

1.10 blLo

1.30

1.50

Figure 6: Normalised optimal pressures (Popt/Pcylinder) versus variable value b/Lo

4 E X P E R I M E N T A L DETAILS Since the load carrying capacity of cylindrical shells under external hydrostatic pressure can be improved by bowing out through generalised elliptical profiles, it is decided to perform several evaluation experimental tests to prove the numerical predictions.

4.1 General Information Four experimental specimens were manufactured, two of them are identical cylindrical shells and two are identical optimal elliptical barrel-shaped shells of b/L0 = 1.0. All specimens were CNC machined with top and bottom internal flanges and stress relief was performed at 650~ for 1 hour under vacuum conditions. The nominal dimension of the cylindrical specimen is L0 = R0 = 100mm and to = 3mm. Fig. 7 shows the pictures of finished specimen S 1-1 (cylinder) and El-1 (optimal elliptical barrel).

714

Figure 7" CNC machined specimens. (a) cylinder; (b) optimal elliptical barrel (b/Lo = 1.0) Wall thickness of all testing specimens was measured along 20 equally spaced circumferentials at 18 ~ intervals and 10 equally spaced meridians of arc-length by ultrasonic probe. Table 3 shows the average, minimum and maximum values of the wall thickness of all specimens. The shape measurement of testing specimens' external surface was conducted by laying each testing specimen on a rotating table and taking relative readings from two gauges. Measurement results revealed that the maximum initial radial variation was 4.1%. TABLE 3 WALL THICKNESS MEASUREMENT tave (mm) 3.02

Specimen SI-1 Sl-2

El-1 El-2

.

.

.

.

3.05

2.57 2.56

tmin (mm) 2.96 2.99 2.50 2.46 ,

tmax (mm) 3.07 3.10 2.67 2.65

. .

,

.

.

.

.

.

Tensile tests on round bars were carried out to determine the mechanical properties of the material. Four coupons were cut from longitudinal direction of the steel tube and test results showed that the average upper and lower yield point of material was 348.13 MPa and 329.40 MPa respectively. During evaluation test, each specimen was mounted in a pressure chamber and external pressure was supplied by hydraulic pump and measured by pressure transducer. Copper pipe connecting inside of the full-oil-filled specimen to an oil container placed on an electronical scale was used to measure the specimen's inside volume change during load applying process. At each pressure increment, the weight of oil being pressurised out from inside of each specimen was recorded when no more oil drops from the copper pipe was observed. In case material plasticity happens, the continuous material deformation would cause continuously oil dropping. It was decided that the reading of the weight should be taken 20s after each incremental pressure was met. Specimen failure (bifurcation or collapse) was considered to have taken place when such following happens: (1) sudden decrease in pressure of the chamber; (2) an audible sound or, (3) sudden large amount of oil being pressurised out. 4.2 Results and Discussion

Experimental data of all testing specimens are given in Table 4 together with numerical predictions from BOSOR5 and ABAQUS. Average lower yield strength, average wall-thickness, average radial diameter and average shell axial length measurement values were used in numerical calculations. Photographs of failed specimens S 1-1 and E 1-1 are shown in Fig. 8. First, let's look at the failure modes of testing specimens. Specimen S 1-1 and S 1-2 failed by bifurcation with 4 wave numbers across the circumferential direction while BOSOR5 predicts 7 waves. By measuring the dimension of each wave of the failed testing specimen, it revealed that the circumferential length of the 4 waves of specimens S 1-1 and S 1-2 exactly accounts for 4/7 of

715 TABLE 4 COMPARISON OF EXPERIMENTAL & ANALYTICAL RESULTS

Sample

BOSOR5 Results

Experimental Results Pexpt (MPa)

PBOSOR5 (MPa)

11.66 11.58

11.17 (7) 11.23 (7)

16.96 16.82

15.80 15.66

SI-1 S1-2 El-1 El-2

Pexpt-PBosOR5 Pexpt 4.2%

ABAQUS Results

PABAQUS (MPa)

3.1%

11.60 11.71

6.8% 6.9%

15.87 15.78

PexP:-PABAQUS Pexpt 0.5% - 1.1%

6.4% 6.2%

Figure 8: View of failed testing specimens. (a) cylinder S 1-1; (b) optimal barrel E 1-1 whole circumferential length. In other words, the above cylindrical testing specimens just failed by bifurcation mode BOSOR5 predicted. The reason why perfect bifurcation mode (full-wave numbers) did not happen is due to the unevenly distributed wall-thickness. As for specimens E 1-I and E 1-2, they all failed by collapse and it could be seen from Fig. 8 that, apart from inward radial deflection due to the applied external pressure, both specimens also deformed along the meridional direction and absorbed large amount of strain energy for the bending deformation. Surface cracks even happened at the outside surface of specimen E 1-1 although it was not a wall-through one. Next, let us consider the difference of the experimental results and the numerical results. It is seen from Table 4 that the percentage errors between experimental results and numerical results vary from -1.1% to 4.2% for bifurcation failure specimens (cylindrical shells) and from 6.2% to 6.9% for collapse failure specimens (optimal elliptical barrelled shells) respectively. Although the overall numerical predictions are a bit conservative compared with experimental results, all predictions from both analytical methods (BOSOR5 and ABAQUS) exhibit direct engineering-usevalue.

5 CONCLUSION Numerical and experimental studies are presented for the bucking analysis of metallic cylindrical and mass equivalent elliptical barrel-shaped shells subjected to external hydrostatic pressure. In the study, numerical software BOSOR5 and FE software ABAQUS were employed to perform bifurcation buckling and collapse analyses and experimental studies were conducted according to numerical results by CNC machined testing specimens. Main results obtained may be summarised as follows: (1) Buckling strength of mild steel cylindrical shells under external hydrostatic pressure can be improved by bowing-out without increasing the material volume (weight) while at the same time keep the original axial length unchanged. For the cylinder geometry of Lo/R0 =1.0, Ro/t0 =33.33,

716 the buckling pressure of the optimal elliptical barrel-shaped shell can be 46% higher than that of the cylindrical shell. (2) The finding procedure of the optimal elliptical shell geometry presented in this paper involved only 2 variables although 3 variables are actually associated to determine the elliptical surface profile. Due to the huge amount of calculation efforts needed to cover all possibilities, appropriate optimisation methods must be introduced and, at this time, optimisation technique called TABU search method is now being tested. (3) Four experimental specimens made of mild steel were CNC machined according to numerical computation result. Experimental result proved that BOSOR5 predicts failure mode more accurate than ABAQUS and agreed well with numerical predictions of bifurcation buckling pressure; for structural collapse, both software predict nearly the same collapse pressures and the predictions are a bit conservative.

ACKNOWLEDGEMENTS The author wishes to thank ORS Award from Committee of Vice-Chancellors and Principles of the Universities (UK) and Hsiang Su Coppin Memorial Scholarship (The University of Liverpool) for their financial supports.

REFERENCES

1. Kruzelecki J. (1997). On optimal barrel-shaped shells subjected to combined axial and radial compression. In: Gutkowski W, Mroz Z, editors. Structural and Multidisciplinary optimisation. IFTR P A N - WE Lublin 1,467-472. 2. Blachut J. (1987). Optimal barrel-shaped shells under buckling constraints. AIAA 25, 186-188. 3. Blachut J. (1987). Combined axial and pressure buckling of shells having optimal positive gaussian curvature. Computers & Structures 26(3), 513-519. 4. Blachut J. and Wang P. (2000). Buckling of barrelled shells subjected to external hydrostatic pressure. ASME PVP Codes and Standards 407,107-114. 5. Zintilis G. M. and Croll J. G. A. (1983). Combined axial and pressure buckling of end supported shells of revolution. Eng. Struct. 5,199-206. 6. Lukasiewicz S. and Wawrzyniak A. (1976). Stability of shells revolution with slightly curved generator under complex load. Mech Teor Stos 14, 535-545. 7. Odland J. (1981). Theoretical and experimental buckling loads of imperfect spherical shell segment. J. of Ship Research 25(3),201-218. 8. Schmidt H. and Krysik R. (1991). Towards recommendations for shell stability design by means of numerically determined buckling loads. In: Jullien J. F., editors. Buckling of Shell Structures, on Land, in the Sea and in the Air, Elsevier Appl. Sci., London NY, 508-519. 9. Blachut J. and Wang P. (1999). On the performance of barrel-shaped composite shells subjected to hydrostatic pressure. In: Toropov V. V., editors. Engineering Design Optimisation, Proc. Of I st ASMO UK/ISSMO Cofer., 59-65. 10. Bushnell D. (1976). BOSOR5-program for buckling of elastic-plastic complex shells of revolution including large deflections and creep. Computers & Structures 6, 221-239. 11. Hibbit H. D., Karlsson B. I. and Sorensen E. P. (1998). ABAQUS User's Manual, Version 5.8, Pawtucket, R102860-4847, USA.

Section XII ULTIMATE LOAD CAPACITY

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Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

719

EXPERIMENTAL TECHNIQUES FOR TESTING UNSTIFFENED PLATES IN COMPRESSION AND BENDING M. R. Bambach 1 and K. J. R. Rasmussen I 1Department of Civil Engineering, Sydney University, Australia

ABSTRACT

Details of a dual-actuator rig developed for testing rectangular plates supported on three sides, with the remaining (longitudinal) edge free, under combined uni-axial compression and in-plane bending are presented. Particular attention is given to ensuring a constant strain gradient at the loaded ends, as opposed to a constant load eccentricity, in order to determine the post-buckling behavior and ultimate load and moment capacities of unstiffened thinwalled elements. Strain gradients varying from pure compression to pure bending are facilitated. Attention is also given to ensuring simply supported boundary conditions, and the methods used for anchoring the tensile stresses that develop at the loaded edges as a result of large plate deflections. Details of the methods for controlling the applied displacements are given, for which a system of four laser displacement devices were employed in order to achieve the required strain gradient. The operation of the rig is verified against established theoretical solutions. KEYWORDS

Plate testing rig, unstiffened plate elements, stress gradients, dual-actuator control INTRODUCTION

Open thin-walled sections consist of stiffened and unstiffened component plates. If these elements are sufficiently slender, the section will locally buckle at a load significantly less than the ultimate load carrying capacity. The section will continue to resist load after local buckling due to the redistribution of longitudinal stress from the most flexible regions. This redistribution may be simplified for design purposes by assuming that certain regions of the cross section remain effective up to the yield point of the material, whilst the remainder is ineffective in resisting load. Current specifications for the design of open thin-walled sections provide equations for determining the effective width of stiffened and unstiffened elements, and the ultimate capacity of the section is calculated from the effective section properties.

720

Extensive experimental and analytical studies on stiffened elements (supported along both longitudinal edges) have been carried out by many researchers and have led to well established equations for the estimation of the effective width of such elements in uniform compression and under stress gradients. In comparison, experimental investigations on unstiffened elements (supported along one longitudinal edge) are quite limited. In the 1970s the applicability of the effective width concept to unstiffened elements under uniform compression was studied in detail by Kalyanaraman et al. (1977), who tested a large number of beams and short columns that contained web elements that were fully effective. Tests of a similar nature previously reported by Winter (1946, 1970), and the experimental and analytical research by Kalyanaraman et al. (1977), led to the adoption of the effective width approach for uniformly compressed unstiffened elements in the 1986 edition of the AISI Specification. Beam tests on sections that contain fully effective webs and simple edge stiffeners subjected to stress gradients have been reported by Desmond et al. (1981) and Winter (1947), and on open channels with inclined flanges under stress gradients by Rhodes (2000). Single plate test data for unstiffened elements with stress gradients have only been reported by Rhodes et al. (1975), where results of 4 tests on individual plates simply supported on three sides are given for varying values of load eccentricity. Due to the lack of test data, current AISI (1996) and Australian specifications (AS/NZS 4600 1996) treat unstiffened elements with stress gradients as if they were uniformly compressed for effective width calculations. Recent studies however, of cold-formed plain channels by Rasmussen (1994) and fabricated Isections by Chick and Rasmussen (1999) have demonstrated a marked conservatism in this method. Experimental and analytical results show that beam-column capacities were typically 30% or more higher than those estimated by the current specifications. The conservatism could be traced directly back to the bending capacities, which were of the order of 50% of the actual strengths, due to excessive conservatism inherent in the design procedures for unstiffened elements with stress gradients. This paper outlines the experimental investigation initiated as a means to establishing more accurate design methods for unstiffened elements under stress gradients. In order to determine the fundamental behaviour of these elements, the tests are performed on rectangular plates simply supported on three sides, as opposed to testing complete sections, so as to avoid unquantified restraints from adjoining elements. LOADING CONDITION- CONSTANT STRAIN ECCENTRICITY

The method for the application of the load is critical for unstiffened plate specimens. Researchers testing stiffened plates under stress gradients commonly use rigs that apply a load eccentrically to the ends of the plate, allowing the ends to rotate in-plane to maintain constant loading eccentricity. This method however, does not facilitate accurate results for unstiffened plates. For example, if a load is applied at the centre of an unstiffened plate (zero load eccentricity), at elastic local buckling the longitudinal stresses in the plate will redistribute towards the supported edge, and the resistance offered by the plate becomes eccentric. If the load point remains at the centre of the plate severe in-plane bending is induced, and consequently little post-buckling strength is achieved. This condition is also not congruent to that of plane sections remaining plane in assemblies of plates, which must be satisfied if accurate design methods are to be produced from the tests, since this is the condition of the nodal lines in a member with a number of locally buckled half-wavelengths.

721

To accurately capture the post-buckling behaviour of an unstiffened plate, and maintain a condition of plane sections remaining plane, a rig has been developed that can apply a constant strain eccentricity to the ends of the plate. El

~

j

Freeedge

El

El

El

El

El

.................. tf Z f !il 1t Figure 1" Strain Gradients for Test Series

DUAL-ACTUATOR RIG

Figure 2: Dual Actuator Rig - Plan and Elevation The rig incorporates the use of two hydraulic actuators, a (primary) 200 ton Dartec actuator applying compressive strains, and a (secondary) 25 ton MTS actuator that is connected to lever arms as shown in Figure 2, applying bending strains to the specimen. The dual action can apply strain gradients varying from pure compression to pure bending as required.

Primary Actuator- Compression Rig The primary actuator is mounted as a compression rig, with high strength Maccalloy tension bars providing the reaction force to the actuator, as shown in Figure 2. The rig is quite flexible compared with those having a fixed reaction frame. As the actuator applies load the tension bars extend elastically, such that the specimen will axially shorten and shift in the longitudinal direction.

722

J11

IIIII

,,5

ilL,

Figure 3: Detailed Schematic of the Rig - Side View

Secondary Actuator- Bending Rig The secondary actuator is seated on support beams and connected to a long SHS as shown in Figure 2. The actuator and the SHS are connected to lever arms that extend to the end platens of the compression rig, where they are rigidy connected. Extension or contraction of the actuator causes the lever arms to rotate about the pin joints shown in Figure 2. The inplane rotation of the end platens can thus be controlled directly. Since the actuator must move relative to its supports, friction is a concem. To minimise this the actuator is seated on nine ball bearings constrained in a grid, denoted J15 in Figure 2, and the SHS is seated on four ball bearings, such that the lever arm rig is essentially floating.

Boundary Conditions Establishing simply supported boundary conditions is essential, particularly for the loaded ends of the plate. The elastic critical buckling stress for slender plates is significantly higher if rotational restraint exists at the simply supported loaded ends. In order to maintain free rotation, the loaded ends of the plate are seated in machined key-ways cut in segments of circular rods, which are fitted into split spherical needle bearings housed in solid bearing blocks, as shown in Figure 4. The rod segments are cut in 20mm lengths, such that the loaded ends of the plate may rotate by varying degrees across the width.

Figure 4: Loaded Edge of a Plate, Rod Segment and End Bearing Block

723

In a slender unstiffened element, the longitudinal stress may redistribute to such an extent after local buckling that tensile stresses may develop at the unsupported edge. If these tensile stresses are not anchored, the loaded edge will deform and the assumption of plane sections remaining plane will no longer be valid. It is important then to restrain the loaded edge, and for this purpose holes are drilled in the rod segments and in the plate specimens where tensile stresses are likely to occur as shown in Figure 4, and pins are inserted such that the end bearings may resist tension while allowing rotation. This condition is important to the operation of the rig, particularly for the case shown in Figure 1 where compressive strain is applied to the unsupported edge and varies linearly to zero at the supported edge. Slender plates under this load condition have large out-of-plane deflections of the free edge in the post buckling range, and as a result the loaded edges are prone to deformation. The pure bending load case in Figure 1, whereby compressive strain is applied to the free edge and tensile strain to the supported edge, presents a unique problem. As the plate is loaded in the post buckling range, the region of the specimen in compression becomes less effective, and the neutral axis shifts. As a result, a large portion of the cross-section yields in tension in order to attain ultimate capacity. Drilling holes through the plate to insert pins for resisting tension is not possible, as the plate will fracture through the line of holes and the ultimate condition cannot be captured. For specimens tested under this load condition end plates are welded to the specimen, such that the boundary conditions of the loaded edges are now fixed, not simply supported. Another effect of the neutral axis shift for this load condition is that the plates develop net tension after local buckling. Stiff compression struts are required between the northern and southem headstocks of the compression rig to facilitate the net tension.

Figure 5: Discreet Finger Support The simple support condition along the unloaded supported edge is achieved by the use of discreet 'finger' supports, as shown in Figure 5, which were originally developed at Cambridge University (Moxham 1971). The plate edge is inserted between the first set of pins only and clamped by tightening the top bolt, thus restraining out-of-plane displacements of the edge while allowing rotation. The use of fingers requires no edge preparation to the plate. The fingers are supported on a rod by means of a spherical bearing, such that the fingers may rotate to allow small in-plane transverse displacements of the edge. The rod has a secondary rod beneath it to assist in minimising bending, as shown in Figure 3, and both rods are seated in linear bearings. The linear bearings allow the entire finger assembly to shift in the longitudinal direction, in accordance with the shift of the specimen as the tension bars of the compression rig extend. The fingers contain a slender section near the base such that they are flexible in the longitudinal direction, thus negligible load is transferred through the fingers to the support. The test setup in Figure 6 shows the use of the end bearings and fingers to provide three sided simple support.

724

Figure 6: End Bearings and Fingers Provide 3-Sided Simple Support CONTROL OF THE APPLIED STRAIN GRADIENT The flexibility of the rig requires that an independent measurement system be used to control the applied strain gradient. Initially LVDTs were mounted on the end bearings on each side of the specimen, as shown in Figure 7b, to measure the longitudinal displacement between the end bearings on the east and west sides. However internal displacements within the end bearings preclude accurate measurements of the axial shortening of the plate. The displacements are twofold; firstly there is slack in the bearing between the rod segments and the spherical bearing, and within the spherical needle bearing itself, and secondly the bearing block elastically compresses. For the case of uniform compression the slack can be excluded from the results, and experiments have shown this to be of the order of 0.2mm. The compression of the bearing blocks causes a change in slope of the load-displacement curve, or apparent change in stiffness, and this too could be removed from the results with careful processing of the test data. The difficulty arises however with the cases where strain gradients are applied. For these cases one longitudinal edge is compressed while the other edge remains unloaded. The individual rod segments settle by different amounts as the slack is only taken up when load is applied, therefore differential settlement within the bearings occurs across the width of the plate. A measurement system is thus required that can measure the displacement of the loaded ends of the plate directly.

Figure 7: Displacment Measurement Systems Various displacement measurement systems have been investigated, and their accuracy determined by comparison with readings from strain gauges that am fixed to both sides of the plates on each longitudinal edge at mid-length, which can be seen in Figure 6. The gauges are accurate to within 5 microstrain. The most reliable system is to rigidly attach a rod across the width of the specimen with epoxy glue, as close to the ends as is practical, as

725

shown in Figures 6 and 7c. LVDTs are mounted on each side of the plate running longitudinally, such that the displacement of the east and west sides is known. The ratio of the values prescribes the strain gradient on the plate (Figure 7a). This method was found to be inaccurate for the more slender specimens however, due to large out-of-plane displacements of the plate causing the rod ends to displace vertically. For slender specimens, targets are mounted on the rods and laser measurement devices are positioned at each rod end (Figure 7d). The mounting arrangement uses bearings to ensure verticality of the target at all stages of plate deflections. The difference between the two readings on each side of the plate gives the displacements of the east and west sides respectively, and the strain gradient may be deduced. If the targets are set carefully only small inaccuracies occur when the rod ends displace vertically (i.e. when large lateral displacements occur). In all tests, the strain gauges are used to determine the applied strain gradient until out-of-plane deflections occur. The gauge readings are not applicable in determining the (average) applied strain after local buckling, since they record the Iocalised strain at the centre and do not account for shortening resulting from plate deflections. The tests are performed by setting stroke speeds on the digital controllers. The controller for each actuator is set to a prescribed control-slave system according to the strain gradient required, whereby one actuator runs at a set stroke speed, and the other actuator runs as a slave at a ratio of that stroke speed. This is achieved by sending the voltage signal from the intemal LVDT of the master actuator to the controller for the slave actuator, and amplifying or reducing the signal accordingly. Due to the flexibilty of the rig, the stroke ratio is determined experimentally and needs to be manually adjusted continuously throughout the test. Continuous monitoring of the displacement measurement system allows the strain gradient to be known at all times during the test, and the stroke ratio is adjusted to produce the required gradient.

OUT-OF-PLANE DISPLACEMENT MEASUREMENT A measurement frame is situated above the specimen as shown in Figure 3, and has two high-prescision rails running longitudinally. A trolley is mounted by means of linear bearings on the rails, and is attached to a timing belt and pre-programmed stepper motor. Transducers are mounted from the trolley and positioned in a line across the width of the specimen. The line of transducers is run along the length of the specimen before and during the test, producing a fine representation of the initial imperfections and buckled shape.

TEST SPECIMENS The test series consists of 80 mild steel specimens, all of nominal thickness 5mm. Plate width to thickness ratios vary from 12 to 35, representing slendemess ratios (Z = ~fi/for ) of 0.75 to 2.17. The aspect ratio is 5 for all specimens, such that the buckling coefficient is close to the asymptotic value of 0.425 for unstiffened elements (Bulson 1970), and such that end effects are minimised. Half the specimens have been heat treated to induce residual stresses approximating those that exist in fabricated sections. Details of the processes and magnitudes of stress induced are given in Bambach and Rasmussen (2001). For each of the four load cases shown in Figure 1, 20 specimens are tested. All specimens have 4 strain gauges attached, one on each side of the plate near the longitudinal edges at mid-length. The 20 specimens that are to be tested under the pure bending case shown in Figure 1 have welded end plates.

726

VERIFICATION

Since the behaviour of unstiffened elements under uniform compression has been more widely reported on than for those under stress gradients, the compression case will be used to verify the rig. The primary concern in establishing that the rig is performing correctly is in the boundary conditions of the loaded edge. A number of pure compression tests were performed on specimens of nominal dimensions 200x600x5mm, for which established theoretical solutions by Bulson (1970) estimate that the elastic critical buckling load will be 60kN if the loaded edges are simply supported, and 97kN if they are fixed. Tests without the pins in the end bearings showed the bearings to be performing well, freely allowing rotation at the loaded edges and the specimens buckled at loads close to 60kN. The tangent to the Load vs. Lateral Displacement Squared method, as described by Venkataramaiah and Roorda (1982), is used to deduce the critical load. It was found that when the pins were used in the end bearings, the bearings had a tendancy to lock and not freely allow rotations. To avoid this, the end bearings must be parallel to the ends of the plates to within a very fine tolerance. To achieve this tolerance, the strain gauges were introduced. During setup, the primary actuator applies a small compressive load and the secondary actuator is adjusted (i.e. the end bearings are rotated in-plane) until the strain gauges read equal compressive strain on each longitudinal edge. Following this procedure ensures that the end bearings are free to rotate, and critical buckling loads show that the plates are effectively simply supported at the loaded ends.

Figure 8: Typical Load - Lateral Displacement Squared Curve

Figure 9: Specimen at Ultimate Load

Figure 8 shows a test result for a three sided simply supported plate under uniform compression, with the pins inserted. The elastic buckling load is very close to that for a plate with simply supported ends (60kN). The photo in Figure 9 shows the plate at the ultimate load carrying capacity, where the buckle has Iocalised at the loaded end shown. Figure 8 verifies that the application of a constant strain eccentricity creates a testing condition whereby the full post-buckling behaviour of unstiffened plates may be captured. The specimen reaches an ultimate capacity of about 2.7 times its elastic buckling load. Some test results for ultimate load carrying capacites of specimens under uniform compression are compared with tests on beams and short columns (Kalyanaraman et al. 1977) in Figure 10, and with the Winter curve for unstiffened elements in pure compression, given by:

727

1.2

I

I

,

I

I

,-, ~ " "~. .

1.0

O

~

0.6

SS

SS

Kalyanaraman et al. (1977) Tests:

. 9

9 Stub column + Beam

o

"'. ~~'~".,.,.;.... x

0.4

I

SS

"'..

0.8

,,

buckling Local

~

"'..~-. +

....... +

-

~ 1 7 6 1 7 6 1 7 6 1 7o6 O O , o o o . , O o o

0.2

- ...... 0 x

0

0.5

~ Winter for u n s t i f f e n e d ~ e l e m e n t s Test results (Bambach and Rasmussen, 2001) - Unwelded specimens Test results (Bambach and Rasmussen, 2001) - Welded specimens I I I I I

1.0

-,

i

1.5

2.0 2.5 3.0 3.5 Slenderness X Figure 10: Comparison of Test Results for Unstiffened Plates in Uniform Compression

CONCLUSIONS Experimental techniques for testing rectangular plates simply supported on three sides, in combined compression and in-plane bending have been presented. Results for uniform compression tests have been verified against established theoretical solutions, and shown to be in general agreement with existing test data of sections. The paper details methods for ensuring simply supported longitudinal and loaded edges, anchoring tensile stresses at the loaded edges, and measuring the applied strains.

REFERENCES

American iron and Steel Institute. (1996) Specification for the Design of Cold-Formed Steel Structural Members. 1996 Edition, Washington D.C. AS/NZS 4600 (1996). Australian/NewZealand Standard. Cold-Formed Steel Structures. 1996 Edition, Sydney Bambach M.R. and Rasmussen K.J.R. (2001). Residual Stresses in Unstiffened Plate Specimens. Proceedings of the Third International Conference on Thin-Walled Structures,Cracow, Poland. Bulson P.S. (1970) The Stability of Flat Plates. Chatto and Windus, London Desmond T.P., Pekoz T. and Winter G. (1981). Edge Stiffeners for Thin-Walled Members. Journal of the Structural Division, ASCE, 107:ST2, pp 329-353 Chick C.G. and Rasmussen K.J.R. (1999). Thin-Walled Beam-Columns.2:Proportional Loading Tests. Journal of Structural Engineering, ASCE, 125:11, pp 1267-1276 Kalyanaraman V., Pekoz T. and Winter G. (1977). Unstiffened Compression Elements. Journal of the Structural Division, ASCE 103:ST9, pp 1833-1848 Moxham K.E. (1971) Buckling Tests on Individual Welded Steel Plates in Compression. University of Cambridge, Report CUED/C-Struct/TR.3 Rasmussen K.J.R. (1994). Design of Thin-Walled Columns with Unstiffened Flanges. Engineering Structures, 16:5, 1994, pp 307-316 Rhodes J., Harvey J.M. and Fok W.C. (1975). The Load-Carrying Capacity of Initially Imperfect Eccentrically Loaded Plates. International Journal of Mechanical Sciences, Vol. 17, pp161-175 Rhodes J. (2000). Buckling of Thin Plates and Thin-Plate Members. Structural Failure and Plasticity, IMPLAST 2000, pp 21-42 Venkataramaiah K.R. and Roorda J. (1982). Analysis of Local Plate Buckling Data. Proceedings of the Sixth International Specialty Conference on Cold-Formed Steel Structures. 1982. Winter G. (1947). Strength of Thin Steel Compression Flanges. Transactions, ASCE. Vol. 112, pp 527-576 Winter G. (1970). Commentary on the 1968 Edition of the Specification for the Design of Cold-Formed Steel Structural Members, American Iron and Steel Institute, New York

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Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

729

EFFECTS OF ANCHORING TENSILE STRESSES IN AXIALLY LOADED PLATES AND SECTIONS M. R. Bambach 1 and K. J. R. Rasmussen 1 1Department of Civil Engineering, Sydney University, Australia

ABSTRACT Thin-walled compression members are commonly designed on the assumption that the loaded edges remain straight. Under this assumption, tensile stresses develop in the most flexible parts of the component plates at advanced local buckling deformation, and thus are assumed to be 'anchored' at the ends. However, current design rules for plate elements, such as the Winter formulae, are based partly on tests in which the load was applied by use of rigid platens that did not permit tensile stresses to develop. Recent literature has pointed to the apparent inconsistency between the assumption of straight loaded edges and the use of a design curve calibrated from tests in which the loaded edges of component plates may not have remained straight. The paper addresses this apparent inconsistency by comparing finite element solutions for the conditions of straight loaded edges and loading by use of a contact surface between the plate edge and a non-deformable rigid body end platen, where there is no constraint for the plate edge to remain in contact with the rigid body. Solutions are provided for a single halfwavelength of unstiffened and stiffened plate elements simply supported along three and four edges respectively. The effect of multiple half-wavelengths is also investigated, as is the effect of interaction between elements in practical sections comprising stiffened and unstiffened elements. KEYWORDS Local buckling, stiffened and unstiffened plate elements, FEM, ultimate capacity, stub columns, tests INTRODUCTION Compression members such as the unlipped channel shown in Figure 1, will locally buckle in a number of half-wavelengths and will retain this mode shape into the post-local buckling range, until Iocalisation occurs in one of the half-waves, propagating failure. It is well-known that loads at failure may be considerably higher than those at which local buckling occurs,

730

due to the redistribution of longitudinal stresses from the flexible to the stiff parts of the component plates. In the stiffened element, supported along both longitudinal edges, the longitudinal stress pattem across the width of the loaded edges changes from a condition of uniform stress prior to local buckling, to that shown in Figure 2a in the post-buckled range. The stresses redistribute from the deformed centre of the plate to the supported edges. Similarly for unstiffened elements, supported on one longitudinal edge, the stresses redistribute from the deformed edge of the plate to the supported edge, as shown in Figure 2b. If the component plates are sufficiently slender, the large plate deflections may cause redistribution of the longitudinal stresses to such an extent that tensile stresses develop at the nodal lines in the regions of maximum deformation. Half-wavelength

(a)

dal planes

Locally

(b) Figure 1: Locally Buckled Member In a uniformly compressed long column, consisting of many local buckle half-waves (cells), the condition of continuity at the nodal lines allows the tensile stresses in each cell to be anchored in its neighbouring cells. In comparison, if a single cell, such as in Figure l a, is uniformly compressed, the nodal lines are now the loaded ends of the short column, and tensile stresses may only develop if the ends are anchored. In a testing situation, anchorage could be achieved by welding stiff end plates to the short column. In the absence of welded end plates, which is the most common situation, the loaded edges will pull away from the end platens at advanced stages of local buckling in the most flexible regions of the cross-section, and as a result the loaded edges will not remain straight. This is shown later in the paper to be the case for unstiffened elements, and to a lesser extent for stiffened elements, (Figures 6 and 7). (a) Stiffened Element (b) Unstiffened Element

~"-Lor~tudnal stresses/ ~:ends Figure 2: Postbuckled Plates

731

Winter(1970) proposed and verified experimentally the following effective width formulae for stiffened (Eqn. 1) and unstiffened (Eqn. 2) compression elements, based on experimental programs (Winter 1947,1970 and Kalyanaraman et al. 1977). b~='/~'II-O'22-/f'~l/b ~l,,t, ~./~

(1)

Many of the tests were conducted on beams where the compression elements locally buckled with several half-wavelengths, and anchorage of tensile stresses could be expected due to continuity of the member at its nodal lines and 'compressed ends' (being the points of application of the beam loads). However, a number of the tests were on stub columns, particularly those tests on unstiffened elemGnts, for which the ends were free to pull away from the end platens. Since the Winter formulation is the basis of the effective width formulae used in the current American specification AISI (1996) and the AS/NZS 4600 (1996) specification, there is an apparent inconsistency between the common practical condition of long columns where tensile stresses can be expected to develop in the area of failure near the centre, and the method used to design it, being calibrated from tests in which loaded edges may not have remained straight. The objective of this paper is to quantify the effect of this inconsistency. From a practical viewpoint, the anchorage of tensile stresses is likely to be important mainly for short members, notably columns forming a single half-wave. For such members, tensile stresses can only develop if the member is welded to stiff adjoining elements, as mentioned above. If a short member is connected through a few points only, such as a stud screwed to the track of a wall frame, tensile stresses cannot develop at the ends and the ultimate load is likely to be affected. However, for long members forming a large number of local buckles and failing near the centre, the ultimate load is unlikely to be significantly affected by the end support conditions. This is because the nodal lines remain essentially straight in the central region, thus allowing tensile stresses to develop. It is noted that in the case of an even number of half-waves, the central nodal line remains exactly straight by symmetry, and evidently, the larger the number of half-waves, the closer the adjoining nodal lines will be to remaining straight. FEM ANALYSIS

In order to examine the effects of the loaded edge conditions on the post-buckling strength of sections, the stiffened and unstiffened components are analysed separately, as four sided and three sided simply supported plates respectively. Finite element analyses using the FEM program Abaqus v5.7 are carried out, using 4-node isoparametric shell elements with five integration points through the thickness. Initially a linear elastic analysis is performed and verified against well established theoretical solutions for elastic buckling stresses (Bulson, 1970), and the buckled shape is then scaled to generate geometric imperfections in the model. A maximum out-of-plane imperfection of 10% of the thickness is used for all models. Non-linear material properties are introduced and a geometric and material non-linear analysis is performed, using the Riks Arc Length method to determine the ultimate load carrying capacity. Two conditions are analysed in the FEM analysis. The 'Straight' condition, whereby the loaded edges are constrained to remain straight by using the geometric loading condition of

732

Figure 3: Contact Models Loaded by Rigid Bodies prescribing all nodes at the loaded ends to displace an equal amount in the longitudinal direction. The second, 'Contact' condition, uses a contact surface between the loaded ends and non-deformable rigid bodies, where there is no constraint for the plate ends to remain in contact with the rigid bodies. In the analysis, the rigid bodies are prescribed to move in the longitudinal direction only (no in-plane rotation is permitted), and bear on the contact surface of the plate. The advantage of this model is that the loaded ends of the plate are allowed to deform, since the contact elements allow the loaded ends to pull away from the rigid bodies. These two conditions reflect the loading of short columns between rigid end platens, with and without welded end plates respectively. Figure 3 shows the Contact models for (a) unstiffened plates and (b) stiffened plates. VERIFICATION WITH TESTS

The non-linear models are compared with experimental data by the authors, and with the Winter Equations 1 and 2. The ongoing experimental program by the authors includes tests on individual steel plates, simply supported on three sides, under various combinations of compression and in-plane bending. The test results from those specimens under uniform compression are discussed here. Further details of the experimental method are given in Bambach and Rasmussen (2001). Uniform compression tests were performed on unstiffened mild steel plates of nominal thickness 5mm, widths of 60 to 125mm, and lengths corresponding to an aspect ratio of five for all specimens. Coupon data showed the plates to have a yield stress of 272MPa, and a Young's modulus of 2.02x105 MPa. Imperfection measurements showed the average out-ofplane imperfection of the free edge to be 0.37mm, which is close to the assumed value of 0.5mm in the model. Two tests were conducted for each specimen dimension. For comparison with the tests, the material properties of the above specimens were included in the FEM model, assuming an elastic perfectly-plastic material stress-strain curve. The FEM results for stiffened elements are compared with Winter Eqn. 1 in Figure 4, and for unstiffened elements are compared with the tests and Winter Eqn. 2 in Figure 5. These results are for a single half-wavelength, as detailed in the next section. The FEM analyses are extended to include all width to thickness (b/t) ratios within the limits as set out in AS/NZS 4600 (1996), being 500 for a stiffened compression element (a slenderness of

733

k=10), and 60 for an unstiffened element (a slenderness of X=3.6).The critical buckling stress

(Iol) and slenderness (~.) are given by:

J;~ = 12(1_o2------~

2_1 = [ f i t

i

o

......

I

I

'

1

'"'

0.75 I~ ~ .

-. , . . ~ "-

Winter for stiffened elements Abaqus-straight loaded edge Abaqus-contact loaded edge

0.50 r~

0.25

0

,

I

0

,.

I

2

I

I

4

6 Slenderness k

8

10

Figure 4" Strength Curve for Stiffened Elements It is noted here that the comparison of the FEM results for unstiffened elements with the Winter Eqn. 2 for unstiffened elements shows a slight optimism in the Winter equation in the slenderness range 0 . 7 5 - 1.55, and a slight conservatism for slenderness values exceeding 1.75. This trend is verified experimentally in the results obtained by Kalyanaraman et al. (1977) and by those of the authors (noting that the authors experimental program has yet to be extended to slenderness values exceeding 1.75). 1.2

r~

i

'

t

I

9

I

'

[

i

O

0.8

Kalyanaraman et ai. (1977) Tests:

9

9 Stub column + Beam

oo~..

.,..t

'1 t~

....

1.0 "'.oo

r.f]

i

0.6

+

~

"

r~

0.4

0.2

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......

W i n t e r for unstiffened elements

---. o

Abaqus-straight loaded edge Abaqus-contact loaded edge Testresults (Bambach and Rasmussen, 2001)

........

I

I

I

I

I

1.0

1.5

2.0

2.5

3.0

I

3.5

Slendemess k

Figure 5: Strength Curve for Unstiffened Elements

734

FEM RESULTS FOR A SINGLE HALF-WAVELENGTH

Figures 4 and 5 show the strength curves based on the section dimensions and ultimate loads given in Table 1. They represent plate specimens which buckle as a single halfwavelength, for which a stiffened element will have an aspect ratio of one. Unstiffened plate elements have one longitudinal edge simply supported, and if this boundary condition is satisfied and no rotational restraint is applied to the edge, the plate will form a single buckle with a half-wavelength equal to the length of the plate. Bulson (1970) has shown that the buckling coefficient for unstiffened elements asymptotes to a value of 0.425 at large aspect ratios, and in order to achieve a similar value for the coefficient in the tests, an aspect ratio of five was used. This aspect ratio was also used in the FEM analysis in Table 1. The influence of the aspect ratio will be discussed further in this paper. STIFFENED W i d t h . Length (mm) (mm) 115 115 155 155 195 195 240 240 335 F 335 465 465 620 620 775 775 1550 1550 UNSTIFFENED Width Length (mm) ( m m ) 60 300 80 400 100 500 115 575 125 ,' 625 150 750 175 i 875 210 ~ 1050 250 1250 300 1500 ,

,

,

,,

,,,

,

,

,

|

EM~MENTS Slenderness

Thickness 3mm Ultimate Load (kN) Difference (~) Straight Contact 0.74 90.5 90.5 0.0% 1.00 103.1 0.0% 103.1 1.26 104.8 0.0% 104.8 0.0% 1.55 109.3 109.3 118.8 0.0% 2.17 118.8 126.5 3.01 126.4 0.1% 1.2% 130.9 129.3 4.01 133.3 - 0.2% 5.01 133.6 141.5 - 2.2% 138.5 10.02 Nominal Thickness 5mm Thickness 5mm "TESTS ELEMENTS Slenderness, Ultimate Load (kN) Difference Width Slendemess Ultimate Load (kN) (~,) , Straight Contact (mm) (;L) 0.0% 61.16 0.74 79.1 0.71 80.7 80.7 0.0% 60.62 0.73 82.5 1.03 110.8 110.8 0.0% 80 1.07 95.7 1.19 101.5 101.5 0.0% 79.92 1.06 103.4 1.37 101.6 101.6 0.1% 100.36 1.21 113.3 1.49 ~ 104.5 104.4 2.1% _- 100.3 1.21 94.5 1.79 , 117.7 115.2 4.0% 125.64 1.52 94.8 2.08 , 132.1 126.8 8.2% = 125.75 1.52 94.3 2.50 i 150.1 137.8 10.6% 2.98 171.3 153.2 3.57 176.2 160.4 .

,

.

.,.

..

....

i

j

.....

=

|

Table 1" FEM and Test Results for 1 Half-Wavelength Stiffened Elements

The strength curves in Figure 4 for the two loaded edge conditions, Straight and Contact, are virtually coincident. Only at high slendemess values, greater than 3, do tensile stresses at the loaded ends have any effect. Values of tensile stress and end deformation for the specimen of slenderness ;k=5, corresponding to a large but still practical bit ratio of 260, are detailed in Figures 6a and 6b. Figure 6a shows the longitudinal stress variation across the width of the loaded ends at ultimate. The stress decreases at the supported edges since these edges are free to pull in transversely. In the central region of the loaded edge of the stiffened plate, for the Straight loaded edge condition, small values of tension are present. The maximum value of tensile stress at ultimate is 2 MPa. For the Contact condition, where

735 there is no restraint to keep the edge straight, the central region pulls away from the rigid body. The longitudinal stress is zero in this region. The pull-in of the loaded edge is very small at ultimate, with a maximum of 4xlO3mm, as shown in Figure 6b. The distribution of longitudinal stress for the two cases are virtually coincident, and as a result the ultimate loads are nearly the same. The simplified stress distribution using the Winter effective width Eqn. 1 is also shown for comparison, and gives a good approximation of the ultimate load capacity for this specimen. It is noted that while the longitudinal stress magnitude is not at yield at the supported edges, the element von Mises stresses are, due to shear stress components. 50 0

15

-50 -100

-200

(][01 .........

a

-150 .~

=::I,E= ;::] ~

~

~'/

!

~-"

Winter Abaqus-co for tnact-ultimate stiffened element s

0 = 0

,,J

0,

,

,

0 1553104656;Z) 775 Width of the Loaded Edges (mm)

(b)

-250 -300

(a)

Width of the Loaded Edges (mm)

Figure 6: Mid-Surface Longitudinal Stress and Displacement at Ultimate Load for Stiffened Element of Slenderness ~.=5 Unstiffened Elements Once the slenderness exceeds a value of ~=1.5 (b/t ratio of 25), the ultimate load carrying capacity is reduced by the Contact condition, and the strength curves for the two cases diverge, as shown in Figure 5. This corresponds to the slendemess after which the specimens in the Straight condition will attain tensile stresses at the unsupported edge, and in the Contact condition this edge will pull away from the rigid loading body. The maximum reduction in strength is 10.6% for a slenderness of ~,=3 (b/t ratio of 50). For this slendemess value, the maximum value of tensile stress at ultimate is 158MPa in the Straight condition, and the pull-in of the free edge in the Contact condition is O.3mm. Figure 7a shows the distribution of longitudinal stress across the width of the loaded edges at ultimate, for a slenderness value of ~.=2.1 (b/t ratio of 35). For this slenderness value, the Contact condition causes a 4% drop in ultimate load carrying capacity. The difference in the stress distributions at ultimate between the two loaded edge conditions is noticeable. The Winter Eqn. 2 for unstiffened elements is conservative in this slenderness range, as seen in Figure 5, and the difference in the assumption for the stress distribution is noticeable also for the Straight condition. The deformation of the loaded edge for the Contact condition is plotted in Figure 7b. For both stiffened and unstiffened elements, the values in Table 1 show that the percentage difference in the ultimate loads between the two loaded edge conditions decreases at very

736

large slendemess values. This is due to a mode change of the deflected shape for very slender elements. i

300

i "

t 200

~ z

' i i Abaqus-straight-utimate Abaqus-contact-ultimate Winter for unstiffened elements

1oo r~

i

a

o

0.5 .... Q4 Q3

-lOO

2-:i0 ~- 02 _=" 0.1

-200

_~ g

50

r~

1o

~

1(

-

111 -!

,.I

-300

-400

/

0

/

/

I

, 0 125 250 Widthof the LoadedEdges (mm) (bl

(a) Width of the Loaded Edges (mm)

Figure 7: Mid-Surface Longitudinal Stress and Displacement at Ultimate Load for Unstiffened Element of Slendemess ~=2.1 FEM RESULTS FOR MULTIPLE H A L F - W A V E L E N G T H S

The FEM results discussed thus far have been for specimens that buckle in a single halfwavelength. For practical purposes, tests for determining section strength will typically be conducted on stub columns, and will generally ensure the formation of 2-3 local buckles. Accordingly, the FEM analyses are extended here to determine the ultimate load carrying capacity of stiffened and unstiffened elements for the two conditions of Straight and Contact loaded edges, when the plates buckle in three half-waves. This mode is achieved for stiffened elements by changing the aspect ratio to three. For unstiffened elements it is achieved by restraining the out-of-plane deflections (only), across the entire width of the plate, at the points corresponding to 1/3 of the length and 2/3 of the length. The length is tripled in order that each half-wave retains the aspect ratio of five. UNSTIFFENED Width (mm) 60 100 125 150 175 250

ELEMENTS

Thickness 5ram Ultimate Load ( k N ) Difference Length Slenderness Straight Contact (mm) (;~) 900 0.71 81.0 81.0 0.0% 1500 1.19 101.6 101.6 0.0% 1875 1.49 104.6 104.6 0.0% 2250 1.79 118.0 116.3 1.4% 2625 2.08 132.3 127.9 3.3% 3750 2.98 171.9 154.6 10.1% 3.57 163.9 6.6%

. . . ........ /

i

I

/J

]

.-]

A .r

~ ~..."

/[

Table 2: FEM Results for Unstiffened Elements with 3 Half-Wavelengths The results for three half-wavelengths are shown in Table 2 for unstiffened elements. As one would expect, the effect of not constraining the loaded edges to remain straight (Contact)

737

diminishes compared to the results shown in Table 1, since there are now two nodal lines along the length at which the tensile stresses of one buckled cell may anchor into the next. APPLICATION TO SECTIONS CONTAINING BOTH STIFFENED AND UNSTIFFENED ELEMENTS

It is well documented that slender sections containing stiffened and unstiffened elements, such as the unlipped channel in Figure 1, will locally buckle at a half-wavelength different to that if the components were treated separately, due to interaction between the flanges and the web. As a result, it is likely that the unstiffened element of the section will buckle at a halfwavelength less than the previously assumed aspect ratio of five. Table 3 shows the ultimate load carrying capacities of unstiffened elements for the Straight and Contact loaded edge conditions, when the aspect ratio is reduced from five to one. The study is for elements buckling in three half-waves. The effect of end loading condition is seen to be strongly dependent on the aspect ratio, eg for a b/t-ratio of 50, the ultimate loads are equal for an aspect ratio of one, while they differ by 10.1% for an aspect ratio of 5. The percentage differences shown in Table 3 are plotted against b/t in Figure 8 for each of the five aspect ratios. UNSTIFFENED ELEMENTS - 3 Half-Wavelen~lths I Aspect ratio of I | Ultimate Load Diff. Width!b/t Length Straight Conlact Length 175 35 525 150.6 15 c.6 0.0% 1050 250 50 750 171.4 171.4 0.0% 1500 Aspect ratio of 3 Ultimate Load Diff, Width b/t Length Straight Contact! Length 150 30 1350 120.8 120.3 0.4% 1800 175 35! 1575 133.4 131.3 1.6% 2100 i

. . . . . . . .

Thickness 5mm Aspect ratio of 2 -Ultimate Straight 135.2 159.3

Load Contact 134.9 156.8 Aspect ratio, of 4 Ultimate Load Straight Contact 119.3 118.0 133.1 129.4

Diff.

0.2% 1.5%

i

Aspect ratio of 5 I Ultimate " .oad Diff. Length Straight C :intact 1.1% 2250 118.0 16.3 1.4% 2.8% 2625 132.3 127.9 3.3% Diff.

Table 3: FEM Results for 3 Half-Wavelengths and Varying Aspect Ratios In practical sections comprising stiffened and unstiffened elements, the stiffness of the web with respect to that of the flanges will cause the section to locally buckle in half-wavelengths producing aspect ratios of the buckles in the web between 1 and 2. In this case, the aspect ratio of the unstiffened element will be in the range from b~/bf to 2b~bf. Given that the b~/bfratio of practical sections typically varies between 2 and 4, the aspect ratios for unstiffened elements in practice are likely to fall in the range from 2 to 8. CONCLUSIONS

The influence of the two limiting conditions of the loaded edges, Straight and Contact, on the ultimate load carrying capacity of stiffened and unstiffened elements has been investigated. These two conditions are analogous to short columns with and without stiff welded end plates respectively. The influence of multiple half-wavelengths, aspect ratio and interaction between stiffened and unstiffened elements comprising a section are investigated. It is shown that the ultimate load carrying capacity of stiffened elements is virtually unaffected by the loaded edge condition. However, for unstiffened elements, the ultimate load can be

738

10.0~ Aspect Ratio of flange local buckle:

Open Sections Locally Buckling In 3 Half-Wavelengths

9.0%

5

8.0~ 7.0% w

ca &O%

0 ,.I

o w Q.

5.0% A s p e c t ratio =

m 4.0% tO 3.0% 2.0% 1.0% 0.(7'/0 25

30

35

40

45

50

Range W i ~ a n g e 1Ndmess

Figure 8: Section Capacity loss if the Loaded Edges are allowed to Deform Three Half-Wavelengths assumed reduced by about 10% when the loaded edges are allowed to pull in (Contact condition). This percentage corresponds to a plate slenderness value (b/t) of 50 and an aspect ratio of 5. The reduction is shown to depend strongly on the aspect ratio and less so on the number of halfwaves. It is concluded then that the Winter formulae provide a satisfactory estimation of the effective width of stiffened compression elements. The Winter formula for unstiffened elements is conservative at high slenderness ratios (;~>2), which may be partly because it was calibrated on the basis of stub column tests which allowed the ends to pull away from the loading platens. REFERENCES

American Iron and Steel Institute. (1996) Specification for the Design of Cold-Formed Steel Structural Members. 1996 Edition, Washington D.C. AS/NZS 4600 (1996). Australian/New Zealand Standard. Cold-Formed Steel Structures. 1996 Edition, Sydney Bambach M.R. and Rasmussen K.J.R. (2001a). Experimental Techniques for Testing Unstiffened Elements in Compression and Bending. Proceedings of the Third International Conference on Thin-Walled Structures. /CTWS (2001), Cracow, Poland. Bulson P.S. (1970). The Stability of Flat Plates, Chatto and Windus, London Kalyanaraman V., Pekoz T. and Winter G. (1977). Unstiffened Compression Elements. Journal of the Structural Division, ASCE 103:ST9, pp 1833-1848 Winter G. (1947). Strength of Thin Steel Compression Flanges. Transactions, ASCE, Vol. 112, pp 527-576 Winter G. (1970). Commentary on the 1968 Edition of the Specification for the Design of Cold-Formed Steel Structural Members, American Iron and Steel Institute, New York.

Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

739

A PROBABILISTIC APPROACH TO THE LIMIT STATE OF CENTRALLY LOADED THIN-WALLED COLUMNS Z. KALA, J. KALA, B. TEPL~" Faculty of Civil Engineering, Bmo University of Technology, Veveri 95, Bmo M. SKALOUD

Czech Acad. of Sciences, Inst. of Theor. & Appl. Mechanics, Prosecka 76, Prague

ABSTRACT The problem of the interaction of local buckling and the loss of stability as a whole is studied taking into account different imperfections. Thin-walled closed section columns in compression are analysed for several models of boundary conditions utilizing the probabilistic approach and by means of a detailed FEA model. A comparison to the load-bearing capacity according to Eurocode is then made.

KEYWORDS Imperfection, limit states, thin-walled cross-sections, finite elements methods, non-linear solution, probabilistic simulation methods. 1. INTRODUCTION It is generally known that in the strength calculations of thin-walled compressed columns it is necessary to include the influence of the interaction of global and local buckling, see for example, [3], [6], [7]. The conception of the normative buckling strength of the compressed columns is based on the analysis of a real structural element that, in contrast with the model of an ideal column, shows a number of initial imperfections affecting its load-carrying capacity. These imperfections are divided into the following three categories: 1. Geometrical deviations: The initial curvature of the bar axis, the eccentricity of the load point, the violation of the theoretical arrangement of the cross-section (the size and shape tolerance of the cross section), etc. 2. Structural imperfections: Dispersion of mechanical properties of a material (the nonhomogeneity of a material that is manifesting itself by dispersion of yield point values, breaking strength values, elasticity modulus values, etc.), the initial state of stress (residual stress as a result of technological production processes), etc.

740 3. Construction imperfections: Imperfections in the execution of joints, connections, bearing and other construction details that manifest themselves in deviations of the action of the real load-carrying system, compared with the theoretical assumptions that are being introduced during the solution of the idealized system. By their nature, the imperfections of the first and the second types are random quantities showing a larger or smaller variability. The analysis of their influence on response may be studied using a number of probability methods (Monte Carlo, Latin Hypercube SamplingLHS, Importance Sampling, etc.)- see, e.g., [1 ]. The influence of the third type of imperfections should be taken into consideration in a more accurate solution of the system. In a number of practical cases when a bar model was used, the construction imperfection cannot always be expressed in greater detail. In this context, let us now concentrate our attention on the thin-walled box-type cross- section where the load-carrying capacity may be considerably affected by the buckling of its slender walls. The most buckled soft centre of the wall transfers relatively smaller loads than the rigid edges (comers) as a result of the irregular rigidity of the thin-walled cross-section. This is also in relation with the modelling of boundary conditions when, for example, by introducing end forces to relatively more rigid comers, higher values of load-carrying capacities may be obtained than in cases when the model mediates the transfer of reactive forces (external forces) uniformly to the whole cross-section or to the soft centres of the walls. The subject of our investigation will be how the change of the type of boundary conditions will affect the load-carrying capacity of the bar. Aiming to make a provision for the effect of the local buckling on the loss of stability as accurately as possible we modelled the problem by means of the shell finite elements. The load-carrying capacity of the bar was evaluated by a geometrically and physically non-linear analysis. The realizations of input imperfections were simulated using the LHS [7] method that makes it possible to simulate the realization of the data from the known statistics of the input values in a similar way as they could be obtained by direct measurements. Based upon the simulated input data, we evaluated three series of load tests for different structure types by introducing the load to the ends of the bar. Thus, the materialisation of the real boundary condition was determined when, due to various structural adaptations (end-stiffening plate, etc.), the load may be irregularly redistributed. The problems of the simplifying relations of EC3 [9] where the load-carrying capacity of the thin-walled column may be evaluated based on the internal forces determined by means of the column model, and where the boundary conditions are considered in a very simple way (the hinge in the cross-section centre of gravity) are also discussed in the paper. 2. CLASSIFICATION OF CLASS 4 CROSS-SECTION Eurocode EC3 [9] takes into consideration the effect of the local buckling with the aid of the coefficient flA = Aeff/A which is determined as the ratio of the so-called effective to the real surface of the cross-section, see Fig. lb. According to [9], the cross-section where the ratios of the breadth and depth of the compressed parts of the cross-section may be put in a class 4 classification at the web or flange plate respectively are

(h-20/t > 42 e,

(b-20/t> 42

c,

(1)

while particular parts of the cross-section may generally be in a different class. Parameter r for steel of S 235 series may be considered according to [9] by value 6 = 1. Then, the influence of local buckling in these types of cross-sections is considered only approximately by means of effective zones- see relation (2) presented hereinafter.

741

@

z

(~)

~ i i

Z

# i

Non-effective zone

.... ~y

.2

Gross

t~

cross-section b

~t

"

Effective cross-section with effective z o n e A.ff

Fig. 1" Class 4 cross-section, axial force In cases when the relation (1) is not fulfilled the column is put in class 3, and the loadcarrying capacity is determined irrespective of the influence of local buckling. Therefore, in our numerical study we focused our attention on the column, which although put in class 4, was analysed by means of a detailed FEA model. 3. CALCULATION MODEL was studied using a column of length L = 12 m, a rectangular shape of the cross-section according to Fig. la, of nominal value of breadth b = 0.5 m, height h = 0.3 m and sheet thickness t = 6 mm. The relative slenderness of the column is 2, = 1. Especially as far as the wall thickness is concerned, this is a very slender column. The reason why this column was selected was to emphasize the effect of the boundary arrangements. To solve the problem by means of FEA, the program system ANSYS was used. With respect to the symmetry of the column solved, and to the exacting character of the numerical calculation, only half of the column length was considered. The symmetry plane in x = 6 was prescribed by preventing the displacement in axis x and by the rotation around the axes y and z (UX, ROTY, ROTZ). The centrically compressed column was hinged in x = 0 m. The hinge was modelled by preventing the displacements UY, UZ at two points in the centre of shorter walls, which made the angular displacement of the end cross-section in plane xz possible. The column geometry was modelled by a four-node shell element provided in [2] as SHELL 43 which makes it possible to solve physical and geometrically non-linear problems. Non-linear material behaviour was included in the calculation by means of a bi-linear kinematics model without stiffening. The density of the elements network had to be selected high enough to enable the description of the local buckling of walls. The model was formed by 2000 finite elements with 2040 nodes. The solution of the problem results in the system of 12 116 equations. Aiming to obtain the information about the contingent error in the determination of the loadcarrying capacity due to different boundary conditions, we modelled the column in four different ways: Variant A) Parabolic distribution of the uniform load at wall edges, the stiffening of the end part- see Fig 2a. Variant B) Constant distribution of the uniform load at wall edges, the stiffening of the end part- see Fig 2b. Variant C)Local transmission of the reaction to the centre of the wall, the stiffening of the end part. Variant D) The same as in C, but the stiffening ring is of a higher rigidity (50 times).

742

9

B

Fig. 2 a, b: Variants of boundary conditions In Variant A, we assumed a redistribution of the end load that approximately corresponds to the stress distribution in remote cross-sections which are not influenced by the boundary condition model, e.g., at the column axis of symmetry. Besides this, the end cross-section was stiffened, which in a way, substitutes the stiffeners or another detail of the column attachment. Variant B is the same as Variant A, the difference rests in the fact that the load was distributed uniformly along the cross-section circumference. Therefore, we may expect that redistribution of internal forces towards the stiffer comers of the cross-section will occur 9 Variant C differs from the previous ones in fact that the load (reaction) was introduced to the centres of shorter walls (in the place where displacements UY, UZ are prevented). As a result of the poor utilization of the material for the transmission of the given load, the load-carrying capacity of this model may be expected to be lower compared with models in Variants A, B. Variant D roughly models a marked stiffening of the end attachment (only a deterministic analysis was carried out). Note that Variant C is probably out of the range of practical reality and serves here (as well the Variant D does) for comparative purposes only showing a certain limiting cases. The loadcarrying capacity was determined as the load under which the determinant of the structure stiffness matrix will equal, with the selected accuracy, to zero. 4. STOCHASTIC CALCULATION MODEL The question is, to what extent the standard relations are truthful. A large number of tests made on real columns (i.e. also with respect to imperfections) were difficult to produce and also from the economical point of view, and would not offer sufficient data for the evaluation of results with respect to the assumed dispersion and heterogeneity of imperfections. Since we know a number of statistically usable data for some columns obtained by measuring or taken from the tolerance standards, the evaluation of the load-carrying capacity based upon numerical simulations carried out by means of the computer is less troublesome [5]. The Yield limit was the first random quantity that was considered in the calculation. The yield point of strength class S 235 steel was expressed by a histogram (according to statistical assessment steel manufacturing in Czech Republic). The mean value of the histogram is 285.74 MPa, the standard deviation is 23.57 MPa and its skewness is practically negligible. Another imperfection that may substantially affect the load-carrying capacity is the initial curvature of the column. The shape of the curvature was introduced in a form of the sine function in interval O - re, both for the initial deflection in the plane of the original bending in direction of axis z (cot) and for the lateral deflection in direction of axis y (co2). With the amplitudes of the maximum initial deflection eol, eo2,, we considered a Gaussion distribution of probability with the mean value 0 (straight bar). The standard deviation for both deflection directions was considered Seol = Seol = 6 mm according to the rule 2 Sx.

743 The initial buckling of the profile walls was introduced to the model by the first proper shape of buckling standardized to the maximum deviation. Its value is assumed as a random quantity eo7, with the mean value meoz = O, and the standard deviation Seoz = 1.67 mm. The initial rotation of the walls of the box- type cross- section is another imperfection affecting, in a way, the load-carrying capacity, see Fig. 2. In random quantities eo3, eo4, we assumed the mean value of the normal distribution to equal to the value of 0 mm and the standard deviation being 1.25 mm. Similarly, in the second direction (eos, eo6) of the shorter wall, we also assumed the mean value of the normal distribution to be 0 mm and the standard deviation to be 0.626 mm. The nominal value of the wall thickness from Table 1 was considered to be the mean value of the normal distribution. The coefficient of variation equalled to the value of 0.07 in all cases, similarly as in ref. [5]. 5. THE STATISTIC ANALYSIS OF THE LOAD-CARRYING CAPACITY By the statistical analysis we understand the problem of determining the main statistical parameters of the construction response, in this case the limit load-carrying capacities of the centrically compressed thin-walled columns of the box-type cross-section. The analysis was made using the Latin Hypercube Sampling method, which in contrast to the Crude Monte Carlo method, offers sufficient accuracy even with a considerably smaller number of runs. This is a crucial and necessary requirement in our task. Here, each run is a non-linear solution of the shell by means of FE! 100 simulation runs were always used. Table 1 summarizes the results of this numerical study. As expected, Variant C leads to an unrealistically low value of the load-carrying capacity. Variant A (in our view the most realistic) leads to the highest load-carrying capacity. This has been verified by the values of statistical moments. However, the design value of the load-carrying capacity that is evaluated in the Table in harmony with Annex A [10] as 0.1% of the quantile for several types of probability distribution is decisive for design. The last of the above types has been selected as the most fitting one (Hermite distribution). The Table also presents detailed values for two distributions determined by means of the relations from Table A3 [10] that only represent the simplification based upon certain weight coefficients. The load-carrying capacity is determined according to EC3 is presented in the last part of Table 1. The difference is remarkable. Table 1 The values of statistical moments of the load-carryinl capacity Variant A Variant B Variant C Arithmetic mean 1304.2kN 1230.2 i~q 924~91 kN Statistical moments Stand. deviation 183.49 kN 176.36 kN 154.42 kN Stand. skewness 0.0130 0.1819 0.3627 Stand. kurtosis 3.0210 2.9575 2.8137 Normal iGauss) 737.191 kN 685.215 kN 447.709 kN 0,1% Lognormal 837.925 kN 783.683 kN 546.508 kN quantile Hermite 737.330 kN 731.956 kN 540.674 kN 0,1% quantile accord Normal (Gauss) 746.39 kN 694.0656 455.4732 to EC 1 table. A3, p. 6,~ Lognormal 850,35 kN 795.6197 556.7675 EC3 938.82 kN ....

In all variants, the hinge at the end cross-section was modelled by preventing the displacements on axes y, z at two points in the centre of the shorter walls. The hinged simple

744 beam whose load-carrying capacity is given in the last line of the Table was determined according to EC3, using relation 5.45, page 105: (2)

Nb.Ra = Z flA A f j, / YM, '

where flA = Aeff/A for the class of cross-section 4 holds, Z is the buckling coefficient, fy is the characteristic yield point and Yu~ is partial reliability coefficient. In relation (2) nothing is said about the detailed location of the external load and its consequent re-distribution along the bar length. The results of the statistical analysis are presented once again in Fig. 3 from which it is obvious that the standard result Nu=938 kN is close to the mean value rated for Variant C, i.e., the variant describing probably in the worst way the arrangement of the hinged support of the thin-wall column. The mean value, however, is not relevant with respect to the required reliability of the design. It is necessary to say that one solution of the imperfect column according to Variant D offered the load-carrying capacity that was identical with the corresponding imperfect column according to Variant B. The necessity of stiffening is pronounced. 0.0027 Relative Frequency 0.0024

Variant C

Variant B Variant A

0.0021 0.0019 0.0016 0.0013 0.0011 0.0008 0.0005 0.0003 0.0000 . 524

.

. 661

. . 798

. 935

. 1073 1210 1347 1484 Load-carrying capacity [kN]

1622

1759

1896

Fig.3 Probability distribution functions of variants A, B, C. Fig. 4 presents the correlation dependence between the results for Variants A and B, i.e., their considerable "uniformity" describes a certain information about the quality of statistical results. Each realization of the random load-carrying capacity plotted on the horizontal and vertical axes represents here a result of two "loading tests" of one column simulated by the computer. This is a great advantage of this procedure since after completing this "loading test", the real destruction of the column does not occur, and the test may be repeated for other boundary conditions as well. With a real bar loaded in the laboratory, the repeated measuring of the load-carrying capacity could not be accomplished on one bar, since after loading the bar to collapse (the maximum load-carrying capacity), a permanent deformation is introduced to the bar, and therefore its initial imperfections are modified.

745 A strong correlation between the results for Variants A and B suggests that the effect of the variability of input quantities upon the variability of the load-carrying capacity is practically the same for both variants. However, the load-carrying capacity of Variant B is lower on average. The extent of the sensitivity of the load-carrying capacity to particular input quantities (especially imperfections) and their statistics are presented in [4]. 31.5

Variant A (parabola)

[kN]

29.9 28.3 26.7

q," .',

9

mm 9

9

9

25.1 :" ,,,u

23.5

.~., ,~g.

21.9

.#,

20.3 18.7 17.1 15.5 15.5

[~] I

'

17.1

I

18.7

I

20.3

I

'1

I

I

21.9 23.5 25.1 26.7 Variant B (constant)

'

I

28.3

'

'

I

29.9

31.5

Fig. 4: Correlation between Variants A, B.

6. THE DISCUSSION It is evident from Table 1 that practically the highest values of the load-carrying capacity were obtained in Variant A for the parabolic distribution of the load where the maximum utilization of the material for the transfer of a given load may be assumed. The cross-section is loaded by such a distribution of the external load that corresponds to the load-carrying capacity of its particular portions. On the contrary, the lowest values of the load-carrying capacity were obtained in Variant C where the introduction of load in the form of a concentrated force in the middle of the shorter walls highly affected the total reduction of the load-carrying capacity compared with Variant A. The reduction of the load-carrying capacity in Variant C is caused by a different rigidity of the cross-section in comers and in the centre of the slender walls. In this way the validity the Bernoulli hypothesis is limited, which in its consequence, would highly affect the state of stress in the welded or bolted joints. Constructional adaptations and additional stiffening of the end cross-section in place of actions of forces are necessary, of course. The above-described results of the statistical analysis suggest that such an analysis may offer interesting data on the consequences of imperfections, on the importance of the truthful modelling of constructions and their details, and may, in this way, reveal the procedure that does not always comply with normative documents. However, these conclusions cannot be generalized since only one element was investigated here.

746 In conclusion we may say that when determining the load-carrying capacity according to the standard relation (2) on the implementation of boundary conditions should be considered with a high degree of caution. It is also necessary to focus our attention on welded and bolted joints through which the element is included in the system of bars. The irregular distribution of rigidity of the welded backing may influence not only the load-carrying capacity of the welded joint itself, but consequently even the distribution of reactive forces, and thus the stress of the basic material. The above studies should be enlarged by more detailed analyses of other boundary conditions and structural arrangements. It would be appropriate to analyse the bars of other cross-sections and other slenderness ratios too. The question is, what would be the effect on the load-carrying capacity of the perfectly rigid distribution plate, which would secure the coaction of all parts of the cross-section, and this way would limit the warping of the end edge.

This study was written during the solution of Project No 103/00/0603 of the Grant Agency of the Czech Republic References

1. O. Ditlevsen, H. O. Madsen: Structural Reliability Methods, Wiley, 1999. 2. ANSYS Element reference, Release 5.5, ANSYS, Inc., 1998. 3. Johnson, C. P., Will, K. M., Beam Buckling by FE Procedure, in Journ. Struct. Div. ASCE, ST3, 1974, pp. 6 6 9 - 685. 4. Kala, J., Kala, Z., Tepl3~, B., Skaloud, M." Probabilistic aspects in the interactive global and local buckling of thin-walled columns, Proceedings of the Third International Conference on Coupled Instabilities in Metal Structures CIMS '2000, Lisabon, 21th23rd.September 2000, pp. 121 - 128, ISBN 1-86094-252-0. 5. Kala, Z., Kala, J., Tepl)~, B." Non-linear analysis of a thin-walled steel beam computed with the random influence of imperfections, Proceedings of the Conference, Bratislava, 2 "d - 3ra October 2000, pp. 117 - 122 (in Czech). 6. Rondal, J., Batista, E., Stsability Problem of Thin-Walled Cold-Formed Steel Columns. in Stavebnicky ?asopis, No. 7, Bratislava, 1988. 7. ~kaloud, M.: The limit state of thin-walled columns with regard to the interaction of the deformation of the columns as a whole with the buckling of its plate elements, Acta technica (~SAV, No 1 1975. 8. Mc Kay,M.D., Beckman,R.J. and Conover, W.J., A comparison of three methods for selecting values of input variables, Technometrics, 2, pp 239-245, 1979. 9. ENV 1993-1-1, EUROCODE 3, Design of steel structures, PART 1-1: General rules and rules for buildings. 10. Eurocode 1 1994. Basis of Design and Actions on Structures, Part 1: Basis of Design. Brussels, CEN

Third InternationalConferenceon Thin-Walled Structures J. Zarag,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

747

R O T A T I O N A L CAPACITY OF I-SHAPED A L U M I N I U M BEAMS: A N U M E R I C A L STUDY G. De Matteis 1, V. De Rosa I and R. Landolfo 2 University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy 2 University of Chieti G. D'Annunzio, Viale Pindaro 42, 65127 Pescara, Italy

ABSTRACT In the current study a non-linear finite element model aiming at obtaining a reliable evaluation of the rotational capacity of I-shaped aluminium alloy beams subjected to a moment gradient loading and prone to both local and global buckling phenomena is established. Such a numerical model is then used for a parametric analysis, so allowing determining an upper and a lower bound to the rotational capacity of the considered aluminium members as a function of the main influencing factors (local slenderness of the cross-section and global slenderness of the member) as well as of some other important factors (moment gradient and web stiffness). The results obtained show the importance, in some circumstances, of taking into account the steepness of the moment gradient and the torsional restraint provided by the web. But, above all, the paramount influence of interactions between flange and lateral-torsional buckling on the values of both the global rotational capacity and its stable part is unquestionably recognized. Therefore, a radical change in the cross-sectional classification approach provided by major structural codes seems to be necessary. KEYWORDS Aluminium Beams, Cross-Sectional Classification, FEM Model, I-Beams, Imperfections, LateralTorsional Buckling, Local Buckling, Rotational Capacity. INTRODUCTION For metal structures, rotational capacity is one of the most significant behavioural parameters of a member under monotonic actions. In fact, a correct evaluation of the member available ductility plays a fundamental role in assessing the available ductility of the whole structure, i.e. its capacity to redistribute plastic strains locally induced by particularly severe loading events, such as earthquake, explosions or structural impacts. Since many decades, in the field of rotational capacity, several works have been carried out for steel. In particular, the remarkable research on simply supported I-shaped steel beams under moment gradient has led to theoretical (Kemp and Dekker, 1991), semi-empirical (Kato, 1990; Mazzolani and Piluso, 1992) and empirical (Akiyama, 1985; Sedlacek and Spangemacher, 1992) models, allowing for the prediction of the rotational capacity of steel I beams by a simplified way. On the other hand, more accurate models based on the finite-element method have been proposed by Greshick et al. (1989), but their results did not fully fit the experimental ones, showing the difficulty of such an approach. Nonetheless, existing modem structural codes (namely, Eurocode 3 and 9, AISC LRFD, CSA S 16-2001) allow for a qualitative estimation of the member inelastic behaviour (with respect to both resistance and ductility) by means of the cross-sectional classification. Such a classification is based upon fixing suitable slenderness limits for each one of the individual structural elements composing the cross-section, so that

748 the whole section is classified according to the less favourable class (i.e. the highest normalised slenderness) among these elements. According to this approach, several geometrical and mechanical factors are generally taken into account as influencing parameters, namely: conventional elastic material strength, stress state on the section and on the considered element (axial/flexural compression), width-tothickness ratio of the element, manufacturing process of the member (welded/unwelded, hot rolled/cold formed), position of the element in the cross section (internal/outstand). On the other side, the simplicity of the approach does not allow considering several other important factors, which are partly common to all metal structures, and partly peculiar to the aluminium ones. Generally, lacks relevant to all metal structures spring from ignoring that the interaction between lateral-torsional buckling and local buckling may produce a relevant decreasing of the ductility of the whole (Earls, 1999). As far as aluminium beams are concerned, the available knowledge is still lacking (Landolfo, 2000). Several experimental analyses have been carried out by Mazzolani et al. (1996) and by Faella et al. (2000), in order to evaluate the ultimate resistance of heat-treated aluminium alloy sections failing in local buckling under uniform compression. Actually, these studies constitute the basis of the cross-sectional classification presently adopted by the Eurocode 9, in a similar approach as for steel. But these results do not consider the effect of stress gradient through the section and moment gradient through the member. Besides, selected specimens do not allow the effect of strain hardening to be suitably investigated (De Matteis et al., 1999). Finally, they do not take into consideration aluminium plastic anisotropy and the possibility of premature tensile failure due to reduced material ductility (De Matteis et al., 2000). Both Opheim (1996) and Moen (1999) have afterwards established numerical models for aluminium beam, in order to: (i) predict the load-deflection curve and so the rotational capacity of experimentally tested aluminium beams; (ii) conduct parametric studies on aluminium beams, aimed at improving the comprehension of their inelastic behaviour and the cross-sectional classification provided by structural design codes. In particular, the numerical model established by Moen et al. (1999b), using the non-linear implicit code ABAQUS (1997), was demonstrated to be capable of accurately predicting both the increasing and the decreasing branches of the behavioural curve, as far as the only local buckling of ductile and compact aluminium beams is concerned; on the contrary, it has shown to be not able to accurately predict the influence of interactions between local (flange and web) and global (lateraltorsional) buckling modes (I beams), so giving rise to considerable discrepancies between the decreasing branch of the numerical behavioural curve and the experimental one. Starting from the above considerations, the aim of the current study is twofold. Firstly, based on the numerical model established by Moen et al. (1999b), an investigation is carried out into the feasibility of establishing a non-linear finite element model, aiming at obtaining a reliable evaluation of the rotational capacity of I-shaped aluminium alloy beams subjected to a moment gradient loading and prone to both local and global buckling phenomena. Secondly, the proposed numerical model, once calibrated, is used: i) to conduct a parametric analysis so to establish both an upper and a lower bound to rotational capacity of the considered members as a function of the applied boundary conditions in the mid-span and of geometrical imperfections; ii) to determine and evaluate any influence of several parameters (moment gradient, web stiffness, local slenderness, global slenderness) on the lower bound to the rotational capacity, in order to assess their impact on the cross-sectional classification presently provided by relevant structural codes. ADOPTED NUMERICAL PROCEDURE

Analysed Scheme Aluminium beams investigated in this study are extruded profiles with I-shaped cross-section, made of AA 6082 alloy Temper T6, characterised by different values of both local and global slenderness. The analysed scheme is the same used for full-scale tests carried out by Moen et al. (1999a). Therefore, beams are simply supported and vertically loaded at the mid-span, by means of an actuator (a compact steel block with a total width of 150 mm) imposing displacements on the top of the upper flange of the beam.

749 At the supports, vertical and lateral displacements of the bottom flange are prevented, while no restraint is provided at the upper flange. As far as lateral restraints in the mid-span section are concerned, they have been experimentally provided by two rigid plates, adjacent to the central region of the beam. In the numerical model, for the sake of simplicity and computational efficiency, such a complex boundary condition is replaced by bilateral restraints set at the web-to-flange intersections. Furthermore, in order to avoid web crippling due to concentrated forces the beam length is extended beyond the end supports.

Basis of the Numerical Model The numerical model has been established by using ABAQUS/Standard (1997) non-linear FEM code. A 4-node shell element with reduced integration (S4R) has been used. A mesh refinement in the central region has been adopted to well interpret local contact and plastic buckling phenomena (Figure 1). The loading device has been modelled using a rigid surface, having the same geometry as the experimental one and imposing vertical displacements to the deformable surface at the upper flange of the beam. The ABAQUS/Standard default algorithm has been used to represent the contact between the master and the slave surfaces. The incremental loading process has been governed through the default automatic arclength control (Riks) procedure provided by the code (Abaqus, 1997).

- ~ 2 elts. ? 1

i

--- 8 elts.

z..

"

~ 50 mm

10 mm ~

q~--~~~

6 elts.

--r

[

1

1I

['i',l',',

l

~....

!

!

1'

i

1 O0 mm

------- 200 mm

Figure 1" The FEM model The multi-component strain-hardening model proposed by Hopperstad (1993) is adopted to describe the uniaxial stress-plastic strain behaviour of the material. The Hill yield criterion (1950) is used to take account for the experimentally observed plastic anisotropy (Moen et al., 1999b). Because of the lateral displacement restraint provided at mid-span, symmetry has been assumed about the transversal vertical plane (xy), so that only one half of the beam has been modelled. However, some numerical simulations have been performed in order to find out possible differences between the halfbeam model and the entire-beam one: the perfect coincidence of results, also in case of beams endowed with geometrical imperfections, has allowed using the simplified, less time-consuming model (see Figure 2a, which is related to the numerical simulation of the beam 'I1-2m' referred in Moen et al., 1999a,b). Some numerical simulations have been also carried out so as to investigate any possible effect of the coarseness of the mesh on obtained results (see Figure 2b, which is still related to the beam 'I 1-2m', but without using any geometrical imperfection in the model). Accordingly to Opheim (1996) and Moen et al. (1999b), it has been found that a very slight influence may appear in the decreasing branch of the behavioural curve of initially perfect systems. On the other hand, it has been noticed that, in case of imperfect systems, coarser meshes often prevent the numerical model from converging on the solution, this letting to prefer the fine-mesh model for performing the following numerical program.

Modelling of Geometrical Imperfections Generally speaking, extruded aluminium beams are characterised by small initial imperfections, really hardly detectable basing on visual inspections (Mazzolani 1995). Recent experimental studies on hollow rectangular thin-walled members (Opheim 1996, Moen et al. 1999a) have indicated that geometrical imperfections are very slight, even though measured data do not allow for general conclusions to be drawn

750 on their exact distribution and amplitude. On the other hand, as far as aluminium 1-section beams are concerned, it has been pointed out that imperfections are no more negligible. In particular, Moen (1999) recognized that beam out-of-straightness (i.e. deflection in the longitudinal planes) is relatively large, if compared to flange straightness (i.e. sinusoidal sweeps of the flange in the longitudinal vertical plane yz) and flange flatness (i.e. deviation of the mid-flange compared to the flange comer in a section). This sort of imperfection is very significant, since it gives rise to a consistent twist between the web and the flange and, as a consequence, to the possibility of influencing the potential buckling mode of the member. Opheim (1996) showed that an initial imperfection of the compression flange of I-sections, involving both flange flatness and beam straightness deviations, reduces the load carrying capacity of the beam obtained from numerical simulations, and the extent of such a decrease is also influenced by the imperfection magnitude. 90 80

,._, 80

70

~ 70

oo 60

~ 6o ~ 50

~o 50

~9 40

~ 40

g 30 ~

20

: ,7.

N 10

Entire-beam model

~ 30

Half-beam model

~9 20

odd [

rt

Coarse-mesh model

10

/ ! i

0 0

a)

50

100

150

M idsp an vertical disp lacement [mm]

200

b)

O

50

100

150

200

M idspan vertical displacernent [mm]

Figure 2: Effect of 'half-beam' modelling (a) and mesh coarseness sensitivity (b) It has been found by several researchers that introducing geometrical imperfection patterns in numerical models of beams, which are not prone to global buckling, has generally limited influence on their global behaviour. As an example, the numerical model established by Moen et al. (1999b) to simulate the inelastic behaviour of RHS and SHS aluminium beams showed a good agreement with the experimental results both in the initial model (where no geometrical imperfections had been introduced) and in the sensitivity study model (where geometrical imperfections at the compressed flange of the box section throughout the beam were applied). Therefore, it may be concluded that imperfections, either absent or characterised by a small magnitude, do not influence significantly the behaviour of hollow section beams in terms of both load carrying capacity and post-buckling behaviour. On the other hand, when dealing with members prone to global buckling phenomena, introducing a relevant imperfection pattern is deemed essential in order to get a reasonable failure mode. In such a case, the global behaviour of the structural member generally depends on both shape and magnitude of the imperfection pattern introduced in the numerical model (Galambos, 1998). Considering all the suggestions from the research works quoted above, it appears clear that a suitable representation of imperfections in I-shaped aluminium beams is generally advisable when dealing with local buckling only, and is mandatory when global failure modes and their possible interactions with the local ones are taken into account. It is generally assumed that an initial imperfection pattern is defined through an assigned shape and a scalar parameter, governing its magnitude. In order to establish such a function, in the current work, as also suggested by Espiga (1996), a linear elastic buckling analysis is firstly performed, aiming at evaluating the eulerian eigenmodes. Bifurcation analyses have shown that, within the range of both local and global slenderness parameters covered by tested beams, the first eigenmode is mainly "global", involving a horizontal translation and a rotation of the beam cross sections. Such eigenmode features can be easily recognised not only for high values of global slenderness, but also for those beam configurations highly prone to local buckling. Therefore, it is here deemed that global

751 imperfection patterns should be appropriately adopted in every numerical simulation. Once ascertained the global nature of the imperfection pattern, a standard shape, proposed by Earls (1999) for I-section high-strength steel profiles, has been adopted. The chosen imperfection (see Figure 1) is characterised by the rigid rotation, which is kept constant throughout the beam length, of both flanges (the same amount both for the upper and the lower one) as respect to the web, which is kept straight. In this way, a tendency for the beam to experience out-of-plane deformations is indirectly created, so giving rise to inelastic flexural modes characterised by coupling of local and lateral-torsional buckling. It has been undeniably pointed out that this sort of imperfection is both effective in order to obtain a suitable numerical model of collapse and post-collapse behaviour of high-strength steel beams under moment gradient, but it has also been shown that this imperfection pattern is the one favouring the most detrimental mode, it being related to the lowest rotational capacity (Earls 1999). Finally, it has been here verified that the imperfection magnitude (i.e. the flange rigid rotation q~),has a negligible influence on the overall response of the beam (Figure 3). Therefore, it does not represent a relevant factor to be taken into account in the performing numerical program. In particular, as imperfection magnitude, a value of q~= b/500 has been assumed for all the analysed cases. 90

90

(

80

q't

70

60

60

~

50

50

"~ 40

"~ 40 Experimental

= 30 .......

~ 20

~9

N lO

9

o

a)

/

.......

~ 20

~ = b/500

50

100

150

M idspan vertical displacement [mm]

200

b)

- - . . . .

-------q~b/lO00

N I0

q~= b / l ~ _ _ _ ~

. . . . .

Experimental "Perfect" mo(

= 30

"Perfect" model

~

"

rl 9

0 0

(

80

;~ 70

~o = b/500 ~ - b/100 w

0

50

100

150

200

M idspan vertical displacement [mm]

Figure 3: Effects of imperfection magnitude and mid-span restraint condition Model Calibration

The numerical model has been calibrated against available full-scale tests concerning 1-section profiles made of different aluminium alloys (Moen et al., 1999a). The FEM inelastic model originally established by Moen et al. (1999b), which did not consider any global geometrical imperfection ("perfect" model), provided an effective simulation for three of the four available experimental results, giving in one case (beam 'I1-2m') numerical results which were remarkably on the unsafe side (Figure 3). In such a case, simulations performed with the numerical model proposed here, endowed with the above initial geometrical imperfection pattern, have shown a very significant influence of the mid-span restraint condition on the slope of the decreasing branch of the behavioural curve. In particular, numerical analyses have been carried out for bilateral restraint preventing lateral displacements located both at both the upper and lower flange (Figure 3a) and at upper flange only (Figure 3b). Conservative result has been provided for the latter mid-span restraint condition. Therefore, for the following numerical test program, an "upper and lower bound criterion" has been applied in order to evaluate the global rotational capacity. In particular, the upper bound is given by the inelastic model adopted by Moen et al. (1999b), which is related to a beam without any geometrical imperfection so that lateral buckling modes are not activated in the numerical model, while the lower bound is yielded by the numerical model defined in the current paper, which is characterised by the above geometrical imperfection and the lateral restraint provided at the upper flange only (Figure 3b).

752 NUMERICAL TEST PROGRAM A parametrical study has been carried out in order to investigate the effect of several parameters (local and global slenderness, web restraining action and moment gradient) on the non-linear response of aluminium beams in bending. Since the analysis is mainly dealing with the effect of lateral-torsional buckling on the rotational capacity of the member experiencing plastic excursions, only class 1 and class 2 cross-sections (plastic and compact, according to the limits provided by present Eurocode 9) have been considered, as the other ones are assumed not to be able to develop conventional plastic moment. Furthermore, all the analysed cases have been devised in such a way the class of the cross-section is always determined by flange slenderness rather than by web slenderness.

2,oc=2eq/20has been accounted for by normalising the equivalent v2)/(7 9p).d/t with respect to 2 o = n'. where d is the plate element

Local (cross-sectional) slenderness slenderness 2eq = ~/12. (1-

width and t its thickness, while coefficients 7 and p depend on the relative boundary conditions and load shape, respectively (Mazzolani, 1995). Global slenderness has been expressed through the parameter

A~r = ~/a i We, "fo.2/Mcr, provided by Eurocode 9, where Mcr is the critical elastic buckling moment, We, is the elastic resistance modulus of the cross-section and a is the generalised shape factor of the crosssection, corresponding to an ultimate value of the curvature (it is equal to 10 times the elastic limit curvature and equal to the ratio Wv/Wet, for class 1 and class 2 cross-sections, respectively). Finally, according to De Matteis et al. (2001), the web restraining action has been controlled by the parameter kw=twS/h, where tw is the thickness of the web and h is the distance between mid-planes of the flanges, while the steepness of the moment gradient has been characterised through the beam compactness ratio L/b, where L is the half-span of the beam and b is the width of its cross-section. As far as the material is concerned, according to tensile tests reported in Moen et al. (1999a) relative to specimen I 1, the AA 6082 alloy Temper T6 considered in this study has been characterised through a conventional elastic strength f0.2=283 Nmm 2, a Young's modulus E=66.7 kNmm 2, a Poisson's ratio v=0.33, hardening parameters (the Hopperstad multicomponent stress-strain curve is considered) C~=2.559, Qt=31.20 Nmm 2, C2 =5.32, Q2=143 Nmm ~. It is worthy noticing that the first results obtained by numerical tests presented here are to be intended as dependent on the material characteristics, since normalisation is carried out with respect to E andfo.2 only, and not with respect to Ci and Qi. On the other hand, previous studies have shown the remarkable effect provided by the material strain hardening (De Matteis et al., 1999, 2001), which is being investigated by further analyses. Study cases have been selected in order to allow for the investigation of all the above parameters in a realistic and relevant range of variation. In particular, three different values have been considered for both local and global slenderness; besides, a specified value of web restraining action and beam compactness ratio parameters has been associated to the former and the latter, respectively. Nevertheless, for the sake of simplicity, for the highest value of the global slenderness (2Lr = 0.90), only the two values of 2tochave m

been considered. All main data of the examined cases (which, according to the above "upper and lower bound criterion", correspond to 16 numerical analyses) are given in Table 1 (tI = flange thickness). TABLE 1 MAIN DATAFOR BEAMSEXAMINEDIN THE NUMERICALTEST PROGRAM Beam No. 1 2 3 "'4" .........

5 6 -'~.......... 8

L

b

h

tw

ty

kw

L/b

Class

~oc

2~r

[mm] 1150 1250 1200

[mm] 105 120 126

[mm] 120 120 125

[mm] 12.0 10.0 15.0

[mm] 22.0 18.0 15.0

[mm2] 14 8 27

11 10 10

(EC9) 1 2 2

0.25 0.35 0.44

0.50" 0.50 0.50

~'6~ .......

i~6 .......

i2:15 .....

"2:~.6 ......

f4" ........

"1"f~ ......

f ..........

"67:~3- .... &:7"6""

"f966.....

2000 120 120 10.0 18.0 8 1950 126 125 15.0 15.0 27 3"6f3"6..... i-6~....... i]Z-6....... i~.15..... -;i2.6 ...... l'/l......... 3050 120 120 10.0 18.0 8

17 2 15 2 ~ ...... i .......... 25 2

0.35 0.70 0.44 0.70 "6~3".... /3.!i6"'0.35 0.90

753 OBTAINED RESULTS The structural response of analysed beams is presented in terms of normalised moment (M/Mo.2)-rotation (0/00.2) curves, where: M = P (L-D~2)~4, Mo.2=fo.2 We, 0 is the rotation at the beam support, 00.2=f0.z/E (L+D/2)/h, D being the extension of the load zone. Furthermore, according to Mazzolani and Piluso (1997), the obtained behavioural curves have been evaluated in terms of both the stable part of rotational capacity (130) and the global rotational capacity (13), defined as follows:

0u

0c-1

130 = 7 0 2 - 1 ;

(1)

13=00.2

where: 0u is the rotation corresponding to the maximum moment resistance, 0c is the rotation corresponding to the conventional collapse moment, which is attained when the decreasing branch of the moment-rotation curve reaches the conventional elastic moment resistance. In Figure 4, for each analysed case, both the behavioural curve relative to the perfect beam (upper bound model) and the one relative to the imperfect beam (lower bound model) are presented. From the examination ofbehavioural curves, it has to be noted that, when analysing beam 8, no solution has been obtained for the imperfect model. Such an event is not uncommon when performing numerical studies dealing with highly non-linear problems (Pekoz and Yogwook, 1999). Secondly, it may be observed that, generally speaking, the interaction between local and global buckling phenomena, while practically unaffecting the peak load-bearing capacity, may have a very great influence on the beam ductility. Nonetheless, the influence of interaction between local and global phenomena is certainly much more significant and extensive when considering the global rotational capacity 13. In fact, the stable part of the rotational capacity (130)often shows no substantial variations when comparing the case where only local buckling phenomena are allowed and the case where also lateral-torsional buckling takes place, while the characteristics of the unstable branch, and therefore the numerical value of 13, always show non negligible reductions when dealing with beams characterized by a relatively high value of the global slenderness. In particular, for the analysed cases, the rotational capacity 13is reduced up to one quarter (and, however, at least to the hal0 of the value provided by the upper-bound numerical model (Table 2). It is worthy noticing that the reduction shown by 130 and 13 as a consequence of the local-global interactions, and the extent of such a reduction, besides of the values of local and global slenderness, depends to some extent also on the other aforementioned parameters (i.e. moment gradient and web stiffness). As far as the steepness of the moment gradient is concerned, its influence can be obtained by examining the cases related to the model without any geometrical imperfection. It may be observed that the stable part of the rotational capacity is not significantly influenced, so as the beam loading bearing capacity, while the post-collapse behavioural branch (and then the global rotational capacity) is fairly sensitive to the L/b ratio (see Figure 5). Also, the effect of the L/b variation on 130 and 13 shows no dependence on the cross-sectional geometry (i.e. kw and Atoc ). m

TABLE 2 COMPARISONBETWEENROTATIONALCAPACITIESFOR THE TWO ADOPTEDMODELS

.

.

.

.

Beam

Modelwithoutimpe~ections

No.

~Q ....

~

~Q

1

4.0

14

2 3 4 5 6 7 8

2.2 2.7 3.2 2.0 2.3 2.6 1.6

14 12 9.0 8.4 7.2 6.1 6.3

3.2 2.2 2.7 2.0 1.8 1.9 1.2

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Modelwithimperfections

.

.

~ = . =

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.

7.8 4.7 6.6 3.6 2.3 2.3 1.5

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754 1.6

1.6

...........

~am

1.4 1.4 l 1.2

1.2

1.0

1.0

0.6

~

i

0.4

-"

0.2 ] ~

~ : model with imperfections

0.0 I. i 0

.

model without imperfections

.

.

3

. 6

9

12

0.8 0.6

;,

0.4

!

0.2

i

0.0

15

- -"

model without imperfections model with imperfections

0

3

6

O/00.2

9

12

O/00.2

1.6

1.6

Baam 4

Beam 3

o

2

1.4

1.4

1.2

1.2

1.0

1.0

0.8

~

0.6

0.8 0.6

0.4 0.2

i

n

0.0

.

0

model with imperfections .

3

.

i

.

6

12

~.

model without imperfections

0.2

[]

model with imperfections

0.0

i

9

0.4

0

15

3

6

9

12

0/00~

O/0o.2 1.6

1.6

14 t

,4

1.2

Beam 6

1.2

1.0 0.8 0.6

~

1.0

~

0.8 0.6

0.4 I~

-

modelwithout imperfections

0.4

_ _ _

0.2 ~

i

0.0

9 modelwithout imperfections

0.2

[]

model with imperfections

0.0 0

3

6

9

12

15

0

3

6

0/00.2 1.6

Beam 7

1.4 1.2

1.6

Beam 8

1.4 1.2

~. 1.0

1.0

~9 0.8

-.. 0.8

0.6

0.6

o

A

0.4 i 0.2 0.0 .' 0

9

0 / 00.2

model without imperfections

0.4

model with imperfections

0.2

,

,

,

,

3

6

9

12

0 1 0o.~

hout imperfections

0.0 15

0

,

,

,

,

3

6

9

12

15

0 1 00.2

Figure 4: Comparison between behavioural curves yielded by the two adopted numerical models

755

--Ik

w=14 mm2;"~toc=O'25f

k w=27 mm 2" ~toc=0.44 [

9

9

~6

~6

r~ {3o

5

l0

15

L/b

20

25

~0

5

,

,

10

15

,

,

20

25

30

L/b

Figure 5' Influence of moment gradient on rotational capacity As far as the web stiffness is concerned, the selected study cases do not allow for significant conclusions to be drawn. Anyway, it has been recognized the slight influence of kw on the peak load-bearing capacity, unless very low values of this parameter are considered. On the contrary, higher values of the web stiffness are able to provide a somewhat decreasing of the slope of the descending branch of the beam behavioural curve, producing a corresponding slight increment of the global rotational capacity. On the other hand, local slenderness has certainly a paramount role in determining the rotational capacity of the beam, but the numericalm tests which have been performed clearly show that, once given the local slenderness parameters (i.e..2~oc and kw), the global rotational capacity is strongly affected by the global slenderness value as well. Figure 6 shows this effect on both 130and 13.For the sake of comparison, in such a figure local slenderness limits for class 1 and class 2 cross-sectional classification according to Eurocode 9 have also been reported. As far as {30is concerned, it may be easily noticed that, up to ALr - 0.50, the analysed case endowed with class 1 cross-section provides a rotational capacity higher than 3.0, which could be considered enough for classification purposes. On the contrary, when the global slenderness parameter exceeds this limit, the stable part of rotational capacity is affected by a relevant reduction, definitely higher that the one concerned with local slenderness (2~oc), emphasising the need to account for the member global slenderness in the definition of cross-section ductility classes. Such an effect is even more evident for the global rotational capacity {3, but it is to be mentioned that the present codified approaches do not account for the unstable part of rotational capacity in defining cross- section ductility classes. 8

8

7 6 5 e~. 4 3 2 1 0 0.20

,

7

O "~Lr = 0.50

6 5

r'l "~t,r = 0.70

9

2Lr

9x'--Lr=

0.70

o

,2Lr = 0.90

~.

9 "~r

4

= 0.90

3

|

D

2

Class 1 0.25

Class 2 0.30

0.35 m

~loc

t

1 0 0.40

0.45

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Figure 6: Influence of global buckling phenomena on rotational capacity CONCLUSIVE REMARKS The results obtained by the performed numerical analyses show that the rotational capacity of aluminium 1-section beams is affected by several influential factors which are not only related to the geometry of the cross-section and to the slenderness of the plate element susceptible to local buckling. In particular, in some circumstances, taking into account the steepness of the moment gradient may be significantly important, since it affects the slope of the decreasing branch of the member behavioural curve. On the other hand, taking into account the effect of the restraint provided by the web on the compressed flange may be significant only in the case of very slender webs. But, above all, the performed numerical tests

756 unquestionably demonstrate the paramount influence of the interaction between local and global lateraltorsional buckling phenomena on both the global rotational capacity and its stable part. Eurocode 9 and also the major current structural codes for steel structures deal separately with these two phenomena, and cross-section ductility classes are deemed not to depend on their combined effect, which, on the contrary, may be noticeably important. Therefore, a radical change in the codified cross-sectional classification approach of metal structures seems to be required. References

ABAQUS/Standard (1997), Theory Manual- version 5. 7, Hibbit, Karlsson & Sorensen Inc., Pawtucket, Rhode Island. Akiyama H. (1985), Earthquake Resistant Limit State Design for Buildings, University of Tokyo Press, Tokyo, Japan. De Matteis G., Moen L., Hopperstad O.S, Landolfo R., Langseth M., Mazzolani F.M (1999). A parametric study on the rotation capacity of aluminium beams using non-linear FEM. Light-Weight Steel and Aluminium Structures (ICSAS'99), Elsevier (P. Makelainen and P. Hassinen, eds.), Oxford (UK), 637-646. De Matteis G., Landolfo R., Mazzolani F.M. (2000). Inelastic Behaviour of Hollow Rectangular Shaped Aluminium Beams. Finite element: Techniques and Developments (CST 2000), Civil Comp Press (B.H.V. Topping ed.), Edinburgh (UK), 373-379. De Matteis G., Moen L.A., Langseth M., Landolfo R., Hopperstad O.S., Mazzolani F.M. (2001). Cross-Sectional Classification for Aluminium Beams - A Parametric Study. Journal of Structural Engineering (ASCE), (under press). Earls C.J. (1999). On the Inelastic Failure of High-Strength Steel I-shaped Beams. Journal of Constructional Steel Research 49, 1-24. Espiga F. (1996). Numerical Investigations on Local and Global Inelastic Buckling Modes of I-Beams, 2"a Int. Conf. on Coupled Instability in Metal Structures, Liege, 85-92. Faella, C., Mazzolani, F.M., Piluso, V., Rizzano, G. (2000). Local Buckling of Aluminium Members: Testing and Classification. Journal of Structural Engineering, ASCE, 126(3), 353-360. Galambos, T. (editor) (1998). Guide to Stability Design Criteria for Metal Structures, 5th Edition, John Wiley & Sons, Inc. Greschik G., White D.W., McGuire W. (1989). Evaluation of the Rotational Capacity of Wide-Flange Beams Using Shell Finite Elements. Proc. Structures Congress (Vol. 3 ), Steel Structures, ASCE ed., New York, 590-599. Kato B. (1990). Deformation Capacity of Steel Structures. Journal of Constructional Steel Research 17:1, 33-94. Kemp A.R., Dekker N.W. (1991). Available Rotation Capacity in Steel and Composite Beams. Structural Engineer 69:5, 88-97. Landolfo, R. (2000). Coupled Instabilities in non-linear materials, 3rd Int. Conf. on Coupled Instability in Metal Structures, Lisbon, 263-272. Lay M.G., Galambos T.V. (1967). Inelastic Beams Under Moment Gradient. Journal of the Structural Division (ASCE) 93:ST1, 381-399. Lukey A.F., Adams P.F. (1969). Rotation Capacity of Wide-Flange Beams Under Moment Gradient. Journal of the Structural Division (ASCE) 96:ST6, 1173-1188. Mazzolani F.M., Piluso V. (1992). Evaluation of the Rotation Capacity of Steel Beams and Beam-Colunms, I s' Cost C1 Workshop, Strasbourg, France. Mazzolani F.M. (1995), Aluminium Alloy Structures (2"dEdition), Chapman & Hall, London, United Kingdom. Mazzolani F.M., Faella C., Piluso V., Rizzano G. (1996). Experimental Analysis of Aluminium Alloy SHS-members Subjected to Local Buckling Under Uniform Compression. 5'h International Colloquium on Structural Stability (SSRC), Brazilian Session, Rio de Janeiro, Brazil. Mazzolani F.M., Piluso V. (1997). Prediction of the Rotation Capacity of Aluminium Alloy Beams. Thin-Walled Structures 27:1, 103-116. Moen L.A. (1999). Rotational Capacity of Aluminium Alloy Beams. Doctoral Thesis, Norwegian University of Science and Technology, Trondheim, Norway. Moen L.A., Hopperstad O.S., Langseth M. (1999a). Rotational Capacity of Aluminium Beams under Moment Gradient. I: Experiments. Journal of Structural Engineering (ASCE) 125:8, 910-920. Moen L.A., De Matteis G., Hopperstad O.S., Langseth M., Landolfo R., Mazzolani F.M. (1999b). Rotational Capacity of Aluminium Beams under Moment Gradient. II: Numerical Simulations. Journal of Structural Engineering (ASCE) 125:8, 921-929. Opheim B.S. (1996). Bending of Thin Walled Aluminium Extrusions. Doctoral Thesis, Norwegian University of Science and Technology, Trondheim, Norway. Pekoz, P., Yogwook, K. (1999). Integrated Numerical Method and Design Provisions for Aluminium Structures, 2"d Aluminium Structures Workshop, Ithaca, New York. Sedlacek R., Spangemacher G. (1992). On the Development of a Computer Simulator for Tests of Steel Structures, Proceedings of the I s' World Conference on Constructional Steel Design, Acapulco, Mexico.

757

AUTHOR

Abruzzese, D. 269 Afonso, S.M.B. 373 Alexandrov, A. 603 Arafath, A.R.A. 277 Aribert, J.M. 161 Arizumi, Y. 161 Awrejcewicz, J. 349 Bakker, M.C.M. 417, 437 Bambach, M.R. 87, 719, 729 Bannikov, D.O. 619 Batista, E. 329 Bednarek, B. 103 Bond, W.F. 357 Bradford, M.A. 153 Cairns, R. 339 "Camotim, D. 329 Cerqueira, N.A. 533 Cheong, H.K. 365 Chusilp, P. 145 Croll, J.G.A. 643 Cui, S. 365 da C. P. Soeiro, F.J. 533 da Silva, J.G.S. 533 Davies, J.M. 3 De Matteis, G. 747 De Rosa, V. 747 Deg6e, H. 171 Dubina, D. 179, 187 Dunai, L. 203 Eisenberger, M. 603 Elbadawy, A. 339 Falco, S.A. 373,549 Falcon, G.S.A. 533 Fan Xuewei, 109 Foschi, R.O. 277 F6ti, P. 203 Fragos, A.S. 3 Fraldi, M. 653 Ftil6p, L. 187

Garstecki, A. 593 Gavrylenko, G.D. 643 Georgescu, M. 187, 195 Ghavami, K. 549

INDEX

Godoy, L.A. 661,669 Gomes, M.P.R.C. 541 Gongalves, P.B. 611 Gr~dzki, R. 469 Guarracmo, F. 653 Guezouli, S. 161 Gtinther, H.-P. 129 Gurba, W. 585 Hafidkowiak, J. 381 Hancock, G.J. 19, 449 Hao, H. 365 Hartono, W. 257 Hasham, A.S. 427 Holanda, A.S. 611 Huang, X. 477 Ikeuchi, T. 391 Imamura, K. 623 Inaba, Y. 209 Isozaki, A. 209 Janusz, L. 103 Jaskuta, L. 483 K~kol, W. 593 Kala, J. 739 Kala, Z. 739 Kasai, A. 145 Kasperska, R. 559 Kazakevitch, M.I. 619 K~dziora, P. 313 Khong, P.W. 499 Kim, Y. 437 Kleiman, P. 407 Kotakowski, Z. 293 Kotetko, M. 225,285, 293 Kowal-Michalska, K. 469 Krasovsky, V.L. 677 Krawczyk, P. 313 Kr61ak, M. 293 Krysko, A.V. 349 Krysko, V.A. 349 Krupa, R. 567 Kubo, Y. 459 Kuhlmann, U. 129 Kuwamura, H. 209 LaBoube, R.A. 241 Landolfo, R. 747

Li, H. 277 Li, Z. 499 Lopez-Bobonis, S. 669 Loughlan, J. 301,507 Lu, G. 477 Macdonald, M. 225 Magnucka-Blandzi, E. 567 Magnucki, K. 567 Mahaarachchi, D. 95 Mahendran, M. 95,523 Manevich, A.I. 575,683 Matsunaga, H. 491 McNiff, W. 225 Mendez-Degr6, J.C. 661 Morozov, G.V. 677 Muc, A. 313,585 Murzewski, J. 693 Nagahama, K. 329 Nagamatsu, H. 459 Nakamura, H. 623 Nanno, Y. 459 Nishimura, N. 391 Obr~bski, J.B. 321 Ostwald, M. 559 Pan, W. 399 Pawlus, D. 515 Pek6z, T. 417,437 Poldaarel, N. 523 Poursartip, A. 277 Raksha, S.V. 575 Rasmussen, K.J.R. 87, 217, 427, 719, 729 Ravinger, J. 407 Rhodes, J. 69, 225 Rogers, C.A. 19, 357 Roman6w, F. 381 Rzeszut, K. 593

Salehi, M. 631 Sarawit, A.T. 437 Shanmugam, N.E. 37 Shimizu, S. 119 Sikofi, M. 313 Silvestre, N. 329

758 Sivakumaran, K.S. (Siva) 233, 399 Skaloud, M. 137, 739 Stephens, S.F. 241 Szabo, I. 179 Szymczak, C. 53

Ungureanu, V. 179, 187 Usami, T. 145

Xuewei Fan 109

Vaslestad, J. 103 Vaz, L.E. 373 Vaziri, R. 277 Vrcelj, Z. 153

Takahashi, K. 623 Tepl3~, B. 739 Thompson, S.P. 301 Tremblay, R. 357 Turvey, G.J. 631

Walentyfiski, R.A. 701 Wang, P. 709 Watanabe, T. 145 Wilkinson, T. 449 Wright, H.D. 339

Yabuki, T. 161 Yamaguchi, E. 459 Yan, J. 249 Yang, D. 19 Yoshikawa, K. 119 Young, B. 249, 257

Zielniea, J. 483 Z~merovfi, M. 137

759 KEYWORDS INDEX actuation forces, 301 adaptive control, 301 ALGOR, 661 aluminum, 437 aluminium beams, 747 analysis, 321 analytical lower bounds, 643 analytical-numerical methods, 469 angle-ply, 491,500 annular, 515 antisymmetric, 507 any cross-sections, 321 arch-shaped corrugated shell roof, 109 arc-spot welds, 357 axial load, 643 beam capacity, 233 beam columns, 427 beam finite elements, 171 beams, 69, 449 bearing capacity, 103 bending, 69, 179 bending strength, 653 bifurcation, 709 bifurcation analysis, 661 bifurcation stresses, 329 bolted connections, 187 box cross-section, 623 box-beams, 241 breathing, 137 bridges, 129, 153 buckling, 3, 69, 249, 257, 277, 293, 301,313,407, 459, 477, 491,515, 549, 603,623,643,661,683 buckling analysis, 437 buckling and post-buckling, 129 buckling behaviour, 329 buckling curves, 195 buckling of composite panel, 499 built-up headers, 241 built-up sections, 187 bunker, 619

calculations exactness, 321 cassettes, 3 channel columns, 249 chaos, 349 characteristic exponent, 381 closed cross-section, 623 cold formed, 225 cold-formed C-profile, 203 cold-formed sections, 179 cold-formed steel, 187, 233,241,249, 257, 437, 449 collapse, 365,669, 709 columns, 69, 225 composite, 491 composite bars, 321 composite construction, 339 composite laminated plates, 469 composite materials, 269, 285,669 composite plates, 301, 313 composite structures, 293 composites, 585 compression, 69, 179 compressive stability, 507 computer algebra, 701 connection flexibility, 233 connections, 357 constitutive relations, 483,701 construction errors, 669 corrugated steel structure, 103 corrugation effect, 109 cost, 119 coupled instabilities, 195,293 coupled instability, 683 critical strength, 693 critical stress, 507 cross-sectional classification, 747 C-sections, 241 cumulative damage, 137 cut-outs, 313 cyclic loading, 145,459 cyclic tests, 399 cylinder, 709 cylindrical shell, 677, 693

760 cylindrical tubes, 653 damage, 313 damping, 365 degree of joint work, 619 design optimization, 499 design strengths, 249, 257 detail category, 129 digester tanks, 669 dimensional analysis, 541 dimensioning, 321 distortion, 171,623 distortional buckling, 153 distortional mode, 329 dual-actuator control, 719 ductility, 19, 145 dynamic buckling criterion, 365 dynamic relaxation, 631 dynamic stability, 407, 515 dynamics, 373 EC.3-Annex Z Procedure, 195 earthquake damage, 391 eccentric loading, 225 effective length, 249 effective width, 209, 523 efficient, 187 efficient points, 549 eigenproblem, 381 elastic, 153,701 elasticity, 631 elastic-plastic behavior, 457 elastic-plastic buckling, 693 elastic-plastic shells, 483 elasto-plastic postbuckling behaviour, 469 elephant-foot buckling, 391 elliptical barrel-shaped shell, 709 energy absorption, 357 energy procedure, 153 erosion, 137, 179 errors, 417 exact stiffness matrix, 603 experimental, 233 experimental investigation, 249, 257 experimental investigations, 683

experimental results, 195 experiments, 37, 95 expert system, 559 extended Kantorovich method, 603

fabricated thin-walled sections, 87 fabrication, 119 farm buildings, 399 fatigue, 129, 313 fatigue cracks, 137 FE analysis, 313, 585 FEA, 709 FEM, 217, 427, 619, 727 FEM model, 747 Ferrocement, 269 fiber reinforced plastics (FRP), 329 fiber-reinforced concrete, 269 fibre-reinforced laminated composite structures, 277 finite difference method, 349 finite differences, 631 finite differences method, 677 finite element analysis, 3,449 finite element mesh, 459 finite element method, 109, 203, 417, 437, 459, 499, 515, 611,739 finite elements, 373,661 finite strip analysis, 217 finite strip method, 507 fixed-ended, 249 fixed-ended columns, 257 flexible plates, 349 flexural buckling, 217 flexural-torsional buckling, 427 foam core, 477 folded plate, 161 foundation subsidence, 677 frames, 321 generalised beam theory (GBT), 329 generalized imperfection factor, 195 genetic algorithm, 499 genetic algorithms, 585 geometric imperfections, 427 geometric non-linear theory, 407

761 high strength steel, 19 higher-order theory, 491 homogeneous, 515 hurricane winds, 661 hybrid dynamic response analysis, 391 I-beams, 241,747 impact load duration, 365 imperfection, 179, 437, 643, 739 imperfections, 195,593,747 Ince-Strutt diagram, 381 initial imperfection, 119 initial imperfections, 407 m-plane loading, 37 Interaction buckling, 217 interactive buckling, 179 intermediate velocity impact, 365 IOF loading, 241 kinematic loading, 677 labour-saving, 119 laminated plate, 491 laminated plates, 507 large deflection, 631 lateral buckling, 567 lateral-torsional buckling, 195, 747 light, 187 limit moment, 653 limit state, 137 limit states, 739 linear complementarity problem, 611 linear-damage accumulation, 129 lipped channels, 225 load-capacity, 285 local buckling, 37, 161,171,209, 217, 427, 449, 523,729, 747 local plate mode, 329 manufacturing imperfections, 277 material model, 437 MA THEMA TICA, 701 MathTensor, 701

membrane-flexural coupling, 293, 507 metal cladding, 399

metal tanks, 661 micro-concrete, 339 mode shape, 661 monosymmetrical open cross-section, 567 multi-criteria minimization, 575 multicriteria optimisation, 559 natural half-wavelength, 507 Nervi, 269 Nickel-Titanium alloy, 301 nonlinear, 701 nonlinear analysis, 417 non-linear problems, 549 non-linear solution, 739 numerical methods, 643 numerical models, 171 optimization of volume, 585 orthotropic cylindrical shell, 477 orthotropic material, 329, 491 ovalization, 653 pallet racks, 233 pareto-optima, 575 pareto-optimum, 559 partial and ordinary differential equations, 349 partial safety factor, 693 penthouse, 187 periodic axial force, 381 perturbation theory, 381 photoelasticity, 313 pipe arch, 103 pipe section, 459 plastic hinges, 693 plasticity, 631 plate, 515 plate elements, 87 plate girders, 119 plate instability, 611 plate testing rig, 719 plates, 69, 585,603,611 polystyrene, 3 polystyrene foam core, 523 portal frame, 203

762 post-buckled reserve of strength, 137 postbuckling, 301 post-buckling, 3, 69, 477 post-buckling analysis, 437 post-buckling of plates, 611 post-failure behaviour, 285 potential energy, 477 pressure, 631 probabilistic simulation methods, 739 process modelling, 277 pull-through failures, 95 push-out tests, 339

railway bridges, 129 rectangular plate, 3685 reinforced concrete, 669 reliability, 277 repair of defects, 119 residual stress, 437 residual stresses, 87, 277 restraints, 153 return lips, 249 reversal cover shell structures, 541 reversal shell structures analysis, 541 reversal thin-walled shell structures, 541 review, 37 RHS, 449 ribbed wall, 619 Ritz method, 483 roof diaphragm, 357 rotation capacity, 449 rotational capacity, 747

sandwich, 515 sandwich panels, 523 sandwich shell, 559 sandwich shells, 483 scalarization, 549 secondary bending stresses, 129 sector, 631 seismic, 357 semi-rigid connection, 203,233 sensitivity, 593

sensitivity analysis, 53 sequential quadratic programming, 373 shape memory alloy, 301 shear, 3 shear buckling, 145 shear center, 623 shear connectors, 339 shear loading, 37 shear stiffness, 399 shear strength, 137, 399 shear wall, 399 sheet steel, 357 sheeting, 187 shell design optimisation, 541 shell design with graphics, 541 shell theory, 483 shells, 585,661,669 simplified method, 549 smart-structures, 301 soil-structure interaction, 103 spatial structure, 109 stability,19, 153, 161,407, 483,593 stability design, 145 stainless steel, 209, 225,257 statics, 53 steel, 3 steel bridge pier, 459 steel pier, 391 steel cladding systems, 95 steel light structure, 109 steel plate girders, 161 steel plates, 523 steel structures, 217, 249, 427, 533 steel telecommunication towers, 533 stiffened and unstiffened plate elements, 729 stiffened cylindrical shells, 683 stiffened plates, 549 storage tank, 677 strain energy, 483,701 strength, 119 stress concentration, 313 stress gradients, 719 stress range spectra, 129 structural analysis, 533 structural connections, 19

763 structural design, 249, 257 structural elements, 37 structural members, 19 structural optimisation, 533 structural optimization, 373,575 struts, 69 stub columns, 729 stub-column, 209, 233 symmetric shells, 643 tapered, 603 tendon force, 87 tensionless foundation, 611 tensor analysis, 701 test based design, 203 test strengths, 249, 257 testing, 3 tests, 217, 729 theory of plasticity, 483 theory of shells, 701 thin shells, 373 thin structures, 269 thin-walled, 37 thin-walled beam, 567 thin-walled column, 623 thin-walled columns, 329 thin-walled cross-sections, 739 thin-walled members, 69, 161,575 thin-walled orthotropic beams and columns, 285 thin-walled section, 209 thin-walled structure, 109

thin-walled structures, 53, 129, 339, 407, 593 through-girders, 153 tubular members, 257 ultimate capacity, 427, 729 ultimate loading, 49 unilateral contact, 611 unstiffened plate, 145 unstiffened plate elements, 719 upright frame tests, 233

valley-fixed, 95 variability, 277 variable thickness, 603 vector optimization, 549 vibration, 407 vibrations, 53 viscoelasticity, 515 von K/lrmgn equations, 349 warping function, 567 washers, 357 web breathing, 129 web crippling, 241 weld shrinkage, 87 wind pressures, 661 wind uplift, 95 wood frame, 399 Z-sections, 217

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