Linear and nonlinear finite strip analysis of bridges

1+1

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Canad!l

LINEAR AND NONLINEAR FINITE STRIP ANALYSIS OF BRIDGES

A thesis submitted to the School of Gradunte Studies and Research in partial fulfillment of the thesis reqllirements fol' the degree of Doctor of Philosophy in the Department of Civil Engineering

PhoDo Candidate: Wenchang Li Thesis SlIpervisor: Mo So Cheung

Department of Civil Engineering Fnculty of Engineering University of Ottawa Ottawa, Canada o

~wenChang Li,

Ottawa, Canada, 1991

•••

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ISBN

Canada

~.

-315-80004-&

~::J UNIVERSITÉ D'OlTAWA UNIVERSITY OF OITAWA

ABSTRACT The annl~'~iR of highway bridges such ns slnb-on-gil'der bl'idges, box-girder bridgcs, cable-stayed bridges etc. is a very complicuted undert,nking. Analyticnl methods are applicable only for the simplest st,ructures. Finite clcmcnt \\lcthod is thc most powerful and versatile tool, which can be applicd to IUlnlyzc /Uly types of bridgc and any load cases. However, thc cfficicncy of that mcthod nccds to be improvcd bccause the finite element solutions usunlJy rcquirc too \\luch computer time, too large core storage and too many input datu. If a structure has a unifonn cross-scction and lillc end support,s (in fnct, a high

proportion of bridges can be simplified to such It st.l'lIctmc), the finite stl'ip method has proven to be the most cfficient numcricnl st.l'lIct.lIml ILnnlysis mcthod, which employs a series of functions to sil11ulate the variation of displacel11flnts in thc longitudinal direction of the structure. Thus, the nllmbcr of dil11ensions of /UlIllysis is reduced by at least one. Consequcntly, the comput,cl' time, storagc and input data are reduced significantly. Since this method

WIIS

first publishcd in 1968, it

has been extensively used for linear and nonlinear, static and dynamic anluysis of rectangu1ar, skew and curved slab bridgcs, slab-on gil'der bridges box-girder bridges etc. In the present study, the following efforts arc made:

1. Extending the finite strip rnethod to the aIlruysis of continuous haullchcd alah-

on-girder bridges and hox-girder bridges.

ii 2. Exl.!mding the splinc finitc strip mcthod to thc Iwalysis of continuous haunched slab-on-girdcr bridges Iwd box-girder bridgc8. :\. Extcnding thc finitc strip mcthod to nonlincar analysis of cable-stayed bridges. 4. hllPIYlving thc cfficicncy of gcomctrically nonlinear finite strip analysis of plates. 5. IJIlproving thc nccuracy of materiaIly nonlinear finite strip analysis of reinforccd concrctc slabs. 6. Combining the finite strip method with finite element method and boundary clement mcthod for analysis of rectangular plates with some irregularities.

A nl1l11bcl' of IlUmerical eXlUnples will show the accuracy and efficiency of the lIIethods devcloped in the present study.

iii

ACKNOWLEDGEMENTS The author wishes to express his sincere appl'cciation to his I'CNCIU'ch N1\pcl'visor, Dl'. M.S. Cheung, for his constructive suggcstions, vllluablc disCIIHsionN lUHI continucd assistance throughout thc coursc of thc study. Sinccre thanks are also exprcsscd to Dl'. L.G .•1ncgcr, Dl'. A.G. Rmmpm, Dr. M. Saatcioglu, Dr. S.F. Ng and Dr. TlUmka for t.heil' important information, valuable advice IUld gcnerous nssistlUlcC in c1lOosing topics, solving c1iflicult qucstions, revicwing the Thesis Proposal ctc. The fimUlcial support from the Natural Scicnccs lUul Engincming Rescarch COl1ndl of Callada is gratefully nckllowledgcd.

I

Contents

ABSTRACT .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . ..

iii

Tahlc of ContcIlts. . . . . . . . . . . . . . . . . . . . . . . • . . . . . .,

iv

List. of Figllrcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

xii

. NOMEMCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xiv

1 INTRODUCTION 1.1

1

ANALYSIS OF HIGHWAY BRIDGES . . . . . . . . . . . . . . ..

1.2 DEVELOPMENT OF FINITE STRIP METHOD 1.3 SCOPE OF STUDY

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

i\'

1 2

6

CONTENTS

2

v

FINITE STRIP METHOD 2.1

CONVENTlONAL FINITE STRlP METHOD

8

· .........

.

8

2.1.1

SERlES PART OF DISPLACEMENT FUNCTION . . . .

2.1.2

DISPLACEMENT FUNCTIONS . . . . . . . . . . . . . "

12

2.1.3

STRAINS............................

13

2.1.4

STRESSES...........................

14

2.1.5

MINIMIZATION OF TOTAL POTENTIAL ENERGY

14

2.1.6

COORDINATE TRANSFORMATION

...........

17

2.1.7

FLEXIBILlTY METHOD . . . . . . . . . . . . . . . . . ..

18

2.2

COMPOUND FINITE STRlP METHOD . . . . . . . . . . . . . ,

ID

2.3

EIGENFUNCTIONS OF CONTINUOUS DEAMS . . . . . . . .,

21

2.4 ANALYSIS OF CONTINUOUS HAUNCHED DRIDGES . . . ..

28

D

2.4.1

STRAIN-DISPLACEMENT RELATIONSHIP .. . . . ..

28

2.4.2

DISPLACEMENT FUNCTIONS . . . . . . . . . . . . . .,

30

2.4.3

SOLUTION PROCEDURES . . . . . . . . . . . . . . . ..

34

vi

CONTENTS

2.4.4

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . ..

35

2.4.5

CONCLUSION

38

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

3 SPLlNE FINITE STRIP METHOD 3.1

47

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . ..

47

3.2 SPLINE FUNCTION INTERPOLATION . . . . . . . . . . . . ..

49

3.3

ANALYSIS OF CONTINUOUS HAUNCHED BRIDGES . . . "

51

3.3.1

STRAIN-DISPLACEMENT RELATIONSHIP . . . . . . .

51

3.3.2

DISPLACEMENT FUNCTIONS . . . . . . . . . . . . . "

53

3.3.3

PENALTY FUNCTION APPROACH . . . . . . . . . . ..

56

3.3.4

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . ..

57

4 NONLINEAR ANALYSIS 4.1

68

NONLINEAR ANALYSIS OF CABLE-STAYED BRIDGES . . ..

69

4.1.1

FINITE STRIP ANALYSIS OF GIRDER . . . . . . . . ..

70

4.1.2

FORMULAS FOR CABLE . . . . . . . . . . . . . . . . ..

72

4.1.3

STIFFNESS MATRIX OF THE PYLON . . . . . . . . ..

77

CONTENTS

4.2

4.1.4

INITIAL-STIFFNESS ITERATION

78

4.1.5

NUMERICAL EXAMPLES ..

80 84

DISPLACEMENT FUNCTIONS AND INITlAL STIFFNESS MATRIX. . . . . . . . . . . . . . . . . . . . . . . . . . ..

87·

4.2.2

GEOMETRICAL NONLINEAR SOLUTION . . . . . . ..

89

4.2.3

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . ..

93

NONLINEAR ANALYSIS OF REINFORCED CONCRETE SLABS 95 4.3.1

MATERIAL MODEL OF CONCRETE . . . . . . . . . ..

96

4.3.2

MATERIAL MODEL OF REINFORCEMENT . . . . . ..

99

4.3.3

FINITE PLATE STRIP . . . . . . . . . . . .. , . . . . .. 101

4.3.4

NONLINEAR SOLUTION . . . . . . . . . . . . . . . . .. 102

4.3.5

NUMERICAL EXAMPLE . . . . . . . . . . . . . . . . . . 105

5 COMBINED ANALYSIS 5.1

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

GEOMETRICAL NONLINEAR ANALYSIS OF PLATES ... " 4.2.1

4.3

vii

118

FINITE STRIP METHOD FOR REGULA R PART .. . . . . .. ll!l

viii

CONTENTS !i.2

5.3

COMJ3JNED WITH FINITE ELEMENT METHOD . . . . . . . , 120 [i.2.1

FINITE ELEMENT METHOD FOR IRREGULAR PART

120

5.2.2

TRANSITION ELEMENT . . . . . . . . . . . . . "

5.2.3

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . . . 122

. . . 121

COMBINED ANALYSIS WITH BOUNDARY ELEMENTMETHOD123 5.3.1

BOUNDARY ELEMENT ANALYSIS FOR IRREGULAR REGION ... ' . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.3.2

TRANSITION STRIP AND COMBINED SOLUTION .. 126

5.3.3

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . .. 128

6 CONCLUSIONS AND RECOMMENDATIONS

138

6.1

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2

RECOMMF.NDATIONS........................ 141

REFERENCES

142

List of Figures

2.1

39

Structure Analyzcd by F.S.M..

o'

••••••••••••

39

2.3 Individual and Common Coordinate System .

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

40

2.4 Continuolls Berun . . . . . . . . . . . . . . . . . . . . . . ; . . . . ,

40

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

41

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

41

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

42

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

42

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

43

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

43

2.11 The Mesh of Shell Elements . . . . . . . . . . . . . . . . . . . . . ,

44

2.2 Folcled Plate Strip· . . . .

2.5 Span i ..

2.6 Support i . . . . . . . . 2.7 Reguli-Fhlsi Iteration . 2.8 Web Strip . 2.9

•••••••••

Shell Strip . . . . . . . . . . . .

2.10 Continuous Box-Girdcr Bridgc

IX

•

x

LIST OF FIGUR.ES

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

44

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

45

2.12 Five Span Composite Box-Girder Bridge . 2.13 Whccl Wcight of Two Trucks

2.14 Division of Strips . . . . . . . . . . . . . . . . . . . . . . ., . . 2.15 Longitudinal Stresscs in Steel Girder at Section X-X (in MPa)

3.1

45 ..

Splinc Function and Its Derivatives .

46

63

........ . .

64

3.3 Will) Strip in Individual System ..

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

64

3.4 Shell Strip . . . ...

•

•

65

3.5 Continl\ous Beanl . . . . . . . . . . . . . . . . . . . . . . . . . . ..

65

3.2 Plate Strip . . . . . . . . . . . . ..

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

o.'

•

•

•

•

3.6 Hmlllched Continuous Bridge . . . . . ..

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

3.7 Haullched Continuous Box-Girder Bridge

.'

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

66

3.8 Division of Strips . . . . . . . . . . . . . . . . . . . . .

67

3.9 Longitudinal Stresses at Cross-Sectioll X-X (in MPa) .

67

4.1

Cablc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2 Pylon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

LIST OF FIGURES

4.3

xi

Initial Stiffncss Method . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4 Single PhUle Cable-Stnyed Bl'idge . . . . . . . . . . . . . . . . . "

108

4.5

Double Planc Cable-Stayed Bl'idgc . . . . . . . . . . . . . . . . .. 100

4.6

Deflection of Girder and Pylon .. . . . . . . . . . . . . . . . . .. 100

4.7 Longitudinal Strcsses at Cl'oss-Section B (in MPII) . . . . . . . .. 110 4.8 Longitudinal Stresses at Cl'oss-Sect.ion F (in MPa) . . . . . . . "

111

4.9 Possible Divergence . . . . . . . . . . . . . . . . . . . . . . . . . "

112

4.10 Equivalent Uniaxial Stress-Strain Model . . . . . . . . . . . . . .. 113 4.11 Biaxial Strength Envelope . . . . . . . . . . . . . . . . . . . . . .. 114 4.12 Material Model of Steel . . . . .

. . . . . . . . . . . . . . . . . 115

4.13 Layers of Strip

. . . . . . . . . . . . . . . . . 115

4.14 Taylor Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 116 4.15 Deflection of Taylor Slab . . . . . . . . . . . . . . . . . . . . . . "

5.1

117

Rectangular Finite Element . . . . . . . . . . . . . . . . . . . . .. 131

5.2 Transition Element . . . . . . . . . . . . . . . . . . . . . . . . . .. 131

LIST OF FIGURES

xii

5.3 Squarc Platc . . . . . . . . . . . . . . . , 5.4

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

132

Platc Supported by Walls and Columns . . . . . . . . . . . . . .. 132

5.5

Dcflcction and Bcnding Moments of Plate in Fig.5.4

5.ü

Double Nodes ..

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

. .. 133 . .. 134

5.7 Transition Strip. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 134 5.8 Simply Supported Square Plate under Uniform Load . . . . . . . , 135 5.0 Plntc wlth Opcning and Skew Corner 5.10 Bcnding Moments along A-B-C

. . . . . . . . . '... 136 . . . . . . . . . . . . . . 137

List of Tables

2.1

Thc Proper Number of Scgmcnts and Gauss Points ..

35

2.2

Longitlldinal !)tresscs in Two Span Box-Gil'dcl' Bridgc . . . . . "

37

3.1

Vallles of Spline F\mction at l(nots . . . . . .

50

3.2 Deflection and Longitudinal Stl'esscs in Continuous Deam

58

3.3 Longitudinal Strcsscs in Two Span Slab-oll-Girdcl' Bridgc

59

3.4

Longitudinal Stresscs in Two Span Dox-Gil'dcr Bridgc . . . . . ..

61

4.1

Deflections (in lllcters) of Girder at Cabl,! Attachment Points and Vertical Forces (in MN) of Cablcs . . . . . . . . . . . . . . . . . .

81

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

81

4.2

Bending Moment (in MN.m) of Girdcl'

4.3

Vertical Forces of Cablcs on Girdcr und Horizontal Forccs of Cublcs on Pylon (in MN) . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

84

DcHcction and Strcsses in Cylindrical Platc Bending

94

4.5 DeHection and Strcsscs in Clamped Squarc Plate . . . .

94

4.6 Cocfficicnt K for Stress of Stecl after Concrete Cracking

. . .. 100

4.4

5.1

DcHcction and Moments in Clampéd Square Plate

. . . . . . 122

5.2 DcHcctioll and Stt-csscs in Simply Supported Squarc Plate . . . .. 129

xiv

NOMENCLATURE {a} b

[B] D

[D] E f~

f:

{F} [F]

h

[K] l

L(y) m

[N]

{P} q T

Til T2

R {R} [T], [t] [T,] u,v,w

U W

displncclllcnt vcctor width of strip strnÍn matrix = Eh 3 j12(1- v 2 ) c1astic matrix Young's modulus uninxial compressivc stl'cngth of concrctc uninxial tensilc stl'cngth of concretc force vcctor flexibility matrix thickness of stl'ip, length of longitudinal scction stiffness matrix length of strip Lagrange interpoIntion cxprcssion nU1l1ber of longiturunal scctions in a strip matrix of shapc function load vcdor loading per unit lU'cn, weight pel' unit length nUlllber of serics tcnns uscd in IUlnlysis curvatl1l'e radii of curvilincnl' coordinatc Iincs curvnturc radius of bot tom fllUlgc vector of redundlUlt forces or resistant forccs coordinnte transfonnntion matriccs for displncclllcnts coordinate transformation mntrices for strnÍns displacements in x,y and z directions strain energy potential energy of exterual londing

xv

Y,,,(y) a

{6}",

l'", lj i

{u}

frec vihration cigcnfullction of heam IUnplification factor of flcxura! stiffness, ratio ut! U2 vcctor of displaccmcnt parruneters cquivalcnt IIniaxial strain strain vcctor free viImItion cigenvalllc of berun Poisson '8 mti o total potcntia! encrgy B3 splinc function ccntcred at Yi vec tor of strcsscs and moments principal strcsscs maximum compressivc stress and corresponding strain of concrete transversc slope 8w / 8x

""

xvi

Chapter 1

INTRODUCTION 1.1

ANALYSIS OF HIGHWAY BRIDGES

In service, highway bridgcs such as slab on gil'clcl' bl'iclgcs, box gil'der bridgcs, cable-stayed bridges etc. undergo not only longitudinni bcnding but al80 transverse bending, torsion, distortion and shcar dcformation. Thc lond distribution runong the girders, the support reactions and cable-tcnsions arc highly static!llly indeterminate and material and gcometricnl nonlincaritics duc to coneretc cracking, cable sagging and p-~ effcet etc. arc oftcn significant. Thercfore, thc Imalysis of a bridge is acomplex undertaking. There exist SOllle analytical methods for bridge analysis, such

liS

the load distri-

bution technique for right simply supported slab-type bridges [1,2,3], thc stiffness method of analysis for stecl orthotropic deck sJ'stellls [4] and thc cxtcnded foldcdplate theory for box-girder bridges [5]. However, bccause thcsc analytical mcthods can only be used to analyze highly simplificd structurcs, their applicability is liml

2

CHAPTER 1. INTRODUCTION

it.ed. The fini te element method is the most powerful and versatile tool for analysis of hridges [6,7J. That mcthod clm be applied to dcal with any specific configuration of bridgc structure and supports. It is suitablc for analysis involving alI types of "t.lltical and dynamical loads and alI kinds of elastic and inelastic deformation. Ncverthdess, the finite clement solution usually requires a

signi~cant

arnount of

computer time, large corc storage and tedious and lengthy input data files. Therefore, t.he efficiency of this method nccds to be improved. In fnct, the simply supported right deck of uniform section ( or a structure which iliny be rClllistically nnalyzed as such ) constitutes a high proportion of the large nU111ber of bridges being built. For analyzing this type of bridge, the finite strip lIIethod has proven to be the most efficient numerical methodj it uses a series of orthogonal functions in the longitudinal direction, y, combined with the conventional finite elcment polynomial shape function in the transverse direction, x, t.o simulatc ali the displaccment components of the structure. In this way, the 1111Inber of dimensions of the analysis is reduced by at least one. Consequently, cOlllputer time, storage and input data requirements are reduced significantly.

1.2

DEVELOPMENT OF FINITE .STRIP METHOD

The fillite strip method was first published by Y.ICCheung [SJ for analysis of simply supported bridge deck structures in 1965. The finite strip method for rectangular

CHAPTER 1. INTRODVCTION

3

slnb-typc bridgc dccks was also suggested independent.\y by Powell IInd Ogden 19J in 1969. Since t.hen, considcrable reseru'ch and development. on thllt method have bcen carricd out in many cOlIntrics. In the late 1960' IUld em'ly 1970', t,he field of rescllr("h cxt,enclcd to nuuly types of bridge mIClloading concli tions~ slIch ll.~:

• rcctanguhu' slabs with end bOllndnry conditions other thllll simple SlIpports,

110J in 1968, • simply supported box girder bridges, I11J in 1969, • curved slab and box girder bridgcs, [12J in 1969 ami [13J in 1971, • slab-type bridgcs with intermediate column slIpport,s Ilsing the Ikxibilit.y lIppronch, 114J in 1970, • rectallgular slabs wi th variable cross section in the splll1\vise directiolI, 114J in 1970, • the frequency analysis of some simple and continuous rcctangulllr slabs, 115J in 1971, • skew slab bridges, [16J in 1972, • skew box girder bridges, [17J in 1975, • the initial buckling analysis of box-type structures, [18J in 1973 and I19J in

1974, • continuous box girder bridgcs with trnnsverse diaphragms, using the HcxibilitY approach, [20J in 1976,

CHAPTER 1. INTRODUCTION

• slab and hox girder

bridg~s

4

continuous over rigid supports, using continuous

bmuu eigeufullctiollS alld a direct stiffness method, [21,22] in 1974 and 1978, • allalysis of general plates, [23] in 1978.

Y.I<.Cheung and Y.C.Loo have published their books [24,25] to summarize the basic theory of the finite strip method and its applications in bridge engineering IlS

the results of research work during that period.

Since the middie 1970's, enormous efforts have been devoted to more complicated . topics. Among them, the main subjects related to the finite strip analysis in bridge engineering may be listed as follows:

• compound finite strip method for analysis of plates continuous over intermediate f1exible beanls and columns using a direct stiffness methodology [2{j,27].since 1983, • post-Imclding bclmv ior and geometrically nonlinear analysis of plate structurcs, [28-33] since 1978, • mntcrialnonlincar analysis of stcel structures,[34-36] since 1986, • mnterinlnonlincnr analysis of reinforced concrete slabs,[37] 1988, • large dcflection clasto-plastic analysis of plate structures [38], 1989.

CHAPTER 1. INTRODUCTION

5

In spite of a muubcl' of advlUltagcs, thc abovc mcntioncd scmi-rull\lyticlII finite strip mcthod experienccd difficultics in dcaling with conccntl'lltcd forces, multipic spans, discrete supports nt strip cn ds ct.c. To ovcrcomc t.hcsc difficultics and to retain the advantagcs of the finitc strip mcthod, a llmthcmnt.icnl tool cnllcd 'Da spline function' was used fol' displacemcnt functions to fOl'm thc splinc finitc stl'ips for analysis of rectangular plates by Y.K.Cheung ct al [39) in 1982. The B3 spline function

ClUl

ensure continuity up to the second dcrivative (thc so-

called C2 continuity). However, in ordcl' to achieve thc slunc continuity condition, the finite element method needs thl'ec times liS many unknowns at the clemcnt nodes. Hence, the use of B3 splines is computationlllly much 11101'C cfficicnt thlLll the finite element method wi th C2 continuity. When using thcBa spline function, the penalty function approllch [6) is I'eadily utilized to imposc lilly type of bou:alary conditions. Thus, the spline finite strip metho.d is more flexible thlLll the scmilUlalytical finite strip method. In addition, the second dCl'ivativc of B3 splincs varies linearly in each longitudinal section, as

II

l'csult of which it can morc casily

simulate peak values of bending moment at the 10lldcd scction Ol' at an intcnncdiatc support. In later years, the spline finite strip mcthod was cxtendcd to box girder bridgcB

[40) in 1983, to skew plates [41,42) in 1984 and 1988, to arbitrary shapcd slabs and arbitrllIj' curved slab bridges [43,44) in 1986, to vibration and stabilityanalysis

[45,46) in 1987 and to postbuckIing lUlalysis [47) in 1989.

ClIAPTER 1. INTRODUCTION

6

huleccl, d1l1'illg the past two decades, a very large number of researchers and engi11""1'8

have made so many important contributions to the development of the fillite

st.rip mt'thod and Hs applicat ion in engineering that it is impossible to list ali of t.heir

ac(~olllplishlIlenl.s

1.3

here.

sc OPE OF

STUDY

The prinHuy objective of this thesis is to extend the finite strip method to more complicat.cd structures and more difficult analysis, thus mnking the method more accurate and efficient. The main topics and efforts arc focussed on the following arcas: A. SEMI-ANALYTICAL FINITE STRIP METHOD: 1. finite striplllethod for continuous structure

2. fini te strip analysis of haunched, continuous bridges 3. finite strip analysis of haunched, continuous box girder bridges D. SPLINE FINITE STRIP METHOD: 1. spline finite strip analysis of haunched, continuous bridges

2. spline finite strip analysis of haunched, continuous box girder bridges C. NONLINEAR FINITE STRIP ANALYSIS OF BRIDGES: 1. nonlinear finite strip analysis of cable-stayed bridges

2. gcol11etrical nonlinear analysis of plates 3. material nonlinear analysis of reinforced concrete slab D. FINITE STRIP METHOD COMBINED WITH OTHER NUMERICAL METHOD:

CHAPTER 1. INTRODUCTION 1. finite strip rumlysis of platc combined with finitc clcmcnt mcthod

2. fillite strip method combincd with boundltl'Y elemcnt mcthod Eltch of thc above topics has formcd a scpcmte clmptcr in thc thesis.

7

Chapter 2

FINITE STRIP METHOD 2.1

CONVENTIONAL FINITE STRIP METHOD

The finitc strip IIlcthod was first published by Y.K.Cheung for analysis of silllply sllpportcd bridgc dcr.k st,ructures in 1968 [8). Since then, this lIlethod has been cxtended to Illany types of bridge such as rectangular, curved and skew box-girder bridges of single SplID or multispan, etc. Y.K.Cheung and Y.C.Lao have published their books [24,25J to sumlllarize the basic theory of the finite strip method and its applications in bridge engineering as the results of research work during thll late 1960' and early 1970'. In this section, the basic concept of the traditional finite strip method for analysis of rectangular box-girder bridges with single or nlllltispan is reviewed briefly. The finite strip method

ClUl

be considered as' a special form of finite element pro-,

cedure using the displacement approach or as the so-called semi-analytical finite clement method. If a structure has constant cross-section and its boundary con8

CHAPTER 2. FINITE STRIP METHOD ditions at both cnds do not challgc trlUlsvcl'scly, thcn fol' its stl'CSS lUlalysis thc struct\ll'e can be divided into a number of finitc strips (Fig.2.1) instcnd of finite elcments. In each strip, the displacemcnt. components nt any point arc cxprcssed in terms of the displacement pammetcrs of nodal lines by mcans uf simpic polynomials in the tl'llnsverse direction and continuously diffel'entiablc slllooth scl'Íes in the longitudinal direction, wi th the stipulation that such scl'Ícs shollld satisfy a priori the boundary conditions at the ends of strips.

2.1.1

SERIES PART OF DISPLACEMENT FUNCTION

The most commonly used series are the bmull cigcnfllnct.ions which

IU'C

dcl'Ívcd

from the solution of the berun vibl'lltion differcntial equntion

(2.1 ) The general form of the berun Cigenfunctions is

Y(y) = with the coefficients

Cl

Cj

sin(py) + C2 cos(py) + C3 sinh(IJY)

+ C~ cosh{Jty)

(2.2)

to be determined by the end conditions.

These have been worked out explicitly for the vnrious cnd conditions and arc listcd below: (a) Both ends simply supported (Y(O)

= ylI(O) = O, Y(l) = ylI(l) = O).

10

CHAPTER 2. FINITE STRIP METHOD

U'm

TTlrr

= -1-)

(h) I30th cnds c1ll1npcd (Y(O) = Y'(O) = O, Y(l) = Y'(I) = O).

Y,,,(y) = sin(/I",y) - sinh(JLmY) - Cl'm[COS(JLmY) - cosh(JLmY)],

sin(JLml) - sinh(JLml) am = cos(JLm l ) - cosh(JLm l) /1",1 ILrc the solution s of cquntion 1 - cos(JLI) cosh(JLI) = O, the first twelvc valucs

al'c:

4.730040744862704, 7.853204624095837, 10.99560783800167, 14.13716549125746, 17.27875!J65739!J48, 20.42035224562606, 23.56194490204045, 26.70353755550818, 2!J. 84 513020910325, 32.!J8672286269282, 36.12831551628262, 39.26990816987241. Fol'

III

grcntcl' thnn 12, I' .. = (m + 0.5)rr/1 ca.n be takcn as a. very elose approxi-

mation. (c) One end simply supported and the other end clamped

(Y(O) = ylI(O) = O, Y( I) = Y'( l) = O). Y,,,(y) = sin(llmY) - Cl'msinh(JLmY), am =

( 1'm l =

? 3.9~66,

sin(JLml) sinh(JLm l )

) 7.0685, 10.2102, ... , 4m+ 4 1 rr.

(d) I30th ends free (Y"(O) = ylI/(O) = O, ylI(l)

= ylI/(l) = O).

Yj(y) = 1,JL! = O,.

CHAPTER 2. FINITE STRIP METHOD

11

r;,,(y) = sin(ll",y) + sinh(IIIIIY) - am[cos(JIIIIY)

+cOHh(JIIIIY))'

. sin(II",1) - sinh(II",I) COS(lt",1 ) - cosh(lt",1 ) 2m-3 (1 /",1=4.7300,7.8532,10.9960, ... , 2 lr,m=3,4, ... ,oo). a", =

(c) One end clamped and the other end free

(Y(O) = Y'(O) = O, Y"(l) = Y"'(I) = O). Ym(y) = sin{t/",y) + sinh(Il",Y) - a",[cos(Jl",y)':'" cosh(lt",y)), sin(Il",I) + sinh(II,,,I) COS(lt",l) + cosh(ll m I) 2m-1 (II", I = 1.875,4.694, ... , 2 lr). Om

=

(f) One end simply supported Imd the other end free

(Y(O) = Y"(O) = O, Y"(I) = Y"'(I) = O). YI(Y)

y

= 1,llI = 1,

Ym(y) = sin(llmY) + am sinh(llmY), am =

sin(llm l) sinh(llml)

(11",1 = 3.9266,7.0685,10.2102,13.3520, ... ,

2m-3 4 lr,

m=2,3, ... ,oo). The beam eigenfunctions possess the valuable properties of orthogonality, i.e.

for mi'n

CIfAPTER 2. FINITE STRIP METIfOD

l' Y,~Y~'dy

12

= O

for m

of n

Ut.ilizntion of thcse properties will rcsult in a significant saving in computation elfort for the clllclllation of stiffncss mlltrices.

2.1.2

DISPLACEMENT FUNCTIONS

In slah on giruer briugcs

IUlU

box-giruer bridgcs, cach finitc strip is subjcctcd to

in-plane strcsses and out-of-planc bending forccs(Fig. 2.2). This type of strip is cnlleu n fokleu plate strip. Thc noual displaccmcnt paramcters and the nodal force vector of the strip corresponding to thc m-th series term arc:

The displncemcnt components at any point (x,y) within the strip are expressed in tcrms of the noull.l displaccment parameters as follows: r tt

= ~)(l - X)Ul m

+ XU2m)Ym(Y)

m=l r

V = L:((l-X)Vl m +XV2m)y':'(Y)/Jlm m=l r

tv = L: (CItvim + C2911• m=l

where,

+ C3tv2m + C4 92m )Ym(Y)

(2.3)

CHAPTER 2. FINITE STRIP METHOD

13

X=:r:/b and r is t.he totnl number of terms considel'ed in the Imnlysis. These expressions can be written in the followillg more compnct form: r

{fl =

{1l,V,w}T =

:E[Nlm {c5}",

(2.4)

m=l

2.1.3

STRAINS

Once the displacement functions are kllown, it is possible to obtnin the stmins, curvatures and twist, through appropriate differentiations with respect to the coordinates x and y in the well-known manner, yieldillg the following:

Ou

er = Ox

Ov

ev - Oy

Ou "'(ru

= Oy

Ov

+ OX 02W

Xr = - OX2 02W

XV = - Oy2 02U)

Xrv

= 2 OxOy

(2.5)

CHAPTER 2. FINITE STRIP METHOD

14

The matrix form of these equations is as follows: r

{fl = {e",ey,'Yru,Xr,X.,Xru}T = L:[B]m{6}m m=l

2.1.4

(2.6)

.

STRESSES

Thc strcslles arc related to the strains by r

{u}

= {Nr, N., Nxy, Mr,M.,Mru}T = [D]{e} = L:[DJ[B]m{6}m

(2.7)

m=l

wherc [D] is the ell\sticity matrix. For an isotropic folded plate, O ] [D] -_ [[D]Pl O [D]b

(2.8)

in which

Eh [D]pl = -1- 2 - v

[1v

Eh3 [D]b = 1?(1 _ 2) -

2.1.5

v

O ] O

v 1

O O l::.!!. 2

[1v

v

O]

1 O O O l::.!!. 2

MINIMIZATION OF TOTAL POTENTIAL ENERGY

The straill energy of a folded plate strip is given by

CHAPTER 2. F1NITE STRIP METHOD By virtue of (2.6) and (2.7), the stra.in cncrgy elUl be expl'esscd

15 IlS

(2.lJ) where [K]m" is t.he stiffness sllbmatl'ix of the stl'ip lmd is given by the following arca integral:

[1\']"", = j[B]!:[D][B]nclA

(2.10)

The potential energy due to the cxternul distribuled lomls {lj} = {qr,lJu,q.}T elm be written simply

IlS

w=

- jU)'/'{q}clA

(2.11)

Substituting (2.4) into (2.11) givcs T

W =-

"L.J

'I' { F}", {6}m

(2.12)

m=1

where {F}m is the equivalent nodal forcc vect.or of the externalload and is givell by the following integral:

{F}m = j[N]~{q}tlA

(2.13)

For a point load the above integral is redueed to thc simple expression of lond multiplied by the eorresponding displacement. The total potential energy of the entire struct.ure,~, is the sum of U and W eontributions from ali the strips. Thus NS

~ = L(U+W) ,=1

(2.14) in which NS is the number of strips.

16

CHAPTER 2. FINITE STRIP METHOD

Accol'ding to the principle of minimum potential energy, the values of the displaccment pILrametcrs must bc such as to make the total potential energy of the stl'llcLure assumc IL minimum valuc. In mathematicnl form this requires that (2.15) SlIbstituting (2.14) into (2.15) and performing the partinl differentiation produces /L sct of lincar I1lgcbraic equationti which can be written in the following matrix form:

[J(){6} = {F}

(2.16)

in whkh [K],{6} and {F} arc the stiffness matrix, the nodal displacement vector /Lnd thc nod al load vector of the whole struct'lre. They are the assembles of the <'Ol'l'cHpondillg matriccs or vectors of alI the strips in the structure. Solving the set nf cquations for the unknown displacement parameters and

~Ilbstituting

them into

(2.4) and (2.7) will give the displaccments and stresses at any point of interest. Bccause of thc orthogonal property of the beam eigenfunctions, for structures with both cnds simply supported, thc derivation shows that alI the terms of the series arc uncouplcd, namely, [J(]mn = O for m =F n. Thus, the stiffness matrix has a very llIU'row half-blUldwidth, and the required computer storage and time consumption arc I'cduccd significlUltly.

CHAPTER 2. FINITE STRIP METHOD

2.1.6

COORDINATE TRANSFORMATION

Ali of the above derivations

IU'C

carricd out. in a local coo\"(liullt..,

syst(~m

, wherdn

the x and y a.'l:es coincide with the mid-smfacl' of a st.tip. lu folded plat., strudurcs, any two plates will in generll.l meet at ml angle, and in order t.o a"scmblc tilt' stiffness matrices, the displaccment vectors mul thc load vcctor" of non-coplnllar strips, a common coordillate system is obviously rcquircd. In Fig. 2.3 the individuru coordinatcs of a strip arc labclled common coordinates

ilS

liS

x',y',z' mul the

x,y,z. y and y' m'c coillcident with cach IUlother mul

also with the intersection line of two adjoilling strips. The trnnsfol"llllltion of displaceme1l.ts

b·~tween

the two sets of coordinate syst.ems is given by

{6'}m = [T]{6}",

(2.17)

in which [T] is the tr311sformation matrix

[T] = [[ot]

O]

[t]

where

cosf3 O - [ -sinf3 O

[t] _

l

O sinf3 O 1 O O O cosf3 O O O 1

(2.18)

(2.19)

Substituting (2.17) into (2.9) and (2.12) leads to the followillg results, it being noted that in (2.9) and (2.12) [K]mn, {F}m, {6}m and {6}n arc ill local coordinatcs and hence arc primed:

[K]mn = [T]T[K1mn[T]

(2.20)

{F}m = [T]T{F'}m

(2.21)

18

CHAPTER 2. FINITE STRIP METHOD

2.1.7

FLEXIBILITY METHOD

ln c()njunction with the flexibility method Ol' force mcthod, the finite strip method elln nlso be used to analyze continuous bridge decks. The bridge deek is first reicIIsed from ali the internal supports and is analyzed by the !init.e strip mcthod. The deflections at alI the internal supports due to external 1'1II<1s elln be elL~ily found, and is here denoted by a column vectol' {a}. Next,a unit vertieILI force is applied at internal support j alone and the deflection fi; of cach int.emnl support i is obtained. ThiR procedure produces the flexibility matrix

[F[ = [fiiJ. Fol' silllplicity, all the internal supports are assumed to be rigid. Thus, the compati bili ty I'equil'ement of displacements yields following equations:

[F]{R}

+ {a} =

O

(2.22)

from which the true reactions {R} at the internal supports can be calculated. Applying externlll loads and the reactions of the internal supports to the released structure will then give the final displaccments and internal forces. For pintes IIDd bridges over continuous line supports, it is also possible to treat each individllal spnn

IlS

simply slIpported, with continuity of slopes over the supports

(nt the nodal line vnly) being subsequently restored. In this case the support lllOlllcnts nre choscn as the redundant forces [24]. The combination of finite strip analysis with the flexibility method is also appli-

CHAPTER 2. FINITE STRIP METHOD cable to box-girder bridgcs wi th intcl'll1cdiatc cliaphragm [25J.

2.2

COMPOUND FINITE STRIP METHOD

Since 1983, a compound finitc st.rip methocl has been devclopc,l for analysis of linear clnstic flat plate systems continuo\ls over deflecting s\lpports s\l('h

liS

Hcxible

beams and columns [26,27J. This approach incorpomtes the effeds of t.he SlIpport clements in a direct stiffness methodology, which makes t.he SOIIlt.iol1 Illore straightforw81'd tlmn with the flexibility method. The so-called compound strip is a finitc plate strip which is composite wit.h t.he supporting clel1;ents such as longit\ldinal beams, t.rllllsverse helUns IIml coltlIlllIS. Ali of thesc elements deform togethcr, mId their displacelTlellts can be expresscd in terms of the nodal displacement pm'anleters of the strip hy mellllS of the cli';plllcement functions of the strip (2.3). The strain energy of It ,!ompo\lnd strip incllldeH not only the strain energy of the plate strip but also t.he following st.miH ellCl'gy of the supporting elements: (a) The flexural and torsional strain energy of the longitudillru beams (2.23) in which EI and GJ arc the flexural and the torsional rigidity of tivelYi NL is the number of longitudinal beams in the strip.

It

heanl rcspec-

CliAPTER 2. FINITE STRIP METHOD

20

(h) TIl
NT EI J éPw GJ J éPw D(-2 )2(/x +? (--)2dx); i= I 2 u:c - uxuy

(2.24)

in which NT is thc lIumbcr of the transversc beruns in the strip. (c) Thc rucial strnin enf'l'gy and the flexural sh'ain energy of columns (2.25)

whcrc KA is the axinl stiffness of columni Ker and Key ru'e the flexural stiffness of colullll1 in the dircctions x and y rcspectivclYi

Ne

is the number of the columns

SlIppol'ting the strip. Tll1IS, thc total strain encrgy of the compound strip, Us, is the sum of following it.clIIs: (2.26) whcrc UI' is the flexural strain energy in the plate strip. SlIbstituting thc expression of displacements (2.3) into (2.26), the strain energy

us can bc cxprcssed in terms of the nodal displacement parameters of the strip in the form:

(2.27) The stilfness submatrix of thc compound strip, [Klmn, is readily obtained. This IIlcthod has been successfully applied to the analysis of a variety of continnaus platc structures.

CHAPTER 2. FINITE STRIP METHOD

2.3 In the

21

EIGENFUNCTIONS OF CONTINUOUS BEAMS anal~'sis

of continllolls foldcd pintcs, til<' (·onvent.iOlml finitl' st.ril' IlIcthn
combined wit.h Hcxibility approach und the C.olllpolln
Because the entire berun vibratcs at the srunc frequcncy, l' is thc smne

[Ol'

nil the

spans. The general solution of this equation for thc i-th span can thcn be expl'csscd

as (2.29)

in which C;I ... C;4 are constants which should be dctcrmined from thc end conditions of the span.

CHAI'TER. 2. FINITE STRIP METHOD

22

If the first spau is simply supported at the end YI = O (Fig. 2.4 a), the end cOllclit.ious are Y1(O)

= Y{'(O) = O, YI(/Il = O.

Equation (2.29) then becomes (2.30)

aud Cl'1 = sin(j.tll)/sinh(pll)' For the lust span n (Fig.2.4.b), in order to simplify mathematic procedures, it is couvenient to use a new coordinate system,

y~

= ln -

Yn, to express the general

solution fol' this span. Tim'.

(2.31) and Cl'n = sin(pln)/ sinh(pln)' Ench intel'111edinte span i (Fig.2.5) can be treated as a coinbination of the abovemcntiolled two situnt.ions, i.e., (2.32) where

IUld

Evidently, the first IlS

SplUl

!lnd the last span of a continuous beam can be considered

the p!l1'ticular case of an intermediate span, with An = O and AnI = O. There-

fore, (2.32)

CIUI

be used to express the solutions for aU spans. The slope, dY;/dy,

CHAPTER 2. FINITE STRIP METHOD

23

anrl thc curvnture, ePJ:i / dy2 of n continuous belUll elUl then bc obtllincd as

eP J:; = ly 1

_/1

2

Ail( sin(/IYi) + Oi sinh(/IYi)) _

/1

2

Ai2( sin(/lyD + Oi sinh(!lyD) (2.34)

For simplicity, thc following notntion is introduccd:

SLi = eos(!tli) - a'i cOSh(/tli)

(2.35)

The continuity conditions betwecn SplUl (i-l) and span i nt support i (Fig.2.6, Yi-I = li-h Y:_I = O, Yi = O, Y: = li) arc 1. The slope of ndjnccnt spnns nmst be cqunl. Thcrcforc, by virhic of (".33) and

(2.35), the following equntion CIUl bc obtnined: (2.36) 2. The bending moment and, conscquently, t.hc curvnturcs must also bc eontillllous over support i. From equations (2.34) and (2.32) thc following cquntioll enn similarly be obtnined: (2.37) The values of p. and Ai; (i=1...n, j=I,2) have to satisfy thc continuity eonditions (equation (2.36) and (2.37)) at ali the intermediate supports. Furthcr, /' must also satisfy the end conditions of the beam : Au

=O ,

Ani

=O

Equation[2.38] states that M~ = O at both ends of thc beam.

(2.38)

CHAPTER 2. FINITE STRIP METHOD

24

The Ai; represent only the mode shape, not its size; hence the following values mny be adopted, without loss of generality,

Au = 1 , At2 = O

(2.39)

For n given p., Ait and A i2 can be evaluated from the known values Ai_I,1 and

Ai-t,2 by using equations (2.36) and (2.37). Doing this span by span, finally, ali the Ai; up to Ant and An2 , which are the functions of p., can be calculated.

If 11.11 the Ai; arc not simultaneously zero, and the value of p. makes (2.40) then p. and Ai; have satisfied both the continuity conditions and the end conditions, thercfol'e these values give a valid solution. In solving equations (2.36) and (2.37), different procedures must be followed, depending on whether or not sin(p.1i)

= O and sin(p.1i_t) = O.

1. If sin(/tli) '" O, from equations (2.36) and (2.37)

(2.41) (2.42) 2. If sin(/11i) = O , /tli =

ln'/!'

(m=1,2 ... ): From equation (2.32), it can then be

seen that ai = O and

li = Ait sin(p.Yi) + Ai2 sin(p.y') = Ai! sin(/IYi) + Ai2 sin (In'/!' - P.Yi)

CHAPTER 2. FINITE STRIP METHOD

25

= (Ail - ( -l)'" Ai2) sin{JIYi)

This means that the cocfficicnts Ait 11m! Ai2 arc dcpclldcllt variablcs. Hcncc, lU1t! again without loss of gcnemlity, .4i2 = O can bc I1SSl1111Cd, so that (2.43)

It is noted, howevcl', that the calcl1lation of .4 i1 still dcpcnds on thc VIUUC of sin(JL1i_l) associatecl wi th thc pl'evious span (i-l), and it lIIUst. be cOllsidcred scp-

arately for the two cascs of sin(/Lli_l) = O I1Jld sin(JL1i_l) a. If sin(JLli_l ) = O, then

Yi-l

i= O.

= Ai_I,1 sin(!tYi_I)' Ai-I,2 = O I1Jld cquation (2.37)

becomes an "identity". Then, from the slopc continl1ity conditionluonc (2.36) wc can obt.ain (2.44) b. If sin(JLli_l)

i=

O, Ai_I,1 must bc

ZC1'O,

othcl'wisc it docs not satisfy equu-

tion(2.37). Thcn from the slope continuity condition alonc (2.36), wc

CM

obtaill (2.45)

However, in the case ofthe first interior support (betwccn SpMS l IUld 2), if sin(JL11) is not equal to zero but sin(JLI2) = O, it cl1Jll'csult in Ai'; = O (i = 2oo.lI,j = 1,2) from equations (2.39) (2.43) (2.44) etc. In this case, although Ani = O, JL is still not a solution of equation (2.28), because it violates the curvature continuity condition (equation (2.37)) at the first interior supports. In the case of the last span, if sin(JLl,,) = O, thc curvature at the last cnd of bcam is automatically equal to zero. Therefore, in this casc, even though equation (2.40) is not satisfied, JL may still be a solution of the equation (2.28).

26

CIiAPTER 2. FINITE STRIP METHOD

TIli! first-order Reguli-Faisi iteration can be employed to solve the p. and Ajj ,(Fig.2.7).

StlLrting with the point /jo, /j is increased by a small increment 8,1 in evel'y step. At IJvm'Y tria! point p., Anl(p.) is determined from A 11

= 1 and AI2 = O by using

elluationH (2.41),(2.42), etc. However in some cases, if Ani changes its sign between two trittl points

p.(0)

= p. -

8p. (the previous trial point) and /-1(1)

= /-I (the present

trinl point), the straight line is drawn between the points (/-1(0), Ant (/-1(0))) and (/L(I), Anl(/-I(I))) as shown in Fig. 2.7. The intersection of the straight line with the /L ruds is nt /-1(2) which is doser to the solution

From /1(2) the point (/I(U), Ani (/L(O)))

(,L(2),

il than /-1(1).

Ani (/-12)) is located and the straight-line is connected to.

or (/1(1), Ant (p.(t))) depending on at which point Ant has different

sign from Ani (/-1(2))). This locates /-1(3) on the /L 8.'Cis,which is doser to the desired resu!\. thllJl /L(2).The process is repeated unti!

(2.46) The itcmtion formula is

(2.47) Continuing to increase /-I by 8/-1 for every step, eventually, the necessary eigenvalues, /1"" and modcs,

Alj)

are identified, these being as many as required by the finite

strip nnalysis. When thc eqllntion (2.41) was applied to determine Ai2 fron. t. ' , both sides of cqllntion (2.37) were divided by sin(/-Ili ) , which might result in missing the solution of sin(lLl j ) = O. Therefare, if during

IL

particular increment of /-I, 5in(/-Ili ) change5

CHAPTER 2. FINITE STRIP METHOD

27

its sign, thcn II = inTr/li nltlst bc used as the t,rial point, t.o l:nknlnt,c .4ij IUlCl scc if this is one of the eigenvalucs. Howcver, in any ot,her ensc, it. must uot I,akc lUly trial value of II equal to inTr/li so as to avoid sihmtions sneh ns 5in(Jl./2) = Owhite sin(pll)

1= O, which will result in IUl C1'1'OnCOU5 vnluc of .4"1(/1).

In the computer program, the following valucs \Vere used: Óp

= Tr /30li",ar

EI

=

lQ-IO

and it is assumed that sin(pli) = O if 1sin(pli)

1< lQ-lO.

If the lengths of alI the spans arc not exactly thc samc, but slightly difl'crcnt from each other, then,p = inTr/li is not a solution, and thc sign of A"I(/L) changcs drastically near p = inTr/li, In this case, snmllcr increment,s of Óli lUust hc uscd l :'

in order to prevent the possibility of missing modcs. For cxlt1nplc, if ali of thc

!

spans havc magnitudc cithcr lmar or Imi,,, and if O < (lm"x - l"'in)/l",ar.

!t

it requircs great care not to miss thc solution ncar rmr / l",ar' Thc rccollllllcnded

~

,,

< 0.05,

remedialllleasure is that if there has bccn mTr/lm"r - /L < Ó/L, thc incremcnt is rcduced to óp' = m11'(l/lmin - 1/lma:)/4 and the ncxt trial value of /L is changcd into p = mTr Ilma: +óp' /2. Aftcr the eigcnvaluc ncar mTr / l..,,: has hecn found, thc original increment ÓIl = mTr /301ima: is resumcd.

28

CIIAPTER 2. FINITE STRIP METIlOD

2.4

ANALYSIS OF CONTINUOUS HAUNCHED BRIDGES

III the

prc~cllt

stllcly, the finitc stri]> method was extended to the analysis of

hlLllllChcd, continuous sIab-on-girdcl' bridges ([49], 1988) and haunched continuaus bux-girder bridges ([50], 1989). In such nn analysis, three types of strip arc used, these being the top flange plate stri]> (Fig.2.2), the vcrtical web strip (Fig.2.8) and the bottom flange shell strip (Fig.2.9).

2.4.1

STRAIN-DISPLACEMENT RELATIONSHIP

Fol' thc top flI1nge plnte strip, the Cartesian coordinate system is used and the strnin-displacement relationships (2.5) are applicable. For the web strip of variable depth, the curvilinear orthogonal coordinate system (~-I/)

is morc convcllient, in which.,., has units of length. The web can be divided

into n numbcr of equal or unequal width strips. The width of cach strip is taken ILS

b = c x b,., where

/1..

is the depth of web, with O < c

~

1. In the web, the

most important deformation is in-plane bending, and thc most significant strain componcllts are thc longitudinal normal strain E~ and the longitudinal shear strain

re./.

According to thc actual proportion of ordinary haunched box-girder bridges,

CHAPTER 2, FINITE STRIP METHOD

29

the relationship bctwcen displaccmcnts nlld stnuns can bc writtcn

Dv

u

f

-

,,-

+1'2 8'1

--

Du

')'(" =

81}

1 Dv

'/I

+ bD~

- ;:;-

18211)

(2.48)

X( = - b2 D~2 X~

!IS:

82 11) = -

1 81 2

2 D211) X(" = bD~ 8,}

where

l',

nlld

1'2

arc the radii of curvaturc of coordinatc lincs (Fig,2,8), Imd thc

following approxilllate geollletrical rclationship can bc takcll with rcusonnble accuraey: 1 1'2

x' cl2 b," = b," cl112 b,"

1't

=

(2.49) (2,50)

~.

1Ir,

For the bot tom flange shell strip, cylindrical surfucc coordinntcs nrc used, and thc relationships between displacelllents and stnuns [59] arc 8u

E:z:=-

8x

f

8v

w

8u

8v

---U - 8y R

')'ru

= -8y +-8 x

82 w

(2,51)

Xr = - 8x2

82 w

W

Xu = - 8y2 - R2

30

Cl1APTER 2. FINITE STRIP METHOD

EJlw Xry = 2 8x8y

28v

+ R8x

whcre R is the mclius of Clll"vature of the middle surface in the longitudinal direct.ion, this is usually a function of y. In ordinary haunched box-girder bridges it can he

!LqSUmcc\

timI. (2.52)

2.4.2

DISPLACEMENT FUNCTIONS

The choice of suitable displacement functions is the most crucial and difficult part of the analysis. The right choice should lead to satisfactory results for maximum deflection

IUlCl

muximum longi tudinal stress using only a very few terms of series

if thc mmlyzed bridge is subjected to uniform load (according to our experience, the rcquh'cd number of terms in series should be less than 5 times the number of spans). Aftcr comprehcnsive research and experiments, the following formulations are. worked out. The strip c1isplucclllcnt paranleters for the m-th term are taken as:

wherc

VI

IUld

V2

mainly represent the longitudinal movements due to in-plane

bcndillg IUld VI IUld V2 mainly represcnt displacements in the y direction caused

CHAPTER 2. FINITE STRIP METHOD

31

by the longitudinal elongation or compression of the stdp. Thc c1isplllcCllIcnts wi thi n a plate strip are as follows: r

II

L «1 -

=

X)lIf'

+ Xu;")Y,,,(y)

m=1 r

V

=

L«l- X)v;" + Xv~')Y,:(y)+ m=) r

L «(1- X)v;" + Xv;T':.•(y)

(2.53)

r

W

=

L «1 -

3X 2 + 2X 3 )w;" + :t:(1 - 2X + X 2 )9;"

m=)

where Y,,, is the eigenfunction of a contilluollS bell.lll which hlls t.he span lengt.hs of the actual briclge, Imcl Y:' (y) = (b,u dY.[m (y

Ym(y) =

_

1

dib," Y,,,) (y

(Ym(y)/R)dy

(2.5<1 ) (2.55)

X =:t:/b. In (2.55) the intcrvnl of integrntion is from the origin of the first Rpml to t.he current coordinate y. If the bottom line of web in each span is a pnl'nbolic curv!!,

R can be assumecl to be a constant(2.52), and the integration cnn be implem!!llted analytically. For the web strip, similal' equatiolls cnn be obtailled by replacing X by {, x by b{ and y by TI in the above equations. These equations can also represent the displaccments in the bottom flange shell strip without any change. However, in this case, x, y and z arc cylindrical surfoce

CHAPTER 2. FINITE STRIP METHOD

"oordillatcs, alld

II,

32

v and w arc the corrcsponding displacement components in

t.his <:oonlinatc system (Fig.2.0) . The cxprcssiolls of u and w meet the boundary conditions at the end and interIIlcdiatc SUppOl'ts which arc asswncd to be diaphragms that arc infinitely stift' in their own plancs, so that u=O and w=O there. Substituting the displaccmcnt functions (2.53) for the web strip into the expression for 'Ye~ in (?.48) and taking into consideration (2.50) and (2.54), 'Ye~ can be cxprcssed in tertns of displacemcnt pal'ameters as follows:

Dlt

1 811

+ b8~ -

"Ie" = Dll =

t

u

;:;-

((1- e)u;"

fII=1

Hu;") d;m c 71

+

-v;" + vr)Y';;/H (-v;"

+ vr)Ym/b}

.

-t ((1- ~)u~' Hu2")1-;n/ !: t((I - ~)u~' + eU2',)(dYm _ dbwYm/bw) d'f1

m=l

=

t {(

m=1

dll

>11= I

d1]

r

+ E {( _v;n + vr)y';;/cbw+ (-v;" + vr)Ym/b} m=l

r

=

E((I- e)u;" + ~u;')y';;/bw m=l r

+ E {( _v;n -I- v2")Y';;/cbw + (-v;" + vr)Ym/b) 1n=1

From the above expression, it can be seen that 'Ye~ = O if u\" = lI~')/C

and

ur, u\"

= (vr -

vj" = v;". This means that the displacement functions (2.53) can

simulate the pure in-plane bending of webs at the term level. Substituting thc displacement functions (2.53) for the shell strip into the expression

CHAPTER 2. FINITE STRIP METHOD for

Ey

in (2.51),

fy Call

33

be expl'cssed in thc following form:

8t, tv --ay R =

tHI - X)v;" 111=1 r

- L ((1 -

+ Xv~,)Dl';~ + t((l- X)ii;" + XfI~,)8f;,,(y) ay

3X 2 + 2X 3 )w;"

8y

m=1

+ x(l -

2X

+ X 2 )9;"

m=l

(2.56) Using (2.55), we have

dY,,,(y) l ey

d l l = -/ (Y,,,(y)IR)dy = Y.. (y) R. ey y

Substituting this exprcssion into (2.56) givcs the following result Ey

=

t ((1 - X)v;" +X v;") a~';~Y + t ((1 - X)ii;" + Xv;")}';,,(y)1 R

m=1

m=1

T

- L ((1 -

3x 2 + 2X 3 )w;"

+ :r(l -

2X + X 2 )9;"

m=l

From t.his expression, i t is not difficult to find that ali the terms of Y,,, (y) 1R can be cancelled out if 9;" = 0,

92' = 0, and wi" = lOr = vj" = v~'.

Similal'
e

also prove that the longitudinal norma.! strains on any givcn curve = con .• tant in the web strip and the entire top Range strip dul' to the arching effcct released under proper combinations of displaccment paramcters.

wI R may bc

Thi~

means that

the displacement shape functions (2.53) arc capablc of reflecting that the lU'dung effect can affect the distribution of longitudinal stress on the cross-scction but may not cause any longitudinal stress at the neutral axis of the cross-section, bccausc in any continuous bridge only the supports at one cross-section arc fixcd in thc longitudinal direction.

CHAPTER 2. FINITE STRIP METHOD

34

If II", displaeelIlenl. fnndiOlis cnn not simuInte the pure in-plnne bending of the web 1-(., = O) Ol' the sl.Jess-fl'ee lungitndinnl expansion of the bridge, the structure will S""1Il II III el ,

2.4.3

stifl'el' than it relllly is, and thc correct IUlswer may nevel' be obtnined.

SOLUTION PROCEDURES

SII],stit.nt.ing the dispIncement functions (2.53) into (2.5),(2.48) and (2.51) respect.ivdy, t.he stmins fol' cach type of stdp cnn be expressed in terms of dispIncement pnrnmet.ers in the following matrix fonn: r

{el =

L: [B] ... {ó} ... m=l

The st.iffness matrix of the strip corresponding to the m-th IUld n-th series terms c/tn t,hell be c)cpl'essed as (2.10):

In the x dil'ection , the integl'ation CIUl be implemented numerically or lUlalytically. However, in the y direction the GaussiIUl integration method has to be used, becnuse of the complexity of the integration in the stiffness matrix. Since fL Y,:.r~ dy and fL Y'::Yndy IIxe not equal to zero when m IUld n are different, the terms of the series arc coupled. In evaluating the stiffness matrix, we should choose the proper number of GaussiIUl integration points to achieve the desired accuracy IUld efficiency. When necessary, every SplUl CIUl be divided into several segments. In each segment, 8 to 10 integration points should be used. The selection of segment size nnd GnussilUl intcgration poillts depends mninly on the number of highest-order hnrmonics in n given SplUl, n~ = Jlmoz/;j7r. Based on our experience the selection

CHAPTER 2. FINITE STRJP lvIETHOD

N"

/1," is constant. JVllt9 Na 1 8 10 1 2 8 10 2 2 10 3 8 10 3 4 10 4 10 10 5 6 10

35 blV is val'iahlc

N IICfJ 1 1 2 2 3 3 3 4 5 5 G

Ne; 8 10 8 10 10 10 10 10 10 10 10

1,2 3,4 5,G 7,8 9 10,11 12-14 15-17 18 19-24 25-30 N# = Ilma~/;/7r N ••o: Number of Segments in SplUl i Na: Number of Gauss Points in Eaeh Segment Table 2.1: The Proper Number of Segmcnts and Gauss Poillts which will best achieve a renson!l.ble balnnce bet.wcclI aecuracy and dfieicney of the solution is givcn in Tablc (2.l).

Following the procedures described in scction 2.1, it is not difIicult to obtain the final solutions required, including the distribution of displaccments and stresscs within the whole structure.

2.4.4

NUMERICAL EXAMPLES

1. Two span haunched continuous concrete hox-girder bridge.

A multi cell box-girder bridge that is continuous over two equal spans is shown ill Fig.2.1O. It is haunched over the interior supports, with c=2.40m. The bridgc span I is chosen equal to 40m, spacing betwcen wchs lPZ = 0.075/ = 3.00rni thickncss of

ClIAPTER 2. FINITE STRIP METHOD

36

top slnh, hl = 0.005/ = O.20m; thickncss of bottom slah, h 2 = 0.004/ = O.16m; thickncHs of wcb, /JI = 0.01125/ = 0.45m; length of hmmehed portion is considel'ed to ,:ovcr

It

hot.t.olll Hang!! is

distnncc of 12.00m. In this pOl·tion, the gencrating line of the iL

pnmbola. Total depth of the box girder(excluding the haunch),

h=1.201ll. Due to Hymmctry, only half II cellneeded to be analyzed; this half was divided into 3 strips, Le. top fllmge strip AB, web strip BC and bot tom Flange strip CD, as sllOwn in the lower part of Fig. 2.10. The applied load is sclf-weight, corresponding to n spccific weight of material of 24[( N 1m3 • The material properties are E=25

GPn,11 = 0.2. Decallse the geollletry of the bridge IUld npplied loads are symmetrical with respect to the intermediate support, the deHection must also be symmetrical. Therefore we t.uke only t.he symmetrical modes as the displacement functions in the longitudin".l ,!irect.ioll. The longitudinal stresses in a munber of cross-sections are given in Table 2.2. These results were obtained by· using 10 symmetrical terms in the series. Fol' compn.rison, the finite clement program ADINA was also used to analyze the snllle structure. In this case, each flange is divided into 1 row of 16-node shell clements; the web is divided into 2 rows. Each row includes 8 elements (Fig.2.11). Thl' results frolll this method arc also given in Table 2.2. It can be seen that thc rcslllts obtll.incd from thc fillÍte strip analysis are in elose agreement with the rl'slIlt.s "btruncd from the ADIN A finite element analysis.

CHAPTER 2. FINITE STmp METHOD Y2 (m)

O 3 6 12 20 30

37

Mcthod

a. (MPn) along nodallinc D D A C ADIN A 4.226 4.292 -3.268 -4.287 FSM 3.792 3.732 -4.075 -4.108 ADIN A 4.373 4.3i8 -4.662 -4.343 FSM 4.335 4.323 -4.869 -4.606 ADINA 4.389 4.610 -5.262 -4.729 FSM 4.596 4.620 -5.353 -4.964 ADINA 1.566 1.403 -1.649 -1.814 FSM 1.733 1.745 -2.061 -2.049 ADIN A -3.264 -3.535 4.115 3.803 FSM -3.142 -3.177 3.703 3.668 ADINA -4.383 -4.637 5.398 5.109 FSM -4.344 -4.367 5.095 5.074

Table 2.2: Longitllclinal Strcsscs in Two SplUl Dox-Girder Dridge 2. Fivc span composite box-girder bridge The briclge is a two-Iane steel-concrete composite box-girder structUl'c and is continllous over five SpI\1lS. There arc two stecl boxes wi th a cMt-in-placc rcinforced concrete cleck. Figure 2.12 shows the clevation of t.hc bridgc, thc depth of thc webs I\1lcl its typical cross sections. The loacls arc two fivc-axle trucks. Their whecl weights and positions arc shown in Figure 2.13., Fol' steel, the material properties are E = 21OGPa, v = 0.3; for rcinforccd concretc ,E = 25GPa, v = 0.2. The structure is divided into 22 strips wi th 21 nodal lines (Fig.2.14). 10 terms of the eigenfunction series arc used in the ,lUllLlysis. The distribution of the longitudinal strcss along cross-scction x-x is depicted ill Fig.2.15. Thc experimentu.1 valucs [60J of thc srunc stress arc also 5hown in this figure for comparison purpos(!. Thc maximum longitudinal stress of web is at the hottom corner of web No.1,its numcrical valuc is 26.25 MPa while thc experimental value is 24.47 MPa. The

CHAPTER 2. FINITE STRIP METHOD

38

IIHlXilll1ll1l longitudinal stress of the flange appears in thc boUom flangc adjaccnt tu web No.l ,its nUIIU!ricllI valll/! is 25.Gi MPa while the expcril1lcntal value is 29.64 MPa.

It. lIIay be concluded that the nUl1lerical results arc in good agreemcnt with the experil1lental reslIlts for all high stress regions. Stress cOl1lparisons for other regions s/!el1l to be less fnvorable. However, this mny be attributed to the accuracy of the experimelltnl mensurcments themselves. Several irregularities of the experimental rcslIlts, as showlI in Fig.2.15, clearly indicnte this possibility.

2.4.5 Havillg

CONCLUSION

dlOS('1I

t.he nppl'Opriate shape functions,the finite strip method can be

cfl'cct.ivl'ly mnploycd fol' !Ulalyzing cOlltilluOUS, hmmched box-girder bridges. The mcthod is simple, nccurate and ensy for ellgineers to use. In addition, because this mcthod dircctly IIses the eigenfunctions of a continuous berun as its shape functions ill the lougitudinal direction, it converges very quickly, especially for distributed loads. Thercfon', it is more effective than both the finite element method and the conventioual finite strip method.

CH.-H'TER 2. FIXITE STRIP METHOD

Noda.l line

30

Hnite StN?

Figul'C 2.1: Stl'UctUl'C Auulyzcd by F.S.M.

y,v

I

e ( .L-_ _-?-~L-_ _ X,U b Z,W Figul'C 2.2: Fulc.lcd Plutc Strip

CHAPTER 2. PINITE STRlP METHOD

40

~---.-----:;~-

z'w'

X. U

Z.W

'Figul'e 2.3: Individuru and Common Coordinate System

(1)

.&

lS

1

(n)

(2)

ZS

h

Á

R

ZS n

2

n~l

b, (l

ln

M'

,

Y1

ct

~

Yn Yn

The Fi rst Span

b.

The LAst Span.

Figure 2.4: Continuous Berun

CliAPTER 2. FIXITE STRIP METHOD

h

I M'

I

Yi

Á'

fE Yi

At

E.

+

M=

,M' f Ji

Y1I

A Y'i

FigUl'C 2.5: Spllll i

It

'1-1

(1)

(i-l)

li

r.

i

4

t-

I

y~

Y1-1 'Y 1- 1 Yi

Figul'C 2.G: SUPPOl't i

1

H

CllAPTER 2. FlJYITE STRIP METHOD

42

A

n,l

I ~\

I '\

t \\ I

I I I

,I"

\ \

\\ \

\

\

,,'" \ \

~'"

\\ 1\ 1 \ d \ ~f

I I I

\

\ .

,1

\ \

\1\1

Figure 2.7: Rcb'1lli-Falsi Itel."a.tioll

top edge of web

l (t -O)

,

'l,V

x'

r2

Figure 2.8: Web Stl"ip

f

2 (t -1)

CHAPTER 2. FINITE STRIP lvIETHOD

'13

y,v

R Figurc 2.9: Shcll Stril>

L=40m

1=40m

_____o

----t

12cL-j

1--- 20Il1--t

.K

l

_-,---::---;,------71 Th. L. ~ m I c=2.4ul ~

~

1--.- Yz A

II

hl .. O.2DI

I

'::=-.=i1!

.

D

i

h

:- b, _0.l.5111

T c

h -O.16m

b,.311

FiglU'C 2.10: COlltilluOUS Dox·Gil·dcr Dridgc

44

Cll.·IJ>TER 2, FISITE STRIP MET.fIOD

6.0

9.0

40.0 Y 2

Figlll'c 2.11: Thc Mc~h of Shell Elcmcllts

X-I l ... í

1. S4

3.H

F ~-r- ~

-;l

=td:

x-1 ~I

I .~4.H&

.44

,I

3&.~6

I

. \I , J.K "

,u. J

~

I

Trm'k<

Wl'b I

I

U,WJ

I.

2

T

I. S4

to

U. Cll ')t,

lI.llIU

'----I 2.~U

- ._-

3.H4

.L

2.~~

x- x Figurc 2.12: Fi\'c SplUl Compositc Dox-Girdcl' Dridge (in M)

CII.-l.PTER 2. FIXITE STRIP METHOD

45

Truck l ~

33.15x2.

43.&4x2 33.711

'ruck l i

3l.Obx2

""-~ 4~.Ulx2 31.4~IKNl

II II I II II

c6~----~L-~L-L-~L-~~----------n6D

Figw'c 2.13: Whcd Wcight of Two Tl'ucks

r web

1

'l

4

3

•

Figul'C 2.14: Division of Stl'ips

Cll.-lI'TER 2. FINITE STRIP METHOD

-lU.SU

. /

~

46

-Y.L6

......

,

/

,

\ 7.Ub

3.53

Web 1

2

. . . .-

=-~

Finit:e Strip

_ .......

~-

Experimert:a1 (ll)

L6.25 (L4.471

16.31 :lU.41

15.52

25.67

~--7

.65

Web 3 11.~

11.55

2.ua 2.48

Figul'C 2.15: LOlIgitudilll\1 Sh'csscs in Stccl Girdcl' at SCctiOll X-X (ill Mpa)

Chapter 3

SPLINE FINITE STRIP METHOD 3.1

INTRODUCTION

The semi-analytical finitc strip method is vcry efficient fol' Il.llItlysis of prisnml.ic structures under distributed loading. Ncverthelcss, thc

lISC

of this Illcthod can

sometimes lead to difficulties. Fol' instlUlcc, bccause the bem n funct.ions arc cuntinuously differentiablc, it is difficult to use such functions to simulatc thc abrupt changes of bending moment at intel'llal supports or at loadcd Cl'Oss-scctions of point forces. In addition, the beam functions must satisfy thc cnr.! conditiulIs of a strip a priori; hence, thcse functions can not deal wil.h discrcte slIjljlorts ILt strijl ends. In order to overcome these difficulties, the mathcm ... ~ical tool callcd 'B3 splinc' was used as the longitudinal displacement functions to form the spline fillite strijlH for analysis of rectangular plates by Y.K.Cheung ct al [39) in 1982.

47

48

CHAPTER 3. SPLINE FINITE STRIP METHOD lu this lIlet.hocl, eaeh Bodai linc is divided into

II

mnnber of sections by evenly

spacecl l(Bots, 111,,1 every knot is taken as the center of a local B3 splinc, which is syllllllet.l'knl to the center nnd has non-zero values ov<'!' four consecut.ive sections (Fig.3.1). Ali t.he B3 splines on a nodal line form

II

series which is utilized to

silllulnte t.he longit.udinal variation of displaccments. The B:\ spline functiolls can ensure continuity up to the second derivative (the so-cnlled C2 continuity). By comparison, in order to achieve the Salne continuity cClJIdit.ion, the tinite elcment mcthad needs three times as many unknowns at the elelIleIIt nodcs. Hence, when C2 continuity is required, the use of B3 splines is (·olllput.atiollnlly much mo!'e efficient than the finite element method. The second clerivative of B3 splines varies linearly in each longitudinal section, as II

reBult of which it can morc easily simulate peak values of bending moment at

the loaded cross-section or at all intermediate support. FlI\'thermol'e, simil81'ly to the finite element method, asplinc finite strip Call easily t.ake up any prescribed external and internal boundary conditions e.g. by a penalty function approach. Tlms, the spline strip method is more flexible tllan the seminnalytical finite strip m~thod in imposing boundary conditions. At.

II

later stage in its development, the spline finite strip method was applied to

the Ilnalysis of skew alul lU'bitral'y shaped plates, arbitrary curved slab bridges, rectllugnlllr box-girder bridges IUld to vibrntion, stabilityand postbuckiing allalysis

[40 through 4iJ.

CH.4PTER 3. SPLINE FINITE STRIP METHOD

·ID

In the prcscnt sllldy, Ihc spliul' finill' strip lIleIlulll is ,'xI"Il,k,1

1.0

Illi'

IInlll~'sis

eontinuous hnllndU',1 si ah on ginl"r hridges 151]lInd hnx-gir,Ic'r hri,lg"s

3.2

"l'

\[,21.

SP LINE FUNCTION INTERPOLATION

"Splinc" was origillally the ualll<' of a snmll Ikxibl,' wlKlllen "tri" (,lupluycd hy draughtsmcn as a tool fol' drawing n eOIlt.illllolls sl11noth

C\ll'V!'

segllll'nt hy scg·

ment. Actually, the splinc fllnction can h" dcrived with any reqllir('(1 ,:nlltinllity (discontinuity) conditions. In the prcsent study, the ll:1 spline of equal sedion length [61] is choscn to reprcsent. t.hc displlleelllcnt. This fllnct.ion is a "i('('('wisl' cubic polynomial with (:O\üiuuity up to the sccond derimtive. In order to usc splinc funct.iolls tn interpolatc lill arbit.rnry fUllet,ioll f(y) on tili' intcrvnl

II

:5

y

:5 II, the interval is divide
spaced knots Yi ( O :5 i

:5

l/l,

Yo

=

(I,

y",

= b, Yi = Yo + i(ll -

II )/m ). The ""lilii'

function <}i(Y) wi th y = Yi lL~ t.he center is defillcd by

(y - Yi_2)3 11 3 + 3h2(y - Yi-d h3 + 3h2(Yi+1 - y) (Yi+2 - y)3 O

Yi-2 :5 II :5 Yi-I + 3h(y - Yi_,)2 - 3(y - Yi-I ri Yi-I :5 11 :5 Yi + 3h(Yi+1 - y)2 - 3(Yi+l - Yrl Yi :5 11 :5 Yi+1 Yi+ I :5 y :5 Yi+2 otherwisc (3.1 )

<}i(Y) lUld its first and second dcrivativcs arc shown grnphically in Fig.3.1, 11\1(1 their vr..lues at knots arc givcll in Tablc 3.1. It elLll be scen thllt. the fundioll ;(y) is twice continuously diffcrcntinblc on the entire interval, and its secoud ,Icrivativc is a Iinear function of y.

CHAPTER 3. SPLINE FINITE STRIP METHOD

'I'i(Y) 'lo:(y) 'lo:'(y)

Ui-'/.

Y,-I

()

1

Yi 4

()

3/h G/h 2

-12/,,2

()

O

50

Yi+2 O -3/h O G/h 2 O Yi+1 1

Tnble 3.1: Values of SplÍ11e Fnllctiun at l(nuts TI", splille fllllctiolls cwtercd nt ali the Imuts complise n series s(y) which is used t.o illtm'polate f(y l. In each sectiun the value of s(y l is related to four spJines whkh I\.l'e cClltercd nt the two ellds of this section and the two knots next to those ellds respcctively. Therefore, two addit.ional knots Y-I and Yon+1 arc needed to illterpolatc f(xl in thc first and the last sections. Thus the series of the spJine fnl1ct.iol1s call he expressed ns follows: ,"+1

L Ci'I>i(Y)

.~(y)=

(J.2)

i=-I

ill which t.he Ci arc cocflicicnts t.o be dctermined by interpolation requirements snch

IL~:

s'(Yo) = f'(yo) S(Yi) = f(Yi) O ~ i ~ m

s'(y.. ) = f'(Ym)

(3.3)

Sllhstituting (3.2) into (3.3) and using the values in Table 3.1 yield a sct of Jinear "'1llnt.ions wit.h Ci IlS nnknuwlls:

~(-C-I + ct! = f'(Yo) Ci_I +4Ci + ci+1

= f(Yi) O ~ i

~(-Cm_1 +c..+1) = f'(y .. )

~ m

CH.4PTER 3. SPLINE FINITE STRIP METHOD Solving

the~(!

51

equntions for Cj gin's tili' \'t'quircd st'des of splint' fundions fnr in·

tl'rpoln tion: m+ I

.'(y) =

L

('j'I'j(U) = f(1/)

i=-1

3.3

ANALYSIS OF CONTINUOUS HAUNCHED BRIDGES

Aspline fillitc st.rip mcthod fol' analysis of contilltlous IlIInndwd slnh·()ll·ginl.,r bridgcs IUld hox·girdcr bddgcs is dcvelopcd in the I>l'e8"nt. "tucly. In this annlysiN, thr!'e t.ypes of strips lU'e tlsed, theNc being t.he t.op nnuge plat.e "t.rip (Fig.3.2), t.he v!'rt.ical web strip (Fig.3.3) and bottom nange shell st.l'ip (Fig.3A).

3.3.1

STRAIN-DISPLACEMENT RELATIONSHIP

Fur thc top fhmgc platc strip, the strain·(üsplacmnent. relat.ionship" (2.5) arc ap· plicablc. Fol' the vcrticnl web stdp, Eq.(2.5) nre nlso valid. Howcvcr, in t.his Cllse, thc x,y and z arc refcrrcd to the !\Xes of nn individuni local Cllrtcsian c:oorclinatc syst.mn (sce scdion 2.1.6). and u,v,w nrc the displaccmcnt componcllts iu the dimdions x,y lUHI z respectivcly. For the bottom flflnge shell strip, thc strain·displac:cUlcnt rdatiollships (2.51)

CILII

be used. It will be noted that Eq.(2.51) nre given in thc cylindrical stlrfacc coordi· nate system; in order to facilitate the assembling proccdurc of structum Ulatric:cs,

52

CHAPTER. 3, SPLINE FINITE STRIP METHOD tbe foJlowing eoo!'dillatc trnllsformations arc rcquired: 'll'

=U

v' = vcos{3 + 10 sin fi 10' = -v sin (3 + 10 cos fi

8 8 8x'!(x,y) = 8x f (x,y)

8

8

-8 f(x,y) = cos{3-8 f(x,y) y'

y

whc!'c x,y,z lU'e thc common Cnrtcsian coordinatcs, IUld U,V,W

!lIC

thc conespond-

ing clisplaccmcnts, whilst thc primccl oncs lU'e refcrred to the cylindrical surfacc wonlinate systcmj f(x,y) represents IUl nrbitrlU'Y fUllction of x and Yj and

fl =

clbw ) ly

CII'etcm( -l-

Pelfn!'ming tbe ahove coordinatc transfnl1uations und doing some simplifications IIcco!'ding t,o tbc theory of shallow shells, the relationsllips betweell strains ill the hot tum /lange shell stdp and its displucement components expressed in the common ClU'tesilUI coordinl\te system

, fr

, fU

CIUl

be ohtained in the following form:

Du - 8x =

Dv' 10' -Dy' R

-

8 10' cos 13 8y (v cos fi + 10 sin 13) - R

-

8v 810 , , 813 10' cos{3{-8 cos{3+ -8 sm(3)+(-vsm{3+lOcos{3)-8 } - R

-

Y

Y

8v 810 , , 8{3 tv' cos{3(-cos{3 + -sm(3) + 10 - - 8y 8y 8y' R 8v 2 810 , 8y cos {3 + 8y cos {3 sm {3

Y

CH.4.PTER 3. SPLINE FINITE STRIP METHOD

,

1'r u

Du' Dv' += DIJ' D",' Du

= ,

Xx

53

{) cos fJ lJ

Dv

Dw

+ {).1' <,us fJ + D;1' sin !~

D2w

= -"fi" X· D 11' = - Dy2 2

,I

\u ,I .\ Xli

=

D2 w - Da:DlI

(3.4)

?--

where I/R = DfJ/Dy' is t.he "IU'Vlltme of the shell strip tuul

ClIlI

be cnlculnted by

means of (2.52).

3.3.2

DISPLACEMENT FUNCTIONS

In IInalysis, eltch nodallin,' is divid"d iut.o ali

I'VI'II

lIIunbel' of secl.iulls by knots,

and thcsc scctions arc gl'OlIped pltil' hy pair (Fig.3.2). TIl(' fl" splinc cxpl'essiolls arc cmployed

IlS

the longitIldinal displllcelllcnt flllldions fol' displllcel11ent compo-

ncnts u and w, and quadmtic intcrpolation in ellch pair of s('diOIlS is used longitudinal displacement function fol'

liS

I.hl'

V.

The displacement parameters of the strip corresponding to the i th sectioll knol. arc

The displacement field wi thi n It top f1ange plate strip or a hot tom fIlUlge shell strip

is: rn+l

U=

2: ((1 -

i=-I

X)Uli + XU2i)
54

ClJAPTER. 3. SPLINE FINITE STRIP METHOD a

II

=

L«(1 - X)vlj + XV2j)Lj(y)

(3.5)

j=1 m+1 /Il

=

L «1 -

3X 2 + 2X 3 )wli + x(l - 2X + X2)ti li

i=-I

wh"re X = x/b,

111

is t.he number of sections, i is the Ba spline expression wi th

y = Yi IL~ the ccnter, j is thc local nwnbcr of knot i in the corresponding pair of

sections, Ilnd Lj l\.I'e quadrntic interpolations in the following form:

LI(Y) = (1 - Y)(l - O.5Y)

La(Y) = O.5Y(Y -1) ill which, Y =

y'/ It (sec Fig.3.2).

Eq. (3.5)

IU'C

x,y,z

local Cl\.I'tcsilUl coordinatcs, IUld u,v,w arc corresponding displacement

IU'C

also applicnblc fol' thc vert.ical web strip. However, in this case, the

componcnts. In addition, X is not only a function of coordinate x, but also of the coordinlltc y in the form:

x=

x-

XI

=

b

X

CI

cbw(Y)

C

where ilu.(Y) is thc Vl\.l'iable dcpth nf thc wcb, b = cbw is the width of the strip and ;rl

=

is thc coordinntc of nodallinc 1, wi th c Rnd CI being constant (Fig. 3.3).

CI" IU

Substituting t.hc displllcclllcnt fundions of the web strip into the expression for in (2.5) yiclds the following relationship: f

U

-

av ay

-

EV

CH1~PTER

3. SPLINE FINITE STRIP METHOD

55

(3.6) It

C!Ul

hc sccn thnt

dcpcnds not only ou thI' first
fu

displacement function ~, but also on the sio!,,, of holt.ol11 lin" of t.h" w"b, ~. In thc actual structure, the slopc of thI' holtom linc of weh lIlay haVI' ahrupt chnngcs, espccially nt intermedinte supports. This lIlay CI\IlSC
fy

nnd significant crror in thc rcsults of t.he lIlust illlpurt.ant

stress componcnt u y if thc Ba spline functions (wit.h un
IU'C

used as the longitu
usc of qundrntic intcrpolntions as the longit.udinal
liS

thc abn,;)t dll\llgcs of the slope

111'1'

loeated at thc

ends of the pnirs, becnuse the first clcrivntivcs of Lj(Y) elUl be discontinuous across these locntions. An alternntivc method of treating this problem is to usc thc splinc function wi th only C" eontimuty

liS

the longitudinal displneclIlcllt functions of Vi

however, this will produce constlUlt longitudinal strnin and, conscquently, poor convergence. Once the displaccment functions for the strips have becn ehosen, the st dp eharacteristies, including stiffness, load etc., elUl be obtained in line with the standard finite strip formulation (sce Section 2.1).

CIIAPTER.1. SPLINE FIN/TE STRIP METHOD

3.3.3

56

PENALTY FUNCTION APPROACH

A lI1\mlll!r of IIwt.hods for imposiug bonncl!u'y cOllditions havc been proposed by Chen IIg et. aI139-44J. III the prcscnt study, thc pcnalty function approach is chosen to treat. end Ilnd intcrJuediatc support conditions of the strips. As ILII eXllJllple, it is IlssulIlcd that thc displllcelllent component w at point (x, y) is prescrihcd to be ze1'O, tlmt is:

1/J

=

L N;(x, y)tv; = O ;

wlWI'I! N;(:r, y) alld

tv;

arc IIssociatcd displncclIlent fUllction and displncclIlcnt pa-

I'lUlll'I..,1' rcspcetivcly. III IIl'dcr t,o impose t.his cOllstraint iniUlIllysis, a fictitious spring with large stiffness ll'

might. he intl'Oduccd iu the corresponding direction at the point (X ,y). Its strain

cnl'rgy or pcualty function is:

Fhlln this exprl'ssiou i t cau be sccn that the stiffncss matrix of this imaginary spri ng is:

Ik;jJ = loN;(x, y)Nj(x, y)J If th .. first del'i'1Itive of dcflection, say ~;, at point (x, y) is prescribed to be zero, thm

Otv =

oy

L oNi(x, y) tv; = O i

oy

CH.4PTER 3. SPLINE FINITE STRIP METHOD

57

Following thI' SlUne prnccdnrc, the required stiffness matrix for this const mini, elUl be obtaincd ilS:

["'d = J

[naN;(:I:,y)aNj(:I"Y)j alJ aY

Asscmbling the stifl'ncss nml.riccs of alJ point constl'lljnts with the structuml stilr· ness nmt,rix will accomplish I,he treat,ment of boundlU'y conditions. An elast.ie support ClUl also be inc\uded by t,he SlUlle method,

ilS

IOllg as the n is

the adun.! stiffness of t.his support.

3.3.4

NUMERICAL EXAMPLES

1. Continuous Bellm wit.h VarilIhle Depth \lnder Point Loud The beam is continll(lIls over t,wo C(llml spllns II = of belUll is 1.0

III

,. ~

t

I.

= 10

\ll

(Fig.3.5). The depth

ut both ends, 2.0 m over t.he intel'llledinte support, and vari!'s

Jinearly in·between. Thc width of the beILlll is 1.0

í,.

l~

lll.

The propert.ies of llIat.erill.l

arc E=10000 MPu and v = O. A vcrtical point force P = 2.0M N iH adillg at. the center of the second spall. The beam is anruyzcd by using only (Jlle weh sl.ril'. The deflection nt the loadcd cross sedion lUld the longitudinw stresses ILt. several points ruong the bot tom nodIlIIinc ILre Iistcd in Table 3.2, ill which the vlLlucs frolll beam theory (shear deformntion illcluded) arc also given fol' comparison. It CILII be seen that this method is accurate and that it convcrgcH very rapidly.

2. 2·SplUl Haunched Continuous Concretc Slub-oll-Girder Bridge The bridge wi th span II = 12m lUld 12 = 16m is shown in Fig.3.6. The end of shorter span is simply supportcd, and the other end is fixcd. The thiclmcss of th"

CHAPTER 3. SPLINE FINITE STRIP METHOD Number of scctions

8 16 24 beam theory

w (m) y=15 III 0.011279 0.D11480 0.D11524 0.011265

58

(MPa) y=1O m y=15 m -4.111 9.998 -3.826 9.995 -3.777 10.002 -3.756 9.994

tr.

y=5m -3.520 -3.370 -3.347 -3.339

Tahle 3.2: Deflection anu Longituuinal Strcsscs in Continuous Beam sllLh is 0.2

III

The bottom

.

ILnu the thickness of ali webs is 0.4 m. The spacing of wcbs is 2.8 m . lillll

of thc web is parabola in cach span. The uepth of web (from its

hottom to the middie plane of slab) is 0.5 m at thc simply supported end, 1.1

III

nver the intermediate support, 0.5 m at the center of longer span, and 0.8 m at lhe fixcd end. Thc material properties arc E=27000 MPa, v = 0.15. In loading el'-~c

l, n unit vertical point lond P=1.0 MN acts on the top of the exterior girder

nt. lh" cenlcl' of the longer SplUlj in case 2, the same load moves to the top of the ('cn t.rnJ girder on the SlUlle CI'OSS section. In analysis, the structure is divided into 12 HIlLb strips, 7 wcb strips and 28 longitudinal sections. The resulting longitudinal st.rCHses on the top and bot tom of wcbs at a number of cross sections are listed in Table 3.3.

If the lond is put on the sccond or third girder, the maximum longitudinal stress ,,"curs nt the bottom of the londcd girder at the loaded cross section and its value is litt.le difl'ercnt from thc one in loading case 2. 3. Two splLn hmmched continuous stcel box-girder bridgc. A Illuiti cell box-girder bridgc is continuous over two equal spam as shown in Fig.3.7. It is haunched over the interior supports, with c=2.40 m. The bridge

CHAPTER 3. SPLINE FINITE STRJP METHOD Loading Cross Case Section l A B C 2

A

B C

59

(T. (MPa) on the tOJland \'ottom of Girdl'r 1 Girdcr 2 Girdcr 3 Girdcr 4 0,01 5.24 1.22 -10.75 -2.76 0.06 -0.3; -15.64 -2.41 45.51 10.75 1.67 11.66 0.55 3.51 -1.15 -29.36 -10.06 1.03 1.86 0.14 -6.20 -0.21 -2.87 -1.59 -6.42 -0.42 8.62 30.19 1.84 3.95 0.62 2.49 -16.10 -1.71 -8.83 -

-

Table 3.3: Longitlldinlll Strcsscs in Two SplUl Slab-oll-Girdcr Bridgc span l is choscn as 40

lll,

the spacing bctwccn wclls h2 = 2.80 Illi thc thiclmcss

of thc flangcs and thc webs is 4.0 Cllli t.hc hlUlllched portion covers a distancc of 12 lll. In this portion, the gcnemting linc of the bottolll Hangc is a pambola. The clepth of thc box girdcr(cxclllding thc hmlllch) is 1.04 m. Thc lond is sclfweight, eorresponding to a speeifie wcight of nmtcrilll of 76.4K N/ m 3 . The material propertics are E=200,OOO MPa,

/I

= 0.3.

Duc to symmetry, only half a cell and onc span necdcd to be IInlllyzcd. For comparison, this structure was first IUmlyr.cd by the finite clelllcnt package ADINA. Each flange was divided into l row of 16-nodc shell clclllcntsi the wcb was divided into 2 rows, with eaeh row including 8 clelIlcnts. Using thc Ml\infrl\llIc AMDAHL-5860, a CPU time of 33.6 scc. was rC'luired for this Ilnalysis. Next, the semi-analytical finitc strip mcthad (scc SCctiOlI 2.4) WI~~ cmploycd to Illl-

CHAPTER 3. SPLINE FINITE STRIP METHOD

60

alyzI! t.hc SIlIIle structure. The top flange, web and bot tom flange were divided into one st.ríp cachj 10 symmctricnl terms were taken in the longitudinal dispincement funet.ion scties. The CPU time needed was 19.3 sec. Finally, the spline tini tc strip method was used. The flanges and web were divided into l strip with 20 longitudinnI sections ench. The CPU time needed was reduced t.o 11.7 sec. The resulting longituclinal strcsscs of the middle surfnce at a number of cross s('ct.ions !U'e listed in Tablc 3.4, in which FSM denotes the semi-analytical tinite strip llIet.hod (Section 2.4) and SFSM represents the spline tinite strip method. ln t.he annlysis of this structure, the time spent on formation of stiffness matrices is t.he Im'gest componenI. of total CPU time consumed. The spline function is rclnt.ivcly simpier than the beam eigenfunction, and it is this featllre that results in its higher cfficiency.

4. Fiv!! SplUI Composite Box-Girder Bridge Thc tivc spml ccntinuous haunclled box-girder bridge, described in Subsection 2.4.4 Exnmplc 2, is mmlyzed by the spline tini te strip method . In mmlysis, the structure is divided into 20 strips with 36 longitudinal sections (Fig.3.S). The resulting longitlldinal stress distribution at the cross-section X-X (SCI'

Fig.2.l2) is depictccl in Fig.3.9. It can be seen that the maximumlongitudinal

st.ress nppl'm'S nt the exterior bottom corner of the box girder under the trucks, whcrc its vallle is 23.02 MPa. The experimental value of the stress at the same

CHAPTER 3. SPLINE FINITE STRIP METHOD

Y2 (m) O

3

6

12

20

30

Methocl

u. (MPa) along nocln.lliu
B ADINA 13.37 FSM 11.72 SFSM 12.46 ADIN A 13.70 FSM 13.58 SFSM 13.88 ADINA 13.92 FSM 13.95 SFSM 14.03 ADIN A 3.53 FSM 4.36 SFSM 5.02 ADINA -10.71 FSM -lD.23 SFSM -10.65 ADINA -14.26 FSM -13.80 SFSM -13.57

C

I

-11.75 -11.16 -11.95 -15.11 -16.69 -14.44 -16.85 -18.04 -16.01 -3.69 -4.46 -4.00 11.19 lD.38 10.65 14.26 13.72 13.57

Tablc 3.4: Longituclinal Strcsscs in Two SpUIl Box-GirdcI' Briclgc

61

GHAPTER :J. SI'UNE FU-UTE STnIP METHOD !,,,illl. is

24.47 MPa [GOJ.

•

62

03

CH.-lPTER 3. SPLINE FlNITE STRlP METlIOD c>

..t ~(y)

c>

..; c>



..: c>

ci -3.0

-2.0

-1.0

0.0

1.0

2.0

3.oh

2.0

3.oh

Y od ......... c>

d~(y)/dy

..t c>



ci

3.0

-2.0

-1.0

c>


c> ~

I

N

od

~

d·~(y)/dya

.;

c>

ci+-----~~----~~--_.--~~~----~----__,

3.0

-2.0

2.0

c>

.; I c>


Figl\l'C 3.1: SplillC FunctiOlI and !ts Dcrh'ativcs

3.oh

64

ClI.-U'1"J::ll:3. Sl'LIXE FIXI TE STRIl' METHOD

hxrn !Jair 1 pair 2

-1 nodal line 1

e (-.--.-.__--l-.-e-e--er--.. y'

•

y,v

knol

-e-~~~~.-

__

--~

___

nodalline 2 Z,w

X,U Figurc 3.2: Platc Stl'ip

web .----r---...------L.--.----r---

1 2

X,

b

x Figlll·c 3.3: Wcb Stl'ip ill Illdividulll Systcm

y

G5

CIBPTER 3. SPLIXE FlXITE STRlP METIWD

1 fl 1! U.IA'

Z

R

XI x'

'W'

W

Z'

Z

v' y

Figur!! 3.4: Shell Strip

P=2MN

1

! I

0.0

Q. I

I

5.0

10.0

I

15.0

Y (M)

Figure 3.5: Continuous Demn

~ I

20.0

G6

CIJ..li'TEH 3. S!'LI,YE FlNITE STm!' METHOD

P=l MN

l

6

*

Q

I

I

A

I

C

lJ

loadillg case 110adillg case 2

~'--"I--I~'--,,--·r,---'1 1

2

3

4

5

O

7

cross secliou D Figul'C 3.G: Haullchcu COlltitlUous Dl'idge

I

l

= dOm

T

I f--/Zm -t- /2m-j

·K

l= 40 m

!

_-r-----:-....,..----f,] I C=2.4m ~

-----Li

!I---Yz

A

D

B

-' JI J

f-

hr=o.04m •

!

h

b,= 0.04 m

. c'I bz =2.8m

hz:O.04m

Figul'C 3.i: HIlW1Chcd COlltitlUOUS Dox-Gil'dcr Dl'idgc

t

.h=1.04m

CH.-lPTER 3. SPLINE FINITE STRIP METHOD

Gi

Figurc 3.8: Division of Stl'Íps

-/.90

-1·70

-15.23

-I. 'll

-0.37

-0.80

-3.3'1

-".38

3.83

23.02

/8.'13

1/.49

2226

7-41

1/.81 9./5

7. 06

/9.66 17.63

FigUl'C 3.9: Longitudinal Strcsscs at Cross-Scction X-X (in Mpa)

-0.06

Chapter 4

NONLINEAR ANALYSIS

The finiLc sLrip IllcLhods have bcen <,xLcnd"d to nonlinear structural analysis such I~~

posL-lmckling bclllLvior /lnd gl'OlIleLricul nonlincar analysis of plate sh'uctures

[28-33[ sincc 1978, LD nmterill.lnonlinear analysis of stecl structll1'es IIDd reinforced concret.! shtbs [34-37J since 1986, to large dcflcction clasto-plllStic analysis of plate sL1'IIctllres [38J since 1989,ctc. Ali of these c\evelopments have confirmed that the advnnLnges of t.he fillitc aLrip methods over finite elemcnt methods, which are higher cfficicncy, lcss requirer.lCnt. for computer storage, CPU time and input, data, hold in rcspcct of n01l1i.l1car st1'l1ctural analysis, just

lIS

they were known to do in Iinear

mmlysis.

In thc prcscnt study, n finitc sLrip method for the nonlincar analysis of cablestnycd bridgcs [53,54J is dcvelopcd. Furthermore, A more efficient finite strip n.cthoc\ fol' gc0111etrically nonlinear analysis of plates [ti5J and a more accurate finitc strip mc'hod for nmtcrially nonlinear analysis of reinforced slahs [56J are IUSO

c\cvclopcd. 68

CH.4.PTER 4. NONLINEAR .4. NA LYSIS

4.1

G9

NONLINEAR ANALYSIS OF CABLE-STAYED BRIDGES

In t.he prt'scnt. sed.ion, the finitc strip IIIct.hod Ilsing thc cigenfllnctions nf n l'lm· t.inllolls bctllll having thc

SIUUC

Sp IUl lengths ns the nct.llnl bridge is

t~lllploycd

fol'

3·0 IUllllysis of t.hc girdcr of the cable-stnyed bridge. In order to improve the aCClIl'Rey of ben ding moments at the cnbl" ntt.nchment points, the eigenfuuctions of a fictitious continuo\ls benm with ndditionnl SlIpports at ali t.he cnble ntt.achnwnt. points Itre added to the sedes of longitudinni displaccmen t fllnct.ions. Using the flexibility approach, the entire girder st.l'Ilct.IlI'C is t.mnsfol1\wd into n substmct.ure hn.ving deflections at the cable att.aclullent point.s as thc only dcgrces of frecdom, so that a matrix of only smnll size is involved in t.hc nonlinClu' solllt.ion. The theory of the catenary is utilizcd in ealeulating the cnblc tension from t.11<' positions of the cablc cnds. In this way, the nonlincarit.y due to sag aud Iluglt' ehallgc of eables is takcn into eonsideration.

-

Thc initinl-stiffness method is taken in the nonlinear itel'lltions. Henee, t.he st.ilfnt'SH ma.trix of the bridge is formed and invelted only onee. Beeause of the high tensionH of the eables, the nonlinearity is not very markcd, and the eonv!!rgence elln be achieved wi thi n very few iterations. In accordanee with the ASCE Task Committel' recommendation [62J the nonlin-

C/IAPTER 4. NONLINEAR ANALYSIS

iD

f'ariti,," of t.II" girdcr and pylon5 generaIly arc uot taken into Ilccount. The effect IIfaxial displacelJleut.s of girder and pylons on cahlc tcnsious is 11.150 neglccted. It is fil 1'1. 11,,1' IL'SIlIJl"d that. ali cables tU'e fixed to the pylons, and that the girdcr is silllply sllpportcd on piers

IL~

it passcs through the pylon legs, which arc fixed to

tlw piers.

4.1.1

FINITE STRIP ANALYSIS OF GIRDER

lu IU051. puhli"ations the girder of a cable-stnyed bridge is treated as n berun

[03,04]. 11ow"""r, 1.0 be morc accurate, the girder shoulcl be considered as a three dilllensiOlml 5t.I'Uctllre, IU1<1 nnlllyzecl by finite element methocl [65]. Because of the Ilolllillelll'

bclmviol' of the cnbles, iterative methods arc normn11y required fol' the

Ilnalysis, Illul if the cnt.irc girder has to be analyzed in every iteration the computer time rC'lllirelllent. is likcly to become unacccptably large. ln lllost mble-stayed bridges, thc girders are continuous over two or three spans. Fol' the three dimensionlll analysis of such a girder, the finite strip method using thc cigenfunctions of n continuous be!Ull which·has span lengths equal to those of thc adual bridgl' (Chapter 2), is preferable to finite elemenI. analysis, because of its higher efficiency. Decause thc second order derivatives of every be!Ull function have smooth variation wit.hin cllch sprul, it is clifficlllt to simulate the peak value of longitudinal bending 1ll00nent at nny cable attachment point. Auually, the deflection of a girder at any cable attachment point is much smaller than the deflection of the S!Ulle girder

CH.4.PTER 4. NONLINE:\R .4.N:\LYSIS

il

without cables Ilndcr I.he same load. Hencc, if IlIl additional support is addcd undl'r every cable attndm:cnt point, the bending momcnt of thc girdcr will uol. chaug" very much, which memls thal. thc lll'ndiug lIIouwnt of a cablc-stnycd bridgc is vcry similnr to thnt of the conl.inllolls bC1U1I wit.h additionnl sllpporl.s. Thc bImding moment of such a continllous bemll is rendi ly representcd by its cigenfunct.ions. Therefore, in nnalysis, the eigenfunctions of such It continllf"~' b(!lUn nrc addcd into the sedes of longitudinnI displacemenl. functions. In this way, the IICI:UI'IICy of t.he bending moments of girder at cable attnclllnent points is improved siglli!icllnt.1y. Another measure to improve the efficiency of the iternl.ive process is 1.0 use the substructuring teclmique, tl'lulsforllling the whole girder by IlIelUIS of the Ilexibility approach into a substructure with only very few nodes at the poinl.s of cnble attnchments. Hcnce, only the substructure with very few dcgrees of freec!olll is involved ill every iteratioll mid the solution time is reduced significant ly. The girder, released from alI stays, is first analyzed by the nbove-llIcntiolled !inite strip mcthod under external loads. The deflections at alI the nodes due to the external loads can be casily found and expressed in terms of

IL

column vedor

{a}. Then a unit vertical force is applied at node j alone, with j being 1,2, ... n, where n is the number of cable attachment points, and the deflection fi; of evCly node i (i=1...n) is obtained. This proccdurc gives the flexibility matrix [Fo] of the substructure:

[Fs] = [Fi;]

CHArTER 4. NONLINEAR ANALYSIS

TIl(' illvenlc matrix of \Fol is thc st.iffncss mat.rix [1\1/1 of thc substructmc,

III order to avoid possiblc singularitics in [Fgl and consequent fwlure to invert it, the lIumber of series terms used in the analysis of the girder must not. be less than . t.he JlIllllber of llodcs on any nodallinc. MIIltiplying thc stiffncss matrix [1\gl by the displacement vector {a} due to extcr1lILll0ILds gives thc cquivnlcnt nodal forccs {P}

{P} = [](g]{a}

4.1.2

FORMULAS FOR CABLE

1. INITlAL STIFFNESS MATRIX OF THE CABLE

The displlLcelllent vcctor of n cable is [tv, vjT (Fig.4.1 a). Its linear elastic stiffncss lIlatrix can be writLen us casasina ] [1\ 1= EA [ sin2~ c So cosasma cos2 a where E is the lIlodulus of elasticitYj A is the area of cross-sectionj So is the free Ilmgth of the cablej and

ct

is the angle of inclination of the cable.

2. BASIC THEORY OF THE CATENARY The shape of n cnble under sclf-wcight and end tensions is a catenary. The formulas used in the present analysis are Jerived as follows.

CH.4.PTER 4. NONLINEAR .4.NALYSIS

i3

If q is the sclf·wcight pcr unit Icngt.hs, ruHI T is the cablc tension with horizont nl eOlllponcnt H and vel-t.iem cOlllponcnt V, the c'luilibriUlll cOlldit.ions of IUl infinitcH' imm eable clement (Fig.4.1 b) ru'e

LX =0,

(4.1)

LY=O,

(4.2)

Fl'om (4.1) it elUl be eoncluded that H 2 = H. = H, which lllCrulS that H is const.lLnt. mong the entire eable. Tnking into considcration that

ely ely IPy V2 - F. = HtruHI'2 - Htancq = H(-I h - H(-I ). = H-I2elx IX la: IX ruld

els = JclX2

ely . d:c

+ ely2 =

1+ (_)2el:l:

(4.2) can be rewritten in thc form: (4.3) (4.3) is the difFcrential eqllatioll of catenary, whosc gcncml solution is

.

I

Y = acosh where

Xo

X -XO

a

+c

(404)

and c arc constallts which arc dcterrnillcd fl'Olll thc cnd conditions and

a= H/q

t ,

For a cable with horizontal projection I and vcrtieal [ll'Ojcction h (Figo4.1 c), the end conditions can be assmned as:

y(O) = O

(4.5)

74

CIIAPTER 4. NONLINEAR ANALYSIS y(l) = h

(4.6)

Substituting (4.4) into (4.5) gives -xo a

acosh-+c=O from which the following expl'esdions can be obtained: Xo

c= -acosha and

x -xo Xo - cosh-) a a

y = a( cosh

(4.7)'

Snbstitution of (4.7) into (4.6) yiclds: 1- Xo Xo a( cosh - - - cosh -) = h a

a

which can then be rcw:itten in the form: n

•

~asm

h l . I 1- 2xo -2 sm l = It a 2a

whencc

h . I 1- 2xo sm l = ::--:--:-.,2a 2a sinh

ia

(4.8)

This cquation ClUl be used to find Xo, giving Xo =

13ccnuse sinh-I(t) = In(t +

. I -I ?1 - sm l ~

(I

n

,.(1

It • l smh 2.

JI + t~), t.he above equation can also be written in the

following form: Xo =

1

2' - aln(t + Jt2'+1)

(4.9)

CHAPTER 4. NDNL1NE.4.R .4.N.4.LYSIS

75

in which

t= Dncc

.1:0

",

2asinh -~CI

is dcfincd, (4.7) can bc utilizcd to find uny values needed in I.he lUlIIlysis,

sueh as the cnble tension T and its vertical componenI. V, t.he curve lengt.ll S or thc cablc and its elongation S - So due 1.0 cable telIsion T as rollows: (a) Cable Tension T and !ts Vertical ComponenI. V:

dy H' 1 :c - Xo l' = H - = SlIIl dx

(4.10)

II

x-

~:o

T = VH2 + 1'2.= H cosh - - -

(4.11)

(I

-

(h) Curved Length of the Cable:

S = = = =

F ' . ,

dy 1+ (_)2lh

r Jo

dx

l , + smh l cash' . 1

o

x -

o

~:u

2 X -

Xu

a . l 1- Xo

l1.dm 1 - - II

dx

ti

rI:c

. -xo

lISIll--

II

. 1 1- 2xo = 2a smh? cash -?-.;;. .... a _a.

. 1 21- 2xo = 2asmh - Slllh 2a

2a

+1

.1

Substituting (4.8) into the above equatioll ')vcntllally givcs I.hc curved length S of the cable as:

5= (c) Elongation of The Cable

s-so

=

d~

l ' Tds o EA

,.-----:h2 + 4(12 sinh2 _I 2a

to Tension:

(4.12)

CH.4.PTER 4. NONLINE.4.R ANALYSIS

_ -l

EA

=

l ,t

E~

= -H

1,' T l + ( rI,,' (/;t, 1,' H cos'I".,. - . l o 1,' cosh - - - cl", dll ...!!.)2

O

. 'I 'l +81111"

'I'

.'

2'1' -

•

n

l (a'

Xo

(I

I 2(1- :1'0) • 2:1'0 _ a n + ?(sm I + slI1h -l) l 1- ?'ro (l + sinh - cosh -')

H (l _ ?EA

-

H .E

,t

'r II

-

(l

-

-

'I''n

I

EA o

2

76

(I.

•

(I

."1

(l

(I

H (l + (l sm . Il -l (?_ sm . Il 2 1-2:1'0 ?EA _ (I 2a

+ l ))

Using (4.8) again and taking into account sinh L = 2siuh 2'II cosh 2'II and ti coth

ia = cosh i.1 sinh fo, t.he above equation can be rewriUen in the form: H h2 l S - So = - - (l + - eoth 2EA (I 2(1

. l + II smh -)

(4.13)

(I

3. SOLUTION PROCEDURE FOR CABLE TENSION Substituting (4.12) into (4.13) gives following equal.ion: H . 2 Iq H qh 2 ql h 2 + 4(-)2smh - - So = --(l + -coth q 2H 2EA H 2H

ql + -H. smh-) q H

(4.14)

Using (4.9) and (4.10), the vertical force VI applied by the cable on t.he girdel' and the horizontal force H applied by the cable on the j)ylon can be l'ehLt.ed to eaeh other as follows:

qh

.

ql

t = 2HI smh 2H Xo

l H = - - -In(t

2

q

+ v't2'+1)

• qxo V;1= Hsmh H

(4.15)

InitiaIly, q, l and h are knownj thence from thc givcn initilu valn!! of V" c'IulLtionH (4.15) can be solved numericaIly for the unknown H, for cXlLlllple by using the

CIIAPTER 4. NONLINEAR ANALYSIS

ii

firHt. order Re)!;uli-FaIHi iterntion (sec Sedion 2.3). In this iteration, fol' any trial value of H, t mil he ohtained hy means of the first of equations (4.15). Then, Huhst.ituting t iuto the second equation gives Xo. Thereafter,

vi

can he computecl

uHin)!; th" third e<[uatiou. Substituting the solution for H into equation (4.14) then giVl!H thc free length SI) of the clible. At /lny defol'llled position, after w Imcl v arc founcl, it is easy to obtain h ancl I. Using a numericaI approach again, H can be obtainecl from equation (4.14). Then

VI is rcadily evaluated by substituting H into equation (4.15). Both

vi

and H

!'eHect the u()nlinear resistance of t.he cable to deformation.

4.1.3

STIFFNESS MATRIX OF THE PYLON

In the present analysis, t.he pylon is treatecl as a cantilever berun, ruld only its deflectioll, not its rotation, is taken into account. For this reason, it is convenient to forlll it.s flexibility matrix first, and then let the computer invert it automatically to obt.n.in i ts stiffness matrix. The flexibility coefficient fij of the pylon fFig.4.2) is definecl by ( 4.16) in which Elp is the bending stiffness of the pylonj and hj ;:: hj. Hence, by assembling I.he fij, the flexibility matrix is obtained as: (4.17)

iS

CHAPTER 4. NONLINEAR ANALYSIS Thc invcrsc matrix of [F,.] is t.hc st,ifl'nf'ss mut.rix [1\,.] nf pylnn,

[1\,.] = [F,.I-'

4.1.4

(4.1S)

INITIAL-STIFFNESS ITERATION

.'\ssembling the linelU' stiffncss nmtriecs of t.he girdcr

slIhst.\'I\(~t.lIre,

cablcs IUld py-

lons togethcl' produces the initiallinclU' stiffncss mat,rix [1\,,1 nf t.he whole st,\'\Ict,lIrc.

1\0 is formed and inverted only onec and thcn is used in evcry it';'at,inn fol' the nonlinear solution [6] without any clllUlge. Because of thc rclativcly high cablc t,ension in ordina.ry enble-stnyed bl'idges, the nonlinenrity is USUltlly

lill!.

vcry significant"

and hence convergence cml be achieved within very few it.erutions. It. is assumed that the st\'\lcturc is dividcd int,o n linelll' part (the girder nnd pylons)

and n nonlinclU' pnrt(cnbles). After k iterntions, the rcsistance of the linenr part to displacelllcnt {a}k is: (4.19)

wherc [/(,] is thc lineur stiffness matrix of this plll't. The rcsistnnce {R,.,} of the nonlineur part to the displaccmen t {a} k is the cnble tension and can be found from {a}k by following the proccdurc clcscribed in Subsection 4.2.2. The resistance of thc entire structure to this deformntion is thc combination of

79

CHAPTER 4. NONLINEAR ANALYSIS t.lw l'eSistallces of hot h parts.

(4.20) If the external loud {P} is not equul to the resistIUIcC {R}k, the structure will dcfOl'1II further under the action of the so-called "unbalanced force" {FJk (the clifrcrcmcc bel,wecn the load and the resistIUIcC ) to the next position {a}k+l, wi th the illitiallinear stiffncss matrix [Ko]lISsumed to be unchangcd (Fig.4.3). Hence (4.21) If the Becond term in the ]>l'Ilcket on the left side of above equation is shifted to

the right side,

thi~

cquation

CIUl

be rewrittcm lIS:

[J\O]{CI}k+1 _ [I\o]{a} k + {P} _ {R}k = ([[(,] + [[(n,]){a}k + {P} - ([K,]{a}k = {P}

+ [1(n,]{a}k -

{R",}k

+ (R"tl k) (4.22)

wherc [[(",] is the initial linea!' stiffness matrix of the nonlinear part (cables), and [Ko] = [1\,] + [Kn'], Thus the cqulltion (4.22)

CIUl

be used to find an improved approximation {aJk+!

fl'om (CI}k. This procedure is rcpeated until convergence is obtained. Convergence lIIay be IIsslImed to

OCCll!'

if th" differcnce in the "alue of every displacement

pnl'mndel' of cvery cablc, determined .from two sllccessive itcrations, does not excccd

1\

chosen small percentagc( say 0.1 percent ).

so

CH.4PTER 4. NONLINE.4.R AN.4.LYSIS

4.1.5

NUMERICAL EXAMPLES

1. Thrcc Sprul Cablc·St.nyel! Bridge undcr Dcnd Lond IUld Livc 101ld. A single pllUle cablc staycd Ll'idgc over tlu'cc sprulS is ShOWll in Fig.4.4. ln ordcr 1.0 compru'c the rcsult.s of analysis wi th bcrun t.hcory morc convcnicnt.1y, it is nsslllnccl that the girdcr has a

IUU'roW

width in compm'ison with its hcighl.. Thc dctn.ils oC

the bridgc arc as follows: For thc girder, E = 2.0

X

105111 Pa, Poisson's ratio

/J

= 0.3, nlldbcnding nlOl11Cnt

of inertin Ig = 1.0m·l . For t.he pylons, Elp = 1.0 x 10"m2 M N. For the cahlcs, Ee = 1.52 X 105 111 Pn, t.he specific weight ic = 0.0764111 N /m 3 , The ;U'CllS oC cross sections AI = A4 = 0.10111. 2 ,

A2 = A3 = 0.05m2 • In load case 1, thc girder is subjcctcd to dead load q" = O.lNI N /m over ali spllns, and the deflections at aU .he cable attachmcnt points arc assumed to bc zei·o. The vertical forces V; in MN (i=I,4) applied to the girder by cablc i IIre listed in Tablc 4.1.The longitudinal bending mOl11cnts in MN.mat given in Table 4.2. In both tables,

III

II

numbcr of cross scctions arc

is the numbcr of cig'!llfunctions associlltcd

with the continuous beam over thrce spansj

112

is the number of eigenfunctious

related to the continuous beam with additional supports at ali of the cablc att.llch· ment pointsj hence the total numbcr of cigenfunctions cmployed in the analysis is n)

+ n2'

CHArTER 4. NONLINEAR ANALYSIS

Load (:iL';C

1

"'+"2 (i+O 15+0 25+0 6+D

lG+!J IOlld case!

2

81

tIIH

top

Wil

0.0 0.0 (J.O (J.O 0.0

0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

v,

V.1

"2

l~,

7.3ii D.375 8.325 7.480 D.080 8.074 7.480 D.060 8.060 7.480 9.041 8.043 7.480 9.046 . 8.049

7.D01 7.D82 7.986 7.D90 7.DDO 7.985 3.802 3.836 3.837 3.838 3.838

Theory 6+0 15+0 23+0 6+D 15+8

0.0 0.0 -0.14DD 0.2761 -0.14D8 0.2736 -0.14D8 0.2735 -0.1500 0.2733 -0.1500 0.2733

0.0 - 9.072 8.075 0.5117 5.473 0.068 4.415 0.5110 5.481 0.049 4.341 0.5111 5.481 0.049 4.338 0.5112 5.482 0.045 4.332 0.5112 5.482 0.045 4.332

Theory

-0.14D8

0.5111

0.2735

-

0.056

4.343

3.837

Tnbl" 4.1: Deflcctions (in 1l1eters) of Girder nt Cable Attachment Poillts !tnd Vert.icIlI Forccs (ill MN) of Cn.bles

ll, load Cll..C;;C

1

+ 112 6+0 15+0 25+0 45+0 6+9 16+D 36+D

Thcory lond 6+0 case 15+0 2 23+0 45+0 6+D 15+8 36+9

MA

MH

0.018 45.10 51.95 44.70 48.27 46.49 46.08

0.021 -34.95 -43.17 -53.05 -61.35 -70.06 -67.74

Me 0.011 20.52 25.30 18.82 22.17 21.43 20.74

46.19 -67.62 21.43 -7.986 -11.30 -8.720 -4.858 -10.18 -12.66 -4.744 -10.01 -13.32 -4.930 -10.04 -13.12 -4.935 -D. 726 -13.22 -5.055 -10.57 -13.28 -4.970 -10.10 -13.06

Thcory -5.054

-10.11

-12.93

ME

MF

Ma

0.005 27.19 30.76 26.10 29.07 28.58 27.36

0.016 -23.68 -32.99 -41.53 -48.90 -55.55 -54.55

0.021 20.93 33.39 24.07 28.23 26.71 26.30

-49.51 28.08 -2.618 -1.290 -15.96 16.39 -14.43 19.07 -15.04 16.09 -14.97 17.76 -14.23 17.64 -15.51 16.79

-54.34 0.8137 -13.42 -18.53 -23.10 -27.12 -29.20 -30.20

26.26 -53.13 4.299 8.052 13.49 -1.265 19.69 -7.063 15.17 -10.90 17.29 -14.59 17.02 -15.80 16.41 -17.24

26.87 9.711 19.89 25.37 21.86 23.78 23.60 22.83

-15.74

-30.11

16.41

22.93

MD 0.001 -47.60 -44.20 -46.54 -44.97 -46.30 -48.45

17.07

Table 4.2: Belldillg Momellt (in MN.m) of Girder

Mil MI 0.020 Om9 -20.73 20.87 -32.23 31.49 -40.20 24.34 -48.24 28.69 -50.91 28.68 -53.46 26.71

-17.07

82

CH.4PTER 4. NONLINE.4R .4.N.4LYSIS

lu lond case 2, the live load q, = 0.05 ll/N /1/1 is illl(loSt'd ou t.11<' <,ut.ire secoud spnu ouly. The v('rt.icnl COlllpOU"Uts of the prestrl'ssiug fo!'l'''s of t.h" V~I

= 7.480

MN, V~2

= 9.071

MN, ,,~,

= 8.004

(~nbles lU','

MN mul ' <101 "" 7.984 MN,

cOllsistent, wit,h the girder haviug zem detlcct.iou nt ever)' cnble ntt,ndnnent point. in lond case 1. The rcsult,s in load case 2 nrc also sllOwn iu Tnble 4.1 nud Tnbl .... 4.2. However, in this cnsc, evel'y qunutity listed iu t.he tnbles is it.s iucrclllcnt due to live load alone. lu fnct, even if ouly thc eigenfunctions of n continuous benm with 3 spans nre used, the results fol' the detlectinu (in \ll) nf t.he girdel' nt cnble nttacInuent. points arc still sntisfactory; their erl'Ors nre less UUU! 1.0 percent. wit.h ou ly O sedes t.erllls used. In this case, the results of the verticnl componellt.s of t.h" cnbl" t.cusious nr" also acceptable, with ermrs of less thnn 3.5 percent. Howcver,t.he a(,clIl"llcy nf til<' bending mOlllcnts is disappointiug,thc errors oC 1ll11..'Cillllllll helllliug 1ll0lllCllt hcing stilllllore thnn 22 percen t iu both lond cnses, dcspite IIsillg 45 series t.enns. Frolll Table 4.2 it cnn be seen that thc illlpmvelllent brought about hy t.he nddit.iollnl eigenfunctions nssocinted with the continuolLq benIll wi th extm fietit.iolls support." at alI the cnble attachment points is significnnt. The erl'Or of 1lI11..'Cimlllu belldillg Illoment is less thllll 5 percent by using

nl

+ 712 = 10 + 9 = 25 "eries terlII"

Cor load cnse 1, and by using 711 + 712 = 15 + 8 = 23 series terIlIs for lond cns" 2. As n compnrison, the error of maximum bending mOlllellt is 3G per"eut wit.h nl +712

= 25+0 = 25 terms for load cnse 1, IlIld38 percent with 711 +112 = 23+0 = 23

terms for load case 2, wben the additional eigenfunctions arc not illcluded. . In both tables, the tbeoretical rcsults arc obtained by means of hCllm tlwory,

CHAPTEn 4. NONLlNEAR ANALYSIS

83

with t.1 ... J,,,am c1dlectionH at the corrcsponding cablc attachment points under the ,,>,I."1'I11L1 appli",1 load. being nssumed to be the same as the actual girdel'. 2. Double- P hUl" Cable-Stayed Concrete Box-Girder Bridge. A douhl"-plallc cnble-stayed concwte box-girder bridge is shown in Fig.4.5. The load. arc the self-weight of thc girdcr and a live load, dis tribu ted uniformlyon the whole deck hctwcen towers, of intensity 4.4881(N/m2. The details of the bridge arc ns follows: For t.hc girdcr, Eg = 25 x 1061(N/m2, Poisson's ratio v = 0.2, specific weight

"fg = 241\ N / nl3 j for the pylons, Elv = 2.5 x 108 m 2J( Nj for the cables, Ee _ Li

X

108 1\N/1I/2, "fc = i81\N/m 3, the arens AI = A4 = 0.04m 2, A 2 = A3 -

0.024m 2 j the vert.ical components of cable prestressing forces under the dead load arc VOI = i849.01(N, V02 = 9066.51(N, V03 = 8121.2J(N,

Vo4

= 8322.7J(N, which

make the deflections of the gil'der at the cable attachment points under the dead load c()ual t.o zero. Because of symmetry, only half of the bridge, which is divided into 6 strips by 5 nodnl lilll!S, needs to be annlyzed. In tbc finite strip analysis of the girder, the lIlIIuber of terms used in the eigenfunction series is nl + n2 = 21 + 9. Convergence is obtlunccl wi thi n 8 iterations. The deflect.ioll of the girdcl' nlollg the noda! line 5 is depicted in Fig. 4.6. The vcrt.icnl forccs V, applicd to the girder by the cables, and the horizonta! components

H of cablc tcnsion arc givcn in Tablc 4.3. Tbc maximum negative bending moment

CH.4.PTER 4. NONLINE.4.R .4.N.4.LYSIS Cnbl" l

2 3 4

84

V H 10.730 25.985 9.831 11.841 11.315 13.419 10.521 25.103

Table 4.3: Vert.icnl Forces of Cnbl!'s on Ginl"r IInd Horizont.al Flln'es Ilf Cablc.: on Pylon (in IvIN) of thc girdcr is nt cross section B , Illul !.ll(! lllnxillllllll posil.iv" hl'llding nl
4.2

GEOMETRICAL NONLINEAR ANALYSIS OF PLATES

If thc deflecl.ion of a platc is large in compariStlll wit.h t.h,· plllt.,· t.hickness, I.he

deflcction may pmduce supplcmcntary stmins and stressl'S in I.h" midelle planc of the platc. Thcse in-planc stresscs eontribute to resistance t.n tldlcction of t.hc platc. As a l'es\ilt, the dcflections of the plate arc 1I1uch Icss th/Ul thosc esl.imatcd by lincar clastic plate thcory. Becausc of the intemctiou bet.wcen ,Idleet.ion Ilnd in-plrulc deformation, thc govcrning diffcrcntial cquations nf platcs h"''''lIlc nonlincnr, and the solution of the proble1l1 hcc01l1es 1I11lch morc c01l1plicatetllú91. EXllct snlutiolJs can be obtrulled only for

it

fcw of thc simplest cnscs. S01l1e ulllnericlLl I!wthnds,

such as the encrgy method, pcrturbat.ion mcthod etc. have beelJ developed, but these are very difficult to apply, because different displacemelJt fllnclions have to be assumed for different loads and plate gcometry. The tinite cleme'Jlt mcthod is

CHAPTER. 4. NONLlNEAR ANALYSIS II

versat.ile tool,

I)(,ill~

85

IIl'plicahle for geomctrical nonlinelll' Ilnalysis of lilly type of

plate; however it.s efliciency nee,ls to he improved beclluse of its rcquircment of too f11l1ny degrees of fl'l'I,(lol1l and extremely large quantity of inpnt data. The finit.e st.dp Illet.hod

IHL~

been effectively used for linear c1astic analysis of

IllIUly types of platcs. For the plate wi th two opposite edges simply supportcd, the displacefllents corresponding to different eigenfunctions arc uncoupled, that is, the loads corresponding to agiven eigenfunction produce only displacelllents corresponding to the SlUlIe eigenfnnction. Hence, the stiffness matrix of this type of plate has a very llIU'row half-bnndwidth. And consequently, the efficiellcy of the finite stdp 1U1Illysis is mnch higher thlUl that of the finite clement method. Since 1078, the finitc strip lllethod hns been extended to geometrical non-linear analysis of platc strnct.l\l'es IUld hns delllonstl'llted that its efficiency is highel' thlUl t.lmt of the finite elel1lent method [28-33]. Nevertheless, hitherto in ali existing references the Newton-Haphson Illethod hus been used, so that at each new iteration t.he f.lmgential stiffness matrix of structure has to bé reformed nnd inverted once again. Fnrtherl1lore, because of the non-linearity of the analysis, the different series t.enns arc conpled in the tlUlgential stiffncss matrix even for simply supported plat($, wi th thc result that the orthogonp.l property between different eigenfullct.ions ('Illl not any long!'r be ntilized· to reduce the half-bandwidth, and hence that tili' !'nluUICe1l1(·nt. uf cfficiency of non-linear lUlalysis by finite strip method is not as significant.

I~'

cxpccted.

An effect.ive l'cmedial mensurc to this problem is to employ Initial Stiffness It-

SG

CHA.PTER 4. NONLINE.J.R ANALYSIS

!'ratiou [GJ. First. th,' iu·plalll' nUll heudiup; displlll·"UIl'ut. flludiolls of ol'llilllu'Y fiuitt~

plat.l· strips m'p applil'd to forl11l1latl' thI' linelll' plllst,it- st.ill·nl'ss Illllt.rix aud

t.he "'lllh'alcnt, nodal forc!'s. Therellft.cr, it, is l'IISY to liml t.h,' linenr <'llll
111'('

oh-

t.aitled frolll t.h"sp st.reSSl'S by 1II"IUIS of the principle of "irt.llal work.

Becallse of th" non-linearil.y of the problel11, the nbove displncclIIl'nt.s Ilrc nol, t. Ill' solution of t.11l' pl'Obll'IIl, and IlII' "ol'l'csponding r,~.istnnces '~lLn not be "'111111 t.o the exterllal loads. Th" so-called "lIulmhuu'"d forces" , i.e.l.he (liffel'eIH'" h,·I.w"clI t.he londs lIud t.I ... uon-liucar \'(!SiSt.ILUl·CS, l11ake the plllte d,,forlll fll\'th'!I·. As a result, new clispla.. eu\l'uts, "o\'l'espoudiug r"sistlluces lIud "uulmla.llced forces" a ... ' obtaiued. This pl'Ocedure is rep"at.ed uul.il I.he non-linem' r,·sist.auces couul.edllLlance the exte1'llal loads. DlI1'iug all the itemtions, the iuit.illl sl.ifrlless ulILl.rix of the plate is assullled to he undtauged. Cousecluently, the ullwuplcd problclII ill linellr-clllStic Ilnalysis will rel11llin uucoupled in the 1I0nlilleILr mllLlysis, I.he Ilill fbllndwidth of the stiffness matrix also renmiuillg unchmlged. Tlms, the IIdvILUI.llgl· of !inite strip 11Iethod will not be lo"t ill lIonlillenr analysis. Also, th" stifru"ss matrix needs not be reforllled and inverted in each new itel'lltioll. Although ti.., 1111Inber of itel'ations will he inCl'ellSed, IUl ovemll econollly CIIII he Ildlicv"d ill lIIosl cascs.

In practice, the transverse stiffne"" of IL plat,! under Illrge deHI!ctioll is lIludl gr,,"ler than its corresponding linear clastic stiffness. This may result in rlivergcllcc of

Si

CIIAPTER 4. NONLINEAR ANALYSIS

it"rntiou if til" iuitilll bcudillg stifflless matrix I'clllaius unchlUlgcd durillg the whol .. itcmtioll pl'Oecss (Fig.4.!J).Thc silllples!. method of lIlodifying the bendillg stifflless Illat ri x is to II\llltiply ali the itellls of t.his mp.trix by IUl amplilication factor which is II

flllldiou of the ratio of eurrcut lI\aximum deflectioll to the thickness of the platc.

IUlHu' <'IL~e,

II

lIlorc dficieut cquivlllellt mcthod is used, which is that of dividing thc

"llulml/Luccd uodal force" cOrl'esponding to every flexura! degree of frecdom by the S/LIIll! fudor. It has bcen shown that this method can ensure effective convergence if the vallIc of thc IUnplification factor is reasonably choscn. SOIl\<: dctailed description" follow.

4.2.1

DISPLACEMENT FUNCTIONS AND INITlAL STIFFNESS MATRlX

Under large deflection, eaeh plate strip is subjected to in-plane stresses and outof-planc bellding forces. The noda! displacements IUld forces of the strip correspoudiug to the m-th terlll of the eigenfunction series are (Fig.2.2):

TIli' displaccmcnt licld wi thi n a phüc strip [24] is T

11

= L (l -

X)lIhn

+ XII2m)Y'::(y)

m=l T

v= L((1-X)vlm+XV2m)Y';:(y) m=l

ss

Ci'lAPTER 4. NONLINE.4.R .4.N.4.LYSIS r

'fl'

=

L (JV1m'llllltl + lV.~m81111 + 1V

am "III'lm

+ JV,lm021J1)

(4.23)

ru=1

whel't' X =

.,,//1, l' is the tot.runumbl'l' of sl'rics t.Cl'1llS used in analysis, nnd ( . műv l ··" '" y) = SIIl - / -

If both ends of the st.l'ip nt·c free to move in t.hc lougitudinal dircetion (11 = O , av

=O

nt y

= O and y '-= / ), then ."( ) műy l '''' y =cos-/-

If both cnds arc immovablc (11 = O , v = O nt. V = O and y V"( ) _ . (m I " , Y - SIll

In thc cxprcs"ion fol'

= / ), tiicn

+/ 1)7I'Y

10 ,

If both cnds are simply snpported in bcnding (UJ

= O, My = O nt y = O UJld

y = / ), then

. n17l'y Ym"'() Y =sm-/ If both ends are clampcd (w

= O , n;; =

O at y

= Oand y = / ),then

"'() =sm-/--smh-/--amcos-/--cos ,JlmY • JlmY [JlmY h -/JlrnY YmY 1

CllAPTEH 4. NONLlNEAR ANALYSIS

89

wi",!,,, (tm

=

sin/,,,, - ainl! I'm I cos fLm - cos 1 /l'fR

It", is t.he lll-th sohlt.ion of cquntion 1 - cos II cosh It

= Oj its values are listed in

SubSf'diOIl 2.1.1. SuhHt.itulillg "
pnramct.crs of t.he platc.

4.2.2

GEOMETRICAL NONLINEAR SOLUTION

After thc displncelllcnts are obtained, the strains and stresses of the plate under 1!U'ge r1eflection cnn be evaluated as follows [59):

8,: 1(8w)2 Er=-+-8a: 2 8a: 8v 18w 2 Ey = 8y + '2( 8y ) "'(ry

8u = 8y

8v

aw8w

+ 8a: + ax 8y 82 w

Xr = - 8X2

82w

Xy = - 8y2 v

82w -?--

.,ry -

-

8a:8y

(4.24)

CH.-\PTER 4. NONLINE.4.R .-\N.-\LYSIS

!JO

Eh Nr = -1-(fz + IJf u) - l j2

N rv = Gll'Yru

Eh 3 M" = 12(1 _ ,,2) (Xr

+ IJXV)

Eh 3 Mu = 12(1 _ ,,2) (Xv + IJXv) Gh3 M"v = 12 X:rv

In the above expressions, E,

/J

(4.25)

and G are the clnst.ic constnnts while h iH t.hc

thickness of the plate.

By substituting equations (4.23) into equations (4.24) and differentint.ing t.he re· sul ting cxpressions, the following equation can be obtained: T

d{e} = cl(e:r,ev ,7"y,X:r,Xu,X:ruf' = L;[Bj",cl{6}",

(4.26)

[B]m = [B] .. + [Bn/(6)] ..

( 4.27)

m=l

n.nd

in which cl{e} and d{6}m arc the infinitesimaI incremcnts of strain and dispince· ment vectors of the strip, respectivclYi [B]m is the linear straill matrix obt.aincd previouslYi [Bn/]m is a fUllction of {6} which reflcc: H thc lIolllillcar efFcct of large

91

CHArTER 4. NONLINEAR ANALYSIS c}('flc!dioll, and is as follows:

O O O O

W,r N 27n,x

lO,%N 1m ,x

w,,,N2m,tl O tv,%N2m ,u+

11J .uNI m,tI

W,xNlm,v+

[nn/J fU -

O O llJ,,,Nhn ,% O O O O O O () O O

w,U N2m ,%

O O O

N4m,z O W ,J/ N3m,,, W,u N 4m,'JJ tv,%N3m,u+ tv,%N4m ,v+ O W,lI N3m ,z w,,,N4m ,% O O O O O O O O O

O O

O O O O

W,x N 3m,x

W,r

(4.28) whcrc t.hc snbsctipts ", x" IUld", y" dcnotc ::c and :u ' respectivclYi and r

10,% = ~ (N1m,;ctvlm

+ N2m,iJlm + N3m,;ctv2m + N4m,:c82")

m=1

r

'/(I,u = ~ (N1m,utolm + N2m,u8lm

+ N3m,utv2m + N4m,v82m)

'11=1

The rcsistanccs {R} to the largc deflections of the plate can then be calculated by mcans of thc princip;. ,)f virtual work [66J. The so-called resistallces {R} are I'cfcrl'cd to thc cxtcl'nal forces requircd to maintain agiven deformation. If any vil't.ual displaccmcnt d{ 5}m occurs from the current eqllilibrium position of astrip, t.hc work done by the external forces dllring the virtual displacement is

and thc work pcrformed by thc intcrnal forccs is

According to the principle of virtual work, dWe:c must be equal to dWin for any possiblc d{c5}",i this rcqllires that

{R}m = /, l[jjJ~{CT} dxdy

(4.29)

CH.-1PTER 4. NONLINEAR .4.N.4.LYSIS

92

where {R}", is the resistance vcctor of a strip cOITesponding t.o lll-th term of cigen-

The integration in (4.29) call be implemcntcd by t.hc Gallssinll illtcgmt,ioll mcthod. If the current resist!mces are 1I0t equal to the extcl'llal Illa ds, the platc clcforms fm·ther under the "unbahmced forces" ,i.e. t,he diffcrellcc bct.wccn t.hc external loads {F} and the resistances {R}, wi th the origillltl !incm' dnst,ic matrix [Kl being assumed to be unchanged:

(4.30) Thence the new displaccments arc reacher.l:

(4.31) This procedure is repeated unti l convergence is nchieved, i.e. unt.il

lL,(6.ó~+1 )2

VE( 8;+1)2


(4.32)

-

where e is a prescribed tolerance. In the prcscnt, study, c = 0.0002. In the last three equations, the superscript "k" or "k+ 1" indicates thc number of current iteration. In order to ensurc convergence, in each itcrat.ion, thc "lInba!;mced forces" correspo.'1ding to the bending degrees of frecclom on the right side of the c<juation

(4.30) are divided by a "flexura! stiffncss amplification factor", o. Aft,er

IL

number

of tria!s, an appropriate va!ue of a has been found as follows: for plates wi th two opposite clamped edges a = 1 + 10.0(~)2

for plates with two opposite simply supported edges

93

CIIAPTER 4. NONLINEAR ANALYSIS

III tJ.csc cxpl'essions, 6"",% is t.he maximum displacemcnt parameter found in the

elll'l'Cllt ilcrnlion.

4.2.3

NUMERICAL EXAMPLES

1. Cylindrical bending of a unifonnly loadcd rectangular plate with simply sup-

porl.c<1 immovllble edges Th(~

plllte is very long in the x direction, so that, the dcflected surface of a por-

tion of such

II

plate at a considerable distance from the ends can be assumed

cylindrical. Thel'efore, the analysis needs only one strip with boundary condit.ioll:

UI

=

U2

= O,

91

=

92

= O along two nodal lines. The length of the strip

iH l = 50in.. The thickness is h = 0.5in.. The elastic constants of the material E = 300000001Jsi, v = 0.3. The intensity of the uniformly distributed load

IU'()

is

lj

= 20]Jsi. The maximum linear elastic deflection is Wo = 4.740 ,wo/h ='9.48.

Dccause of symmetry of deformation, only the eigenfunctions symmetrical to the midsplul are used, i.e. m = 1,3,5,.... The numericaI results of the deflection, 1/lm"%

(in inches), the longitudinal stress of the middie plane,

(1~

in psi, and the

longitudinw bending moment, My in Ib-in, at midspan are Hsted in Table 4.4 .For ('omplU'ison, the theoreticw answers [59J are also given in this table, in which m....% is the highest order of eigenfunctions used in these analysis, N it • r is the number of iterntions up to convergence.

2. LIU'gc dcHections of a uniformly loaded square plate with clamped edges A squlU'c plllte of side 2a and thickness h, clamped along ali the immovable edges,

CH.4PTER 4. NONLINE.4R I~NALYSIS l1l. mllr

·W mor

1

0.69i2 0.6827 :I 5 0.6841 Thcory 0.688

(I.

94

M.

15820. 945.20 15i70. ii8.iO 15840. 825.60 15830. 830.14

lViter

li 4i 48

.

Tnble 4.4: Deflcdion IUld Strcsscs in Cylindrical Plat.e Dending

qa"/Dh

100/h

wmnr/h

(I,.a~(1- I/~)/ EI/ 2

JVjrrr

50 100 150 200 250

1.008 2.016 3.024 4.032 5.040

0.767 1.158 1.418 Uil8 1.782

4.22 7.09 9.37 11.33 13.05

i 13 17 21

25

1II",.z/ It

[59]

0.76 1.15 1.41 1.60 1.78

Table 4.5: Deflcction and Strcsses in Clampcd Squal'e Plnt.c is subjected to

It

unifol'mly distl'ibutcd load q. Usiug sylllmctl'y, only half th"

iJlate is IUlalyzedj it is divided into five equal width stl'ips anclllllluy7J!d by mcnus of five eigenfunctions symmetl'ical to the midspan (

lll",,,,.

= 9 ). The llIullel'icnl

l'esults are Hsted in Table 4.5, in which D is the flexural l'igidit,y of the plnt.e,

D = Eh3 /12(1 - v2 )j

Wo

is the lincar clastic deflectiou nt. t.he center of plat,cj

is the result obtained from nonlinear analysisj

17,.

tII"... r

is the maximum uormlal bending

stress at the center of each edge. For compnrisoll, thc IUISWel'S obtaillcd hy

IUl

energy method with 11 parameters ([59J,pp421.433) ILre ILlso giVCll iu the same table. In every case, the whole load is imposed within oue st.ep. The number of iterations up to convergence,

NilcT!

is 7 to 25, dependillg on thc ratio of Illo/It.

In ordinary structures, this ratio will not be too hll'gej it. is conchlded that the analysis decs not need too many iterations.

CHAPTER 'J. NONLINEAR ANALYSIS

4.3

95

NONLINEAR ANALYSIS OF REINFORCED CONCRETE SLABS

Thc JJlaterial nnnlinenr ~tIIaly"is nf reinforced concrete slabs is an important subject fol' inves!.igat.ing thc bclmvior of mlUly civil engineering structures such as floors, roofs and bridgc clecks. During the past two decades, finite element methods based on n number ol cliffcl'l ilt material models have been used successfully to simulate the Illaterialnonlinenr response of Re structures. However, the finite strip method 11Il.~

t.he ndvantnge of considernbly reducing the computation, storage requirement

ancl input dnta prcpamtion. Recently, Guo et al.[37J have developed a layered lilli!.c strip method for the nonlinear elasto-plnstic analysis of static Md dynamic I'csjlonse of concrete shtbs t1sing the Von Mises yield criterion. Their results have shown that their method needs nUlch less storage capacity, Md spends much less cOlnputer time than the fillite clement method does. However, because they used a I'clatively simple material model, the discrepMcy between their solution and experimentl1.1 results for concrete slab seems larger than might have been expected. In this thesis, the model based on orthotropic nonlinear elasticity is used to represent thc property of plain concrete under biaxial stresses. This model was proposed by Darwin ct n1.[67J, modified by Kabir [68J Md is in close agreement with experimcntnl I'cstllts [69J. For reinforcement , the bilinear elasto-plastic uniaxial stress st.nun relationship is employed. Thc finite strip is divided into imaginary concrete layers. In addition, each layer of rcinforcement is replnced by n smeared steel layer with unchMged original

CHAPTER 4. NONLINEAR .4NALYSIS

96

l'cinfol'ccment arca having stiffncss only in thc dircction of l'ciuforcr.l1lcnt.. In eVC1'y Ncwton-Raphson itcrlltiou, usiug fillitc strip intcrpolat.ion funcLions lIud thi n plate theOl'Y [24,59], thc strnin statc iu cuch laycr CI\1l bc found, Aftcr which its stress st.ate can be casi ly obtaincd by mC!Uls of thc abovc-lIlcutioucd models. Next, une C?II calculatc thc tangcutial st.iffucss Illatrix, uubahulccd fOl'l'cS I\1ld corresponding displacclllcnt inCl'ClJlCnts [70].This proccclurc is rcpcntcd unti! convergence occurs.

In order to prevent possible oscillations, !UICI to cnsure convergence of Ilonlincar solution, the rclmcation techniquc is applied, that is, in every it.cration, thc displaccment increments are multiplicd by a rclaxation factor less thlUl unity befol'c being addcd to the accumulated displacemcnts.

-

FUrther details are given bclow.

4.3.1

MATERIAL MODEL OF CONCRETE

Concrete is treated as an incrementally ort,hotropic !inear clastic material [67-69].

(4.33) where

[Dc]

1

= 1 _ 112

in which Eh E2 and

(EI

(4.34)

sym II

= "';111112 are stress-dependent material properties, fj = 1.0

prior to cracking and O <

f3 :5 1.0 after cracking.

Material axes 1 and 2 coincide

9i

CIIAJ'TER 4. NONLINEAR .4NALYSIS

with the eurrcnt principllI stress axes. TIli! couccpt of "()!juivnlcut uniaxial strain" is introdllccd in ordcr to tl'llllsfol111 the nd,ual biaxial strcss statc to two "indcpcndcnt uuiu..xial stress statcs" . It is defincd Wi : f::,.u j

fi" =

""" l.J

olt load incrcmenl/l

with

f::,.Uj

= incrcmcntlll

g

(4.35)

I

clumgc ill principal stl'essj and

stiffncss for load incrcmcnt correspolldillg to

Ej

= varying

tangent

f::,.Uj.

Thc relationship betwccll the prillcipal stress Uj and the equivalent unirocial strain fj"

is showll ill Fig.4.10.

Fol' comprcssivc loadillg prior to the maximum stl'ess, the compressive principal st.rcss Ilncl elnstic 1Il0dulus can be calculatcd as follows: (4.36)

(4.37)

whl'rc

Eu

= thc trulgcnt

modulus of elnsticity at zero strcssj

sl'cIInt IllodulllS Ilt thc point of maximum compressive stress

Ec

= Ujc/ fjc

Ujc,

and

fjc

= the = the

c!jllivalcnt unin..xial str/l.Í1l at the maximum compressive stress. Ujc

and fjc depcnd on the principal stress ratio ct

= U1/U2'

The analytical maximum

st.J'(·nv;th ellvelopc may be cmployed to detemlÍne the depelldence of (Fig.4.11).

Ujc

on

ct

CH.-\PTER 4. NONLINE.4.R .4.N.-\LYSIS

!J8

Fol' tl'1lsile strl'SS prior t.o crllcking, t.he linenr dllstil'

" j ' fIl.

I'l'lat.ionship is applicd: (4.38)

The folll' regions of t.hc strengt,h .,nvclopc with the IIccompnnying "'1IlIlt.ions for the maximum strcsscs "je IUld corrcsponcling cquivnlent str;uns

for III'C

stllll1llllriz.,d

IlS

follows [71]:

1. For "t = cOlllpressioll,

U2

=

cOlllpl'es.~ion.

O ::;

(I' ::;

1

1 + 3.65(1' I e UZ = (1 + (I')Z J"

\Vhcre Jlt

= UIc/J; , Jlz = UZ./ J; and J; = the unin.'Cinl compressive strcngth.

2. For Ut = tcnsioll, Uz = cOlllpressioll. -0.17::; o ::; O

_ 1 + 3.280 fl UZ e - (1 + o)Z J.

whcrc

99

C/lAI'TER 4. NONLINEAR ANALYSIS

3. Fol'

I1J

= t.CIIHioll,112 =

comprcs.~ion. -00 ::; o' ::;

"2c

= 0.65f~

Uli = f2c

=

fc{ 4042

])2

I1J

= tcnsion,

U2

=

U2c/ f~

=

::; 0.65

= uli/Eo

= tcnsion. 1 ::; o'

fil

in which

f:

- 8.38p2 + 7.54p~ - 2.58p~)

fu

4. Fol'

-0.17

f2t

::; 00

=

fU Eo

J: is unirudnl tcnsilc strcngth.

Crncks nrc n"sumcd to opcn pcrpcndiculnr to thc highcst principnl tcnsilc strcss dirccLion whcll thc fnilure cnvclopc in FigA.11 hns becn rcachcd, and UJ

nnd

4.3.2

EJ

nrc I\ssmllcd to bccomc zcro ns soon ns thc crncking occllrs.

MATERIAL MODEL OF REINFORCEMENT

For thc strcss-strnin relationship of reinforccment, the unirudnl bilillear elnstoplnstic model (Fig.4.12) is cmploycd and the following notations arc used: E. = Young's ModIIIIIS, E.h

Illul

f ••

= strain hardening modulus, U u = yield stress

= IIltimnte strnin.

In concrete, whcll

UJ

reachcs Uli nt some point, primary cracks form at finite

spncing in thc sllrrollnding region. The totnl tensile force is transferred across

CHA P TER 4. NONLINEAR :\NALYSIS

f./e,

100

1.0-1.5 1.5-3.0 3.0-5.0 5.0-8.0 8.0-11.0 .) 4.0 2.0 1.6 1.15

11.0-14.0 olhel' 1.0 1.05

_.,-

K

Table 4.6: Coctficicnt K fol' Stress of Stecl after Concrete Cl'IIfking cnch cmck by the tensil!' stcd, hllt. hct.wt'en t.he cracks the tensile I:oncrct.e clu'ries stress locally, nminly in t.he dir!'ction of the stt'CI blU'S, due 1.0 I.he bond betwccn the concretc and the rcinforcelllent. This is clulcd the t.1:nS:,lll stiffening effect. As the 10lld increllSes, secondary crack" form 1\1'01111.1 I.he reinforcement. IUld betwccn th.: prinmry cmcks so that the 10lld clU'ried hy the concrete IUld the average stress in the concrete decrcllSe progrcssively. In order to incorporate this effect, aft.er the slIrrollnding concretc hlls cracked, the stress of steel,

17.

is magnified

liS

follows 172J: (4.30)

where 17~ is the stress of reillforcellleut. found by the bi1inell.r model in Fig.4.12 !UICI K is a coefficient shown in Table '1.6, ill which

IUId

Et

is the lIniruda! stmin at concrct.e crncking,

Et

=

E.

is the stmin of "tecl

fl/Eo.

The concretc is

assumed to carry no stress normlu to a crnck but IUl additiollal stress will be carried at the steellevel. This additiomu stress represcnts the total intel'llll.l tensile force in fact carried by the concretc between the cmcks, .:onvellielltly lllmped at the level of the tensile reinfol'cement and oriented iu the dil'ection of the bar. E. and E.h are also multiplicd by the same cocfficicnt K in fOfUlIIlatiou of the strip stiffness matrices after cracking of the sllrrotmding concrete.

CIlAPTER 4. NONLINEAR ANALYSIS

4.3.3

101

FINITE PLATE STRIP

Eadl platc strip is subjcctcd to in-planc stresscs and to out-of-planc bcnding fuJ'ccs. The nodal displacclllcnts and forccs of thc st.rip corrcS)londing to the III-th t.eJ'1II of eigcnfunctions arc (Fig.2.2):

If both cnds arc simply supportcd (II

= O, W = O,

(Ty

= O, MN = O at y =

°

and y = / ), thcn the displaccllIcnt cOlllpuncnts in the lIIiddle planc of the stJ'ip arc: r

L «l -

II =

X)lIlm

+ XII2m)Y... (y)

m=1

~

V = L.. m=1 r

II!

= L «l fil

=l

«l - X, )v.... +.'\:, V2m)

i 7n1r

dl'~,(y)

d

y

3X 2 + 2X 3 )Wl m+ x(l - 2X + X 2 )81m + (3X 2 - 2X3)W2m (4.40)

whcl'c X =

x/b, r is thc total nlUnbcr of series terms userl in analysis, and •

7n1ry

Y..(y) = sm-/ln t.hc vcrticnl dircction,z, the finite strip is divided into several concrete laycl'sj cach reinforcelllcnt layer is replneed by a slllcared steellayer with undll\nged origillal reinforcelllent arca and having stiffness only in the direction of the I'cinforcement (Fig.4.l3).

CH ..l.PTER 4. NONLINE.4R .4NALYSIS

102

According to the theúl'y of thin plates and shclls [59[,

t~le

st.rnin nt any point

of eac.h layer elUl be expresscd in terms of nodaI c\isplnccmcnt pnrlUnctcrs aN follows:

r

Bt·

{ej = ( :: ) _

fÍJ -

8'1IJ

z Dy'

- 'L [D]", {8} '"

(4.41)

m=1

'Yru

whcre -bI y'nt

[D]", ==

(1 - 10)_1 1':" b tn1l' 'Jn -z(1 - 3~:

O

(1 - fjY,:. Iv

bIm

O

O

'" 1 y" iim;; 711

Ily' -bm; na

( :ll! 6'

(:llb - !!5)Y 62 '"

(6.!. 62 - lla)y' b3 nl

O

+ 2~ )Y,::

l?.... zbf-b3 (" "')Y.'UI

+ lla)V ~3.lm

(_ 6,:;=

-z (:r. - ?~ - b

+ .'bT lY"

? (1 - 4";; -_z

+ 3"W )1"'m

iti

+ 2b' )y;" (4.42)

ty~ L~~y~ -12z(;.% - ~)Y,:. -2z(3 b: - 2~)Y,:. in which 1':'m-dy - dl'" and 1':" - d'Vm m-dll'·

4.3.4

NONLINEAR SOLUTION

The loads arc applied in a number of steps. In each loading step, the NewtonRaphson iteration method [6] is utilized for nonlinear solution. For eaeJ. iteration an incrementallinear stress-strain relationship is ruJsumed, the dis, placement increment calculated, the strain and stress state determincd !Lnd the material matrix updated. The following steps [71] arc used to determine the states of strain and stress at any point in a concrctc layer of II atrip for the k-th iteration:

ClJAPTER 4. NONLINEAR ANALYSIS

103

l. From thc clisphu:cmcnt incrcmcnt vector .ó.{Ó}k obtaillccl from thc previO\lS

it"ratioll. thc strnin incremcnt vector is computed as: r

.ó.{E}k = (.ó.E! • .ó.E~ •.ó."'f;"f =

L [B]m.ó.{ó}~

(4.43)

m=l WhCl'C

[B]", is strain matrix in Eqllatioll (4.42).

2. Using Equatioll (4.33) and performillg coorclillate transformation

[66]. the

appl'Oximatc strcss illcremellts can be calclllated as:

(4.44) whcrc [T.] is thc str'lin cOOl'dinate trlUlsformatioll matrix from x-y axes to t.hc principnl u.xes. as given beIolI':

[T,] =ill which

COS2 e sin 2 [ -2sin

e ecose

e is the angle between UI

sin 2e cos 2 2 sin cos

e

e e

e e ] e- e

sin cos -cosesine cos 2 sin 2

(4.45)

and x, obtained from the previous itera-

tion. Thc strcss incremcllts are only approximate because of the linearized material matrix. Thc current total approximate stresses are then obtained as: (4.46) 3. From the totnl appl'Oximate stresses the approximate principal stresses u~ (i=1,2) alld

ek are calcull.ted.

4. Thc equivulcnt unill-'l:ial strains are then evaluated from (4.35): (4.47)

CHAPTER 4. NONLINEAR AN•.t.LYSIS

104

5. The bia..'I(ial stress ratio is dcterlllined fl'OlII (4.48) Then, afc and

Efc Ol' ait

lU'e obtained from the bia..'I(inl st.rcngth cnvdope in

Subscction 4.3.1. 6. The principal strcsses

af

IUld clastic 1II0duli

Et corrcspondillg to Ef

cnn

then be computed from (4.36-4.38) and Fig.4.10. Thc c\Il'rent totnl st.ress vectol' {a}k is obtained by trlUlsforming thc pl'incipal strcsscs to thc x-y IL.'I(/!S. The material matrix [D~J is also updnted to reHecl. the new tlU~gent lIloduli. The contribution of concrete layers to the tnngentilll stifrness Ilmt.rix of t.he strip in the k-th itemtion, correspollding to the III-th nml n-th t.enus of eigenfunctions, is

[I~.-kJ nm=

f[ BonT. J1' [ kJ1'[ DckJ[ T.kJ[ B"dv J

(4,40)

Similru'ly, the stress state of rcinforcelllent lUHI its contributiOll to the tangentiAI stiffness matrix of the strip can also be computed us ing the correspoll,ling material model. Aecording to the principle of virtual work, in each plate strip, the resistllllce vect.or corresponding to the m-th term of cigenfllnetions is:

(4.úO) For the two integrations given abovc, the GaussiBll inteb't'ation method is employed in both x Blld y directiol1s. lu eaeh strip, 4 Gauss poil1ts arc used in the x direetion, whilst in the y direction the atrip is dividcd into

IL

CHAPTER 4. NONLINEAR ANALYSIS

105

nUlllbcr of scgmcnts, and 4 Gauss points arc used in eaeh segments. The numbe!' of sC!gmcnts uscd must bc sufficicnt to cnsure satisfactOl'y accuracy of iutcgl'ation for the highcst order series term. Ncxt., thc incrclllcntal displacclllcnts arc cvaluated as follows:

(4.51) IUld the ncw displaccments are reachcd:

(4.52) whcrc {F} is thc vcctor of equivalent nodal forces of extemal loads, and ,\ is thc rclaxation coefficien t. In order to prevent possible oscillation and divcrgcnce in iterntions, the value of ,\ must be less than unity, ,\

< 1.0 (,\ =

0.8 in the prescnt study). This procedure is repeated unti! the convergence is achicvcd:

.,JE(6.67+1)2 < .,JE(67+ J )2 - e in which e is a prcscJibcd allowable error (e=0.002 in the present study).

4.3.5

:NUMERICAL EXAMPLE

A simply supported two-way square reinforced concrete slab is subjected to IInifonn lond. This slab, wi th a thiclmess of 2 in. and a span of 72 in., is rcinforcccl in bot.h directions wi th uniform spacing. Due to symmctry, only a quarter of the slab needs to be analyzed. Therefore, half of slab is modcled by 4 strips, and the integration in calculating the

lOG

CHAPTER 4. NONLINE.4.R ANALYSIS

tangcntinl st.iffncss matrix !lll
0111.

over

only half the span. 4 synllnel.rical series tcrms me IIsed ill this analysis. Thc cross scction is divided int.o 10 conCl'l'te layers and 2 orthogonlllrcinforcing stecllayers of 3/lG in.diameter bars with thc splicing shmvn in FigA.14. Thc following mutcrilll propertics me uscd fol' thc analysis:

= 4700 ksi, f~ = 5940 psi, f! = 550 psi, = 0.0025, ,j = 0.18. ReinfOl'cing Steel: E. = 30000 ksi, E.h = 2000 ksi, fy = M.5 ksi, f ... = 0.1.

Concretc: Eo

fc

The Newton-Raphson mcthod is used. The c1efiections at. th" ccntcr of thc slah undcr incrcasing ulliform load arc sl10wn in FigA.15. Fol' compm'ison, the resu1ts obtaincd from cxperimcnt [73], finitc e1cment IUmlysis [71] and finite strip annlysis [37] arc ruso shown in the SlUllC fig\ll'c. It can be scen that the prcsent research impl'oved the accu!'acy of nonlincar finitc strip annlysis of the RC slab significantly.

107

Cll ..U'TEH 4. ['WXLINEAR ANALYSIS

y

if. H

l

T,

;+.....,.+<;-_ _ _ _...-1._... X

V,

'W. V, (11.)

Cb) . Figul'e 4.1: Cable

Figul'C 4.2: Pylon

li

V,

~)

CHAPTER 4. NONLINE.-l.R ANALr"SIS

lOS

R Pl----~--....,.,...---::::=,....:::::.

Figm·c 4.3: l1útinl Stifl"ncss Mcthod

o'"

I

" 80

FigUl·C 4.4: Singlc Plrulc Cablc"Staycd Bridgc

111

X

Z

109

ClI.·ll'TER 4. j\;OXLIJYE.-tR .-tNALl"SIS

~

li

A

B

C

D

E

som x 2

to'"

I

F

. 6.5 m

6.5'"

6'"

Solti X 2.

5

x

6'('

.

~

0.2 m

o

5

I

O.16

m

0.3/11

2

4

~I

Figure 4.5: Duuble Plmle Cable-Stayed Dl'idge

0./30

A

D

E

F

0·592 FigtU·c 4.6: Defledioll uf Gh'der mId Pyloll

}-

CI:f..l.l'TER 4. XUXLIXE.-lR .-lX.-lLl·SIS

8.3'19

9.564 8.757

r--..,..----I~..::..-

+

·11.849

110

___

Ó.166

+

-/4: 524

-11.25-:7- - - _ - ' -/3.916 -lJ. 578

Figlll'c 4.i: LOllgitudillul Strcsscs ut CI"O~s·ScctiOll II (ill 11pa)

III

CJl.·ll'TER.J. J\"OXLIJ\"E.-I.R .-I.NAL1·SIS

~ -6.989

-6.7/4

-7.154

- 6.577

-6.925

\ /0.395

10.354

10.295

Figmc 4.S: LOllgitudillnl Sh'CSliCS at Cl·oss·Scctioll F (ill Mpa)

CI:l.-l.PTER 4, JYONLINE.-l.R .-l.N.-l.L1·SIS

112

R

--~~-----------+-------8 S2

8°

68'

S'

OS2

FigUl'C 4,9: Possiblc Dh'cl'gcncc

Cl1..l.PTER 4. NONLINEAR ANAL1"SIS

113

-a I -a10 I------:::=-"..

-02f • c

-e Ic

Figul'C 4.10: Equivalcnt Uniaxial Stl'ess-Strain Model

Cli.,lPTER 4, XOXLINE.-I.R A,N.-I.Ll·SIS

-,

114

- ' - '-,_,

/4

aI 0:=-00

fl

----~~--------------------~~~4_~

/',

/

o:=a l /Ca 2"

/ ,/'

/' /'

0:=1.0 ,/'

,/

/ ,/' / ,/'

, \3 \ ,

\ '

\

2

,

\ 0:=-0.17

0:=0 FigUl'C 4,11: Dinxinl Stl'cllgth Ell\"clopc

115

C/UPTER ·1. JY()SLISE.-tR AN.Ul·SIS

Figul'C 4.12: Matcl'ial Modcl of Stcc!

:x ..

Z

ts = As ILs lb

Figul'C 4.13: Laycl's of Strip

CIH1'1'ER 40 I'iONLINE.-lR .-lN.-lLl·SlS

110

------------~-----------,

I I I I I I I I I

! Io

~o--o-.+_o-r-o-

o I I I I I I I I I I

u)

p,.

.....

0:0

I

E-<

u)

"'" -----------r-----------

I I I I I I I I I I I I I I I I I I I I I I I

;.....

.;.....

M ID

o•

-rL On.-_0

0- N _o_

o

~

I

\

d=3/16"@2.5" d-3/16"@3"

72"

Figurc 4014: Taylor SiaD

117

ClI ..U'TER 4. JYOXLIXEAR ANALYSIS "! c

N

_ _ o ~.---.~.-

,..- . ./ '

j'

c

...

ID

/ -c

/

Cll'

Z:!

O

Eo-<

/

I .............

/

................... :~.~.~.~.~.~.~~.~ ~ ~ ~ ~ ~ ~ ~

.

..•... .-.- ...... ...... ,.,;0;.;':'_---- ...

/. /~~/ , -' .'

.'

I

I

.. I

EXPERlMENTAL

j ../

. ..,.', ;/ .. i/~ ,,~ : .,

• I

o

ID

~~~

",

, ,.; •• '

/

/

FlNlTE ELEMENT

~

.•

.........

FlNlTE STRlP PRESENT STUDY

c

o.-------.--------r------~------~------~r_----__, 0.0

0.5

1.0

1.5

2.0

CENTRAL DEFLECTION (IN) Fig\ll'c 4.15: DcflcctioIl of Taylor Slnb

2.5

3.0

Chapter 5

COMBINED ANALYSIS If a plate structurc hM constant cross-scdion and it.s end support ('OIldi-

tion docs not chaJlgc tl'lU1Svcrsely, thc finit.c strip IlIcl.lwd hlL' p\'Clvcn t.o Iw the most cfficicnt Iltnllcrical structural I\.nalysis IlIct.hod. However, if t.he structmc hM any irregllhu'ities, e.g. a rectlUlg1l1ar platc with opcnings, t.he finite strip method is no longer applicnblc on its own and t.he finitc cIcment. method or the boundary clement mcthad has to bc used. In this

CIL~C,

how-

ever, if these methods call be combined together, with thc finitc stl'ips hdng used for the regular part of the plntc and thc finitc elcments or boumlary elcments modelling the irregular part, then the efficicncy of the finit.c strip mcthad and the univcrsality of thc latter mcthocls arc both lItili1.cd to their full advantage. In the present study, special tnmsition elcments arc nsed to conned the two different regions. One side of snch atransi tion elcment coincides wi th the nodalline of adjaccnt finitc strip and has the Sanle degrces of frcedomj on the opposite side there are a number of nodes which are conncctcd with thc finit!!

118

CHAPTER 5. COMDINED ANALYSIS

"'mIlelIt.

Ol'

119

boundnry elelllent.. In"ide the transition elcment the deflection is

expre"s"d in terllls of t.he degree" of freedom of the finite strip nodalline and the llode" of t.he fiuit.e element or bOllndary clement by their corresponding slmpe fllnct.ions. Using thc principle of minimum total potential energy, the st.i/rll"ss Illatrix and load vector of the transition elcment can be obtained.

5.1 FINITE STRIP METHOD FOR REGULARPART Thc regular region of t.he plate is sllbdivided into a number of finite strips. Within each "trip, the deHection is expressed as: T

//I

= ~(f1W1/Jh"

+ b91(081m + f2(OW2m + b92(082m )l';"(y)

(5.1)

m=1

where l and b arc the length Imd width of the strip respectively: ,ft(~), h(~), 91(0 and g2(~) arc transvcrse shape functions which are 'Hermitian cubic polynomials:

f1(e)

= 1 - 3e 2 + 2e,

gl(O = l~,,(y)

e- 2e + e,

M~)

= 3~2 -

g2(e)

2ea,

= ea - e,

(e = x/b)

(5.2)

is the m-th series term of beam vibration eigcnfunctions, its expression

is listed in Subscction 2.1.1 according to the end conditions ofthe plate strip. Following the stlUldard procedure in Section 2.1, it is not difficult to obtain the stiffncss matrix alld load vector of such a strip.

CHAPTER 5. COMBINED .4NALYSIS

120

5.2 COMBINED WITH FINITE ELEMENT METHOD 5.2.1 FINITE ELEMENT METHOD FOR IRREGULARPART In this analysis, thc irrcguhu' I'cgion of plat.e is anu.\yzed by

It

conforming rectangular plate clcmcnts (Fig.5.1) [74] wit.h f01l1'

number of degrcc~

of

freedom at each nodc which arc

{S} = (tv, Ow, Ow, 02W O:c Oy 0.1:011

f

The shape function of this clemcnt. is: tv

=

fl(~)fl(ll)Wi + b9l(~)fl(I/)~Vi + (!fl(O!II(I/)~~Vi + IIbg ~()gl(I/) u:/: ~210Viy uX uy Ow· ow· 021f! . +fl(Oh(TJ)wj + bgl(of2(TJ)~ + (!f,(~)g2(1/)~ + (!I'YIWg 2(I/l ~J oxU1l u:C uy OWk

+f2(~)!I(TJltvk +bg 2(Ofl(TJ) Ox ~

.

+f2(~)h(TJ)tvl + bg2(e)h(TJ) Ox

+ IIh (e)g I(1/) .

OWk 02Wk Oy +(!b92(~)!JI(I/)0:/:(')1I

~

~~

+ ah(OY2(TJ) Oy + (!b!I2(~)!I2(!iJ 0:/:011 (5.3)

where ~ = x/b and II = y'/a. This type of elemcnt meets the continuity rcquirt':ncnt betwccll any adjaccnt elements not only for the dcflection tv and the tangcntinJ dcrivativc ~: but also for the norma! deriV/Ltivc

a;::.

Therefore, this clcmcnt is

IL

couforming

plate bending element, and it achicvcs considcrablc improvcrncnt in ILCCllrllCy in comparison with rectangular clemcnts having only 3 dcgrccs of frccdolU at each node.

121

CHAl'TER li. COMBINED ANALYSIS

5.2.2

TRANSITION ELEMENT

A row of rectangular transitioll elcments is used to connect the finite element region with the finite strip region. One side of each transition element coillcides wi th the nodal line of adjacent strip and has the same degrees of freedom. On the opposite side there arc two corner nodes which are attaehed to the nodes of adjacent finite elements and have the same degrees of freedom as these finite element nodes (Fig.5.2). Insidc the transition element, the deflection w is expressed in terms of degrees of freedom of nodalline and two nodeB by their corresponding shape functions in the form: r

111 =

where

L; (fl (OWlm + bYI(09Im )Ym(Y)

e= :x/b IUld 1/ = y'/a.

It should be noted that the llodalline

~

and nodes k, l have the same type

of slmpe functioll in the direction x, which are HermitiIUl cubic polynomials. This is in order to simulate certain basic deformation patterns more accuratcly, such

IlS

one dimensional bending in the direction y, torsion about the

I,-"is y, etc. The stiffness matrix and the load vector can be obtained in accordance with standard fillite strip formulation. Simply assembling the stiffness matrices and load arrnys of these transition elements with those of finite strips and

CHAPTER 5. COMBINED AN.4LYSIS

No.of Strips 2 No.of Sylll.Tcrllls 3 Mesh of Clellll'nts 2 by 4 0.0113105 WmaZ (lll) Mr at A (MN-Ill/lIl) -4.390 Mr at B (MN-Ill/m) -4.710 My at C (MN-lll/lIl) -5.111 2.496 1.1" at E (MN-Ill/Ill) My at E (MN-Ill/m) 2.445

122

4 3 Thcory 5 3 3 by 6 4 by 8 0.D111349 0.0110848 0.0110074 -4.747 -4.896 -5.001 -4.916 -5.130 -4.915 -5.071 2.380 2.331 2.376 2.322 2.375

Table 5.1: Dcflection and Momcnts in Chuupcd S<]lInrc Plat.e finite elelllents willlcad immcdiately to thc forllllllntioll fol' t.he whol" plat.t' structure.

5.2.3

NUMERICAL EXAMPLES

1. Cla.mped s<]uare platc under unifol'm load This exalllple is cllOsen to verify the prcscnt method. Thc

S'IUlU'C

plat.c is

clampcd along four edges and is subjcctcd to uniform load. Thc length of each side is 10.0 m, the thickncss is 0.5 m. Thc lIlnterilll properties are

E = 1000001.1 Pa and v = 0.3. Thc intcnsity of the load is q = LOM N 1m2 • Half thc platc is dividcd into finitc strips, thc othcr lmlf into finitc clclllcnts. The mcsh is shown in Fig.5.3 and thc rcsults arc Hstcd ill Tablc 5.1. TIl!! analytical results are also givcn in this table for comparison. From this tllblc it can been secn that the method achievcs satisfactory IlCCttrllCy in cVllhmting maximum deflection arid bcnding momcnt. 2. Square plate supportcd by walls and columns A square plate with a rectangular opening is supportcd by walla and columlla

CIíAPTER .5. COMBINED ANALYSIS

123

(Fig.5.4). The length of eaeh side is 9.0 m and the thiclmess is 0.2 m. The material properties arc E=25000 MPa, and v = 0.15. The plate is subjected t,o unifol1n load of intensi ty q = 10kN/m2. Thc port slIpported by colllmns is divided into 6 by 18 conforming rectangular plntc clcmcnts with 4 dcgrecs of frccdom at each node; thc rest of thc platc is dividcd into 11 strips !lI1d onc row of transition elements that connect the strips to the finite clcmcnts.Five symmctrical series terms are used. For simplicity, the walls arc regorded as simple supports, and the columns ILS

point slIpports.

The rcslllting deflection along line C-C and the bending moment Mx along the lincs B-B Dml C-C are shown in Fig.5.5. The CPU time spent for the analysis is 4.2 sec. on a Ma.infrume AMDAHL-5860. Thc structure WILS also Il.I1alyzed using an 18x 18 array of the above-mentioncd clemcnt,s at a cost of 12.1 sec. of CPU time. However, the results show no disce1'1lible diffcrence from thc prcvious solution when the results of both mcthods arc drawn on the samc figure.

5.3 COMBINED ANALYSIS WI TH BOUNDARY ELEMENT METHOD 5.3.1 BOUNDARY ELEMENT ANALYSIS FOR IRREGULAR REGIO N . In this analysis, the boundary of the irregular region is, .

, into a number

of boundary clements. Each element has between 2 an ... 11 nodes,and ali the

CHAPTER 5. COMBINED ANALYSIS

124

boundru'Y functions nlong thc clcmcnt lUC cxprcsscd in tCl'ms of their nodlll values by Lagrangc int,crpolat,ion [i5J. A SOUl'CC point is locntcd nt a little distnnce from thc boundll.ry on the outside llornll\llillC pllssil1g through etlch nodc. Applyillg II unit force and

1\

unit normalmomcnt, at eaeh SOUl'CC point

respectively ruul implementing t,hc Maxwell-Betti theorem provide the two • integration equations:

where A ruld s dcnote the plate tlrcn and thc bouml!try coordinlltc, w is deflection in d011lain,

W IUld

e are boundary deflection ruul nonnal rot,ation,

M,V IUld C are boundary moment, equivalent shcllr Ilnd comcr force, f IUld m are superscripts identifying thc fundruncntal solutions for thc unit force IUld the unit moment respectively,

Ne is the number oÍ corners. All four boundary functions W,

e, M and V are unkl1owl1s on the interface

bctween the trlUlsition strip and the ncighboring boulldary clemcllt, but among them only two are unknown on the othcr bouIIdaries aftcr imposing displacement IUld force boundary conditions. At the ends of interface, doublc nodes arc used [76] (Fig.5.6). Both llodcB A

125

CHAPTER 5. COMBINED ANALYSIS

"illi B have the samc coordinatcs, but may havc diffcrcnt boundary conditiOBS. lu ordcr to makc thc numbcr of unknowns at nodcs A and B match the uumber of thc boundary intcgration equations, the corner forces at these two nodcs lIlay bc exprcsscd in tcrms of the normnl l'otations of their respectivc clclIlcnt in thc form:

8210 N, 8L(1) CA = D(l - v)(-)(1) = D(l - v) L: _ 3 ej 8s8n j=1 8s 8210 . N, 8L(2) CB = -D(l- 11)(_)(2) = -D(l- v) L: - j ej 8s8n j=1 as

(5.7)

(5.8)

whcJ'C (l) and (2) arc thc superscripts idcntifying thc elel!lents cnding at node A and starting at nodc B respectivclYi N, IIlld N2 arc the nl1mbel's of nodes on bOl1ndary elcments l and 2 respectivelYi Lj

is the LagraJlge shape function of node j aud is expressed as: N

Lj(S)

= TI

i = l

S -

Si

Sj - Sj

(5.9)

i=h Substitl1ting the cxpressions for fundnmental solutioIlS into the boundary integmi cquations and implementing the boundary elcment discretization tcchniquc [75,76] produce the following boundary elcment matrices:

(5.10) wherc U dcnotes the values of deflection and normal rotation at ali the nodes,

CHAPTER 5. COMBINED .4.NA.LYSIS

126

P represcnt,s t.he values of equivalcnt shelU' IUld nonllal 1ll0lllent nt nl\ t.he nodcs, B is thc effect of thc loads in domuin, 1 is the superscript identifying the boundnxy clcmcnt region, I is the subscl"ipt defining the intcrfacc bctwccn thc trallsit,ion st,rip ami tht' boundury cIcments. For cach nodc on thc intcrfacc thcre arc 4 unknowns (W,e,V und M) versus 2 cquations. Thcrcforc, two more cquutions IU'C rcquircd fOl' each of thes," nodes.

5.3.2 TRANSITION STRIP AND COMBINED SOLUTION A transition strip is inscrtcd bctwccn thc finitc strip rcgion IUld t.he bounc.lIl.l'y clcmcnt region and connccts thc two rcgions togethcr. Onc side of U1I! trll.llsition strip coincidcs with thc nodal linc of the Ildjnccnt strip and IUL, thc same dcgrees of freedom

HS

this nodal line. On the 0pJlosite side t.her"

at·c a number of nodcs which Il.l'e attachcd to the nodes of thc neighboring boundll.l'Y elements and havc the same displllccmcnt pnrmneters (dcll,dioll W Il.lld normal rotatiOll e) as they do (Fig.5.7). Inside the tmnsition sl,rip the deflection tv is expresscd in tenns of thc dcgrccs of frcedom of I,he lIocht! line Il.lld the nodes by their corresponding shapc functions in the forlII: N

r

tv =

L m=l

(fl

({)tvl m

+ bg l ({)8 Im )Ym(Y) + L LiCh({)tvj +ag2(~)8j) ;=1

whcre N is the number of nodes of this strip along thc intcrfacc,

(5.11)

CHAPTER 5. COMBINED ANALYSIS

127

Lj is lh" Lagrange shnpe function of node j.

It shoul<1 he noted that the nodnl line and nodes 1 to N have the same type of shnpe functions in the direction x, which are Hermitian cubic polynominls, in order to simulate certain basic deformation patterns more accurately as notcd earlier. Using the Jlrinciple of minimum totnl potentinI energy, the stiffness matrix and the load vector of the transition strip are generated. Assembling the stiffness matrix and load vector of the transition strip with those of the finite strips and applying condensation technique [6] effect a tl'lUlsfommtion of the finite strips and transition strip into a substructure having only the degrees of freedom of the nodes on the interface. The matrix equation of this substmcture is of the form:

(5.12) where

FI is the vec tor of unknown interaction nodnl forces, II repl'esents the effects of externnI load, 2 is the superscript identifying the finite strip and transitiori strip region. At nil nodes on the interface, the displacement on both sides must be the SlUlle, lUul the interaction force must be the same in magnitude and opposite in dircction.These requirelllents give the following relationships

{Un = {UJ}

(5.13)

{Fl} = -[M]{PJ}

(5.14)

CHAPTER

5.

COMBINED ANALYSIS

128

where [M] is the transformation matrix frolll thc distributcd boundltry fOI'ces on the interface boundary elemcnt to the nodnl forct)s nt t.he ntl
Mij =

1

Lj(y)Lj(Y) ely

Substituting Equations (5.13) and (5.14) into (5.12) givcs It set of equations:

(5.15) Combining matrices (5.10) and (5.15) t.oget.her will yield a sct, of cquations that are just sufficient in number to salve fol' ali the unknownH in the houndary element region, including the unlmowns nt the nodes on t,he interface; these laUer can be substitutcd Imck int.o the nlat,rix of t,he finitc strip and transition strip region, and ali the unknown nochtl pal'l\.mctcrs in this rcgioll can then be rendily obtnincd. Thcrcafter it. is compute the displacelJlent

5.3.3

lUHI

It

straightforwlLrd proccdurc to

stress cOlllponcnts at rUly points of intcrest..

NUMERICAL EXAMPLES

1. Simply supported square plate un,lcr uniform load This example is chosen to verify the prescnt lllethod. A square plate is hinged along four edges and subjected to uniform load. Thc length of eaeh side is L, Poisson's ratio is v = 0.3, Flexural rigidity is D, aud the iutensity of the load is q. Three dift'ercnt mcshes are used to analyze the plate (Fig.5.8). The upper part of each mesh is dividecI iuto finite strips wi th equal width O.lL. Five symmetrical terms are used in the series of longitudinal shape functions.

129

CIlAPTER 8. COMBINED ANALYSIS 1VJ/ltJx

4

M!!sh

= nqL /D

m-3 m=5 m=7 Theory

(Mr )",ar = fiqU

(>

f3

0.004003 0.004002 0.004063 0.00406

0.04804 0.04790 0.04789 0.0479

R nqL 2 Ó n 0.4205 0.0618 0.4205 0.0646 0.4205 0.0654 0.420 0.065

Vmar =óqL

TabI!! 5.2: DeHection and Strcsscs in Simply Supported Square Plate Thc boundnxy of the lower part is simulatcd by 4 boundary clements with equal nodal spacing O.lL. Both parts arc connected by a transition strip with IL

widl·h of O.lL. The llUmcrical rcsults are Hsted in Table 5.2. In this table,

1/) ..

or and (Mr)mor arc the deHection and bending mOIT'.ent at the plate center

rcspectivcly, Y..ar is the cquivalcnt shcax force at the center of the bottom cdge, and R is the corner force nt the lower corners. In comparisoD with plate thcory [59], it can bc seen that the accuracy of the present method is satisfactory. 2. Simply supported square plate with an opening and a skew corner A square platc is simply supported along alI four sides and subjected to unifoI111 load q = 10 1\li 1m2 • The length of each side is 10.0 m, and the thickncss 0.2

Ill.

Thc material properties are E = 25000 MPa and

II

= 0.15.

One corncr is cuI, off at 45 degrees, and a square opening is located near IUlothcr corner,

IlS

shown in Fig. 5.9.

The part inc! uding the opcning and the skew corner is analyzed by 9 boundlU'y clc111cnts with 50 nodes whilst the rest of the plate is analyzed by 5 finite strips IUld onc trallsition strip with 10 series terms.

CHAPTER 5. COMBINED AN.4.LYSIS

130

The resllltin b bcnding momcnts Mx IUlcI My along line A-B-C IU'C depictl'd in Fig. 5.10. The platc \Vas also IUlalyzed by the bOIlIl(hu'y clemen!. Illcthod alone, with II clements !md 58 nodcs. The results of this analysis if plotlt'd on Fig. 5.10 would be indistingllishable from thosc already shnwn.

131

ClI.J.PTER 5, COlviDINED ANALYSIS

I

J

Y b K 1

I

a y'

L

X

F,iglU'C 5.1: RcctulIgulu.r Finitc Elclllcnt

nodalline 1

.------'i"~=.::.=-r--Y

b

node K

a

-_o Y'

node L

.1-1

x

Figlll'C 5.2: Tt'ullsitioll Elcmcnt

CHAPTER 5, COMBINED ANAL1"SIS

132

'A

C

E

D

F

B

Figul'e 5,3: Squal'o Plato

90M

B

A

C

"___ J _.-r

---- ..

_J _________________

-t

I

FINITE - : STRIPS- :,

I

I

I

I

I

I

I

y

,, ,, , ,,, , ,,

I

::il

o

<ó

TRANSITION -: ELEMENTS -:, ,. , ,

,

"

, ,

,

, "

',I,',r,'I','I',I,', ' I I I I I

,

J

I

III I I I I

i

i

i

A

B

C

FINITE ELEMENTS I I ILL II

;:Il

o

..;

FigUl'e .5.4: Plate Supported by Walls and Colul1Ins

133

CIIAPTER 5. COMBINED ANAL1'SIS

o o,-______________ t"l

,

~

______________________________________,

Mx ALONG C-C .................. o,,

o o

,,'

o

tN

.....

0,°

I

cll

OPENINfi

o

.........o

~

..... 0

~

I

....

I

/

o~~____________________________________~.-____~~,'--.~____~~__~~~~r~

Z ci \

"

Ö " ~ ~ \ ~Z I

"

···;1 .... /

\\

O~ ~I

\

' \

.

6, Z

~ or::l • ...:l

"

CII~

W ALONG C-C \

~~

/

\

\

I

"

o

,

,"

'

?o

/

,"

I

'..

O•

r::! ~

" I

\

Zo ~

o

.....

"

\\

S?

:::s

," \

0~ ZO

o

~O

I

\\

~.

o"

,

\

~o

../.

.......

'---~'

,,'"

ci

o

r~------r-----~------,------,-------r------r-----~------,-------t~ 0.0

1.0

2.0

3.0

4.0

x

....

5.0

6.0

7.0

Figul'c 5.5: DcHcctiou null Dcnding Momcnts of Plntc in Fig.5.4

8.0

9.0

CHAPTER 5. COlI'IBINED ANALYSIS

13'1

A

B·

ELEMENT (1)

•

•

•

•

Figure 5.6:' Double Nodes

Z,W

L

ee.

nodalline

y,v

wJ node j X,U

Figure 5.7: TransitiolI Strip

2

1

CIIAPTER 5, COMBINED ANALYSIS

135

r------------------------------·--------~I I I I

FINITE STRIPS

::::::::::::'TRAi.isIrIóN' STRÍP' ::::::::::::::: ••

o

••••

o

••••••••••••••••••••••••••

~

El

BOUNDARY ELEMENTS

~ ....

.

o

~

---

---

-

__ •

___ 0 __ -

______

O.lLxlO

---,---- ---

Figure 5.S: Simply Supported Square Plate uudel' Uuiform Load

CHAPTER 5. COj)'IDINED .4.NALl"SIS

13ll

10.0

I'

'I

•r---------------------------------------

y

FINITESTRIPS

o

có

:::::::::::: 'sriúp '::::::::::::::: ..... TR,Ü~siTioN ............................ "

1"8-. EL~MEN~ BOUNDARY

o

ru o

...;

A

,,,

-

o

C

... . .. -------------- ... ... ,

ru _L

~

1.Q". 2.0 ..J.,

I-

2.0

'I

x

Figw'c 5.9: Platc with Opcllillg RlldSkcw Corucr

137

ClIrll'TER ii. CO.ulJINED ANAL1'SIS

--~q .."..

\

::?l 0I~

Z

~

.

'-'-'-

-o

,--------,'

Z

W

.."..

0 _ ""'o 0

,,,.,.'"

~

Z

OPENING

O

~ oZ ...

./

~ I~-" 0.0

A

",-~==

, ,, ,

::?lN

~

'---'"",

My _____."'-.. --'~~

cn 0E-ot?

-

Mx

, 1.0

2.0

, ,,, , ,,, , ,, 3.0

4.0

5.0

6.0

,

,

r

7.0

8.0

9.0

(M)

Figul'C li.10: Dcutliug l\Iolllcuts aloug A-D-C

B

-,

C

10.0

Chapter 6

CONCLUSIONS AND RECOMMENDATIONS 6.1

CONCLUSIONS

In the present study, the objcctives pl'oposecl carliCl' havc becn accomplishcd. The main achievemcnts arc listed as follows: (1) The semi-analytical finite strip mcthocI has bcen extendecl to the analysis of continuous haunchecl slab-on-girclcr and box-girder bridges. In this analysis, three types of finite strip arc c1evclopccl, which being top flangc plate strip, vertical web strip and bot tom Hange shell strip. For cl\ch type of the above strip, appropriate curvilincar coordinatc system is choscll, stmindisplacement relationships and displaccmellt functions are derived. This method gives high efficiency for distributed loading, improved Ilccuracy for the longitudinal stresses over intermediate supports. The results obtailled are in reasonable agreement with those from finite element analysis or experimental analysis. (2) The spline finite strip method ha been extended to the analysis of con138

CIfAPTER

a.

CONCLUSIONS AND RECOMMENDATIONS

139

tinuous haunched slab-on-gil'del' and box-girder bridges. In this analysis, three types of strip are developed, which being top Hange plate strip, verticlu web atrip and bot tom Hangc shell strip. For alI of the above strips, Cartesian coordinate system is chosen and some coordinate transformations arc performed, strain-displacement relationships and displacement functions are derived. This method is more Hexible than semi-analytical finite strip method in dealing with point loads, continuity and vnriety of boundary condi tions, Iwd shows higher efficiency in most llJ.'.d cases. Satisfactory results are obtained in numericaI examples. (3) The finite strip method.has been extended to nonlinear analysis of cablestayed bridges. By using the finite strip method for continuous structures combined with t.he flexibilityapproach, the girder is transformed into a substructure with ti.o.f at cable attachment points only, thus, only very few unknowns are involved in nonlinear iterations. In the analysis, cables are calculated according to theory of catenary, therefore, the nonlinearities due to sag Dud angle change of cables are taken into consideration accurately. Furthermore, initial stiffness iteration is employed for nonlinear solution. Consequently, the present rnethod is more accurate than those based on simpIer thcory, and more efficient than finite element method. (4) The efficiency of geometrically nOlllinear finite strip analysis of plates is improvcd. In this analysis, modified Newton-Raphsoll method is used, Le. the initial stiffness matrix is forrned and inverted only once during alI the iteratiolls. Besides, the different series terms remain uncoupled if they are

CHAPTER 6. CONCLUSIONS .4.ND RECOMMENDATIONS

140

not coupled in linem' mmlysis. This enlUUlces the eHicicney of nonlinelu' finitc strip mtalysis significMtly. In order to avoid possible divergence, the flmmral stiffness is Mlplified by a factOI', which depends on the current mll.'dmum displacement Md hl\S bccn worked out by computational expel'imcnts. Numcrical cxamplc shows that this mcthod does not need too mnJly itemtions even for very large dcflection, mICI its convergency is dependablc. (5) The accuracy of materiaIly nonlinear fini te strip lumlysis of rcinfol'ced concrete slabs is improvcd. The concrctc is treated

l\S

a incremcntnlly or-

thotropic linear ell\Stic materinl. By introducing thc concept of cquivalent unirucial stra.in, the actual biaxial strcss stat" is trnnsfonncd into two in: dependent uniaxial stress statcs, in each of which the c1l1stic modulus lUld . principal stress depend on the current cquivalcnt unill.'(in] stmin and can be determined by diagram mICI strength cnvclope bascd on experiments. Fol' reinforcement, bilinear elasto-plastic material model is used, Md the tensilc stiffening effect is included. In lUlnlysis, the finite strip is dividcd into concrete layers Md each layer of reinforccmcnt is rcplaced by a smcarcd stccl layer. Newton-Raphson method is employed for nonlinear solution in order to reduce the number of itcrations required Md to ensure converging to correct Mswer. NumericaI example shows the satisfnctory accuracy of this analysis. (6) The finite strip method is combined with finite clcmcnt method in order to analyze a rectangu.lar plate with some irregularities, such as opcnings Md ch ange in boundary conditions. The plate may be divided into two

CHAPTER G. CONCLUSIONS AND RECOMMENDATIONS

141

pnrts, nlllnely a regular part, which is analyzed by finite f trip method, and n irregular part, which is analyzed by finite elcment method. A type of tmnsition clement is developed in order to connect both parts together. In the present study, conforming rectangular plate elements with 4 d.o.f at ench node arc chosen, and displacement funCtions of the transition element arc derived. The combined analysis shows a better efficiency and the same accuracy in comparison with conventional fiilite elcment analysis. (7) The finite strip method is combined with boundary element method in order to analyze a rectangular plate with some irl'egularities. The plate may bc dividcd into two parts, namely a regular part simulated by finite strips and an irregular part analyzed by boundary elcment mdhod. A type of transition strip is devcloped in order to connect both parts together. The displaccment functions are derived in this thesis. This combined analysis gives satisfactory

6.2

l'esu!t~

in al! the numericaI examples.

RECOMMENDATIONS

The following topics might be recommended for future work: (1) Hil!;her order displacement functions in the transverse direction may be cmploycd for ali types of finite strip developed in this thesis in order to im pl'OVC nccuracy and efficiency further, especially for materially nonlinear IUlnlysis, in which the accuracy of stresses is very importantj (2) Ali the lIlethods introduced in the present study may be applied to skew, curved or arbitrary shaped bridgesj

(3) The spline finitc stI;P method may bc cxtcndcd t,o lIlutcriully nunlincul' analysis of al'bitrnry shaped rcinforced concrctc slabs; (4) Some ncwly dcveloped methods may be upplicd to rumlysis of othcr t,ypcs of structurc, such as roofs, floors of buildings ctc ..

142

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i45

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