Stability of rock structures: proceedings of the 5th International Conference on Analysis of Discontinuous Deformation, ICADD-5, Ben-Gurion University of the Negev, Beer Sheva, Israel, 6-10 October 2002

STABILITY OF ROCK STRUCTURES

Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands

PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ANALYSIS OF DISCONTINUOUS DEFORMATION, ICADD-5/BEN-GURION UNIVERSITY OF THE NEGEV/BEER SHEVA/ISRAEL/6–10 OCTOBER 2002

Stability of Rock Structures

Edited by Yossef H. Hatzor Ben-Gurion University of the Negev, Beer Sheva, Israel

A.A. BALKEMA PUBLISHERS

LISSE/ABINGDON/EXTON (PA)/TOKYO

Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands

Cover: Upper terrace of King Herod’s Palace, Masada/photo by Yael Ilan Curtesy of Israel Nature and Parks Authority Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure the integrity and quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: A.A. Balkema, a member of Swets & Zeitlinger Publishers www.balkema.nl and www.szp.swets.nl

ISBN 90 5809 519 3

Printed in The Netherlands

Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands

Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Table of contents

Preface

VII

Organisation

IX

Sponsorship

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Keynote lecture Single and multiple block limit equilibrium of key block method and discontinuous deformation analysis Gen-Hua Shi

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Plenary lectures Realistic dynamic analysis of jointed rock slopes using DDA Y.H. Hatzor, A.A. Arzi & M. Tsesarsky Numerical models for coupled thermo-hydro-mechanical processes in fractured rocks-continuum and discrete approaches L. Jing, K.-B. Min & O. Stephansson

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Grand challenge of discontinuous deformation analysis A. Munjiza & J.P. Latham

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High-order manifold method with simplex integration M. Lu

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Case studies in rock slope and underground openings in discontinuous rock Experimental investigations into floor bearing strength of jointed and layered rock mass D. Kumar & S.K. Das

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Stability analysis for rock blocks in Three Gorges Project W. Aiqing & H. Zhengjia

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Some approaches on the prediction of hillsides stability in karstic massif E. Rocamora Alvarez

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Analysis of displacement and stress around a tunnel S. Chen, Y.-N. Oh, D.-S. Jeng & L.-K. Chien

107

Analysis, response, prediction and monitoring of existing rock and stone monuments A parametric study using discontinuous deformation analysis to model wave-induced seabed response Y.-N. Oh, D.-S. Jeng, S. Chen & L.-K. Chien

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113

Simulations of underground structures subjected to dynamic loading using the distinct element method J.P. Morris, L.A. Glenn, F.E. Heuzé & S.C. Blair

121

Numerical analysis of Gjøvik olympic cavern: a comparison of continuous and discontinuous results by using Phase2 and DDA T. Scheldt, M. Lu & A. Myrvang

125

Earthquake site response on hard rock – empirical study Y. Zaslavsky, A. Shapira & A.A. Arzi

133

Numerical simulation of shear sliding effects at the connecting interface of two megalithic column drums N.L. Ninis, A.K. Kakaliagos, H. Mouzakis & P. Carydis

145

On determining appropriate parameters of mechanical strength for numeric simulation of building stones N.L. Ninis & S.K. Kourkoulis

153

Validation of theoretical models Experimental validation of combined FEM/DEM simulation of R.C. beams under impact induced failure T. Bangash & A. Munjiza

165

A study of wedge stability using physical models, block theory and three-dimensional discontinuous deformation analysis M.R. Yeung, N. Sun & Q.H. Jiang

171

Shaking table tests of coarse granular materials with discontinuous analysis T. Ishikawa, E. Sekine & Y. Ohnishi

181

Pre-failure damage, time-dependent creep and strength variations of a brittle granite O. Katz & Z. Reches

189

Dynamic block displacement prediction – validation of DDA using analytical solutions and shaking table experiments M. Tsesarsky, Y.H. Hatzor & N. Sitar

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Theoretical developments in modelling discontinuous deformation Crack propagation modelling by numerical manifold method S. Wang & M. Lu

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Continuum models with microstructure for discontinuous rock mass J. Sulem, V. de Gennaro & M. Cerrolaza

215

Development of a three-dimensional discontinuous deformation analysis technique and its application to toppling failure H.I. Jang & C.I. Lee Three-dimensional discontinuity network analysis (TDNA) on rock mass X.-C. Peng & H.-B. Tang

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Preface

Rock is typically discontinuous due to the presence of joints, faults, shears, bedding planes, and foliation planes. Stability analysis of structures in rocks must therefore address deformation along block boundaries and interaction between rock blocks, which may be impossible to address properly by imposing a continuum framework model. Over the past decade we have seen significant progress in development of methods of analysis for discontinuous media. Of note in particular are the DEM (distinct element method), DDA (discontinuous deformation analysis), block theory, combined FEM/DEM, and the new manifold method. Four international meetings on analysis of discontinuous deformation have been held, in Taipei in 1995, Kyoto in 1997, Vail in 1999, and Glasgow in 2001. The objective of this ICADD-5 is to focus on the application potential of discontinuous analysis methods to the stability evaluation of structures in rock, including both modern engineered rock slopes and underground openings, as well as ancient monuments in fractured rock. Nevertheless, this proceedings volume also contains original, high quality theoretical papers which explore issues such as fracture mechanics modeling in the new manifold method, continuum models with microstructure for discontinuous rock mass, coupled thermo-hydromechanical processes in fractured rocks, and recent developments in three dimensional DDA. It is believed that the collection of papers in this volume demonstrates the directions in which theoretical developments in analysis of discontinuous deformation should proceed, the validity and limitations of existing codes, and the range of engineering problems to which discontinuous analysis can be applied. Yossef H. Hatzor Chair, ICADD-5 Organizing Committee President, Israel Rock Mechanics Association

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Organisation

INTERNATIONAL ADVISORY PANEL Bernard Amadei (USA) Nenad Bicanic (UK) Yossef H. Hatzor (ISRAEL) Ante Munjiza (UK) Yuzo Ohnishi (JAPAN) Friedrich Scheele (SOUTH AFRICA) Gen-Hua Shi (USA) Nicholas Sitar (USA) Chung-Yue Wang (TAIWAN, ROC) Aiqing Wu (CHINA) Man-chu Ronald Yeung (HONG KONG, CHINA)

ORGANISING COMMITTEE Yossef H. Hatzor Chairman John P. Tinucci Website and peer-review site manager Michael Tsesarsky Secretary Avner A. Arzi, Chaim Benjamini, Tsvika Tzuk Members

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Sponsorship

The conference is sponspored by The President of Ben-Gurion University The Rector of Ben-Gurion University The Kreitman Foundation Fellowships – Ben-Gurion University The Faculty of Natural Sciences – Ben-Gurion University

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Keynote lecture

Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands

Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Single and multiple block limit equilibrium of key block method and discontinuous deformation analysis Gen-hua Shi Belmont, CA, USA

ABSTRACT: Limit equilibrium is still the basic method to do the stability analysis including slope stability, tunnel stability and dam foundation stability. Key blocks are the blocks very possible to reach limit equilibrium first. Newmark key block method is a dynamic limit equilibrium method. The latest version of delimiting key blocks on the unrolled tunnel joint trace map is also used. Two and three dimensional dynamic DDA is used to compute multiple block multiple step dynamic limit equilibrium. The recorded seismic loads are input. Block system statics with large displacements is the results of stabilized dynamic DDA method. Statics is infinite long time and stabilized dynamics therefore is much more difficult. In the following, key block limit equilibrium and limit equilibrium related to the multiple block and time depending multiple step DDA computations are studied: one step static computation, non-stabilized dynamics computation, and stabilized dynamics computation.

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LIMIT EQUILIBRIUM ANALYSIS OF BLOCKS

For small displacement and continuous computation, the normal computation solve the equation only once. It is one time step computation. Its assumption is the single time step is very large. Due to the long time and the small displacements in this single step, the velocity and the acceleration are nearly null. In this case, the inertia is neglected. Most traditional limit equilibrium, key block and structure mechanics computations belong to this case. However this kind of simple statics computation cannot compute the large displacements, large deformation and discontinuous cases. It even has substantial difficulties for the majority of material non-linear computation. For the large displacement, large deformation or discontinuous cases, both statics and dynamics use time steps. Statics is infinite long time and stabilized dynamics. Therefore the general statics is much more difficult than dynamics. Based upon the two or three dimensional block kinematics, the loading conditions and the friction law, limit equilibrium analyses were performed. The following results are required at the end of each time step from the analyses: 1. 2. 3. 4. 5. 6. 7. 8.

Normal forces perpendicular to the sliding interface Resisting force from the shear strength at sliding interface Sliding forces along with the sliding interface Dynamic or static equilibrium Satisfying friction law at all interfaces Inertia forces for multiple step dynamic computation Convergence of contact forces for multiple step static computation Modes of failure including sliding surfaces and sliding direction

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LIMIT EQUILIBRIUM OF SLOPE KEY BLOCKS

In the following case, the slope has double free planes. Four joint sets and two slope planes are listed in Table 2.1. Figure 2.1 and 2.2 are the upper hemisphere stereographic projection of dominant joint sets A, B, C and D. The two dashed circle are the side slope and top slope. From Figure 2.1 and 2.2, the region entirely included in the union of two dashed circles (projections of two slope planes) are regions 0000, 0010, 0110 and 0100. Also, regions 0020, 0200, 0120 and 0220 are in the same union of two dashed circles. In the field, only one key block with JP code 0220 has been found. Each joint and slope plane is located by one point: • • • • •

the key blocks of JP code 0220 is on the upper side of joint set A on the upper side of joint set D on the lower side of slope side plane on the lower side of slope top plane

In order to compute the block shape and block volume, one point is chosen in each face of this specific block. There are four faces in this block. In the following table, the intersection point of first three faces can be chosen as the coordinate system original (0,0,0). A point (−2.30, 6.77, 14.00) is chosen in the fourth face (Table 2.2). Table 2.1.

Figure 2.1.

Input data.

Joint set

Dip angle

Dip direction

Friction angle

Joint set A Joint set B Joint set C Joint set D Side slope Top slope

73◦ 58◦ 70◦ 32◦ 67◦ 10◦

108◦ 20◦ 219◦ 225◦ 195◦ 195◦

33◦ 25◦ 25◦ 33◦

Key blocks and sliding forces.

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Figure 2.3 shows the block shape. From Figure 2.1, 2.2 and 2.3 the following results can be read from Table 2.3. This block is stable under gravity load. This slope is stable under gravity load. Figure 2.4 is the upper hemisphere equal angle projection of the joint sets and slope planes. The following section 3 and 4 are for earth quake stability analysis of this block 0220. 3

STEREOGRAPHIC PROJECTION SOLUTIONS OF LIMIT EQUILIBRIUM WITH EARTH QUAKE LOADS

P. Londe stereographic projection solution is well known for dam foundation and dam abutment stability analysis. This method is suitable for earth quake loads. As the limitation of this method, cohesion c = 0 is assumed. The input data are given in Tables 3.1 and 3.2.

Figure 2.2.

Sliding planes and factor of safety. Table 2.2.

X Y Z of a point on each face of the block.

Coordinates

X (feet)

Y (feet)

Z (feet)

Joint set A Joint set D Slope Top slope

0 0 0 −2.30

0 0 0 6.77

0 0 0 14.00

Table 2.3.

Block stability.

Block JP code Block volume Sliding force per unit weight Factor of safety (under gravity only)

0220 2742 cubic feet −0.28 1.61

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Figure 2.3.

Key block JP = 0220.

Figure 2.4.

Equal angle projection of joint sets and slopes. Table 3.1.

Joint data.

Joint set

Dip angle

Dip direction

Joint set A Joint set D

73◦ 32◦

108◦ 225◦

Table 3.2.

Stereographic projection.

Contour step 5◦ Stereographic projection lower hemisphere Resultants weight and earth quake accelerations

P. Londe pictures are contours of friction angles. The contour value of the projection point of a resultant force is the required friction angle for limit equilibrium. This friction angle corresponds factor of safety 1.0. Figure 3.1 shows the required friction angles of different time intervals from the whole earth quake time history loads.

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Figure 3.1.

Friction angle contours of key block limit equilibrium.

Results Figure

Time interval

Stable friction angle

Figure 3.1 Figure 3.1 Figure 3.1 Figure 3.1 Figure 3.1 Figure 3.1

0–50 seconds 0–10 seconds 10–20 seconds 20–30 seconds 30–40 seconds 40–50 seconds

70◦ 53◦ 70◦ 52◦ 43◦ 23◦

The Figures show 70 degrees of friction angle makes the block stable under the given time depending earth quake loads. The solid circle in the center of the picture represents the loads corresponding all possible 0.6 g earth quake loads. It can be found from the Figures, 55 degrees of friction angle is stable for 0.6 g earth quake. 4

NEWMARK DISPLACEMENT SOLUTION OF LIMIT EQUILIBRIUM WITH EARTH QUAKE LOADS

Since the block 0220 has two sliding joint faces of dominant joint sets A and D respectively, double face sliding Newmark displacement algorithm is needed. A C-language code including PostScript graphics is written for

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this special purpose. Due to the variant earth quake load, the changing of sliding planes and sliding modes is considered. The results and the formulations are consistent with the previous P. Londe graphic solutions. The input data of Newmark approach is simply the joint information, the block geometry and the earth quake acceleration data are given in Table 4.1 and 4.2. The earth quake data are given by Figure 4.1 to 4.2. The curves of Figure 4.3 gives the accumulated displacements and the instant velocity during the 50 second earth quake process. This computation is single block three dimensional dynamic limit equilibrium analysis without rotations. The final displacement is 2.2 feet under 33 degrees friction angle and 0 cohesion. The all-time maximum sliding velocity is 2.10 feet per second. The cohesion is sensitive relative to the results. If cohesion 4.0 psi or 0.576 kips/ft2 is applied to the joint set D, using 25◦ friction angle for both sliding joint sets A and D, the cohesion of joint set A remain to zero, there are no permanent displacement in the entire 50 second earth quake process.

Table 4.1.

Joint information.

Joint set

Dip angle

Dip direction

Friction angle

Joint set A Joint set D

73◦ 32◦

108◦ 225◦

33◦ 33◦

Table 4.2.

Block geometry.

Block volume Sliding force per unit weight Rock unit weight Area of side slope face Maximum distance to side slope Sliding force on side slope Area of top slope face Maximum distance to top slope Sliding force on side slope

2742 cubic feet 0.5528 0.15 kips per cubic feet 293 square feet 28.1 feet 0.7754 kips per square feet 645 square feet 12.8 feet 0.3474 kips per square feet

Table 4.3. Time depending accelerations.

Figure

Time interval seconds

Direction g

Direction g

Figure 4.1 Figure 4.1 Figure 4.2 Figure 4.2

0–25 25–50 0–25 25–50

X X Z Z

Y Y resultant resultant

Table 4.4.

Results.

Figure

Time interval seconds

Displacement Maximum velocity feet feet/second

Figure 4.3 Figure 4.3 Figure 4.3 Figure 4.3 Figure 4.3

0–10 10–20 20–30 30–40 40–50

0.5444 1.7895 2.2131 2.2369 2.2369

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2.10 1.15 1.75 0.22 0.00

Figure 4.1.

X and Y components of earth quake forces.

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Figure 4.2.

Z component and resultant of earth quake forces.

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Figure 4.3.

Displacements and velocity of the key block.

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LIMIT EQUILIBRIUM OF TUNNEL KEY BLOCKS

The joint sets are given in Table 5.1. Figure 5.1 is the upper hemisphere equal angle stereographic projection of the joint sets. Table 5.1.

Figure 5.1.

Joint set data of joint system.

Joint set

Dip angle

Dip d.

Friction angle

Cohesion

Joint set 1 Joint set 2 Joint set 3

79◦ 81◦ 5◦

270◦ 230◦ 45◦

39◦ 39◦ 39◦

0 ton/m2 0 ton/m2 0 ton/m2

Equal angle stereographic projection of joint sets.

Figure 5.2. Total key block sliding force for all tunnel directions.

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The equal angle stereographic projection has the following three advantages: 1. The projection of any plane is a arc or a segment of a circle. 2. The intersection angle between two projection arcs is the true intersection angle between the two joint set planes. 3. This simple diagram shows all of the angular relationship between joint sets. The volume and sliding forces of key blocks are sensitive to the tunnel directions. The following is the study: If the given tunnel direction is safe in terms of the key block sliding force. This tunnel direction study only based on the tunnel directions and the joint set directions. The assumptions are: 1. The joints are very long. 2. The joint spacing are very small. 3. The key blocks in between two parallel joints are not considered. Figure 5.2 shows the contours of the sliding forces of maximum key blocks for all tunnel directions. The contours is equal area projection of the tunnel axis inside of the reference circle. In this case, the tunnel direction N 75◦ E is marked as a small circle in the drawing. The maximum sliding force of this direction is about 20% to 30% of the over all largest sliding force for all tunnel directions. Due to the two vertical joint sets, the tunnels with nearly 90 degrees rise angle or shafts have relatively large key block sliding force. 6

KEY BLOCK ZONES REACHES LIMIT EQUILIBRIUM FIRST

For a given tunnel direction, Figure 6.1 shows the zones of maximum key blocks for each joint pyramid (JP). The maximum key block zones are the projections of the maximum three dimensional key block on the tunnel

Figure 6.1.

Key block zones with JP codes and sliding forces.

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Figure 6.2.

Key block zones with sliding planes and factor of safety.

section plane. Also all the key blocks of the same joint pyramid (JP) are in the corresponding key block zone. The numbers under the JP codes are the sliding forces with unit g. The zone marked “011” means JP = 011, where “0” means upper side of the joint set and “1” means lower side of the joint set. All key blocks of JP = 011 are in the upper side of joint set 1, lower side of joint set 2 and lower side of joint set 3. Under the number “011” is “+0.83”. The sliding force of the JP = 011 key blocks is 0.83 times the key block weight. Figure 6.2 shows the sliding joint sets of the maximum key blocks. The zone marked “1” means all of the key blocks of this JP slide along joint set 1. The zone marked “13” means all of the key blocks of this JP slide along the intersection line of joint set 1 and joint set 3. The second number under the sliding joint set number is the factor of safety of all the key blocks of the corresponding JP. The factor of safety of JP = 011 key blocks is 0.16. If the factor of safety is greater than 9.99, 9.99 is printed. All of the computations of sliding forces and factor of safety are based on limit equilibrium. The key blocks are these blocks which very likely reach limit equilibrium first. As the joints have limited lengths and wide spacing, the key blocks can occupy only a part of the maximum key block zone. Most of the real key blocks will lie near the tunnel surface. Therefore, relatively smaller key block region can be considered in the stability analysis. From Figure 6.1 to 6.2, only key blocks of JP = 011 and JP = 101 can fall. The blocks of JP = 100 and JP = 010 are only removable. 7 THREE DIMENSIONAL VIEW OF MAXIMUM KEY BLOCKS Figure 7.1 shows the three dimensional view of the maximum key block JP = 011. Here the key block can be in between two parallel joints of joint set 1 or JP = 311 (Table 7.1). Figure 7.2 shows the three dimensional view of the maximum key block JP = 011. Here the key block can be in between two parallel joints of joint set 2 or JP = 031 (Table 7.2).

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Figure 7.1.

Maximum key block JP = 311.

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Figure 7.3 shows the three dimensional view of the maximum key block JP = 101. Here the key block can be in between two parallel joints of joint set 1 or JP = 301 (Table 7.3). Figure 7.4 shows the three dimensional view of the maximum key block JP = 101. Here the key block can be in between two parallel joints of joint set 2 or JP = 131 (Table 7.4). For the maximum three dimensional key blocks, the assumptions are: the joint length in each joint set is sufficiently large and the joint spacing in each joint set is sufficiently small. Under these extreme assumptions, the maximum key blocks are drawn. Any actual key block can not be larger than these maximum key blocks. The actual key blocks could be much smaller due to the limited length and substantial spacing.

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KEY BLOCK AREAS ON TUNNEL SURFACE JOINT TRACE MAP

The joint sets are given by Table 5.1. Based upon the joint statistics, the joint geometric parameters are given by the following: From the joint direction, joint spacing and joint length of Table 5.1 and Table 8.1, the joint traces on the curved tunnel surface are produced statistically. Here for a practical reason, the joint bridges are all assumed to be 0.1 m. It has been assumed here that the joints having traces in the tunnel surface extend sufficiently far behind the tunnel surface as to form blocks by their mutual intersections. It has been proved that if the joints do thus extend behind the tunnel surface, the three dimensional key blocks of the tunnel can be delimited by operating only with the joint traces exposed on the tunnel surface. Then using key block theory, the key block zones are delimited from the curved polygons of the unrolled joint trace map. Figure 8.1 is the diagram which shows the way the tunnel, including the joint sets, is unrolled. Figure 8.2 is the statistically produced unrolled joint trace map of the whole tunnel. Figure 8.3 is the key blocks on the unrolled joint trace map of the whole tunnel.

Table 7.1.

Maximum key block JP = 311.

Key block JP code Key block volume Area in tunnel surface

Table 7.2.

011 or 311 0.77 m3 2.16 m2

Maximum key block JP = 031.

Key block JP code Key block volume Area in tunnel surface

Table 7.3.

011 or 031 0.64 m3 1.84 m2

Maximum key block JP = 301.

Key block JP code Key block volume Area in tunnel surface

Table 7.4.

101 or 301 0.39 m3 1.47 m2

Maximum key block JP = 131.

Key block JP code Key block volume Area in tunnel surface

101 or 131 0.47 m3 1.68 m2

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Figure 7.2.

Maximum key block JP = 031.

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Figure 7.3.

Maximum key block JP = 301.

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Figure 7.4.

Maximum key block JP = 131.

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Table 8.1.

Statistical joint set data.

Joint set

Spacing (m)

Length (m)

Bridge (m)

Joint set 1 Joint set 2 Joint set 3

0.30 0.30 0.50

1.8 2.4 1.8

0.30 0.30 0.50

Table 8.2.

Key block zones.

JP code

Key block area (m2 )

Sliding joint set

311 031 301 131

9.81 7.70 11.19 5.88

1 1 2 2

Figure 8.1. Tunnel wall unroll.

Figure 8.4 is the three dimensional far side view of statistically produced joint traces. Figure 8.5 is the three dimensional far side view of key block. Figure 8.6 is the three dimensional near side view of statistically produced joint traces. Figure 8.7 is the three dimensional near side view of key block.

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Figure 8.2.

Joint trace unroll map.

Figure 8.3.

Key blocks on unroll map.

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Figure 8.4.

Far side view of joint traces.

Figure 8.5.

Far side view of key blocks.

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Figure 8.6.

Near side view of joint traces.

Figure 8.7.

Near side view of key blocks.

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DYNAMIC LIMIT EQUILIBRIUM OF DDA

The discontinuous deformation analysis (DDA) computes deformable block systems. In the current version, the block displacements are complete linear functions of the coordinates. Each block is assumed to have constant stresses and strains. In spite of the complex shape of DDA blocks, DDA method uses analytic integrations for all of its matrices. This analytic integration is simplex integration. The simplex integration can compute ordinary integrations without subdividing domains to simple elements. Using simplex integration, the integration of any polynomials can be represented by the coordinates of boundary vertices of generally shaped two or three dimensional blocks. DDAcomputation offers the movements, stresses and strains of each block. The computed block displacements are often large enough to be visible, the modes of failure and the final damage can be seen directly. On the other side, the DDA codes can perform traditional limit equilibrium analysis for whole block systems. When large deformation are involved, the static solution is the stabilized state from the dynamic solution due to friction or real damping. The current DDA program treats the damping in a simple manner: the dynamic computation inherits the full velocity at the end of the previous time step. The static computation inherits only a part of the velocity at the end of the previous time step as the initial velocity at the beginning of this time step. The DDA computation must satisfy the following conditions at the end of each time step: 1. Each degree of freedom of each block has an equilibrium equation. The simultaneous equilibrium equations are derived by minimizing the total potential energy at the end of each time step. All external forces acting on each block, including loads and contact forces with other blocks, reach equilibrium in all directions and reach moment equilibrium for all rotations. Equilibrium is also achieved between block stresses and external forces on the block. 2. Entrance theory is used to identify all possible first entrance positions. Contacts occur only on the first entrance position, interpenetrations are prevented on the first entrance positions and sliding is controlled by the friction law. 3. Within each time step, if the tensile force from the normal contact spring exceeds the limit, this normal spring will be removed. If interpenetration occurs in a entrance position, a normal spring is applied. The global equations have to be solved repeatedly while selecting the closed entrance positions. This procedure for adding or removing springs and solving equilibrium equations is referred to as an open-close iteration. The open-close iteration will continue until all tensile force and all interpenetrations are within set limits over all the entrances.

10 THE GEOMETRY AND MECHANICAL DATA OF DYNAMIC DDA COMPUTATION The joint sets are given by Table 5.1. Based upon the statistics, the joint geometric parameters are given by Table 8.1. Based on the geometric data of Table 5.1 and Table 8.1, DDA-DL program produces the joints and tunnel boundary lines. From the joint and tunnel boundary lines, DDA-DC program produces the block system. The block system is the geometric input of DDA-DF program. The mechanical parameters of both rock masses and joints are the following: Based on the mechanical data of Table 10.1, the program DDA-DF computes the time depending block movements and block stresses. The process of block falling can be shown. Table 10.1.

Mechanical data. 2.27 ton /m3 3000000 ton /m3 0.21 20000 0.0010 second 20 second 39◦ 0 ton /m3

Unit weight E of rock mass ν of rock mass Number of time steps Time step Earth quake duration Joint friction angle Cohesion

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For this computation, the earth quake acceleration data are from California Department of Transportation. The original data are 50 seconds, our computation only uses from 10 second to 30 second. However these 20 second data are the main part of the strong earth quake. In DDA computation, as an extension of Newmark method from one block to multiple blocks, the earth quake accelerations are applied as body forces. Figure 4.1 show X and Y components of the time depending earth quake acceleration data. Figure 4.2 also show Z components and the resultants of the time depending earth quake acceleration data.

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CASES OF ROCK FALLING DDA COMPUTATION WITH EARTH QUAKE LOAD

Figure 11.1 and 11.5 show the joint maps statistically produced on the tunnel section plane based upon the joint direction, joint length, joint spacing and joint bridge on Table 5.1 and Table 8.1. Figure 11.2 and 11.6 show the block meshes on the tunnel section plane produced from statistically produced joint maps Figure 11.1 and 11.5 respectively. Figure 11.3, 11.4, 11.7 and 11.8 show the rock falling of the meshes Figure 11.2 and 11.6. In the computation, the earth quake load is applied.

12

DYNAMIC THREE DIMENSIONAL DISCONTINUOUS DEFORMATION ANALYSES

The joint polygons and block systems are three-dimensional. Most of joints are not perpendicular to a given two dimensional cross section. Therefore, the two-dimensional computations of jointed rocks or block systems are of limited reliability and accuracy. The three-dimensional analyses of block systems are important. Threedimensional discontinuous deformation analysis (3-D DDA) forms blocks directly from general polygons. The blocks can be convex or concave. Also, the blocks can have any numbers of polygonal faces. The 3-D DDA program computes three-dimensional deformable block systems. In the current version, there are 12 degrees of freedom per block: displacements on X, Y, Z directions, rotations around axis X, Y, Z and six 3-D strains. The block displacements are complete linear functions of the coordinates. Each block is assumed to have constant stresses and strains. The discontinuous contacts between 3-D blocks are the main part of 3-D DDA algorithms. There are more ways for blocks to contact in three-dimensions compared with two-dimensional block contacts. For the friction law, the two-dimensional sliding directions form a line, while the three-dimensional sliding directions form a plane. The example of Figure 12.1 is a free falling block. g = 9.8 m/s2 , step time = 0.009035 second, total 50 steps. Theoretical falling distance is: 1 s = gt 2 = 1.0000 m. 2 Result of 3-D DDA is: s = 1.0000 m Figure 12.2 A block slides on one plane Friction angle of the sliding plane is 0. The sliding plane has 45◦ friction angle. g = 9.8 m/s2 Step time = 0.0080812 second, total 50 steps t = 100 × 0.0080812 second = 0.80812 second Vertical sliding distance is 1 sv = gt 2 = 1.6000 m 4 Result of 3-D DDA is: s = 1.5998 m

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Figure 11.1.

Statistically produced joints of case 1.

Figure 11.2.

Block mesh formed by joints of case 1.

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Figure 11.3a.

Case 1 rock falling after 0 steps.

Figure 11.3b.

Case 1 rock falling after 100 steps.

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Figure 11.4a.

Case 1 rock falling after 200 steps.

Figure 11.4b.

Case 1 rock falling after 2000 steps.

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Figure 11.5.

Statistically produced joints of case 2.

Figure 11.6.

Block mesh formed by joints of case 2.

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Figure 11.7a.

Case 2 rock falling after 0 steps.

Figure 11.7b.

Case 2 rock falling after 100 steps.

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Figure 11.8a.

Case 2 rock falling after 200 steps.

Figure 11.8b.

Case 2 rock falling after 2000 steps.

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Figure 12.1a.

Block free falling after 0 steps.

Figure 12.1b.

Block free falling after 50 steps.

Figure 12.3 A block slides on two planes The force along the sliding line is 1 mg cos (45◦ ) = √ mg = ma, 2 1 a = √ g, 2 where a is the acceleration along the sliding intersection line.

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Figure 12.2a. A block slides on one plane after 0 steps.

Figure 12.2b. A block slides on one plane after 200 steps.

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Figure 12.3a. A block slides on two planes after 0 steps.

Figure 12.3b. A block slides on two planes after 55 steps.

The sliding distance along the intersection line is: 1 1 s = at 2 = √ gt 2 . 2 2 2 Theoretical sliding distances sh and sv along horizontal and vertical directions are: 1 1 sv = sh = √ s = gt 2 = 0.4000 m. 4 2

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Figure 12.4a. Three block simultaneously sliding after 0 steps.

Figure 12.4b. Three block simultaneously sliding after 70 steps.

Results of 3-D DDA are: sh = 0.3998m, sv = 0.4007m. Figure 12.4 is the dynamic displacements of four blocks. In the process of three block sliding, the middle block separate the other two blocks.

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Figure 12.5a. Arch block displacements after 0 steps.

Figure 12.5b. Arch block displacements after 110 steps.

The sliding modes are the same with two dimensional results and the base friction machine test. Figure 12.5 is the dynamic displacements of seven arch blocks. It shows the three dimensional block mesh of the arch and the three dimensional block movements after 105 time steps. The movements keep symmetries in all directions and consistent with two dimensional DDA computations. Figure 12.6 is a 30000 cubic meter key block computation. From key block theory, this is a single face sliding. However, three dimensional DDA shows rotations.

13 THE STATIC STABILITY ANALYSIS OF DAM FOUNDATIONS Figure 13.1 to 13.4 are dam foundation cases. The mechanical parameters of rock masses are given in Table 13.1. Figure 13.1 and 13.2 are the dam foundation with resisting blocks. If the two major horizontal joints have friction angle 45◦ and 29◦ respectively, the dam is stable. This is statics case. The statics is computed by time depending dynamics. In the end of each time step, set the velocity to zero. In case the friction angles are 0◦ and 17◦ respectively, the dam will slide together with the resisting blocks as shown in Figure 13.2.

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Figure 12.6a.

One block complex sliding after 0 steps.

Figure 12.6b.

One block complex sliding after 700 steps.

Table 13.1.

Mechanical data.

Reduce step velocity E of rock mass Contact spring ν of rock mass Number of time steps Cohesion

1.00 432000 ton /m 3 9216000 ton /m 0.20 100 0 ton/m 3

Figure 13.1 The dam and rock foundation with resisting blocks. Figure 13.2 Dam, foundation and resisting blocks movement. Figure 13.3 and 13.4 are the dam foundation. If the two major horizontal joints have friction angle 32◦ , the dam is stable. This is statics case. The statics is computed by time depending dynamics. In the end of each time step, set the velocity to zero. In case major horizontal joints have friction angles 29◦ and 17◦ respectively, the dam will slide as shown in Figure 13.4.

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Figure 13.1.

Dam and foundation movement case 1 after 0 steps.

Figure 13.2.

Dam and foundation movement case 1 after 400 steps.

Figure 13.3 The dam and rock foundation block mesh. Figure 13.4 Dam and foundation movement.

14 THE GEOMETRY AND MECHANICAL DATA OF TUNNEL STATICS DDA COMPUTATION The rock block movement near the tunnel is basically controlled by existing joints. The joint sets of joint system and mechanical parameters of the tunnel given in Table 14.1. Based upon the statistics, the joint geometric parameters are given in Table 14.2. The geometry of the tunnels are given in Table 14.3. Based on the geometric data of Table 14.1, 14.2 and 14.3, program produces the joints and tunnel boundary lines.

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Figure 13.3.

Dam and foundation movement case 2 after 0 steps.

Figure 13.4.

Dam and foundation movement case 2 after 400 steps.

Table 14.1. Angle data of joint system. Joint set

Dip angle

Dip d.

Friction angle

Cohesion

Joint set 1 Joint set 2 Joint set 3

82◦ 82◦ 14◦

288◦ 229◦ 40◦

39◦ 39◦ 39◦

0 ton/m2 0 ton/m2 0 ton/m2

From the joints and tunnel boundary lines, program produces the block system. The block system is the geometric input of the mechanical analysis program. The total block number is 3784.

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Table 14.2.

Statistical length data of joint system.

Joint set

Spacing: m

Length: m

Bridge: m

Joint set 1 Joint set 2 Joint set 3

2.6 m 2.6 m 2.6 m

16.45 m 14.9 m 25.8 m

−1.0 m −1.0 m 0.4 m

Table 14.3. Tunnel data. 75◦ 0◦ 5.5 meter circular

bearing angle of tunnel axis rise angle of tunnel axis tunnel diameter tunnel shape

Figure 15.1.

Rock displacements after 2000 steps.

The mechanical parameters of both rock masses and joints are given in Table 14.4. Based on the mechanical data of Table 14.4, the program computes the time depending block movements and block stresses. The process of block falling can be shown.

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Table 14.4.

Mechanical data.

Reduce step velocity E of rock mass Contact spring ν of rock mass Number of time steps Time step Computation duration Joint friction angle Cohesion

Figure 15.2.

15

0.01 3000000 ton /m3 6000000 ton /m 0.21 20000 0.0010 second 2 second 39◦ 0 ton /m3

Rock block stresses after 2000 steps.

STATIC DDA TUNNEL COMPUTATION USING DYNAMICS WITH UNIT MASS DAMPING

Here in this following case, the dynamics with unit mass damping is used. The relative displacements are reduced near zero following the time steps. When the step time is 0.001 seconds, the next step uses 0.99 (normally 0.95– 0.99) of the velocity from the end of the previous time step. If this number is 0.97, the relative displacement reduces much faster. Figure 15.1 The block boundary after 2000 time steps (2.0 seconds). Figure 15.2 The principle stresses of the blocks after 2000 time steps (2.0 seconds).

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Table 15.1. Average relative displacement. Time step

Relative displacement

Open-close iteration

1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

0.00083 9 0.00041 5 0.00032 8 0.00030 5 0.00016 5 0.00028 5 0.00038 5 0.00047 5 0.00055 5 0.00062 5 0.00103 5 0.00106 5 0.00086 4 0.00054 4 0.00023 3 0.00025 3 0.00046 2 0.00049 2 0.00039 1 0.00014 1 0.00001 1 0.00001 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 Equation solving iteration: 40–50 Factor of SOR: 1.25

The following table is the computation of the statics using dynamics. The relative displacements are reduced following the progress of the time steps. The damping is made by reducing 0.01 times of velocity after each 0.001 seconds (time interval). It is also can be noticed, the open-close iterations are all 1 after time step 100. This means all contacts keep the same after time step 100. The computation is stable. REFERENCES [1] Gen-hua Shi, “Applications of Discontinuous Deformation Analysis (DDA) and Manifold Method” The Third International Conference on Analysis of Discontinuous Deformation, pp. 3–15 Vail, Colorado (1999) [2] Gen-hua Shi, “Block System Modeling by Discontinuous Deformation Analysis” Computational Mechanics Publications, Southampton UK and Boston USA (1993) [3] Gen-hua Shi and Richard E. Goodman, “Generalization of Two Dimensional Discontinuous Deformation Analysis for Forward Modeling,” International Journal for Numerical and Analytical Methods in Geomechanics. Vol. 13, pp. 131–158 (1989)

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[4] Gen-hua Shi and Richard E. Goodman, “The Key Blocks of Unrolled Joint Traces in Developed Maps of Tunnel Walls,” International Journal for Numerical and Analytical Methods in Geomechanics. Vol. 13, pp. 359–380 (1989) [5] Gen-hua Shi and Richard E. Goodman, “ Stability analysis of infinite block systems using block theory,” Proc. Analytical and computational methods in engineering rock mechanics, E.T. Brown, London: Allen and Unwin, pp. 205–245 (1987) [6] Richard E. Goodman and Gen-hua Shi, “The Application of Block Theory to the Design of Rock Bolt Supports for Tunnels,” Felsbau 5 Nr. 2, pp. 79–86 (1987) [7] Gen-hua Shi and Richard E. Goodman, “Two Dimensional Discontinuous Deformation Analysis,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 9, pp. 541–556 (1985) [8] Richard E. Goodman and Gen-hua Shi, “Block theory and its application to rock engineering,” Englewood Cliffs, NJ: Prentice-Hall (1985) [9] Gen-hua Shi and Richard E. Goodman, “Keyblock Bolting,” Proc. of International Symposium on Rock Bolting, pp. 143–167 Sweden (1983) [10] Gen-hua Shi and Richard E. Goodman, “Underground Support Design Using Block Theory to Determine Keyblock Bolting Requirements,” Proc. of the Symposium on Rock Mechanics in the Design of Tunnels, South Africa (1983) [11] Richard E. Goodman and Gen-hua Shi, “Geology and Rock Slope Stability – Application of a Keyblock Concept for Rock Slopes,” Proc. of Third International Conference on Stability in Surface Mining, pp. 347–373, (SME) (1983) [12] Gen-hua Shi, “A Geometric Method of Stability Analysis of Discontinuous Rocks,” Scientia Sinica, Vol. 25, No. 1, pp. 125–148 Peking, China (1982) [13] Gen-hua Shi and Richard E. Goodman, 1981. “A New Concept for Support of Underground and Surface Excavation in Discontinuous Rocks Based on a Keystone Principle,” Proc. 22th U. S. Symposium on Rock Mechanics, pp. 290–296 MIT (1981) [14] Gen-hua Shi, 1977. “The Stereographic Projection Method of Stability Analysis of Rock Mass,” Scientia Sinica, Vol. 3, pp. 260–271 Peking, China (1977)

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Plenary lectures

Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands

Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Realistic dynamic analysis of jointed rock slopes using DDA Y.H. Hatzor, A.A. Arzi, and M. Tsesarsky Department of Geological and Environmental Sciences, Ben Gurion University of the Negev, Beer–Sheva, Israel

ABSTRACT: A fully dynamic, two dimensional, stability analysis of a highly discontinuous rock slope is demonstrated in this paper using DDA. The analytically determined failure modes of critical keyblocks are clearly predicted by DDA. However, application of a fully dynamic analysis with no damping results in unrealistically large displacements that cannot be confirmed by field studies. With introduction of dynamic damping the calculated results can be made to match historic evidence. Our study shows that introduction of at least 5% dynamic damping is necessary to predict realistically the earthquake damage in a highly discontinuous rock slope with about 400 individual blocks. The introduction of dynamic damping is necessary to account for 2D limitations as well as for various energy loss mechanisms, which are not modeled in DDA.

1

INTRODUCTION

Mount Masada is a table mountain, having a comparatively flat summit surrounded by steep slopes, rising about 480 meters above the nearby Dead Sea. The uppermost tens of meters of the slopes consist of nearly vertical cliffs. About two thousand years ago, King Herod fortified the mountain and built a major palace based on three natural rock terraces at the northern tip of the summit (Figure 1). Mount Masada was the site of heroic Jewish resistance against the Romans. It is a national historic monument. The Israel Nature and Parks Authority commissioned this study of the stability of the upper rock terrace of Herod’s Palace under earthquake loading, as part of its preservation work. We carried out the stability evaluation using a fully dynamic version of DDA, with inputs based on a comprehensive field and laboratory study.

2

GEOLOGICAL – SEISMOLOGICAL SETTING

The upper portion of Mount Masada consists of essentially bare hard rock. The rock is mainly bedded limestone and dolomite, with near vertical jointing. Structurally, the entire mountain is an uplifted block within the band of faults which forms the western boundary of the Dead Sea Rift. The Dead Sea Rift is a seismically active transform (Garfunkel et al., 1981; Garfunkel and Ben-Avraham,

Figure 1. A photo of the North Face of Masada showing the upper terrace of King Herod’s Palace.

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Locally it was even founded slightly beyond the rim, on a somewhat lower edge of rock. On the aforementioned three palace terraces, jutting at the northern tip of the mountain top, construction was again carried out up to the rim and beyond in order to achieve architectural effects and utilize fully the limited space. Thus, the remaining foundations effectively serve to delineate the position of the natural rim of the flat mountain top and associated northern terraces about 2000 years ago. Missing portions along such foundation lines indicate locations in which the rim has most probably receded due to rockfalls, unless the portions are missing due to other obvious reasons such as local erosion of the flat top by water or an apparent location of the foundation on fill beyond the rim. Our inspection of the entire rim of the top of Masada, aided by Hebrew University archeologist Guy Stiebel, reveals that over almost the entire length of the casemate wall, which is about 1400 m, the rock rim has not receded during the past two thousand years more than a few decimeters, if at all. Only over a cumulative total of less than 40 m, i.e. about 3% of the wall length, there are indications of rockfalls involving rim recessions exceeding 1.5 m, but not exceeding 4.0 m. Since the height of the nearly-vertical cliffs below the rim is in the order of tens of meters, these observations attest to remarkable overall stability in the face of the recurring earthquakes. On King Herod’s palace terraces there has been apparent widespread destruction, mostly of walls and fills which were somehow founded on the steep slopes. However, in the natural cliffs themselves there are few indications of rockfalls involving rim recessions of more than a few decimeters. Remarkably, most of the high retaining walls surrounding the middle and lower terraces are still standing, attesting to the stability of the rock behind them. In the upper terrace, on which this study is focused, there appears to be only one rockfall with depth exceeding several decimeters. It is a local rockfall near the top of the 22 m cliff, in the northeast, causing a rim recession of about 2.0 m. It is notable that this particular section of the terrace cliff was substantially modified by the palace builders, perhaps de-stabilizing the preexisting natural cliff. We have also inspected rare aerial photographs of Mount Masada dated December 29th 1924, i.e. predating the 1927 earthquake. Our comparison with recent aerial photographs would have been capable of detecting rim recessions exceeding about one meter, if any had occurred in the northern part of the mountain. None were found, suggesting that the 1927 earthquake did not cause any significant rockfalls there (the southern part was less clear in the old photographs). The information presented above essentially constitutes results of a rare rock-mechanics field-scale “experiment”. Two thousand years ago the Masada cliff top was marked by construction. The mountain was later shaken by several major earthquakes, with

1996). According to the Israel building code – Israel Standard 413, based on research by the Geophysical Institute of Israel Seismology Division, under the direction of Dr. A. Shapira, the Dead Sea region has been classified as a region in which an earthquakeinduced peak ground acceleration. (PGA) exceeding 0.2 g at the deep bedrock level is expected with a 10% probability within any 50 year window. This is analogous to a 475 year average recurrence interval for such acceleration. In this paper we repeatedly refer to PGA for simplicity, which is adequate in the present context, although PGA is not generally the best measure of destructive potential (Shapira, 1983; Shapira and van Eck, 1993). Inspection of the historic earthquake record (BenMenahem, 1979; Turcotte and Arieh, 1988; Amiran et al., 1994) suggests that the strongest shaking events which have actually affected Mount Masada within the past two thousand years were due to about ten identified earthquakes with estimated magnitudes in the range of 6.0 ± 0.4 and focal distances probably in the order of several kilometers to a few tens of kilometers from the site. With these parameters, it is highly likely that some of these earthquakes have caused at Mount Masada bedrock PGA’s reaching and even exceeding 0.2 g, in general agreement with predictions for a 2000 year period based on the aforementioned building code assumptions. One of the most notable historic earthquakes in this region occurred probably in the year 362 or 363, with a magnitude estimated at 6.4 (Ben-Menahem, 1979) or even 7.0 (Turcotte and Arieh, 1988). Reported effects included a tsunami in the Dead Sea and destruction in cities tens of kilometers from the Dead Sea both east and west. This is probably the earthquake identified by archeologists as “the great earthquake which destroyed most of the walls on Masada sometime during the second to the fourth centuries” (Netzer, 1991). The most recent of the major historic earthquakes near Mount Masada occurred on July 11th, 1927. This earthquake was recorded by tens of seismographs, yielding a magnitude determination of 6.2 and an epicenter location 30 ± 10 km north of Masada. It also caused a tsunami in the Dead Sea and destruction in cities tens of kilometers away (Shapira et al., 1993) 3

OBSERVED HISTORICAL STABILITY

The fortifications built by King Herod on Mount Masada about two thousands years ago (Netzer, 1991) included a casemate wall surrounding the relatively flat top of the mountain. Clearly, because of its defensive function, the outer face of this wall was built so as to continue upward the face of the natural cliff, as much as possible. The outer wall was therefore founded typically on the flat top within several decimeters from its rim.

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Table 1. Discontinuity data for the foundations of King Herod’s Palace – Masada. Set #

Type

Dip

s m

C MPa

φ◦

1 2 3

Bedding Joints Joints

5/N 80/ESE 80/NNE

0.60 0.14 0.17

0 0 0

41◦ 41◦ 41◦

J1 Spacing

Frequency

25 15 10 5 0

MECHANICAL BEHAVIOR OF THE ROCK AT MASADA

40

0

80 120 160 200 Spacing (cm)

10 20 30 Spacing (cm)

12

40

Length

N = 69 Mean = 16.8 cm

8 4 0

50 Frequency

Rock mass structure

Herod’s Palace, also known as the North Palace, is built on three terraces at the north face of Masada. The rock mass structure at the foundations consists of two orthogonal, sub vertical, joint sets striking roughly parallel and normal to the NE trending axis of the mountain, and a set of well developed bedding planes gently dipping to the north (Table 1). The joints are persistent, with mean length of 2.7 m. The bedding planes, designated here as J1 , dip gently to the north with mean spacing of 60 cm. The two joint sets, J2 and J3 , are closely spaced with mean spacing of 14 cm and 17 cm respectively (Figure 2). 4.2

10

J3 Spacing 16 Frequency

4.1

N = 80 Mean = 14 cm

20

0 0

4

J2 Spacing

30

N = 59 Mean = 60 cm

20

Frequency

deep bedrock accelerations certainly exceeding 0.1 g and probably even exceeding 0.2 g. Due to the topography affect, motions at the top are substantially amplified at frequencies about 1.3 Hz (Zaslavsky et al., 2002). Observations at the present stage of the “experiment” show that all the cliffs surrounding the top of Mount Masada essentially withstood the shaking, with some relatively minor rockfalls at the top of the cliffs. The above is a substantial result of a full-scale “experiment” on the real rock structure. Therefore, a fundamental test of any model of this structure is that it must essentially duplicate the above “experiment”. As shown in the sequel, we subjected our DDA model to this test, obtaining instructive results.

N = 100 Mean = 2.7 m

40 30 20 10 0

0

10 20 30 Spacing (cm)

40

0

2

4 6 8 10 12 14 Spacing (cm)

Physical properties

The rock mass consists of bedded dolomites with local karstic voids between beds. The bulk porosity of intact samples ranges between 3% – 12% and the dry unit weight is 25 kN/m3 . The bedding planes are generally clean and tight. 4.3

Strength and elasticity of intact rock

The elastic behavior of the rock was studied using a stiff, hydraulic, closed-loop servo controlled load frame with maximum axial force of 1.4 MN, and stiffness of 5 ∗ 109 N/m (Terra-Tek model FX-S-33090). Testing procedures are described by Hatzor and Palchik, 1997, 1998) and Palchik and Hatzor (2002). Tests were performed at a constant strain rate of 5 ∗ 10−6 s−1 . In Figure 3 the result of a load – unload loop of uniaxial compression performed on a solid cylinder from the dolomite at Masada is shown. This result indicates that the uniaxial compressive strength of the tested dolomite sample is greater than 315 MPa, that the elastic modulus is 43 GPa, and that Poisson’s ratio is 0.18 (radial strains are not shown). These data

Figure 2. Discontinuity length, spacing, and orientation distribution at the foundations of King Herod’s Palace, Masada. Upper Hemisphere projection of poles.

fall within the range of strength and elasticity values determined experimentally for other dolomites in Israel (Hatzor and Palchik, 1997, 1998; Palchik and Hatzor, 2002). 4.4 Residual shear strength of discontinuities The residual friction angle of joints was determined using tilt tests performed on saw-cut and ground surfaces of dolomite, assuming the joint planes are

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1.2

MNP3 - Natural bedding plane 1.38 MPa

300 200 100

0.8

1.2 MPa

0.4

1.03 MPa 0.86 MPa 0.7 MPa 0.52 MPa

Shear Stress (MPa)

E = 42,909 MPa v = 0.1844 Shear Stress (MPa)

Axial Stress (MPa)

400

1.2

Failure Envelope

0.8 0.4 0 0

0.4 0.8 1.2 Normal Stress (MPa)

0.35 MPa

0

Normal Stress 0.17 MPa

0

0.1

0.2

0.3

0.4

0.5 0

Axial Strain (%)

0

Figure 3. Result of a load – unload cycle under uniaxial compression performed on an NX size solid cylinder of the dolomite in Masada.

Shear strength of filled bedding planes

2

0 < σn < 0.5 MPa : τ = 0.88σn (R2 = 0.999) 0.5 < σn ≤ 12 MPa : τ = 0.083 + 0.71σn (R2 = 0.998)

Shear strength of rough bedding planes

These results indicate that for low normal load (up to 0.5 MPa) the peak friction angle for the bedding planes at Masada is 41.3◦ . For higher normal loads the peak friction angle is 35.3◦ . The residual friction angle may be taken from the triaxial tests of the filled discontinuities as 23◦ . The maximum height of the terrace at the North Palace is 25 m and therefore the normal stress acting on bedding planes at the North

The shear strength of rough bedding planes was determined using real bedding plane samples from the foundations of the North Palace. The upper and lower sides of the mating planes were kept in contact with no disturbance and were transported to the lab at natural water content. The two samples were cast inside two 200 mm ∗ 200 mm ∗ 150 mm shear boxes while the mating surfaces were kept intact. The gap between

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1.6

the rock and the box frame was filled with Portland cement. Direct shear tests were performed using a hydraulic, close loop servo-controlled, direct shear system with normal force capacity of 1000 kN and horizontal force capacity of 300 kN (Product of TerraTek Systems Inc.). The stiffness of the normal and shear load frames is 7.0 and 3.5 MN/m respectively. Normal and horizontal displacement during shear were measured using four and two 50 mm LVDT’s with 0.25% linearity full scale. Axial load was measured using a 1000 kN capacity load cell with 0.5% linearity full scale. Shear load was measured using a 300 kN load cell with 0.5% linearity full scale. The direct shear tests were performed for two samples (MNP2, MNP3) under constant normal stress and shear displacement rate of 1mil/sec (0.025 mm/sec). In Figure 4 shear stress vs. shear displacement is shown for sample MNP-3 that was loaded, unloaded, and reloaded in eight cycles of increasing normal stress from 0.17 MPa to 1.38 MPa. In each cycle the sample was sheared forward, in the first cycle a distance of 1.3 mm, and then additional 0.5 mm of forward shear displacement in each consecutive segment. Plotting the peak shear stress vs. normal load for the two segment tests (Figure 5) reveals a bilinear failure envelope with the following failure criterion:

The shear strength of filled bedding planes was estimated using a segment triaxial test performed on a right cylinder containing an inclined saw cut plane at 55◦ to the axis of the cylinder, filled with crushed dolomite. Seven different segments were performed, with confining pressure values ranging between 2.2 and 16.2 MPa. A linear Coulomb – Mohr failure criterion was found, with zero cohesion and a residual friction angle of 22.7◦ . The test procedures and results are discussed by Hatzor (in press). The similarity between the result of tilt tests on ground surfaces (23◦ ) and the segment triaxial test on a filled saw-cut plane (22.7◦ ) suggests that during shear the infilling material crushed all remaining asperities in the saw-cut sample resulting in a failure envelope representing residual conditions. The residual friction angle value of 23◦ may therefore be applicable for very large blocks where some initial shear displacements have already taken place due to historic cycles of seismic loading (see Hatzor, in press). However, for dynamic analysis of smaller blocks with high static factor of safety the strengthening effect of initial asperities ought to be considered. 4.6

0.8 1.2 Shear Displacement (mm)

Figure 4. Shear stress vs. shear displacement for a natural bedding plane sample from the foundations of Herod’s Palace in Masada.

clean and tight. 20 tilt tests performed on saw-cut and ground surfaces provided a mean friction angle of 28◦ and 23◦ respectively. The 5◦ difference is attributed to roughness resulting from saw-cutting. 4.5

0.4

Direct Shear of Natural Bedding Planes Triaxial Shear of Filled Saw-cut

Shear Stress (MPa)

12

8

4

0 0

5

10 15 Normal Stress (MPa)

20

25

Figure 5. Failure envelope for rough bedding planes – direct shear tests.

Figure 7. A photogeological trace map of the northern face of Herod’s Palace upper terrace.

Figure 8. A deterministic joint trace map of the terrace prepared using the photogeological map (Figure 7) and the block cutting algorithm (DC) of Shi (1993). Figure 6. A synthetic joint trace map of the upper rock terrace of Herod’s Palace in Masada using the statistical joint trace generation code (DL) of Shi (1993).

individual blocks. Kinematic, mode, and removability analyses confirm these intuitive expectations.

Palace cannot be greater than 682 kPa. Therefore, in light of the experimental results, the low normal load criterion should be used for dynamic analysis.

5 5.1

5.2 Deterministic joint trace generation While it is quite convenient to use mean joint set attitudes and spacings in order to generate statistically a synthetic mesh, the resulting mesh is quite unrealistic and bears little resemblance to the actual slope. The contact between blocks obtained this way is unrealistically planar, thus interlocking between blocks is not modeled. Consequently the results of dynamic calculations will be overly conservative. In order to analyze the dynamic response of the slope realistically a photo-geological trace map of the face was prepared using aerial photographs (Figure 7), and the trace lines were then digitized. Thus, the block-cutting (DC code) algorithm of Shi (1993) could be utilized in order to generate a trace map which represents more closely the reality in the field (Figure 8).

MESH GENERATION METHODS AND EXPECTED FAILURE MODES Synthetic joint trace generation

Two principal joint sets and a systematic set of bedding planes comprise the structure of the foundations of Herod’s Palace (Figure 2). An E-W cross section of the upper terrace is shown in Figure 6, computed using the statistical joint trace generation code (DL) of Shi (1993). It can be seen intuitively that while the East face of the rock terrace is prone to sliding of wedges, the West face is more likely to fail by toppling of

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7

The same failure modes would be anticipated in the deterministic mesh shown in Figure 8. However, the interlocking between blocks within the slope is much higher and therefore the results of the dynamic analysis should be less conservative and more realistic. 6

7.1 The numerical discontinuous deformation analysis The DDA method (Shi, 1993) incorporates dynamics, kinematics, and elastic deformability of geological materials, and models actual displacements of individual blocks in the rock mass using a time-step marching scheme. The formulation is based on minimization of potential energy and uses a “penalty” method to prevent penetration or tension between blocks. Numerical penalties in the form of stiff springs are applied at the contacts to prevent either penetration or tension between blocks. Since tension or penetration at the contacts will result in expansion or contraction of the springs, a process that requires energy, the minimum energy solution is one with no tension or penetration. When the system converges to an equilibrium state however, there are inevitable penetration energies at each contact, which balance the contact forces. Thus, the energy of the penetration (the deformation of the springs) can be used to calculate the normal and shear contact forces. Shear displacement along boundaries is modeled in DDA using the Coulomb-Mohr failure criterion. The fixed boundaries are implemented using the same penalty method formulation: stiff springs are applied at the fixed points. Since displacement of the fixed points requires great energy, the minimum energy solution will not permit fixed-point displacement. The blocks are simply deformable: stresses and strains within a block are constant across the whole region of the block. In this research a new C/PC version of DDA, recently developed by Shi (1999), is used. In this new version earthquake acceleration can be input directly in every time step. A necessary condition for direct input of earthquake acceleration is that the numerical computation has no artificial damping, because damping may reduce the earthquake dynamic energy and the damage may thus be underestimated (Shi, 1999). In DDA the solution of the equilibrium equations is performed without damping (Shi, 1999) for the purpose of a fully dynamic analysis in jointed rock masses. As we shall see below however, application of dynamic DDA with no damping returns unrealistically high displacements and is therefore overly conservative.

INPUT MOTION FOR DYNAMIC ANALYSIS

In order to perform a realistic dynamic analysis for the rock slopes at Masada, we prefer modeled input motion representing ground motions for the Dead Sea rift system. In this research we chose to use the recorded time history of the Mw = 7.1. Nuweiba earthquake which occurred in November 1995 in the Gulf of Eilat (Aqaba) with an epicenter near the village of Nuweiba, Egypt. The main shock was recorded at the city of Eilat where the tremor was felt by people, and structural damage was detected in houses and buildings. The city of Eilat is located 91 km north from the Nuweiba earthquake epicenter and 186 km south of Masada, on the northern coast of the gulf of Eilat (Aqaba). Figure 9 shows the vertical and EW components of the accelerogram that were recorded in Eilat. The horizontal Peak Ground Acceleration (PGA) of the Nuweiba record was 0.08 g. The Eilat accelerograph station was on a thick fill of Pleistocene alluvial fan deposits. The recorded accelerogram therefore represents the response of a site situated on deep fill rather than on sound bedrock. However, we regard this as a secondary issue in the present context. As shown in the sequel, we utilized as input both the 0.08 g PGA accelerogram as well as the same accelerogram normalized to a 0.18 g PGA, so as to explore a range of PGA’s. As explained in chapters 2 and 3 above, the Masada cliffs, including Herod’s Palace upper terrace, have withstood historic earthquakes in this PGArange with only minor rockfalls. 0.12

Vertical

Accl. (g)

0.08 0.04 0 -0.04 -0.08 0.12

FULLY DYNAMIC ANALYSIS USING DDA

E-W

Accl. (g)

0.08

7.2

0.04 0 -0.04

Hatzor and Feintuch (2001) demonstrated the validity of DDA results for fully dynamic analysis of a single block on an incline subjected to dynamic loading. First the dynamic solution for a single block on an incline subjected to gravitational load (constant acceleration), a case which was investigated originally by MacLaughlin (1997), was repeated using the new

-0.08 0

10

20

30

40

50

60

Time (sec)

Figure 9. Time history of the Mw = 7.1 Nuweiba earthquake (Nov. 22, 1995) as recorded at the city of Eilat.

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Validation of dynamic displacement prediction by DDA using analytical solutions

dynamic code (Shi, 1999). For a slope inclination of 22.6◦ , four dynamic displacement tests were performed for interface friction angle values of 5◦ , 10◦ , 15◦ , and 20◦ . The agreement between the analytical and DDA solution was within 1–2%. Next, Hatzor and Feintuch investigated three different sinusoidal functions of increasing complexity for the dynamic load input function, and checked the agreement between DDA and the derived analytical solutions. A very good agreement between analytical solution and DDA was obtained in all cases, with errors between 5% to 10% (Figure 10). It was found by Hatzor and Feintuch that in order to get a good agreement between DDA and the analytical solution, the maximum size of the time step (g1) had to be properly conditioned. For example, with increasing block velocity the maximum time step size had to be reduced in order to obtain good agreement with the analytical solution. The best method to estimate the proper time step size would be to check the ratio between the assumed maximum displacement per time step (g2) and the actual displacement per time step calculated by DDA. In order to get good agreement between DDA and the analytical solution that ratio should be as close as possible to 1.0. Such an optimization procedure however is only possible for single block cases. 20

The dynamic displacement problem of a block on an incline was studied by Wartman (1999) using shaking table experiments that were performed at the U.C. Berkeley Earthquake Engineering Laboratory. The same tests were repeated numerically by Tsesarsky et al. (2002) using dynamic DDA, and some results are shown in Figure 11 (see complete paper in this volume). The results of Tsesarsky et al. suggest that with zero dynamic damping DDA overestimates the physical displacements by as much as 80%. However, with as little as 2% dynamic damping the results of DDA match the experimental results within 5% accuracy. This result suggests that realistic application of dynamic DDA must introduce some measure of damping in order to account for energy loss mechanisms that are not modeled by DDA, the first of which is energy consumption due to irreversible deformation during block interactions. The results of Tsesarsky et al. (2002) pertain to a single block on an incline. A multi-block problem was studied by McBride and Scheele (2001), using a slope with a stepped base consisting of 50 blocks that undergo sliding failure under gravitational load. Their conclusion was that as much as 20% dynamic damping was necessary in order to obtain realistic agreement between the physical model and DDA. Perhaps better conditioning of the control parameters would have reduced the required dynamic damping by a significant amount.

Acceleration Velocity Displacement DDA displacement

15 a (m/s2), v (m/s), s (m)

7.3 Validation of dynamic displacement prediction by DDA using shaking table experiments

10

5

8

8.1 Numerical details

0 0

2

1

3

4

5

In all DDA simulations the complete record was computed for the entire 50 seconds of earthquake duration (see Figure 9). The numerical input parameters used in this work are listed in Tables 2 and 3 (for explanation of each control parameter see Shi, 1993). DDA computations were performed on a P41.5 GHz processor with 128 Mb RAM. To complete the required 25000 time steps (earthquake duration of 50 seconds) approximately 42 hours of processor time were requires, namely a computation rate of approximately 600 time steps per hour. The mesh consists of 344 blocks.

a(t) = 2sint + 3sin2t

-5

Time (sec)

Figure 10. Validation of dynamic DDA using analytical solutions (after Hatzor and Feintuch, 2001). 0.1 Measured Block Displacement DDA (k01 = 1)

0.08 Displ. (m)

RESULTS OF DYNAMIC ANALYSIS

DDA (k01 = 0.98)

DDA solution for g0 = 500*106 N/m g1 = 0.0025 sec g2 = 0.0075

0.06 0.04 0.02

8.2 Realistic damage prediction by DDA

0 0

1

2

3

4

In Figure 12 the computed response of the upper terrace is shown with various amounts of dynamic damping. Figure 12A shows the computed damage with zero dynamic damping, namely the initial velocity in every time step is inherited from the previous

5

Time (sec)

Figure 11. Validation of dynamic DDA using shaking tabel experiments from Berkeley (after Tsesarsky et al., 2002).

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Table 2.

Numeric Control Parameters.

Total number of time steps: Time step size (g1): Assumed maximum displacement ratio (g2): Contact spring stiffness (g0): Factor of over-relaxation:

Table 3.

25,000 0.002 0.0015 5 ∗ 106 kN/m 1.3

Material Properties.

Unit weight of rock (γ ): Elastic Modulus (E): Poisson’s Ratio (v): Friction angle of all discontinuities (φ): Cohesion of all discontinuities (C): Tensile strength of all discontinuities (σt ):

25 kN/m3 43 ∗ 106 kN/m2 0.184 41◦ 0 0

time step. After 50 seconds of shaking, with 0.08 g PGA, the upper terrace seems to disintegrate completely. In light of the historic observations this result is clearly unrealistic. In Figure 12B the response of the terrace is shown again after 50 seconds of shaking but with 5% dynamic damping, namely the initial velocity of each block at the beginning of a time step is reduced by 5% with respect to its terminal velocity at the end of the previous time step. The model predicts onset of toppling failure at the foot of the west slope and minor sliding deformations at the east slope. The failure modes predicted by the model are similar to the expected modes from both field and analytical studies. The extent of damage in the terrace and the depth of the loosened zone in the west slope are reduced significantly with comparison to the undamped analysis. The performance of the slope with 10% dynamic damping (Figure 12C) is roughly the same as with 5% dynamic damping and therefore the justification for more than 5% dynamic damping seems questionable. Assuming that 5% damping is the correct amount necessary to account for energy loss mechanisms ignored by DDA, we studied the response of the terrace slopes to the same accelerogram when normalized to a 0.18 g PGA, which is still within the range of historic earthquakes as discussed in chapters 2,3 and 6 above. As shown in Figure 13, after 50 seconds of shaking the damage is not much different than that which was modeled for the original time history (Figure 12B). Both Figures 12B and 13 indicate the expected depth of the loosened zone in the slope due to the seismic loading. We believe that the graphical output in Figure 13 is still very conservative because it does not compensate fully for: a) various real energy dissipation mechanisms, b) reinforcing potential of the third, in slope, dimension.

Figure 12. Results of dynamic DDA calculation of the original Nuweiba record (PGA = 0.08 g) after 50 seconds of shaking. A) No dynamic damping in DDA, B) 5% damping, C) 10% damping.

Nevertheless, a graphical result such as the one presented in Figure 13 can be used as an aid for support design. Both the spacing and length of the support elements (anchors or rock bolts) can be dimensioned using the graphical output. The required capacity of the anchors may be estimated using a pseudo-static analysis for a representative block with the peak horizontal acceleration taken for the pseudo-static inertia force.

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slope for which exists a good historic record of stability under recurring strong earthquake shaking. We found that in order to obtain realistic predictions for this multi-block analysis, at least 5% dynamic damping was required. • Only when proven to be realistic, the graphical output of such an analysis may be used to estimate the depth of the loosening zone following the earthquake, and the spacing, length and capacity of support elements – if required.

ACKNOWLEDGEMENTS Figure 13. Results of dynamic DDA calculation (with 5% dynamic damping) after 50 seconds of shaking using the Nuweiba record normalized to PGA = 0.18 g.

9

This research was funded by Israel Nature and Parks authority and partially by the Bi-national Science Foundation (BSF) through grant 98-399. The support of the two agencies is hereby acknowledged. A. Shapira and Y. Zaslavsky from the Geophysical Institute of Israel are thanked for the Nuweiba record. Gen-Hua Shi is thanked for making his dynamic DDA code available for this study. Finally, Guy Stiebel is thanked for guiding us on Masada with archeological insights and for providing the 1924 aerial photographs.

SUMMARY AND CONCLUSIONS

In this paper a highly jointed rock slope, which withstood many events of strong seismic ground motions in historic times, is modeled using dynamic DDA. The field observations are compared with the results of the numerical model. The following are tentative conclusions:

REFERENCES

•

We find that for a realistic calculation of dynamic response the modeled joint trace map (the mesh) must be as similar as possible to the geological reality. We recommend using digitized photogeological trace maps in conjunction with the DC block-cutting algorithm of Shi (1993) in order to generate the realistic mesh, rather than a statistical joint trace generation algorithm such as the DL code of Shi (1993). Such a deterministic approach will capture some block-interlocking mechanisms active in the modeled slope because of dissimilarities in joint attitudes and variations in dip angle along the surface of the discontinuities. • Previous studies have shown that dynamic DDA with zero dynamic damping will match analytical solutions for a single block on an incline with great accuracy. However, when results of dynamic DDA are compared with shaking table experiments for a single block it is found that at least 2.5% of dynamic damping is necessary for accurate displacement predictions. We believe that the dynamic damping is necessary in order to account for energy loss mechanisms, which are abundant in the physical reality but are ignored by the linear – elastic approach taken by DDA. Also, the dynamic damping may partially compensate for the two dimensional formulation which does not allow modeling the strengthening effect of the third, in slope, dimension. • Using a real time history from the Dead Sea rift system we modeled the response of the jointed rock

Amiran, D.H.K., Arieh, E. and Turcotte, T. 1994. Earthquakes in Israel and adjacent areas: Macro – seismicity observations since 100 B.C.E. Israel Explor. J., 41, 261–305. Ben – Menahem, A. 1979. Earthquake Catalog for the Middle East. Bollettino di Geofisica Teorica e Applicata, v. XXI, pp. 245–313. Garfunkel, Z. and Ben-Avraham, Z. 1996. The structure of the Dead Sea basin. In : Dynamics of extensional basins and inversion tectonics. Tectonophysics, 266, 155–176. Garfunkel, Z., Zak, I. and Freund, R., 1981. Active faulting in the Dead Sea Rift. Tectonophysics, 80, 1–26. Hatzor, Y.H. and V. Palchik, 1997. The influence of grain size and porosity on crack initiation stress and critical flaw length in dolomites. International Journal of Rock Mechanics and Mining Sciences, Vol. 34, No. 5, pp. 805–816. Hatzor, Y.H. and Palchik V., 1998. A microstructure-based failure criterion forAminadav dolomites – Technical Note. International Journal of Rock Mechanics and Mining Sciences, Vol. 35, No. 6, pp. 797–805. Hatzor, Y.H. and Feintuch, A. 2001. The validity of dynamic displacement prediction using DDA. International Journal of Rock Mechanics and Mining Sciences, Vol. 38, No. 4, pp. 599–606. Hatzor, Y.H. in press. Keyblock stability in seismically active rock slopes – the Snake Path cliff – Masada. Journal of Geotechnical and Geoenvironmental Engineering,ASCE. MacLaughlin, M.M. Discontinuous Deformation Analysis of the kinematics of landslides 1997. Ph.D. Dissertation, Dept. of Civil and Env. Engrg., University of California, Berkeley.

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McBride, A. and Scheele, F. Investigation of discontinuous deformation analysis using physical laboratory models. Proceedings of ICADD-4, 4th International Conference of Analysis of Discontinuous Deformation (N. Bicanic Ed) Glasgow, Scotland, UK. Netzer, E. 1991. Masada III – the Yigael Yadin Excavations 1963–1965. Final Reports – The Buildings Stratigraphy and Architecture. Israel Exploration Society, The Hebrew University, Jerusalem, Israel, 665p. Palchik, V. and Y.H. Hatzor, 2002. Crack damage stress as a composite function of porosity and elastic matrix stiffness in dolomites and limestones. Engineering Geology. Vol. 63, pp. 233–245. Shapira, A. 1983, A probabilistic approach for evaluating earthquake risk with application to the Afro-Eurasian junction, Tectonophysics, 95:75–89. Shapira, A., Avni, R. and Nur, A. 1993. A new estimate for the epicenter of the Jericho earthquake of 11th July 1927. Israel Journal of Earth Science, Vol. 42, No. 2, pp. 93–96. Shapira, A. and van Eck, T. 1993. Synthetic uniform hazard site specific response spectrum, Natural Hazard, 8: 201–205 Shi, G.-H. 1993. Block System Modeling by Discontinuous Deformation Analysis, Computational Mechanics Publications, Southampton UK, p. 209.

Shi, Gen-Hua, 1999. Application of Discontinuous Deformation Analysis and Manifold Method. Proceedings of ICADD-3, Third International Conference of Analysis of Discontinuous Deformation (B. Amadei, Ed) Vail, Colorado, pp. 3–15. Tsesarsy, M., Hatzor, Y.H. and Sitar, N. 2002. Dynamic block displacement prediction – validation of DDA using analytical solutions and shaking table experiments. Proceedings of ICADD-5, 5th International Conference of Analysis of Discontinuous Deformation (Y.H. Hatzor, Ed) Beer-Sheva, Israel. Published by Balkema Rotterdam. Turcotte, T. and Arieh, E. 1988. Catalog of earthquakes in and around Israel, Appendix 2.5A in: Shivta site Preliminary Safety Analysis Report, Israel Electric Corp. LTD. Wartman, J. 1999. Physical model studies of seismically induced deformation in slopes. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. Zaslavsky, Y., Shapira, A. and Arzi, A.A. 2002. Earthquake site response on hard rock – empirical study. Proceedings of ICADD-5, 5th International Conference on Analysis of Discontinuous Deformation (Y.H. Hatzor, Ed) Beer Sheva, Israel. Published by Balkema, Rotterdam.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Numerical models for coupled thermo-hydro-mechanical processes in fractured rocks-continuum and discrete approaches L. Jing, K.-B. Min & O. Stephansson Department of Land and Water Resources Engineering, Royal Institute of Technology, Stockholm, Sweden

ABSTRACT: This article reviews the two main approaches of numerical modeling for the coupled thermohydro-mechanical (THM) processes in fractured rocks – the discrete and the equivalent continuum models, respectively. The presentation covers the governing equations derived from the conservation laws of mass, momentum and energy of the continuum mechanics, with focus on the FEM formulations and explicit blocfracture system representations for discrete models. Two applications related to nuclear waste repository design and performance assessment are presented as examples. Special attention is given to the DEM approach of homogenisation and upscaling for deriving equivalent continuum properties of fractured rocks based on the REV concept, with realistic representations of fracture systems. A brief summary of the trends, characteristics and outstanding issues in numerical modeling of fractured rocks is given at the end to highlight the advances and remaining difficulties.

1

INTRODUCTION

be crossed so that their interactions be fully expressed in the resultant mathematical models and computer codes. The coupled THM process is mainly described by mechanics of porous geological media, such as soils, sands, clays and fractured rocks. The first theory is von Terzaghi’s 1-D consolidation theory of soils (Terzaghi, 1923), followed later by Biot’s phenomenological approach of poroelasticity (Biot, 1941, 1956), which was further enriched by the mixture theory (Morland, 1972, Bowen, 1982). Non-isothermal consolidation of deformable porous media is the basis of coupled THM models of geological media, using either the averaging approach as proposed by Hassanizadeh & Gray (1979a, b, 1980, 1990) and Achanta et al. (1994), or an extension to the Biot’s theory with a thermal component (de Boer, 1998). The former is more suitable for understanding the microscopic behavior of porous media and the latter is better suited for macroscopic description and computer implementation. The subject has attracted very active research activities because of its wide reaching impacts in the fields of both mechanics and engineering, and generated extensive publications. The fundamentals are systematically presented in many volumes, e.g. Whitaker (1977), Domenico and Schwartz (1990), Charlez (1991), Charlez and Keramsi (1995), Coussy (1995), Sahimi (1995), Selvadurai (1996), Lewis and Schrefler

Many rock engineering projects, such as radioactive waste disposal in underground repositories, Hot-DryRock geothermal energy extraction, oil/gas reservoir exploitation and oil/gas underground storage caverns, require understanding of interactions among different physio-chemical processes in various geological media. In most of the cases, mainly the mechanical process of rock stress, deformation, strength and failure (M), hydraulic processes of fluid flow and pressure (H), thermal processes of heat transfer (T) and chemical processes (C) of contaminant transport related to different fluid-rock interaction mechanisms are important. These processes are coupled, meaning that one process affects the initiation and progress of others. Therefore, the rock mass response to natural (such as in-situ stresses and groundwater flow) and engineering (such as excavation, fluid injection and extraction, etc.) perturbations cannot be predicted with acceptable confidence by considering each process independently. The requirement to include coupling of these processes depends on the specifics of the engineering design and performance/safety requirements. To gain a proper understanding of coupled behavior of rock masses, the boundaries of traditional fields of research, e.g. rock/soil mechanics, heat transfer, hydrogeology and geochemistry must

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In this review, only the models and applications for THM coupling in fractured rocks are presented, using FEM and DEM approaches.

(1987, 1998) and Bai & Elsworth (2000), with focus on multiphase fluid flow and transport in deformable porous continua under isothermal or non-isothermal conditions. The volume edited by Tsang (1987) is more focused on fractured rocks, especially for nuclear waste disposal applications. Extension to poroplasticity and rock fractures are reported in Pariseau (1999) and Selvadurai and Nguyen (1999), respectively, and comprehensive reviews on applications for geothermal reservoir simulations are given by WillisRichards & Wallroth (1995) and Kohl et al. (1995). The physics of the coupled THM processes in continuous porous media are well understood, and the mathematical theory, numerical methods, computer implementation, code verification and applications for practical soil mechanics and reservoir engineering problems have become widely accepted. On the other hand, coupled THM phenomena in fractured rocks are more complex in almost all aspects, from processes, material models, properties, parameters, geometry, initial/boundary conditions, computer methods, down to code verifications and validations. Comprehensive studies, with both continuum and discrete approaches, were conducted in the international DECOVALEX projects for coupled THM processes in fractured rocks and buffer materials for underground radioactive waste disposal. The results were published in a series of reports (Jing et al., 1996, 1999), an edited book by Stephansson et al. (1996), and two special issues of Int. J. Rock Mech. Min. Sci. (32(5), 1995 & 38(1), 2000). Applications in other areas have also been reported, such as for reservoir simulations and non-Darcian flow (Sasaki & Morikawa, 1996, Nithiarasu et al., 2000), mechanics of soils and clays (Gawin & Schrefler, 1996, Thomas & Missoum, 1999, Masters et al., 2000) and tunneling in cold regions (Lai et al., 1998). The interactions between chemical and THM processes for geological media, however, are not well understood and less reported. Tsang (1991) summarized some fundamental THMC issues regarding applications for nuclear waste disposal. Some recent developments are reported by Zhao et al. (2000) for numerical modeling of fluid-water interaction for transport in porous media, Yang (2001) for reservoir compaction with mineral reaction, Yeh et al. (2001) for reactive flow and transport, and Sausse et al. (2001) for change of fracture surface and permeability due to fluid-rock interaction process. Consideration for chemical alteration on mechanical properties of geological materials is reported in Renard et al. (1997), Loret et al. (2002) and Hueckel (2002), respectively. The most well known code for THC coupling is perhaps the TOUGH2 code (Pruess, 1991), with wide applications in geothermal reservoir simulations and nuclear waste repository design and performance assessments.

2

The equivalent continuum approach means that the macroscopic properties of the fractured media have their corresponding supporting volumes, or representative elementary volume (REV). The governing equations and FEM formulations given below are perhaps the most basic and common for porous continua, but they vary with specific requirements for processes, properties and parameters. 2.1

The governing equations

Assuming that the porous medium is a mixture of solid phase of homogeneous, isotropic and linear elasticity (characterized by Lame’s constant µ and λ), and fluid phase (water and gas) with saturation degree S (0 ≤ S ≤ 1) and porosity φ, the primary variables are the displacement vector u (relative) temperature T and fluid pressure P, the governing equations are derived based on the basic laws of momentum, mass and energy conservations of the mixture and the individual phases (s-solid, l-liquid and g-gas), as given below. a) Linear momentum conservation equation of the mixture:    ∂
Dijkl εkl − αl Pl + αg Pg + γ Ts δij ∂xj 
+ (1 − φ)ρs + φS(ρl + ρg ) gi = 0

(1)

where Dijkl is the elasticity tensor of solid phase, εkl = (uk,l + ul,k )/2 the solid strain tensor, gi the acceleration vector by gravity, απ the Biot parameter for phase π(π = s, g, l) of density ρπ , γ = (2µ + 3λ)β the thermo-elastic constant, and β the linear thermal expansion coefficient, respectively. b) Gas (dry air) mass conservation equation: φ

 ∂
∂ (1 − S)ρga + (1 − S)ρga (∇ · u) ∂t ∂t  ∂  d + =0 ρg vg + ρg vˆ gw ∂xi

(2)

where ρga is the mass concentration of dry air in the gas phase, vg is the velocity vector of the gas phase, d and vˆ gw is the average diffusion velocity of the dry air species, respectively.

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FEM SOLUTION OF COUPLE THM PROCESSES IN FRACTURED ROCKS – THE EQUIVALENT CONTINUUM APPROACH

c) Water species (liquid and vapor) mass conservation equation: φ

∂S ∂ [(1 − S)ρgw ] + φl ∂t ∂t

∂ + [(1 − S)ρgw + Sρl ] (∇ · u) ∂t   d + ∇ · ρgw vg + ρg vˆ gw − ρl vl = 0

where dots over primary variables indicates their rate of change with time. The explicit expressions of the integrals for the coefficient matrices in (5) are given in Lewis and Schrefler (1987, 1998), Schrefler (2001), and Schrefler et al. (1997), with small variations. The equation can be symbolically written as ˙ + CX = F X

(3)

(6)

Then a standard staggered scheme for temporal integration with time-marching leads to

where ρgw is the mass concentration of water vapour in the gas phase, and vl is the velocity of the liquid water phase, respectively. d) Energy conservation (enthalpy balance):

(I + tC(k+1) )X(k+1) = tF(k+1) + X(k) = (x based on mid-point rule x˙ an arbitrary scalar variable x. (k+1)

∂T + (ρw Cpw vl + ρgw Cpg vg ) · ∇T ∂t  ∂S ∂ −∇ · (KmT ∇T ) = hvap φρl + Sρl (∇ · u) ∂t ∂t  ∂εkk −∇ · (ρl vl ) − T0 γ (4) ∂t

k+1

(7)

− x )/t for (k)

(ρCp )m

2.3 An example of application-Kamaishi in-situ experiments One of the problems studied in the DECOVALEX II project is the numerical simulation of an in-situ THM experiment carried out in Kamaishi Mine, Japan, where field scale tests of a heating-and-cooling period of more than one year was conducted to verify numerical modeling techniques for coupled THM analysis. Measured results of temperature, water content, stress, strain and displacements at numerous monitoring points were used for prediction and calibration of FEM models and material parameters. This work is continued in DECOVALEX III project for better understanding of the physics of the rock-buffer system, based on a simplified axi-symmetric psuedo-2D model as shown in Figure 1. Temperature, water contents, radial strains and radial stresses are calculated at four points at positions r = 0.52 m, 0.685 m, 0.85 m and 1.45 m, respectively, where values of these parameters were measured during the experiments. Four FEM codes were applied to investigate the problem: ROCMAS (KTH, Sweden), FRACON (CSNC, Canada), THAMES (JNC, Japan), and CASTEM 2000

where (ρCp )m = (1 − φ)ρs Cps + φρl Cpl + φρg Cpg is the composite heat capacity of the medium, Cpπ , (π = s, g, l) are the heat capacity of the phase π, KmT = (1−φ)KsT +φKlT +φKgT is the effective thermal conductivity of the medium, and hvap is the enthalpy of vaporization per unit mass. 2.2

FEM formulation and solution

Applying standard Galerkin FEM spatial discretization approach leads to the following set of matrix equations:  Kuu u + Cuw Pw + Cug Pg + CuT T = Fu    C u˙ + H P + P P˙ + C P˙ + C T˙ = F wu ww w ww w wg g wT w  Cgu u˙ + Cgw P˙ w + Hgg Pg + Pgg P˙ g + CwT T˙ = Fg    CTu u˙ + CTw P˙ w + CTG P˙ g + PTT T˙ + HTT T = FT

(5)

Test cavern Heater

Steel bars

Steel Bentonite

r = 0.47 m Concrete

r=0

r = 0.85 m

Rock

r = 1.0 m

r = 0.52 m

1-D Axi-symmetric model Bentonite

Figure 1.

Heater

Geometry of the simplified axisymmetric model-BMT1A-DECOVALEX III project (Jing and Nguyen, 2001).

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(CEA, France). The governing equations and constitutive models used in these codes are similar but different to that presented in section 2, especially regarding thermo-hydraulic behavior of the bentonite used as buffer for the experiments. The details of the

Temperature at Point 1

Temperature (degree)

120

Water content (%)

Measured JNC KTH CNSC CEA INERIS

100 80 60 40 20 0

0

50

100

150

200 250 Time (days)

Water content at Point 1

18 16 14 12 10 8 6 4 2 0 0

Figure 2.

background, code formulations, initial/boundary conditions and material properties can be seen in Jing and Nguyen (2001). The results at Point 1 are shown in Figures 2 and 3. It illustrates that the FEM codes applied can predict very accurately the temperature

50

100

150

200 250 Time (days)

300

350

Measured KTH CEA

300

400

450

JNC CSNC INERIS

350

400

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Calculated and measured temperature and water content at Point 1 (r = 0.52 m) (Jing and Nguyen, 2001).

Radial stress at Point 1 (tensile stress negative) DDA

1000

Stress (KPa)

800

BBC

JNC

KTH

CNSC

CEA

600 400 200 0

Figure 3.

0

50

100

150

200 250 300 Time (days)

350

400

Calculated and measured radial stress at Point 1 (r = 0.52 m) (Jing and Nguyen, 2001).

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450

evolution, in general. The models can also predict reasonably well about fluid flow and saturation evolutions. The prediction to stress behavior, on the other hand, is much less reliable, due mainly to the unknown effects of the fractures and rock-bentonite interfaces presented in the test area and the changes the fractures caused in the initial hydro-mechanical conditions.

3

each “domain” (fracture sections for flow calculations determined by contact locations) is given by



 t V P = P 0 + Kf Q − Kf (9a) V Vm V = V − V 0 ,

The most representative discrete numerical method for coupled THM processes in fractured rocks is perhaps the UDEC code (ITASCA, 1993), with the assumption that fluid flow is conducted only through fractures. The THM coupling algorithm is therefore split so that in the rock matrix (blocks), only the one-way TM coupling, e.g. the impact of thermal stress increments and thermal expansion on stress and displacement of rock matrix by heat conduction, is considered, and in fractures, the effective stress (in full saturation sense) and convective heat transfer by fluid flow are considered. Therefore, the equations are partially uncoupled and can be solved separately with updated primary variables (displacement, fluid pressure and temperature) as the results of solution of independent equations for deformation/stress, flow and heat transfer, through a time-marching process. No special coupling parameters (such as Biot’s coefficients απ ) are needed as for the continuum case where coupling parameters must form an integral part of the constitutive laws. This is possible since the fractures are explicitly expressed in the model and flow is limited only in fractures. Therefore the constitutive laws are much simplified.

or in discrete form

 t t ∂qxt ≈− T = − cp ρ ∂xi 2cp ρ 3     qit,a + qit,b ni S k × qit = −kit

e = e0 + un

∂T ∂xi

σij = −δij Ks βT

(11b)

(12)

where Ks is the bulk modulus of the solid matrix. Heat convection due to fluid flow along fractures can be considered (Abdaliah, et al., 1995), but partial saturation and fluid phase change have not been incorporated yet since no fluid in matrix is assumed. 3.2 HM coupling of block systems-implicit DDA approach The coupling of rock block deformation and fluid flow (through connected fracture systems) was incorporated in DDA by Kim et al. (1999) and Jing et al. (2000), respectively, with the more general block deformation and discretization compatibility considered in the latter. The fluid flow in fractures are governed by the Cubic law 

ρf ge3 ∂(h − bx x) (13) q= − 12µ ∂x

(8)

where e0 is the residual hydraulic aperture of fracture of length L, and un is the normal displacement of the fracture, respectively. The fluid pressure at

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(11a)

where the heat flux follows the Fourier’s law with kit being the thermal conductivity of the rocks and qit,a , qit,b are the heat fluxes at two nodes (a and b) along a side of triangle elements in a block. The thermal stress increment by a temperature increment T is given by

Partial THM coupling by the UDEC code-explicit approach

e3 (P/L) , 12µ

(9b)

k=1

For the explicit DEM approach such as the UDEC code, the equations of motion for blocks and fracture deformation are solved by the standard dynamic relaxation approach through the contact-deformationmotion loop scheme (Hart, 1993). The flow through connected fractures are simulated by parallel plate model (or Cubic law) based on the fracture aperture, e, with the flow-rate given by q=

V + V0 2

where P 0 is the domain pressure at previous time step, Q the sum of flow rate into the domain from all surrounding contacts, Kf the bulk modulus of the fluid, t the time step, and V 0 and V the domain volumes at previous and current time step, respectively. The heat conduction equation for rock blocks is written as  

 1 ∂T ∂T ∂ ∂T ∂ = kx + ky (10) ∂t cp ρ ∂x ∂x ∂y ∂y

DEM SOLUTION OF COUPLED THM PROCESSES IN FRACTURED ROCKS – THE DISCRETE APPROACH

3.1

Vm =

the pressure information. For unconfined flow field, an iterative procedure with initial pressure is used to determine the final geometric locations of the free piezometric surface (water table). The standard DDA equation for deformable block systems can be written simply as (Jing, 1998)
    kij dj = fj (21)   where kij is the global stiffness matrix of the block system, consisting of contributions from  elastic deformation and block contacts, etc., dj is the global   nodal displacement vector and fj is the resultant global RHS vector, with contributions from mainly the boundary conditions. The combination of the equations (18) and (21) then leads to the coupled HM equation for DDA 
      T d p = qˆ
ij  j  j  j (22) kij dj = fj pj

where a correction for equivalent aperture for unparallel fractures is given by (Iwai, 1976)  1/3 em 16r 2 = em (1 + r)4 F

e=

em = (ea + eb )/2,

r = ea /eb

(14) (15)

where ea and eb are the hydraulic apertures at the two ends of the wedge-shaped fracture, respectively. Assuming that there are ni fracture segments connected at intersection i, where there exists also an external resultant recharge (or discharge) rate qis . From the law of mass conservation, the sum of total inflow and outflow rates should be equal to the recharge (positive) or discharge (negative) rate, i.e.  3 ni  ρf g eij pi − pj = qis 12µ Lij j=1

(16a)

or ni  j=1

 3 pi − pj 12µf s = q eij Lij gρf i

The solution of this equation is through a time marching process using properly selected time step t. The coupled analysis requires to perform two tasks at the end of each time step:
  1. updating the conductivity matrix Tij dj according to current values of nodal displacements, by re-calculating equivalent aperture eij and length Lij of the fracture connecting i and j.   intersections  2. updating the load vector fj pj according to the pressure distribution along the boundaries of blocks or boundary edges of elements.

(16b)

where pi and pj are the pressures at intersections i and j (j = 1, 2, …, ni ), eij and Lij the equivalent hydraulic aperture and length of the fracture segment between intersection i and j (j = 1, 2, …). The collection of all similar equations at all intersections (including the ones at boundaries with known values of pressure or flux) results in a simultaneous set of algebraic equation
    Tij pj = qˆ j (17)

The full THM coupling for DDAhas not been developed at present. However, since FEM is the basis of DDA formulation, incorporation of THM coupling using the standard FEM algorithms presented above inside each block is then much more straightforward without pausing any additional difficulty, and the heat convection along fracture due to fluid velocity can be considered using the same model as developed in Abdaliah et al. (1995). Special attention, however, is needed to matrix-fracture interaction in terms of fluid flow.

after moving the terms with known pressures
into the right-hand-side of the equation. The matrix Tij is called the conductivity matrix, with its elements defined by ni  3  eij Tii = Lij j=1

(18)



3 e Tij = − ( Lijij) , i = j, but are adjacent Tij = 0, i = j, and not adjacent i 12µ s  (eik )3 qj + pˆ k ρf g Lik

3.3 An example of application – derivation of equivalent properties of fractured rocks using DEM approach

(19)

When FEM models are used for simulating THM behavior of fractured rocks, equivalent deformability and permeability of the rock masses often need to be established with their supporting REVs, especially for large scale practical problems. Closed-form solutions exist rarely except for problems with regular fracture patterns and numerical solution must be used for general cases. Because of the fact the effect of fracture systems is the central issue of homogenisation/upscaling

n

qˆ j =

(20)

k=1

where pˆ k (k = 1, 2, …, ni ) is the known pressure at intersection k adjacent to intersection i. Solution of equation (17) will lead to values of pressures at all intersections, and the rest of unknowns (piezometric heads and flow velocity, etc.) can be obtained with

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process for deriving equivalent properties, discrete models are more natural choices for such works. The example given below contains some results from the ongoing DECOVALEX III project for deriving equivalent hydro-mechanical properties of three geological formations (Formations 1 and 2 and a fault zone) (Fig. 4) of fracture rocks using UDEC code, with data from Sellafield, UK, for fracture statistics and hydro-mechanical properties. The derived equivalent properties will then be used by the FEM code ROCMAS to investigate the impact of THM processes on transport of nuclides from a hypothetic repository to

the sea or ground surface (Fig. 4). The details of the background, material properties, fracture geometry statistics, and their treatment and results can be seen in Mas-Ivars et al. (1999), Min et al. (1999) and Min et al. (2002). In this section we only report the approach of the homogenization and results of hydraulic permeability for Formation 1. Figure 5 shows the stochastic generation of fracture system realizations for the problem, with increasing domain side lengths from 1 m to 15 m, and 10 realizations for each size according to the fracture statistics. The boundary conditions are that two constant pressures P1

Recharge 100 m asl Sea Vertical fault zone Detailed model area

500 m

Formation 2 50 m

100 m 1 km

100m Formation 1

10 m 50 m 100 m 5m

20 m Repository block

Not to Scale!

5 km

Figure 4. The global model for investigating impact of THM processes on performance assessment of a hypothetical nuclear waste repository.

Figure 5.

Stochastic generations of fracture system realisations for Formation 1.

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permeability,kxx (m2)

3.0E-13 1 2 3 4 5 6 7 8 9 10

2.5E-13 2.0E-13 1.5E-13 1.0E-13 5.0E-14 0.0E+00 0

1

2

3

5

4

6

7

9

8

10

side length of square model (m) Figure 6. Variation of permeability component Kxx with side length of the test domains.

permeability (m2)

1.5E-13 1.0E-13 5.0E-14

kxy

0.0E+00

kyx -5.0E-14 -1.0E-13 -1.5E-13 0

1

2

3

4

5

6

7

8

9

10

side length of square model (m) Figure 7. Variations permeability components Kxy and Kyx with side length of the test domains.

Probability Density

side length10 m

5m

1m

0.5 m 0.25 m

0.0E+00

5.0E-14

1.0E-13

1.5E-13

2.0E-13

2.5E-13

3.0E-13

directional permeability, kyy (m2) Figure 8.

PDFs of the Kyy with increasing side length of testing domain.

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3.5E-13

4.0E-13

5.E+06 0 330

5.E+06 0 330

30

300

60

270

0.E+00

300

90

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120 210

240 210

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90

0.E+00

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5.E+06 0

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0.E+00

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120 150

210

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side length 2 m

Figure 9.

180

30

150 180

150

side length 1 m

0

330 60

210

120 210

side length 0.5 m

30

90

0.E+00

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side length 0.25 m

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5.E+06 0 330

90

0.E+00

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5.E+06 0 330

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0.E+00

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120 150

210 180

180

side length 5 m

side length 10 m

√ Elliptical form of permeability (values are expressed as 1/ Kyy).

and P2 (P1 > P2) are described for the two opposing side of the test domain and the other two sides have linearly variable pressure conditions. Thereafter the model is rotated clockwise by a 30◦ interval to investigate the anisotropy of the permeability. The resultant values of permeability parameters, their statistics and their homogenization process with test domain size are shown in Figures 6–8, respectively. It should be noted that the permeability parameters (Figs. 6, 7) do not converge to single values but a band after reaching an approximate REV size. Therefore, distributions instead of single values of the permeability components need to be established for flow analysis by the large global model, using stochastic FEM approach. Figure 8 shows the probabilistic density functions (PDF) thus established for Kyy, assuming a Gaussian distribution. The evolution of the directional permeability, plotted as inverse of the square roots of Kyy in Figure 9, confirms that a permeability tensor can be determined with a supporting REV of a side length of 5 m, thus justifying applicability of continuum mechanics for fluid flow analysis. The work is continued currently for obtaining equivalent mechanical properties and relationships between the permeability tensor and stresses.

been impressive – especially in numerical methods, based on both continuum and discrete approaches. The progress is especially significant in the areas of representation of fracture systems, comprehensive constitutive models of fractures and interfaces, discrete element methods and coupled THM or THMC models. It appears that continuum and discrete model are more linked than before, especially when homogenization/upscaling processes are needed for characterization of fractured rock masses. Many well-verified FEM and DEM codes are developed and applied to practical problems where full or partial THM coupling is required, often with reliable results. Despite all the advances, our computer methods and codes can still be inadequate when facing the challenge of some practical problems, especially when adequate representation of rock fracture systems and fracture behavior are a pre-condition for successful modeling. Some of the issues of special difficulty and importance are: •

• • •

4 ADVANCES, TRENDS AND OUTSTANDING ISSUES

• •

Over the last three decades, advances in the use of computational methods in rock mechanics have

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Systematic evaluation of geological and engineering uncertainties and how to represent them in numerical models. Understanding and mathematical representation of large rock fractures (e.g. fracture zones). More realistic constitutive models of rock fractures with special attention to roughness effects. Quantification of fracture shape, size, connectivity and effect of fracture intersections for DEM models. Time effects (e.g. fracture creeping). Scale effects, and representation of rock mass properties and behavior as an equivalent continuum and

De Boer R. The thermodynamic structure and constitutive equations for fluid-saturated compressible and incompressible elastic porous solids. Int. J. Solids & Structures, 1998; 35(34–35): 4557–4573. Domenico PA, Schwartz FW. Physical and chemical hydrogeology. John Wiley & Sons. New York. 1990. Gawin D, Schrefler BA. Thermo-hydro-mechanical analysis of partially saturated materials. Engineering Computations, 1996; 13(7): 113–143. Hart RD. An introduction to distinct element modeling for rock engineering. In: Comprehensive Rock Engineering, J. A. Hudson (Ed. inchief), Vol. 2, Pergamon Press, Oxford, 1993, 245–261. Hassanizadeh M, Gray WG. General conservation equations for multiphase systems: 1. Averaging procedures. Adv. Water Res., 2 (1979), 131–144. Hassanizadeh M, Gray WG. General conservation equations for multiphase systems: 2. Mass momenta, energy and entropy equations. Adv. Water Res., 2 (1979), 191–203. Hassanizadeh M, Gray WG. General conservation equations for multiphase systems: 3. Constitutive theory for porous media flow. Adv. Water Res., 3 (1980), 25–40. Hassanizadeh M, Gray WG. Mechanics and theormodynamics of multiphase flow in porous media including interphase transport. Adv. Water Res., 13 (1990), 169–186. Hueckel T. Reactive plasticity for clays during dehydration and rehydration. Part 1: concepts and options. Int. J. Plasticity, 18 (2002), 281–312. ITASCA Consulting Group Lt. UDEC Manual. 1993. Iwai K. Fluid flow in simulated fractures. American Institute of Chemical Engineering Journal, 2 (1976), 259–263. Kim Y, Amadei B, Pan E. Modeling the effect of water, excavation sequence and rock reinforcement with discontinuous deformation analysis. Int. J. Rock Mech. Min. Sci., 1999; 36(7): 949–970. Kohl T, Evans KF, Hopkirk RJ, Rybach L. Coupled hydraulic, thermal and mechanical considerations for the simulation of hot dry rock reservoirs. Geothermics, 1995; 24(3): 345–359. Jing L. Formulations of discontinuous deformation analysis for block systems. Int. J. Engineering Geology, 49 (1998), 371–381. Jing L, Stephansson O, Tsang CF, Kautsky F. DECOVALEX – mathematical models of coupled T-H-M processes for nuclear waste repositories. Executive summary for Phases I, II and III. SKI Report 96:58. Swedish Nuclear Power Inspectorate, Stockholm, Sweden. 1996. Jing L, Stephansson O, Tsang CF, Knight LJ, Kautsky F. DECOVALEX II project, executive summary. SKI Report 99:24. Swedish Nuclear Power Inspectorate, Stockholm, Sweden. 1999. Jing L, Ma Y, Fang Z. Modeling of Fluid Flow and Solid Deformation for Fractured Rocks with discontinuous Deformation Analysis (DDA) Method. Int. J. Rock Mech. Min. Sci., 2000; 38(3): 343–356. Jing L, Nguyen ST (eds.). Technical Report of BMT1A. DECOVALEX III Project. 2001. Lai YM, Wu ZW, Zhu YL, Zhu LN. Nonlinear analysis for the coupled problem of temperature, seepage and stress fields in cold-region tunnels. Tunneling and Underground Space Technology, 1998; 13(4): 435–440. Lewis RW, Schrefler BA. The finite element method in the static and dynamic deformation and consolidation of

existence of the REV with complexity in aperture, width, size and shape behaviors. • Representation of interfaces (contact zones of different materials or system components, such as rock-reinforcements, rock-buffer, rock-soil, etc). • Numerical representation of engineering processes, such as excavation sequence, grouting and reinforcement. • Large-scale computational capacities. The numerical modeling for coupled THM and THMC processes played a very significant role in extending rock mechanics from an art of design and analysis of rock construction works based on “empirical” concepts of stress, failure and strength to a more “scientific” branch of engineering mechanics based on conservation laws, with integrated understanding and treatment of diverse information about geology, physics, construction technique, the environment and their interactions. Linking up with geo-chemical processes will further enhance the field of rock mechanics and rock engineering, with numerical modeling as the basic platform of development. Further extension to include biochemical, electrical, acoustic and magnetic processes have also started to appear in the literature and are an indication of future research directions.

ACKNOWLEDGEMENT The funding organizations of the DECOVALEX III project and EC supported the example works presented in this article.

REFERENCES Abdaliah G, Thoraval A, Sfeir A, Piguet JP. Thermal convection of fluid in fractured media. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 1995; 32(5): 481–490. Achanta S, Cushman JH, Okos MR. On multicomponent, multiphase thermomechanics with interfaces. Int. J. Engng. Sci., 32 (1994), 1717–1738. Bai M, Elsworth D. Coupled processes in subsurface deformation, flow and transport.ASCE Press. Reston, VA. 2000. Biot MA. General theory of three-dimensional consolidation. J. Appl. Phy., 12 (1941), 155–164. Biot MA. General solution of the equation of elasticity and consolidation for a porous material. J. Appl. Mech., 23 (1956), 91–96. Bowen RM. Compressible porous media models by use of theories of mixtures. Int. J. Engng. Sci., 20 (1982), 697–735. Charlez PA. Rock mechanics. Vol. 1-Theoretical fundamentals. Editions Technip. Paris. 1991. Charlez P, Keramsi D (eds.). Mechanics of porous media (Lecture notes of the Mechanics of Porous Media summer school, June 1994). Balkema, Rotterdam. 1995. Coussy O. Mechanics of porous media. John Wiley & Sons. Chichester. 1995.

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porous media. 2nd edition. John Wiley & Sons. Chichester. 1998. Lewis RW, Schrefler BA. The finite element method in the deformation and consolidation of porous media. John Wiley & Sons. Chichester. 1987. Loret B, Hueckel T, Gajo A. Chemo-mechanical coupling in saturated porous media: elastic-plastic behavior of homoionic expansive clays. Int. J. Solids and Structures. 2002 (in press). Mas-Ivas D, Min KB, Jing L. Homogenization of mechanical properties of fracture rocks by DEM modeling. In: Wang S, Fu B, Li Z (eds.), Frontiers of rock mechanics and sustainable development in 21st century (Proc. Of the 2nd Asia Rock Mechanics Symp. Sep. 11–14, 2001, Beijing, China). Balkema, Rotterdam 322–314. 2001. Masters I, Pao WKS, Lewis RW. Coupling temperature to a double-porosity model of deformable porous media. Int. J. Numer. Anal. Meth. Geomech., 49 (2000), 421–438. Min KB, Mas-Ivars D, Jing L. Numerical derivation of the equivalent hydro-mechanical properties of fractured rock masses using distinct element method. In: Rock Mechanics in the National Interest. Elsworth, Tinucci & Heasley (eds.), Swets & Zeitlinger Lisse, 1469–1476. 2001. Min KB, Jing, Stephansson O. Determination of the permeability tensor of fractured rock masses based on stochastic REV approach, ISRM regional symposium, 3rd Korea-Japan Joint symposium on rock engineering, Seoul, Korea, 2002. (in press) Morland LW. A simple constitutive theory for fluid saturated porous solids. J. Geophys. Res., 77 (1972), 890–900. Nithiarasu P, Sujatha KS, Ravindran K, Sundararajan T, Seetharamu KN. Non-Darcy natural convection in a hydrodynamically and thermally anisotropic porous medium. Comput. Methods Appl. Mech. Engng., 188 (2000), 413–430. Pariseau WG. Poroelastic-plastic consolidation. analytical solution. Int. J. Numer. Anal. Meth. Geomech., 23 (1999), 577–594. Renard F, Ortoleva P, Gratier JP. Pressure solution in sandstones: influence of clays and dependence on temperature and stress. Tectonophysics, 280 (1997), 257–266. Pruess K. TOUGH2 – A general purpose numerical simulator for multiphase fluid and heat flow. Lawrence Berkeley Laboratory Report LBL-29400, Berkeley, CA. 1991. Sahimi M. Flow and transport in porous media and fractured rock: from classical methods to modern approaches. VCH Verlagsgesellschaft mbH. Weinheim. 1995. Sasaki T, Morikawa S. Thermo-mechanical consolidation coupling analysis on jointed rock mass by the finite element method. Engineering. Computations, 1996; 13(7): 70–86.

Sausse J, Jacquot E, Fritz B, Leroy J, Lespinasse M. Evolution of crack permeability during fluidrock interaction. Example of the Brézouard granite (Vosges, France). Tectonophysics, 336 (2001), 199–214. Schrefler BA. Computer modelling in environmental geomechanics. Computers and Structures, 79 (2001), 2209–2223. Schrefler BA, Simoni L, Turska E. Standard staggered and staggered Newton schemes in thermo-hydro-mechanical problems. Compt. Methods Appl. Mech. Engng. 144 (1997), 93–109. Selvadurai APS (ed.). Mechanics of poroelastic media. Kluwer Academic Publishers. Dordrecht. 1996. Selvadurai APS, Nguyen TS. Mechanics and fluid transport in a degradable discontinuity. Engineering Geology, 53 (1999), 243–249. Stephansson O, Jing L, Tsang CF (eds.). Mathematical models for coupled thermo-hydro-mechanical processes in fractured media. Elsevier, Rotterdam. 1996. Thomas HR, Missoum H. Three-dimensional coupled heat, moisture and air transfer in a deformable unsaturated soil. Int. J. Numer. Meth. Engng., 44 (1999), 919–943. Tsang CF(ed.). Coupled processes associated with nuclear waste repositories. Academic Press Inc. 1987. Tsang CF. Coupled thermomechanical hydrochemical processes in rock fractures. Rev. of Geophys., 29 (1991), 537–551. von Terzaghi, K. Die berechnug der Durchlässigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen. Sitzungsber. Akad. Wiss., Math.Naturwiss, Section IIa, 1923; 132(3/4): 125–138. Whitaker S. Simultaneous heat, mass and momentum transfer in porous media: a theory of drying. Academic Press, New York. 1977. Willis-Richards J, Wallroth T. Approaches to the modelling of HDR reservoirs: a review. Geothermics, 1995; 24(3): 307–332. Yang XS, 2001. A unified approach to mechanical compaction, pressure solution, mineral reactions and the temperature distribution in hydrocarbon basins. Tectonophysics, 330 (2001), 141–151. Yeh GT, Siegel MD, Li MH. Numerical modeling of coupled variably saturated fluid flow and reactive transport with fast and slow chemical reactions. J. of Contaminant Hydrology, 47 (2001), 379–390. Zhao C, Hobbs BE, Mühlhaus HB, Ord A. Numercal modeling of double diffusion driven reactive flow transport in deformable fluid-saturated porous media with particular consideration of temperature-dependent chemical reaction rates. Engineering Computations, 2000; 17(4): 367–385.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Grand challenge of discontinuous deformation analysis A. Munjiza Department of Engineering, Queen Mary, University of London, UK

J.P. Latham Department of Earth Sciences and Engineering, Imperial College of Science Technology and Medicine, London, UK

ABSTRACT: It is now accepted that many processes in nature and industry, and problems of science and engineering, cannot be modeled using the assumption of continuum properties. In recent decades a number of formulations have been developed, based on the a priori assumption that these processes are best modeled by considering interacting discontinua. These have been accompanied with approximate computational methods such as the discrete element method (DEM) combined discrete-finite element methods (FEM/DEM) and discontinuous deformation analysis (DDA). In practice, the applicability and success of these methods, up until recently, has been greatly limited by the CPU power available. In this article the feasibility of large-scale computer simulations of discontinuum problems is investigated in the light of our recent algorithmic developments and ever-decreasing hardware constraints. We also report on the implications of our latest algorithm, which in principle enables us to model a whole range of particulate processes using real particles. A virtual game of 3D snooker (or pool) in which we can introduce a million arbitrary-shaped snooker “balls” of all sizes and as many simultaneous players as one likes is now within our reach. 1

INTRODUCTION

size of the sample. From the constitutive law and the conservation laws (for energy, momentum, mass, etc.) a set of governing equations defining the physical problem is derived. Solution of these equations was first sought in analytical form. Later, approximate numerical techniques were developed and in the last few decades, computational techniques have been applied to almost all conceivable physical systems for which continuum models have the slightest chance of success. Continuum models can only be an approximation of the real physical process or problem, and thus are only as valid as the underlying assumptions on which they are based. The most important assumption necessary for the validity of a continuum model is the assumption that the scale of the problem or the scale of the part of the problem of practical significance is much larger than some characteristic length defined by the microstructure of the material. For elastic analysis of isotropic and homogeneous solids this characteristic length is a few orders of magnitude greater than the size of molecules making up the material. For example, practical problems invoking the theory of elasticity are well represented by the continuum model. On the other hand, there exists a completely different class of problems, an example of which is the

Early in 2001, an algorithmic breakthrough was achieved which now sits on top of a decade of computational developments, allowing 3D transient dynamic modelling of particulate systems of realshaped particles1 . In short, we now have the essential algorithms to be able to model a game of 3D snooker (or pool) with a million arbitrary-shaped real particles as snooker “balls” of any size and as many simultaneous players as one likes. Whereas with spheres it is relatively simple to establish whether particles are in contact from the position of their centres and their radii, and to establish forces and trajectory paths associated with collisions, considerable algorithmic sophistication is required for collisions of rock fragment-shaped particles. 2

CONTINUA VERSUS DISCONTINUA

Formulations for solutions to problems of continuous media (for simplicity, called continuum models) are based on what are termed constitutive laws. These are that the physical properties of matter are described from the premise that the underlying microstructure of the physical matter is the same irrespective of the

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such as DEM, FEM/DEM, DDA, etc. A diverse research community is now spawning leading-edge particulate modelling tools based on these methods. The teams include collaborations with physicists, powder technologists2 , geotechnical engineers3 , mineral processing engineers4 , and petroleum reservoir engineers5 . A broad cross-section of particulate modelling applications in powder technology is given by Thornton6 .

process of filling a cube-shaped container with pebbles. One might pose the question: how many pebbles can fit into a single container? (The detailed solution will of course need to know whether the filling is by pouring, by dynamic dumping or involving vibrocompaction). In the text that follows, this problem is referred to as the container problem. The point to note about the container problem is that the size of the container (edge length) might be, say, only five times larger than the size (diameter) of a single pebble. It is therefore evident that the scale of the problem is similar to the characteristic length of the microstructure of the material. Indeed, this is also the case if the container size is 500 times the pebble size because details of the granular micromechanics between pebbles would reveal localised shear behaviour and flow cells extending perhaps over hundreds of particle diameters. 3

4 THE CHALLENGE AND THE OPPORTUNITIES In here we now introduce our approach and examine the opportunities for many such teams facing the challenge of how to model systems with many millions of particles, while at the same time accommodating particle characteristics such as realistic shapes. Up to the present time, numerical models of particulate systems that track motions have mainly used discs and superquadrics in 2D or spheres, bonded spheres and ellipsoids in 3D. These discrete approaches have struggled to approximate real behaviour for the more angular particulate systems normally encountered in nature or during mineral processing. (There will be notable examples of particulate materials such as flint beach shingle and other very rounded granular media for which reported methods have proven effective.) Simulations like those in Figure 1 are now feasible only because of a series of recent algorithmic breakthroughs which include linear search algorithms7,8 , potential contact force interaction strategies9,10,11 , discretised contact solution strategies9 , and crucially, 3D irregular body transient motion solutions1 . It is emphasised that the animation shown in Figure 1 is meant to

MODELLING DISCONTINUA

Particulate problems requiring microstructural examination similar to the container problem are of practical importance in many branches of science and engineering. The common feature of these process phenomena is that the representative volume of the physical matter whose behaviour is being modelled is either much larger or of a similar order of magnitude to the physical problem to be analysed. Thus, the necessary conditions for successful modelling using a continuum-based governing set of equations are not satisfied. Researchers and engineers have long recognised this fact. Historically the solution to problems characterised by the container problem was sought through either experimental investigations or phenomenological analytical approximations. In recent decades, a set of modelling approaches based on the a priori assumption of discontinua, has also been developed. The common feature of all these discontinua-based approaches is that no constitutive law is formulated. Instead, they take into account the physical characteristics of the building blocks of the material which may include for instance: shape of individual particles, interaction among particles, friction among particles, adhesion between particles, transient dynamics of individual particles, deformability of individual particles, etc. Sets of governing equations from such discontinuabased formulations are then solved. For very simple microstructures and shapes a set of analytical solutions is readily available (for instance some problems of packing of spheres). Unfortunately, these are only special cases and in general numerical (spatial and temporal) discretisations of the physical problem are necessary. These discretisations usually necessitate use of digital computers. The set of computational methods developed for this purpose include methods

Figure 1. Numerical simulation of initial motion, collision, bouncing off container walls, pirouetting, rocking and final rest state of real-shaped particles (3D laser-scanned pieces of rock aggregate) propelled into a container.

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era. This is not the case with the problems of discontinua because in most cases the scale of the problem was computationally prohibitive in terms of the CPU time. This is best illustrated by comparing a linear elastic analysis of a 3D elastic block of edge size measuring 1 m. The total number of finite elements needed ranges from 1000 to 30000 elements. In contrast discontinua-based analysis of a similar size container problem (container measuring 1 m filled with particles with an average size of 1 mm) would involve over one billion separate particles. In terms of CPU time many of the problems of this type are therefore in the realm of what are often described by numerical modellers as grand-challenge problems. There are two necessary conditions to make grand-challenge problems solvable in practical timescales: (1) algorithmic breakthroughs must be made and (2) large CPU power at low cost must be available. Before leaving algorithms, it is worth dwelling on the contact search problem. Up until recently the most CPU efficient search algorithms required total CPU time proportional to N log2 N , where N is the total number of elements comprising the grid. The new generation of no binary search, or “NBS-based” search algorithms7,8 requires CPU time proportional to N (i.e. are as fast as is theoretically possible). These contact search algorithms together with those described earlier for dealing with complex shapes combined with limited hardware of today has enabled us to address real problems involving granular material. This is demonstrated in Figure 2, where transient dynamic simulation of a container problem comprising over 3 million degrees of freedom is shown. Again the transient motion and final state of rest are as a result of motion and interaction of all individual particles making up the system. Cube packing experiments were also performed and these provided an ideal

illustrate the combined potential of this latest generation of algorithmic solutions. To deal with the irregular shapes, the discretised contact solution strategy we use is based on discretisation of the complex surface shape of irregular bodies into sets of simplex contact geometries. In the context of the combined finite-discrete element method, these discretisations are also used for deformability analysis, while in the case of rigid particles it is enough to discretise the surface layer only. It is often assumed implicitly that such discretisations increase CPU time. In fact the opposite is true. The number of contacts, and thus solutions per time step, is a function of the geometry of the discrete system and is not a function of contact discretisation, which is in any case meant only to simplify contact geometry. The actual situation is that discretisation speeds up contact procedures and this is the reason why systems comprising millions of degrees of freedom can be handled effectively1,7 . Algorithmic solutions for dynamically interacting discontinua1 , even ones incorporating fracturing elements, e.g. by combined discrete-finite elements12,13 have been developed. As discussed at ICADD-41,8,14,15 in June of 2001,the problems of how to detect16,7,8 and how to represent the contact interactions17,18,9 between sufficient numbers of bodies and how to include the effects of realisticshaped particles1 and display results in a manageable CPU time3,15 remain outstanding challenges for future research. The common feature of both continuum-based and discontinuum-based computational methods is the spatial grid. The size of the spatial grid for continuumbased problems (say linear elasticity) is in essence governed by the geometry of the problem. For instance, an elastic beam of 5 m span is analysed using the same number of elements as a similar beam of 0.5 m span. In contrast, the size of the spatial grid for discontinuumbased problems is governed by the microstructure of the material. For the container problem, the size of the spatial grid is largely defined by the total number of individual pebbles and their shapes. This means that if the size of the container in the container problem is increased by 10 times, the size of the grid increases by at least 1000 times. Grid size is the most important and the most challenging aspect of discontinua representation and modelling. Problems of continua do not in general involve length scale as the defining factor in representation of the physical matter and usually result in spatial grids comprising a relatively small number of elements. Such coarse grids still provide approximations of sufficient accuracy. The problems of discontinua, however, do involve this length scale and in most cases result in grids comprising millions or even billions of elements. Computational solutions for problems of continua have been developed since the dawn of the computing

Figure 2. A 3-million degrees of freedom dynamic deposition simulation showing a collapsing “cloud” of variously coloured particles and their final rest state. Animation and analysis of key diagnostics indicate the pulsating or bouncing nature of the dumping and settling process as the bulk packing density oscillates before stabilising.

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required depends on the information to be collected. In general there is a limit to what information one can collect and what one cannot. For instance it is not feasible to measure interaction forces between all the particles. In short, there is a limit to what can be observed and how much data can be recorded. In this light it would be beneficial to be able to perform a numerical experiment instead. With numerical or “virtual” experimentation, any key diagnostic information required is readily available and new perhaps simplified theories can then be developed to more efficiently approximate the physics involved.

benchmark problem from which initial confidence in the modelling technology was established. In 1998 the Royal Society held a discussion meeting on the mechanics of granular materials in engineering and earth sciences19 . Researchers with interests as diverse as pyroclastic flows on Montserrat to flow in cement and grain storage silos were assembled. One objective was to bring researchers a step closer to developing strategies that can deal with the dual solidlike and fluid-like rheologies of granular systems. We were struck by a conclusion in the summing up, that much of the particulate behaviour is related to effects occurring as a result of grain shape and that more work was required in this area if scientists were to accommodate the effects of real particle shapes within new computational tools for granular media. Scientists and engineers can now look towards building on these powerful algorithmic tools in the search for a deeper understanding of the vexing and often unpredictable behaviour of granular systems. Hitherto has been difficult because of the limitations in our ability to understand and model the complex graingrain interactions. The numerical experimentation that now seems possible will be invaluable in advancing subjects such as packing, mixing and segregation, avalanching, and spontaneous stratification20 . These subjects are considered vital for precision mixing in the pharmaceutical industry, for explaining particulate behaviour under mechanised sieving in the minerals industry and a host of natural phenomena such as sedimentation and avalanches including the enigmatic long-runout rockslides21 . Many researchers are working on simple theoretical models of these processes that would be greatly enhanced by an ability to model realistic systems and to test the validity of the more simple models. Improving an understanding of particulate behaviour and especially concepts in particulate packing has in fact triggered breakthroughs in many disciplines e.g. aeronautics, agriculture, biology, ceramics, chemical engineering, chemistry, civil engineering, composites, electrical engineering, foods, geology, mechanical engineering, medicine, metallurgy, nuclear, paint technology, pharmaceuticals, physics, polymers22 . The international research community is witnessing an explosion in the development of computer modelling for particulate systems designed to tackle their complexity. The activity is a consequence of developments in algorithmic solutions for handling systems of discrete elements and increasingly affordable CPU/RAM power, which in turn have resulted in an increasing range of science and engineering problems that appear amenable to discontinuum numerical modelling approaches (DEM, FEM/DEM, DDA). At present, problems exemplified by the container problem mentioned above can be addressed most readily by experimental means. The instrumentation

5

For many engineering and scientific applications involving discontinua, the problems at present of going down the numerical route are the massive CPU times required. For instance, only 1 CPU second per 1 mm particle in the one-metre cube container problem translates into about 32 CPU years. When these massive CPU times are translated into cost, the sums obtained are unreasonable. The encouraging fact is that the cost of CPU is going down and performance is going up. With future computer technologies and future computer architectures one can easily estimate that problems of the above scale may require very short CPU times on a 2020 generation PC. On a “1 kg ultimate laptop”23 the above mentioned container problem would only take 32 · 365 · 24 · 3600 · 10−40 = 10−31 seconds. If it is possible to extrapolate Moore’s law into the future, it would take 250 years for such a laptop to become available. It may just become available in a shorter time than it would take to solve the above problem on a present day PC set running today. The point is that, although at present some of these problems appear to be of a grand-challenge type, it is evident that in the near future some of these problems will become relatively small-scale in terms of both CPU time and computational cost.

ACKNOWLEDGEMENTS We gratefully acknowledge the Engineering and Physical Sciences Research Council of Great Britain, for their support under GR/L93454. REFERENCES [1] A. Munjiza, J.P. Latham, N.W.M. John. Transient motion of irregular 3D discrete elements. Proceedings of the Fourth International Conference on Analysis of Discontinuous Deformation (ICADD-4), 23–33,

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CONCLUSION

[2] [3]

[4] [5] [6] [7] [8] [9] [10]

[11]

Ed. Nenad Bicanic, Glasgow, Scotland UK, 6–8 June 2001 (Full paper: A. Munjiza, J.P. Latham, N.W.M. John. Transient motion of irregular 3D discrete elements. Int. J. Num. M. Eng. submitted June 2001.) Z.P. Zhang, A.B. Yu, et al. A simulation study of the effects of dynamic variables on the packing of spheres. Powder Tech. 116 (2001) 23–32. P.A. Cundall. A discontinuous future for numerical modelling in geomechanics? Proceedings of the Institution of Civil Engineers, Geotechnical Engineering. 149 (2001) 41–47. P.W. Cleary. DEM simulation of industrial particulate flows: dragline excavators, mixing and centrifugal mills. Powder Tech. 109 (2000) 83–112. P.-E. Oren, S. Bakke, O.J. Arntzen. Extending predictive capabilities to network models. SPE J. 3, (1998) 324–336. C. Thornton (Editor) Special Issue: Numerical simulations of discrete particle systems, Powder Tech. 109(1–3) (2000) 1–298. A. Munjiza, K.R.F. Andrews. NBS contact detection algorithm for bodies of similar size. Int. J. Num. M. Eng. 43 (1998) 131–149. P. Perkins, J.R. Williams. Cgrid: Neighbor searching for many body simulation. (ICADD-4, p427–438, see ref 1) A. Munjiza, K.R.F. Andrews. Discretised penalty function method in combined FEM/DEM analysis. Int. J. Num. M. Eng. 49 (2000) 1495–1520. A. Munjiza, D.R.J. Owen, N. Bicanic. A combined finite-discrete element method in transient dynamics of fracturing solids. Int. J. Engineering Computations, 12 (1995) 145–174. A. Munjiza, D.R.J. Owen, A.J.L. Crook. Energy and momentum preserving contact algorithm for general 2D and 3D contact problems. Proc. Third Intnl. Conf.

[12] [13]

[14] [15] [16] [17]

[18] [19] [20] [21] [22] [23]

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on Computational Plasticity: Fundamentals and Applications, 829–841, Barcelona, April (1995). A. Munjiza, et al. Combined single and smeared crack model in combined FEM/DEM. Int. J. Num. M. Eng. 44(1999) 41–57. A. Munjiza. Fracture, fragmentation and rock blasting models in the combined finite-discrete element method, Chapter in Fracture of Rock, Computational Mechanics Publications 1999. G.-H. Shi. Three-dimensional discontinuum deformation analysis. (ICADD-4, p1–21, see ref 1) D.R.J. Owen, Y.T. Feng. Parallel processing strategies for particulates and multi-fracturing solids. (ICADD-4, p299–313, see ref 1) Z.H. Zong, L. Nilsson. A contact search algorithm for general 3-D contact-impact problems, Comp. Struct. 34 (1990) 327–335. G.-H. Shi. Discontinuum deformation analysis – a new numerical method for the statics and dynamics of block systems, PhD Thesis, Dept. Civil Engng., Univ. of California, Berkeley (1988). J.R. Williams, G. Mustoe. Proc. 2nd U.S. Conference on Discrete Element Methods. MIT, MA (1993). D. Muir Wood, G.S. Boulton J.M. Rotter. Mechanics of granular materials in engineering and earth sciences. Phil. Trans. R. Soc. Lond. A 356 (1998) 2451–2452. H.A. Makse, S. Havlin, P.R. King, H.E. Stanley. Spontaneous stratification in granular mixtures. Nature 386 (1997) 379–382. Nature Editorial, News & Views editorial by J. Fineberg, Nature 386 (1997) 323. R.M. German, Particle Packing Characteristics, Metal Powder Industries Federation, Princetown, NJ (1989). S. Lloyd. Ultimate physical limits to computation, Nature 406 (2000) 1047–1054.

Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

High-order manifold method with simplex integration Ming Lu SINTEF Civil and Environmental Engineering, Trondheim, Norway

ABSTRACT: The original MM program developed by Genhua Shi uses a constant cover function, leading to constant strains and stresses for the triangle elements. Experience indicates such a cover function is not adequate for applications in which accurate computation of stresses is required. It is particularly the case for simulating hydraulic fracturing tests. This paper presents formulation of a complete N-order manifold method and verification examples. Also given in the paper is a potential problem associated with high-order manifold method as well as a possible solution.

1

INTRODUCTION

identical, particularly for high order MM. For instance, the basic unknowns of FEM are nodal displacements, whilst for the high order MM they are coefficients of the polynomial cover function. The nature of the right-hand “loading” vector is also different. In MM it is no longer “nodal force”, rather than the “loading terms” corresponding to the relevant displacement coefficients. Consequently, the way of handling initial stress, boundary conditions and various types of loading is different. Various terms have been used in publications for describing geometry in MM, such as mathematical cover, physical cover, manifold element, node, vertex and etc. In this paper four terms are used and defined as follows:

The original numerical manifold method (MM) invented by Shi (1997) and corresponding computer programs use a constant cover function, leading to constant strains and stresses for the triangle elements. The programs give satisfactory results for problems with crack dominating failure mode. However, experience indicates such a constant cover function is not adequate for applications in which accurate computation of stresses is required. It is particularly the case for simulating hydraulic fracturing, in which very accurate evaluation of displacements and stresses at the crack tip is absolutely essential. Chen et al (1998) proposed a high-order manifold method. In order to develop a MM program for simulating hydraulic fracturing tests, following the ideas of Shi and Chen et al, numerical manifold formulation with a complete N-order cover function is worked out at SINTEF Civil and Environmental Engineering. A computer program in Fortran is written to implement the computation and the code is verified by means of comparing to the closed form solutions. During the development, however, some problems associated with small blocks at the model boundaries are also revealed. Attempts are made to solve the problem and it is found that at least one of the alternative solutions may work in certain conditions. This paper presents the formulae of the complete N-order manifold method and a verification example. Also given in the paper is the potential problem associated with the high-order manifold method as well as a possible solution. The governing equations of the MM are similar to those of FEM. However, the basic formulations are not

•

MM element: Basic geometry generated by the code (triangle); occupied with material either fully or partly; may be cut by joint(s) such containing more than one element blocks; similar to FEM. • Element block: A part of an element fully occupied by material and cut by joints; may also occupy an entire element if the element is not cut by any joints; may be triangle or polygon. • Node: Connection points between elements; same as FEM; a geometrical point may be associated with more than one node if one of the elements connecting to the node is cut by any joint; also called physical cover in other publications. • Vertex: Apex of element block. Figure 1 illustrates the definitions. The N-order MM formulation presented in this paper is based on triangle elements. It should be mentioned that numerical integration is commonly used for FEM, whilst simplex integration

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Figure 2. Illustration of weighting function for triangle element (after Chen et al, 1998).

 a1 = x2 y3 − x3 y2 a2 = x3 y1 − x1 y3 a = x y − x y 2 1 3 1 2  b1 = y2 − y3 b2 = y3 − y1 b = y − y 3 1 2  c1 = x3 − x2 c2 = x1 − x3 c = x − x 3 2 1

Figure 1. Illustration of MM element, node, element block and vertex.

is used for Shi’s original MM program. It is true that the simplex integration gives analytical solution and is more accurate than the numerical integration. All formulae presented in this paper are based on the simplex integration. The difficulty involved in high-order MM is greatly enhanced by adopting simplex integration since all integrands have to be expressed explicitly in the polynomial form.

2 2.1

Element displacement

 u = ui1 + ui2 x + ui3 y + ui4 x2 + ui5 xy   i + ui6 y2 + · · · + uim yN v = vi1 + vi2 x + vi3 y + vi4 x2 + vi5 xy   i + vi6 y2 + · · · + vim yN

In each MM element, the displacements of a point (x, y) are computed from the weighting function wi and the displacements of three nodes of the element ui and vi :   3    w (x, y)u (x, y)       i i   u(x, y) (1) = i=1 3 v(x, y)        wi (x, y)vi (x, y)   

m=

Weighting functions are smaller than or equal to unity and their summation is always equal to unity. For the triangle elements the weight function at the nodal points is 1 and it is zero at the outer edges, with linear variation, see Figure 2. Having such defined, the weight functions for the triangle elements become      a1 b1 c1    w1  1  1   w2 = (2)  a2 b2 c2  x   w  y    a b c  3 3 3 3

(4)

(N + 1)(N + 2) 2

(5)

Coefficients of the cover function, uij and vij , are the basic unknowns. Eqn (1) can be rewritten as: U = TD   u(x, y) U= v(x, y) T
D = D1 D2 D3   ui1       vi1       ui1         ui2      ui2  Di = vi2 = .. ..           .  .      u   im    uim   vim

(3.1)

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(3.4)

m is the number of terms of the cover function which is related to the order of the cover function N as follows:

i=1

" y1 "" " y2 " " y3 "

(3.3)

xi , yi are coordinates of the three nodes of the element. For a complete N-order polynomial cover function the nodal displacement can be expressed as:

BASIC FORMULATIONS

where " "1 x1 " "  = "1 x2 " "1 x3

(3.2)

(6) (7) (8)

(9)

 uik =

uik vik



Then the element stiffness matrix for the elastic material can be computed as follows:

(10)


T = T1 T2 T3

Kij = ∫ BTi EBj dA

(11)



1 0 x 0 y 0 x 0 xy 0 y 0 · · · y 0 Ti = wi 0 1 0 x 0 y 0 x2 0 xy 0 y2 · · · 0 yN 2

2

N

E is the elastic matrix, the same as FEM. The global displacement vector can be expressed as:
T U = U1 U2 U3 · · · Un (19)



(12) 2.2


T Ui = ui1 ui2 ui3 · · · uim

Strain matrix

∂ ∂x        ε x   εy = 0    γ   xy ∂ ∂y

 0     ∂   u(x, y) ∂y   v(x, y)  ∂  ∂x 



∂  ∂x   0 =   ∂ ∂y

0

  ∂   TD = BD ∂y   ∂ 



(21) Its location in the global stiffness matrix is: Row – (i − 1) m + j; Column – (k − 1) m + l. Here index i and k are in the global system. For the explicit expressions of the integrand of elements of [Kij,kl ] and their coefficients, see Lu (2001).

Strain matrices B can be written as: 
B = B1 B2 B3

(14)


 Bi = Bi1 Bi2 Bi3 · · · Bim

(15)

2.4 Initial stress matrix In MM the equivalent force of stress at the end of a time step is transferred to the next time step. This includes the first time step in which the initial stress is the in-situ stress. The stress is computed as follows:



σ = Eε = ENL = SL (16)

  N11 N12 N13 · · · N1m N = N21 N22 N23 · · · N2m  N31 N32 N33 · · · N3m

Element stiffness matrix

 S11 S12 S13 · · · S1m S = S21 S22 S23 · · · S2m  S31 S32 S33 · · · S3m

(17)

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(24)



Similar to the FEM the global governing equilibrium equation is KU = F

(22)

where stress vector σ , strain distribution matrix N, stress distribution matrix S and coordinate vector L are defined as:
T σ = σx σy τxy (23)

Exponents mj and nj are evaluated from the location index j. For explicit expressions of general terms see Lu (2001). 2.3



(13)

∂x

 ∂(xmj ynj wi ) 0   ∂x     mj nj ∂(x y wi )     0 Bij =   ∂y      ∂(xmj ynj w ) ∂(xmj ynj w )  i i ∂y ∂x

(20)

n is the total number of nodes and m is the total number of DOFs of each node in one dimension. So the total number of DOFs for a 2-D problem is 2nm. The basic sub-matrix [Kij,kl ], which is the stiffness of jth DOF of ith node to lth DOF of kth node, is a 2 × 2 matrix and computed as follows: # [Kij, kl ] = BTij EBkl d #  # kij,kl (1, 1) dx dy kij,kl (1, 2) dx dy     # =  # kij,kl (2, 1) dx dy kij,kl (2, 2) dx dy

The strain vector can be expressed as: 

(18)

(25)

T
L = 1 x y · · · yN

2.5 Point loading matrix

(26)

Different from FEM, a point load can be applied on any location of the material in MM, not necessarily $ Px on a node. When a point load is acting on the Py location (x, y), it’s contribution to the load vector is:   P fp = T T x (34) Py

After the displacement coefficients D have been computed matrix N can be computed from Eqn (22), and then matrix S is obtained. Derivation of matrix N is tedious and interested readers are referred to Lu (2001). The stress-equivalent load vector fs is computed as follows: # (27) fS = BT σ dA

where TT is computed from Eqn (12). On the element level the force vector is corresponding to the displacement vector defined in Eqns (9) and (10) T
fS = fS1 fS2 fS3 (28)   fsxi1      fsyi1           fsi1   fsx  i2        fsi2  fSi = fsyi2 = .. .       ..    .        fs   im   fsx   im   fsyim Substituting Eqn (22) into (27) leads to # # fS = BT SL dA = GL dA

2.6 Body force matrix 

$

gx acts on an element gy block, the equivalent loading vector fg is   # g TT x dA (35) fg = gy A

When constant body force

(29)

Since T(1,2) = T(2,1) = 0, Eqn (35) can also be written as:    #  f Tij (1, 1) gx fg,ij = gij, x = dA (36) Tij (2, 2) gy fgij, y A Explicit expressions are needed that are given in Lu (2001).

(30)

where G = BT S

2.7 (31)

The contributions of the inertia force to the global stiffness matrix and load vector are given below: # 2ρ TT Tkl dA (37) [Kij,kl ] = t 2 A ij

Elements of S are constants, but elements of BT are functions of (xm yn ). The final form of matrix G can be written as Eqn (32). The explicit expressions of Gij,k can be found in Lu (2001). For the first time step, the in-situ stress is used, which is assumed to be linearly distributed, as given in Eqn (33).   G11,1 G11,2 · · · G11,m  G12,1 G12,2 · · · G12,m     ··· ··· ··· ···   G G · · · G 1m,m   1m,1 1m,2  G  21,1 G21,2 · · · G21,m   G22,1 G22,2 · · · G22,m   G= (32)  ··· ··· ··· ···    G2m,m G2m,2 · · · G2m,m     G31,1 G31,2 · · · G31,m    G · · · G G  32,1 32,2 32,m   ··· ··· ··· ···  G3m,1 G3m,2 · · · G3m,m  0 σx = σ0 (1, 1) + σ0 (1, 2)x + σ0 (1, 3)y σ 0 = σ0 (2, 1) + σ0 (2, 2)x + σ0 (2, 3)y  y0 τx = σ0 (3, 1) + σ0 (3, 2)x + σ0 (3, 3)y

fρ =

  #  2ρ fρij, x TijT Tkl dx dy Vkl = fρij, y t A

(38)

where ρ is mass density of the material, t is the time step and Vkl is the “velocity” term at the end of previous time step. Explicit expressions required by the simplex integration are given in Lu (2001). 2.8 Fixed point matrix The fixed points are handled by applying hard springs. The same as the concentrated forces, the fixed points are not necessarily at the nodal points. The contribution of the springs to the element stiffness matrix of jth DOF of ith node to lth DOF of kth node is: [Kij,kl ] = kTijT (x0 , y0 )Tkl (x0 , y0 )I

(33)

(39)

where k is the stiffness of the spring(s) and (x0 , y0 ) is the location of the fixed point. The matrix I is

Array σ0 is input data.

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Inertial force matrix

 1     0    1  0  

    0 0



0 1



0 0

if fixed in 

x&y x direction(s) y

Element I

(40)

0 1

P1(x1,y1)

The corresponding load term of jth DOF of ith node is:   f ffptij = fptij, x = kTijT (x0 , y0 )ufptij (41) ffptij, y

Figure 3.

ufpt, x is computed displacement of the ufpt, y

Illustration of block contact.

where i,k = 1,2,3 and j,l = 1–m. HIij and HIkl are for element I and J, respectively and computed as:

fixed point. 2.9

P3(x3,y3) Element J

$



where ufpt =

P2(x2,y2)

  1 y − y3 Tij (x1 , y1 ) 2 (48) x3 − x2 l     Tkl (x3 , y3 ) y1 − y2 Tkl (x2 , y2 ) y3 − y1 + HJkl = x1 − x3 x2 − x1 l l (49) % (50) l = (x2 − x3 )2 + (y2 − y3 )2 " " "1 x1 y1 " " " S0 = ""1 x2 y2 "" (51) "1 x y " 3 3

Normal contact matrix

HIij =

Assume the stiffness of the normal spring is kn , the contributions of the spring to the global stiffness matrix are: T [Kij,kl ]ii = kn HIij HIkl

(42)

T [Kij,kl ]ij = kn HIij HJkl

(43)

T [Kij,kl ]ji = kn HJij HIkl

(44)

T [Kij,kl ]jj = kn HJij HJkl

(45) 2.10 Shear contact matrix

where [Kij,kl ]ii

Sub-stiffness matrix (2 × 2) of jth DOF of ith node of element I to lth DOF of kth node of element I

Similar to the normal contact matrices, the shear contact matrices and forces are computed as follows:

[Kij,kl ]ij

Sub-stiffness matrix (2 × 2) of jth DOF of ith node of element J to lth DOF of kth node of element I

T [Kij,kl ]ii = ks HIij HIkl

(52)

T [Kij,kl ]ij = ks HIij HJkl

(53)

T [Kij,kl ]ji = ks HJij HIkl

(54)

T [Kij,kl ]jj = ks HJij HJkl

 S0 I HIij fsij = −ks l

 S0 J fsjk = −ks HJjk l

(55)

[Kij,kl ]ji

[Kij,kl ]jj

Sub-stiffness matrix (2 × 2) of jth DOF of ith node of element I to lth DOF of kth node of element J Sub-stiffness matrix (2 × 2) of jth DOF of ith node of element J to lth DOF of kth node of element J.

The normal contact force terms for nodes of element I and J are computed from Eqns (46) and (47)

 S0 I fnij HIij = −kn (46) l

J = −kn fnjk

S0 l

(47)

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(57)

where ks is the stiffness of the shear spring. HIij and HIkl are for element I and J respectively and computed as:   1 x − x2 HIij = Tij (x1 , y1 ) 3 (58) y3 − y2 l

 HJjk

(56)

Element I

P1(x1,y1)

P2(x2,y2)

P0(x0,y0)

P3(x3,y3)

Element J Figure 4.

HJkl = l=

Illustration of shear contact.

  1 x − x3 Tkl (x0 , y0 ) 2 y2 − y3 l %

(x2 − x3 )2 + (y2 − y3 )2

(59) (60)

S0 = (x1 − x0 )(x3 − x2 ) − (y1 − y0 )(y3 − y2 ) (61) 2.11

Figure 5. Two MM models: (a) coarse mesh and (b) fine mesh.

Contact friction force

polynomial integrand are derived and a computer subroutine is written to implement the integration. Detailed description can be found in Lu (2001).

The force terms resulting from the sliding of the boundary contact for nodes of element I and J are computed from following equations (see Figure 4): I = −HIij kn ds tan φ ffij

(62)

= −HJij kn ds tan φ

(63)

J ffij

4 VERIFICATION The high order MM formulation and the corresponding computer program have been tested with a number of examples and verified by means of comparing to the closed form solutions. Given below is one of the testing examples, which is a cylinder subjected to internal pressure. As shown in Figure 5, two MM models are used, one has a coarse mesh and the other has a finer one, consisting of 116 and 478 triangle elements, respectively. The basic geometrical and mechanical data are:

where kn is the stiffness of the normal contact spring, d is the penetration distance, φ is friction angle,  1 x>0 if x = 0 s = sgn(x) = 0 −1 x<0 x is the movement of point P1 relative to point P0 in the direction of P2 to P3 . HIij and HJij are calculated from Eqns (58) and (59), respectively. 3

• • • • •

SIMPLEX INTEGRATION

In FEM numerical techniques are widely used for computing integration. Alternatively, one can also chose simplex integration, which gives analytical solutions when the integration domain is a triangle or a polygon. Shi (1997) presented a general formulation of simplex integration in both 2-D and 3-D and gave explicit equations for 2-D simplex integration with the second order polynomial integrand. In order to meet the requirements of high-order Manifold, formulations for 2-D simplex integration with an arbitrary N-order

Figures 6–8 show the radial displacement, radial stress and tangential stress computed from the two models with four different cover functions. The testing indicates the MM formulation with high order cover function provides more accurate results of displacement and stress, especially the stresses. Also can be seen is that the second order cover function accompanied by a reasonably fine mesh is capable of producing adequate predictions in both displacement and stresses.

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Inner radius: 1.0 m Outer radius: 5.0 m E-modulus: 10.0 GPa Poisson’s ratio: 0.2 Pressure: 100 MPa

Figure 6.

Comparison of MM prediction with CFS, displacement. (a) Coarse model and (b) fine model.

Figure 7.

Comparison of MM prediction with CFS, radial stress. (a) Coarse model and (b) fine model.

Figure 8.

Comparison of MM prediction with CFS, tangential stress. (a) Coarse model and (b) fine model.

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Figure 9.

Figure 10.

Illustration of the problem associated with small boundary blocks: (a) MM model and (b) Model geometry at failure.

Illustration of computed geometry of elements (mathematical mesh) at various time steps.

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5 A POTENTIAL PROBLEM AND A POSSIBLE SOLUTION During development of the high order MM method a problem was encountered that is for some examples when program running to a certain number of time steps distortion of small blocks at the model boundary appears, develops and finally terminates the program execution. An example is given in Figure 9, which is a simply supported composite beam with a few preexisting vertical cracks under body force loading. The program runs smoothly until about time step 50 when distortion of two small blocks at the left boundary of the model appears. Finally, the program stopped at time step 59 due to large distortion and separation of the two blocks. The common features of the problem include:

Figure 11.

Final model geometry at time step 160.

•

Every thing seems normal for the first several (maybe tens) time steps until distortion of small boundary blocks occurs. • The distortion develops and the problematic blocks may separate from the model and finally leading to a negative block area and consequently program execution terminates. • The problem appears only on small blocks located at the model boundary. Even when failure takes place, deformation of the major inner part of the model looks normal.

steps are computed without any problems, as shown in Figure 11.

6

The MM formulation with a complete N-order cover function presented in this paper accurately computes displacements and stresses such providing a sound base for further development for simulating hydraulic fracturing tests with the numerical manifold method. The problem of small boundary blocks associated with high order MM is a fundamental and crucial problem. Despite a possible solution has been proposed in this paper further study is definitely still needed.

A great effort has been made trying to figure out the reasons that cause the problem and possible solutions. Detailed investigation reveals that by the time of computation fails the geometry of elements (mathematical covers), to which the problematic blocks belong to, is severely distorted. Figure 10 illustrates the model element geometry at some time steps. It is believed after discussion with Shi (2002) that the problem is caused by the numerical illconditioning resulting from significant discrepancy in material associated with boundary nodes to the material associated with the neighboring nodes. Detailed analysis is given in Lu (2002). Efforts have been made trying to find a way out. The following alternative might be a solution for some problems that is to merge the small boundary blocks to the neighboring “big” elements, such the problematic nodes and elements will disappear. The contribution of the small blocks to the global stiffness will now go to the nodes of the neighboring elements. The example given in Figure 9 is reanalyzed after such a treatment and 160 time

REFERENCES Chen, G., Ohnishi, Y. & Ito, T. 1998. Development of high-order manifold method. International Journal for Numerical Methods in Engineering 43: 685–713. Lu, M. 2001. Complete N-Order Polynomial Cover Function for Numerical Manifold MethodSINTEF report, STF22 F01139. Lu, M. 2002. Numerical manifold method with complete N-order cover function, Part 4 – Handling of small boundary blocks. SINTEF report, STF22 F02124. Shi, G.H. 1997. The numerical manifold method and simplex integration. Working forum on manifold method of material analysis. Volume 2, USArmy Corps of Engineers. Shi, G.H. 2002. Private communication.

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CONCLUSIONS

Case studies in rock slope and underground openings in discontinuous rock

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Experimental investigations into floor bearing strength of jointed and layered rock mass D. Kumar & S.K. Das Indian Institute of Technology, Kharagpur, W.B, India

ABSTRACT: The floor bearing strength of rock strata is markedly influenced by both the presence of layers and joints with respect to their location, strength and orientation. The paper depicted the research works carried out for the case of surface footing using physical modeling technique. In order to determine the floor bearing strength, plate loading test on simulated floor strata are carried out for varying geotechnical conditions of weak floor strata. The entire tests for floor bearing strength were conducted under confinement in the self developed testing blocks resembling the in-situ condition. The load exerted by the plate on the floor strata was assumed to be uniformly distributed. The analysis were carried out by considering the various parameters like orientation of joints and layers with respect to the direction of loading, variation in the joint sets i.e. joint spacing and the layer thickness as a function of footing plate width. The result indicates that bearing strength is significantly related to the location and orientation of joints and the layers. Bearing strength of jointed rock mass is almost 0.6 times the bearing strength of intact rock mass for joint located vertically i.e. in the direction of applied load. The ratio reduced to almost 0.3 for joint located at an angle of 80◦ with the direction of loading. With the increase in the weak sandwiched layer thickness, there is a considerable decrease in bearing strength. Further the bearing strength in the case of the joint located at the center of the footing is always higher than the joint located at the edge of footing.

1

INTRODUCTION

provide a realistic approach to the problem defined earlier, physical modeling using simulated material based on equivalent material modeling technique is considered to be a direct perceptional methodology for mining geo-mechanics study (Wang, 2000). The paper attempts to investigate the influence of joints and layers parameters on the stability of the floor strata.

Floor bearing strength plays an important role in design of strata control system. Weak floor strata are considered to be an important factor responsible for instability of the supports apart from the roof and pillar element. The rigidity and the effectiveness of a support system is based on the ability of the floor strata to take the pressure transferred by the roof rock, without yielding and letting the support system penetrate into the floor rock at the point of contact. For a weak or highly fragmented rock condition, there is a need for more rigorous methods to establish the bearing strength of the floor strata for the sake of stability and safety of the underground workings. So far floor bearing strength analysis on the layered and jointed rock mass was restricted mainly to analytical approaches primarily based on numerical modeling (Miscevic & Jasarevic, 1995) and theoretical approaches based on Hoek and Brown’s failure criteria (Serrano & Olalla, 1998). If ultimate bearing strength is estimated by numerical modeling, the results will depend on the range of load increment in steps. All these methods have their own limitations. In order to

2

Scale model studies are carried out for both the jointed and layered rock mass condition. Physical scale modeling technique (Chugh et al. 1992) has been used for the present investigation. While preparing of model floor strata following factors were taken into consideration: 1. Rock mass strength of floor strata as compared to lab strength of the equivalent material. 2. Floor strata thickness (both composite and individual layers) for model preparation. 3. Modeling of weakening planes i.e. joints.

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MODEL STUDY

Table 1. tions.

17 cm 20 mm

1 5.

2

25 mm

S. No.

Parameters

Variables

1. 2.

Shape of Floor Strata Size of Floor Strata: (a) Length (L) or width (W) (b) Thickness (t) Shape of Footing Size of Footing Circular (d) Square (l × b) Cast material for the sample preparation

Square Section

210 mm 6.

3. 4. 8.

5. 4

(a)

20 mm

Parameters and variables for laboratory investiga-

3

17 cm 8 cm (massive) Circular and Square 5 cm 5 × 5 cm2 Sand, cement, fly ash mixtures as required (Mica for intrusions wherever necessary)

210 mm

Figure 1. (a) Plan showing the testing block used to confine the model floor strata. (b) Section along X-X of the experimental setup, 1&2. Sliding frames. 3. Bolts to fix the sliding frames. 4. Fixed frames. 5. Model floor strata confined by frame. 6. Square footing of mild steel plate. 7. Bottom Platen of Compression Testing Machine. 8. Joint plane induced by mica sheet. 9. Uniform load applied to the footing plate through CTM.

Figure 2. Experimental setup arrangement for the test of ultimate bearing strength of floor strata.

load, strain at failure and the vertical settlement of footing at the time of floor strata failure, were recorded. To accomplish these purpose, strain gauges are fixed at the mid-thickness of the scaled model strata both in lateral and longitudinal direction. To record the footing settlement, magnetic dial gauge were installed as shown in the Figure 2. The tests for floor bearing strength are carried out in order to investigate the influence of following parameters on the floor bearing strength:

The model floor strata were simulated using the mixture of sand, Portland slag cement and some accelerating materials. Prior to that composition and properties of material used for the casting the model floor strata were studied in details. The similarity conditions for intact rock modeling used by Mostyn et al. (1995) were taken into consideration. Test frames are specially fabricated for confining the model floor strata (Fig. 1). The size of samples representing floor strata were decided based on the radius of influence of footing plate load and space available in the compression testing machine. The parameters and the variables used for the laboratory investigations purpose are shown in Table 1. For each test of bearing strength, parameters like longitudinal and lateral strains as a function of applied

•

Location of joint and position of the weak layers present in the floor strata under the following conditions: 1. Located at the middle of the footing. 2. Located on the edge of the footing. • Inclination of the joint and position of layers with respect to the direction of applied load. • Variation in the joint sets i.e. the joint spacing, and • Variation in the layer thickness with respect to footing width (b).

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3

GEOTECHNICAL MODELING AND EXPERIMENTAL WORK

Table 2. Characteristics of model strata and the corresponding rocks. Physico-mechanical Properties

Typical floor strata of the underground mine (Morgon Pit Mine, S.C.C.L, INDIA) was selected for the simulation. The geotechnical conditions of the floor rock as given below were used as a guideline to choose proper representative model material for simulating weak underground floor strata. Detailed geotechnical conditions of the floor strata are: Floor rock type:

coarse grained sandstone (CG SST) Uniaxial compressive strength: 15 MPa RQD: 60% Thickness of immediate floor strata: 2.7 m Poisson’s ratio (ν): 0.20 Angle of internal friction (ϕ): 40◦ Bulk density (γ ): 2000 kg/m3 The geotechnical modeling scale (C) determined was 1:25 which assisted to model floor strata up to 2 m in thickness and 4.25 m x 4.25 m in size. The bulk density (γm ) of simulated floor strata (σc = 0.60 MPa) was 1785 kg/m3 . The density scale factor Cγ =

Therefore the stress scale factor 1 1 × = 0.0357 25 1.12

The compressive strength of model rock mass is calculated as per the following Equation: (σc )m = Cσ · (σc )p

(1)

where ρm = density of model rock mass, ρp = density of prototype rock mass, (σc )m = compressive strength of model rock mass, (σc )p = compressive strength of prototype rock mass. 3.1

Square 17 × 17 8 Massive

Square 425 × 425 200 Massive 15 2000

Cement: sand (ratio) Cement: water (ratio)

0.6 1785 Sand, Cement, Water 1:1.70 1:1.80

Joint properties

Model

Prototype

Joint surface Joint aperture, mm Joint spacing Joint filling

Smooth 0.2 Remote Soft

Smooth 5 Remote Soft

Shape: Dimension (L × W), cm Thickness (t), cm Strata condition Uniaxial compressive strength (σc ), MPa Bulk density, kg/m3 Strata composition

–

a) When the joint located at the middle of surface footing. b) When the joint located at the edge of surface footing.

Effect of location and orientation of weakness plane (Joint) on the bearing strength of the floor strata

The results are shown in the Table 3. Figure 3 shows the graph of bearing strength of jointed rock versus joint inclination with respect to direction of loading for central and edge footing (surface footing). It is evident from the graph that the bearing strength (Bsj ) of jointed rocks for both case of footing location (central and edge footing) is decreasing considerably. The value is minimum for the joint inclination of β = 60◦ . This may be due to the shear failure of rock strata along the direction of weakness plane. Further

The tests for knowing impact of presence and location of weakness plane (joint) on the bearing strength of the floor strata were carried out on various sets of simulated floor strata having the same physico-mechanical properties. For the simulation of jointed rock, mica filled open joint was used. The single set of joint was introduced into the floor strata with the help of thin mica sheets at the time of casting. This is achieved

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Prototype strata.

by placing mica sheets inside the raw model material prior to compaction. An inclined rammer is used in order to make specimens with variably oriented finite size of open joint. The mould is partially filled with the mixture and is initially compacted using the inclined rammer of required orientation. Mica sheets are then placed and fixed on the artificially created and compacted surface. Further the mould is completely filled up with the cast materials and again compacted with the flat rammer. The strength and the opening of the joint were kept constant for all the tests. The same circular footing of size 5 cm was used for all the tests. The orientations of joint were varied in step of 20◦ with respect to the direction of loading i.e. vertical. The physico-mechanical properties of model strata used are given in the Table 2. The tests were carried out for both the following cases of joint location:

ρm 1785 1 = = ρp 2000 1.12

Cσ = C × Cγ =

Model strata.

Table 3. Variation in bearing strength (Bs) with respect to inclination and location of weak plane (joint). Orientation of joint with respect to loading direction i.e. vertical (Degree)

Bearing strength (MPa) Central footing

Edge footing

Central footing

Edge footing

Without joint 0 20 40 60 80

7.9 4.9 4.3 3.3 2.0 2.3

7.6 3.6 3.3 2.8 1.5 2

– 0.612 0.548 0.419 0.253 0.290

– 0.466 0.433 0.366 0.190 0.253

Bsj /Bsi

0.7 0.6

Central footing

Bsj/Bsi

0.5 0.4 0.3 Edge footing

0.2 0.1 0 0

10

20

30

40

50

60

70

80

90

Joint dipping (degree) w.r.t loading direction

Figure 3.

Influence of joint and their location on floor bearing strength.

a rising trend is seen at joint angle above 60◦ which may be due to both the direction of acting load on the floor strata and the joint plane reaches close to perpendicularity. Bearing strength (Bsj ) for jointed rock is about 0.6 times the bearing strength of the intact rock (Bsi ) when the joint is located almost vertical (β = 0◦ ). The said ratio (Bsj /Bsi ) decreased to about 0.26 at β = 60◦ in case of footing located at the center. For the joint located at edge of the footing the ratio (Bsj /Bsi ) is reduced to almost 0.45 at β = 0◦ and it is 0.2 for β = 60◦ . 3.2

Table 4.

Joint Spacing (cm)

Joint set (no. of joints present)

Floor bearing strength (MPa)

7 5 3 2 1

1 2 3 4 5

4.9 4.1 3.8 3.6 3.0

figure that there is a considerable increase in the bearing strength as the number of joints set decreases.

Influence of the joint spacing on the floor bearing strength

3.3 Influence of variation in weak layer thickness and its location

In order to find the influence of the joint spacing, bearing strength of a set of simulated rock-mass (physico-mechanical properties as depicted in Table 2) were determined. The joint spacing was varied from 1 cm to 7 cm. The strength and aperture of joint was kept constant. The same circular footing of size of 5 cm was used as surface footing located at the centre. The results are shown in the Table 4. Figure 4 shows the influence of joint spacing on bearing strength of floor strata. It is clear from the

For this purpose weak layer of fly ash was introduced in the model floor strata at the time of casting. The square plate of size 5 cm was used as surface footing for both the case of footing location i.e. when footing located at the central axis of weak layer and at the edge of the weak layer. The layer thickness were varied with respect to the footing plate width (b) i.e. b/4, b/2, 3b/4, and b. The physico-mechanical properties of model strata used for the purpose are

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Effect of joint spacing on the floor bearing strength.

6

Bearing Strength (MPa)

5

4

3

2

1

0 0

1

2

4

3

5

6

7

8

Joint spacing (cm)

Figure 4.

Influence of presence of number of joint set on floor bearing strength for centrally located footing.

shown in the Table 5 and the results are shown in Table 6. The test setup is shown in the Figure 5.

Layer properties Uniaxial compressive Strength (σc ), MPa Bulk density, kg/m3 Composition (fly ash, cement, water) Fly ash: cement (Ratio) Fly ash: water (Ratio) Layer thickness (cm)

Table 5. Characteristics of model strata and the corresponding rock. Physico-mechanical properties

Model strata

Prototype strata

3.73

Shape Dimension (L × W), cm Thickness of the strata, cm

1700 –

Strata condition

Square 17 × 17 8 (Composite) Layers are present 0.6

Square 425 × 425 2 (Composite) Layered

1785

2000

Model strata

Prototype strata

0.125 1421 1:0.250 1:0.530 1.25, 2.5, 3.75, 5

Uniaxial compressive strength of the block (σc ), MPa Bulk density of the block, kg/m3 Strata Composition Sand, Cement, Water Cement: sand (ratio) Cement: water (ratio) Footing plate (Square), cm Footing width/strata thickness (d/t), ratio

31.2, 62.5, 93.7, 125

Figure 6 shows the effect of the layer thickness and its location on floor bearing strength. It is clear from the figure that there is a decreasing trend of floor bearing strength for an increase in the layer thickness. For a narrow layer, floor-bearing strength of central footing is much more compared to the edge footing but as the layer thickness increases considerably, the trend shows quite reverse characteristics. 3.4

– 1:1.70 1:1.80 5 0.625

125 0.625

Table 6. Variation of bearing strength (Bs) with respect to layer thickness.

Influence of the layer inclination with respect to the direction of applied load

Bearing strength (MPa)

In order to find out the influence of the orientation of the layers present in the floor strata, the model strata having same properties as depicted in Table 5 was used to determine the bearing strength for the different location of layer with respect to direction of the load. The layer inclination were varied in step of 30◦ i.e. 0◦ , 30◦ , and 60◦ . The layer thickness was kept constant i.e. 2.5 cm. The same square footing plate of 5 cm was

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15

Layer thickness (cm)

Central footing

Edge footing

5.00 3.75 2.50 1.25

1.2 2.2 4.2 5.4

2.6 3.4 3.6 4.4

1800 1600

Uniformly distributed load on footing plate

1400 1200

Strain

Square footing plate (5 cm)

800 600 400

8 cm

17 cm

Lateral Strain

1000

Longitudinal Strain

200 0 17 cm

-200

Layer width = b, 3b/4, b/2, b/4

0

1

2

3

4

5

6

7

-400

Stress (MPa)

Figure 5. Diagram showing the loading on model strata with the presence of the weak layers sandwiched in it.

Figure 8. Stress vs Strain at a depth of 2 cm from the top of the floor strata in the presence of joint located in the direction of applied load (central circular footing of size 5 cm).

6

200 Lateral Strain

4

Edge Footing

100

3

0 0 2 1 0

1

2

3

4

5

6

7

Central Footing

Strain

Bearing Strength (MPa)

5

0

1

2

3

4

5

-100 -200

Longitudinal Strain

-300

6

Layer width (cm) -400

Figure 6. Influence of layer width on the floor bearing strength for both the central and edge footing.

-500

Stress (MPa)

Figure 9. Stress vs Strain at a depth of 6 cm from the top of the floor strata in the presence of joint located in the direction of applied load (central circular footing of size 5 cm).

Table 7. Variation of bearing strength (Bs) with respect to orientation of the weak layer. Orientation of weak layers with respect to Bearing strength loading direction i.e. vertical (in degree.) (MPa) 0 30 60

4.2 2.0 1.6

4.5

Bearing Strength (MPa)

4 3.5 3 2.5 2 1.5 1 0.5 0

0

10

20

30

40

50

60

70

Figure 10. Mode of failure of model weak floor strata in the presence of single joint oriented at an angle of 60◦ from the vertical i.e. the direction of applied load (for the case of edge square footing).

Layer Inclination (degree)

Figure 7. strength.

Influence of layer orientation on floor bearing

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•

•

•

Figure 11. Mode of failure of model weak floor strata in the presence of weak layer sandwiched to the floor strata in a direction parallel to the applied load (central square footing).

used as central footing for the test. The result is shown in the Table 7. Figure 7 indicates the influence of layer orientation on the floor bearing strength. A sharp decrease in the floor bearing strength was observed with an increase in the layer inclination from 0◦ to 30◦ . Further an increasing trend in bearing strength in a lesser magnitude was seen for an increase in layer orientation above 60◦ . Figure 8 and Figure 9 show the boundary of influence of footing load. It can be seen from the figures that the lateral strains are predominant in the vicinity of the footing load. As the depth increases the longitudinal strain will increase which plays significant role for the further failure of floor rock. Figure 10 and Figure 11 show the mode of failure of strata in the presence of weak plane and layers. 4

•

REFERENCES Chugh, Y.P. Singh, T.N. Singh & V.K. Ober, S. 1992. Development of an equivalent material facility, Department of Mining Engineering, Southern Illinois University, Carbondale, IL 2901: 9–83. Miscevic, P. & Jasarevic, I. 1995. Ultimate bearing capacity of strip surface footing on layered rock mass by numerical modeling, Mechanics of Jointed and Faulted Rock, Rossmanith (ed.), Balkema, Rotterdam : 633–638. Mostyn, G.R. & Bagheripour, M.H. 1995. New model material to simulate rock, Mech. of Jointed and Faulted Rock, Rossmanith (ed.), Balkema, Rotterdam: 225–230. Ohtsuka, S. 1997, Bearing capacity analysis of rock structure including cracks, Computer Methods and Advances in Geomechanics, Balkema, II: 740–745. Serrano, A. & OIalla, C. 1998, Ultimate bearing capacity of an anisotropic discontinuous rock mass, Part I basic modes of failure, Int. J. of Rock Mechanics and Mining Science, 35 (3): 301–324. Wang, C. 2000, The optimal support intensity for coal mine roadway tunnel in soft rocks, Int. J. of Rock Mechanics and Mining Science, 37: 1155–1160.

CONCLUSION

Following conclusion may be drawn from above analysis: •

Presence of a weakness plane partially controls the floor strata failure. At the same time it also causes a reduction in bearing strength. The bearing strength (Bsj ) for jointed rock mass is about 0.6 times the bearing strength of the intact rock mass (Bsi ) when joint is located almost vertical (β = 0◦ ). The ratio Bsj /Bsi decreased to about 0.3 at β = 80◦ in case of

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footing located at the center. The value is minimum for the joint inclination of β = 60◦ . This may be due to the shear failure of rock strata along the direction of weakness plane. Further a rising trend is seen at joint angle above 60◦ which may be due to both the direction of acting load on the floor strata and the joint plane reaches close to perpendicularity. For the joint located at edge of the footing, the ratio (Bsj /Bsi ) is reduced to almost 0.45 at β = 0◦ and 0.25 for β = 80◦ . The presence of joint at a close spacing causes reduction in the floor bearing strength. In such cases the behavior of the strata completely changes to plastic. There is a decreasing trend of floor bearing strength for an increase in the layer thickness. For a narrow layer floor-bearing strength of central footing is much higher compared to the edge footing but as the layer thickness increases considerably, the trend shows quite reverse characteristics. Influence of the orientation of weak layer on the floor bearing strength is also significant. The bearing strength reduced drastically for an increase in layer inclination from 0◦ to 30◦ with respect to loading direction.

Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Stability analysis for rock blocks in Three Gorges Project Wu Aiqing, Huang Zhengjia Yangtze River Scientific Research Institute, Wuhan, China

ABSTRACT: The Three Gorges Project (TGP) is the largest project in China which is being under construction today. The permanent shiplock is located in left bank of the dam, and there exist rock slopes with the maximum height more than 170 m which will be formed completely by rock cut in Tan Ziling mountain in left bank. There are more than 300 rock blocks appeared in the vertical wall of the rock slopes during the Shiplock Excavation. The TGP has a layout with an underground power house in capacity of 4,200 MW, which is located in the right bank of the dam, and it will be lately under construction after the completion of dam construction. With usage of block theory, this paper presents the stability analysis of rock blocks in the permanent shiplock and the underground powerhouse, and some improvements of Block Theory for its engineering applications have been studied.

1

INTRODUCTION TO TGP

papers. Some improvements of block theory for its engineering applications have been studied.

The Three Gorges Project, one of the greatest water conservancy projects in the world, located in Sandouping of the Yangtze River, is a multi-purpose hydro-development project producing comprehensive benefits mainly in flood control, power generation and navigation improvement. It consists of concrete main dam, two power plants behind the dam, and a double ways permanent shiplock in left bank. The main dam is a 2,331 long gravity one with a maximum height 185 m.The total storage capacity of the reservoir is 39.3 billion m3 .Two power plants, located in each side of spillway, house 26 turbine generating units with a total installed capacity of 18,200 MW which produce electricity 84.68 TW.h annually. On left bank of the river, a double ways and 5 stage continuous shiplock with 1,631 m long and 113 m total water head is constructed for navigation. It is completely cut in rock at Tan Ziling mountain and there exist high rock slopes with the maximum height more than 170 m. There are inclined rock slopes above the top of shiplock chamber and a vertical wall with height of 50–70 m below it. A middle isolated rock mound with 60 m widen exists between the two shiplock lines. Located in the right bank of river, an underground power house with capacity of 4,200 MW will be lately constructed after the completion of dam construction. The sizes with 30–32 m span and height of 88.6 m for underground house are designed. With use of block theory, the stability analysis of rock blocks in the permanent shiplock slopes and the underground powerhouse is presented in the

2

2.1 Geometric characteristic of concave block According to the shape of blocks exposed on the rock excavation surface in engineering, there are two types: convex and concave. For convex block, any apex must be at the same side of one surface. But in the concave block apexes may be at the two sides of concave surface. Although the convex analysis method is not suitable for concave, the method is utilized for analyzing the apexes of concave block. Defining the apex of concave blocks is the key problem for geometric analysis of them. There are two steps for solving this problem. First, because any concave block is made up of several convex blocks, the apexes of concave block are union sets of all apex sets of convex block if every apex set of convex block is exactly defined. Second it is judged whether the line connecting two apexes is the side of concave block or not. The volume of any three dimensional convex block can be calculated by divided it into several tetrahedron blocks and summing their volumes directly. In the same ways, the volume of a concave block which is made up of several convex blocks is the algebraical sum of volume for convex block. 2.2 Water pressure acting on block surfaces The underground water is one of many important factors affecting the block stability. But previous works

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IMPROVEMENTS ON BLOCK THEORY

and is satified with below equation S Vk > 0

(2)

Where S is slide direction, R is the exterior force and Vk is the normal vector toward block. Eq (1) represents slide direction is at the same direction with the exterior force. Eq (1) represents the block must be departed or take off the all joint surfaces. (2) Single slide Figure 1.

When block slides along joint surface i, the slide direction

Convex and concave block.

S=

(ni × R) × ni |ni × R|

(3)

and R Vi 0 S Vk > 0

(4) ki

(5)

where ni is the up normal of slide joint surface i. (3) Double slide

Figure 2. Water pressure distribution model.

If a block slides along joint surface i with joint surface j, the slide direction

or studies is only restricted to the underground water distribution model for tetrahedral blocks. Since the shape of many block is nontetrahedral, below is a new underground water distribution model for general shape block (Fig 2). The polygon in figure 2 represents a joint planar surface of one block. Because lines A5A1 and A5A4 are free surfaces, the pressures of apex A1 , A4 and A5 are zero. The water pressure is according to static water pressure distribution in the line section A1A2 , increasely linear distribution in the line section A2A3 and decreasely linear distribution in the line section A3A4 . The pressure of apex A2 and A3 is calculated by above pressure distribution. With linear interpolating function the pressure force along the joint surface is exactly defined by the pressure of apex. Then the water press on the joint surface is calculated and the direction perpendicular to the planar surface is towards the block. 2.3

S=

ni × nj sign[(ni × nj ) · R] |ni × nj |

(6)

and Si Vj 0

(7)

Sj Vi 0

(8)

S Vk > 0

kij

(9)

(4) Self lockage Each removable block must be in one of the three slide modes under one exterior force. If a block is not satisfied the one of the three slide modes, the slide mode is called self lockage. When one block is in self lockage slide, it is stable even if the friction and cohesion coefficient of joint surface are zero. Based on the results of slide mode, the stability safety factor for block is taken different formula. The stability safety factor is zero if the block is sloughage off and a block is stable if it is self lockage. The stability safety factor formula for single and double slide are found out in reference [2].

Slide mode and stability analysis

The first step for block stability analysis is judging the block slide mode and then calculating the safety stability factor. The block slide mode is divided into 4 types, that is sloughage slide, single slide, double slide and self lockage.

3

(1) Sloughage slide

ROCK BLOCKS IN ROCK WALLS OF THE PERMANENT SHIPLOCK IN TGP

One block is sloughage mode, the slide direction S = R/|R|

The excavation for main shiplock begins in spring 1996 and basically ends in spring 2000. Although

(1)

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45.6% 140

30

120

25

N Slope S Slope

20 15

Mound

10

Block Num

Block Num

40 35

80

17.0%

60

17.7% 11.1%

8.5%

40

5 0

100

20

1

2

4 Chamber No. 3

5

0 100-300

300-500

500-700

700-1000

>1000

Volume

Figure 3. The block distribution along chamber in shiplock slope.

Figure 5. Volume distribution in shiplock slope.

250

Block num

200 Gravity

150

Cable 100

water

50 0

Figure 4.

Concave block in shiplock slope.

Figure 6.

784 blocks are found out under construction, there are only 317 blocks whose volumes are greater than 100 m3 . In these blocks, there exist 171 blocks in the left shiplock, 146 ones in right and 167 ones in isolated middle mound. Because the shiplock are five stage lock, each lock chamber has different block and figure 3 is the block distribution according to lock chamber. Since there are 12 blocks which have been slided off or excavated during the construction periods, the remaining 305 blocks are studied and calculated with block theory and its code on detail in the papers. The shape of blocks which are made up of faults or joints and excavation face are mainly tetrahedron, nontetrahedron with more than 4 surfaces which consist of them and concave. Most blocks are nontetrahedrons in the entire 305 blocks. There are only 111 tetrahedron blocks with 36.4 percent and one is concave block (Fig 4). The geometric characteristic of block is mainly considered the volume and depth of block. Figure 5 is the statistic distribution diagram for volume. In the permanent shiplock block, most block volumes are distributed between 100 m3 and 500 m3 , having 62.0 percent of the entire block, and there are 54 huge blocks with a volume greater than 1,000 m3 , among which the largest volume is up to 29,658.6 m3 . The depth for 49.2 percent block is 5–10 m and there are 185 blocks whose depth is less than 10 m. It is

double

sloughage self lockage Slide

Statistic slide mode in shiplock slope.

explained that the length for the design bolt with 8–10 m is reasonable. In the gravity case, most blocks are double slide modes with 70.5 percent of the entire ones. There are 59 single slide mode blocks with 19.3 percent, and 10.2 percent blocks are in self lockage. Because of the pre-stressed cable and different exterior forces, the block slide mode is changed, for example, the slide mode from single to double and from double to self lockage. In pre-stressed cable case, the block number of single slide mode is decreased from 59 to 32 and the self lockage one is increased from 31 to 64. With the change of block slide mode, the safety stability factor of block is enhanced. In the water pressure with prestressed cable case, there are 3 blocks which take in sloughage mode due to the water pressure. Figure 6 is the statistic diagram of block slide mode. The block stability factor Kc considering the slide joints cohesion is different in 3 cases that is in the gravity, prestressed cable and water pressure with cable case. First the block is basically stable in the gravity case because the slope with stability factor greater than 1.3 is stable according to design standard value of stability factor in the high slope of shiplock in TGP. Second, nearly all block is stable under the prestressed case because the stability factor Kc of 99 percent of block is greater than 1.3. It is favourable for

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single

Table 1.

300

Joint dip dip dir. in underground house. Strike◦

Block num

250 200

Gravity cable water

150 100 50

Type

steep

0

Kc<1.

Figure 7.

1.3
2.0
Statistic stability in shiplock slope.

gentle

block stability with pre-stressed cable support. Third, 16.6 percent block is unstable in the total water head pressure with cable case. Since more block is unstable compared with the second case, the water pressure is unfavourable for block stability. Figure 7 is statistic diagram for block stability factor.

4

4.1

Dip

Dip dir.

Percentage

NNW

250 260 60 80

60 80 60 80

13.5%

NEE EW NNE

330 360 150 180 275 310 90 120 90 130

60 80 60 80 60 80 60 80 20 35

NNE

18.7% 9.3%

0.45 0.4

Probability

0.35 0.3 0.25 0.2 0.15

ROCK BLOCKS IN SURROUNDINGS OF THE UNDERGROUND POWER HOUSES IN TGP

0.1 0.05 0

Random blocks

According to joint or fissure survey in site of the underground house, the main joint sets is steep dip ones for which the percentage is 60.2 in the entire joints, and the second is middle dip ones for which the percentage is 22.7, and the third ones is the gentle having 17.1 percent. The average length for joints is generally less than 5 m, but a small number of joints have a possible length greater than 10–20 m. According to joint developed degree, the steep and middle dip joints is divided into three sets that is NNW, NEE-EW and NNE set. The gentle dip joints strike 90–130◦ and dip 20–35◦ . The strike and dip of entire joints in underground house for random block analysis are below (Table 1). For each joint set, the average dip and dip direction represent it and it is divided into two opposite joint set due to two dip directions. The axis direction of underground house is the same direction of dam that is NE43.5◦ . In calculation the upstream and downstream wall, the inner end wall, the vault arch and the face slope of underground house are simplified as a plane that all are called free excavation surface. Combined the joint sets with excavation surface, the type of keyblock is found out in the whole space stereoprojective diagram. For the dimension of keyblock, the 15 m is given up as limit length of joint. If the volume of keyblock is less than 1 m3 ,the keyblock will not include in the statistics because they are cleared away in the course of smoothing the excavation surface. Figure 8 shows the statistic distribution for

Figure 8.

10

30

50

70 90 Volume

110

130

Statistic volume of random block.

volume random keyblock in the underground house. In the gravity case, the slide mode of keyblock is studied and stability of keyblock is calculated with the same slide mode. Figure 9 is the statistic stability factor distribution of random keyblock in the underground house. According to joint investigation at present, the volume of most random keyblocks which is made up of joints is less than 100 m3 , and a small number of ones have a volume of 100–150 m3 . The depth of keyblocks is generally 2–8 m in the upstream and downstream wall, a small number of ones might have a depth greater than 8 m. The factor of keyblocks is greater than 3.0, but in the vault arch, there are a few keyblocks which will slough off or are unstable. The length of the design supported bolt must be greater than 8 m since the depth of random block is 2–8 m.

4.2 Located position blocks According to geological data in survey or exploratory adit, there exist two fault or joint sets striking NNW-SW and NEE-NW, whose intersection line will be exposed in the down wall of the house. It is

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Joint set

unfavourable for rockmass stability. On another occasion,there are 6 densely developed fissures band which have a gentle dip in the top part of downstream wall. Judged by these geological data, there are 4 located position blocks which is made up of faults and densely developed fissure bands and which are unfavourable for rockmass stability. On basis of the strike and position of faults and densely developed fissure bands, using the block theory method and its code, the 3D diagram of located position block is drawn and the geometric characteristic of block is calculated. Figure 10 is 3D diagram of the block I whose volume is 36,527 m3 .

With the force conditions, the slide mode of block is judged by usage of slide vector method, and then analysed the stability factor of block. In the gravity case, the block must be reinforced for keeping them stable. There are two ways to enhance the block stability, changing the exterior force and enhancing the cohension of sliding plane. Since the volume of block I is huge and in the gravity case the stability factor is only 0.30, various reinforcements have been taken, for example the prestress cable with each having 3000 kN plus the exchanged concrete method in which the concrete is filled the joint surface, to enhance the stability of block I. It is to enhance the cohension of the sliding joint plane,in which the cohension of the exchanged joint is the average one of joint and concrete. Table 2 is the geometric characteristic and stability analysis results of block. Since the volume of block I is up to 36 km3 and the safety of underground rockmass is directly depended on block stability. So in the design stage, a new plan in which the axis of underground house from the original site is parallelly moved down 20 m, called scheme 156, is presented. The block
and
of the 4 original blocks is fundamentally disappeared. The volume of the other two blocks is diminished and the block can keep stable with a few 3000 kN prestress cables.

0.35 0.3 Probability

0.25 0.2 0.15 0.1 0.05 0

1

3

5

7

11

18

5

Factor

Figure 9.

Figure 10.

The local stability is very important in the rock slope and tunnel engineering.The block theory is suitable for this problem on engineering rock slope and underground tunnel because there exist many joints or faults which intersect with each other in excavation planar surface. With bolts or pre-stress cables reinforcement the stability safety factor will be enhanced to meet the design stability need of unstable block. The calculating method for geometric characteristic of concave block and a vector analysis method is presented and the concrete example is given in the high rock slope of the Three Gorges Project. These are the improvement on block theory because the classic block theory is mainly the convex block.

Statistic stability factor of random block.

3D diagram of block I.

Table 2.

CONCLUSION

Geometric characterstic and stability results.

Name

Depth (m)

Volume (m3 )

Slide

Stability (f KC )

Block I

27.3

36,527

Single

0.30

Block II Block III Block IV

12.2 17.5 22.1

2,740 7,580 13,396

Double Double Double

1.41 1.77 1.22

Cable

KC

250 120∗ 15 15 45

2.02 2.07 2.64 2.26 2.05

∗ denote cable plus 50 percent exchanged concrete area of the slide joint surface.

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A general water distribution model on the surface for any shape block is advanced, which is generalization on the tetrahedronal water pressure distribution suggested by professor Hoek. With the block theory, the block stability under water pressure is studied and water pressure is unfavourable to the block stability. REFERENCES

Wu Aiqing, Zhu Hong, Li Xingguang. A Method for Block Stability Analysis Considering A General Water Pressure Distribution Model Acting on The Block surfaces [J] (in chinese), Chinese Journal of Rock Mechanics and Engineering, Vol 19(supp.), June 2000. Huang Zhengjia, Wu Aiqing, Sheng Qian, Usage of Block Theory in The Three Georges Project [J] (in chinese), Chinese Journal of Rock Mechanics and Engineering, Vol 20 No 5, June 2001.

Goodman R E, Shi G H. Block Theory and Its Aplication to Rock Engineering[M], Englewood Cliffs, N J, PrenticeHall, 1985.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Some approaches on the prediction of hillsides stability in karstic massif E. Rocamora Alvarez Group of Terrestrial Water of Geophysics and Astronomy Institute

ABSTRACT: In the westerns part of Cuba, province of Pinar del Río, is located the more important karstic region of our country, that is represented for karstics Jurassic limestone, with the singularity that the hillsides of these massif have vertical slope. Starting from the analyses of the possibility of occurrence of landslides and block fall, they were carried out engineering geologic and geotechnic studies, which dedicate their attention to the factor of underground waters and their influence in the relationship of force balance of massif. The study of the distribution and geometry of the regional and local fracturation, supplemented with the massif hydrodynamic, allowed establishing very vulnerable areas to the occurrence of movements of masses phenomenon, while other areas are more resistant to this phenomenon. This first prediction has represented the bases for detailed future studies in engineering works that are built in the area.

1

INTRODUCTION

The necessity to determine the stability of hillsides and slope, is an approach not only associated to the engineering and the construction, but at the same time it is also an environmental indicator of the massif quality. These approaches of stability depend on many factors, among those that stand out the lithology, the fracturation, the layers distributions and the presence of water, condition the movements of masses dynamics. In the analysis of the landslides and rock fall phenomenon, when they are associated to the karstic massif, is important to consider several particularities of the behavior of these mentioned factors, which influence significantly on the balance forces of the massif stability. Taking in to account that 66% of the territory of Cuba is occupated by carbonated rocks, with diverse morphological features, is very necessary that during the engineering geological investigations, is detailed on these particularities and its influence on the conditions of stability, especially in the mountainous areas, where the predominant morphology in the elevations are mogotes, with vertical hillsides.

2

GENERALITIES

In the western part of Cuba, there is an extensive territory where rocks represented for Jurassic limestone, very fractured and karstic, in where have been formed

the biggest caverns of our archipelago. In numerous areas of this territory, different engineering projects have been executed for the construction of dams, civil works, tunnels, bridges and roads, and in those places the topic of the prediction of the hillsides stability and of the slope, reaches a considerable weight inside the structural approaches of the basic engineering. In the geological engineering and geotechnic investigations carried out previously to the projects, have included the study of all rocks properties and their state conditions. The results of these analyses have established a series of aspects on the resistance properties of the rocks, the geometry and distribution of the fracturation, and the relationship fracturation-karstification, which will condition the massif hydrodynamic, as the factor in the approaches of stability. However, these aspects have a high heterogeneity, varying to local scale inside the massif, like one of the characteristics of the anisotropy of karst. In this work, we will analyze these aspects for central north sector of the Sierra de Quemado massif, in the territory of Sierra de los Organos (Picture 1).

3 ANALYSIS OF THE STRUCTURAL FACTORS OF MASSIF The resistance properties of the karstic rocks are a factor very discussed by their variability in the results and the tendency to interpretation errors. It is known that, in the karstic massif, the approach of the rocks sresistance parameter depends on the augers

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Picture 1. The East Hillside of the Sierra de Quemado gf gf max gf min

␴m i n

␴m a x

␴

Figure 1. Relationship among the parameters of γ f y σ in a study sector.

hole samples, based on physical properties or for the analyst’s estimate. Our experience is based on a primary analysis of the porosity (n), the wet density (γ f ) and the resistance parameters (dry axial compression σ or cohesion c) for a sector to evaluate, where leaving of the determination of the maximum rock resistance σmáxima with nmínima and γ fmáxima , and continuing for a much bigger number of samples that the habitual ones, we can establish a tendency of these properties, and of it to estimate the resistance (Fig. 1). The results of this procedure are applied to very local scale, without considering analogy to other areas. The estimate of a resistance value for this procedure, allows us an approach of this property to use it in the calculations of stability of hillsides and slope, eliminating this way, the possibility of false interpretations for properties that are derived of non representative samples. The most common cases, for example, are the obtaining of a low value of the softening coefficient starting from the σ dry and saturated of rocks with a different structure, the erroneous relationship for these rocks of σ dry <σ saturated and in the worst in the cases false values of resistance capacity. The results of the use false values of these rocks properties in the traditional methods of calculation of stability, could allow misinterpretation of the prediction of masses movement in the massif, those which

commonly are of estimates of landslides in massif sectors where the real phenomenon are the slab break down. In the semiquantitative procedures (Rocamora, 2001), these erroneous interpretations could be alter to the direct approaches, like to the indirect ones. In the karstic massif, the study of the relationship fracturation-karstification is more and more significant to define the occurrence of natural dangers or physical-geologic phenomenon. In the case presented, this relationship means much more than the structural stability and the balance of forces, and it represents the way of groundwater circulation. The rocks karstification processes follow an unequal way of patterns orientation of the fractures, that which establish preferential directions. According to the studies of Rocamora 1995, 1997 and 1998, in the Sierra de Quemado massif this fracturationkarstification relationship is represented by five main groups with direction 0–10◦ , 30◦ , 60◦ , 90◦ and 150◦ . When considering that the massif longitudinal axis, which are parallel to its walls, it has a direction 20–25◦ , they call the attention to the points where they can be intercepted, at least the groups with direction 60◦ , 90◦ and 150◦ that in dependence of the distribution of these cracks, they will form blocks, with a high possibility to fall . This phenomenon of blocks fall in the study area, is one of the main problems associated to the underground engineering works and in hillsides and slope, which are accentuated by the action of external factors among those that stands out the water in the cracks. In the superior sector of massif hillsides, the slabs break down not very frequent, due to a equilibrium between the drainage through this not saturated area is very quick and liberate, following the cracking and karstification system. Until the inferior levels of the massif, the slab break down happened with more frequency, but not is a common phenomenon, and in general is related to periods of intense rains and local floods. However, in the hillsides are observed areas where these described structural factors have unfavorable characteristics, and they coincide with the permanent sinkhole of fluvial water to the massif (Fig. 2). Here, the possibility of occurrence of masses movements phenomenon is high, as it has been verified by semicuantitative procedures of prediction of hillsides stability (Rocamora, 2001). In these “weak” areas of the massif, the morphology demonstrates the occurrence of collapses or landslides that are considerable volumes of rocks fall, and where the hydrodynamic factor has played a hegemonic roll (Picture 2 and Fig. 3). Also, other areas have been identified with a similar morphology, where at the present moment the superficial water flows don’t exist, but that however, the chronological reconstruction

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Sierra del Infierno

St Tomas Valley

massif

Valley cavern in superior level of karstification

superficial water flow cavern in active level of karstification

Quemado Valley

Alturas de Pizarras del Sur

massif

Valley cavern in superior level of karstification

Sierra de Quemado

superficial water flow

cavern in active level of karstification

Figure 3. Profile of the massif, where the hillsides morphology is observed in the slabs break down areas. 1 km

Curve 1

Figure 2. Map of Sierra de Quemado, where are show the areas with high possibility of occurrence of the masses movement. Curve 2

Rock

Clay of valley

Figure 4. Calculation of dejection cone stability and of the massif hillside in the area 1.

Picture 2.

Dejection cone at the vertical hillside foot.

demonstrates that they are areas associated to ancient bed river. As main objective of our geological engineering and geotechnic studies, the calculation of the stability of these areas in the massif hillside (Fig. 4), it was carried out considering to each one of them as a case refugee and evaluating several displacement surfaces. The results demonstrated the acceptable stability for the dejection cone to landslides by their base (curve 1 Fig. 4), with a security coefficient (FS) > 1.72, and

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analyzed for the normal conditions in the environment of the massif. In the analyses of surfaces of superficial landslides, the FS decreased at 1.30, which suggests that they are also stable earth masses (curve 2 Fig. 4). However for these last analyses, in the valuation of the phenomenon indicators (semicuantitative procedure) some of them took insufficient values, what could understand inside of the phenomenon dynamics, as slight haulage of blocks or local stones flows. The calculation of hillside stability, in the point where the fluvial waters sink to massif, it becomes extremely complex with the necessity of including a cavern in the theoretic model. Here were considered two potential landslides surfaces, considering an approximation to the real conditions, and the results of FS are between 3.50 and 4.30, a sufficiently stable approach for the study case. In summary, the hillsides are stable for the established normal conditions, and according to the results of semicuantitative prediction the possibility of occurrence of masses movements are low to locally moderate. These normal conditions in the massif, vary for extreme hydrometeorological events, giving place to three critical situations with an unfavorable influence of the hydrodynamic factor on the forces balance of the massif stability. These situations are: A. During events of extreme rain, the small hydrological basins in the south sector contribute the flooding into the massif producing their internal saturation. Under these conditions, the flooding of the principal basin (in the north sector) arrives which is, alone partly, assimilated by the massif that is saturated. In these conditions, the water flows is invests directional with a turbulent circulation, that provocate a critical ascent of the saturation level in the fractures and conduits at the hillsides base. B. Under these conditions of intense rains, for effect of their unequal distribution in the region, can and not to take place the previously described phenomenon, and then the flood of the valley is in dependence of the massif capacity to absorber the superficial runoff. In these cases the permanency of flood conditions in the valley causes saturation in the hillside foot. C. When the excavation takes place and the materials haulage of saturated hillside, as a result of the hydrodynamic under the conditions of the phenomenon explained previously. Under these extreme conditions, the massif properties are affected considerably, and the possibilities of occurrence of masses movements are very high. The calculations of the FS for each described critical state (A, B, C), are difficult in correspondence with the complexity to modeling the action of the factors that impact in the stability. An approximation to this problem allowed to have FS quantitative values for the massif under different conditions.

We carried out the calculations for the conditions of the case A and B (Fig. 5), where the action of the waters in inverse flow inside the massif structure was established starting from considering different physical–mechanical properties in the rocks and soils layers. The lowest values in FS (Case A 0.90–1.20, Case B 1.03–1.25) are related, in a first group, to the landslides surfaces those go by the contact between the materials of the collapse cones and the structural rock of the massif, and that they don’t arrive to its base; while a second group, are associate to the surface of structural sector of the massif. These values suggest the unstability of these massif sectors, although with dependence of the lithology layers distributions, natural compaction and local fracturation factors. In the case C, the surfaces with FS critic go by the base of the mass of socavate material (Fig. 6) that in saturation state, decrease to 40% of the values under normal conditions. The obtained FS has maximum value of 0.80 (surface 1 in figure 6), while under normal conditions of humidity were of 1.1 (surface 2 in figure 6), considering that this hillside without undermining had a FS = 1.72.

Rock

Saturation level Case A Saturation level Case B

Clay of valley

Figure 5. Landslides surfaces with more unfavorable FS for the case A and B. surface 1

Rock

Saturation level

Clay of valley

Clay of valley

Figure 6. Landslides surfaces with more unfavorable FS for the case C.

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surface 2 Rock

Although the soils and rocks saturation, in general is a condition that influences negatively about the massif stability, the analysis in the karstic massif should keep in mind the way of groundwater circulation. For the study case, given the lithologic characteristic of the Jurassic rocks, are not saturated by the sporadic conditions of flood, but rather the dynamics of the waters through conduits and open cracks, will demands from an approach of different hydrodynamic analysis and for it, the landslides surfaces that are traced in the massif don’t indicate unstability. This approach is a particularity of the karstic massif, where although the flows in cracks can mean cases of masses movements and it doesn’t have the characteristics neither the same influence level that in the non-karstic massif. However, these approaches are different in the dejection cones, like we have seen in the previous analyses, getting the attention to the base of these morphologies where after they recover the circulation way of water to the massif, and its stability is increased under saturation conditions. These questions are represented in the previous cases, if it has been noticed that the critical surfaces spread to not arriving at the massif bases, where the way of free circulation are established; associating at the levels where the clogging conditions and the fracturation-karstification width, can have variation for saturation conditions during the sporadic floods. These approaches have a wide use for the projection of engineering works, especially in the slope design, considering that the approaches of stability for slabs break down, are not the same ones that for the landslides and stones flows. 4

geotechnic approaches, based on the particularities of the physical-geologic phenomenon. The interpretation of the influence of the natural and antropics factors on the dynamics of these phenomena, are the way to minimize their development and obtaining optimal engineering solutions. BIBLIOGRAPHY Rocamora Alvarez, E. (1995): Caracterización ingenierogeológica del carso. Experiencia práctica regional. Technique report Empresa de Proyectos e Investigaciones Hidráulicas Habana, INRH. 15. Molerio León, L. F., Flores Valdés, E., Guerra Oliva, M. G, Menéndez Gómez, A, Bustamante Allen, C and Rocamora Alvarez, E. (1997): Evaluación, Uso y Protección de las Aguas subterráneas in áreas montañosas de Cuba. Technique Report Centro de Hidrología y Calidad de las Aguas, INRH. 60. Rocamora Alvarez, E. y Portuondo López, Y. (1997): Relaciones y particularidades de la fracturación y la carsificación en la Sierra de los Organos. Pinar del Río. Cuba, en Arellano, D.M; Gómez-Martín, M.A. y Antigüedad, I (eds): Investigaciones Hidrogeológicas en Cuba. País Vasco, España. pp. 155–164. Rocamora Alvarez, E. y Portuondo López, Y. (1998): Los fenómenos físico-geológicos en la evolución del relieve regional. Caso de estudio Sierra de Quemado. Proceeding of the I International Workshop de Grandes Sistemas Subterráneos de Cuba, El Caribe y Centro América, Cuba, pp. 16–20. Rocamora Alvarez, E. (2001): Pronóstico de riesgo de ocurrencia de fenómenos físico-geológicos a partir de su evaluación ingeniero-geológica. Proyecto técnico de la Agencia de Medio ambiente de Cuba, 83.

FINAL NOTE

The projection of engineering works, every time demands more than the geological engineering and

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Analysis of displacement and stress around a tunnel S. Chen Department Of Environmental Design, Hua-Fan University, Taipei, Taiwan (R.O.C)

Y.-N. Oh and D.-S. Jeng School of Engineering, Griffith University Gold Coast Campus, Gold Coast, Australia

L.-K. Chien Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan (R.O.C)

ABSTRACT: The mining industry in Australia is undergoing a transition from being dominated by open cut mining, to an increasing focus on underground mining and tunneling. Early signs of high stresses have been observed in several of the newly developed mines and this has stimulated a general interest in stress analysis for underground tunnel. The objective of this study is to investigate the stability of underground tunnel using Discontinuous Deformation Analysis (DDA). DDA is a useful tool to determine the displacement and stability of tunnel that were created by the intersection of joints in rock blocks. In the analysis of underground tunnels with lining, stability problem arises with several parameters. The first is the material behavior of the lining and the surrounding rock or soil. The second is the large deformations effect. The result presented in this paper will enable mines industry to select the appropriate stress assessment in the future.

1

INTRODUCTION

Tunnel excavation is facing many challenges since the rapid development in mining and construction engineering. Especially when it involves stability of tunnel in underground rock excavations. Rock blocks from a geological structure viewpoint can be regarded as a system of blocks cut by planes in space. The geological planes, which serve as discontinuities, could be faults, joints, or cracks. Major structural planes such as faults and large joints are the most important discontinuous planes for rock systems. The geometrical distribution and physical properties of discontinuities considerably act the mechanical behavior of a jointed rock system (Chappell, 1979; Hoek and Bray, 1977). Such rock systems are the subjects of study for several computational rock mechanics methods, including distinct element method (Cundall, 1971), block theory (Goodman and Shi, 1985) and discontinuous deformation analysis (Shi and Goodman, 1989). A fundamental deficiency of using the conventional FEM in dealing with jointed rock masses is the assumption of displacement continuity across elements. Releasing this displacement continuity assumption by using the interface element alone is not always sufficient to provide a reasonable solution. The discrete

element method starts by considering continuity, which is only maintained within an element. The discrete element method considers displacement function to be discontinuous across element boundaries (Cundall 1971). Distinct element method remains the main tool for the discrete analysis of rocks. In 1988, a new numerical method “Discontinuous Deformation Analysis (DDA)” was introduced by Shi (Shi and Goodman (1984); Shi and Goodman (1985); Shi (1988); Shi and Goodman (1988)). DDA is an implicit method. It chooses the displacements as variables and solves the equilibrium equations in the same way as FEM does. The DDA method is similar to the DEM because it also considers the rock mass as an assemblage of discrete blocks. DDA differs from the DEM in that it is displacement-based and an implicit method. A number of researchers have also applied the DDA method to a wide variety of geomechanical problems. Ke (1995) developed a DDA solution scheme to solve a rigid particle system consisting of circular discs. Shyu et al. (1997) refined the DDA program and presented the analysis of deformable particles. Koo and Chern (1998) presented a rigid body version of the DDA method and its application in rock fall simulation. Onhishi (1996) et al. described the application

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Figure 1.

Illustration sketch of tunnel in DDA blocks.

of DDA using a linear displacement function and a post-correction technique in rock fall analysis. Yeung and Goodman (1995) present the simulation of several rock slope failure problems. Lin and Chen (1997) presented applications to the soil slope stability problems. Chen et al. (1995) applied DDA to study the performance of a tunnel due to slope instability. In this study, the modeling of tunnel excavation process resulting from mining activity is presented. The approach is done by adopting DDA with concepts of effective stress and earth pressure. To start out, a tunnel is excavated from blocks elements (as shown in Fig. 1).

2 2.1

NUMERICAL FORMULATION

shear strain. These parameters describe the displacements of the centered of the block, from which the corresponding locations of the block vertices are determined. The kinematical constraints of the block system are imposed using the penalty method. Contact detection is performed in order to determine which block vertices are in contact with edges and vertices of other blocks. Numerical penalties analogous to stiff springs are applied at the contacts to prevent interpenetration of the blocks. Tension or penetration at the contacts will result in expansion or contraction of these “springs,” which adds energy to the block system. Thus, the minimum energy solution is one with no tension or penetration. When the system converges to an equilibrium state, there are very small penetrations at each contact. The energy of the penetrations can be used to calculate the contact forces, which are in turn used to determine the frictional forces along the interfaces between the blocks. DDA is developed with two-dimensional stability analyses of rocks. The rock medium is considered as an assembly of linear elastic blocks. The displacement and rotation at the center of gravity (u0 , v0 , r0 ) as well as the strain tensors εx , εy and γxy for each blocks are treated as unknowns. For the ith block, these 6 unknowns are induced in the matrix [Di ]. [Di ] = (u0 v0 r0 εx εy γxy )

Individual blocks are connected and form a system by contacts between blocks and by displacement constraints on each block. For a system having n blocks, the simultaneous equilibrium equations have the form: 

Discontinuous deformation analysis (DDA)

Discontinuous DeformationAnalysis (DDA) is similar to Finite Element Method (FEM) where the equations are listed in the similar manners (Shi and Goodman, 1989). However, DDA has a discontinuous boundary among the elements. Therefore, the discontinuous deformation behaviors among the soil particles can be simulated. In DDA, considering the equilibrium of forces, moment and kinematics for all blocks, a unique solution in displacement and deformation for each block is obtained. DDA method is based on the minimization of the total potential energy of the system of blocks. The mechanical interactions of the blocks and their surroundings are formulated in terms of the displacement parameter set. These interactions include various loadings, block inertia and elastic deformability, and displacement constraints due to block contacts and boundary conditions. The formulation of DDA incorporates a six-member displacement parameter set for each block that consists of the x- and y-displacements, rotation, x- and y-components of normal strain, and

[k]11

[k]12

··· ···

[k]n1

[k]22 .. . ···

 [k]21   .  ..

··· ···

 [k]1n     [F]1 ..  [d]1  .  .   .   ..  =  ..  (2) ..   . [d]n [F]n [k]nn

Where [F]i is a submatrix which includes the loading condition applied on the ith block. It considers body forces as well as other applied forces. [K]ij is a submatrix which defines the properties of block i. In this paper, the block are consider as linear elastic. The properties are defined by Young’s modulus, E and Poison’s ratio υ. In (2), [K]ij characterizes the contacts between blocks i and j. The contact properties include the friction angle and stiffness ratios which are related to the amount of shear and normal forces that could develop between blocks. 2.2 Stress analysis Denote (u(t)v(t))T as the time dependent displacements of element i and M as the unit mass. The force

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(1)

of inertia per unit volume can be defined as  



 ∂ 2 d0i ∂ 2 u(t) fx = −M 2 = −M [Ti ][Xi ] fy ∂t v(t) ∂t 2

500

(3)

␪ Tunnel

  where d0i are the element displacements at the beginning of time step. Assuming that constant acceleration is through current time step and the initial element displacements are zeros (begins with the updated configurations). Let  be the time interval of current step and {di } be the element displacements at the end of the time step. {di } = 

    ∂ d0i 2 ∂ 2 d0i + 2 ∂t 2 ∂t

Elastic Modulus, E=1000 kPa Elastic Modulus, E=1250 kPa Elastic Modulus, E=1500 kPa

Moment (kPa)

Angle (␪)

0 -200

-100

0

100

200

-500

-1000

(4)

-1500

Therefore, the potential energy of the inertia force of element i is given by

-2000

(a) Moment distribution around the tunnel

 fx t dx dy fy R ##  ∂ 2 {d0i } = M {di }T [Xi ]T t [Ti ]T [Ti ] dx dy [Xi ] ∂t 2 R  ## [Ti ]T [Ti ] dx dy = {di }T [Xi ]T t R 

2M 2M  0  v (5) {d } − × [Xi ] i i 2  ##

m = −

Earth pressure (kPa)

(uv)

Elastic Modulus, E=1000 kPa Elastic Modulus, E=1250 kPa Elastic Modulus, E=1500 kPa

1000

␪ Tunnel

800

600

400

200

The equilibrium equations are obtained by taking the derivatives from the potential energy m . # #  ∂m = −[Xi ]T t [Ti ]T [Ti ] dx dy − ∂{di } R

 2M 2M  0  × [Xi ] vi {di } − (6) 2  From the above equations (equation (1) to equation (6), a {fi } can be formed a 6 × 1 local force submatrix and is added to the global force matrix {F}. The earth pressure is calculated from the force submatrix with the self weight of each block. # #    2M [Xi ]T t {fi } = [Ti ]T [Ti ] dx dy [Xi ] v0i (7)  R

3

NUMERICAL EXAMPLES

In this study, the soil block is considered to be homogeneous with a unit weight of 25 kN/m3 . The Poisson ratio is 0.3. The undrained strength property along the surface is modeled by cohesion of 30 kPa. The tensile

-100

0

100

200

-200

(b) Earth pressure distribution around the tunnel Figure 2. Analysis of forces around the tunnel.

strength of soil is considered to be uniform 15 kN/m2 . Plane strain condition is considered. As shown in Fig. 1, the tunnel is excavated in the soil blocks, where, d is the diameter of tunnel, h is the soil depth above the tunnel, H and L are the height and length of the boundaries respectively. In this study, for the case example, d is 6 m, h is 9 m, H is 50 m and L is 80 m. As shown in Fig. 2, the moment and earth pressure for the tunnel. Different elastic modulus of soil from E = 1000 kPa, E = 1250 kPa and E = 1500 kPa are adopted for analysis. Those elastic modulus can be said to be tangent modulus and initial elastic modulus for soil during the excavation stage. As shown in the figure, the moment and earth pressure decrease as the soil elastic modulus increases.

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Angle (␪)

0 -200

4

CONCLUSIONS

This paper demonstrates the feasibility of modeling the process of a tunnel excavation from mining activity using Discontinuous DeformationAnalysis (DDA). The influence of soils elastic modulus is discussed with the moments and earth pressures. These results, though simplified and preliminary, do point to the potential of the approach undertaken. Further studies will be conducted with failure evolution and pore water pressure included.

REFERENCES Chappell, B.A. 1979. Deformational response in discontinua. J Rock Mech Min Sci and Geomech Abstr, 16(6):377–390. Chen, S., Chern, J.C. and Koo C.Y. 1995. “Study on Performance of Tunnel near Slope by DDA”. Proceeding of First International Conference on the Analysis of Discontinuous Deformation, Chung-li, Taiwan, 109–123. Cundall, P.A. 1971. Computer model for simulating progressive large-scale movements in blocky systems. Proceedings of the Symposium of the International Society of Rock Mechanics, Nacy, France, Vol. 1, Paper No. II-8. Duncan, J.M. and Chang, C.T. 1970. “Nonlinear Analysis of Stress and Strain in Soil”. Journal of the Soil Mechanics and Foundations Division, ASCE, 96(SM5):1629–1653. Goodman R.E. and Shi, G.H. 1985. Block theory and its application to rock engineering. Prentice-Hall. Hoek, E. and Bray, J.W. 1977. Rock slope engineering. 2nd ed. London: The Institution of Mining and Metallurgy. Ke, Te-Chih and Bray, J. 1995. “Modeling of Particulate Media Using Discontinuous Deformation Analysis,” Journal of Engineering Mechanics, 121(11): 1234–1235.

Koo, C.Y. and Chern, J.C. 1998. “Modification of the DDA Method for Rigid Block Problems”. International Journal of Rock Mechanics and Mining Sciences & Geomechanics, 36(6):683–695. Lin, J.S. and Chen, S. 1997. “Soil Slope Stability Analysis Using DDA”. Proceeding of Second International Conference on the Analysis of Discontinuous Deformation Kyoto, Japan, 239–244. Ohnishi, Y., Yamamukai, K. and Chen, G.Q. 1996. “Application of DDA in Rockfall Analysis”. Proceeding of Second North American Rock Mechanics Symposium Montreal, Quebec, Canada, 1996. 1:2031–2037. Proceedings of Computer Methods and Advances Geomechanics, Vol. 1, 469–472. Shi, G. (1988): Discontinuous deformation analysis – a new numerical model for the statics and dynamics of block systems. PhD. Dissertation, Department of Civil Engineering, University of California at Berkeley, USA. Shi, G. and Goodman R.E (1988): Discontinuous deformation analysis, a new method for computing stress, strain and sliding of block systems. Proceeding of 29th U.S. Symposium Rock Mechanics, 381–393. Shi, G. and Goodman R.E. (1985): Two dimensional discontinuous deformation analysis. International Journal of Numerical and Analytical Method in Geomechanics, 9, 541–556. Shi, G., Goodman R.E. and Tinucci, J.P. (1985): The kinematics of joint interpenetrations. Proceedings of the 26th U.S. Symposium on Rock Mechanics, 121–130. Shi, G.H. and Goodman, R.E. 1989. Two-dimensional discontinuous deformation analysis. Int J Num Analy Mech Geomech. 9(6):541–56. Shyu, K. and Salami, M.R. 1997. “Foundation Analysis with Deformable Particles Using DDA”. Yeung, R. and Goodman R. 1995. “Use of Shi’s Discontinuous Deformation Analysis on Rock Slope Problems”. Proceedings of 1995 Engineering Mechanics, 1:461–478.

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Analysis, response, prediction and monitoring of existing rock and stone monuments

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

A parametric study using discontinuous deformation analysis to model wave-induced seabed response Y-N. Oh and D-S. Jeng School of Engineering, Griffith University Gold Coast Campus, Gold Coast, Australia

S. Chen Department of Environmental Design, Hua-Fan University, Taipei, Taiwan (R.O.C)

L-K. Chien Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan (R.O.C)

ABSTRACT: In this study, a Discontinuous Deformation Analysis – Effective Stress Model (DDA-ESM) is proposed to investigate the wave-induced seabed response. Unlike previous investigations for the wave-seabed interaction, the deformation of soil block is considered to be discontinuous. This model is particularly useful for the case with large deformation. With the proposed model, deformations in foundation soil can be simulated. In this study, modifications are made to improve Discontinuous Deformation Analysis (DDA) and examples are presented in this paper. The behavior of soil under wave action is analyzed. KEYWORDS:

1

Discontinuous Deformation Analysis (DDA), seabed.

INTRODUCTION

Caisson type structures are commonly employed as breakwaters and seawalls. The main geotechnical failure features related to breakwater include circular slides, flow slide, consolidation, piping under soil, and cyclic compaction due to wave impact. Therefore, the main geotechnical mechanisms involved should be evaluated, and a more accurate method is needed to analyze the wave-induced response in porous seabed. The problem of the wave-seabed interaction has been widely studied by coastal engineers in recent years. It has been well known that gravity water waves propagating over the ocean generate significant dynamic pressure on the seabed. These pressure fluctuations further induce effective stresses and pore pressure within the soil matrix. Once the pore pressure becomes excessive with accompanying decreases in effective stresses, a sedimentary bed may becomes unstable, leading to seabed instability such as liquefaction and shear failure. Numerous offshore installations (such as pipelines, soil storage tanks, piers, breakwaters etc.) have been damaged by the wave-induced seabed instability, rather than from construction causes (Lundgren et al, 1989).

Based on the assumptions of compressible pore fluid and soil skeleton, leading to Biot’s consolidation equation, numerous investigations have been carried out in the past. Some of them have considered the seabed as an infinite medium (Madsen, 1978). Few have attempted to consider non-homogeneous soil characteristics and anisotropic soil behavior (Jeng, 1997; Kitano and Mase, 1999). All these have based on continuous deformation of the porous seabed. However, the soil deformation will become discontinuous, when the failure occurs. Under such a condition, the previous continuous model is no longer valid. Thus, a discontinuous deformation analysis model is desired. The wave-induced pore pressure may develop soil failure in the seabed. Development of a more accurate analysis technique for seabed has been a major concern of current research. This paper is aimed at developing a discontinuous deformation analysis (DDA) model for the wave-seabed interaction problem. In the model, the concept of effective stresses and pore pressure will be employed into the previous DDA model. A comprehensive comparison between the previous analytical solution (Yamamoto, 1978) and proposed model will be performed and discussed in detail.

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2

DISCONTINUOUS DEFORMATION ANALYSIS (DDA)

The concept of “Discontinuous Deformation Analysis” (DDA) was introduced by Shi (1988). DDA has a discontinuous boundary among the elements. Therefore, the discontinuous deformation behaviors among the soil particles can be simulated. In DDA, considering the equilibrium of forces, moment and kinematics for all blocks, a unique solution in displacement and deformation for each block is obtained. DDA is based on the minimization of the total potential energy of the system of blocks. The mechanical interactions of the blocks and their surroundings are formulated in terms of the displacement parameter set. These interactions include various loadings, block inertia and elastic deformability, and displacement constraints due to block contacts and boundary conditions. The current first-order formulation of DDA incorporates a six-member displacement parameter set for each block that consists of the x- and z- displacements, rotation, x- and z-components of normal strain, and shear strain. These parameters describe the displacements of the centered of the block, from which the corresponding locations of the block vertices are determined. The formulation is transient and fully dynamic. Large displacements and deformations are the result of accumulation of displacements and deformations over a number of small time steps. Fixed boundary conditions are implemented in a manner consistent with the penalty method formulation. Stiff springs similar to those at the contact points are applied at the fixed points; displacement of the fixed points adds so much energy to the block system that the minimum energy solution will be one where the fixed points do not change their position, after a small initial displacement to balance the total load on the system and the weights of the blocks. The kinematical constraints of the block system are imposed using the penalty method. Contact detection is performed in order to determine which block vertices are in contact with edges and vertices of other blocks. Numerical penalties analogous to stiff springs are applied at the contacts to prevent interpenetration of the blocks. Tension or penetration at the contacts will result in expansion or contraction of these “springs,” which adds energy to the block system. Thus, the minimum energy solution is one with no tension or penetration. When the system converges to an equilibrium state, there are very small penetrations at each contact. The energy of the penetrations can be used to calculate the contact forces, which are in turn used to determine the frictional forces along the interfaces between the blocks.

The concept of DDA developed by Shi (1988) was mainly for two-dimensional stability analyses of rocks. The rock medium is considered as an assembly of linear elastic blocks. The displacement and rotation at the center of gravity (u0 , v0 , r0 ) as well as the strain tensors εx , εz and γxz for each blocks are treated as unknowns. For the ith block, these 6 unknowns are induced in the matrix [Di ]. [Di ] = (u0 v0 r0 εx εz γxz )

Individual blocks are connected and form a system by contacts between blocks and by displacement constraints on each block. For a system having n blocks, the simultaneous equilibrium equations have the form: [k]

11

[k]12

 [k]21   ..  . [k]n1

···

[k]22 .. . ···

··· ··· ···

[k]1n      [F]1 ..  [d]1 .  .   .   . . ..   .  =  .  (2)  . [d]n [F]n [k]nn

Where [F]i is a submatrix which includes the loading condition applied on the ith block. It considers body forces as well as other applied forces. [K]ij is a submatrix which defines the properties of block i. In this paper, the block are consider as linear elastic. The properties are defined by Young’s modulus, E and Poison’s ratio υ. In (2), [K]ij characterizes the contacts between blocks i and j. The contact properties include the friction angle and stiffness ratios which are related to the amount of shear and normal forces that could develop between blocks. DDA has been rapidly developed and applied to several engineering problems. Chen (1993) applied DDA to slope stability to calculate the safety factor of slope. Huang and Ma (1992) also applied DDA to slope stability by demonstrating the failure mechanisms. Ke (1993) made some modifications in DDA with artificial joints for rock mass analysis. DDA has been applied to a wide variety of problems, ranging from tunneling (Yeung, 1991; Chen et al, 1996; Shyu et al, 1997) to dams (Chang and Monteiro, 1995; Kim et al, 1999; Kottenstettee, 1999) to soils and granular media (Thomas et al, 1995; Huang et al, 1995; Ishikawa et al, 1997) to beams and structures (Yeung, 1995; Chiou et al, 1995, Hatzor, 1999) to blasting and impact (Cai et al, 1995; Mortazavi and Katsabanis, 1999). All aforementioned investigations have been based on total stress analysis, which is suitable for rock mass, but not convenient for soil mass. In this study, we will apply the DDAto the wave-seabed interaction problem by employing the concept of effective stresses and pore pressure.

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(1)

3

METHODS AND STEPS FOR ANALYSIS

Discontinuous Deformation Analysis (DDA) has been widely used for several years to compute the response of rock and soil slopes. In this study, DDA has been adopted for wave-seabed analysis, where, a numerical model based on DDA is proposed and named as DDAESM. The analyze steps are described as follows. 3.1

Governing equation for the porous seabed

The closed-form expressions for fluid pressure terms in DDA were derived for block elements. A fluid flow method was introduced into DDA, so that water flow problems with moving boundary or free surfaces could be readily solved. The basic assumptions are: (1) The block matrix is continuous, homogenous, isotropic, linearly elastic and impermeable. (2) The mechanical process is truly dynamic with the inertia terms incorporated, but can be made quasistatic by using artificial damping techniques. (3) The flow is assumed to be laminar, in a steady state. (4) The problem is assumed to be two-dimensional for stresses and the fluid flow is essentially onedimensional along the block.

In this study, we consider a standing wave propagating over a porous seabed, as shown in Figure 1. Based on conservation of mass, the governing equation for the wave-induced seabed response that is −

∂p ∂γ 1 ∂ 2p − =B Kn ∂z 2 ∂t β∂t

(3)

where, p is the pore pressure, n is the porosity, τ is the shear stress, γ is the shear strain, B is a soil material parameter, K is the permeability coefficient, and β is the compressible modulus. 3.2 Boundary conditions for the porous seabed For a porous seabed of finite thickness, as depicted in Figure 1, the evaluation of the wave-induced pore pressure requires appropriate boundary conditions. First, zero displacements and no vertical flow occurs at the impermeable horizontal bottom (z = −h), i.e., u=v=

∂p =0 ∂z

(4)

Second, the vertical effective normal stress and shear stress vanish at the seabed surface (z = 0), i.e., σz = τxz = 0

(5)

z

and the pore pressure on the upper soil boundary is given by

L

p= d 0

3.3 Analysis of loading on blocks

h

x

P2

Block i

Figure 1.

(6)

Note that only dynamic wave pressure is included in (4). The static wave pressure is excluded in this model, because it is a constant load on the seabed.

seabed surface

P1

γw H cos kx cos ωt = po cos kx cos ωt 2 cosh kd

In this study, the seabed is characterized by blocks (as shown in Figure 1). As the pressures acting on the seabed are calculated, the pressures at each sections of the seabed are applied as loading on blocks. The loading force (Fx , Fz ) acts on point (x, z) of block i. From equation (1), the displacement of point (x, z) for block i can be written as:

 t t t t t u = [Ti ][Di ] = 11 12 13 14 15 t21 t22 t23 t24 t25 v

Definition of wave-seabed interaction.

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  d1i    d2i   t16  d3i   t26 d4i   d5i  d6i

(7)

where, (u, v) is the displacement at a point, [Ti ] displacement matrix of block i, [Di ] is the unknowns of block i and dri is the displacement variables of block i. Since the apertures vary with block movements and displacements, the load matrix of blocks should change simultaneously with pore pressure. The analysis requires updating the load matrix according to the pressure distribution along the boundary or edge of blocks. This would require extra formulations of load matrix with pore pressure. The formulation is similar to linear loads. Here, the pore pressure is always pointing towards the boundary of blocks, and is linearly distributed over a length L, in which is defined by two points (x1 , z1 ) and (x2 , z2 ). The formulation are given as x = (x2 − x1 )t + x1 ,

(8)

z = (z2 − z1 )t + z1 , % L = (x2 − x1 )2 + (z2 − z1 )2 ,

(9)

Equation (16) is the product matrix of a 6 × 2 matrix and a 2 × 1 matrix. The resulting sub matrix is then added in the matrix in equation (2). Recall equation (2), the solution of equation (1) by DDA method results in an equilibrium equation, and can be re-written as [Kij ][dj ] = [Fi { pi }]

where, the load matrix becomes a function of pore pressure ({ pi }). As the pressures acting on the seabed are calculated from equation (6), the pressures at each sections of the seabed are applied as loading on blocks. The loading force (Fx , Fz ) acts on point (x, z) of the i–th block. The wave pressure P is determined from equation (6), which is in a form suitable to apply on the seabed. The effective stress and pore pressure on seabed can be determined by superposing the result from equation (6) for the equations (8) to (18).

(10)

where, 0 ≤ t ≤ 1. The line segments with pore pressure can be expressed as a function of t, and are illustrated as follows px (t) = ( px2 − px1 )t + px1 ,

(11)

pz (t) = ( pz2 − pz1 )t + pz1

(12)

3.4 Analysis procedure The transformation of wave loading through the structure to the subsoil involves changes in soil stress

Assuming that there is a constant uniform loading {F  } = (Fx Fz )T distributing on a straight-line from point (x1 , z1 ) to point (x2 , z2 ) on i-th block. The potential energy (p ) of the point loading (Fx , Fz ) can then be simplified as p = −(Fx u + Fz v)

 F p = −(u v) x Fz p = −[Di ]T [Ti (x, z)]T

(13) (14)

 Fx Fz

(15)

To minimize p , the derivatives are computed as:

 ∂p (0) ∂ Fx = [Di ]T [Ti (x, z)]T (16) fr = − Fz ∂dri ∂dri fr , r = 1, . . ., 6, forms a submatrix as follows:  t11 t21 t12 t22     t13 t23  Fx (17) → [Fi ]  t  14 t24  Fz t15 t25  t16 t26 

Figure 2.

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(18)

Computation of wave-seabed interaction.

(pore pressure and effective stress). Particularly, in coastal area, the stress changes in soft soil will gradually develop during a long period of time. Due to these changes in soil stresses, the underlying soil layers will deform vertically and horizontally while the shear strength of the soil may reduce. As a consequence, structure built on top of the soil will deform or lose its stability. By using the mentioned calculations and measurements, a simple stability computation for seabed can be proposed. The calculation steps are shown in Figure 2.

4

NUMERICAL EXAMPLES

4.1 Input data for case study As mentioned previously, the major difference between the previous work (e.g., Yamamoto et al, 1978; Jeng, 1997) and the proposed model is the concept of discontinuous deformation. In the previous work, the deformations of all soil particles were assumed to be continuous. Thus, the wave-induced soil response under large deformation (especially near

0

-0.2

-0.2

-0.4

-0.4

z/L

z/L

0

-0.6

-0.6

-0.8

-0.8 Coarse Sand, Sr=1.0 Yamamoto et al. (1978) DDA-ESM (present solution)

Coarse Sand, Sr=1.00 Yamamoto et al. (1978) DDA-ESM (present solution)

-1

-1 0

0.2

0.6

0.4

0.8

1

p/p0

(a)

0.1

0.2

0.3

0.4

σ'x/p0 0

-0.2

-0.2

-0.4

-0.4

z/L

z/L

0

-0.6

-0.6

-0.8

-0.8 Coarse Sand, Sr=1.0 Yamamoto et al. (1978) DDA-ESM (present solution)

-1 0

(c)

0

(b)

0.2

0.4

σ'z/p0

Coarse Sand, Sr=1.0 Yamamoto et al. (1978) DDA-ESM (present solution)

-1 0

0.6

(d)

0.1

0.2

0.3

0.4

τxz/p0

Figure 3. Wave-induced seabed responses in saturated coarse sand. (a) Vertical distribution of the wave-induced seabed response ( p/p0 ) versus soil depth in a sandy bed. (b) Vertical distribution of the wave-induced seabed response (σx /p0 ) versus soil depth in a sandy bed. (c) Vertical distribution of the wave-induced seabed response (σz /p0 ) versus soil depth in a sandy bed. (d) Vertical distribution of the wave-induced seabed response (τxz /p0 ) versus soil depth in a sandy bed.

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Table 1.

Input data for case study.

be conducted with non-linear wave theory, where, the influence of wave non-linearity will be discussed.

Wave characteristics Wave period Water depth Wave steepness H/Lo

10 sec 20 m 0.05

Soil characteristics Shear modulus, G Permeability, kz Poisson ratio, µ Porosity, n Degree of saturation S

107 N/m2 10−2 m/sec 1/3 0.35 1.0

REFERENCES

the failure status) cannot be estimated. On the other hand, the soil particles are assumed to be discontinuous deformable in the proposed model. Thus, the present model can also apply to the case with large deformation. Based on the numerical model proposed in section 3, we will further compare the previous solution and the present model for the waves-induced seabed response (including pore pressure and effective stresses) in seabed. The material properties of the soil parameter and wave parameters adopted in this study are listed in Table 1. 4.2

Wave-induced soil response

Since the mechanical part of the DDA code was developed and verified long ago, the discussion in this paper is focused on only the new seabed algorithms with the pore pressure for determining the seabed response. Due to the sizes of the blocks and the model, the effect of self-weights of the soil block on seabed was included. Figure 3 illustrates the vertical distribution of the wave-induced soil response versus the soil depth (z/Lo ) in a saturated seabed. In the figure, the solid lines denote the results of previous analytical solution by Yamamoto et al. (1978), while the dashed lines are the present solution by DDA-ESM. The comparison of the present measured results and the previous calculated results (Yamamoto et al, 1978) shows an acceptable agreement.

5

CONCLUSIONS

In this paper, a new numerical model for wave-induced seabed response is proposed. Significant differences of wave-induced seabed response have been demonstrated in the numerical examples. In this paper, only some preliminary results have been presented. More advanced results such as the estimation of shear failure and detailed parametric study will be available in the near future. In addition, this paper only presents the results from linear wave theory. Further research will

Cai, Y., Liang, G-P., Shi, G. and Cook, N.G.W. Cook (1995): Studying an Impact Problem by Using LDDA Method, Proceedings of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, Berkeley, California, 288–294. Chang, C-T. and Monteiro, P.J.M. (1996): Reassessment of the St. Francis Dam Failure Using Finite Element Meshed Discontinuous Deformation Analysis, Proceedings of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, Berkeley, California, 295–301. Chen, S. (1993), “Slope Stability Assessment Based Upon Discontinuous Deformation Analysis”, Ph.D. Dissertation, University of Pittsburgh, Pittsburgh, USA. Chen, S., Chern, J.C. and Koo, C.Y. (1996): Performance prediction of tunnel excavation in clean cobble-gravel deposits by DDA method, Proceedings of the Second North American Rock Mechanics Symposium, Montreal, Canada, 2017–2024. Chiou, Y-J., Tzeng, J-C., and Lin, M-J. (1995): Discontinuous Deformation Analysis for Masonry Structures, First International Conference on Analysis of Discontinuous Deformation, Chungli, Taiwan, 288–297. Hatzor, Y.H. (1999): The Voussoir Beam Reaction Curve, Third International Conference on Analysis of Discontinuous Deformation, Vail, Colorado, 117–126. Huang,A-B., Hsiao M-C., and Lu, Y-C. (1995): DDASimulation of a Graded Particulate Assembly Under Shear, First International Conference on Analysis of Discontinuous Deformation, Chungli, Taiwan, 360–372. Huang, A.B. and Ma, M.Y. (1992), “Discontinuous Deformation Slope Stability Analysis”, Stability and Performance of Slopes and Embankments II, Geotechnical Special Publication No.31, ASCE, pp. 479–490. Ishikawa, T., Ohnishi, Y., and Namura, A. (1997): DDA applied to deformation analysis of coarse granular materials (ballast), Second International Conference on Analysis of Discontinuous Deformation, Kyoto, Japan, 253–262. Jeng, D.S. (1997), “Wave-Induced Seabed Response in Front of a Breakwater”, PhD thesis, The University of Western Australia. Ke, T-C. (1993), “Simulated Testing of Two-Dimensional Heterogeneous and Discontinuous Rock Masses Using Discontinuous Deformation Analysis”, Ph.D. Dissertation, University of California, Berkeley, USA. Kianto, Y. and Mase, H. (1999), “Boundary layer approximation for wave-induced pore pressure of anisotropic seabed”, Journal of Waterway, Port, Coastal and Ocean Engineering, A.S.C.E., 125(1), 187–194. Kim, Y.I.,Amadei, B., and Pan, E. (1999): Modeling the effect of geology on uplift in concrete gravity dam foundations with the Discontinuous Deformation Analysis, Proceedings of the 37th U.S. Rock Mechanics Symposium, Vail, Colorado, 527–534. Kottenstettee, J.T. (1999): DDAAnalysis of the RCC Modification for Pueblo Dam, Third International Conference on

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Analysis of Discontinuous Deformation, Vail, Colorado, 127–132. Lundgren, H., Lindhardt, J.H.C., Romhild, C.J. (1989), “Stability of breakwaters on porous foundation”. Proceeding 12th International Conference on Soil mechanics and Foundation Engineering, 1, 451–454. Madsen, O.S. (1978), “Wave-induced pore pressures and effective stresses in a porous bed”, Geotechnique, 28(4), 377–393. Mortazavi, A., and Katsabanis, P.D. (1999): Application of Discontinuous Deformation Analysis to the modelling of rock blasting in mining, Proceedings of the 37th U.S. Rock Mechanics Symposium, Vail, Colorado, 543–550. Shi, G-H. (1988), “Discontinuous Deformation Analysis – A New Numerical Model for the Statics and Dynamic of Block System”, Ph.D. Dissertation, University of California, Berkeley, USA. Shyu, K., Chang, C-T. and Salami, M.R. (1997): Tunnel engineering applications using discontinuous deformation analysis with finite element mesh, Second International

Conference on Analysis of Discontinuous Deformation, Kyoto, Japan, 218–237. Thomas, P., Bray, J.D., and Ke, T-C. (1996): Discontinuous Deformation Analysis for Soil Mechanics, Proceedings of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, Berkeley, California, 454–461. Yamamoto, T., Koning, H.L., Sellmejjer, H. and Hijum, E.V. (1978): On the response of a poro-elastic bed to water waves. Journal of Fluid Mechanics, 87, 193–206. Yeung, M.R. (1991): Application of Shi’s Discontinuous Deformation Analysis to the Study of Rock Behavior, PhD. Dissertation, Department of Civil Engineering, University of California, Berkeley. Yeung, M.R. (1995): Analysis of Three-Hinged Beam Using DDA, Proceedings of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, Berkeley, California, 462–469.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Simulations of underground structures subjected to dynamic loading using the distinct element method J.P. Morris, L.A. Glenn, F.E. Heuzé, S.C. Blair Geophysics and Global Security Division, Lawrence Livermore National Laboratory, Livermore, U.S.A.

ABSTRACT: We present results from a parameter study investigating the stability of underground structures in response to ground shock. Direct simulation requires detailed knowledge of both the facility itself and the surrounding geology. In practice, however, key details of the geology, rock properties and reinforcement may not be available. Thus, in order to place bounds upon the predicted behavior of a given facility, an extensive series of simulations representing different realizations may be required. We have performed simulations of subterranean facilities using the Livermore Distinct Element Code (LDEC). This paper presents a description of the method, with emphasis on techniques for achieving improved computational efficiency, including the handling of contact detection and approaches to parallelization. In addition, some continuum approaches to the simulation of underground facilities are discussed along with results from underground explosions. Finally, results of LDEC simulations of dynamic loading of generic subterranean facilities are presented, demonstrating its suitability for this application.

1

INTRODUCTION

Continuum mesh-based methods have been applied successfully to many problems in geophysics. Even if the geology includes fractures and faults, when sufficiently large length scales are considered a continuum approximation may be sufficient. However, a large class of problems exist where individual rock joints must be taken into account. This includes problems where the structures of interest have sizes comparable with the block size. In addition, it is possible that while the structure may experience loads which do no measurable damage to individual blocks, some joints may fail. A continuum, mesh-based treatment of such systems is usually inappropriate. We employ the Distinct Element Method (DEM), as defined by Cundall & Hart (1992). By nature, the distinct element method can readily handle large deformation on the joints. In addition, the method detects all new contacts between blocks resulting from relative block motion. The Lagrangian nature of the DEM simplifies tracking of material properties as blocks of material move. It is also possible to guarantee exact conservation of linear and angular momentum. The joint models can be very flexible and can incorporate experimentally observed effects such as cohesion, joint dilation, friction angle, and hysteresis (Heuzé, Walton, Maddix, Shaffer & Butkovich, 1993).

The DEM has been applied to a wide range of problems in geomechanics. For example, Antonellini & Pollard (1995) simulated the formation of shear bands in sandstone using the DEM. Morgan (1999a, 1999b) applied the DEM to the mechanics of granular shear zones. Heuzé, Walton, Maddix, Shaffer & Butkovich (1993) used the DEM to analyze explosions in hard rock. Cundall (2001) reviews the application of the DEM to simulation of granular material and rock. 2

We use the “Common-Plane” (Cundall 1988) approach to reduce the complexity of the contact detection algorithm. The iterative procedure of the common-plane approach is easy to implement and is very efficient for many classes of problem. This is because the common-plane orientation from the previous time-step typically provides a good initial guess of the current orientation. Provided the appropriate normal of the contact has not changed much between time steps, the iterative procedure converges rapidly. The number of distinct elements used in a single simulation is limited by the available computational power (both processor speed and available memory). We chose to use an approach similar to Cleary & Sawley (1999) and parallelized the DEM through

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OUR DEM IMPLEMENTATION

Figure 1. An excavation, reinforced with rockbolts within tuff, collapsed at low stress.

spatial domain decomposition. The entire problem domain is divided into nearest neighbor cells which are used to identify neighboring blocks which are potential contacts. Each processor is assigned a contiguous region of nearest neighbor cells. Communication occurs via message passing (MPI) at the start of each time step. All blocks within neighboring cells are copied between processors. To reduce the amount of time wasted during communication, each processor performs calculations on blocks which do not directly interact with neighboring processors while communication occurs. Duplicate calculations are performed on each processor in the region of overlap where blocks are copied back and forth. Consequently, speedup is best for larger problems where the region of overlap between processors is a smaller fraction of the total work performed. 3

SIMULATING BURIED STRUCTURES

To predict damage sustained by underground structures, several coupled regions must be modeled. In the immediate vicinity of an explosion, the ground shock is sufficient to rubblize the rock, material strength is irrelevant, and the material behavior is hydrodynamic. Further from the explosion, material strength becomes important. Finally, in the vicinity of the facility, the detailed structure of the rock mass and the excavation itself are important. Traditionally a rock mass is deemed to fail when the strength of the material is exceeded. Failed rock

is no longer able to withstand load without undergoing inelastic strains. However, hard rock strength increases markedly with increased pressure and yet it has been observed that functional damage or even complete tunnel collapse can occur at stress levels far below those previously thought to be required. For example, Figure 1 shows the collapse of an excavation in tuff subjected to loads significantly lower than the compressive strength of the rock. In this example, the discrete nature of the rock mass is evident and failure has occurred through block displacement. Clearly, the orientation, spacing, and shear strength of geologic discontinuities (joints) can control the behavior of a tunnel. Hard rock joints dilate strongly before reaching peak strength, after which the strength drops rapidly with increased loading. As a result of the controlling effects of the joints it is not possible to estimate tunnel response via continuum based analysis alone. Our approach is to combine continuum and discrete numerical methods by applying each in different regions of the problem. Typically, the depth of the tunnel is large compared with the size of the blocks making up the rock, and continuum approaches have been very successful in reproducing measured attenuation rates from the source. Lomov, Antoun & Glenn (2001) present an approach for accurately modeling projectile penetration and explosions in rock media. Using an Eulerian code (GEODYN) Lomov,Antoun & Glenn (2001) fit a constitutive model (Rubin, Vorobiev & Glenn, 2000) to peak velocity and displacement attenuation data from tamped (buried)

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Figure 2. A simulation of a tunnel in jointed rock subjected to a free-field peak particle velocity of 4 m/s. The simulation predicts that a substantial portion of the tunnel roof will collapse, making the tunnel unusable.

Figure 3. (a) A hypothetical underground facility in a jointed rock island simulated by distinct elements. (b) Results indicate extensive damage to the facility when subjected to a free-field peak particle velocity of 4 m/s.

nuclear explosions in granitic rock. This continuum treatment was able to reproduce peak velocity and displacement from tamped explosions in granitic rocks to within a factor of two over ten orders of magnitude in yield. The velocity or stress history predicted by GEODYN at a given point can be used to provide boundary conditions for a DEM simulation of the response of the underground facility. For example, Figure 2 shows a tunnel in jointed rock. The average block size is approximately 1 m. The jointed rock mass is confined by 7.5 Mpa of lateral and vertical stress. The rock island was subjected to a free-field peak particle velocity of 4 m/s. The simulation predicts substantial caving of the tunnel roof. We have also performed simulations of more complete underground structures. Figure 3a shows a generic underground facility in a jointed rock island. The code predicts extensive roof collapse and substantial block motion in the floor (Figure 3b). The joint structure is realistic, with non-orthogonal joint planes. Blocks are free to move and make new contacts with other blocks in the simulation.

In practice, however, only limited knowledge of local fault zones may be available. To provide bounds on the response, one must study a range of probable fault geometries. That is, a stochastic analysis with many realizations is required to obtain adequate statistics to bound results. Future work will include parameter studies to investigate the range of tunnel responses for given variability of joint properties. 4

We have seen that continuum approaches can provide some details of potential damage to underground structures. Peak velocities and displacements are well predicted to within a factor of two over ten orders of magnitude in yield. However, this information alone cannot provide complete damage estimates. In particular, it has been observed that substantial damage, or total collapse of a tunnel can occur at stresses well below the strength of the rock. The mechanism for this damage is thought to be key-block displacement. Distinct element methods can simulate collapse of tunnels for realistic fracture set geometry including

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DISCUSSION

the effects of this block motion. In practice, however, we only have limited knowledge of local fault zones. To provide bounds on the response, we must study a range of probable fault geometries. That is, we must perform stochastic analysis with many realizations required to obtain adequate statistics to bound results. Current three-dimensional scalar codes take days to simulate the dynamic loading of structures containing ∼104 −105 distinct elements. We have presented our approach to parallelization of the DEM by employing domain decomposition. The common-plane approach to contact detection is both efficient and easy to implement. However, under some circumstances the method of Cundall (1988) gives misleading contact classifications. The contact type is based upon counting the number of penetrating vertices. Thus, it is possible that a contact which would be better approximated by two interpenetrating faces will be misclassified as an edge to face contact. We will be investigating more accurate contact detection algorithms which build on the commonplane approach. Other future development of our DEM code will include coupling with other codes and the inclusion of structural elements. ACKNOWLEDGMENTS This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48. REFERENCES

Cleary, P. W. & Sawley, M. L. (1999). Three-dimensional modelling of industrial granular flows. In Second International Conference on CFD in the Minerals and Process Industries, Melbourne, Australia, pp. 95–100. CSIRO. Cundall, P. (1988). Formulation of a three-dimensional distinct element model – Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 25, 107–116. Cundall, P. A. (2001). A discontinuous future for numerical modelling in geomechanics? Proc. Inst. Civ. Eng. – Geotech. Eng. 149(1), 41–47. Cundall, P. A. & Hart D. H. (1992). Numerical modelling of discontinua. Eng. Comput. 9, 101–113. Heuzé, F. E., Walton, O. R., Maddix, D. M., Shaffer, R. J., & Butkovich, T. R. (1993). Analysis of explosions in hard rocks: The power of discrete element modeling. In J. A. Hudson, E. T. Brown, C. Fairhurst, & E. Hoek (Eds.), Comprehensive Rock Engineering, Vol. 2, Analysis and Design Methods, pp. 387–413. Pergamon Press. Lomov, I., Antoun, T., & Glenn, L. (2001, June). Explosion in the granite field: Hardening and softening behavior in rocks. In Proceedings of 12th APS Topical Conference, Shock Compression of Condensed Matter, Atlanta, Georgia. Morgan, J. K. (1999a). Numerical simulations of granular shear zones using the distinct element method 1. Shear zone kinematics and the micromechanics of localization. J. Geophys. Res. 104(B2), 2703–2719. Morgan, J. K. (1999b). Numerical simulations of granular shear zones using the distinct element method 2. Effects of particle size distribution and interparticle friction on mechanical behavior. J. Geophys. Res. 104(B2), 2721–2732. Rubin, M. B., Vorobiev, O. Y., & Glenn, L. A. (2000). Mechanical and numerical modeling of a porous elasticviscoplastic material with tensile failure. Int. J. Solids Struct. 37, 1841–1871.

Antonellini, M. A. & Pollard, D. D. (1995). Distinct element modeling of deformation bands in sandstone. J. Struct. Geol. 17, 1165–1182.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Numerical analysis of Gjøvik olympic cavern: a comparison of continuous and discontinuous results by using Phase2 and DDA Therese Scheldt, Ming Lu, Arne Myrvang Norwegian University of Science and Technology, Trondheim, Norway

ABSTRACT: In this paper continuous and discontinuous analyses are performed for calculating stresses and displacements of Gjøvik Olympic Cavern which was built for the 1994 Winter Olympic Games in Norway. Phase2 , a two-dimensional non-linear finite element program, and the Discontinuous Deformation Analysis (DDA) are used as representative tools for respectively continuous and discontinuous programs. For the continuous analyses both linearly elastic and non-linear analyses are carried out. A construction sequence of five stages is also simulated. The results from the Phase2 calculations are compared with the results from the DDA calculations. The reason why Gjøvik Olympic Cavern was used as a case study is the large amount of available input data through a comprehensive stress and deformation monitoring program during the excavation. The analysis results show good agreement with the reality and also with analyses carried out in UDEC-BB and the finite element program COSHWAN, by the Norwegian Geotechnical Institute and SINTEF respectively.

1

INTRODUCTION

Compared with other engineering materials, the rock mass stands out because it is featured with the existence of discontinuities. Whether, these discontinuities should be allowed for, is a primary decision to be made for a particular rock mechanics problem. If the answer is yes a further question is how, explicitly or implicitly? According to Brown (1986), the fundamental consideration to be done in order to select a suited model for a particular rock mechanics problem, is the relation between the discontinuity spacing and the size of the problem. Gjøvik Olympic Cavern was built for the XVII Winter Olympic Games 1994 at Lillehammer. The main cavern of the complex, which was created primarily to house the ice hockey games, has a span of 62 m, a length of 91 m and a height of 25 m. The spectator capacity is currently 5300, and today the cavern is widely used for both sport arrangements and concerts. The cavern is constructed by a five stage excavation process in Precambrian gneiss and the overburden fluctuates from 25 to 50 m. The pre-investigations consisted of geological field mapping, mapping in nearby caverns, core drilling, refraction seismics and seismic tomography, stress measurements, laboratory testing and numerical analyses (Thidemann & Dahlø 1994). Therefore Gjøvik Olympic Cavern has been a realistic case study for

Lillehammer Gj⵰vik Oslo

Figure 1.

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Map of Norway.

verifying the validity of the discontinuous deformation analysis (DDA) and Phase2 models. 2

GEOLOGY

The precambrian gneiss at the site, has a composition varying from granitic to quartzdioritic. Due to tectonic effects, the rock has developed a network of microjoints which are filled or coated with calcite and epidot. The result is a well jointed rock mass with average rock quality designation (RQD) of about 70%. RQD is a common method for characterising the degree of jointing in bore hole cores, described by Deere (1966). The joints are generally rough and well interlocked, and have rather irregular orientations. Foliation is poorly developed, but generally strikes approximately E-W with a dip of 35◦ to 55◦ towards S. The jointing is perhaps more frequent than in Norwegian basement rocks in general, but is normally irregular, rough walled and with quite large variations in dip and strike. The spacing of the more persistent jointing is often several meters. The general joint character is one of low persistence, moderate to marked roughness and without clay filling (Barton et al. 1992). 3 3.1

NUMERICAL ANALYSES Continuous modeling by using Phase2

the E-modulus close to the ground surface are reduced due to weathering so, material 3 from surface (level 220) to the crown of the cavern (level 175), has E-modulus 20 GPa. Material 2 (level 150–175), and material 1 (level 100–150) has 30 and 40 GPa, respectively (Fig. 2). During the investigation period both overcoring and hydraulic fracturing measurements were undertaken in order to determine the in situ rock stress. The measurements showed quite high virgin horizontal stresses in the cavern area, and the different measurement methods applied for the project gave horizontal principal stresses with a magnitude of 3.5–4.5 MPa (Hansen & Kristiansen 1994). Differences in situ stress models are therefore carried out. However, the vertical stress is identical for all models and is only a result of the overburden: σv = ρgh. In the first group (1) of models the ratio between the horizontal and vertical stress is constant from surface (level 220) to the bottom of the model at level 100. A second group (2) has one ratio from the surface to the crown of the cavern (level 175), and ratio 1:1 from this level to the bottom. A third group (3) has the same in situ stress distribution as (2), but the values of the strength parameters are increased. Group (1) to (3) are all linearly elastic analyses. The corresponding non-linear analyses are represented by group (4), (5) and (6). Table 1 summarises the properties of the six different groups. The cavern is situated close to the surface with uneven terrain topography and the NW side higher

Phase2 is a two-dimensional non-linear finite element program for calculating stresses and displacements around underground openings, and it can be used to solve a wide range of mining and civil engineering problems (Rocscience 2002). 3.1.1 Numerical model In the continuous models the rock joints are allowed for implicitly by using a Mohr Coulomb model for the rock mass. Both linearly elastic and non-linear analyses are carried out. Usually laboratory triaxial tests or direct shear tests are required for evaluating the Mohr Coulomb parameters: Cohesion, c, and friction angle φ. However, for some reasons the test was not performed. Therefore, the parameters are derived from Barton-Bandis peak shear strength envelope by linear fitting (Lu 1994). It was estimated that the cohesion is 0.3 and the friction angle fluctuates between 40◦ and 45◦ for the different materials. The mean values of E-modulus and the Poisson’s ratio (ν) measured in laboratory tests of the drill cores are 51.2 GPa and 0.21 respectively. Due to the size of specimens, these cannot represent the rock mass. Thus, the corresponding elastic parameters for the rock mass are obtained from Bieniawski’s Geomechanics Classification system: E = 30 GPa and ν = 0.21. The density of the rock mass is 2650 kg/m3 . In the model

Table 1.

Properties of Phase analyses.

Group

Linearly elastic

1 2 3 4 5 6

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Phase2 model for Gjøvik Olympic Cavern.

Figure 2.

Nonlinear

x x x

Constant stress ratio

Increased strength Parameters

x x x x x

x x

than the SE side. The 2-D model is taken at the caverns central cross section, and to eliminate the influence of the applied boundary conditions and correctly simulate the effect of the ground topography the Finite Element mesh stretches to the ground surface and extends in lateral directions three times the cavern span (62 m). The mesh consists of 1171 elements and 3609 nodes. 3.1.2 Modeling procedure Phase2 The modeling starts with generating the initial stress field and this is followed by simulating a five stage excavation process (Fig. 3). In one of the models rock bolts are installed. The bolting consists of alternate 6 m fully bonded bolts and 12 m plain strand cable in a 2.5 by 2.5 m pattern or by 5 by 5 m respectively. The former has a diameter of 25 mm and capacity of 220 kN, while the latter has a diameter of 12.5 mm each and total capacity of 334 kN at yield. 3.2

Discontinuous modeling by using DDA

Since the development of DDA several new theories and extensions to the method have been advanced by both academic and practical engineers in order to solve different kinds of problems within the field of rock mechanics and geotechnics. The DDA parallels the finite element method (FEM), since it solves a finite element type of mesh where all the elements are real isolated blocks, bounded by pre-existing discontinuities. It is, however, more general since the elements or blocks used by the DDA method can be of any convex or concave shape, whereas the FEM uses only elements of standard shape. When blocks are in contact, Coulomb’s law is applied to the contact interface. Further simultaneous equilibrium equations are selected and solved for each loading or time increment. The number of unknowns is equal to the sum of degrees of freedom for all blocks, while for the FEM method, the number of unknowns is the sum of the degrees of freedom of all nodes. Large displacements and deformations in the DDA method are the accumulation of small displacements and deformations at each step (Shi 1989). In Norway the universal distinct element code (UDEC) has traditionally been used as a tool for discontinuous modeling and the utilisation of DDA

Figure 3.

Cavern construction process.

is therefore quite new among the engineering geology and rock mechanics scientists. 3.2.1 Source codes of DDA According to Ma (1999) several DDA codes have been independently developed since Shi introduced his first DDA computer code in 1986. Principally two different DDA codes are made use of in this work. The first one, written in C is named kyotoDDA:20020206 and the second one is Shi’s code updated March 2002 compiled with the NDPC compiler. The Kyoto code consists only of three programs, the block producing program DDA CUT (DC), the analysis program DDA FORWARD (DF) and the graphic output program DDA GRAPH (DG), while Shi’s code also includes the line producing program DDA LINES (DL). Thus, the lines representing joints can be generated statistically and then make a more realistic model of the jointed rock mass. However, the main difference is connected to the DF programs. The Kyoto code makes it possible to analyse problems where the construction process is divided into different stages. Further universal equations for defining the horizontal and vertical initial stress distribution are introduced. A third difference is that material lines are substituted with areas which reduce the data amount of the DF’s input file. 3.2.2 Numerical model In the discontinuous models the joints are allowed for explicitly. Totally five different joint models are developed, but only two of them are presented in this paper. The first one has two regularly joint sets (Fig. 4). The number of blocks are 987 and the size and shape of the blocks are almost identical except for the blocks along the outer edge. Also the latter has two joint sets, but these are generated statistically by using line producing program in Shi’s code DL (Fig. 5). Thus, the shape and size of the blocks are varying. The number of blocks is 1407.

Figure 4.

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DDA model 1.

Cavern crown subsidence [m]

0,03500 0,03000 0,02500

Non-linear calculation results (4)

0,02000

Linearly elastic calculation results (1)

0,01500

Measured cavern crown subsidence

0,01000 0,00500 0,00000

0

Figure 5.

Table 2.

DDA model 2.

2

3

4

5

Stress ratio

Figure 6. and (4).

Properties of DDA analyses.

Model

Name

ϕ1

ϕ2

1 1 1 1 1 2 2 2 2 2 2

1 2 3 4 5 a b∗ c d e∗∗ f

32 32 32 32 32 32 32 32 32 32 32

26 26 30 24 28 26 26 30 24 26 28

∗

1

Sequential excavation

3.2.3 Modeling procedure DDA Program DF is used for analysing the models. The line producing program DL is only used for model 2, while model 1 is a result of directly input lines in program DC. For model 1 a five stage excavation process is introduced. However, one of the model 1 cases excavates the whole cavern in one stage in order to be able to explore if there is a noticeable difference in the excavation processes. As Figure 5 shows the cavern in model 2 is already excavated in the blck file.

x x x x

4 5000 step. E-modulus 30 Gpa.

∗∗

RESULTS

4.1 Phase2 calculations

In conformity with the continuous analyses, the discontinuous analyses are also performed with varying values of the input parameters. The vertical stress is gravitational for both model 1 and 2, while the horizontal stress is twice as the vertical for model 2. For model 1, the horizontal stress is 4.5 MPa at the crown of the cavern and from there the stress ratio between the horizontal and vertical is one. The intact rock properties used in the DDA modeling, model 1, take into account the degree of rock weathering, so a variation of E-modulus between 20 and 40 GPa is applied. For model 2 the E-modulus is constant like 40 GPa and 30 GPa for one of the cases. The Poisson’s ratio and the rock density are identical for the two models, 0.2 and 2650 kg/m3 respectively. The joint properties in DDA are friction angle, cohesion and tensile strength. The friction angle for joint set one is 32◦ while for the second joint set the value varies from 24 to 30. The cohesion is 0.2 and 0.3 respectively and the tensile strength is zero. The number of time steps for the calculation is 2500. Table 2 summarises the properties for discontinuous models.

A total of 39 cases, respectively 19 linearly elastic and 20 non-linear, have been analysed for various in situ stress and Mohr-Coulomb parameter values. Figure 6 shows the cavern crown subsidence for group (1) and (4) after the last excavation step. A total of 16 different stress models are analysed with stress ratio from 0.5 to 4.0. Figure 7 shows the corresponding results for group (2), (3), (5) and (6). All the four groups consist of five different stress models. Since the stress ratio from level 100 to 175 is 1:1, the x-axis represents the stress ratio from level 175 to the surface. The measured cavern crown subsidence is also plotted in the figure. Rock bolts were only introduced in one of the 39 cases (non-linear analysis). The reduction of the cavern crown subsidence is estimated to be only 7,32%.

4.2

DDA calculations

A total of 13 different cases are analysed, 6 for model 1 and 7 for model 2. All cases show that the large span

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Stress ratio vs cavern crown subsidence group (1)

Cavern crown subsidence [m]

0,04000 0,03500 0,03000 0,02500

Non-linear calculation results (5)

0,02000

Non-linear calculation results (6)

0,01500

Linearly elastic calculation results (2) and (3) Measured cavern crown subsidence

0,01000 0,00500 0,00000

0

1

2

3

4

5

Stress ratio

Figure 7. Stress ratio vs cavern crown subsidence group (2), (3), (5) and (6).

Figure 9. a) and b) Block movement model 1, friction angle of joint sets 32◦ and 26◦ , varying Poisson’s ratio.

Figure 8. Block movements model 1, friction angle of joint sets 32◦ and 30◦ .

cavern is stable, which is in accordance with both the reality and the calculations performed in Phase2 . Change of the friction angle for model 1 has no considerable influence on the stability. The results are almost identical, except for the case named 4, where two-three blocks near the excavation surface indicate movement (Fig. 8). Sequential excavation has neither influence on the stability. The case with friction angle 32◦ for joint set 1◦ and 26◦ for joint set 2 is performed both with and without sequential excavation, but shows no difference in the results. The case named 1 in Table 2 was because of a mistake first calculated with Poison’s ratio 0, and the result shows movements of a couple of blocks at the right side wall. For Poisson’s ratio like 0.2, there is no block movement. The two different situations are presented in Figure 9. The results from the model 2 cases do not differ from the model 1 results. The influence of changing the joint set 2 friction angle shows exactly the same tendency as in model 1, a little block movement for the case with friction angles like 32◦ and 30◦ . The cases named a and e are identical except for the number of time steps. The block movement for the 5000 step calculations are noticeable compared with the 2500 time

Figure 10.

Model 2, 2500 time steps.

Figure 11.

Model 2, 5000 time steps.

step calculation, but the cavern is still characterised as stable. Changing of the E-modulus from 40 to 30 GPa gives a less stiff rock mass and it is therefore native to think that the deformation will make the block movements increase. However this is not the situation. The results are almost identical.

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5 5.1

DISCUSSION Phase2 calculations

For the linearly elastic calculations higher horizontal in situ stress produces lower cavern crown subsidence. This is in coincidence with a rule of thumb which says that a high horizontal in situ stress is important to maintain stability of a cavern with large span. Except for the last excavation stage, the crown subsidence increases during the excavation process. This is due to the establishment of fully span at stage 4. Tensile stress of small magnitude has developed in limited areas around the cavern. For the case with stress ratio 0.5, tensile stresses are located at the floor and the right side wall, while for stress ratio 4.0 tensile stresses are only registered at the right side wall. However, the situation is opposite for the nonlinear calculations, where higher horizontal in situ stress results in higher crown subsidence and higher deformations. This can be explained by the rock mass yielding, which is caused by the differential principal stress, σ1 − σ3 , in this case σh − σv . Thus, high horizontal in situ stress is not always favourable to maintain stability of a large span cavern when joint shearing takes place. Further, it can be seen from figure 7 that the crown subsidence is sensitive to the Mohr-Coulomb parameters, especially when the stress ratio exceeds 2.5. For stress ratio 2.9 the cavern crown subsidence was reduced from 26.8 mm to 16.10 mm by increasing the tensile strength with 2 MPa and the friction angle with 5◦ for material 1–3. No tensile stresses are registered for the non-linear analyses. 5.2

DDA calculations

DDA is a block analysis program and the post processing possibilities are still limited compared with for example UDEC. However, the possibilities are satisfactory for localising potential failure blocks. The block geometry seems to have surprisingly small effect on the calculation results. Both model 1 and model 2 are stable after 2500 time steps and the variation of joint set 2 friction angle results in rather insignificant differences, except for the case with friction angles like 32◦ and 30◦ where some block movements are registered. This is caused by the orientation of the joint set compared to the size of the friction angle. The cases with Poisson’s ratio 0.2 and 0.0 show results in conformity with the practise. A rock mass with Poisson’s ratio 0.2 will be able to take up higher stresses than the 0.0 situation and with that bear more before block movements arise. Change of the rock mass E-modulus from 40 to 30 GPa for model 2 gives no marked results. This is probably because of the relative high value in both cases.

Model 2 is analysed for both 2500 and 5000 time steps. The reason why only one of the cases is calculated for 5000 step is that the calculation process is fairly time-consuming. The calculations are performed on a portable computer, Pentium III, 600 MHz, 128 MB, and take for 2500 number of time steps one to two days. 5.3 Preliminary comparison of continuous and discontinuous modeling This paper shows that both continuous and discontinuous modeling can be used in order to analyse the general cavern stability. However, there are important differences connected to input data and presentation of calculation results. The crucial input data for discontinuous modeling include joint patterns and strength parameters of the joints and these are not easy to obtain. The essential input data for continuous modeling are strength and deformability parameters of the rock mass and they are also difficult to get. Discontinuous modeling gives potential of rock falls and details of local stability, while continuous modeling gives a better picture of stress and deformation distributions. 6

A number of parametric studies have been conducted for Gjøvik Olympic Cavern. Both the results from the Phase2 analyses and the DDA analyses show that the large span cavern is stable under the given conditions. The results also show good agreement with the UDEC and COSHWAN calculations, performed by the Norwegian Geotechnical Institute and SINTEF and with the field measurements. ACKNOWLEDGEMENTS The authors are grateful to Gen-hua Shi, professor Yuzo Ohnishi, Takeshi Sasaki and Jian-Hong Wu for their comments and suggestions connected to the DDA modeling. REFERENCES Barton, N. et al. 1992. 60 m span Olympic Cavern in Norway, Towards New Worlds in Tunneling, Vieitez-Utesa & Montaez-Cartaxo (eds) ©1992 Balkema, Rotterdam. ISBN 90 5410 050 8. Brown, E.T. 1986. Analytical and computational methods in engineering rock mechanics. Allen & Unwin, London. Chryssanthakis, P. 1994. Numerical Modelling Using a Discontinum Approach. Final Report, The Research and Development Programme, Rock Cavern Studium, NGI, SINTEF & Østlandsforskning.

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CONCLUSION

Goodman, R. 1989. Introduction to Rock Mechanics, Second Edition, John Wiley and Sons. Hansen, S.E & Kristiansen, J. 1994. In Situ Stress Measurements. Final Report, The Research and Development Programme, Rock Cavern Studium, NGI, SINTEF & Østlandsforskning. Lu, M. 1994. Finite Element Modelling. Final Report, The Research and Development Programme, Rock Cavern Studium, NGI, SINTEF & Østlandsforskning. Lu, M., Kjørholt, H. & Ruistuen, H. 1994 Numerical study of the Gjøvik ice-hockey cavern. Computer Methods and Advances in Geomechanic, Siriwardane & Zaman (eds), ©1994 Balkema Rotterdam, ISBN 90 5410 380 9.

Ma, M. 1999. Development of the discontinuous deformation analysis the first ten years (1986–1996). Proceedings ICADD-3, Colorado, USA 1999. Morseth, B. & Løset, F. 1994. Preinvestigations – Decision Base – Excavation. Final Report, The Research and Development Programme, Rock Cavern Studium, NGI, SINTEF & Østlandsforskning. Shi, G.H. 1989. BLOCK SYSTEM MODELING by Discontinuous Defromation Analysis, Department of Civil Engineering, University of California, Berkeley. Thidemann, A. & Dahlø, T.S. 1994. Engineering Geological Investigations. Final Report, The Research and Development Programme, Rock Cavern Studium, NGI, SINTEF & Østlandsforskning.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Earthquake site response on hard rock – empirical study Y. Zaslavsky, A. Shapira & A.A. Arzi Seismology Division, Geophysical Institute of Israel, Lod, Israel

ABSTRACT: A bedrock site on a high plateau near the escarpment top, in the seismically active Dead Sea rift system, showed an amplification by a factor of 5, between 2 and 3 Hz, due mainly to an EW oscillation of the NS topographic feature. Mount Masada, also located on the Dead Sea rift, exhibited amplifications near 1.3 Hz, by a factor of 3.5 for EW motion and 2 for NS motion. The weathered and cracked granite bedrock near the Red Sea port of Eilat showed amplifications by a factor of about 4 in the frequency range of 6 to 7 Hz, which is within the range of engineering interest as regards low-rise buildings. Recorded ground motion in a hard rock tunnel near the Dead Sea shore showed amplification by a factor 2, near 3 Hz. We explain this effect as being due to the interference between incident (upgoing) and surface–reflected (downgoing) waves.

1

2

INTRODUCTION

Rock structures, whether natural or man-made, are usually long-life structures and, therefore, their mechanical stability typically means stability under earthquake loading. Such dynamic loading is defined by the regional earthquake characteristics at the deep bedrock level, as well as by possible modifications of the motion due to local site effects, which may be very significant. Amplifications due to soft near-surface sediments are widely recognized site effects; however, significant amplifications due to various mechanisms may also occur at hard rock sites. Such amplifications must be included in the earthquake loading input for the stability analysis of rock structures. These rock structures could be both man-made constructions built on or excavated within the hard rock site, as well as natural rock structures protruding over the site or contained within it, such as pinnacles, cliffs and caverns. We show site effects as determined empirically for seismic hazard analysis at several hard rock sites. These effects are caused by several mechanisms, as follows: Topography effects – amplifications near an edge-ofplateau escarpment top and on a mountain top. Weathering/cracking effects – amplifications due to near-surface weakening of the bedrock. Underground effects – amplifications and attenuations due to wave interference effects at a subsurface location (tunnel).

The site response functions are best determined from recorded ground motion during an actual strong event by comparison with recordings at a nearby reference site located on rock (Jarpe et al., 1989). In most cases, mainly in regions where the seismic activity is relatively low as in Israel, this type of analysis is usually impractical. Many investigators evaluated site response functions from moderate to weak earthquakes motion (for example, Field and Jacob, 1992; Carver and Hartzell, 1996; Zaslavsky et al., 2000). Nakamura (1989) hypothesized that site response could be estimated by dividing horizontal component noise spectra by vertical component noise spectra. Results obtained by implementing the Nakamura technique (Field and Jacob, 1995; Mucciarelli, 1998; Zaslavsky and Shapira, 2000) support such use of microtremor measurements to estimate the site response for surface deposits. In this study, we focus on four previously cited approaches. S-wave spectral ratio with respect to reference site The most common technique used for estimating site response is the standard (classic) spectral ratio procedure first introduced by Borcherdt (1970). This approach considers the ratio (RB ) between the

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METHODS USED TO DETERMINE SITE AMPLIFICATIONS

spectrum of a seismogram recorded on a site of interest (Ss ) and the spectrum of a seismogram from the same source recorded at a reference site, which is usually a nearby outcrop of rock (Sr ), Rb (ω) =

|Ss (ω)| |Sr (ω)|

(1)

Noise spectral ratio with respect to reference site Kagami et al. (1982) proposed that the ratio of the horizontal components of the velocity spectra at the investigated site to those of a reference site can be used as a measure of the site response function: Rk (ω) =

|Hs (ω)| |Hr (ω)|

3

Horizontal-to-vertical noise spectral ratio Nakamura (1989) proposed the hypothesis that the site response function under low strain can be determined as the spectral ratio of the horizontal versus the vertical component of motion observed at the same site. He hypothesized that the vertical component of microtremors is relatively unaffected by the softer near-surface layers. Hence, the site response is the spectral ratio between the horizontal component of microseisms (Hh ) and vertical component of microseisms (Hv ) recorded at the same location: |Hh (ω)| |Hv (ω)|

(3)

In other words, the vertical component of the microtremors on the surface retains the characteristics of horizontal microtremors at the bedrock (reference site).

This technique is based on Nakamura’s hypothesis for S-wave (Lermo et al., 1993): |Ssh (ω)| |Svs (ω)|

Ground motions were recorded using a multi-channel, PC-based, digital seismic data acquisition system (see Shapira and Avirav, 1995) designed for site response field investigations. The seismometers used were sensitive velocity transducers with a natural frequency of 1.0 Hz. Each of the stations was equipped with one vertical and two horizontal seismometers (oriented north-south and east-west). Digital recordings were made using a 0.2–25 Hz band-pass filter with a sampling rate of 100 samples per second. Prior to and during the measurements we checked and determined the transfer functions of the instrumentation in order to facilitate transformation of the recorded signals into ground motion data, i.e., particle velocity. Based on many previous site investigations, we concluded that a window length of 25–30 sec for spectral calculations is sufficient to provide stable results. The selected time windows were Fourier transformed using cosine tapering before transformation. The spectra were then smoothed with a triangular moving Hanning window (0.4 Hz). After data smoothing and in order to obtain spectral ratios, the spectra of an EW or NS channel at a site were divided by the spectra of the corresponding channel of a reference site (Borcherdt and Kagami ratios) or by the spectra of the vertical channel at the same site (receiver function and Nakamura estimate). The arithmetical average of each individual ratio was also computed. We observed that there was practically no difference between arithmetical and geometrical averaging.

4 TOPOGRAPHY EFFECTS

Horizontal-to-vertical S-wave spectral ratio (Receiver Function)

Rs (ω) =

(4)

Macroseismic observations have shown that the effects of local topography on ground motion might be of great importance. A dramatic topographic effect was observed during the Northridge, California earthquake (Spudich et al. 1996; Bouchon and Barker, 1996) where ground accelerations reaching 1.7 g were

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DATA ACQUISITION AND PROCESSING

(2)

where Hs and Hr denote spectral amplitudes of the horizontal components of motion at the investigated site and those of the reference site, respectively. This approach is appropriate when the source and path effects of the recorded motions of the two sites are identical. In practice, however, these conditions can very seldom be met, especially when measuring microtremors.

Rn (ω) =

where Ssh and Ssv , respectively, denote horizontal and vertical amplitude spectra computed at the same investigated site, from S-waves. Receiver function was introduced by Langston (1979) to determine the velocity structure of the crust and upper mantle from teleseismically recorded P-waves. Langston made the assumption that the vertical component of motion is not influenced by local structure, whereas the horizontal components, owing to the geological layering, contain the P to S conversion. In the spectral domain this corresponds to a simple division of the horizontal spectrum by the vertical (equation 4).

recorded on top of a small hill (about 15 m high) in Tarzana. The results obtained by Celebi (1987) from the Chili earthquake (3 March 1985) show the relationship between damage to buildings in Vina del Mar and topographic site effects. During the 18 October 1989 Loma Prieta earthquake (Hartzell et al. 1994), significant structural damage to modern wood-frame homes and ground cracking were observed on Robinwood Ridge, California. The effects of topography on surface ground motion were observed and studied by Davis and West (1973), Tucker et al. (1984), Chavez-Garcia et al. (1996) Zaslavsky and Shapira (2000) and several other authors. Their studies show that amplification up to factor ten can be expected at the ridge top. Simulations of topographic amplifications have been performed using various theoretical methods (see Bard and Tucker, 1985; Sanchez-Sesma and Campillo, 1993 and others). We have measured topography effects at two locations at the Dead Sea rift, as follows: – The Parsa site near the edge of a plateau terminated by an escarpment. – The Masada site, at and slightly below the summit of an isolated table mountain. The Parsa edge-of-plateau site The Parsa area is situated on the western side of the Dead Sea rift valley, at the zone of the rift boundary faults. As shown in Figure 1 it straddles the main fault escarpment of the western side of the valley, which is over 400 m in height, separating a high bedrock plateau to the west above sea level (according to the project datum, sea level is designated as elevation 2000 m) from the sediment filled valley below to the east. An exploratory tunnel has been driven westward from the

foot of the escarpment. The aim of the seismological study was to provide data for the seismic design of a major project. As shown in Figure 1, the deployed seismic stations included Site 1 – located inside the tunnel and Site 2 – on the plateau. The geology of the area is known mainly from detailed geological mapping to a scale of 1 : 2.500, from the exploratory tunnel, from a 500 m deep borehole on the plateau, from several high resolution seismic reflection and seismic refraction lines and from numerous shallow borings. Essentially, the strata marked with the letters B through F in Figure 1 represent a hard rock sequence of limestone and dolomites, including some marls. This sequence is down faulted to the east and buried by younger sediments in the valley. The velocity structure in the materials underlying the area has also been exhaustively investigated. P and S velocities have been determined by numerous refraction lines and uphole surveys throughout the area (Shtivelman, 1996). For the limestone and dolomite of the upper plateau (F member in Figure 1), one uphole survey at Site 2 yielded Vs = 1300 m/sec in the 4–16 m depth range and a refraction line about 600 m south of this site yielded Vs = 1970 m/sec for a refraction at a depth of 2 m. In Figure 2a we present classical spectral ratios for Site 2 computed with respect to reference Site 1 from S-wave windows of three Gulf of Eilat earthquakes. These curves clearly show high amplification up to 5 in the frequency range between 2 to 3 Hz for horizontal ground motions oriented approximately perpendicular to the long axis of the escarpment (EW direction). For motions parallel to the escarpment (NS direction) there is no such amplification. Figure 2b presents average spectral ratios (Kagami’s ratio) calculated from simultaneous recordings of microtremors at Site 2 and at the reference Site 1 (both are hard rock sites spaced

Figure 1. East-west cross section of the Parsa area (fault escarpment of the Dead Sea rift valley) along the exploratory tunnel (arbitrary elevation datum – 2000 is Mean Sea level).

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Figure 2. Empirical transfer function of the Parsa edge-of-plateau site (Site 2): (a) Spectral ratios of S-wave from three earthquakes with respect to Site 1; (b) Average spectral ratios of microtremors with respect to Site 1; (c) Individual horizontal-to-vertical spectral ratios of microtremors (Nakamura estimates).

about 0.7 km apart). The main difference between the EW and NS component of motions is in the frequency band about 1.5–4 Hz. In this band, an amplification factor about 3 is shown only for the EW component, which is approximately perpendicular to the axis of the topographic escarpment. The large amplifications above 4 Hz for both NS and EW components are attributed to radio telemetry noise in the Site 2 data. Figure 2c displays individual horizontal-to-vertical spectral ratios for Site 2 based on microtremors (Nakamura’s ratio). These plots demonstrate considerable similarity among individual estimates, with a peak about 3 Hz which is much stronger for the EW component. The Masada mountain-top sites Figure 3 shows a map of the study area, topographic profiles and the locations of the seismic stations. Stations 1 and 2 were deployed near the summit of

the mountain; Station 3 was installed on the slope of Mt. Masada and Station 4 was located near the foot of the mountain. Figure 4 shows individual and average horizontal-to-vertical spectral ratios for Sites 1 and 2 obtained from microtremors. The dominant feature of all spectral ratios is the high spectral ratio level at a frequency of about 1.4 Hz. At this frequency we also observe differences between the EW and NS components. Such differences are characteristic of topography effects. At the summit of Mt. Masada, the average spectral ratios reach maxima of about 2.5 in the EW direction and about 2 in the NS direction. We should point out here that the Nakamura method provides, in general, a relatively reliable estimate of the predominant frequency of the site (resonance frequency) but it is less reliable for estimating the amplification level, especially at other frequencies. Figure 5 shows spectral ratios for Sites 1 and 2 with respect to reference Site 4. These are calculated from

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Figure 3. Topographic map and profiles along sections A and B of Mt. Masada.

seismic waves of an earthquake (southeast Cyprus, October 13, 1998 at 12:28, ML = 2.9; the epicentral distance is 545 km). There are only small variations in the site response of the two sites. The ratios show a prominent peak at about 1.3 Hz. Here the horizontal ground motion oriented EW is amplified by a factor of about 3.5, while it is about 2.0 in the NS direction, i.e., Mt. Masada exhibits a preferential direction of resonance motion. Plotted in Figure 6 are the

horizontal-to-vertical spectral ratios for Sites 1 and 2 for the S-wave window (receiver function). Again, the receiver function clearly exhibits the resonant peak in the frequency range 1.2–1.4 Hz with amplification values of about 3.5. Figure 7a shows the horizontal-to-vertical spectral ratios for the NS component obtained from microtremors at Site 3. As shown, the average Nakamura site response estimate has a predominant peak near

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Figure 4. Individual and average (heavy lines) horizontal-to-vertical spectral ratios for Mt. Masada obtained from an microtremors recorded at Sites 1 and 2.

Figure 5.

Spectral ratios for Sites 1 and 2 at Mt. Masada computed from earthquake with respect to Site 4.

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Figure 6.

Horizontal-to-vertical spectral ratios obtained from earthquake data for Sites 1 and 2 at Mt. Masada.

Figure 7. (a) Horizontal-to-vertical spectral ratios from microtremors (EW component); (b) spectral ratios to Site 4 from an earthquake; all for Site 3 at Mt. Masada.

1.4 Hz with amplification up to 2.2. Another dominant feature of the average spectral ratio is the high in the frequency range near 6 Hz. Figure 7b presents spectral ratios calculated from the earthquake near Cyprus with respect to reference Site 4. The main differences between Sites 1, 2 and 3 are in the frequency band at about 1.4–1.8 Hz and about 4.0–5.0 Hz. For Site 3, the average spectral ratio obtained by the reference station shows a prominent peak near 1.4 Hz with amplification of only 2.0 whereas at about 4.0 Hz the amplification factor is about 4.0.

5

In Figure 8a we present three components of scaled seismograms from a Gulf of Eilat earthquake (200101-14 18:31, ML = 3.2, epicentral distance R = 35 km) recorded on granite. The seismograms are plotted on the same scale and demonstrate the large differences in amplitudes of the horizontal and vertical components of motion. In terms of peak velocity, amplitudes of the horizontal components are about 2.5 times greater than the amplitudes of the vertical component. The

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EFFECTS ON WEATHERED BEDROCK

Figure 8. (a) Records of three-components of a Gulf of Eilat earthquake (2001-01-14 18:31, ML = 3.2, epicentral distance R = 35 km) recorded at outcrop of granite near Eilat; (b) the corresponding Fourier spectra and (c) spectral ratios obtained for that event.

Fourier spectra of the S-waves are shown in Figure 8b. Increases in the spectral levels of horizontal components are clear in the frequency range 6.0 to 7.0 Hz. This effect is more pronounced in the spectral ratio. Figure 8c shows spectral ratios of the two horizontal components of the earthquake motions computed with respect to the vertical component (receiver functions). These curves show prominent peaks at about 7 Hz, with an amplification factor of about 4. Figure 9 presents individual and average receiver functions obtained from explosions and earthquakes. Again, we observe dominant peaks at about 7 Hz with an amplification factor of about 2.5. These amplifications are most probably the effect of softening

of the granite near the surface due to cracking and weathering. 6

Observations indicate that seismic motion in tunnels can be either amplified or de-amplified. Such a site effect is interpreted as interference between up-going and down-going waves. It has been noted that such underground site effects may be significant for underground facilities. Examples illustrating effects for receivers at the surface and inside boreholes at depth have been presented by other investigators, who

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UNDERGROUND EFFECTS

Figure 9. Individual and average (heavy lines) receiver functions obtained at outcrop of granite near Eilat from: (a) explosions; (b) earthquakes.

Figure 10. Horizontal-to-vertical spectral ratios (EW component) for Parsa Site 1 located in the tunnel obtained from: (a) earthquakes and (b) microtremors.

found that the borehole seismometer site response is frequency dependent, owing to interference between up-going and down-going waves.As already presented above, at the Parsa site Station 1 was installed inside an exploratory tunnel on hard rock about 400 m from the portal and at a subsurface depth of about 200 m.

Figure 10 displays individual horizontal-to-vertical spectrum ratios at Site 1 (EW component) for earthquakes (S-wave window) and for microtremors. Clearly, the shape of all curves is quite similar. All horizontal-to-vertical spectral ratios show amplifications near 3.4 Hz and de-amplifications at about 1.7

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Figure 11. Average horizontal-to-vertical spectral ratios for Parsa Site 1 located in the tunnel obtained from different date bases: S-wave (solid line), coda S-wave (dashed line), and microtremors (dotted line).

and 5.1 Hz. In Figure 11, mean spectral ratios (NS and EW components) are compared, based on data from S-wave windows (20 earthquakes), S-wave coda windows (7 earthquakes) and microtremors (30 samples). Again, all curves are similar as regards both frequency and amplitude. It will be shown in the following that the persistent shape of the horizontal-to-vertical spectral ratios as shown above is the result of constructive and destructive interference of incident and surface-reflected waves mainly in the vertical component of motion, i.e., the horizontal components are less affected.

DISCUSSION AND CONCLUSION The empirical transfer function for the Parsa edgeof-plateau (Station 2), obtained from earthquakes, showed prominent peaks within the 2.0–3.0 Hz range with amplification factors up to 5. Analysis based on the recorded microtremors yielded similar results. This station is located on hard rock in an area where the local topography is well developed and, therefore, the observed amplification is probably due only to a topographic effect. Indeed, the amplification in the EW component, perpendicular to the escarpment, was far greater than the small amplification in the NS component. The results of numerical studies of topographic effects on the seismic response of steep slopes (Ashford et al., 1997) show that topographic amplification of vertically propagating SV-waves has a peak at wavelengths greater than the escarpment height by a factor 5. In our case, the escarpment height is a little over 400m and a borehole measurement at Site 2

yielded a shear velocity about 1400 m/sec. The calculated resonance frequency is, therefore, 0.7 Hz. This does not agree with the observed amplification peak about 2.5 Hz, suggesting that the applicable model is more complicated. The site response obtained at Mt. Masada with respect to the reference station shows a well-defined peak at about 1.4 Hz. The horizontal ground motion oriented EW is amplified by a factor up to 3.5. Similarly, this peak is present in the receiver functions and in the average horizontal-to-vertical spectral ratios of microtremors. If the amplification effect is caused by topography, the frequency involved should correspond to a wavelength equivalent to the horizontal relief dimension (Geli et al., 1988 and others], which here is about 1300 m (see Figure 3). Assuming an S-wave velocity of 1400 m/sec for the uppermost layer of Mt. Masada, (Zaslavsky et al., 2000) we should expect the topographical effects to be observed in the frequency of 1.1 Hz, in arguable agreement with the experiment. Average horizontal-to-vertical spectral ratios at Site 1 obtained from the S-wave windows, S-wave coda windows and microtremors (Figure 11) show troughs at about 1.7 and 5.1 Hz and a peak at about 3.4 Hz. To discover the cause of the peak and troughs, we computed the average vertical-to-vertical spectral ratio at Site 1 with respect to Site 2, for S-wave windows, as shown in Figure 12. This curve has peaks about 1.7 and 5.1 Hz and a trough about 3.4 Hz. Similar curves were obtained for Site 1 with respect to several other sites, representing a variety of topographical and geological conditions over distances up to 1.2 km from Site 1. Therefore, the peaks and trough of Figure 12 are all explained as peculiarities of Site 1, i.e.,

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Figure 12. Average vertical-to-vertical spectral ratios obtained from earthquakes at Parsa Site 1 located in the tunnel with respect Site 2 (edge-of-plateau).

the tunnel, being due to interference of incident and surface-reflected waves in the vertical component at this subsurface site. The subsurface depth at Site 1 is about 205 m and the average S wave velocity in the overlying bedrock is believed to be about 1400 m/sec, based mainly on borehole measurements at Site 2. Thus, at 1.7 Hz, 3.4 Hz and 5.1 Hz, the reflection twoway travel depth of 410 m represents 0.5 wavelength, 1.0 wavelength and 1.5 wavelength, respectively. Data collected from earthquakes and microseisms on weathered and cracked granite bedrock to the north of Eilat city show that surface rock weaknesses can modify a site response. The records show a spectral amplification in the frequency range 6 to 7 Hz, with a factor about 4. Thus, the near-surface weathering and cracking affect the recorded ground motion at frequencies of engineering interest. These results show that hard-rock response is very sensitive to fine topographic details and various elusive parameters of theoretical models. Accurate determination of hard-rock site effects should therefore be done using experimental techniques.

REFERENCES Ashford, S.A., Sitar, N., Lysmer, J. & Deng, N. 1997. Topographic effects on the seismic response of steep slopes. Bull. Seism. Soc. Am. 87: 701–709. Bard, P.Y. & Tucker, B.E. 1985. Underground and ridge and site effects: comparison of observation and theory. Bull. Seism. Soc. Am. 75: 905–922.

Borcherdt, R.D. 1970. Effects of local geology on ground motion near San Francisco Bay. Seism. Soc. Am. 60: 29–61. Bouchon, M. & Barker, J.S. 1996. Seismic response of a hill: the example of Tarzana, California. Bull. Seism. Soc. Am. 86: 66–72. Carver, D. & Hartzel, S.H. 1996. Earthquake site response in Santa Cruz, California. Bull. Seism. Soc. Am. 86: 55–65. Chavez-Garcia, F.J., Sanchez, L.R. & Hatzfeld, D. 1966. Topographic site effects and HVSR – a comparison between observations and theory. Bull. Seism. Soc. Am. 86: 1559–1573. Celebi, M. 1987. Topographical and geological amplifications determined from strong-motion and aftershock records of the 3 March 1985 Chile earthquake. Bull. Seism. Soc. Am. 77: 1147–1167. Davis, L.L. & West, L.R. 1973. Observed effects of topography on ground motion. Bull. Seism. Soc. Am. 63: 283–298. Field, E.H., Jacob, K.H. & Hough, S.E. 1992. Earthquake site response estimation: a weak-motion case study. Bull. Seism. Soc. Am. 82: 2283–2306. Field, E.H. & Jacob, K.H. 1995. A comparison and test of various site-response estimation techniques, including three that are not reference-site dependent. Bull. Seism. Soc. Am. 85: 1127–1143. Hartzell, S.H., Carver, D.L. & King, K.W. 1994. Initial investigation of site and topographic effects at Robinwood Ridge, California. Bull. Seism. Soc. Am. 84: 1336–1349. Jarpe, S.P., Hutchings, L.J., Hauk, T.F. & Shakal, A.F. 1989. Selected strong- and weak-motion data from the Loma Prieta earthquake sequence. Seism. Research Letters. 60, No. 4: 167–176. Kagami, H., Duke, C.M., Liang, G.C. & Ohta, Y. 1982. Observation of 1- to 5-second microtremors and their application to earthquake engineering. Part II: Evaluation of site effect upon seismic wave amplification deep soil deposits. Bull. Seism. Soc. Am. 72: 987–998. Langston, C. 1979. Structure under Mount Rainier, Washington, inferred from teleseismic body waves. J. Geophys. Res. 84: 4749–4762. Lermo, J. & Chavez-Garcia, F. J. 1993. Site effect evaluation using spectral ratios with only one station, Bull. Seism. Soc. Am. 83: 1574–1594. Mucciarelli, M. 1998. Reliability and applicability of Nakamura’s technique using microtremors: an experimental approach. Journal of Earthquake Engineering. 4: 625–638. Nakamura, Y. 1989. A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface. QR of RTRI. 30, No.1: 25–33. Sanchez-Sesma, F.J. & Campillo, M. 1991. Diffraction of P, SV and Rayleigh waves by topographic features: a boundary integral formulation. Bull. Seism. Soc. Am. 81: 2234–2253. Shapira, A. & Avirav, V. 1995. PS-SDA Operation Manual, IPRG report Z1/567/79, 24pp. Shtivelman, V. 1996. Uphole and refraction seismic survey (P and S waves) at the Parsa site, Dead Sea area. IPRG Report no K805/78/96, 25pp. Spudich, P., Hellweg, M. & Lee, W.H.K. 1966. Directional topographic site response at Tarzana observed in

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aftershocks of the 1994 Northridge, California, earthquake: Implications for mainshock motions. Bull. Seism. Soc. Am. 86: 193–208. Zaslavsky, Y., Shapira, A. & Arzi, A.A. 2000. Amplification effects from earthquakes and ambient noise in Dead Sea Rift (Israel). Soil Dynamics and Earthquake Engineering. 20/1–4: 187–207. Zaslavsky, Y. & Shapira, A. 2000. Questioning nonlinear effects in Eilat during Mw = 7.1 Gulf of Aqaba

earthquake. Proceedings of XXVII General Assembly of the European Seismological Commission (ESC), Lisbon, Portugal, September 10–15, 343–347. Zaslavsky, Y. & Shapira, A. 2000. Experimental study of topographic amplification using the Israel seismic network. Journal Earthquake Engineering. 4, No. 1: 43–65, Imperial College Press.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Numerical simulation of shear sliding effects at the connecting interface of two megalithic column drums N.L. Ninis Civil Engineer, Archaeological Museum of Epidauros, Ligourio, Hellas, Greece

A.K. Kakaliagos Dr. Ing., Engineering Consultant, Athens, Greece

H. Mouzakis Dr. Civil Engineer, Laboratory for Earthquake Engineering, NTUA, Athens, Greece

P. Carydis Professor of Earthquake Engineering, Director of the Laboratory for Earthquake Engineering, NTUA, Athens, Greece

ABSTRACT: This work concerns the problem of simulating numerically the behaviour of dry joints in megalithic structures, and specifically the stone monuments of ancient Greek architecture. In principle, the design of an ancient Greek temple approximates that of a rock structure with big voids and individual structural members of discontinuous rock mass with dry joints, in which the major deformation takes place along the joints. The work uses experimental data from full-scale tests at the Laboratory for Earthquake Engineering of NTUA, involving slow cyclic sliding motions along the interface of a two-drum system, and describes the formulation of a mathematical model using SAP2000 with 3-D solid elements and non-linear links. Using the recorded P-δ relationship the detailed stress distribution across the base of the column drum was evaluated. Stress concentrations revealed a corresponding shear-slide failure mechanism of the experimental set-up.

1

INTRODUCTION

The key to understanding and analyzing the behaviour of megalithic structures lies in the behaviour of their dry joints. They act as energy dissipators for both sliding and rocking movements imposed on the structure. Thus, a discretisation of the overall behaviour takes place, based on the response of each particular joint. A change in one of them produces in effect a new, slight variation of the initial structure. It is therefore of great importance to understand how they behave in all their details. This paper concentrates on the sliding aspect of their behaviour. It is part of ongoing research collaboration between the Committee for the Preservation of the Epidauros Monuments and the Laboratory for Earthquake Engineering of NTUA for investigating the behaviour of monuments under seismic action. Its particular aim was to evaluate the expected response to sliding in terms of stress distribution and concentration across the contact interface of joints, and its practical

implications regarding their optimum configuration. The problem was studied using full-scale laboratory experiments and parallel numerical simulation of the observed behaviour. 2 TEST SET-UP AND SPECIMEN Experiments were carried out in a stiff loading frame especially designed for slow cyclic shear tests. The setup comprised two specimens symmetrically placed in respect to the point where the horizontal displacement of the upper half of the frame is applied. Ageneral view of the experimental set-up, complete with specimens, is shown in Figure 1. The vertical load applied to all tests consisted of the dead weight of the frame top plus that of the upper half of the specimen, equal to 49.6 kN. Horizontal displacement was applied at a constant rate through a hydraulic jack MTS244. The jack had a capacity of ±500 kN and its maximum possible single amplitude stroke

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3

EXPERIMENTAL RESULTS

The experiments resulted in a series of horizontal load-horizontal displacement curves. Figure 2 shows a typical diagram for a complete loading cycle. As it can be seen from it, after an initial sharp increase (corresponding to the development of static friction to the value of kinetic friction), the horizontal force becomes practically constant. The friction coefficient was measured equal to 0.73 corresponding to a friction angle of 36◦ . The friction coefficient varied between 0.60 (for relatively rough contact surface) and 0.75 (for smooth contact surface). There was significant difference between the horizontal displacement inferred from the jack’s movement and that measured by displacement transducers (2.5 mm as compared to 0.4 mm for the initial part of the curve corresponding 40.0

2

3

1

20.0

FORCE (kN)

was ±200 mm. Different displacement rates were used from 0.16 mm/sec to 1.6 mm/sec. Displacements between the bases were measured using LVDTs mounted on the frame in three directions – vertical, horizontal and diagonal. Each specimen consisted of a pair of stone column drums. Each drum was manufactured from a single stone block, part of which was shaped as a cylinder of 1 m diameter and 0.45 m height, with a rectangular base, having a cross-section of 1×1.1 m and a height of 0.30 m, in order to facilitate anchoring to the testing frame. The horizontal movement was applied to the line of the interface. Great care was taken to ensure that the base-drum interface was in very close contact and aligned to the horizontal. The final stage of stoneworking consisted of using a metal fitting plate, to check the still existing small irregularities and eliminating them by hand grinding. The stone employed in the experimental work was a porous limestone from Cyprus. The particular stone was chosen because it is relatively uniform and easy to curve. It belongs to the group of stones usually termed “porolithoi”, and to a certain extent is quite representative of the Kenchreae stone that has been used in the Epidaurean Asklepieion. Its basic mechanical properties, used in the subsequent numerical analysis, are given in the table 1.

0.0

-20.0

-40.0 2.0

3.0

4.0

5.0

6.0

DISPLACEMENT (mm)

Figure 2.

Figure 1.

Table 1. stone. Property →

Horizontal load-displacement curve.

General view of the experimental set-up.

Mechanical properties of Cyprus porous lime-

Unconfined strength (MPa)

Young’s modulus (N/mm2 )

Poisson’s ratio

35

7000

0.26

Figure 3. Another example of the fluting phenomenon.

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7.0

Figure 5a. Development of a close surface contact along the perimeter.

Figure 4a. View of a top drum with a flute-like spauling.

Figure 5b. Detail of the above with sign of a new flute-like rupture developing.

Figure 4b.

Detail of the flute-like exfoliation.

to the mobilisation of kinetic friction). It is a measure, on one hand of the compliance of the testing set-up and on the other hand of the fact that initially there is more straining of the material, and rather less movement, along the interface. With regard to the specimens, two interesting observations were made. First, there was surface breaking of the specimen at the very beginning of the test with hardly any noticeable movement. As load increased and displacement progressed, sizeable chunks of stone broke away. We must note that the size and shape of these failure (slip) surfaces remind the pattern of flutes in ancient Doric columns. With further sliding,

this breakage subsequently developed into an irregular coarse surface. We could then safely assume that, if we had allowed repeated sliding of the two column drums in all directions, they would have developed a rough exterior, vaguely resembling that of a fluted column drum. The second interesting observation, after dismantling the specimens following repeated slow cycling sliding tests it was discovered that they had formed a zone along the perimeter (Figures 5), indicating that sliding was actually taking place only (or predominantly) there, and not along the rest of the interface. We should remind here that initially the two surfaces were touching across the whole section of the interface uniformly. 4

The shear-sliding interface between the two column drums was modeled using the general-purpose computer program SAP2000 non-linear version 7.1. The

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COMPUTER MODEL

computer model described in this section was constructed using an assembly of 3D-solid and frame elements together with nonlinear links, hereafter referred as Nllinks (Reference, SAP2000, the Nllink Element). In all, 1698 joints, 440 Nllinks and 25,600 solid elements were deployed in the computer model. Computer runs were executed on a PC-PentiumIII800 MHz. Storage requirement on hard disk for a typical run was 2.8 GB, associated with a total execution time of approximately 6 hours. Each individual drum had a total height of 450 mm and a radius of 500 mm. Drum material properties reflected linear isotropic behaviour. Elastic modulus of 7000 N/mm2 and Poisson ratio of 0.26 were used, reflecting actual data obtained from material testing (Ninis & Kourkoulis, 2001). The two drums were modeled using 8-node 3Dsolid elements following a cylindrical generation (Figure 8). Elements were generated following the typical radial pattern found in the base of column drums in various monuments of the Athenian Akropolis (Penrose 1973, Orlandos 1994). The exact configuration for the drum base of Parthenon columns taken from Orlandos (1994) is shown schematically in Fig. 6. Zone A along the perimeter, perfectly manufactured as plane surface, was in direct tight contact to the adjacent column drum, above or below. Zones B and C, on the other hand, had a rough surface reducing the active contact area to the adjacent column drum, to a percentage of the whole, thus effectively reducing friction mobilisation. Their difference lie in theroughness, with zone C being more roughly worked. Finally, a narrow zone  of smooth, tight contact was surrounding the typical polos-empolion arrangement (polos in Greek = the connecting pivot, empolion in Greek = the encasement of polos) in the centre of the section, centrally connecting the two drums. Solid element generation was set to closely reflect the circular arrangement of the Parthenon column section, whereby the four distinct zones A, B,  and  shown in Figure 6 were realized. In the computer model developed, all these zones are reflected in the mesh, however they are all realized on the same plane. The polos-empolion connecting element with its typical hole in the centre of the section was not realized in this phase of the research. The task of the analysis presented here was the investigation and computer modeling of the column drum shear-sliding interface under horizontal excitations. The typical section of the Parthenon column drum bases presented above seems to suggest a preferential shear sliding activation at column perimeter and center. This latter effect had to be checked by subsequent computer analysis. Another aspect taken into account in the mesh generation, and the subsequent analysis, it was the presence of flutes (usually 20) along the column perimeter,

Figure 6. Schematic configuration of a Parthenon column drum base in plan and section.

running over the whole column height. Investigations of the flute section shape (Zambas, 1998) has shown that in fact it was following an elliptical shape, a tripartite curve with increased curvature towards the arrises. There are cases, however, of other monuments, where it was circular. The depth of flutes also varied from monument to monument of fluting from 3 to 6 cm depending on the size of the column. It is commonly assumed that fluting is present due to aesthetical reasons only, being one of the many architectural refinements we find in ancient Greek architecture. Simple engineering judgment suggests that fluting might as well be associated to material cracking and spauling under cyclic shear action at column drum connection interface. Hereby, under horizontal shear action at column drum shear-sliding interface, shear stress concentration should occur at column perimeter. Typically, maximum stress should occur at column section symmetry axis in the major shear force direction. Under progressively increasing shear action, material failure would occur at latter location. Using the analogy to slope failure mechanism under horizontal force with slide circle analysis, only a distinct part of the slope should fail, where the combination of stresses along a circular/cylindrical failure surface a critical value. Comparing to the column drum

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shear sliding interface, the region at column section symmetry should fail, thus, forming a cylindrical failure plane defined by the intersection of the column drum cylinder and the failure surface cylinder. In order to assess the behaviour of shear sliding mechanism and thus, effectively verify stress concentrations, it was decided to model the fluting of the column section. A division of the drum finite element mesh at 18◦ intervals was adopted corresponding to 20 flutes (Fig. 9). Detailed fluting geometry was realized by adequate mesh arrangement. In fact, corresponding solid elements were not eliminated, to thus, model the fluting. In as far as the task of the analysis presented in this paper deals with the identification of stress concentration, it was decided to model detailed flute geometry inside the drum solid body. This decision was taken in order to facilitate the identification of potential stress concentration at flute location (Figure … . .). Admittedly, stress evaluation in finite element analysis is directly related to mesh geometry and layout. It was realized, that mesh arrangement at flute location with typical mesh density close to fluting edge (arris), would potentially affect stress evaluation. To effectively eliminate the latter effect, it was decided to “rotate” the flute mesh at 9◦ . Thus, at column drum symmetry plane, the fluting edge (arris) should be present. This decision was taken in order to evaluate stress concentration independently from mesh geometry and mesh density. In case the assumption of stress concentration as described earlier was true, region of maximum stress should lie at 9◦ below and above the column’s horizontal section axis (Figure 5). Consequently, finite element analysis results should yield a true, and mesh independent, picture only if maximum stress should occur in the middle of each corresponding flute, above and below the horizontal symmetry axis. The two column drums were placed one atop the other, however, leaving a 5 mm gap. This gap was provided to effectively place Nllinks for shear sliding computer modeling. In as far as, 3 mm was the typical joint offset in SAP2000, large enough not to activate the automatic weld-constraint (Reference SAP2000), and thus, rigidly connect two adjacent joints, additional 2 mm were provided to the gap, to thus, effectively model the gap size. Gravity loads resulting from mounting test frame were introduced atop top drum. Horizontal structure loads were introduced to the two-drum Nllink assembly at the shear-sliding interface. Hereby, the top level of the top drum was used to rigidly attach a stiff frame. The latter structure, consisting of 3D-frame elements was attached to the top nodes of the top drum (Figure 7). This arrangement permitted the assessment of shear sliding interface mechanism under horizontal shear force only.

Figure 7. Computer model geometry – finite element mesh with 3D solid elements.

In order to access P-Delta effect introduced by structure horizontal displacement, a pair of selfequilibrating bending moments was introduced at the shear-sliding interface. Latter moments were determined as the product of total axial load acting at the shear sliding interface times the total structure displacement. Displacement degrees of freedom of bottom drum base nodes were restrained, to effectively simulate lab tie-down slab restraints. Solid elements deployed in the analysis had 3 displacement degrees per node. To effectively connect solid elements to the Nllinks, a grid of frame elements was superimposed to the top nodes of bottom drum. Latter frame elements had 1 mm2 section area and were assigned typical mild steel mechanical properties. The same procedure as described previously was executed for the bottom nodes of top drum. This arrangement was necessary to capture the unrestrained rotational degrees of from at the 3D-solid elements. Stiffness properties and material quality of latter frame elements were negligible compared to Nllink and solid element stiffness and did not affect computer model load-displacement capacity. Nllink members deployed in the computer model had shear and axial stiffness. Axial Nllink compression was preset at 5.000.000 KN/m, and kept constant throughout all ensuing analysis runs. Latter axial stiffness reflected approximately rigid member axial behaviour. On the other hand, shear stiffness was set equal in both horizontal directions at the shear-sliding interface. It must be emphasized that adequate Nllink shear stiffness was found by iteration. The secant stiffness resulting from full scale testing of the shear sliding interface between the two column drums was evaluated in Figure 2 and numerically presented in Table 2. The task of individual computer runs was to numerically determine adequate Nllink shear stiffness compatible with the applied horizontal force

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Table 2.

Specimen and NLLINK stiffness.

Load step

1

2

3

Target Displ. [mm] Specimen Global Secant Stiffness [N/mm] NLLINK Stiffness [N/mm]

0.18

0.91

3.78

111,111

40,659

9,788

250,000

226

28

and corresponding displacement. In general, three significant load-displacement points were selected to effectively evaluate stress distribution at the shearsliding interface. These points have been marked on the load-displacement curve recorded experimentally (Figure 2). Point 1 reflected the yield limit, point 2 the plastic limit and point 3 the point at maximum horizontal drift accordingly (Figure 2 and Table 1). From the above considerations it is evident that adequate Nllink shear stiffness was evaluated by numerical iteration. A series of computer runs was performed, whereby Nllink stiffness was numerically adjusted to establish equilibrium with the applied force and corresponding displacement. In general, it was evaluated that Nllink stiffness did not correspond to the global test secant stiffness divided by the number of Nllink elements (Table 2). 5

Figure 8a. Von Mises stress concentration at the circumference of top column drum.

NUMERICAL RESPONSE

To check the validity and adequacy of the computer model presented in the previous sections, solid element stress contours at the shear-sliding interface were investigated. Hereby, the von Mises criterion to identify maximum stress and potential yield initiation was employed. In as far as, drum material behavior was assumed isotropic and linear elastic, the von Mises yield criterion could be applied. Typically, the von Mises combined stress was employed using the equation below, whereby, σ1 , σ2 , and σ3 were the principal stresses: ,
 σvm = 12 (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 The computations yielded numerical results describing von Mises stress concentrations at the shear sliding interface, thus, identifying regions of maximum stress where potentially fracture and/or material spauling can occur. To present in detail stress distribution at the shearsliding interface, solid element von Mises stress contours above and below the sliding/contact surface

Figure 8b. Von Mises stress concentration at the circumference of bottom column drum.

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Figure 9. Von Mises stress concentration at shear sliding interface. (a) Top drum. (b) Bottom drum.

between the drums are plotted in Figures 9a and 9b. The von Mises stresses corresponding to load step 3 (Table 2) are presented. The maximum von Mises stress identified at this step was 1.5 N/mm2 . A thorough inspection of the numerical results verified stress concentration along drum circumference. Bottom drum stress concentration was present in the direction of horizontal load (Figure 9a), while for the top drum stress concentration was present in the opposite direction (Figure 9b). Initially, at load step 1 stress concentration was identified along a 2 mm wide ring along drum circumference. With increasing horizontal load the zone expended and propagated inwards until, ultimately, at load step 3, it developed into a 100 mm wide ring, whereby von Mises stress concentration was clearly evident (Figures 8 and 9).

Detailed inspection of stress distribution at the region of maximum stress at drum symmetry axis in the action direction of applied horizontal load, showed that maximum stress appears in the middle portion of the flute realized by the finite element mesh (Figs 8a, b). Thus, the numerical simulation revealed a sector corresponding approximately to 18◦ where maximum stress was present. Consequently, it can be concluded that the stress distribution evaluated numerically is independent from finite element geometry. In addition, the location and shape of the regions with maximum stress along drum circumference (Figs 8a, b) reflect closely the arrangement of fluting in the drum section of a typical ancient doric column, as is also shown in Figure 9. Alongside the stress concentration along drum circumference, a region around drum central region was identified to participate in the shear sliding mechanism, however, in the latter region, von Mises stresses were approximately at 50% magnitude compared to the stresses at drum circumference. Remarkably, the central portion of the stone drum was not participating in the load carrying mechanism (Figs 9a, b). This corresponds to the point where the vertical wooden pole is inserted. In addition to the observation made previously, it was verified, that a ring, 250 mm wide, between the outer drum region and the central drum region, where stress concentration was present, did not participate in the load carrying mechanism (Figure 5b and 5c). These observations correspond well to the experimental behavior observed. Similar stress concentration as presented in Figure 5 was detected for load steps 1 and 2 (Table 2), although at lower stress magnitude, as it was to be expected. Computed von Mises stress distribution for load steps 1, 2 and 3 was compatible to the load-displacement curve recorded experimentally. As far as the typical stress concentration is concerned, its numerical evaluation verified location of stone failure during testing. It can, therefore, be concluded that formulation of the computer model presented in the previous sections was adequate in predicting the stress concentration in the stone material and in identifying also the formation of typical fluting along the drum circumference.

6

A thorough survey of the analytical results obtained from the computer model presented, as well as their comparison to the experimental results obtained from full scale static cyclic testing led to the following conclusions: 1. Stress concentration at the shear-sliding interface between the two column drums was located along drum circumference. Maximum stress concentrations

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CONCLUSIONS

in the bottom drum were identified in the direction of horizontal load, while for the top drum, latter concentrations were present in the opposite direction. In addition to the latter observation, a region around drum center was also found to participate in the load transfer mechanism, whereby, the rest of drum surface was inactive, and hence, von Mises stresses were extremely low compared to the peak values recorded at drum circumference and centre. 2. The horizontal sliding stress is not distributed evenly across the interface of the two-column system. The area is divided in alternating areas of high and low stress concentrations. These follow closely the configuration found in the drum bases of ancient columns with radial zones of varying roughness and friction activation. In addition to that it was shown that the typical number of flutes, i.e. 20, appears to be a design optimum. 3. To effectively model the shear-sliding mechanism between the two column drums, Nllinks could be placed at drum circumference and around drum center only. Such procedure can be used for future computer analysis, whereby, required number of Nllink elements would be drastically reduced. 4. The numerically calculated distribution of stress concentrations along the perimeter of the stone drum during sliding corresponds to the typical fluting present in columns of ancient Greek and Roman architecture. Evidently, fluting of the columns was a conscious decision made by their builders and stone masons based on previous evidence that such regions were prone to stone cracking and spauling. Hence, in order to prevent and control possible material deterioration at the surface these areas were removed, resulting in a shape that acts as a reinforcement of the perimeter surface. 5. Numerical simulation of the shear sliding interface revealed regions of stress concentration in close agreement with the experimental behaviour observed. The latter effect strongly supports the conclusions made previously. In as far as the analysis presented in this paper was a linear computation, the execution of potential future computer runs, with material non-linearity can definitively improve the above results.

6. It appears that the design of columns in ancient classical architecture reflects faithfully and closely the way the material functions, or expected to function, in the specific structural configuration. The aforementioned tentative conclusions have to be taken into proper account in decisions aiming to restore the structural and aesthetic integrity of damaged stone column drums.

ACKNOWLEDGMENTS The authors would like to thank the Committee for the Preservation of the Epidauros Monuments for its financial assistance and especially Prof. Lambrinoudakis for his initiative to support the research. They would also like to acknowledge the willing contribution to the experimental work of all the staff in the Laboratory for Earthquake Engineering at NTUA.

REFERENCES Bathe, K.J. 1996. Finite Element Procedures. New Jersey: Prentice Hall Inc. Ninis, N. & Kourkoulis, S. 2001. On selecting a compatible substitute for the Kenchreae poros stone used in the Epidaurean Asklepieion. In Proc. 6th National Congress of Mechanics,Hellenic Society of Theoretical and Applied Mechanics Thessaloniki, 19–21 July, Greece. Orlandos, A. 1994. The building materials of ancient Greeks and their way of application. Publication of the Archaeological Society in Athens (in Greek). Penrose, F.C. 1973. An investigation of the principles of Athenian Architecture McGrath Publishing Company. SAP2000. Three Dimensional Static and Dynamic Finite Element Analysis and Design of Structures. Computers and Structures Inc., Berkeley California USA. SAP2000. Three Dimensional Static and Dynamic Finite Element Analysis and Design of Structures, Analysis Reference, The Nllink Element. Computers and Structures Inc., Berkeley California USA. Zambas, C. 1998. The refinements of the Parthenon columns Ph. D. Thesis, National Technical University of Athens.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

On determining appropriate parameters of mechanical strength for numeric simulation of building stones N.L. Ninis Archaeological Museum of Epidauros, Ligourio, Hellas, Greece

S.K. Kourkoulis Department of Mechanics, School of Applied Sciences, National Technical University of Athens, Zografou Campus, Athens, Hellas, Greece

ABSTRACT: The present work concerns the determination of the mechanical behaviour of building stones in relation to the restoration of stone monuments. The study is focused on the particular case of the Kenchreae poros stone, the main building material in the construction of the Epidaurean Asklepieion. It is characterized as limestone of low strength and relatively ductile behaviour, with a distinct microscopic nature, distinguished by its ability to sustain substantial deformation without developing high stresses in areas of concentrated strain. The mechanical behaviour of the Kenchreae stone is compared to that of a hard, brittle limestone, the Alfopetra of Crete, as a possible substitute of the ancient material. Composite specimens of simple geometry have also been tested in an attempt to evaluate the interaction between old and new material, as well as specimens with specially configured bases in order to investigate the influence of contact conditions at the boundaries.

1

INTRODUCTION

The investigation of the mechanical behaviour of natural building stones presented here is associated with an on going restoration program concerning three of the most important monuments of the Epidaurean Asklepieion, the most celebrated healing center of the ancient world: The circular building of Tholos (the assumed subterranean dwelling place of the healing god), the Avaton or Enkoimeterion (a large stoa used for the incubation and cure of the sick) and finally the Propylon of the Gymnasium (a building complex used for the sacred meals). The main part of the upper structure of the above monuments is built of a local natural building stone, commercially known under the names “the Kenchreae stone” or “the porolithos of Kenchreae”. It is a porous oosparite of sandstone-like appearance, due to its high porosity. The constitutive mechanical behaviour and the mechanical properties of this material, as well as of a series of other natural building stones considered as possible substitutes of it, have already been well established in preliminary investigations by Kourkoulis et al. (2000) and Ninis & Kourkoulis (2001). However, it is widely accepted that the proper use of the values of mechanical properties, as they

are determined from standardized Strength of Materials experiments, demands good understanding of the function of the specific building element made from each material, as well as clear description of the boundary conditions of the structural element in the particular structure. In other words, the material properties from a laboratory test are to a certain degree more akin to a structure than nearly an element. This is due to boundary constrictions, and it cannot be ignored in case numerical analyses and simulations are carried out. As a typical example of the dependence of the mechanical properties on the boundary conditions, one could mention the well-documented influence of the lubrication of the loading platens and the bases of the specimens, which changes not only the values of the compressive strength of natural building stones (Read & Hegemier 1984, v. Vliet & v. Mier 1995), but also the failure mode itself. Such an observation should be carefully considered, especially in case of the restoration practice, where the co-operation and transfer of loads between the various structural elements is usually achieved with the aid of dry friction. A typical case is shown in Figure 1, depicting details of the dry joint between ancient and new material in a recently restored pillar of the Avaton of the Epidaurean Asklepieion.

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5

σ [MPa]

4

3

2

1

0

0

0.01

0.02

0.03

0.04

0.05

ε Figure 2. Stress–strain curves for ancient Kenchreae stone obtained from standardized uniaxial compression tests. Figure 1. Detail of the contact between ancient and new material in a restored pillar of the Avaton.

Towards this direction an effort is described in the present work, aiming to the understanding of the limitations in the use of the mechanical properties due to the boundary conditions. The aim of the work is three-fold: First to draw conclusions concerning the use of the appropriate values of the mechanical properties in numerical models, second to develop criteria permitting suitable selection of substitute stones as compatible as possible with the ancient material and, finally, to prepare mortars adequate for use alongside with authentic ancient material.

2 THE MATERIALS 2.1

The Kenchreae stone

The authentic material used by ancient Greeks for the erection of the monuments of the Epidaurean Asklepieion comes from the region of Kenchreae in northern Peloponnese. It is an oosparitic limestone of relatively high porosity (about 40%), layered structure and sandstone-like appearance. It is a more or less macroscopically homogeneous material, with complex networks of internal pores and surface vents, sometimes running through the whole width of the specimens, rendering the influence of “size-effect” extremely pronounced and the wide scattering of the experimental results unavoidable. Frequent calcite veins or small calcite accumulations appear within it. The texture ranges from compact looking to very thin-bedded and the colour varies from a whitishgray to grayish-beige and rarely to light yellow. The

dimensions of the pores and the vents vary between a few millimeters and a few centimeters. The above-described kind of structure imposes to the Kenchreae stone a strongly anisotropic character. It has been concluded after series of preliminary tests that it has two axes of anisotropy, one perpendicular to the material layers and one parallel to them, with completely different properties along them. In other words, the Kenchreae stone can be classified as a transversely isotropic material that is described with the aid of five mechanical constants: Two elasticity moduli with respect to directions lying in the plane of isotropy and perpendicular to it, two Poisson’s ratios characterizing the transverse reduction in the plane of isotropy for tension in the same plane and in a direction normal to it, respectively, and the shear modulus for the planes normal to the plane of isotropy (Lekhnitskii 1963). The material appears to be extremely friable and it is very difficult to prepare specimens suitable for accurate Strength of Materials tests. Its mechanical properties exhibit strong variation depending on the exact point of sampling and the age of the material (Ninis & Kourkoulis, 2000). Some typical axial stress – axial strain curves for ancient Kenchreae stone under uniaxial unconfined compression are shown in Figure 2. The specimens were formed from amorphous architectural remains of the archaeological site with the permission of the respective authorities (Kourkoulis & Ninis 2002). It can be seen that after a more or less linearly increasing portion up to a strain level of about 0.004 the curve exhibits an abrupt drop. Then it rises again, sometimes exceeding the initial peak. From this point on, the curve follows a smooth path almost horizontal up to a strain level equal to about 0.015. After this

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point it starts to drop with very small slope until the final destruction of the specimen. Figure 3 shows the same stress–strain curves plotted for specimens made from recently quarried blocks or drilled cores. Despite the variation of the samples due to sampling point and depth it can be seen from Figures 2,3 that the overall behaviour of the ancient and fresh Kenchreae stone is very similar, notwithstanding the larger scattering of the maximum strength and the ultimate strain. At this point it could be argued that results from small size specimens are doubtful due to the macropores, layering and inhomogeneity of the material. However, the tests so far have shown that this type of influence is restricted to mechanical parameters such as the peak load and the extent of the post-peak deformation. On the other hand, in qualitative terms the appearance of the stress–strain curves is very consistent, thus implying that it represents the true material behaviour irrespectively of the specimen size. The failure mode can be described as a combination of axial cracks with parallel crushing of weak-material layers. The familiar Mohr’s cone was not detected, indicating that the conventional failure theories cannot be applied for such type of materials. The first 6

5

σ [MPa]

4

3

2

1

0

0

0.01

0.02

0.03

0.04

0.05

ε

Figure 3. Stress–strain curves for freshly quarried Kenchreae stone, obtained from standardized uniaxial compression tests. Table 1.

visible cracks appear at strain levels equal to about 0.016. However, the final destruction of the specimens (depending on the lubrication conditions) takes place at strain levels corresponding to a height reduction of about 5%. The average values of the modulus of elasticity, E, compressive strength, σC , failure strain and Poisson’s ratio for both ancient and fresh Kenchreae stone are listed in Table 1. The same table also includes values for the Alfopetra stone, which is mentioned in the next section. 2.2 The “Alfopetra stone” Having established the peculiar character of the authentic material of the monuments with respect to its mechanical behaviour, it was considered appropriate to compare this behaviour to that of another stone exhibiting the brittle behaviour mode, most commonly associated with natural building stones such as marble, sandstones, hard limestone etc. For this purposeAlfopetra was chosen, a micritic porous limestone from Crete, the physical and mechanical properties of which, from commercial standard testing, appeared to be fairly close to the respective ones of the Kenchreae stone (Sakellariou 2001). It must be noted at this point, that the above results were obtained from quasi-static tests, i.e. for strain rates not exceeding 10−2 mm/min. However, it has been observed that in unconfined compression experiments under load control conditions and strain rates around 0.5 MPa/sec, the respective values for the failure stress vary in the range of 13–18 MPa (Sakellariou 2001). The above difference is clearly illustrated in Figure 4: The strength of Alfopetra is significantly increased under quasi-static tests, from 23 MPa to 37 MPa, while that of Kenchreae stone is decreased from 16 MPa to about 4 MPa. Such a behaviour could be attributed to the different response of the respective microstructures to creep. The two different curves for Alfopetra, shown in Figure 4, correspond to data obtained from external and internal strain measurements (dial gauges and electrical strain gauges, respectively). The discrepancy between the two measurements is a welldocumented problem in the literature of Rock Mechanics and it is beyond the scope of the present work, which studies the bedding effects as part of

Mechanical properties of ancient and fresh Kenchreae stone and of Alfopetra.

Property → Material ↓

Failure stress [MPa]

Young’s modulus [GPa]

Poisson’s ratio [-]

Failure strain [-]

Specific weight [kN/m3 ]

Ancient Kenchreae Alfopetra

3.5 4.2 34.2

1.8 2.0 12.5

0.26 0.26 0.27

0.050 0.035 0.004

14.8 15.3 17.8

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40

3 mm Alfopetra stone

σ [MPa]

30

Kenchreae stone

134 mm

20

10

0

0

0.01

0.02

0.03

0.04

(b)

3 mm

(a)

0.05

ε

Figure 4. Stress–strain curves for Alfopetra and Kenchreae stone as obtained from “quasi-static” compression tests.

the base contact configuration. For that reason all the results presented here are based on measurements with dial gauges from base to base.

50 mm

50 mm

70 mm

70 mm

Figure 5. The configuration and dimensions of the first class of specimens (geometrical modification of the bases).

3

EXPERIMENTAL PROCEDURE

The aim of the present experimental work was the investigation of the influence of the boundary conditions at the bases of the specimens, as well as the behavior of composite specimens with various portions of ancient and substitute material. For this purpose two classes of specimens were prepared: In the first the contact was modified in two ways internally antisymmetric to each other, by special geometrical configuration of the two bases, as shown in Figure 5. The idea for the first modification (Fig. 5a) came from the way the bases of column drums in Greek architecture are configured. In them, near perfect contact between adjacent drums is secured only along a perimeter rim, where surface is made as smooth as possible, while the rest of the base is left roughly worked having a greater roughness and a true contact area that can be less than 50%. In order to also study the complementary effect the reverse configuration was tested (Fig. 5b). Following a preliminary work with composite specimens, the second class of specimens introduces a different way of modifying the bases: One base was substituted by a thin disk of different material. Two cases were studied: Kenchreae stone specimens with a base made of Alfopetra and specimens made of Alfopetra with a base made of Kenchreae stone. In effect, using this class of specimens the friction conditions

Figure 6. The configuration and dimensions of the second class of specimens (material modification of the bases).

at the base are changed while the mass of the specimen remains essentially the same, as it can be seen in Figure 6. The above-described two classes of specimens were tested in uniaxial unconfined compression using a very stiff hydraulic loading frame of maximum loading capacity 1000 kN. The load was statically applied at a rate not exceeding 10−2 mm/min.

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Specimens with specially configured bases

The special configuration of the bases with contact along a perimeter rim (thickness 10 mm and height 3 mm) resulted in an interesting change of the overall behaviour of the Kenchreae stone, as it can be seen in Figure 7. It is emphasized at this point that the stresses were obtained by dividing the externally applied load with the true contact area, i.e. that of the perimetrical contact ring. Specimens A and B were of the “ring contact” type (Fig. 5a), where specimen C was of the same type but slightly modified since the corners of the ring were smoothened slightly resulting to a “tapered ring contact” specimen. In all three cases, the stress–strain curve increases to a peak strength (at a strain level equal to about 0.0024, roughly 40% less than in

12 9 6 3 0

0

0.01

0.02

0.03

0.04

0.05

ε

Figure 7. The influence of the ring type contact on the true stress–strain curve of Kenchreae stone.

the normal specimen) after which it drops following local yielding of the material of the ring. The initial response becomes significantly stiffer, by roughly 40%. Specimen C exhibits the most dramatic change: Its maximum strength is practically tripled without affecting the extent of the post-peak region. Characteristic tested specimens of the “ring contact” type are shown in Figure 8. It is worth mentioning the development of vertical calcite veins during loading at the upper and lower bases of the specimens, which indicate load transfer paths (Fig. 8a). These veins are gradually transformed to axial cracks as the specimen is led to failure. The respective curves for Alfopetra are plotted in Figure 9. As it is seen from this figure, specimens with “tapered ring contact” yield a more stiff initial

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C B A

Figure 8. Tested Kenchreae specimens of the “ring contact” type, showing the development of calcite veins and the failure mode.

4 TEST RESULTS AND INTERPRETATION 4.1

15

σ [MPa]

The dimensions of the specimens are shown in Figures 5,6. Special care was taken to ensure that the bases of the cylinders were parallel to each other and perpendicular to the load direction. A semi-spherical head interposed between the loading plate and the moving piston further ensured the coaxiality between load and specimen. For the internal measurement of axial displacements, three dial gauges of sensitivity equal to 10−6 m were used, placed at 120◦ to each other, in order to check the symmetry of the loading. The detected rotation of the end-platens was negligible. In some tests, a system of three electrical strain-gauges of the orthogonal rosette type was used, placed again at 120◦ to each other, for comparison reasons and in order to determine Poisson’s ratio of the materials. Concerning the interface between end-platens and specimens, the majority of tests were carried out using non-lubricated surfaces, since the material is used in dry joints. However, some preliminary tests were carried out with carefully lubricated surfaces, since Drescher and Vardoulakis (1982) and Read and Hegemier (1984) have pointed out that little can be inferred from non-lubricated uniaxial compression tests. Minimization of friction was achieved by interposing two sheets of PTFE between the base of the specimen and the respective platen. Additionally, the internal surfaces of the two sheets were covered with an amount of stearic acid, which has been proved very efficient in reducing friction under high loads (Vardoulakis et al. 1998). The results of these preliminary tests verified the conclusions of previous research concerning the paramount importance of lubrication. However, since the present study was motivated by the needs of experts working for restoration projects and using exclusively dry joints, the respective conclusions are omitted from the analysis.

40

σ [MPa]

30

20

Intact specimen

10

Tapered ring contact Central contact Figure 10. The failure mode of “ring-contact” and “tapered ring-contact” types of specimens made from Alfopetra stone.

0 0

0 .001

ε

0.002

0.003

Figure 9. The influence of various contact types on the true stress–strain curve of Alfopetra.

Figure 11. The failure mode of the “central-contact” type specimens made from Alfopetra stone. The punch-type action of the central portion of the specimen is clearly shown.

40 Intact specimen 30 σ [MPa]

response and produce a 20% decrease in terms of maximum strength and ultimate strain. The “reverse” configuration of the specimens’ bases that is the “central-contact” type specimens (Fig. 5b), yield a less stiff initial response, an additional decrease in terms of maximum strength and about 10% decrease of the ultimate strain. However, the most important out-come of the particular configurations for the case of Alfopetra, is the change in the mode of failure, with the total suppression of the familiar Mohr-Coulomb friction cone in the specimens, which failed instead by forming vertical cracks on the outer lateral surface in both variations of “ring-contact” and “tapered ring-contact” specimens, as it is clearly seen in Figure 10. For the case of the “reverse” configuration of the bases or “central-contact” type specimens the failure mode is slightly more complex: A friction cone was produced in the central part of the base, which acted as a punch, splitting the outer cylinder of the specimen into full height big vertical chunks (Fig.11). In the case of Alfopetra it is very instructive to compare Figure 9 with Figure 12 in which the nominal stresses are plotted, i.e. the load divided by the total area of the central cross section of the specimen. Here it is seen that the maximum strength obtained (associated with the mid-section) reaches the value of about 15 MPa, which corresponds to the yield stress of the material as it is obtained from the curve of the intact Alfopetra specimen.

'Central-contact' type specimens 20

10 'Tapered ring contact' specimens 0 0

4.2

In order to illustrate problems arising from forcing two mechanically incompatible stones to work together

0.002

0.003

Figure 12. The nominal stress–strain curves for Alfopetra and for the various types of specimens.

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0.001

ε

Composite specimens

Figure 13. A tested composite specimen consisting of two halves with adhesion plane parallel to the load direction. The different failure modes are clearly visible.

and study the resulting behaviour, tests on composite specimens are performed. Tests on composite specimens of simple geometry have been reported in previous work (Ninis and Kourkoulis 2001), in a first attempt to evaluate the interaction between old and new material in restoration works. The specimens consisted of Kenchreae stone (low strength, high ductility) and Alfopetra (low to medium strength, brittle nature) respectively. Two types of composite cylindrical specimens were tested. The first consisted of equal parts of Kenchreae stone and Alfopetra, with the adhesion plane either parallel or perpendicular to the loading direction (Fig. 13). In the specimen consisting of two halves connected along a vertical interface, failure originated at the weaker material, by crushing of the weak-material layer, almost perpendicularly to the loading axis, at an overall stress level equal to about 5.5 MPa. Then at a stress level equal to about 6.0 MPa axial cracks appeared, again in the Kenchreae stone, and finally at a stress level equal to 11.2 MPa the failure propagated in the substitute material in the form of surface axial cracks. Taking into account the ratio of the elasticity moduli of the two materials tested (EAlfopetra /EKenchreae ∼ = 6.2), one concludes that the first failure of the Kenchreae stone takes place at a stress level equal to 3.2 MPa and the axial cracks appear at a stress level of 3.5 MPa. On the other hand, the failure of the substitute material takes place at about 35.0 MPa. It is thus seen that the failure of the two materials takes place at their respective failure stresses and the substitute material (Alfopetra) cannot improve on the strength of the mechanically weaker material (Kenchreae stone), at least for the specific configuration. A characteristic fractured specimen of this type is shown in Figure 13. For the specimens of equi-dimensional parts with horizontal interface, as it was expected, the weaker material (Kenchreae stone) failed first at 5.6 MPa. However, it is very important to note that this stress is relatively higher compared to the failure stress of the pure Kenchreae stone. The failure of Alfopetra takes

Figure 14. Tested composite specimens consisting of two parts with join plane parallel to the load direction. The change of the failure modes are clearly visible in both cases.

place at about 7.7 MPa, considerably lower compared to its failure stress. For the specific configuration, i.e. load perpendicular to the adhesion plane, the substitute material (Alfopetra) constrained the generation of cracks within the mass of the authentic material increasing its apparent strength. However, once the first cracks appeared within the mass of the Kenchreae stone, they propagated within the mass ofAlfopetra the strength of which was reduced to less than one fourth of its true failure stress. The previous work is extended here by testing another type of composite specimens having the configuration shown in Figure 6: One of the bases of the specimen was substituted by a horizontal disk from a different material with an approximate thickness of 1 cm, i.e. about 1/7th of the overall height of the specimen. The above configuration is in effect another kind of modification of the boundary conditions. Where a soft, ductile end is provided the mode of failure of the brittle material changes to that of vertical splitting. In a way it functions as the base with “ring-type” contact. It acts as an elastic spring and energy absorber, which absorbs the energy by yielding and failing, thus protecting the rest of the specimen. The base friction effects are accommodated within the soft base, and the load is distributed evenly across the section. The result is failure by vertical tension cracks along the perimeter of the specimen with little or no bulging at all. All the same, the Mohr cone is totally suppressed (Fig. 14b). The opposite takes place in the case of a hard, brittle base (Alfopetra) to a specimen from a ductile rock (Kenchreae stone). Here the small patch of alfopetra at the top adds to the rigidity of the specimen and reinforces friction effects. As a result the vertical cracks of Kenchreae stone are forced to form the familiar Mohr–Coulomb cone, as it can be seen in Figure 14a. Figure 15 shows the corresponding stress–strain curves for the two types of composite specimen tested

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15

Kenchreae stone Alfopetra

σ [MPa]

12

9

6

3

0

Figure 16. Exfoliation and fracture of ring-type contact specimens concentrated near the surface and generating finally vertical tension cracks.

0

0 .01

0 .02

0 .03

0 .04

0 .05

ε Figure 15. Stress–strain curves for both types of composite specimens tested in the present work.

here. It can be seen from this figure that: 1) In the case of the Alfopetra specimen with its top base made from Kenchreae stone the overall appearance of the curve is seriously altered compared to that of pure Alfopetra specimens. In fact and up to a strain level equal about ε = 2.5% the curves approximate qualitatively that of pure Kenchreae stone. From that point onwards as the top base is practically crushing under the applied load the load starts to increase rapidly towards the pure Alfopetra strength, without reaching it due to the cracks which were generated. 2) In the case of the Kenchreae stone specimen with a base made of Alfopetra stone the respective curve is almost identical to that of pure Kenchreae material, transposed simply to higher load levels. However, in both cases it is clear from Figure 15 that the weakest material plays the dominant role in the recorded behaviour irrespectively of its relatively small size. The configuration of a hard specimen with a soft top base produces a stress–strain curve resembling the initial part of the typical curve for metals, however at an overall lower stress level. It consists of the typical linear elastic portion up to the yield, followed by a nearly horizontal flow region leading finally to the hardening region up to the peak load. In effect, a soft base of Kenchreae stone assigns to a very brittle material, as Alfopetra, non-element ductile characteristics by modifying only the boundary conditions where the load is applied. The same can be effected by the geometrical configuration which was reported in previous paragraph, although to a far lesser degree.

5

The main objective of the present investigation was to study the mechanical strength of some natural building stones, considering the contact between the end-platen and the specimen base as a dry joint and changing the conditions across this boundary so as to better approximate real life situations.Adirect analogy can be drawn, for example, between the cylindrical specimen used in unconfined compression and the column drum of an ancient temple, which under static conditions is also subjected to the same kind of loading. It appears that by restricting the contact surface to a narrow ring along the perimeter of the specimen base, the non-uniform distribution of stresses across the base is minimized. The load is transferred along vertical linear paths in the surface. The lateral surface of the specimen is being stressed as a cylinder under plane strain conditions. The results show that exfoliation and fracture are concentrated near the surface and take the form of vertical tension cracks some-times running the full height of the specimen. The two photographs of Figures 16a and 16b clearly illustrate the above point. The exfoliation seen in Fig.16b bears a cunning resemblance to the flutes of a doric column: A number of vertical cracks (usually between seven and ten), developing at the base at equal distances along its perimeter, have systematically recorded during the present series of tests (Fig. 17). By introducing the ring contact at the bases it seems that one is able to control the mode of failure and control the “how” and “where” the specimen is going to crack. The effectiveness of such a geometrical modification of the bases is more pronounced in case of brittle materials. In fact the modification of the

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DISCUSSION

Figure 17. Vertical cracks (seven to ten), developed along the perimeter of the base at almost equal distances from each other.

bases constitutes a method for eliminating the nonuniform distribution of contact stresses in cases where lubrication cannot be applied. Recapitulating it is not implied that the composite stress–strain curve should be used generally, however in case an overall qualitative picture of the expected behaviour of a structural element (e.g. restored architectural member) is required the curve of the composite specimen provides useful guidance. The effect of the aforementioned base configuration depends also on the material used. In a brittle material it functions by making the contact more ductile, able to absorb energy as this outer ring is free to expand laterally without failing due to its shape and dimensions since it is a very thin annulus. On the contrary, in materials like the Kenchreae stone, which possesses a cellular microscopic nature and fail by crushing internally, the effect is rather reverse: In that case the initial response becomes stiffer. The substitution of the bases of a specimen by a different material functions, also, along the aforementioned logic: The new material will introduce a layer of increased ductility or brittleness altering thus the boundary condition. The situation is relevant to restoration work where many times the restorer will have to introduce a new material to the structure. Knowing how the composite structure is likely to behave qualitatively the … likely disadvantage of having two mechanically incompatible materials working together, could be turned to an advantage. The use of composite specimens is a way of illustrating, notwithstanding the scale effect and the other

mutually related parameters involved in the problem, the expected behaviour of a stone restored with new material. The example of the specimens with bases made of different material makes it clear. The stress– strain curve changes both qualitatively and in absolute terms (maximum strength and strain). The geometrically modified bases, in the context of compression test, constitute an effective way of eliminating the effect of friction, other than the lubrication, by reducing the contact area to a narrow zone along the perimeter. That way the material loaded to high stresses and forced to failure is very small relatively to the size of the specimen, and of such a shape that can be assumed under plane strain conditions. As a result, the mechanism of tension cracks formation is amplified, with the load transfer apparently taking place along vertical linear paths and with most of the damage concentrated in the lateral surface thus leaving almost intact the main part of the specimen. The ensuing overall behaviour of the specimen approaches the element behaviour of the material.

6

The work presented has examined the determination of the mechanical behaviour of natural building stones in relation to the restoration of monuments. Several aspects have been highlighted: The importance of the material nature (brittle versus ductile) as an important evaluation and compatibility criterion for stones, the use of qualitative estimation of the behaviour of composite specimens (and structures) and the need for clear understanding and controlling the boundary conditions. On the basis of the experimental observations reported here the following specific conclusions could be drawn: •

In evaluating the mechanical strength of natural building stones it is important to take into account the full stress–strain curve including postpeak behaviour of the material. Strain rate dependency of the various mechanical constants should also be examined as an indirect measure of the microstructure. • Boundary conditions at the loading interfaces, such as friction, non-uniform stress distribution at the loading surfaces as well as the detailed geometrical configuration of the contact areas are critical for the correct interpretation of experimental results and of paramount importance in simulating the real conditions in a structure using data from element tests. • The Mohr–Coulomb cone is demonstrated to be a friction effect. Whether it forms or not depends on the material tested. Lubrication is one way to reduce

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CONCLUSIONS

friction. It seems that an equally effective way is to modify the geometry of the bases. • Concerning the behaviour of composite specimens, the weaker material, which fails first, is the controlling factor of the overall behaviour, irrespectively of its relative size. It is also important to know the exact nature of each constituent (brittle or ductile), when it comes to predicting the combined behaviour. • The study of composite specimens demonstrated that the mechanical behaviour of each material involved depends on the kinematical restraints imposed on it by the geometry and way of loading. • The completion of ancient architectural members with different material, mechanically incompatible to the authentic, has to be considered carefully based on the size, shape and position of the patch in the particular member, as well as the position of that member in the overall structure. ACKNOWLEDGEMENTS The financial support of the “Committee for the Preservation of the Epidauros Monuments” is gratefully acknowledged. The willing assistance of Mr G. Karydas and Mr Th. Gerakis during the experimental procedure is, also, acknowledged. REFERENCES Drescher, A. & Vardoulakis, I. 1982. Geometric softening in triaxial test on granular materials. Geotechnique, 32: 291–303.

Kourkoulis, S.K. & Ninis, N. 2002. Mechanical properties and compatibility of natural building stones used in the Epidaurean Asklepieion. Final report of the project NTUA-CPEM-1999. Athens: NTUA. Kourkoulis, S.K., Vardoulakis, I. & Ninis, N. 2000. Evaluation and theoretical interpretation of mechanical properties of porolithoi used in the restoration of the Epidaurean Asklepieion. In G. Lollino (ed.), Geological and Geotechnical Influences in the Preservation of Historical and Cultural Heritage; Proc. intern. symp., Torino, Italy, 7–9 June 2000. CNDCI Publishing. Lekhnitskii, S.G. 1977. Theory of elasticity of an anisotropic body. Moscow: Mir. Ninis, N. & Kourkoulis, S.K. 2001. On selecting a compatible substitute for the Kenchreae poros stone used in the Epidaurean Asklepieion. In E. Aifantis and A. Kounadis (eds.), Proceedings of the 6th National Congress on Mechanics. Thessaloniki 17–20 July 2001. Thessaloniki: Giahoudi & Giapouli. Read, H.E. & Hegemier, G.A. 1984. Strain softening of rock, soil and concrete – A review article. Mechanics of Materials 3: 271–294. Sakellariou, A. 2001. The physico-mechanical characteristics of the natural building stones of the Epidaurean Asklepieion. Final report of the project PPC-CPEM-2000. Athens: PPC Testing, Research and Standards Center (in Greek). Van Vliet, M.R.A. & Van Mier, J.G.M. 1995. Concrete under uniaxial compression. Report 25/5-95-9. TU Delft, Faculty of Civil Engineering. The Netherlands:TU Delft. Vardoulakis, I., Kourkoulis, S.K., & Skjaerstein, A. 1998. Post-peak behaviour of rocks and natural building stones in uniaxial compression. In R. de Borst and E. van der Giesen (eds), Material Instabilities in Solids: 207–226. New York: John Wiley & Sons.

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Validation of theoretical models

Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands

Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Experimental validation of combined FEM/DEM simulation of R.C. beams under impact induced failure T. Bangash & A. Munjiza Department of Engineering, Queen Mary, University of London, United Kingdom

ABSTRACT: A combined finite/discrete element model for the pre-failure and post failure transient dynamics of a reinforced concrete structure has been developed. The numerical model uses a simple element already successfully tested for the case of static loading. In this paper, the accuracy of this simple element is assessed under dynamic loading conditions using beam impact experiments undertaken at the Swiss Federal Institute of Technology. The results obtained from these experiments include deflection-time curves for various points along the beam and the deflection pattern for the beam throughout the loading period. The results of this validation process have indicated that this relatively simple element is able to accurately predict both failure mode and failure criteria for dynamic loading conditions. In previous work it has already been proved that the same applies to static loading conditions. Thus this element is suitable for both static and dynamic analysis of failure, fracture and/or fragmentation of a beam or column type structural elements due to hazardous loading conditions. It is worth mentioning that the work is a part of a larger research undertaking aimed at the combined finite/discrete element modeling of structural failure and collapse due to hazardous loads.

1

INTRODUCTION

Hence the equation of motion is represented by

The finite-discrete element method is a numerical method used in the failure and fracture analysis of solids. A typical FEM/DEM system comprises of any number of separate and distinct solids interacting with each other in a defined space. Each of these bodies occupies only one single point in that space at any particular point in time. In such a system contact interaction and contact detection are important factors in the interacting behaviour of each of the bodies. In the case of 2D contact interaction is understood by integrating the governing equations and solving for translation and rotation about the centre of mass. For 3D discrete element systems the task is complicated by the finite rotations about the centre of mass of the discrete element (2–6). A simple two noded element is adopted (1). Each node has three rotational as well as three translational degrees of freedom. The rotational degrees of freedom are eliminated through static condensation. It should be noted that where one of the element nodes is rotated, the bending moment on the other element node is kept at zero. This eliminates the use of matrix inversion when applying static condensation (1).

ˆ +M ˆ u¨ + Cu˙ f (t) = Ku

ˆ is the condensed stiffness matrix, M ˆ is the where K lumped mass matrix and C is the damping matrix. These are fully expressed as     u f .  ..   .  .  .     (2) f (t) =  M  u = θ      .. .. . . 

 m1 0 0 0 0 m 0 0   2 ˆ = M 0 0 m3 0  0 0 0 m4   Ktt  K= ···  Kθu

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(1)

.. . .. . .. .

 Kuθ    ···   Kθ θ

(3)

where, matrix K is the stiffness matrix and u and θ represent the translational and rotational degrees of freedom respectively. In addition matrix Kθ θ is a diagonal matrix. This element has been presented and successfully tested for the case of a reinforced concrete beam, which is subjected to static loading conditions (1). The aim of the research here presented is to further assess the accuracy provided by the use of this simple element when used under the milieu of dynamic loading. Experiments, which have been carried out at the Swiss Federal Institute of Technology (7–10) using a variety of beams, are employed for the purposes of comparison and validation.

2

EXPERIMENTAL PROCEDURE

Beam B1 is chosen for the numerical simulation. The properties of this beam are described in Table 1. The experimental set up is shown in Figure 1. The beam is raised by means of an overhead crane and then dropped from a height of 3.75 m onto a shock absorber. Only one single drop takes place. The authors of the experimental reports (7–10) state that this way of simulating impact loading has the advantage of low cost and mechanical simplicity.

Table 1.

A ringfederpuffer (Spring) type shock absorber was used in the case of this particular beam. This shock absorber can endure a maximum force of the order of 3000 kN. A small plate of mass 57 kg was placed at the base of the shock absorber. The impact force of the beam was measured using three force transducers, positioned underneath the plate. The properties of this shock absorber were investigated using a truss with a concrete block at the free end. This truss was raised and released in the same way as the beams prior to landing on the shock absorber. The mass of the block and the truss were 997 kg and 342 kg respectively. Drop heights for the truss were limited up to 0.75 m. Once beam B1 had landed on the shock absorber bending was induced almost instantaneously and vibration ensued. After the vibration passes the maximum plastic deflection is measured as 590 mm and the ultimate bending moment is found to be 90 kNm. Cameras using high-speed exposures were employed to photograph the beam at the rate of 1000 exposures per second. This is used to plot the deflection of the beam and the deflection at certain points along the beam as a function of time. Maximum deflection is reached at a time of 0.164 seconds after release and occurs approximately 4 m from the hinge. The energy dissipation of the beam is found to be about 70% and that of the shock absorber was found to be between 15–30%. The total angle of rotation is measured as 0.422 radians.

Beam Properties.

Beam B1

H=3.75 m

Beam Depth (m) Beam Breadth (m) Beam Length (m) Beam Mass (kg/m) Upper Reinforcement Lower Reinforcement Youngs Mod x 2nd Mo Inertia EI (kNm2 ) Concrete Strength (N/mm2 ) Concrete Cube Strength (kg/mm2 ) Reinforcement Yield Strength (N/mm2 )

0.3 0.4 8.15 300 3φ10 3φ10 32988 30 35.3 at 5 days 550

Load cell 0.15 m x Load cell

Hinge z

Shock Absorber

y 7.85 m

Figure 1.

Experimental set up for beam B1.

0.15 m

3

The first task of the numerical simulation was to accurately simulate the properties of the shock absorber found in the experimental investigations. Simulation of the impacting truss is performed using a computer code based on the finite-discrete element method. The steel truss, of length 5.36 m, was modelled as a rigid beam. The end element was given greater mass, thus representing the mass of the concrete block. A velocity field is applied along the truss with maximum velocity at the free end descending to zero velocity at the hinge. Drop heights of 0.30, 0.65 and 0.75 m were simulated. The results obtained for the force-time, displacement-time and force displacement properties for the shock absorber are compared in Figures 2–4 with those obtained from the experiment. The same computer code is then used to model the beam impact. The beam is discretized into 57 nodes and 55 elements as shown in Figure 5. The dropping action performed in the experiment is modelled similarly to the truss by applying a velocity field along the beam, as is illustrated in Figure 6.

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NUMERICAL SIMULATION OF EXPERIMENT

z 53

27 0

54

x

L/2

L/2

55 x 0.151 m

Figure 5.

Discretized beam B1 with boundary conditions.

H

Figure 2. Force-time relationship of the shock absorber for various drop heights.

v = vmax

v=0

Figure 3. Displacement-time relationship of the shock absorber for various drop heights.

ε Figure 6.

Figure 4. Force-displacement relationship of the shock absorber for various drop heights.

The beam is assumed to fall as a rigid body and as such the elastic energy prior to impact is negligible in comparison to the total kinetic energy. The net deflection of the beam is calculated by subtracting the deflection due to rigid body motion from the total deflection. The deflection at points 2.5, 4.0 and 5.0 m from the beam is enumerated and compared with those of the experiment. This is shown in Figure 7. The deflected form of the beam is shown in Figure 8. The dotted line shows the movement of the plastic hinge during the time-span of the loading. This output is reproduced at time 0.022s, 0.059s and 0.164s and compared as shown in Figure 9.

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Initial conditions for beam B1.

4

Figure 7. Deflection-time relation at various points from the hinge.

CONCLUSIONS

A simple element has been tested in combination with dynamic loading conditions. The results displayed in the above figures show good agreement with the experimental observations. Figure 8 validates the conclusion of the authors of the Swiss Instititute’s report that the plastic hinge moves along the beam as the deflection increases with time. This shows that the simple element already successfully tested for the static loading case is equally effective when used in combination with dynamic loading conditions. Static condensation can be performed without matrix inversion, thus the CPU processing time is kept to the minimum. The simple element has been shown to work whilst taking into account membrane theory, finite rotations and axial and bending loading. The non-linearity seen in the experimental forcetime relation for the shock absorber using a height of 0.65 m may be due in part to freeplay in the plate fixings.

ACKNOWLEDGEMENTS A sincere debt of gratitude is owed to, Prof. Bachman, and Mrs N Ammann in appreciation of their most generous assistance in the provision of the experimental reports. The assistance provided by Dr W Ammann and Dr S Heubbe-Walker in the interpretation of the report is also gratefully appreciated.

REFERENCES Figure 8.

Deflected form of beam B1.

Figure 9.

Deflected form of beam B1 at various times.

[1] Munjiza, A., Bangash, T., John, N., 2001. Analysis of Structural Failure using Combined Finite-Discrete Element Method, Advances in Fracture and damage Mechanics II, Hoggar: Milan, pp. 405–411. [2] Munjiza, A., Owen, D.R.J., Bicanic, N., 1995. “A combined finite-discrete element method in transient dynamics of fracturing solids”, Int. J. Engineering Computations, 12, 145–174. [3] Munjiza, A., Andrews, K.R.F., 1998. “NBS contact detection algorithm for bodies of similar size”, Int. J. Num. Methods Eng, 43. [4] Munjiza, A., Owen, D.R.J., Crook, A.J.L, 1995. “Energy and Momentum Preserving Contact Algorithm for General 2D and 3D Contact Problems”, Proc 4th Int. Conf Computational Plasticity- Fundementals and Applications, Barcelona, pp. 829–841, 3–6 April. [5] Munjiza, A., Andrews, K.R.F., White, J.R., 1997. “Discretized Contact Solution for combined finite-discrete Method”, 5t ACME Conf. London UK, pp. 96–100. [6] Munjiza, A., Andrews, K.R.F., 2000. “Penalty function method for in combined finite-discrete element systems comprising large number of separate bodies”, Int. J. Num. Methods Eng., 49, pp. 1377–1396.

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[7] Ammann, W., Bachmann, H., Muhlematter, M., 1982. “Versuche an Stahlbeton und Spannbetonbalken unter stossartiger Beanspruchung Teil 2- Konzeption und Durchfuhrung der Balkenversuch, Zusammenfassung der Versuchresultate”. Birkhauser Verlag Institure fur Baustatik und Konstruction, Zurich. [8] Ammann, W., Bachmann, H., Muhlematter, M., 1982. “Versuche an Stahlbeton und Spannbetonbalken unter stossartiger Beanspruchung Teil 3- P1, P2 und B1 bis B8”. Birkhauser Verlag Institure fur Baustatik und Konstruction, Zurich.

[9] Ammann, W., Bachmann, H., Muhlematter, M., 1982. “Versuche an Stahlbeton und Spannbetonbalken unter stossartiger Beanspruchung Teil 4- B9 bis B21”. Birkhauser Verlag Institure fur Baustatik und Konstruction, Zurich. [10] Ammann, W., Bachmann, H., Muhlematter, M., 1982. “Versuche an Stahlbeton und Spannbetonbalken unter stossartiger Beanspruchung Teil 5- Verhalten von zweifeldrigen Stahlbetonbalken bei Ausfall einer Unterstutzung”. Birkhauser Verlag Institure fur Baustatik und Konstruction, Zurich.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

A study of wedge stability using physical models, block theory and three-dimensional discontinuous deformation analysis M.R. Yeung & N. Sun Department of Civil Engineering, The University of Hong Kong, Hong Kong, China

Q.H. Jiang College of Water Conservancy and Hydropower Engineering, Wuhan University, Wuhan, China

ABSTRACT: A study of wedge stability has been carried out using three different methods, namely, physical modeling, block theory and 3D DDA. Physical models of two different wedges were constructed. The orientation of each wedge was varied systematically and the behavior of the wedge at each orientation observed and recorded. All the physical modeling cases were then analyzed using block theory, and four cases involving four different failure modes were analyzed using 3D DDA. The results show that physical modeling and block theory give the same failure mode in all but a few cases. Torsional sliding was observed in one physical model test. For this case, block theory gives a mode of translational sliding on plane2 because it does not consider torsional sliding. The 3D DDA results agree well with the physical modeling results, including for the case involving torsional sliding. This shows that 3D DDA can potentially be used as a general wedge stability analysis because it considers dynamic equilibrium and general failure modes including rotational ones.

1

2 WEDGE STABILITY ANALYSIS METHODS

INTRODUCTION

A discontinuous rock mass consists of discontinuities and intact rock blocks bounded by the discontinuities. Because the intact rock is usually competent, the behavior of a discontinuous rock mass is controlled by the discontinuities. The discontinuities intersect to form rock blocks of different shapes, and the stability of these rock blocks must be analyzed to assess the stability of the rock mass itself. The stability of a block or wedge of rock in a discontinuous rock mass has been studied analytically by many researchers, mainly based on the limit equilibrium approach (e.g. Hendron et al. 1980, Hoek & Bray 1981, Warburton 1981, Priest 1985, Goodman & Shi 1985, Mauldon & Goodman 1996, and Tonon 1998). A review of existing limit equilibrium analysis methods shows that (1) they do not consider dynamic equilibrium, (2) most cannot handle rotational modes, and (3) none can handle complicated rotational modes such as torsional sliding. To address these limitations, a study of wedge stability has been carried out. In this study, three different methods, namely, physical modeling, block theory and three-dimensional discontinuous deformation analysis (3D DDA) are used to analyze the stability of a tetrahedral wedge. This paper presents some results and findings from this study.

2.1 Physical modeling Three-dimensional physical models of rock masses containing joints, satisfying similitude with respect to all important physical quantities, provide a practical means for engineers to assess and design for rock weaknesses due to discontinuities. Moreover, threedimensional physical models have great visual impact. In this study, physical models are constructed to systematically study the failure modes of tetrahedral wedge blocks. 2.2 Block theory Block theory is a three-dimensional geometrical method that allows a rigorous inventory and analysis of rock blocks that can be formed by intersecting rock mass discontinuities and free surfaces (Goodman & Shi 1985). Through performing removability analysis, mode analysis and stability analysis using block theory, one can arrive at the “keyblocks,” blocks that would fail without support. The essence of the theory is to determine rigorously if a block is removable when exposed by free faces, and to evaluate its state of equilibrium.

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(a)

(b)

(c)

Kn Ks Figure 1.

Figure 2. planes.

Safe region (shaded).

Figure 3.

Safe regions for different friction angles of joint

In block theory, stereographic projection is used to obtain graphical solutions to the wedge stability problem. In this study, PT Workshop version 1.0, a software for block theory analysis produced by PanTechnica Corporation, is used to analyze the stability of a wedge. Figure 1 shows a typical “safe region” (shaded) on the stereonet for the “active resultant” of a wedge. If the active resultant is plotted inside the safe region, the factor of safety is greater than 1. Figure 2 shows the safe regions for the wedge for different friction angles of the joint planes. This chart can be used to find the friction angle required for limiting equilibrium, and hence the factor of safety. 2.3

3D DDA

The original DDA developed by Shi and Goodman (1984) is a two-dimensional numerical model for the statics and dynamics of discontinuous rock systems. Significant recent development of DDA has been mainly focused on extending the capabilities of the two-dimensional model, e.g. by Cheng and Zhang (2000) and Kim et al. (1999). While we understand various researchers are working on 3D DDA, only

some preliminary work on this subject has been published (Shi 2001). The key to 3D DDA is a rigorous kinematic theory that governs the interaction of many three-dimensional blocks. This theory must provide algorithms to judge contact locations (first entrance positions) and the appropriate state of each contact (open, sliding or locked). As a part of this kinematic theory, a point-toface contact model for 3D DDA has been developed. Figure 3 shows examples of how different types of contacts can be reduced to point-to-face contacts. Normal spring, shear spring and friction force submatrices are derived by vector analysis and the penalty function method. In addition, open-close iteration criteria and operations performed for different changes in contact state are defined. This point-to-face contact model has been implemented into a 3D DDA computer program. The stability of a wedge can be analyzed using this program. 3

CASES STUDIED

In this study, tetrahedral rock blocks are considered because they dominate in many rock types (Windsor & Thompson 1992) and because it has been observed in tunnels that most of the failed blocks are tetrahedral in shape (Hatzor 1992). Physical models of tetrahedral wedges are constructed and tested, and then block theory and 3D DDA are used to analyze the stability of the tetrahedral wedges in the physical models. The physical models are constructed of plaster.As shown in Figure 4, the model consists of a base block, two fixed side blocks and the wedge block. The wedge block is placed on the two side blocks and can move freely. The base block is used as a supporting block to which the two side blocks are attached. It also allows the wedge block to slide or fall freely without hitting the tilt table. An assembled model is secured to a tilt table. The angle of inclination of the tilt table, α, and the angle β that characterizes the orientation of the model

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Point-to-face contact model.

Table 1. block 1.

Block theory & physical modeling results for

Block theory

Figure 4.

Physical model of wedge.

7cm

16cm

7cm 7cm Plane 2 45º

Plane 2 45º

Plane 1 45º

Plane 1 45º

Block 1

Block 2

Figure 5. Wedge blocks.

with respect to the dip direction of the tilt table (see Figure 4) are varied systematically. In this tudy, the angle β varies from 0◦ to 360◦ in 20◦ increments, and the tilt angle α varies from 0◦ to 90◦ in 10◦ increments. To perform a test, a model is secured in the desired position corresponding to the chosen α and β values, with the wedge block held in place. Once the desired position is fixed, the wedge block is released and its behavior observed and recorded. Each test is videotaped by a digital video camera for later analysis. In this study, two different models are used. The dimensions of the two wedge blocks, named block 1 and block 2, are shown in Figure 5. Each wedge block is bounded by two joint planes (plane 1 and plane 2) and two free surfaces, one horizontal and one vertical. The free-surface faces are isosceles triangles.

4 4.1

RESULTS Physical modeling

From the physical model tests, the behavior of the two different wedge blocks was observed and recorded. Some representative results are given in Table 1 for block 1 and in Table 2 for block 2. Four different failure modes were observed. They are sliding on a single plane (plane 1 or plane 2), sliding on two planes simultaneously (wedge sliding), free falling and torsional sliding. For block 1, when β is constant and α varies from 0◦ to 90◦ , the block changes from being stable to unstable

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Physical modeling

β

α

Factory of safety

Failure mode

Failure mode

60 60 60 60 60 60 60 60 60 60 60 80 80 80 80 80 80 80 80 80

0 10 18.25 20 30 40 50 60 70 80 90 0 10 17.17 20 30 40 50 60 70

1.98 1.37 1.04 0.99 0.72 0.51 0.36 0.28 0.21 0.13 0.05 1.97 1.31 1.00 0.92 0.65 0.44 0.26 0.12 0.04

80

80

0

80

90

0

Stable Stable Wedge Wedge Wedge Wedge Plane 2 Plane 2 Plane 2 Plane 2 Plane 2 Stable Stable Wedge Wedge Wedge Wedge Wedge Plane 2 Plane 2 Free Falling Free Falling

Stable Stable Wedge Wedge Wedge wedge Plane 2 Plane 2 Plane 2 Plane 2 Plane 2 Stable Stable Wedge Wedge Wedge Wedge Wedge Plane 2 Plane 2 Free Falling Free Falling

Table 2. block 2.

Block theory & physical modeling results for

Block theory

Physical modeling

β

α

Factor of safety

Failure mode

Failure mode

320 320 320 320 320 320 320 320 320 320

0 10 10.25 20 30 40 50 60 70 80

0.78 0.98 0.99 1.26 1.70 2.61 5.89 5.39 2.24 1.33

Wedge Wedge Wedge Stable Stable Stable Stable Stable Stable Stable

320

85.00

1.07

Plane 2

320

90

0.89

Plane 2

Wedge Wedge Wedge Stable Stable Stable Stable Stable Stable Stable Torsional Sliding on Plane 2 Plane 2

Sliding on plane 1, wedge sliding or free falling

Sliding on plane 1 or wedge sliding

Sliding on plane 1, wedge sliding or free falling

Sliding on plane 2, wedge sliding or free falling

Sliding on plane 2, wedge sliding or free falling

Sliding on plane 2 or wedge sliding

β

β Sliding on

Sliding on plane 1

Sliding on plane 1 or wedge sliding

plane 2

Sliding on plane 1 or wedge sliding

Sliding on plane 2 or wedge sliding

Sliding on plane 2 or wedge sliding Wedge sliding

Wedge sliding

Figure 6.

Figure 7.

Failure mode distribution chart for block 2.

Failure mode distribution chart for block 1.

at a critical angle. The failure modes that are observed for a given β are shown in a failure mode distribution chart for block 1 in Figure 6. This chart shows the failure modes that are possible when the block is unstable. It can be seen from this chart that wedge sliding is possible when β is between 40◦ and 140◦ and between 220◦ and 320◦ . Free falling occurs when β = 80◦ –100◦ and α = 80◦ –90◦ . Sliding on plane 1 is possible when β is in the range of 90◦ –260◦ . Sliding on plane 2 is possible when β is in the ranges of 280◦ – 360◦ and 0◦ –90◦ . For β = 260◦ –280◦ , the only failure mode that is possible is wedge sliding. Block 2 is unstable at any value of α for β = 0◦ – 180◦ . When β is constant and in the range of 200◦ – 340◦ and α varies from 0◦ to 90◦ , block 2 changes from being unstable (wedge sliding) to stable and then to being unstable (wedge sliding, plane sliding or free falling). Therefore there are two critical angles for these cases. The failure mode distribution chart for block 2 is shown in Figure 7. It can be seen from this chart that wedge sliding is possible for any value of β. Free falling occurs when β = 80◦ –100◦ and α = 60◦ –90◦ . Sliding on plane 1 is possible when β is in the range of 90◦ –260◦ . Sliding on plane 2 is possible when β is in the ranges of 280◦ –360◦ and 0◦ –90◦ . For β = 260◦ – 280◦ , the only failure mode that is possible is wedge sliding. In general, the failure mode changes from wedge sliding to plane sliding as α becomes larger.

Torsional sliding of block 2 on plane 2 was observed when β = 320◦ and α = 85◦ . It occurred at a critical angle in a particular test in which a vertex of the tetrahedral wedge appeared to be somehow stuck to the supporting plane, causing rotation about the vertex. Many subsequent trials of the same case failed to reproduce the torsional sliding mode but produced the mode of sliding on plane 2. This may be due to the fact that the block was almost equally likely to slide on plane 2 in a translational manner and a torsional manner. The rotation may have been triggered by the way the block was held before it was released during the test. 4.2 Block theory For the wedge, the safe region on the stereonet for the “active resultant” was obtained using block theory. Different orientations of the same wedge were considered conveniently by changing the orientation of the “active resultant,” which was the gravitational force in this case. In this way, the factor of safety and the failure mode of the wedge at different orientations were obtained. For block 1 (see Table 1), when β is fixed and α varies from 0◦ to 90◦ , the factor of safety generally decreases from larger than 1 to less than 1. So the block changes from being stable to unstable (wedge sliding, sliding on plane 1 or sliding on plane 2). There is one critical angle (cases in bold in Table 1). For block 2, when β = 0◦ –180◦ , the factor of safety is less than 1, and it decreases as α increases from

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0◦ to 90◦ . So the block is unstable for these cases, and the failure mode changes from wedge sliding to plane sliding as α increases. When β is in the range of 200◦ –340◦ and α varies from 0◦ to 90◦ , the factor of safety changes from less than 1 to larger than 1, and then back to less than 1 (see Table 2). The wedge changes from being unstable to stable and then back to being unstable. So there are two critical angles (cases in bold in Table 2). The stability charts obtained from block theory can be used to explain why there are two critical angles for some cases of block 2 and only one critical angle for block 1. For example, for the case of β = 260◦ , the stereographic projection of the resultant force (gravity) vector changes when α varies from 0◦ to 90◦ as shown in Figure 8. Figure 9 shows the path of the gravity vector. The gravity vector moves from inside the unsafe region (non-shaded region) to inside the safe region (shaded region) and then out into the unsafe region again, which shows why there can be two critical angles for some cases. Similarly, Figure 10 shows why for block 1 there can only be one critical angle for any β when α varies from 0◦ to 90◦ . 4.3

3D DDA

As there were four kinds of failure modes observed in the physical modeling tests, one case for each observed failure mode was analyzed using 3D DDA. For all 3D DDA analysis, the friction angles of planes 1 and 2 are 32.5◦ ; the mass density of the block is 1400 kg/m3 ; the Young’s modulus is 1×107 N/m2 ; and the Poisson’s ratio is 0.49. Figure 11 shows the 3D DDA results of a free falling block 1. For this case, β = 80◦ and α = 80◦ (case included in Table 1). Figure 11 shows the wedge block positions at different elapsed times of 0.04 s, 0.08 s, 0.12 s and 0.16 s. Figure 12 shows the 3D DDA results of block 1 sliding on plane 2. For this case, β = 60◦ and α = 60◦ (case included in Table 1). Figure 12 shows the wedge block positions at different elapsed times of 0.08 s, 0.16 s and 0.24 s. Figure 13 shows the 3D DDA results of block 1 failing by wedge sliding. For this case, β = 60◦ and α = 30◦ (case included in Table 1). Figure 13 shows the wedge block positions at different elapsed times of 0.08 s, 0.16 s, 0.24 s and 0.28 s. Figure 14 shows the 3D DDA results of block 2 failing by torsional sliding. For this case, β = 320◦ and α = 85◦ (case included in Table 2). Figure 14 shows the wedge block positions at different elapsed times of 0.20 s, 0.40 s, 0.60 s and 0.76 s. For this case, to simulate the torsional sliding mode observed in the physical model, the vertex of the wedge that appeared to be stuck during the test was fixed artificially in the 3D DDA, inducing the rotation about this vertex.

Figure 8. Stability charts for different active resultant directions.

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Figure 9.

Figure 10.

Figure 11.

Path of gravity vector for block 2.

Figure 12.

3D DDA results for sliding on plane 2.

Figure 13.

3D DDA results for wedge sliding case.

Figure 14.

3D DDA results for torsional sliding case.

Path of gravity vector for block 1.

3D DDA results for free falling case.

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5

DISCUSSION

The results from the three different methods of wedge stability analysis are compared and additional comments on the methods given in this section. 5.1

Comparison between physical modeling and block theory results

From the physical modeling, the failure mode and the critical angles are obtained. Using block theory, the failure mode and factor of safety are calculated. Comparing these results, partially included in Tables 1 & 2, it can be seen that the failure modes predicted by block theory agree well with those observed from the physical model tests. Disagreement occurs for only a small number of cases. One case is for block 2 when β = 320◦ and α = 85◦ . The failure mode from physical modeling is torsional sliding on plane 2, while from block theory, it is (translational) sliding on plane 2. This is because the block theory does not consider torsional sliding as a possible failure mode. Therefore, torsional sliding on plane 2 is regarded as plane 2 sliding. The other four cases of disagreement are for block 2 when β = 240◦ , 260◦ , 280◦ and 300◦ , and α = 10◦ . As the factors of safety from block theory are larger than 1, the block is stable in these cases. However, the physical modeling results indicate wedge sliding. This is probably because of errors in measuring the friction angle. The real friction angle may be less than the one obtained from tests; therefore, the factor of safety in theory may be a little larger than in the real case. Comparing the critical angles obtained from physical modeling with those from block theory, we found that they agree well for most cases except the cases when β = 240◦ , 260◦ , 280◦ and 300◦ , and α = 10◦ . Theoretically, the critical angles in these cases should be in the range of 0◦ –10◦ ; however, the experimental results give critical angles of about 10◦ . The cause of the differences is probably because of errors in measuring the friction angles and critical angles. 5.2

Comparison between physical modeling and 3D DDA results

As the physical modeling tests were videotaped, the progressive movement of the wedge block recorded in the video can be compared with that given by 3D DDA. The same cases analyzed by 3D DDA and described in Section 4.3 are considered here. For the free falling case, Figure 15 shows the block’s positions in the physical model and also as given by 3D DDA at the elapsed times of 0.04 s, 0.08 s, 0.12 s and 0.16 s. It can be seen from this figure that the position of the wedge block given by 3D DDA is very close to that in the physical model at each of the elapsed times.

Figure 16 shows the 3D DDA computation result along with the state of the physical model for the case of sliding on plane 2 at an elapsed time t = 0.16 s; Figure 17 shows a similar comparison for the wedge sliding case at t = 0.20 s; and Figure 18 shows one for the torsional sliding case at t = 0.60 s. These comparisons of 3D DDA results with physical modeling results at the same elapsed time show that the two methods give similar results. For the cases analyzed involving four different failure modes, 3D DDA results agree well with physical modeling results in terms of the time history of block movement. This shows that the 3D DDA developed not only can handle translational sliding modes and the free falling mode, but it can also handle the more complicated torsional sliding mode. 5.3 On the three methods Physical models can be constructed to study practical rock engineering problems. They are threedimensional and have great visual impact. However, it takes time and effort to make such models, especially for complicated problems. Block theory is a powerful method that can be used to analyze the stability of a block. However, this method only considers sliding modes and some special rotational modes and cannot handle general modes of simultaneous sliding and rotation. In addition, block theory does not consider dynamic equilibrium. 3D DDA can potentially be used as a general stability analysis of a wedge, and the three-dimensional graphical output from it has great visual impact. The advantages of 3D DDA are that it considers general modes of failure including rotational modes and that it considers dynamic equilibrium. 6

A study of wedge stability has been carried out using three different methods, namely, physical modeling, block theory and 3D DDA. Physical models of two different wedges were constructed. The orientation of each wedge was varied systematically and the behavior of the wedge at each orientation observed and recorded by a digital video camera. All the physical modeling cases were then analyzed using block theory, and four cases involving four different failure modes (free falling, sliding on one plane, wedge sliding and torsional sliding) were analyzed using 3D DDA. For the wedge stability problem in this study, physical modeling and block theory give the same failure mode in all but a few cases. One case of note is the case in which torsional sliding on plane 2 was observed in the physical model. Because torsional sliding is not considered in block theory, block theory gives a mode of translational sliding on plane 2 for this case.

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CONCLUSIONS

Figure 15.

Physical modeling and 3D DDA results for free falling case at different elapsed times.

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Figure 16. Physical modeling and 3D DDA results for plane 2 sliding case (t = 0.16 s).

Figure 18. Physical modeling and 3D DDA results for torsional sliding (t = 0.60 s).

The 3D DDA results agree well with the physical modeling results in all four cases, including the case involving torsional sliding. This shows that 3D DDA can potentially be used as a general stability analysis of a wedge because it considers general failure modes including rotational modes and because it considers dynamic equilibrium.

ACKNOWLEDGEMENTS The authors would like to thank Mr. W. M. Tang for his help in the physical modeling work and block theory analysis and to thank Mr. N.C. Poon for his assistance with the physical modeling work. REFERENCES

Figure 17. Physical modeling and 3D DDA results for wedge sliding (t = 0.20 s).

Cheng, Y.M & Zhang, Y.H. 2000. Rigid body rotation and block internal discretization in DDA analysis. Int. J. Numerical and Analytical Methods in Geomech. 24(6): 567–578. Goodman, R.E. and Shi, G.-H. 1985. Block theory and its application to rock engineering. London: Prentice-Hall.

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Hatzor, Y. 1992. Validation of block theory using field case histories. Ph. D. Dissertation, Department of Civil Engineering, University of California, Berkeley. Hendron, A.J., Cording, E.J. & Aiyer, A.K. 1980. Analytical and graphical methods for the analysis of slopes in rock masses. Tech. Rep. GL-80-2. U.S. Army Engineers Nuclear Cratering Group, Livermore, California. Hoek, E. & Bray, J.W. 1981. Rock slope engineering, 3rd Ed. London: Inst. of Min. and Metallurgy. Kim, Y. I., Amadei, B. & Pan, E. 1999. Modeling the effect of water, excavation sequence and rock reinforcement with discontinuous deformation analysis. Int. J. Rock Mech. Min. Sci. 36(7): 949–970. Mauldon, M. & Goodman, R.E. 1996. Vector analysis of keyblock rotation. J. Geotech. Engng.122(12): 976–987. Priest, S.D. 1985. Hemispherical projection methods in rock mechanics. London: George Allan and Unwin. Shi, G.-H. 1988. Discontinuous deformation analysis: a new numerical model for the statics and dynamics of

block systems. Ph.D. Dissertation, Department of Civil Engineering, University of California, Berkeley. Shi, G.-H. 2001. “Three dimensional discontinuous deformation analysis,” Proc. 38th U.S. Rock Mech. Symp, Washington, D.C. Shi, G.-H. & Goodman, R.E. 1984. Discontinuous deformation analysis. Proceedings of the 25th U.S. Symposium on Rock Mechanics. Illinois: Evanston. Tonon, F. 1998. Generalization of Mauldon’s and Goodman’s analysis of keyblock rotations. J. Geotech. Engng. 124(10): 913–922. Warburton, P.M. 1981. Vector stability analysis of an arbitrary polyhedral rock block with any number of free faces. Int. J. Rock Mech. Min. Sci. and Geomech. Abstrac. 18(5): 415–427. Windsor, C.R. & Thompson, A.G. 1992. Reinforcement design for jointed rock masses. Proc. 33rd US Symp. Rock Mech.: 520–521.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Shaking table tests of coarse granular materials with discontinuous analysis T. Ishikawa & E. Sekine Railway Technical Research Institute, Tokyo, Japan

Y. Ohnishi Kyoto University, Kyoto, Japan

ABSTRACT: This paper describes a new analytical method to elucidate the dynamic response of railroad ballast under high seismic loads. A series of numerical simulations of shaking table tests for single grained crushed stone were performed with DDA. The applicability of discontinuous analysis to the dynamic behavior of coarse granular materials and the seismic bearing capacity of coarse granular materials were discussed in terms of the mechanics of granular materials. As the result, it was revealed that discontinuous analysis was an effective method to simulate the dynamic behavior of coarse granular materials in earthquakes, and that the seismic bearing capacity was closely related with the mobility of individual particles. These results indicate that the evaluation of the seismic bearing capacity and ductility of coarse granular materials needs to estimate the mobility of constituent particles.

1

INTRODUCTION

After the Hyogoken-Nambu Earthquake of January 17 1995, a two-stage seismic design procedure was newly proposed by JSCE (the Japan Society of Civil Engineers) in 1996. The reason for this is that many important engineering structures designed by the old design procedure were seriously damaged including railway structures. Because of historical particulars, a number of existing structures have been designed to possess the seismic durability against the static horizontal seismic force calculated by multiplying the horizontal coefficient of 0.2 by self-weight of the structure in Japan. Accordingly, most of the existing structures will not be able to withstand the acceleration level over 800 gal like the Hyogoken-Nambu Earthquake in the future. As for the railway structures, therefore, a new design code, namely “Seismic Design Code for Railway Structure (RTRI, 1999)”, was established in accordance with the two-stage seismic design procedure. (The code is called “the Railway Code” for short in the rest sentences.) The Railway Code is a method to evaluate the stability level of the railway structures in addition to the damage level of members against two different levels of seismic load, namely L1 earthquake and L2 earthquake (JSCE, 2000). The distinctive features of the new seismic design method is to predict

earthquake-induced permanent deformation in order to check whether the response calculated through dynamic analysis satisfies the seismic performance of railway structures. The design method employs a relationship of soil between dynamic stress and permanent strain, which is determined from cyclic loading laboratory tests. However, the design method has not been adopted for railway track structures though it was adopted for most of railway structures such as bridges, viaducts, tunnels, embankments, foundations and retaining walls. Especially, there are few methods to assess the seismic performance of ballasted track (Fig. 1) despite nearly 80 percent of Japanese railway track being ballasted track. The reason for this situation is that it is difficult to assess the ductility of ballasted track in intense earthquakes because non-linear behavior of “railroad ballast”, a track component of ballasted track composed of coarse crushed stone, is complicated.

Ballast

Railroad Ballast Subgrade Figure 1.

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Rail

Sleeper

Ballasted track structure (cross-section).

Therefore, it becomes necessary to establish a new seismic design and evaluation method of ballasted track under high seismic loads. In general, the mechanics of granular materials is commonly employed in analyzing the complex behavior of coarse granular materials such as ballast. From the viewpoint of the mechanics of granular materials, non-linear behaviors of railroad ballast are caused by the movement of ballast particles. Accordingly, discontinuous analysis seems to be effective in simulating the dynamic behavior of ballasted track because it regards a ballast particle as an element of discontinuous analysis. This paper describes the fundamental study to examine the applicability of discontinuous analysis to the dynamic behavior of ballasted track and to evaluate its seismic bearing capacity under high seismic loads in terms of the mechanics of granular materials.

2

OBJECTIVES OF RESEARCH

The objectives of this paper are: •

to examine the applicability of discontinuum analysis to the dynamic behavior of coarse granular materials in earthquakes.

•

to evaluate the seismic bearing capacity and ductility beyond failure of coarse granular materials in terms of the mechanics of granular materials.

In this paper, the numerical simulations of shaking table tests are performed with discontinuous analysis regarding a ballast particle as a block (Here, the term block is used in the same way as the term element.). In the simulations, a trapezoid specimen piled with hundreds of polygon blocks was shook by several sine waves changing their acceleration amplitude or frequencies in order to discuss the above assignments.

3 3.1

OUTLINE OF ANALYSIS Modeling

The simulations of shaking table tests were performed with two-dimensional DDA models under plane-strain conditions. DDA (Discontinuous Deformation Analysis, proposed by Shi & Goodman, 1985) is a kind of discontinuous analysis. In DDA, each block is separated by its boundaries and moves individually. Figure 2 shows the size, dimension and boundary condition of analytical models. Figure 3 shows the element meshes of DDA models before shaking. The DDA

2000mm

Ballast block for observation

Analytical specimen

Shaking ⫺ Direction

⫹

3000mm

Figure 2.

Schematic section of DDA model.

Figure 3a.

Element mesh of model A.

Figure 3b.

Element mesh of model B.

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34.4˚ Shaking table block

400mm

100mm

mass (γ ), Young’s modulus (E) and Poisson’s ratio (ν). As for the material properties, assuming the ballast blocks made of andesite and the concrete shaking table, the parameters were set referring to the material properties of andesite and past experimental results. On the other hand, the interface properties of block edges are characterized by block friction angle (φµ ) and cohesion of surface (Cµ ). Here, the reason why φµ between ballast blocks was set equal to 55◦ is that the internal friction angle derived from the analytical results in case of φµ =55◦ indicated the best fit to experimental results in simulating triaxial tests with Table 2.

Table

Unit Mass γ Young’s Modulus E Poisson’s Ratio ν Cohesion Cµ Friction Angle φµ

2.77 t/m3 20.0 GPa 0.10 0 55.0◦

2.38 t/m3 30.0 GPa 0.20 0 37.0◦

displacement(cm)

Ballast

Frequency = 1Hz

4.0 0 -2.0 -4.0

displacement(cm)

4

6

8

10 12 time(sec) (b)

0.2 0 -0.2

Frequency = 3Hz

-0.4

0.2

1

2

3

4

5 6 time(sec)

Frequency = 5H z

(c)

0.1 0 -0.1 -0.2

Features of DDA model.

2

0.4

0

The deformation property of all blocks employed in this paper is liner elastic. Table 2 shows the material properties of blocks and the interface properties of block edges. In this paper, when two DDAblocks come in contact, springs and a slider are created at contact points as shown in Figure 4. Accordingly, the material properties of DDA blocks were characterized by unit

(a)

2.0

0

3.2 Analytical conditions

Table 1.

Material properties of DDA model.

Property

displacement(cm)

model is composed of some polygon blocks, named “ballast blocks”, which represent ballast particles and a rectangular block, named “a shaking table block”, which represents a shaking table as shown in Figure 3. Furthermore, the element shape of a ballast block is a regular hexadecagon instead of a circle in view of the angularity of actual ballast particles as the block shape has a strong influence on the stress-strain behavior of granular materials (Kohata, 1999). In the simulations, two types of analytical specimens which differ in the size of ballast blocks were employed as shown in Figure 3. Table 1 shows the feature of the specimens in comparison with experimental ones. Here, the term “model B” is used to refer to the specimen which constituent particle has smaller mean grain size (D50 ) of 40 mm, and the term “model A” is used to refer to the other in this paper. Seeing Table 1, the mean grain size of model B is nearly as large as that of test specimens, and the mean grain size of model A is larger than that of model B. However, the porosity of analytical specimens is much smaller than that of experimental specimens as shown in Table 1, though there is little difference between the porosity of both analytical specimens. This phenomenon seems to be mainly caused by the reason that our DDA models are two-dimensional models though the laminated state of experimental specimens is three-dimensional. Furthermore, analytical specimens of ballast blocks were made as follows. First, some ballast blocks were piled up in the tetrahederal arrangement in order to model the cross section of the ballasted track. Here, analytical specimens were assumed to be the densest state compacted by vibrator as well as test conditions. Second, stability analysis was done by gravity force of 1.0 G. In this paper, the state of DDA models after stability analysis is called “the initial loading state”. Figure 3 shows the initial loading state of each model.

0

1

2

3

4 time(sec)

Property

Model A

Model B

Ballast

Number of Ballast blocks Uniformity Uc Mean Grain Size D50 (mm) Porosity n (%)

204 1.00 55 9.3

386 1.00 40 9.3

– 1.70 41 38.2

Shaking table Lower layer

Figure 4. Time histories of response displacement; under horizontal vibration of 200 gal; a) loading frequency of 1 Hz; b) 3 Hz; and c) 5 Hz.

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Middle layer Upper layer

Frequency Acceleration Displacement (Hz) amplitude (gal) amplitude (mm) 200 200 200 1000 1000 1000

49.6 5.5 2.0 248.2 27.6 9.9

Model B 1.0 3.0 5.0 1.0 3.0 5.0

200 200 200 1000 1000 1000

49.6 5.5 2.0 248.2 27.6 9.9

DDA (Ishikawa et al. 1997). As for φµ between a ballast block and a shaking table blocks, the value was set equal to 37◦ referring to conventional studies. Besides, as for Cµ between all materials, the value was set equal to zero considering that ballast is coarse granular material. Seismic loads were applied by shaking the shaking table block horizontally with the regular incident wave which waveform is sinusoidal. The vibration of the shaking table block occurred in the minus direction as shown in Figure 2. The loading number was 10 cycles, and the loading frequency of 1.0 Hz or 3.0 Hz or 5.0 Hz was selected. The single acceleration amplitude of the incident wave was set to 200 gal or 1000 gal. Furthermore, the gravity force was applied to analytical specimens through the simulation. Table 3 shows analytical conditions of all simulations together.

4 ANALYTICAL RESULTS 4.1

Influence of vibration conditions

The influence of vibration conditions on the dynamic behavior of coarse granular materials under horizontal vibrations is discussed on the basis of the analytical results of model A. Figure 4 and Figure 5 show the time histories of the horizontal response displacement at each ballast block for observation inside analytical specimens as shown in Figure 2, compared with those of the incident wave on the shaking table. The relations at the single acceleration amplitude of 200 gal in the respective loading frequencies of 1 Hz, 3 Hz and 5 Hz shown in Figure 4, and the relations at the single acceleration amplitude of 1000 gal show in Figure 5. Here, the sign of the horizontal response displacement was defined so that it agrees with the sign of the shaking direction as shown in Figure 2. Seeing Figure 4 and Figure 5, if the loading frequencies are different even at the same acceleration amplitudes, the movements

(a)

2.0 0 -2.0 -4.0

4.0

2

4

6

8

10 12 time(sec)

Frequency = 3Hz

(b)

2.0 0 -2.0 -4.0

0

2.0

1

2

3

4

5 6 time(sec)

Frequency = 5Hz

(c)

1.0 0 -1.0 -2.0

0

1

2

3

4 time(sec)

Shaking table Lower layer

Middle layer Upper layer

Figure 5. Time histories of response displacement; under horizontal vibration of 1000 gal; a) loading frequency of 1 Hz; b) 3 Hz; and c) 5 Hz.

of ballast blocks are different. The displacement of ballast blocks increases with the decrement of loading frequency. Moreover, in case of the same loading frequencies, the displacement of ballast blocks increases with the increment of acceleration amplitude. Figure 6 shows the relations between the amplification factor of the maximum response displacement and the height along the center line of analytical specimens from the upper surface of shaking table block at the 10th cycle. Here, the marks in Figure 6 show the analytical results of the ballast blocks located in the center of analytical specimens (Fig. 2). Moreover, the term “amplification factor of response displacement” can be defined as the ratio of the response displacement to the incident displacement. Seeing Figure 6, the response of incident wave at each part of analytical specimens is different from others. The amplification factor decreases with going toward the upper layer of the specimen. Furthermore, seeing Figures 4 and 5 again, the waveform of the time histories falls into disorder with the increment of acceleration amplitude and the followability of the response displacement to the incident wave goes down. One of reasons for this is

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Frequency = 1Hz

4.0

0 displacement(cm)

Model A 1.0 3.0 5.0 1.0 3.0 5.0

displacement(cm)

Model

displacement(cm)

Table 3. Analytical condition.

40

2.0

1H z 3H z 5H z

30

residual displacement(cm)

height from base line(cm)

1000gal

200gal 1H z 3H z 5H z

20

10

0

0.1

0.2

0.3

10waves 1Hz 3Hz 5Hz

1Hz 3Hz 5Hz -2.0

0

2.0

displacement amplitude(cm)

Influence of grain size

The influence of grain size on the dynamic behavior of coarse granular materials under horizontal vibrations

displacement(cm)

Property of plastic deformation. Frequency = 1Hz

4.0

(a)

2.0 0 -2.0 -4.0 0

displacement(cm)

the plasticification of analytical specimens because the growth of plastic flow inside granular assemblages seems to cause the deterioration of interlocking between particles and disturb the transmission of exciting forces. For example, in Figure 4, both the displacement amplitude and the residual displacement are small at every part of analytical specimens, and the deformation characteristics of ballast blocks are still quasi elastic. On the other hand, in Figure 5, both the displacement amplitude and the residual displacement are large, and the deformation behavior of ballast blocks is plastic. Therefore, it is considered that the plasticification of analytical specimens has a considerable influence on the dynamic behaviors of granular assemblages. Accordingly, the tendency of plastic deformation in analytical specimens is discussed next. According to Ishikawa & Ohnishi (2001), the plastic axial strain was proportional to the amplitude of axial strain in the cyclic plastic deformation of coarse granular materials. Figure 7 shows the relations between the residual displacement and the double displacement amplitude at the 1st cycle and the 10th cycle of all vibration conditions. Seeing Figure 7, it is recognized that a first degree equation can give a close approximation to the relations regardless of vibration conditions such as acceleration amplitude of incident wave, loading frequency and loading number. These results indicate that the movement of individual particles has a strong influence on the dynamic response of an overall granular body because the plastic deformation of coarse granular materials in earthquakes is closely related with the response displacement amplitude of constituent particles.

Figure 7.

displacement(cm)

Figure 6. Distribution of amplification factor of maximum response displacement.

1.5 1.0 0.5 0 -0.5 -1.0 -1.5

2

4

6

8

10 12 time(sec) (b)

Fr equency = 3H z

0

0.2

1

2

3

4

5 6 time(sec)

Freq uency = 5H z

(c)

0.1 0 -0.1 -0.2

0

1

2

3

4 time(sec)

Shaking table Lower layer

Middle layer Upper layer

Figure 8. Time histories of response displacement; under horizontal vibration of 200 gal; a) loading frequency of 1 Hz; b) 3 Hz; and c) 5 Hz.

is discussed on the basis of the analytical results of model B. Figures 8 and 9 show the time histories of the horizontal response displacement at each ballast block for observation inside analytical specimens like Figures 4 and 5. Figure 10 shows the relations between the amplification factor of the maximum response displacement and the height along the center line of

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1wave

-2.0

-4.0 -0.4

0.4

amplification factor of displacement

4.2

0

0 -2.0 -4.0

displacement(cm)

0

displacement(cm)

2.0

(a)

2.0 residual displacement (cm)

displacement(cm)

Frequency = 1Hz

4.0

4.0

2

4

6

8

10 12 time(sec)

Frequency = 3H z

(b)

2.0

1waves 1Hz 3Hz 5Hz

-2.0

0 -4.0 -4.0

-2.0 -4.0

0

2.0

1

2

3

4

Figure 11.

0 -2.0 0

1

2

3

4 time(sec)

Middle layer Upper layer

Figure 9. Time histories of response displacement; under horizontal vibration of 1000 gal; a) loading frequency of 1 Hz; b) 3 Hz; and c) 5 Hz. 40 1000gal 1Hz 3Hz 5Hz

30 200gal

1Hz 3Hz 5Hz

20

10

0

0.1

0

2.0

Property of plastic deformation.

(c)

1.0

-4.0

-2.0

10waves 1Hz 3Hz 5Hz

displacement amplitude (cm)

5 6 time(sec)

Frequency = 5H z

Shaking table Lower layer

height from base line (cm)

0

0.2

0.3

0.4

lower layer of analytical specimens in comparison with that of model A. The reason for this seems to be that the increase in total number of ballast blocks due to the decrease of grain size causes the exciting forces transmitted from the shaking table block to be easily attenuated inside analytical specimens. Furthermore, Figure 11 shows the relations between the residual displacement and the double displacement amplitude at the 1st cycle and the 10th cycle of all vibration conditions. Seeing Figure 11, it is recognized that the residual displacement at the 10th cycle is hardly related with the double displacement amplitude though a first degree equation can give a good approximation to the relations at the 1st cycle as well as Figure 7. The reason for this seems to be that the mobility of individual particles under horizontal vibration increases with the increment of total number of ballast blocks due to the decrease of grain size. These results indicate that the evaluation of the seismic bearing capacity and ductility of coarse granular materials needs to estimate the mobility of constituent particles correspondent to damage degree of granular assemblages because a large movement of constituent particles causes local plastic flow in an overall granular body. 5

amplification factor of displacement

Figure 10. Distribution of amplification factor of maximum response displacement.

analytical specimens from the upper surface of shaking table block at the 10th cycle like Figure 6. Seeing Figures 8 and 9, there is little influence of grain size on the time history response of analytical specimens. However, seeing Figure 10, the amplification factor of the maximum response displacement decreases from

The following conclusions can be obtained: 1. Discontinuous analysis is an effective method to simulate the dynamic behavior of coarse granular materials in earthquakes because the movement of individual particles has a strong influence on the dynamic response of an overall granular body. 2. The seismic bearing capacity and ductility is closely related with the mobility of individual particles because a large movement of constituent particles causes local plastic flow in an overall granular body.

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CONCLUSIONS

These results indicate that the evaluation of the seismic bearing capacity and ductility of coarse granular materials needs to estimate the mobility of constituent particles. We presume that this study will produce significant and novel results to propose an evaluation method of seismic bearing capacity and ductility correspondent to damage degree of railroad ballast. REFERENCES Ishikawa, T., Ohnishi, Y. & Namura, A. 1997. DDA applied to deformation analysis of coarse granular materials (ballast): Proc. of ICADD-2, Kyoto, 10–12 July 1997: 253–262.

Ishikawa, T. & Ohnishi, Y. 2001.Application of discontinuum analysis to cyclic plastic deformation of coarse granular materials: Proc. of the 10th IACMAG, Tucson, 7–12 January 2001: 321–324. Japan Society of Civil Engineers 2000. Earthquake Resistant Design Codes in Japan: 3-1–3-30. Tokyo: JSCE. Kohata, Y., Jiang G.-L. & Sekine, E. 1999. Deformation characteristics of railroad ballast as observed in cyclic triaxial tests: Poster Session Proc. of the 11th Asian regional conference on soil mechanics and geotechnical engineering, Seoul, 8–16 October1999. Railway Technical Research Institute 1999. Seismic Code for Railway Structures. Tokyo: Maruzen (in Japanese).

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Pre-failure damage, time-dependent creep and strength variations of a brittle granite O. Katz Geological Survey of Israel, Jerusalem, Israel

Z. Reches Institute of Earth Sciences, Hebrew Univ., Jerusalem, Israel

ABSTRACT: We experimentally analyze the effects of stress-induced damage and time-dependent creep on the brittle failure of Mount Scott granite of Oklahoma. Fourteen dry granite samples were tested at room temperature and under 41 MPa confining pressure; they were loaded to a pre-selected axial stress and held under a constant stroke for periods of up to six hours before unloading. The majority (80%) of the microfractures mapped in thinsections belong to two groups: tensile fractures that subparallel the loading axis, and shear fractures trending 15◦ – 40◦ off the loading axis. The analysis indicates (a) that samples loaded above a critical stress (about 0.96 the mean strength) creep under constant stroke and relaxing stresses during the hold period and eventually fail spontaneously; and (b) that the strength variations of the samples fit Weibull distribution with profound weakening due to the creep. 1

INTRODUCTION

Stressing brittle rocks leads to the development of distributed damage long before the rock fails unstably. The damage is commonly manifested by microfractures and dilational microcracks (Tapponier & Brace, 1976). Typically, these microfractures are smaller than the grain size and they are often quasi-uniformly distributed prior to faulting (Hadely, 1976; Lockner et al. 1992). Local, non-uniform distributions of microfractures are related to fault nucleation and growth (Reches & Lockner, 1994). The microdamage was used to explain the reduction of seismic wave velocity, seismic anisotropy, the reduction of elastic moduli and strength, and the mechanics of rock failure (Ashby & Hallam, 1986, Reches & Lockner, 1994, Lyakhovsky et al. 1997; Lockner, 1998). Further, the stress-induced damage may facilitate time-dependent creep driven by stress corrosion and subcritical crack growth (Lockner, 1998). This creep strongly affects the long term strength and failure stability. For example, granite samples subjected to one month of constant, uniaxial stress could fail under stress of ∼0.65 the instantaneous strength (Schmidtke & Lajtai, 1985). Or, “delayed fractures” could develop days to years after the applied loads were removed (Salganik et al. 1994). We examine here the pre-failure damage and rock strength in triaxial experiments of brittle granite samples. The stress-induced damage was determined

from both rheological parameters and microfracturing analysis (Katz, 2002). The load-hold method is applied here to recognize the time-dependent damage in the tested brittle granite. The stress distribution results are analyzed following Lawn (1993, Ch. 10) who discussed the lifetime of material under load below the inert strength level in terms of fatigue and crack growth velocity function. The present experimental work was conducted at the Rock Mechanics Institute, University of Oklahoma, Norman, Oklahoma, and the experimental details appear in Katz (2002). In this paper, we briefly outline the experimental procedure and describe the macroscopic rheology and microstructural observations. Then, we discuss the effects of the instantaneous damage and time-dependent damage on the strength of the granite. 2

2.1 Experimental set up We used samples of Mount Scott granite (MSG) of the Wichita Mountains, southwestern Oklahoma. MSG has anorthoclase phenocrysts in a matrix of alkali feldspar and quartz with small amounts of hornblende, biotite and iron oxides (Price et al. 1996). It is a fine- to medium-grained rock with mean grain size of 0.9 ± 0.2 mm and dry density of 2,645 kg/m3 .

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EXPERIMENTAL PROCEDURE

Katz et al. (2001) conducted a series of thirteen uniaxial and triaxial loading-to-failure tests under confining pressure up to 66 MPa. They found that the Young’s modulus, E, increases from 75 GPa for the uniaxial tests, to 82 GPa at 66 MPa confining pressures, and the Poisson’s ratio is 0.21–0.31. The Coulomb strength of MSG is σ1 = 270 + 8.7σ3 (in MPa), and the measured angle θ between the normal to the major faults and the sample axis (σ1 ) is 68◦ –75◦ (Katz et al. 2001). The present tests were performed on 25.4 mm diameter cylinders with length-to-diameter ratio of 2.5–3.9. We used a 69 MPa pressure vessel and the axial load was supplied by a servo-controlled hydraulic load frame (MTS 315). Load was monitored with an internal load cell, and the displacements were monitored with two LVDT (axial), and a chain extensometer (lateral). All tests were performed under the confining pressure of 41 MPa for which the Coulomb strength is 586 ± 16 MPa (Katz et al. 2001). We define the “Normalized Differential Stress”, NDS, as NDS = (σ1 − σ3 )/586. While ideally the maximum NDS = 1.0, in our tests 0.96 < maximum NDS < 1.05, reflecting the inherent inhomogeneity of the samples and deviations from mean strength.

2.2 Loading procedure We use three loading procedures: 1. Load-to-failure at axial strain rate of 1·10−5 s−1 , after the confining pressure loading at a constant rate of 0.023 MPa/s. This procedure was used for tests 101, 103, 112. 2. Load-hold procedure applied in 14 tests (Table 1), each consists of four steps: (a) Confining pressure loading at a rate of 0.023 MPa/s; (b) Axial loading to a pre-selected load that ranges from NDS = 0.54 to NDS = 1.05. Axial shortening was at a strain rate of 1·10−5 s−1 ; (c) Once the pre-selected load was achieved, the specimen was held at a constant stroke for up to six hours; (d) Eleven samples did not fail during the hold time and three failed spontaneously. After the hold period, the unfailed samples were unloaded. 3. Cycle-to-failure procedure was applied on three unfailed samples (105, 124 and 125). The axial load was unloaded (to the confining pressure) after the holding period and the sample was reloaded to failure in one or two cycles.

Table 1. Experimental loading data. Loading procedure includes: load to failure tests (specimens 101, 103, 112); load-hold tests (specimens 102, 104, 105, 106, 108–110, 113–117, 123, 125); cyclic loading to failure tests (specimens 105, 124, 125); Hold time: the time elapsed from start of stroke holding to unloading or to failure; Hold stress is the maximum stress at the start of holding; Maximum NDS: is the normalized differential stress at hold point or at failure.

Test # 101 102 103 104 105 106 108 109 110 112 113 114 115 116 117 123 124 125

Hold time (min)

Hold stress (MPa)

Failure stress (MPa)

Max. NDS

Comments

– 95 – 61 180 – 1.25 180 180 0.03 – 180 180 360 180 180 180 – – – 180 –

– 601 – 613 467 – 592 505 546 564 – 563 518 534 460 318 334 556 562 – 546 –

613 – 595 528 – 636 517 – – 561 573 – – – – – – – – 657 – 617

1.05 1.03 1.02 1.05 0.80 1.09 1.01 0.86 0.93 0.96 0.98 0.96 0.88 0.91 0.78 0.54 0.57 0.95 0.96 1.12 0.93 1.05

load to failure load-hold load to failure spontaneous failure load-hold cycle 1 reload to failure spontaneous failure load-hold load-hold spontaneous failure load to failure load-hold load-hold load-hold load-hold load-hold load-hold load-hold cycle 1 reload cycle 2 reload to failure load-hold cycle 1 reload to failure

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3 3.1

EXPERIMENTAL OBSERVATIONS

during which the sample failure occurred by initial stable stress decrease followed by unstable stress drop.

Stress-strain relations

The stress-strain curves of test 101 (Fig. 1) are characteristic for the present experiments. Figure 1 displays the axial strain curve (with diamonds indicating the pre-selected holding stresses), the experimental volumetric strain, V/V, and the permanent volumetric strain, CVS. The later is associated with crack dilation where CVS = [(experimental volumetric strain)(elastic volumetric strain)], or CVS = (σ1 − σ3 )(1 − 2ν)/E. The curves display several stages that are similar to previously recognized stages (Wawersik & Brace, 1971). Stage I, at the range of 0 < NDS < 0.15, includes the nonlinear stress increase associated with cracks closure. Stage II, at 0.15 < NDS < 0.40, displays apparently linear elastic curve. Stages III and IV start at NDS ≈ 0.4 (Cci – crack initiation stress, Martin & Chandler, 1994) and NDS ≈ 0.85 (Ccd – crack damage stress, Martin & Chandler, 1994), respectively, are characterized by first stable (stage III) and then unstable (stage IV) crack growth and dilation. Stage V is the failure stage (NDS ≈ 1.0)

3.2 Time-dependent creep Time-dependent effects of damage evolution are recognized in the holding periods. During the holding period the sample length was maintained constant and the axial stresses could relax spontaneously. In this respect, the present procedure differs from typical creep test in which the stress level is constant and the sample is allowed to shorten. The holding periods did not exceed six hours due to limited availability of the loading frame. All eleven samples with maximum loading of NDS < 0.96 did not fail spontaneously during the hold periods. These samples exhibit similar variations of the axial stress, volumetric strain and crack volumetric strain that are similar to the relations in Figure 1. During the hold period, the axial stress of these samples relaxes first by 2–3% and remains approximately constant thereafter (test 113 in Fig. 2). Different behavior is observed for the three samples loaded with NDS ≥ 0.96 in the holding stage (tests 104, 106 and 110, Table 1). In these tests, the axial stress relaxed during the hold time until they fail spontaneously (Fig. 2). The irregular relaxation curves indicate poorly constrained creep processes that are probably associated with stress corrosion or subcritical crack growth. 3.3 Strength and time-to failure Figure 3 displays for each sample the holding times versus its maximum axial load. Four groups are

Figure 1. Stress-strain relations of test 101. Stress axis is marked by Normalized Differential Stress (NDS, see text). The shown curves are: axial load, total volumetric strain and permanent, crack volumetric strain (CVS, see text). The curves display several stages (after Wawersik and Brace, 1971): I- nonlinear stress increase associated with cracks closure; II- quasi-linear elastic stage; III- nonlinear stress increase associated with crack growth and dilation; IV- failure stage with increase of crack growth; V-failure. Cci is the crack-initiation stress, where dilation begins; Ccd is the crack-damage stress, where failure initiates. Diamonds represents the maximal NDS of each of the load–hold test in the present series.

Figure 2. Differential-stress variations during the holding period of three samples that failed spontaneously (104-upper curve, 106-middle curve, 110-lower curve, all with open arrow) and sample 113 that did not fail.

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plotted: (a) samples loaded to failure (solid squares) for which the time is arbitrarily selected as 0.1s and the shown stress is the ultimate strength; (b) load-hold samples that did not fail (solid diamonds) shown by experimental hold time and maximum load; (c) loadhold samples that fail spontaneously marked by a line connecting the stress at the start of the holding (small solid dot) and the stress during failure (large, solid dot); and (d) cycle-to-failure tests (solid dots) for which the time is arbitrarily selected as 0.1s and the shown stress is the ultimate strength.

Failing

Stable

Time (s)

10,000

100

Load to hold- no failure Load to failure Cycled to failure C y Spontaneous Spontaneous failure failure

1

0.01 300

400

3.4 Microdamage

500

600

700

Hold stress or failure stress (MPa) Figure 3. The hold time that is the elapsed time from start of holding period to unloading or to spontaneous failure, as function of the pre-selected load. Vertical dotted line is the critical stress (0.96 the strength, see text). Three inclined lines indicate the stress relaxation of the spontaneously failed samples (104, 106, and 110).

Frequency

10000

157 123

1000

114 113

100

110

10

Our microstructural analysis covered the mode, dimensions, density and distribution of stress-induced microfractures in the deformed samples as reported by Katz (2002) and Katz & Reches (2002). The microfractures were mapped in four samples subjected to load-hold testing with maximum NDS values of 0.57, 0.88, 0.96 (unfailed) and 0.96 (failed), and one unstressed sample (123, 114, 113, 110 and 157). The deformed specimens display two dominating microfracture groups that account to more then 80% of the mapped fractures. One group includes tensile microfractures trending subparallel to the loading axis, and the other group includes shear microfractures trending in the interval of 15◦ – 40◦ off the loading axis. We note a general lengthening of the microfractures (Fig. 4a) and nonlinear increase of their density (Fig. 4b); density is defined as the cumulative length of mapped microfractures per unit area (mm/mm2 )

a 1 0.01

0.1

10

1

4

DISCUSSION

Length interval (mm)

4.1 A critical stress for spontaneous failure Maximal fracture density (mm/mm2)

2.5 2.0 1.5 1.0 0.5

b 0.0 0

0.2

0.4

0.6

0.8

1

Maximal NDS

Figure 4. Measured microfracture length and density in deformed and undeformed samples of MSG. (a) Frequency of microfractures length as function of length interval for samples 157 (NDS = 0.00), 123 (NDS = 0.57), 114 (NDS = 0.88), 113 (NDS = 0.96), 110 (NDS = 0.96, and failure). (b) Maximum fracture density as function of maximum load (the same samples as in a).

The present experiments indicate that for hold periods up to six hours, spontaneous failure occurs only above a critical stress of NDS ≈ 0.96 (Fig. 3), and this failure is preceded by the time-dependent creep and stress relaxation (Figs. 2, 3). The behavior above this critical stress is highly nonlinear as portrayed by the wide range of the time-to-failure (Fig. 3), the irregular style of stress relaxation (Fig. 2), the wide range of crack volume strain, CVS, for samples loaded to NDS ≥ 0.96 (Katz, 2002), and the nonlinear increase of microdamage (Fig. 4b). The delay in spontaneous failure (Figs. 2, 3) is apparently a self-induced process that requires no additional external energy (note the stress relaxation in Fig. 2). The long-term strength of granite samples was experimentally analyzed by Schmidtke and Lajtai (1985). They conducted 140 unconfined creep tests on Lac du Bonnet granite for up to 40 days. While they concluded that the granite has a finite, longterm strength of about 0.45 the instantaneous strength,

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they also show that a zero long-term strength cannot be rejected. Martin and Chandler (1994) used the same data set to show that samples loaded above 0.7 the instantaneous strength would fail in less than one day. They related this critical stress to the irreversible microdamage associated with the Ccd load of trend reversal of the experimental volumetric strain curve in Figure 1. Lockner (1998) showed that within the time limits of experimental creep data, Westerly granite has zero long-term strength. The results of these studies indicate that the critical stress decreases with the increase of holding time. Thus, the observed critical stress of NDS ≈ 0.96 of the MSG samples is limited to loading period of few hours. 4.2

Strength variations

We examined the strength variations of MSG with the Weibull distribution. While originally developed for the analysis of tensile strength in brittle solids (Lawn, 1993, Ch. 10), this distribution has been applied to analyze the shear strength of rocks (Gupta & Bergstom, 1998) and other phenomena. This distribution predicts that the probability P for a sample to fail under differential stress US is P = 1 − exp[−(US /σ0 )m ] where the Weibull modulus, m, and the scaling stress σ0 , are adjustable parameters. Strength values of 11 failure tests are used here (nine from the present work, Table 1, and two from Katz et al. 2001). One should note that this is a small sampling size for typical Weibull analysis (Lawn, 1993, p. 340).

Failure probability, P

Time-dependent failure: P = 1 - exp [-(US / 618)13 ]

0.3

"Instantaneous failure: P = 1 - exp [-(US / 622)22 ]

0.1

500

525

550

575

600

CONCLUSIONS

(1) The spontaneous failure of Mount Scott granite occurs above a critical stress of about 95% of its ultimate rock strength for the present conditions. Above this stress the damage increases nonlinearly even when the load spontaneously relaxes and the sample creeps. (2) The pre-failure damage includes shear and tensile microfractures in approximately equal amounts. The shear microfractures are significantly longer in the later stages of the deformation. (3) The Weibull distribution parameters of the strength data of Mount Scott granite indicate a well-behaving damage population.

The laboratory work was conducted at the Rock Mechanics Institute, University of Oklahoma, Norman, with the help and advice of J-C. Roegiers, Gene Scott and Pete Keller. The study was supported, in part, by Eberly Family Chair funds of M. Charles Gilbert, the Rock Mechanics Institute, University of Oklahoma, Norman, the US– Israel BiNational Science Fund, grant 98–135 and the Geological Survey of Israel project 30255.

0.7 0.5

5

ACKNOWLEDGEMENTS

Time-dependent failure "Instanteneous" failure Linear (Time-dependent failure) Linear ("Instanteneous" failure)

0.9

We follow Lawn (1993) to calculate the Weibull probability for two sets of data. For the first we use the strength values during actual failure, namely after the time-dependent creep of the samples (solid diamonds and linear fit for “Time-dependent failure” in Fig. 5). For the second we use the strength values before the time-dependent creep (open squares and linear fit for “Instantaneous failure” in Fig. 5). For this case, the values of maximum load at the start of the holding period are used. The calculated Weibull modulus have high values (m ≈ 13 and m ≈ 22 for the first and second option, respectively, in Fig. 5), which are typical to reliable solids (Lawn, 1993). Finally, the fairly clear linear fit for both options (Fig. 5) suggests a well-behaving population of pre-failure microfractures (Lawn, 1993), in agreement with our microstructural observations (Fig. 4) (Katz & Reches, 2002).

625

REFERENCES

650

Us - Strength (MPa)

Figure 5. Strength variations of MSG samples plotted on Weibull diagram (after Lawn, 1993). The two sets of data that are plotted are described in the text.

Ashby, M. F., Hallam, S. D. 1986. The failure of brittle solids containing small cracks under compressive stress states. Acta Metall., 34, 497–510. Hadley, K. 1976. Comparison of calculated and observed crack densities and seismic velocities in Westerly Granite. J. Geophys. Res., 81, 3484–3494.

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Gupta, V., Bergstrom, J. S. 1998. Compressive failure of rocks by shear faulting. J. Geophys. Res., 103, 23875– 23895. Katz, O. 2002. Mechanisms of faults nucleation in brittle rocks. Ph.D. dissertation, Hebrew University, Jerusalem. Katz, O., Gilbert, M. C., Reches, Z., Roegiers, J. C. 2001. Mechanical properties of Mount Scott granite, Wichita Mountains, Oklahoma. Oklahoma Geology Notes, 61 (2), 28–34. Katz, O., Reches, Z. Microfracturing, damage and failure of brittle granites. Submitted to J. Geophys. Res. (May, 2002). Lawn, B. 1993. Fractures of brittle solids-Second edition. Cambridge University press. Lockner, D. A. 1998. A generalized law for brittle deformation of Westerly granite. J. Geophys. Res., 103, 5107–5123. Lockner, D. A., Moore, D. A., Reches, Z. 1992. Microcracks interaction leading to shear fracture, in Tillerson and Wawersik (eds), Rock Mechanics, Rotterdam: Balkema. Lyakhovsky, V., Reches, Z., Weinberger, R., Scott, T. E. 1997. Non linear elastic behavior of damaged rocks. Geophys. J. International, 130, 157–166.

Martin, C. D., Chandler, N. A. 1994. The progressive fracture of Lac du Bonnet granite. Int. J. Rock Mech. Min. Sci. Geomec. Abst., 31, 643–659. Price, J. D., Hogan, J. P., Gilbert, M. C. 1996. Rapakivi texture in the Mount Scott granite, Wichita mountains, Oklahoma. European Journal of Mineralogy, 8, 2, 435–451. Reches, Z., Lockner, D. A. 1994. The nucleation and growth of faults in brittle rocks. J. Geophys. Res., 99, 18159–18174. Salganik, R. L., Rapoport, I., Gotlib, V. A. 1994. Delayed fracture in brittle wear: an approach. International J of Fracture, v. 68, 65–72. Schmidtke, R. H., Lajtai, E.Z. 1985. The long-term strength of Lac du Bennet granite. Int. J. Rock Mech. Min Sci. Geomech. Abstr., 22, 461–465. Tapponnier, P., Brace, W. F. 1976. Development of StressInduced Microcracks in Westerly Granite. Int. J. Rock Mech. Min Sci. Geomech. Abstr., 13, 103–112. Wawersik, W. R., Brace, W. F. 1971. Post-failure behavior of a granite and diabase. Rock Mechanics, Supplementum, 3, 2, 61–85.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Dynamic block displacement prediction – validation of DDA using analytical solutions and shaking table experiments M. Tsesarsky & Y.H. Hatzor Department of Geological and Environmental Sciences, Ben Gurion University of the Negev, Beer – Sheva, Israel

N. Sitar Department of Civil Engineering, University of California, Berkeley, USA

ABSTRACT: The accuracy and validity of Discontinuous Deformation Analysis (DDA) is tested using analytical solutions and shaking table experiments. The displacement history of a single block on an inclined plane subjected to a sinusoidal loading function, with input frequencies ranging from 2.66 Hz to 8.6 Hz, is studied. DDA predicts accurately the measured displacements and it’s evolution with time. Sensitivity analyses of the numeric control parameters show: 1) Artificial numeric damping is required if better accuracy is sought. For the case of a single block on an incline a reduction of the dynamic control parameter by 2% is recommended; 2) Accurate DDA solution is attained for high penalty (numeric spring stiffness) values, provided that the chosen time step is small enough to assure diagonal dominance of the global stiffness matrix; 3) For a given time step size a sensitivity analysis of the numeric spring stiffness should be performed to detect ill conditioning or loss of diagonal dominance.

1

INTRODUCTION

Discontinuous Deformation Analysis (DDA) (Shi, 1988; 1993) is a numeric model for analyzing statics and dynamics of discontinuous block systems. Successful application of the DDA method to various engineering problems requires rigorous validation. The accuracy of DDA has been tested by many researchers. Yeung (1991) and MacLaughlin (1997) tested the accuracy of DDA for applications ranging from tunneling to slope stability, using problems for which analytical or semi-analytical solutions exist. Doolin and Sitar (2001) explored the kinematics of a block on an incline for sliding distances of up to 250 meters. Hatzor and Feintuch (2001) validated DDA using direct dynamic input. Analytical integration of sinusoidal functions of increasing complexity was compared to displacements prescribed by DDA for a single block on an incline subjected to the same acceleration functions as integrated analytically. The necessity for DDA validation using analytical solutions is evident if the method is to be adopted by the engineering profession. However, analytical solutions are only valid for the inherent underlying simplifying assumptions. This limitation can be overcome by comparison between DDA prediction and

experimental results of carefully planned physical models. Up to date, such attempts have been limited, or practically non-existent for the dynamic problem. O’sullivan and Bray (2001) simulated the behavior of hexagonally packed glass rods subjected to bi-axial compression, showing the advantages of DDA in the study of soil dynamics. McBride and Scheele (2001) validated DDA using a multi-block array on an incline subjected to gravitational loading, and a bearing capacity model. Validation of DDA using analytical solutions (Yeung, 1991; McLaughlin, 1997; Doolin and Sitar, 2001; Hatzor and Feintuch, 2001) showed that DDA accurately predicts single block displacements, up to tens of meters. However, validation using physical models proves less successful. In particular, it is found that kinetic damping is required for reliable prediction of displacement (McBride and Scheele, 2001). In this paper we study the displacement history of a single block on an incline subjected to dynamic loading. The following issues are addressed: 1. Comparison between DDA solution and results of a physical model. 2. Sensitivity analyses of the numeric control parameters: numeric spring stiffness (g0), time step size

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(g1), assumed maximum displacement ratio (g2), and the dynamic control parameter (k01). 3. The nature and evolution of the computational error.

EXPERIMENTAL SETTINGS

45 in

1

5

11.37˚

Inclined plane

3 4

Shaking Table 48 in No. 1 2 3 4 5

Instrument accelerometer accelerometer accelerometer displacement transducer displacement transducer

Direction of Measurement parallel to plane parallel to plane horizontal horizontal parallel to plane

Figure 1. General view of the inclined plane and the sliding block (top); Sliding block experimental setup and instrumentation location (bottom). Source: Wartman (1999). 20

la rg e d is p l. R 2 = 0 .8 8

0.36

s m a ll d is p l. R 2 = 0 .3 3 m e d . to la rg e d is p l. s m a ll (< 0 .0 5 in .) d is p l.

18

0.32

16

0.28

14

0.24

12 0 .0 0 1

n e a r s ta tic

0 .0 1 0 .1 1 average sliding velocity (inches/sec)

10

Figure 2. Back analyzed friction angles as a function of average sliding velocity for the rigid block tests, from Wartman (1999).

upper bound velocity, the corresponding friction angle is φav < 17◦ , while φav = 16◦ is the most likely value. In this study sinusoidal input motion tests were used for validation. A typical sinusoidal input motion is

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2

Rigid Block

friction coefficient (tan φ)

The physical modeling used in this research was performed by Wartman (1999) at the Earthquake Simulation Laboratory of the University of California at Berkley. The tests were performed on a large hydraulic driven shaking table, producing accurate, well controlled, and repeatable motions to frequencies up to 14 Hz. The table was driven by a 222.4 kN (50 kip) force, 15.24 cm (6 in.) hydraulic actuator range manufactured by MTS. The system was closed loop servo controlled. A Hewlett Packard 33120A arbitrary function generator produced the table command signal. An inclined steel plane was fitted to the shaking table. The plane inclination was set to 11.37◦ during the rigid block tests. The steel rigid block was 2.54 cm (1 in.) thick, with area of 25.8 cm2 (4 in.2 ), and weight of 1.6 kg (3.5 lbs). Linear accelerometers were fitted on top of the sliding block and the inclined plane. Displacement transducers measured the relative displacement of the sliding block, and of the shaking table (Fig. 1). Geotextile and geomembrane were fitted to the face of the sliding block and the inclined plane respectively. The static friction angle (φ) of the interface was determined using tilt tests and a value of φ = 12.7◦ ± 0.7◦ was reported. Kim et al., (1999) found that the geotextile–geomembrane interface friction exhibited pronounced strain rate effects, and reported an increase by 20% over a log-cycle of strain rate. Wartman (1999) showed that the friction angle of the interface was controlled by two factors: 1) amount of displacement; and 2) sliding velocity. For the range of velocities and displacements attained at the shaking table experiments the back-calculated friction angle was in the range of φ = 14◦ − 19◦ (Fig. 2). At its present stage of development DDA accepts a constant value of friction angle. Therefore a representative value of friction angle (φav ) should be chosen for the validation study. The value of φav was determined as follows. First, the measured displacement of the block was differentiated with respect to time and hence the velocity record was attained. Next, the velocity content was computed. Taking as an example, the 2.66 Hz input motion frequency test showed that the velocity upper bound value was bellow 10 cm/sec (4 in/sec), refer to Figure 3a. This value was attained only for short periods of time during the test. The velocity content chart shows that 70% of the velocities fall under the value of 2.54 cm/sec (1 in/sec), refer to Figure 3b. Taking the value of 2.54 cm/sec as the

friction angle (φ)

2

Table 1. Input motion summary: ω is the input motion frequency, dT is the shaking table displacement, dB is relative block displacement, and ah is maximum horizontal table acceleration.

(a) displacement velocity

Displ. (cm) / Vel. (cm/sec)

12 10 8

4

0

-4 0

1

2 3 Time (sec)

4

5

(b) 1

200

0 .8

150 0 .6 100 0 .4

50 0

Cumulative (*100%)

250

Frequency

dT

dB

ah

Test

Hz

cm

cm

g

1 2 3 4 5 6 7 8

2.66 4 5.33 6 6.67 7.3 8 8.66

0.889 0.559 0.305 0.254 0.254 0.228 0.228 0.019

5.367 6.604 3.341 3.647 3.410 3.353 3.937 2.882

0.28 0.25 0.19 0.19 0.22 0.22 0.23 0.21

2

-2

0 .2 0

1

2

3

4

5 6 7 8 velocity (cm/sec)

9

11

10

Figure 3. a) Displacement derived velocity, 2.66 Hz frequency sinusoidal input test; b) Velocity content of the 2.66 Hz frequency sinusoidal input test. A c c e le r a tio n ( g )

ω

6

independently without interpenetration. In the DDA method the formulation of the blocks is very similar to the definition of a finite element mesh. A finite element type of problem is solved in which all elements are physically isolated blocks bounded by pre-existing discontinuities. The blocks used in DDA can assume any given geometry, as opposed to the predetermined topologies of the FEM elements. DDA first order displacement approximation assumes that each block is a constant strain/stress element. The displacements (u, v) at any point (x, y) in a block i, can be related in two dimensions to six displacement variables

0 .2

[Di ] = (u0 v0 r0 εx εy γxy )T

0 .1

(1)

0 -0 .1 -0 .2

0

1

2 3 T im e (s e c )

4

5

Figure 4. Shaking table typical sinusoidal input motion, 2.66 Hz frequency (Test 1 at Table 1).

shown in Figure 4. The motion was ramped up linearly for 1.5 seconds to insure shaking table stability, followed by full amplitude for duration of 2 seconds, and finally ramped down for 1.5 seconds. Eight different tests were used for validation (Table 1).

where (u0 , v0 ) is the rigid body translations of a specific point (x0 , y0 ) within a block, (r0 ) is the rotation angle of the block with a rotation center at (x0 , y0 ), and εx , εy and γxy are the normal and shear strains of the block. For a two-dimensional formulation of DDA, the center of rotation (x0 , y0 ) coincides with block centroid (xc , yc ). Shi (1988) showed that the complete first order approximation of block displacement takes the following form

 u = [Ti ][Di ] v . 0 ( y − y0 )/2 1 0 −(y − y0 ) (x − x0 ) [Di ] = 0 1 (x − x0 ) 0 ( y − y0 ) ( x − x0 )/2 (2)

3 3.1

DDA FUNDAMENTALS AND NUMERICAL SETTINGS DDA formulation

DDA models a discontinuous material as a system of individually deformable blocks that move

This equation enables the calculation of displacements at any point (x, y) of the block when the displacements are given at the center of rotation and when the strains are known. In DDA individual blocks form a system of blocks through contacts among blocks and displacement

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constrains on a single block. For a block system defined by n blocks the simultaneous equilibrium equations are K

K12  K21 K22  K31 K32   . .  . .  . . 11

K13 K23 K33 .. .

Kn1 Kn2 Kn3

    · · · K1n  D1  F1       · · · K2n   D2  F2              · · · K3n  D3 F3  = ..  . . ..    ..   . .  .  .                  Dn Fn · · · Knn

(3)

where Kij are 6×6 sub-matrices defined by the interactions of blocks i and j, Di is a 6×1 displacement variables sub-matrix, and Fi is a 6×1 loading submatrix. In total the number of displacement unknowns is the sum of the degrees of freedom of all the blocks. The diagonal sub-matrices Kij represent the sum of contributing sub-matrices for the i-th block, namely block inertia and elastic strain energy. The off diagonal sub-matrices Kij (i = j) represent the sum of contributing sub-matrices of contacts between blocks i and j and other inter-element actions like bolting. The simultaneous equations are derived by minimizing the total potential energy  of the block system. The i-th row of (3) consists of six linear equations ∂ = 0, ∂dri

r = 1, . . ., 6

(4)

where dri are the deformation variables of block i. Full detail of stiffness matrix and load vector assembly is found in Shi (1993). The solution to the system of equations (3) is constrained by inequalities associated with block kinematics, no penetration and no tension condition between blocks. The kinematic constrains on the system are imposed using the penalty method. Contact detection is performed in order to determine possible contacts between blocks. Numerical penalties analogous to stiff springs are applied at the contacts to prevent penetration. Tension or penetration at the contacts results in expansion or contraction of the “springs”, which adds energy to the block system. Thus the minimum energy solution is one with no tension or penetration. When the system converges to an equilibrium state the energy of the contact forces is balanced by the penetration energy, resulting in inevitable very small penetrations. The energy of the penetrations is used to calculate the contact forces, which are in turn used to calculate the frictional forces along the interfaces between blocks. Shear displacement along the interfaces is modeled using Coulomb–Mohr failure criterion. Fixed boundary conditions are enforced in a manner consistent with the penalty method formulation. Stiff springs are applied at fixed points. Displacement of the fixed points adds considerable

energy to the block system. Thus, a minimum energy solution satisfies the no displacement condition of the fixed points. The solution of the system of equations is iterative. First, the solution is checked to see how well the constrains are satisfied. If tension or penetration is found along contacts the constrains are adjusted by selecting new position for the contact springs and a modified versions of [K] and {F} are formed for which a new solution is attained. The process is repeated until each of the contacts converges to a constant state. The positions of the blocks are then updated according to the prescribed displacement variables. The large displacements and deformations are the accumulation of small displacements and deformations at each time step. DDA time integration scheme adopts the Newmark (1959) approach, which for a single degree of freedom can be written in the following manner: 

1 ui+1 = ui + t u˙ i + − β t 2 u¨ i + βt 2 u¨ i+1 2 (5) u˙ i+1 = u˙ i + (1 − γ )t u¨ i + γt u¨ i+1 where u, ¨ u˙ , and u are acceleration, velocity, and displacement respectively, t is the time step, β and γ are the collocation parameters defining the variation of acceleration over the time step. Unconditional stability of the scheme is assured for 2β ≥ γ ≥ 0.5. DDA integration scheme uses β = 0.5 and γ = 1, thus setting the acceleration at the end of the time step to be constant over the time step. This approach is implicit and unconditionally stable. 3.2 Numerical implementation of DDA Computer implementation of DDAallows control over the analysis procedure through a set of user specified control parameters. The control parameters are: 1. Dynamic control parameter (k01) – defines the type of the analysis required, from static to fully dynamic. For static analysis the velocity of each block is set to zero at the beginning of each time step, k01 = 0. In the case of the dynamic analysis the velocity of each block at the end of a time step is fully transferred to the next time step, k01 = 1. Different values from 0 to 1 correspond to different degrees of damping or energy dissipation. 2. Penalty value (g0) – is the stiffness of the contact springs used to enforce contact constrains between blocks. 3. Upper limit of time step size (g1) – the maximum time interval that can be used in a time step, should be chosen so that the assumption of infinitesimal displacement within the time step is satisfied. 4. Assumed maximum displacement ratio (g2) – the calculated maximum displacement within a time

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step is limited to an assumed maximum displacement in order to ensure infinitesimal displacements within a time step. The assumed maximum displacement is defined as g2 · (h/2), where h is the length of the analysis domain in the y-direction. g2 is also used to detect possible contacts between blocks.

A c c e le r a tio n ( g )

0 .2

(a)

a(t) = 0.1sin8πt + 0.05sin4πt + 0.025sin2πt

0 .1

0

-0 .1

In this study the recently developed C/PC version of DDA (Shi, 1999) is used. In this version, dynamic acceleration can be input directly, and updated at every time step. A necessary condition for direct input of dynamic acceleration is that the numerical computation has no artificial damping because artificial damping may lead to energy losses. In DDA the solution of the equilibrium equations is performed without damping.

A ll D D A s im u la tio n s fo r: g 0 = 1 * 1 0 9 N /m g 2 = 0 .0 0 7 5 B lo c k E la s tic M o d u lu s (E ) = 5 0 0 0 *1 0 9 N /m

-0 .2 0 .8 0 .7

2

3

F u ll S y m b o ls - D D A o u tp u t fo r g 1 = 0 .0 0 2 5 s e c E m p ty S y m b o ls - D D A o u tp u t fo r g 1 = 0 .0 0 5 s e c F u ll L in e s - A n a ly tic a l S o lu tio n

0 .6 D is p la c e m e n t (m .)

(b) ω1 = 8π = 2ω = 4ω

0 .5

ω1 = 10π = 2ω = 4ω 2

0 .4 0 .3

ω1 = 15π = 2ω = 4ω 2

3

3

0 .2 0 .1

4

RESULTS OF VALIDATION STUDY

0 0

4.1

DDA calculation vs. analytical model

A Fourier series composed of sine components represents the simplest form of harmonic oscillations, in general notation: a(t) =

n 

ai sin(ωi t)

(6)

i=1

where ai and ωl are the amplitude (acceleration in this case) and frequency respectively. The displacement of a mass subjected to dynamic loading is attained by double integration of the acceleration record (Eq. 6) from θ to t: d(t) =

n  ai [−sin ωi t + sin ωi θ 2 ω i i=1 + ωi (t − θ) cos ωi θi ]

(7)

where θ is the time at which yield acceleration ay is attained. Goodman and Seed (1965) showed that for frictional sliding of a single block on a cohesionless plane the down slope horizontal yield acceleration is ay = tan (φ − α)g, where φ is the friction angle and α is the plane inclination. Hatzor and Feintuch (2001) showed that for an acceleration function consisting of sum of three sines DDA prediction are accurate within 15% of the analytic solution, provided the numeric control parameters g1, g2 are carefully optimized, and without application of damping. Moreover, they argued that the influence of higher order terms in a series of sine function is negligible. Hatzor and Feintuch demonstrated their validation for a1 = ω1 = 1, a2 = ω2 = 2, a3 = ω3 = 3. The prescribed values produce a low frequency dynamic input assuring nearly constant block

2

3 T im e ( s e c .)

4

5

6

Figure 5. a) The loading function a(t) = a1 sin (ω1 t) + a2 sin (ω2 t) + a3 sin (ω3 t); b) Comparison between analytical and DDA solution for block displacement subjected to a sum of three sines loading function.

velocity, which was attained at the beginning of the analysis (ca. 20% of elapsed time). In order to attain a better understanding of the frequency effect upon the numerical solution we have extended the analysis to higher frequencies, constraining the peak horizontal acceleration to 0.15 g. A typical input motion of sum of three sines is presented in Figure 5a. The analysis was performed for a single block resting on a plane inclined α = 15◦ to the horizontal. The block material properties were: density = 2700 kg/m3 , E = 5000 MPa, and v = 0.25. The friction angle of the sliding plane was set to φ = 15◦ , thus the yield acceleration (ay = 0) was attained immediately at the beginning of analysis (θ = 0 sec). Three different sets of frequencies were modeled (Table 2). Constant values of numeric spring stiffness g0 = 1000 MN/m, assumed maximum displacement ratio g2 = 0.0075, and dynamic control parameter k01 = 1 were used. Each set was modeled twice, first the time step was set to g1 = 0.005 sec, then the time step was halved to g1 = 0.0025 sec. Comparison of analytical solution and numerical estimate of the total displacement are presented in Figure 5b, generally showing excellent agreement between the analytical solution and the DDA solution, regardless of the frequency set chosen. The absolute numeric error was defined in a conventional manner " " " d − dN  " " " (%) EN = " (8) " d

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1

Table 2. Frequency sets for sum of three sines input function. ω1 (π ), a1 (g)

ω2 (π ), a2 (g)

ω3 (π ), a3 (g)

1 2 3

8, 0.1 10, 0.1 15, 0.1

4, 0.05 5, 0.05 7.5, 0.05

2, 0.025 2.5, 0.025 3.75, 0.025

a y ( φ = 1 7 o ) = 0 .1 g 0 .1 A c c e le ra tio n (g )

Set

(a)

0 .2

0 -0 .1 a y ( φ = 1 6 o ) = 0 .0 8 g

-0 .2 fric tio n a n g le (d e g .) ; k 0 1 16 ; 1 1 6 ; 0 .9 8 5 1 6 ; 0 .9 8 1 6 ; 0 .9 7 5 17 ; 1 m e a s u re d

-0 .3

0 .1 N u m e ric e rro r (% ) g 1 = 0 .0 0 2 5 g 2 = 0 .0 0 7 5 g 1 = 0 .0 0 5 g 2 = 0 .0 0 7 5 g 1 = 0 .0 0 7 5 g 2 = 0 .0 1 g 1 = g 2 = 0 .0 1

0 .0 8 Displacement (mm)

10000 1000 100

0 .0 6 ty (φ = 1 6 o ) 0 .6 4 s e c

0 .0 4

ty (φ = 1 7 o ) 1 se c

E

N

(b)

10

0 .0 2

1

0 0

1

3

2

4

5

T im e (s e c )

0.1 100

1000

10000

Numeric spring stiffnes (106 N/m)

Figure 6. Absolute numeric error of DDAultimate displacement prediction as a function of spring stiffness, for a sum of three sines loading function.

where d and dN are the analytical and the numeric displacement vectors respectively. · is the norm operator, which for a 2-D displacement vector is % d = u2 + v2 . The numeric error for g1 = 0.005 sec simulations is within 4.5% (Figure 5). Halving the time step reduces the numeric error to 1.5%. We have further investigated the interrelationship of the numeric control parameters using the input function of set 2 (Table 2). Figure 6 shows the dependence of the numeric error on the choice of the numeric control parameters g1, g2 and the numeric spring stiffness g0 (penalty value). It is found that for an optimized set of g1 and g2 (g1 = 0.0025 sec and g2 = 0.0075) the DDA solution is not sensitive to the penalty value, which can be changed over a range of two orders of magnitude. Within this range the numeric error never exceeds 10% and in most cases approaches the value of 1%. Naturally, stiffer contact springs reduce the magnitude of displacement until a certain minimum is reached. Further increase in the spring stiffness results in an introduction of a large numeric error into the DDA solution. Departing from the optimal g1, g2 combination results in increased sensitivity of the DDA solution

Figure 7. a) Physical model sinusoidal input function, 2.66 Hz frequency; b) Comparison of measured displacement and DDA solution.

to the penalty value. The departure from the analytical solution occurs at lower penalty values with increasing time step size.

4.2 DDA calculation vs. shaking table experiments It has been showed that there is a very good agreement between the DDA and analytic solutions for the e block on an incline problem. However, the analytical solution is only an approximation of the physical problem with various simplifying assumptions including: perfectly rigid block, constant friction, and complete energy conservation. Comparison between DDA results and physical modeling can help us probe into the significance of these assumptions. The frictional properties of the geotextile– geomembrane interface are strain rate dependent as discussed above. Based on the criteria described earlier the upper bound for interface friction angle was φav < 17◦ , with the φav = 16◦ being the most likely value. Consequently, the DDA analyses were performed for friction angle values of φav = 17◦ and φav = 16◦ . The numeric control parameters for the two friction configurations were: penalty value g0 = 500∗ 106 N/m, time step size g1 = 0.0025 sec, assumed maximum displacement g2 = 0.005. 2.66 Hz input motion is discussed here in detail and the comparison results are shown in Figure 7.

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EN at ultimate displacement (%)

100000

10000

Friction angle 16 (deg.) 17

Block Penetration Region

1000

Block Elastic Modulus (E) = 5*109 N/m2

100

10

1 1

1000

100

10

to 1.5% velocity reduction, reduces the error to below 10%. Furthermore, reduction of k01 improves the tracking of the displacement history by DDA. Plotting the relative numeric error (a non absolute version of Eq. 4) against the input motion frequency (Fig. 9) shows that in general DDA accuracy increases with higher frequencies, with the exception at 6 Hz. For φav = 16◦ and k01 = 1 the numeric error is always conservative, with the exception at 6 Hz. Reducing k01 to 0.98 shows a similar effect for all frequencies, reducing the numeric error below 10%.

Numeric Spring Stiffness (106 N/m)

Figure 8. Absolute numeric error of DDA ultimate displacement prediction as a function of spring stiffness, for a sinusoidal input function. All DDA solutions for time step g1 = 0.0025 sec, and assumed displacement ration g2 = 0.005, dynamic control parameter k01 = 1.

40

Numeric Error (%)

Unconservative 0

Conservative -40

φ (deg.); k01 16 ; 1 16 ; 0.98 17 ; 1

-80

-120 2

4

6

8

10

Input motion frequency (Hz)

Figure 9. Numeric error of DDA ultimate displacement prediction as a function of input frequency, for a sinusoidal input function

With dynamic control parameter k01 = 1 the DDA solution for φav = 17◦ falls within 20% of the measured displacement (Figure 8). Furthermore, the DDA solution captures the major features of the displacement history. The onset of displacement for φav = 17◦ is at ay = 0.0985 g according to the analytical solution of Goodman and Seed (1965). This result is contemporary with both the DDA calculation and the measured record of displacement. When the acceleration falls below the yield value the block eventually stops. This behavior is captured by the DDA computation as well. Setting φav = 16◦ reduces the accuracy of the DDA solution and the numeric error increases to approximately 80% (Fig. 8), but the ultimate displacement values are close, 0.055 m measured displacement compared to 0.093 m of calculated solution. Introducing some kinetic damping by reducing k01 below 1 improves the agreement between DDAand the physical test. Setting k01 = 0.985, corresponding

5

The implicit formulation of DDA guarantees numerical stability regardless of time step size. However, it does not guaranty accuracy. Where the time step is too large or too small relative to the numeric spring stiffness, loss of diagonal dominance and/or ill conditioning error may result, interfering with convergence to an accurate solution. With the penalty method, employed to prevent block penetration or tension between blocks, the theoretical solution is approached only when the penalty value approaches infinity. Nevertheless, too large penalty values may result in errors due to lack of diagonal dominance and/or illconditioning. The numeric implementation of DDA utilizes the SOR Gauss–Seidel equation solver. The convergence of the SOR equation solver is guarantied for diagonally dominant matrices: " " " " " " n " " " |Kii | > " Kij "" (9) " j=1 " " " j=i Larger inertia terms on the diagonal of the global stiffness matrix increase the stability of the computation. A small time step size is needed to increase the inertia terms, which are inversely proportional to the square of time step. This effect can be seen in Figure 6. For small time steps (0.0025 sec) the numeric error does not exceed 10% for increasing penalty values up to 5 ∗ 1010 N/m, higher values introduce significant error as the off diagonal sub-matrices become larger, resulting in loss of diagonal dominance. Enlarging the time step results in reduction of the inertia term in the diagonal sub-matrices. Thus, for a given value of time step size the loss of diagonal dominance will occur at lower penalty values. Figure 8 shows the accuracy of the DDA solution for different penalty values, for a given values of g1 and g2. When the penalty is lower than 5 ∗ 106 N/m inter-block penetration occurs. For penalty values of 5 ∗ 106 N/m and up to 600 ∗ 106 N/m the accuracy of

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DISCUSSION

10000

EN(%)

1000

100

k01 = 1

10

All DDA simulations for: g1 = 0.0025, g2 = 0.005

0.985 0.975

g0 = 500*106 N/m, φ = 16˚

k0.98

1

0

0.2

0.4 0.6 Elapsed time/Total time

0.8

1

Figure 10. Numeric error evolution of DDA solution for sinusoidal input function at 2.66 Hz input frequency.

the solution is well confined between relatively narrow error margins. With φav = 17◦ the error is reduced from 110% to 20% over the studied range of penalties. Similarly, with φav = 16◦ the error is reduced from 120% to 80% over the same penalty range. When the penalty is exceedingly high an abrupt accumulation of error occurs due to loss of diagonal dominance of the global stiffness matrix, or due to matrix ill conditioning. Most of the error is accumulated at the beginning of the analysis and it declines with time, a phenomenon known as algorithmic damping (Figure 10). Similar observations were reported by Doolin and Sitar (2001) for the case of a gravity driven block. The maximum error value is a test artifact associated with the transition from a ramped motion to a steady sine input motion in the shaking table experiment and can be ignored. This trend is maintained here for all values of k01 selected; greater accuracy is attained when k01 is optimized. Algorithmic damping is typical to implicit time integration schemes. In DDA, a Newmark type implicit, time integration scheme (collocation parameters are β = 0.5, δ = 1) assures unconditional stability of integration and high algorithmic damping (Wang et al., 1996). Thus, damping is performed without the introduction of energy consuming devices. The amount of algorithmic damping depends on the time integration method, the time step size, and the natural period of the system. In this study we have limited the duration of the analysis to 5 seconds, in conjunction with the physical model. It has been shown that algorithmic damping reduces the numeric error as calculation evolves. Doolin and Sitar (2001) showed that error reduction is evident for sliding distances of up to 250 m over 16 sec. Thus, for larger time spans the error will decline with calculation progress to a certain minimum value, further improving solution accuracy. Dynamic formulation of DDA is essentially undamped, thus for evolving systems the only way to

dissipate energy is by frictional resistance. The physical model is however more complicated, energy losses through structural vibrations, heat radiation, drag, and other physical mechanisms are present, and not accounted for by DDA. Reduction of the transferred velocity at each time step reduces the overall dynamic behavior of the discrete system without imposing illconditioning of the stiffness matrix (Wang et al., 1996). In a similar manner a quasy-static analysis is performed by setting k01 = 0. Thus we recommend that for full-scale simulations a certain amount of kinetic damping should be applied. McBride and Scheele (2001) showed similar effect for a gravity driven multi–block structure, showing that optimal results were achieved for k01 = 0.8. It is reasonable to assume that higher kinetic damping is required for multi – block structures, to account for a large number of contacts and block interactions. However, this estimate should be examined in conjunction with the time step size and the penalty value.

6 •

The results of the validation study show that DDA solution of an idealized system for which an analytical solution exists, is accurate. The DDA intra-block contact algorithm is therefore a true replication of the analytical model for frictional sliding. • The accuracy of DDA is governed by the conditioning of the stiffness matrix. DDA solution is accurate provided that the chosen time step is small enough to assure diagonal dominance of the global stiffness matrix. • Numeric spring stiffness should be optimized in conjunction with the chosen time step size to assure accurate solution and to preclude ill conditioning of the global stiffness matrix. • Comparison between a shaking table model and DDA calculation shows that the DDA solution is conservative. For accurate prediction of dynamic displacement of single block on an incline a reduction of the dynamic control parameter (k01) by 2% is recommended.

ACKNOWLEDGMENTS This research is funded by the US–Israel Binational Science Foundation through grant 98–399. The authors wish to express their gratitude to Gen-hua Shi who kindly provided his new dynamic version of DDA. Shaking table data were provided by J. Wartman of Drexel University, R. Seed, and J. Bray of University of California, Berkeley, and their cooperation is greatly appreciated.

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SUMMARY AND CONCLUSIONS

REFERENCES Doolin, D. & Sitar, N. 2001. Accuracy of the DDA method with respect to a single sliding block. In: Rock Mechanics in the National Interest, proceedings of the 38th U.S. Rock Mechanics Symposium, Washington D.C., July 5–7, 2001. Balkema, Rotterdam. Goodman, R. E. & Seed, H. B. 1965. Earthquake induced displacements in sand embankments. J. of Soil Mech. and Foundations Div. ASCE. 92(SM2): 125–146. Hatzor, Y. H. & Feintuch, A. 2001. The validity of dynamic block displacement prediction using DDA. Int. J. of Rock Mech. and Min. Sci. 38: 599–606. Kim, J., Bray, J. D., Reimer, M. F. & Seed, R. B. 1999. Dynamic interface friction properties of geosyntetics. Unpublished report, University of California at Berkeley, Department of Civil Engineering. MacLaughlin, M. 1997. Discontinuous Deformation Analysis of the kinematics of rock slopes. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. McBride, A., Scheele, F. 2001. Investigation of discontinuous deformation analysis using physical laboratory models. In: Bicanic, N. (ed.). Proc. of the Fourth International Conference on Discontinuous Deformation Analysis. 73–82. Glasgow, 6–8 June. Newmark, N. M. 1959. A method of computation for structural dynamics. J. of the Eng. Mech. Div. ASCE. 85(EM3) O’Sullivan, C. & Bray, J. D. 2001. A comparative evaluation of two approaches to discrete element modeling to

particulate media. In: Bicanic, N. (ed.). Proceedings of the Fourth International Conference on Discontinuous Deformation Analysis. 97–110. Glasgow, 6–8 June. Shi, G-h. 1988. Discontinuous Deformation Analysis – A new model for the statics and dynamics of block systems. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. Shi, G-h. 1993. Block system modeling by discontinuous deformation analysis. In: Brebbia, C. A. & Connor, J. J. (eds). Topics in Engineering, Vol. 11. Computational Mechanics Publication. Shi, G-h. 1999. Applications of Discontinuous Deformation Analysis and Manifold method. In: Amadei, B (ed.). Third International Conference on Analysis of Discontinuous Deformation. 3–16. Vail, Colorado, 3–4 June. Wang, C-Y., Chuang C-C. & Sheng, J. 1996. Time integration theories for the DDA method with Finite Element meshes. In: Reza Salami, M. & Banks, M. (eds.). Proceedings of the Fifth International Forum on Discontinuous Deformation Analysis (DDA) and Simulation of Discontinuous Media. 97–110. Berkeley, 12–14 June. TSI Press: Albuquerque. Wartman, J. 1999. Physical model studies of seismically induced deformation in slopes. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. Yeung, M. R. 1991. Application of Shi’s DDA to the study of rock behavior. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley.

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Theoretical developments in modelling discontinuous deformation

Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands

Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Crack propagation modeling by numerical manifold method Shuilin Wang Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, P. R. China

Ming Lu SINTEF Civil and Environmental Engineering, Trondheim, Norway

ABSTRACT: Manifold method, proposed by Dr. Genhua Shi about 10 years ago, catches the attention of many scholars around the world. It employs cover systems to form elements similar to the finite element method using meshes. In this paper, manifold method is employed to simulate crack propagation for the reason that the mathematical meshes can be kept unchanged during the simulation processes. The theory of linear elastic fracture mechanics (LEFM) is chosen to judge whether a crack extends or not. Stress intensity factors in LEFM are calculated by a path independent contour integral to avoid singularity at crack tip. High order cover functions are used on physical covers to improve the accuracy of the crack propagation simulation. An algorithm to deal with the crack tip’s stopping at any place of an element is presented. Finally, test examples are given to validate the method and the corresponding program.

1

INTRODUCTION

Crack propagation is encountered in many engineering problems and simulation of crack propagation has been a challenging problem. As far as the authors’ knowledge, there are 4 kinds of numerical methods that can be used to model crack growth. They are finite element method (FEM) (Lu & Bostrøm 1999), boundary element method (BEM) (Scivia 1995), element free Galerkin method (EFGM) (Belytschko et al 1996) and manifold method (Wang 1998). Some FEM programs (Bittencourt et al 1996, Lu & Bostrøm 1999) for simulating crack propagation have been developed. But remeshing is an overburden for problems in which the crack path is not previously known, especially when problem is extended from 2-D to 3-D. BEM has advantages in crack propagation simulation for simplifying the problems (reducing the dimension by 1). But it also has its own limitations and requires simple material property and geometry. EFGM is proposed recently and produced on the basis of moving least square interpolation. No mesh is needed in this method except that a background rectangular mesh is sometimes added during numerical integration. However, according to our experiences, further studies are needed on the numerical stability of the method. Numerical manifold method, a newly proposed method, is similar to FEM in some respects. But the element shape in this method is not as restricted as in FEM. It can be irregular. The initial meshes (i.e.

mathematical meshes) can be kept unchanged during the simulation process. Therefore, manifold method is especially suitable for simulating crack propagation in solids. In this paper, MM is chosen to model crack propagation. Numerical procedure about simulating crack propagation by the method is described. In the procedure, theory of linear elastic fracture mechanics (LEFM) is used for dealing with crack propagation. Related equations for computing stress intensity factors (SIF) and criterion for judging crack propagation are given. An algorithm to make the crack tip stops at any place is presented. Meanwhile, corresponding program was written as a part of the hydraulic fracturing simulation code (Lu et al 2001). The programs are tested with two numerical examples.

2 THEORETICAL BACKGROUND 2.1 Criterion of judging crack propagation Only 2-D problems are discussed in this paper and the crack propagation model is based on LEFM theory. Generally speaking, once relative displacements occur on the crack plane with cracks, stress singularity will appear at the crack tip. As shown in Figure 1, local Cartesian coordinate XoY is located at the crack tip with X-axis in the crack plane.

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Thereby, the classical strength theory is not applicable to judge whether failure will occur at the crack tip. In this situation, LEFM theory, in which SIFs KI and KII are the key parameters, is utilized to evaluate the potential of crack extension. In mixed mode problems, the maximum circumferential tensile stress criterion is simplified to compare the equivalent SIF Keq , which is a function of KI and KII and θ, with the material toughness. When the following condition is met 3 θ θ Keq = cos [KI cos2 − KII sin θ] = KIc 2 2 2 Figure 1. Crack tip, integration contour 2 and its corresponding encompassed domain.

Stress and displacement fields around the tip can be written in the following analytical form (Whittaker et al 1992):

 3θ KI θ θ 1 − sin sin σx = √ cos 2 2 2 2πr

 3θ θ θ −KII 2 + cos cos sin +√ 2 2 2 2πr

 3θ KI θ θ 1 + sin sin cos σy = √ 2 2 2 2πr θ 3θ KII θ +√ sin cos cos 2 2 2 2πr θ 3θ KI θ τxy = √ cos sin cos 2 2 2 2πr

 3θ θ KII θ 1 − sin sin +√ cos 2 2 2 2πr /   3θ θ KI 2r (2k − 1) cos − cos u= 8G π 2 2 /   3θ θ KII 2r (2k + 3) sin + sin + 8G π 2 2 /   3θ θ KI 2r (2k + 1) sin − sin v= 8G π 2 2 /   3θ θ KII 2r −(2k − 3) cos − cos + 8G π 2 2

(1)

where KI and KII are mode I and II stress intensity factors, respectively. G is the shear modulus. k = (3 − v)/(1 + v) and k = 3 − 4v for plane stress and plane strain conditions, respectively. v is Poisson’s ratio. r and θ are local polar coordinates originating from the crack tip. Equation (1) shows that the stresses will be infinite at crack tip under loading, even if the load is small.

the crack will be considered to extend in the direction θ0 , in which Keq takes its maximum value. θ0 satisfies the following equation ∂Keq = 0; ⇒ KI sin θ0 + KII (3 cos θ0 − 1) = 0 ∂θ θ = θ0 (3) Now the problem is to evaluate SIFs KI and KII . 2.2 Computation of stress intensity factors In LEFM, stress singularities exist at crack tip. Because SIFs are used to evaluate crack, it is important to evaluate them accurately. In practice, analytical solutions are available only for few simple problems. Numerical methods are usually needed. SIFs can be computed directly from the displacements on the crack plane near the crack tip. Usually fine meshes are required due to the high stress gradient around the tip. Alternatively, a kind of singular element can be constructed to model the stress singularities at crack tip. One can also use contour integration away from the crack tip. Then SIFs are computed from the displacements and stresses along the contour. In such a way, stress singularities can be avoided. In this paper, the contour integration method derived based on Betti’s work reciprocal theorem is used. In the method, an auxiliary displacement and its corresponding stress fields represented by u, v, σx , σy , τxy are constructed. They can be written in analytical expressions. Equation (4) is a closed form solution derived from the complex functions proposed by Muskhelishvili (Yang 1996). cI and cII are constants similar to KI and KII . We utilize Betti’s work reciprocal theorem. i.e., the work the true stress fields σx , σy , τxy do on auxiliary displacements uˆ , vˆ equals to what auxiliary stresses σˆ x , σˆ y , τˆxy do on true displacements u, v. 1 σˆ x = √ 2πr 3

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(2)

 cos

+

3 5θ 3θ − sin θ sin 2 2 2

−2 sin

 cI

3 5θ 3θ − sin θ cos 2 2 2



 cII

σˆ y = √

1 2πr 3

 cos

3 5θ 3θ + sin θ sin 2 2 2

 cI

  5θ 3 sin θ cos cII 2 2   5θ 3 1 sin θ cos cI τˆxy = √ 2 2 2πr 3  

5θ 3θ 3 − sin θ sin cII + cos (4) 2 2 2   3θ θ 1 cI (1 − k) cos + sin θ sin uˆ = √ 2 2 2G 2πr    3θ θ + (1 + k) sin + sin θ cos cII 2 2   3θ θ 1 cI (1 + k) sin − sin θ cos vˆ = √ 2 2 2G 2πr    3θ θ cII + (k − 1) cos + sin θ sin 2 2 +

As illustrated in Figure 1, domain encompassed by contour  = 2 + AB + (−1 ) + CD is considered. 2 = DEFGA, and 1 = CE1 F1 G1 B. Betti’s work reciprocal theorem is applied on the domain and we can have # (ui ˆti − uˆ i ti ) d = 0 (5) 

i is a dummy index and i = 1, 2. Subscripts 1 and 2 denote axis x and y in local coordinates. u1 and u2 represent the displacements in x and y directions, and uˆ 1 and uˆ 2 represent the auxiliary field displacements of x and y directions. t1 = σx • nx + τxy • ny . t2 = τxy • nx + σy • ny . (6) ˆt2 = τˆxy • nx + σˆ y • ny (7)

ˆt1 = σˆ x • nx + τˆxy • ny ,

where nx and ny are unit outer normal along the contour in local coordinate XoY. Equation (5) can be further written as # # (ui ˆti − uˆ i ti ) d = (ui ˆti − uˆ i ti ) d (8) 1

2

for the reason that the sum of integration along AB and CD will disappear. After a tedious manipulation on the integration along path 1 , the left side of Equation (8) is simplified as an analytical expression. # k +1 (KI cI + KII cII ) (ui ˆti − uˆ i ti ) d = (9) 2G 1 Integration along path 2 , is evaluated by numerical means. Substituting numerical solutions of σx , σy , τxy ,

u and v into the right side of Equation (8) results in # (10) (ui ˆti − uˆ i ti )d = m1 cI + m2 cII 2

m1 and m2 are coefficients obtained by numerical computation. Equating (9) with (10) leads to k +1 (KI cI + KII cII ) = m1 cI + m2 cII 2G

By comparing the coefficients of cI and cII , we can get KI and KII . Even if there is pressure acting on the plane, the above equation still holds. See Wang (1998) for a detailed explanation. 2.3 Numerical integration As discussed in section 2.2, the computation of SIFs comes down to the integration along a contour starting from a point on one crack plane and ending at a point on the other crack plane. In our program, a nonclosed circle similar to path 2 in Figure 1 is chosen as the integration contour. The center of the circle is located at crack tip. The arc of the circle is divided into N sections. Numerical integration is performed section by section. The sum of the integration of all sections gives the coefficients m1 and m2 . Then KI and KII are obtained from Equation (11). During the computation, displacements u, v and stresses σx , σy , τxy at the end points of each section are obtained by numerical analysis. The auxiliary displacements uˆ , vˆ and stresses σˆ x , σˆ y , τˆxy along the contour are calculated from Equation (4). 2.4 An algorithm for managing crack tip ending at any places In MM, only when an element or edge of an element is penetrated completely by a crack, the crack will be considered to exist in that part of the element. In reality, the crack tip may stop at any place of the domain after extension. In order to handle the possible termination of crack tip in the element, a penalty method is adopted. In dealing with crack propagation, two cases will occur as showed in Figures 2 and 3. For case 1, crack tip may end within an element. Here we suppose that crack tip ends at node T as shown in Figure 2(a); for case 2, crack extends along the boundary of two elements and stops at any point of the boundary. Here we assume that crack tip ends at node T as shown in Figure 3(a). If two penalty springs with stiffness ks and kn in tangential and normal directions are applied at the “true” crack tip T with coordinate (xt , yt ) as illustrated in Figures 2 and 3, then the potential of the springs is     1 k 0 uu − ul w = {uu − ul vu − vl } s (12) 0 kn vu − vl (x ,y ) 2 t t

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(11)

kn

displacement function of physical cover (12 , 21 , 31 ). Displacement function of manifold element (11 21 31 ) is

ing

ing

lty

a pen

spr

spr

alty

pen ks

Yt

X t,

3 31 31 31 21 12 T 12 31

Y X

Yt

X t,

3 31 31 31 21 21 2

12

11 11

21

12

21 21 2

31

(14)

11 11 1

1

(a)

(b)

Figure 2. Crack propagating across an element edge and ending at a point within an element (Case 1). ing

ing

alty

spr

alty

spr

pen

pen kn

ks Yt

Yt

X t,

X t,

3

3 31

31 21 2 α 21

Y

where wi (x, y) (i = 1,2,3) is the weight function of physical cover (11 , 21 , 31 ). They are the same as those of manifold element (12 21 31 ) because both manifold elements have the same mathematical covers. u11 , v11 , u21 , v21 and u31 , v31 displacement function of physical cover (11 ,21 ,31 ). Therefore,

12 1 11

T

1

12 11

w=

4

4

(a)

Equation (12) can be rewritten as 41

41

X

(15)

21 2 21 T

(b)

Figure 3. Crack propagating along the boundary of two elements and ending at a point on the boundary (Case 2).

uu

vu

where and are tangential and normal displacements along the crack plane on the upper element; ul and vl are tangential and normal displacements along the crack plane on the lower element. ks and kn are tangential and normal penalty stiffness, respectively. (xt , yt ) are the coordinates of the current crack tip where the penalty springs are inserted and displacements take values at that node. For case 1, displacement function on manifold element (12 21 31 ) is

(13) where wi (x, y) (i = 1,2,3) is the weight function of physical cover (12 , 21 , 31 ). ui and vi (i = 12 , 21 , 31 ) are

t

t

Equation (16) can be further written in the following form   1 T T T T T D11 w= D11 D12 (F C W KWCF)(xt ,yt ) (17) D12 2 where  K =



ks 0 , 0 kn

 cos α sin α 0 0 0  −sin α cos α 0 C= 0 0 cos α sin α 0 0 −sin α cos α 

 T11 0 , F = 0 T12   0 w1 (x, y) 0 −w1 (x, y) W = , 0 −w1 (x, y) 0 w1 (x, y) 

Di (i = 11 , 12 ) are general unknown variable vectors of physical cover 11 and 12 . Ti (i = 11 , 12 ) has the same meaning as in report (Lu 2001) and is the function of coordinates. C is a transform matrix, α is the angle between crack plane and x-axis as shown in Figures 2

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 1 w1 (x, y)(u12 − u11 )w1 (x, y)(v1 2 − v1 1 ) 2    w1 (x, y)(u12 − u11 ) ks 0 × (16) 0 kn w1 (x, y)(v1 2 − v1 1 ) (x ,y )

and 3, and K is the local stiffness matrix. For case 2, we can write the same potential of springs as Equation (12) and get an equation similar to Equation (17). If we want to let crack tip stop at any place of the crack plane within the extending element or along the boundary, a penalty stiffness matrix Kp = [FT CT WT KWCF](xt,yt) must be added in the global stiffness matrix. Note that this algorithm will work only when high order (≥1) cover functions on physical cover are used. 2.5

σY0 10 m

1m

σx0

10 m

P

σx0 0.6 m

Simulation of crack propagation

At each step, the general procedure for the numerical modeling process is as following: (a) The mixed mode SIFs KI and KII are computed from Equation (11) after displacement and stress fields are obtained. (b) The maximum equivalent Keq and its corresponding angle θ0 are determined from Equations (2) and (3). (c) If Keq is greater than or equal to toughness KIc of the material, the crack will propagate. The crack tip extends in the direction of θ0 and ends temporarily at a node or at a boundary of an element. The crack length is measured from the tip. If the crack length is less than the length specified by the user, the crack will continue extending and end at another node or at a boundary of another element. This process goes on until the extending length is greater than or equal to the inputted crack length. Then the true crack tip is determined. Meanwhile, physical covers and manifold elements are added and updated. (d) If Keq is less than the toughness KIc , increase the load until Keq is equal to fracture toughness KIc . Then the crack starts to propagate as what is described above. (e) The procedures are repeated from (a) to (d) for the subsequent steps.

σY0

Figure 4. A plate with a circular hole and an initial crack subjected to internal pressure and outside uniform compression.

Given below are two test examples, in which the first order cover function is used and crack propagation is modeled following the procedure.

Figure 5.

3

Table 1. Relationship between internal pressure and crack extension (case 1, σx0 = 0.0 and σy0 = 0.0).

NUMERICAL EXAMPLES

Firstly, the model is applied to simulate propagation of an initial crack (0.6 m) in a plate with a circular hole. The geometry and loading conditions are shown in Figure 4. Figure 5 presents the finite element mesh used as mathematical covers. Plane stress condition is adopted. The material properties are E = 10 GPa, V = 0.23 and KIc = 0.52 MPam1/2 . Three loading cases are discussed and their results are given in Tables 1, 2 and 3. The size of crack increment is set to be 0.8 m, meaning that the new displacement and

Internal pressure P (MPa) Crack extension (m)

0.67 0

0.68 0.8

0.71 1.6

0.70 2.4

Table 2. Relationship between internal pressure and crack extension (case 2, σx0 = 0.375 MPa and σy0 = 0.25 MPa). Internal pressure P (MPa) Crack extension (m)

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Initial finite element mesh.

1.27 0

1.58 0.8

1.85 1.6

1.95 2.4

Table 3. Relationship between internal pressure and crack extension (case 3, σx0 = 0.25 MPa and σy0 = 0.25 MPa). Internal pressure P (MPa) Crack extension (m)

1.35 0

1.63 0.8

1.88 1.6

2.05 2.4

Figure 8.

Initial finite element mesh.

Table 4. Relationship between the applied forces and crack extension. Px = Py (103 KN) Crack extension (m)

Figure 6.

0.60 0.0

0.22 0.8

0.30 1.6

0.17 2.4

Predicted crack trajectories. 54˚

P x

3m

44˚

Py

7m

2˚

34˚

2m

1m

45

8m 6m

Figure 9.

Crack path.

10 m

Figure 7.

Structure geometry, cutting force and restriction.

stress field will be computed after the crack extends 0.8 m at each step. Three steps are computed for each loading case. The development of crack propagation at the last step is shown in Figure 6, in which the deformation is scaled up a little so that crack path can be seen clearly. Following are observations from the results. (1) Crack extension is stable for all of the 3 cases. After the crack extends a certain length, it will stop if the internal pressure doesn’t increase. (2) Compared with cases 2 and 3, it is easy for the crack to extend in case 1 because no outside

compression is applied. In case 2, the horizontal compression σx0 is greater than the one in case 3. It becomes easier for the crack to extend in case 2 than in case 3. The results sound reasonable. Additionally, in order to make the crack keep going, higher internal pressure is needed in cases 2 and 3 than in case 1. (3) Theoretically, the crack will extend in a straightline due to the symmetry in geometry and loading. Trivial computation error makes the crack route deviate from the horizontal line. This will be corrected in the next step. The other example is to simulate the rock cutting process due to drag picks. The geometry, loadings and boundary conditions of the model are shown in Figure 7. There is an initial crack of 1 m long in

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processes. This will get rid of the overburden of remeshing as in FEM. An algorithm to deal with the crack tip’s stopping at any place of an element is presented so that the crack extension can be controlled. Test examples are given and their results look reasonable. REFERENCES

Figure 10. 2.4 m.

Deformed geometry at crack extension length

the structure. The material properties are E = 10 GPa, V = 0.25 and KIc = 1.2 MPam1/2 . Plane stress condition is assumed. Figure 8 presents the finite element mesh used as mathematical covers. Relationship between the applied forces and crack extension is presented in Table 4. The crack path and the deformed geometry are shown in Figures 9 and 10, respectively.

4

DISCUSSION AND CONCLUSION

The presented methodology gives an effective approach to simulate crack extension in solids. Its advantage is that the mathematical meshes (initial mesh) can be kept unchanged during the simulation

Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. & Krysl, P. 1996. “Meshless methods: An overview and recent developments” Comput. Meth. Appl. Mech. Engng., 139: 3–47. Bittencourt, T., Wawrzynek, P., Ingraffea, A. & Sousa, J. 1996. Quasi-automatic simulation of crack propagation for 2D LEFM problems. Engineering Fracture Mechanics. 55(2): 321–334. Lu, M. 2001. Complete N-order polynomial cover function for numerical manifold method. SINTEF report 2001 F01139. Lu, M. & Bostrøm, B. 1999. Investigation of capacity of existing computer programs for simulating crack propagation. STF22 A99105. Lu, M., Bostrøm, B. & Svanø, G. 2001. Hydraulic fracturing simulation with numerical manifold method. ICADD-4: 391–401, Univ. of Glasgow, Scotland, UK Scavia, C. 1995. A method for the study of crack propagation in rock structures. Geotechnique. 45(3): 447–463. Wang, S. 1998. Numerical manifold method and simulation of crack propagation. [Ph.D. dissertation]. Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Whittaker, B., Singh, R. & Sun, G. (ed.) 1992. Rock fracture mechanics: principles, design and applications. Amsterdam: Elsevier. Yang, X., Fan, J. & Kuang, Z. 1996.Acontour integral method for stress intensity factors of mixed-mode crack. Chinese Journal of Computational Mechanics. 13(1): 84–89.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Continuum models with microstructure for discontinuous rock mass J. Sulem & V. De Gennaro CERMES, Ecole Nationale des Ponts et Chaussées-LCPC, Paris, France

M. Cerrolaza IMME, Facultad de Ingeniería, Universidad Central de Venezuela, Caracas, Venezuela

ABSTRACT: The discontinuous structure of layered or blocky rock is approximated by an equivalent Cosserat continuum. This implies the introduction of couple stresses and internal rotations which model the relative rotations between blocks and the bending stiffness of the layers. The advantage of the Cosserat homogenisation is also that various failure modes such as inter-block slip, block tilting, layer bending can be easily described through a multi-criteria plasticity model. This model is illustrated by two applications to geotechnical problems such as foundation in blocky rock and slope stability in foliated rock.

1

INTRODUCTION

The numerical analysis of discontinuous rock mass can be dealt with (in most cases) using off the shelf discrete or finite element codes. In the latter case, special interface elements are needed in order to account for the unilateral kinematics of the rock joints. The major drawback of these discrete analyses is that they are very computer time intensive when the number of layers is large. Moreover, detailed information on the geometry and the properties of the individual layers is needed for such models. The interest of developing continuous models for discontinuous rock mass or discrete structures is that for practical applications, a homogenized continuum model would provide a large-scale (average) response of the medium. However the validity of the approximation is restricted to the case where the characteristic size of the recurrent cell of the periodic medium (e.g. layer thickness or block size) is small as compared to the characteristic size of the problem (e.g. the wavelength of the deformation field). An other important limitation of the homogenization of layered or blocky structures with classical continuum theories is that they cannot account for elementary bending due to inter-layer or inter-block slip and may thus considerably overestimate the deformation. In order to overcome these limitations and to expand the domain of validity of the continuum approach one has to consider the salient features of the discontinuum within the frame of continuum theories with microstructure (Vardoulakis & Sulem 1995). The Cosserat theory has been used with some success in the recent years for analyzing blocky

and laminated systems (e.g. Mühlhaus, 1993, 1995, Adhikary & Dyskin, 1996, Sulem & Mühlhaus 1997). The enriched kinematics of the Cosserat continuum allows to model systems of microelements undergoing rotations which are different from the local rotations of the continuum. For blocky rock various failure modes such as inter-block slip and block tilting can then be easily described. In this paper we present several geotechnical applications of Cosserat continuum for layered and blocky rock as encountered in slope stability and foundations problems. It is shown that for toppling failure of rock slopes a Cosserat model provides the necessary link between the slipping mechanism along the layers and the subsequent bending of the rock columns that may lead to tensile breakage. The Cosserat continuum modeling can then be coupled to a discrete approach of block stability. Zones of bending failure are identified in the material to determine the failure surface of the slope that will intersect the foliation discontinuities. This allows to overcome one of the major shortcomings of the limit equilibrium approach: the arbitrary and sometimes unrealistic assumption of a pre-defined failure plane roughly normal to the set of discontinuities of the foliation. 2 A COSSERAT CONTINUUM FOR BLOCKY ROCK In a rock mass the continuity of the material is generally interrupted by a system of bedding planes, faults or joints. Among discontinuous rocks with “regular”

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network of discontinuities one can mention stratified rock mass where the rock mass is divided into parallel bedding planes and blocky rock for which the rock mass is jointed in a collection of separate blocks in such a way that from the macroscopic it gives the impression of a dry brickwork. In Cosserat theory a material point of the continuum has three additional rotational degrees of freedom as well as the three translations of a classical continuum. In a regular block structure, one can consider the influence of relative rotations between blocks by means of additional Cosserat rotations. The relative rotations cause moments and consequently, material parameters with dimension of length (here the block dimensions) appear in the constitutive relationship. 2.1

The Cosserat elastic model

In a two-dimensional Cosserat continuum each material point has two translational degree of freedom (u1 , u2 ) and one rotational degree of freedom ωc . The index is used to distinguish the Cosserat rotation from the rotation ω=

 1 u2,1 − u1,2 ; 2

(·),i =

∂(·) ∂xi

i = 1, 2

(1)

The state of deformation is described by the four components of the rate of the so-called “relative deformation” (Schaefer 1962) γ11 = ∂u1 /∂x1 ;

γ12 = ∂u1 /∂x2 + ωc

γ22 = ∂u2 /∂x2 ;

γ21 = ∂u2 /∂x1 − ω

c

(2)

and the two components of the gradient of the Cosserat rotation which is called the curvature of the deformation κ1 = ∂ωc /∂x1 ;

κ2 = ∂ωc /∂x2

(3)

The six deformation quantities (equations 3 and 4) are conjugate in energy to six stress quantities. First we have the four components of the non symmetric stress tensor σij which is conjugate to the non symmetric deformation tensor γij and second we have two moment stresses (moment per unit area) m1 and m2 , which are conjugate to the two curvatures κ1 and κ2 . Force and moment equilibrium at the element (dx1 ,dx2 ) lead to

relationships for a 2D anisotropic Cosserat continuum are σ11 = C11 γ11 + C12 γ22 σ22 = C21 γ11 + C22 γ22 σ12 = [G + Gc (1 − α)] γ12 + [G − Gc ] γ21

where α is a parameter of anisotropy. We consider here a simple model for blocky rock (Fig. 1). Each block is surrounded by six others. We are mainly concerned with the accuracy with which the continuum model reflects the domain of rigidity set by the size of the blocks. The elasticity of the blocks and the joints elasticities are lumped at the block edges for simplicity. We assume fully elastic joint behavior. We assume that the interaction between the blocks is concentrated in six points of the edges as shown on Fig. 1. Normal and shear forces are written as Qkl = cQ ukl

(6)

Nkl = cN vkl

where cQ and cN are the elastic shear and normal stiffness respectively and u and v at various contact points are given by ui±1, j±1 = ui±1, j±1 − ui, j ±

where 2a and b are the dimensions of the block, ui,j , vi,j and ϕi,j are the displacements and rotation of the block number (i,j). The continuum model is derived by identifying the elastic energy of the equivalent Cosserat continuum with that of the discrete structure leading to

N

i-2, j

Q

i+1, j+1 i, j

i-1, j-1

i+2, j i+1, j-1

Figure 1. The blocky structure.

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(8)

 a ϕi±1, j±1 + ϕi, j (9) vi±1, j±1 = vi±1, j±1 − vi, j ∓ 2   (10) vi±2, j = vi+2, j − vi, j ∓ a ϕi±2, j + ϕi, j

(4)

In the above equations dynamic effects are included through inertial forces and moment. The stress-strain

 b ϕi±1, j±1 + ϕi, j (7) 2

ui±2, j = ui±2, j − ui, j

i-1, j+1

σ11,1 + σ12,2 − ρ u¨ 1 = 0 σ21,1 + σ22,2 − ρ u¨ 2 = 0 m1,1 + m2,2 + σ21 − σ12 − I ω¨ c = 0

(5)

σ21 = [G − Gc ] γ12 + [G + Gc (1 + α)] γ21 m1 = M1 κ1 m2 = M2 κ2

b 2a

the following expressions for the elastic constitutive parameters of the Cosserat continuum (Sulem and Mühlhaus, 1997) a  b C11 = cQ + 2cN ; C22 = cN ; C12 = C21 = 0 b a   a 1 b  cQ + cN + 2cQ G = Gc = 4 a b   2 2 a cN + 2cQ − cQ b  (11) α = 2 2 a cN + 2cQ + cQ b2 

2  a2 b a + 2a2 + cQ cN M1 = 4 4 b  2 2 a b b + cQ cN M2 = a 4 4 The domain of validity of the above representation of a blocky structure by a Cosserat continuum is evaluated by comparing the dynamic response of the discrete and the homogenized structures. The dynamic response of a structure is characterized by its dispersion function which relates the wave propagation velocity to the wave length of the input signal. For elastic behavior it is possible to derive analytical solutions for the dispersion function of the discrete and the continuous systems by using 2D discrete and continuous Fourier transform (Sulem and Mühlhaus, 1997). It was obtained that the Cosserat model is appropriate for wave-lengths greater than 5 times the size of the block (Fig. 2). 1.0

2.2 Extension to elasto-plastic joints The above elastic Cosserat continuum can be extended to an elasto-plastic Cosserat continuum (Mühlhaus 1993). Two different plastic mechanisms can in a blocky structure: block sliding along the joints and/or block tilting. Consequently several yield conditions have to be examined simultaneously. The state of joint slip is defined by a simple Mohr-Coulomb yield condition (compression is assumed to be negative) F 1 = |σ12 | + tan φσ22 − c ≤ 0

(12)

where c and φ are the joint cohesion and friction angle respectively. It is physically acceptable to assume zero dilatancy for friction mechanism so that the corresponding plastic potential is expressed as Q1 = |σ12 |

(13)

Statically admissible force/moment states of a volume element of the block structure are characterized by the tilting conditions (Fig. 3) F (1,2) = −N (1,2) +

2 "" (1,2) "" M ≤0 a

(14)

where for incipient gap opening we have F = 0. The tilting yield criterion for the corresponding Cosserat continuum is expressed as " " b 2" b " F (2,3) = σ22 ± σ21 − ""m2 ± m1 "" ≤ 0 a a a

(15)

with normality flow rule V-Cosserat/V-discrete

0.8

Q

(2,3)

=F

(2,3)

" " b 2 "" b "" = σ22 ± σ21 − "m2 ± m1 " a a a

(16)

0.6

N (2) N (1)

0.4 wave in x-direction wave in y-direction

(2)

M

(2)

M

(1)

(1)

∆␻c

(1)

0.2

∆u2

(0) 0.0 2

1

3

4 5 6 7 89

2

3

(3)

4 5 6 7 89

10

(4)

100

w1/2a ; w2 /b a/2 Figure 2. Dispersion function for continuous and discrete approach.

Figure 3. Tilting conditions.

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(0)

2.3

Example of application: foundation on blocky rock

We consider here the 2D-example of a half-space under uniform normal load over an extent L. This simple configuration can represent the classical problem

Figure 4.

of a strip-footing on a blocky rock mass (Fig. 4). For numerical applications the following set of geometrical and mechanical data is considered: – Block characteristics: length: a = 1 m, width: b = 0.5 m – Joint characteristics: joint stiffness: cN = cQ = 1 GPa friction angle: φ = 20◦ cohesion: c = 1 KPa dilatancy angle ψ = 0◦ – Length of the footing: L = 10 m. For symmetry reasons, only half of the domain is discretized. On Fig. 5a (respectively 5b) the plastic zones for the sliding criterion (respectively the tilting criterion) are represented in dark color. These results show that the tilting criterion is reached at the surface of the half-space on a limited extent at the vicinity of the side of the footing. The sliding criterion is reached deep inside the rock mass with an orientation of about 30◦ with respect to the horizontal axis. These results are compared to those obtained with a classical isotropic elastic-plastic Mohr-Coulomb yield surface with the geomechanical characteristics of a gravel (Young’s modulus = 25 MPa, Poisson’s ratio = 0.3, friction angle = 40◦ , zero cohesion). In the latter case, the classical result of standard soil mechanics is retrieved: an elastic cone is formed under the footing and plastic yield occurs under it (Fig. 6).

Strip-footing of a periodic block structure.

Figure 5. Uniform normal loading on a blocky structure, (a) sliding zones (p = 8.4 KPa), (b) tilting zones (p = 4.5 KPa).

Figure 6. Plastic zones for a uniform normal loading on an isotropic Mohr-Coulomb half-space.

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3 ANALYSIS OF TOPPLING FAILURES IN JOINTED ROCK SLOPES: AN EXAMPLE OF TRANSITION FROM CONTINUOUS TO DISCRETE APPROACH Toppling failures can develop in natural slopes as well as in artificial cuts (e.g. open pit walls) provided that the spatial distribution of fractures gives rise to a layered (foliated) rock mass with a principal system of parallel discontinuities that dip deeply into the slope at a given angle. According to existing published researches two main toppling failure mechanisms are usually considered: (a) flexural toppling, (b) block toppling (among others: Hoek & Bray 1977). Both type of failures have a common characteristic: slender blocks may tend to topple thrusting forwards downhill elements. In case (a) the triggering mechanism inducing slope instability is often the flexion of slender rock columns due to joint slipping and gravity force. At a critical depth the induced tensile stress due to bending exceeds the tensile strength of rock material, and fractures progressively develop in the cross section of blocks leading to the progressive ruin of columnar array. In case (b) rock mass has a well defined structural pattern, resulting from the system of pseudo-vertical discontinuities intersecting at right angles a system of bedding planes dipping towards the slope surface. In this configuration, even if the inclination of bedding planes is lower than the angle of friction mobilized at the interface between the base of block and the basal surface, toppling of slender columns causes, in turn, toppling and sliding of blocks below.

Figure 7.

In a reference system with directions 1 and 2 as in Figure 6, for this layered rock, the elastic constitutive parameters of equivalent Cosserat continuum can be expressed as follows (Zvolinskii and Shkhinek, 1984, Mühlhaus, 1995, Adhikary and Dyskin, 1996) C11 = C22 =

1−

−

ν 2 (1+ν)2 1−ν 2 +E/(kn )

(1 − ν)E (1 + ν)(1 − 2ν) + (1 − ν)E/(kn )

νE (1 + ν)(1 − 2ν) + (1 − ν)E/(kn ) (17)   E 5ks + E/(2(1 + ν)) G= 8(1 + ν) ks + E/(2(1 + ν))

Gc =

E ; 8(1 + ν)

α=2

  E/(2(1 + ν)) Eh2 ; M1 = 12(1 − ν) ks + E/(2(1 + ν))

M2 = 0

A simplified analysis is developed along the following lines: (a) when no sliding occurs the homogenized continuum reduces to a classical anisotropic elastic one; (b) when sliding occurs (i.e. when the frictional resistance of joints is reached), the corresponding parts of material fall in the Cosserat state with zero-joints stiffness. We emphasize the fact that in that case the sliding can be restrained by the bending rigidity and thus, the Cosserat model can address also the situation of zero-joints stiffness as opposed to the conventional homogenization which breaks down. In the parts of material in Cosserat state, the distribution of bending moments can be computed. Bending moments induce a “microscopically” non-uniform distribution of normal stress in the individual layers which may reach

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E ν2

C12 = C21 =

3.1 A Cosserat model for foliated rock In the models based on classical continuum approach, the layered material is replaced with a homogeneous anisotropic medium characterized by “effective” elastic moduli (e.g. Salamon 1968). If sliding between the layers can occur, the equivalent continuum is viewed as elasto-plastic as for example in the “multi-laminate” model proposed by Zienkiewicz & Pande (1977). Such models provide a good approximation of discontinuous material behavior when the shear stiffness along the joints is comparable to the individual layer shear modulus. In that case, joint slip is small enough to neglect layer bending. However if it is not the case, bending rigidity has to be included in the model otherwise the homogenized model may considerably overestimate the deformation. Bending stiffness of layers can be accounted for by using Cosserat theory where bending moments are considered in addition to conventional stresses. In order to verify the reliability of such an approach we present on Figure 7 the geometry of a slope in a foliated rock mass susceptible of toppling failure.

Slope geometry

the material tensile resistance. Zones of bending failure are thus identified in the material to determine the failure slope surface that will intersect the foliation discontinuities. This allows to overcome the major shortcomings of limit equilibrium approach: the arbitrary and sometimes unrealistic assumption of a pre-defined failure plane roughly normal to the set of discontinuities of foliation. 3.2

Finite element analysis

For the numerical application the following set of geometrical data has been considered: – slope characteristics: height, H = 80 m; slope face inclination, α = 45◦ ; – foliation characteristics: orientation with respect to the horizontal axis, β = 70◦ , thickness = 2 m.

Figure 8.

Sliding zones for layered rock slope.

Figure 9.

Distribution of bending moment.

Material data are as follows: – rock mass characteristics: Young’s modulus = 20 GPa, Poisson’s ratio = 0.2, friction angle = 30◦ , cohesion = 1 MPa; – joints characteristics: shear stiffness = 0.1 GPa/m, normal stiffness = 1 GPa/m, friction angle = 22◦ , cohesion = 10 kPa. The finite element analysis for Cosserat material is performed, using the code COSSBLPL (Cerrolaza et al. 1999). A first computation is performed to determine the zones where the sliding criterion is reached (Fig. 8). A second computation is then performed for which, it is assumed that the shear stiffness along the joints is zero for the part of model where sliding occurs. In this zone, the effect of bending stiffness as introduced by the Cosserat model is thus more important. The results for the bending moment are presented on Figure 9. As mentioned above, bending moments will result in a microscopically non-uniform distribution of normal stress in the layers. In a first approximation the microscopic stress distribution is assumed to be linear within the layer. From elementary beam theory, the normal stress can be evaluated as σN = −

M N + y A I

(18)

where N is the axial force in the layer, M is the bending moment, A is the cross-sectional area and I is the second moment of inertia. The maximum value of tensile stresses acting in the layer can be thus estimated as σtensile = 6

m1 + σ11 b

(19)

where m1 is the Cosserat couple stress, b is the layer thickness and σ11 is the microscopic stress.

Figure 10.

If one assumes that the rock tensile strength is 1 MPa, the zones of possible tensile failure are represented on Figure 10. 3.3 From flexural toppling to block toppling As depicted in Figure 10, if one assume that 1 MPa is the rock tensile strength, a well defined pattern gives the failure surface direction expected. Once that this surface is localized, the system of parallel discontinuities located inside the wedge shaped zone delimited by the slope face, the ground level (Fig. 11) and the failure surface, gives rise to a system of interacting unstable blocks.

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Zones of tensile failure.

Figure 12. Limit equilibrium analysis: (a) slope geometry, (b) forces and geometry of i-th block (Mongiovì et al. 1995).

Figure 11. analysis.

Slope geometry used for limit equilibrium

The problem of flexural toppling failure is then reconducted to a block toppling failure. For the latter, the basal failure plane, usually assumed as pre-existing, is obtained from a preliminary computation based on an equivalent continuum approach, accounting for microstructure. The analysis of block toppling is usually performed in a more conventional way, adopting the well known limit equilibrium approach (Goodman & Bray 1976). It seems then licit to investigate whether the wedge obtained in Figure 10 is unstable also in the context of limit equilibrium analysis. In this respect a theoretical 2D model based on the original method proposed by Goodman & Bray (1976) is used. The slope response is analyzed with reference to the geometry at the onset of failure induced by flexural toppling (Fig. 10). The new slope geometry is presented in Figure 11. All blocks have the same width b and form a regular system on a basal surface stepped upwards, as schematically represented in Figure 12a. The system is delimited by the slope face (α3 ), the ground level (α1 ) and the basal surface (α2 ). The inclination of bedding planes is αo . The inclination of basal surface has been depicted from the pattern of tensile stress shown in the same figure. The value retained is α2 = 30.2◦ . This value corresponds to the basal surface inclination obtained joining the point at the slope toe and the point on the ground level where the tensile stress reaches the rock tensile strength. Assuming that blocks have constant width (b = 2 m), the basal surface inclination involves about 80 blocks in the calculation. The other angles (all calculated versus the direction of X-axis) are as follows: αo = 20◦ , α1 = 0, α3 = 45◦ . The block i is subjected to gravity force (Pi ), side forces (Si ,Ti , Si−1 , Ti−1 ) and basal forces (Qi , Ri ) (Fig. 12b). It is assumed that Mohr-Coulomb cohesionless failure criterion holds true for the basal and

side forces, with a friction angle ϕ = 22◦ as in previous finite element analysis. The force distribution on each element is determined by solving simultaneously the three equations of equilibrium and assuming the less favorable configuration kinematically compatible among the following conditions: stability, downhill sliding, downhill toppling, simultaneous uphill sliding and downhill toppling. Four compatibility conditions are introduced, related to the failure frictional criterion given at the interfaces between blocks and to the points of application of base and side forces. Considering the scheme in Figure 11b, they are as follows: |Ri | − Qi tan ϕ ≤ 0

(20)

|Ti | − Si tan ϕ ≤ 0 " " " " " fi − di " ≤ di " 2" 2 " " " " "ei − b " ≤ b " 2" 2

(21)

(23)

where b is the block width and di is the block height. The condition of simultaneous uphill sliding and downhill toppling was proposed by Mongiovì et al. (1995), who verified that often no solutions of equilibrium equations exist and satisfy the kinematic compatibility for downhill toppling. The analysis starts from the block at the top, progressing down to the toe. The response of each block is determined selecting among the solutions related to the four modes of behavior considered the only acceptable one that: (a) yields the highest value for the force Si , transmitted by block i to block i+1 below, and (b) fulfills the compatibility conditions (20)–(23). Obviously, being So the force acting at the left side face of block at the slope toe, the slope is considered:

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(22)

– stable if: So < 0; – at limit equilibrium if: So = 0; – unstable if: So > 0.

3.4

Numerical results

The results of calculations performed in order to verify the consistency of equivalent continuum approach and the limit equilibrium analysis, are presented on Figure 13. The dimensionless value S on Figure 13 is the ratio between the forces Si , transmitted by block i to block i + 1, and the total weight of the unstable region defined in Figure 11. As it can be observed, S increases moving from the top (Lo /L = 0) to the toe (Lo /L = 1), and for Lo /L = 1 is So > 0. Consequently, the inclination α2 considered, issued from the finite element calculation for Cosserat material, leads to an unstable block system. We present in the same figure the evolution along the slope of the kinematics of blocks. As expected, the small blocks at the top of the slope are stables. In fact, being ϕ > αo , sliding on the surface of bedding planes will never occur. Moreover, neither downhill toppling nor downhill toppling and simultaneous uphill sliding are possible due to their reduced slenderness. Progressing towards the toe the height of blocks increases and blocks start topple. Finally, when approaching the toe of the slope complex failure mechanisms are activated. The occurrence of complex mechanisms is analyzed in Figure 14. If only downhill toppling or sliding are accounted in the calculations (see for instance: Goodman & Bray 1976, Zanbak 1983), by inspecting the evolution of the normal force Qi and the tangential force Ri at block base is easy to verify that condition (20) is violated at Lo /L = 0.65. Moreover, being Ri < 0, this implies simultaneous block uphill sliding. If this incompatibility is ignored, further calculations will lead also to negative values of Qi forces (dot line

in Figure 14), corresponding to an hypothetical and rather unrealistic block lifting. The problem can be regularized assuming that a new mechanism is developing, where the downhill toppling is accompanied by a simultaneous uphill sliding (Mongiovì et al. 1995). In this case, due to uphill sliding, the unstable block lower corner may knock against the stepped base riser, and extra forces are generated at the uphill side of the block in order to restore the equilibrium. The implications related to the evaluation of the action at the toe of the slope are quite significant. Indeed, as presented in Figure 15, the computed So in the hypothesis of simple mechanisms is of about 0.04

Q, R 0.02

0

-0.02

-0.04

Dimensionless forces Q and R on bedding planes Q simple mechanisms R simple mechanisms Q complex mechanisms R complex mechanisms

-0.06

-0.08 0

0.2

0.4

0.6

0.8

1

Lo

/L

Figure 14. 20

Dimensionless forces Q and R at block bases.

25 250 Dimensionless force S ␣2 = 30.2° (80 blocks)

S

Dimensionless force S S

16

simple mechanisms complex mechanisms

20 200

3

3

12

15 150

2

MECHANISM 0 = stable

1 = downhill sliding 2 = downhill toppling 3 = downhill toppling & uphill sliding

4

1

MECHANISM

2 8

10 100

50

0 0

0 0

0.2

0.4

0.6

0.8

0

1

0

Lo

Figure 13. blocks.

0.4 0

0.6 0

0.8 0

1

/L

Dimensionless force S and mechanisms on the

Figure 15. Influence of block kinematics on the dimensionless forces S.

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0.2 0

Lo

/L

10 orders of magnitude higher than the force at the toe of the slope as obtained admitting complex mechanisms. 4

CONCLUSION

In this paper we have discussed the representation of a rock mass with a regular network of fractures, using a Cosserat continuum. For layered and blocky rock the enriched kinematics of the Cosserat continuum allows to account for individual block rotation and bending of individual layer. Geometrical characteristics of the discontinuities such as block size and orientation, layer thickness are thus introduced directly in the constitutive relationships of the equivalent continuum. It is shown that this representation is valid even for a structure with a relatively small number of blocks or layers. Various failure mechanisms such as sliding or tilting can be considered with appropriate multi-criteria plasticity model. The practical relevance of using continuum models for discontinuous structures is that it is extremely flexible when used with numerical methods since no interface elements are needed and since the topology of the finite element is independent of block size and geometry (one mesh can be used to study several different structures). The homogenization procedure is however restricted to the case of a fixed and regular structure of orthogonal joints and it is the major limitation of the approach. It is shown that for the analysis of toppling failure of layered rock slopes, the Cosserat continuum approach provides the link between the slipping mechanism along the joints and the formation of a tensile failure surface which intersects the foliation discontinuities. The continuous approach is thus coupled with subsequent discrete approach of block stability.

Cerrolaza, M., Sulem, J. & El Bied, A. 1999. A Cosserat non-linear finite element analysis software for blocky structures. Int. J. of Advances in Eng. Soft. (30): 69–83. Goodman, R.E. & Bray, J.W. 1976. Toppling of rock slopes. In ASCE (ed.), Proc. Specialty Conf. on Rock Engineering for Found. and Slopes. Boulder, Colorado: 201–234. Hoek, E. & Bray, J.W. 1981. Rock slope engineering. The Institution of Mining and Metallurgy. Cambridge. Mongiovì, L., Bosco, G. & De Gennaro, V. 1995. Analysis of complex rotational and translational failure mechanisms in jointed rock slopes. In Rossmanith (ed.), Proc. Mech. Of Jointed and Faulted Rock. Wien, Austria: 617–622. Rotterdam: Balkema. Mühlhaus, H.-B. 1993. Continuum models for layered for layered and blocky rock. In: Comprehensive Rock Engng., Vol. 2 (Charles Fairhurst ed.) Pergamon Press: 209–230. Mühlhaus, H.B. 1995. A relative gradient model for laminated materials. In H.B. Mühlhaus (ed.), Continuum Models for Materials with Micro-Structure: 450–482, J. Wiley. Salamon, M.D.G. 1968. Elastic moduli of stratified rock mass. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr. (5): 519–527. Schaefer, H. 1962. Versuch einer Elastizitätstheorie des zweidimensionalen ebenen Cosserat-Kontinuums. In: Miszellannenn der Angewandten Mekanik, Akademie Verlag, Berlin: 277–292. Sulem, J. & Mühlhaus, H.B. 1997. A continuum model for periodic two-dimensional block structures. Mechanics of cohesive-frictional materials. (2): 31–46. Vardoulakis, I. & Sulem, J. 1995. Bifurcation analysis in geomechanics. Blackie Academic & Professional. Zanbak, C. 1983. Design chart for rock slopes susceptible to toppling. ASCE, Journal of Geotechnical Engineering. (109, 8): 1039–1062. Zienkiewicz, O.C. & Pande, G.N. 1977. Time dependent multilaminate model of rocks – a numerical study of deformation and failure of rock masses. Int. J. Numer. Anal. Meth. Geomech.(1): 219–247. Zvolinskii, N.V. & Shkhinek, K.N. 1984. Continual model of laminar elastic medium. Mechanics of Solids. (19): 1–9.

REFERENCES Adhikary & D.P., Dyskin. 1996. A Cosserat continuum model for layered materials. Computers and Geotechnics. (20): 15–45.

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Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Development of a three-dimensional discontinuous deformation analysis technique and its application to toppling failure H.I. Jang Research Institute of Engineering Science, Seoul National University, Korea

C.I. Lee School of Civil, Urban & Geo-system Engineering, Seoul National University, Korea

ABSTRACT: We developed a three-dimensional discontinuous deformation analysis (DDA) theory and computer program to analyze the deformation of rock blocks influenced by discontinuous planes in rock masses. In this, a first order deformation function was made and the potential energy and sub-matrices of a single block and contacts were obtained using this. Contacts between the blocks were classified into 4 categories (vertex-vertex, vertex-edge, vertex-triangle, edge-edge contact) according to their distance and position criteria. Using input parameters such as center, normal vectors and the radius of the discontinuous plane the three-dimensional block generation program was developed. In the verification test, two cases were analyzed using the three-dimensional DDA program: two-block sliding (one sliding face), and wedge sliding (two sliding faces). The results showed a good match when compared with those derived from an alternative theoretical analysis. The toppling mechanism (domino effect) was also analyzed and similarly gave good results.

1

INTRODUCTION

Since the development of Discontinuous Deformation Analysis (DDA) by Shi (1984), there has been much improvement in the theory and programs. These, however, are all based on the assumption of a twodimensional plane strain or plane stress state; and because a rock block system is a three-dimensional problem a two-dimensional analysis has limited application. A three-dimensional analysis required in the design of rock slopes and underground spaces where three-dimensional discontinuities dominate stability. In this paper, Shi’s two-dimensional DDA theory is extended to encompass three-dimensional theory. The three-dimensional DDA program, thus developed, was verified against three cases that had been solved by alternate means.

2 THREE-DIMENSIONAL DDA THEORY 2.1

Block deformation function and simultaneous equations

DDA calculates the equilibrium equations by minimization of the potential energies of single blocks and contacts between two blocks. To calculate the

simultaneous equilibrium equations, deformation functions must be defined. The deformation function calculates the deformation of all the blocks using the displacement of each block centroid. This function is similar to the shape function of Finite Element Method and can represent the potential energy of the blocks and the simultaneous equilibrium equations simply. Assuming all displacements are small and each block has constant stress and constant strain throughout, the displacement (u, v, w) of any point (x, y, z) of a block can be represented by 12 displacement variables. In the 12 variables, (u0 , v0 , w0 ) is the rigid body translation of a specific point (x0 , y0 , z0 ), r1 is the rotation angle (radians) of block around z axis, r2 is the rotation angle of block around x axis, r3 is the rotation angle of block around y axis, εx , εy , εz , γxy , γyz , γzx are the normal and shear strains in the block. The displacement of any point (x, y, z) in the block can be represented by Eq. (1). In DDA, the equilibrium equation is established by differentiation of the potential energy of the block as with FEM. The stiffness matrix is constructed using the potential energy of a single block and the contacts between two blocks. As DDA is a displacement method like FEM, the equilibrium equations are established by transposition of the constant to the right side, which is calculated by the differentiation of the

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total potential energy with respect to the displacement variables. Assuming there are n blocks in the defined block system, the simultaneous equilibrium equation has the same form as Eq. (2). In Eq. (2).

3.1 Two blocks with a single sliding face Two blocks with a single sliding face were analyzed. Table 1 shows the input variables of the blocks and

  0 0 0 (z − z0 ) (x − x0 ) 1 0 0 −(y − y0 ) u  v  = 0 1 0 (x − x0 ) −(z − z0 ) 0 0 (y − y0 ) 0  w 0 0 1 0 (y − y0 ) −(x − x0 ) 0 0 (z − z0 )

Kij (i = 1, . . ., n, j = 1, . . ., n) is a stiffness matrix which is a 12 × 12 matrix calculated for a single block and its contact with two other blocks; Di represents the displacement variables; Fi is the loading on the blocks, distributed to the 12 displacement variables.      D1 K11 K12 K13 · · · K1n F1 K K K · · · K  21 22 23 F2  2n  D2  K K K · · · K  D  F  3n   3  =  3   31 32 33 (2)      . . . .  . . . . . ..   ..   ..  . . . . . . Kn1 Kn2 Kn3 · · · Knn Dn Fn

2.2 Analysis algorithm of DDA

0

(x−x0 ) 2

(z−z0 ) 2

0

(y−y0 ) 2

the discontinuity properties used in this verification. Fig. 1 shows the initial state, before sliding of the two blocks. The slope angle between the bottom and upper block is arctan(1/2), 26.57 degrees. These two blocks have 4 contacts, which are divided into two categories; the upper two vertex-edge and lower two vertex-edge contacts. Fig. 2 shows the blocks after 100 time steps. Once the centroid of the upper block passed the edge of the bottom block, the upper block fell down while rotating in a counterclockwise direction. There were only two edge-edge contacts in Fig. 2. Fig. 3 shows the z-axis value (height) of the point “a” on block A Table 1.

The algorithm of the three-dimensional DDA is the same as that of the two-dimensional DDA. First, the block generation program generated the block data based on the discontinuities, whether they were fixed or sliding, and the loading point data. The block data, loading data, properties of the blocks, and discontinuity data were then saved. The next step was the contact finding process, which is crucial to discontinuous analysis; for this, a sub-matrix for each single block was calculated by adding or subtracting a normal spring or shear spring according to the contact condition (sliding or fixed). Subsequently, all the contacts were reviewed. If the no penetration and the no tension conditions between each block were not satisfied, the sub-matrices were recalculated until they were satisfied, and the results then saved.

(y−y0 ) 2

  u0 v0    w0 r  1 (z−z0 )   r  2  2   r3    0   · εx  (1)   (x−x0 ) εy    2  εz    γxy γ  yz γzx

Properties of blocks and discontinuities in 3.1.

Spring stiffness Block stiffness Unit mass

2 GN/m 1 GPa 2.7 t/m3

Time step Total steps Poisson’s ratio

0.1 (second) 200 (step) 0.24

3 VERIFICATION Using the three-dimensional DDA program, two cases were used for verification. The first had two blocks with a single sliding face, and the other, a wedge analysis. The unit of length in this chapter is meter (m).

Figure 1.

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Sliding of block A over block B (initial state).

Figure 2.

Sliding of block A over block B (after 100 steps). Figure 4. Deformed block B of wedge sliding analysis (initial state).

Figure 3. Position of block (z-coordinate of point a) as a function of friction angle (after 100 steps).

(shown in Fig. 1) at different friction angles after 50 time steps. For this slope, the z value of “a” was at the initial value of 7.5; this value changed with different friction angles. There was however, no movement when the friction angle was greater than the slope angle. Conversely when the friction angle was smaller than slope angle, the measured point moved down (i.e. z got smaller), and the smaller the friction angle the greater the movement. This is illustrated in Fig. 3 the measured point does not move with friction angle of 26.57◦ or larger but remains constant at a value of 7.5. Thus, this shows that analysis program calculates the sliding between two blocks exactly. 3.2

Wedge sliding

This analysis was performed to determine whether the three-dimensional DDA program could calculate an

Figure 5. Deformed block B of wedge sliding analysis (after 200 steps).

exact value when sliding occurs along two sliding planes. The mechanical properties were the same as those given in table 1, chapter 3.1. Fig. 4 shows the blocks before analysis. The overall block was 10 m on each side and was divided into 4 blocks by two planes whose center points were (0, 5, 5), (0, 5, −1.25) with dips of 40◦ and 60◦ , and dip directions of 130◦ and 200◦ , respectively. The cases in Figs 1–3 were analyzed for a single slope angle whereas the cases in Figs 4–6 were analyzed for two different slope angles. In Fig. 4 all the blocks were fixed except the wedge shaped block B, so only block B could move as the friction angle was changed. This case was also modeled for comparison using 3DEC, the three-dimensional

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Figure 7. Three-dimensional blocks used in toppling analysis. Figure 6. Position of block (z-coordinate of point b) as a function of friction angle (after 200 steps).

discrete element code; this model has the same geometry and properties as the “sliding wedge model” in the manual “Verification Problem” 3DEC version 2.0 (Itasca, 1998). Fig. 5 shows the displaced position of block B after 200 steps with a friction angle of 10◦ . In the initial state there were a total 6 contacts (two sets of 3 vertex-vertex contacts) between block B and the other side-blocks; after sliding down, the contact state changed to a different 6 contacts, including two sets of two vertex-triangle contacts and two edge-edge contacts. In the analysis of Fig. 6, the z-axis value of point “b” of block B is measured against a changing friction angle varied from 31◦ to 36◦ , in order to calculate the lowest friction angle when the wedge begins to slide. The critical friction angle of wedge block calculated theoretically by Hoek & Bray (1979) was 33.36◦ and the calculated value by 3DEC was 33.19◦ , which is about 0.5% smaller than theoretical value. The calculated value (Fig. 6) using the three-dimensional DDA was 33.36◦ , exactly the same as the theoretical value.

Figure 8. Deformed blocks in toppling analysis after 30 seconds.

4 TOPPLING FAILURE SIMULATION To simulate toppling failure the simplified model in Fig. 7 was tested. There were ten blocks on the plate and two 0.1 MN forces were applied at the upper corners of the No. 10 block as in Fig. 7. The forces were applied for 90 seconds out of the total time of 120 seconds using a friction angle of 10◦ . Fig. 8 shows the results after 30 seconds. All the blocks rotated in a counterclockwise direction. Fig. 9 shows the results after 50 seconds. No. 1 and No. 2 blocks began to separate after more rotation. After 80 seconds, No. 1 block was contacting the lower plate. Because the forces were loaded at two points only up until 90 seconds,

Figure 9. Deformed blocks in toppling analysis after 50 seconds.

Fig. 11, which is after 120 seconds, has no external force associated with it. Accordingly No. 2 and No. 3 blocks have separated and No. 10 block is shown sliding down the face of the No. 9 block. The state of ten blocks in Fig. 7 was analyzed changing on angle between the blocks from 20◦ to 23◦ , the

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all contacts change to sliding state and all blocks start moving. This means that 21.8◦ is the critical friction angle for this toppling failure analysis. 5

Figure 10. Deformed blocks in toppling analysis after 80 seconds.

Figure 11. Deformed blocks in toppling analysis after 120 seconds.

Figure 12.

Shi’s discontinuous deformation analysis (DDA) has been developed in many parts, but until now DDA analysis has been based on two-dimensional plain strain or plain stress. Two-dimensional analyses are limited because discontinuities are basically a three-dimensional problem. We developed a three-dimensional discontinuous deformation analysis theory and computer program, to analyse the deformation of blocks made by discontinuous planes in the rock mass. To develop the three-dimensional DDA theory, a first order deformation function was created and the potential energy and sub-matrices of a single block and contacts were obtained using this. In the verification test, the program calculated the two exact theoretical friction angles at the onset of sliding, which were 26.57◦ for the two-block sliding case and 33.36◦ for the wedge sliding case. In the toppling mechanism analysis, the program calculated the block deformation after 30, 50, 80, 120 seconds and critical friction angle and gave good results. REFERENCES Hoek, E. and Bray, J.W., 1979, Rock slope engineering, Institute of Mining and Metallurgy, London. Shi, G.-H. and Goodman, R.E., 1984, Discontinuous deformation analysis, Proceedings of the 25th U.S. Symposium on Rock Mechanics, pp. 269–277. Shi, G.-H. 1988, Discontinuous deformation analysis: a new numerical model for the static and dynamics of block systems, PhD thesis, Civil Eng., University of California, Berkeley. Yeung, M.R., 1991, Application of Shi’s discontinuous deformation analysis to the study of rock behavior, PhD. Dissertation, Civil Eng., U.C. Berkeley Ohnishi, Y., Chen, G. and Miki, S., 1995, Recent development of DDA in rock mechanics, Proceedings of the First International Conference on Analysis of Discontinuous Deformation, Chunghi, Taiwan, pp. 26–47. Cundall, 1998, 3DEC User’s manual, Itasca consulting group.

Stability analysis of toppling blocks.

result is showed in Fig. 12. Fig. 12 shows the change of the contact number which can be fixed or sliding according to the change of the friction angle. Below the friction angle 21.8◦ , all contacts remain fixed state and blocks show no movement, but above that angle,

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CONCLUSION

Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Three-dimensional Discontinuity Network Analysis (TDNA) on rock mass Xiao-Chu Peng & Hong-bo Tang Mid-south Institute of Investigation Design Research, Changsha, Hunan Province, PRC, China

ABSTRACT: TDNA is of the discontinuity mechanics. It is a new three-dimensional discontinuity numerical analysis method for analyzing the structural features of rock mass. TDNA will firstly vividly show the space attitude, location, size and mechanical characteristics of the actual structured planes system (which is fully random but with regularity) in the analyzed zone by using distinct multi-dimensional Montre-Carlo method; secondly, search out the network system of the structural planes; thirdly, analyze the topologic, geometric, kinetic and mechanical characteristics of the system; and finally, display the moving tendency of the rock mass by stepped animation frames. Based on this, the process and the final formation of the failure of the rock mass will be inferred, and the specific zone and depth of the failure of the rock mass will be defined. So, the fairly accurate parameters such as the depth, orientation and anchoring force of the anchorages can be obtained.

1

INTRODUCTION

Structural planes such as joints, fissures, beddings and faults intersecting in the rock mass in versatile direction is an important structural characteristics of rock mass. While existing of a lot of structural planes make the deformation mechanism and the mechanic mechanism of the rock mass have substantial difference from those of the continuous media, the pattern of and the principles for the rock disruption also have large difference form those of the continuous media, and mainly depend on the density, the combination pattern and the mechanical characteristics of the structural planes. Therefore, numerical analysis on the stability of the rock mass requires to accurately express the structural planes in density, dimensions, attitude, location and mechanical characteristics. In this sector, experts as Priest, S.D. & Hudson, J.A. (1)(2)(3) have done much work and comparatively mature conclusions have been made. This paper will set forth the way to establish the mathematical model of the discontinuity system closer to the actual conditions, and judge the stability of the rock mass and provide the supporting parameters for the unstable blocks on the basis of calculation of the internal forces in the system after obtaining the discontinuity system. TDNA method, as described in this paper, supplements the Block Theory and has overcome the following disadvantages of “Block Theory”: 1. As a numerical method of discontinuity mechanics, the accuracy of the formation of the discontinuity

system directly determines the reliability of the analysis results. During the simulation of the discontinuity system by “Block Theory”, the structural planes are attitude-simplified in groups and then are simulated, that undoubtedly results in the inconsistency between the simulation result and the actual random discontinuity system; so, it is naturally difficult to gain final analysis results approximate to the actual results. 2. The Block Theory of “Complete Three Dimensional Analysis” established by Dr. Shi Genhua is, in fact, a combined analysis on trace lines on the three-dimensional structural planes (or polygonal planes). Hence, it is quasi-three-dimensional. Therefore, the Block Theory can only consider the self gravity of each block but not any external force (however, the external forces such as, crustal stress, seepage pressure and ex-system agent, are the decisive factors for stability of rock mass), and neither the internal force of the structural planes system, so that many dangerous factors are ignored and cases such as stability of anti-dip structural planes can not be distinguished. All these result in distortion of the analysis results. 3. Block Theory can not provide the process, extent and degree of the failure as well as the final pattern and the total volume. It can only distinguish the key block, but can not estimate the impacts of the failure of the key block in the rock mass. 4. Block Theory can not provide the supporting forces required for engineering stability, since it only

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considers the self-gravity of each block but can not consider the impacts on the whole system, therefore, the supporting forces provided by Block Theory can not reflect the supporting forces actually required by the rock slope engineering. TDNA has resolved the above problems in a relatively good way. TDNA has stronger pre-processing and postprocessing functions and less input data, showing the designer in vivid three dimensional graphs the output results that make the analysis results visibly clear. The principles of TDNA are introduced as follows. 2 2.1

NETWORK SIMULATION OF STRUCTURAL PLANES Simulation of the attitudes of random structural planes

If the attitudes of structural planes are expressed in pole density graph, its density distribution can hardly be expressed by a simple function. Figure 1 shows the statistical density diagraph of poles of small faults in an adit of a slope works. It is obvious that the distribution of pole points is discontinuous and it’s impossible to get the bivariate function expressing the attitude density distribution by means of fitting. Therefore, the author suggests to gain the subsamples, which are for a relatively accurate simulation of attitude of each fault, by means of the distinct two-dimensional Montre-Carlo method. The distinct two-dimensional Montre-Carlo method is described as follows: Suppose that there is two dimensional variable (x, y) and its probability density function is f(x, y), a ≤ x ≤ b, c ≤ y ≤ d, and its accumulative distribution is F(x, y). In accordance with the definition of probability, there shall be F(b,d) = 1 Step one: calculate the total distribution of x, F1 (x): # d f (x, y)dy F1 (x) = c

inf

F1(t)≥r2n−1

t

where, ζxn is the independent subsample sampled in the nth turn and in accordance with the distribution function F1 (x), and r2n−1 is the (2n − 1)th pseudo-uniform random number, 0 ≤ r2n − 1 ≤ 1 Step three: calculate subsample of y, ζyn ζyn =

inf

F(ζxn,t )≥r2n

Contour diagram of joint poles of insite.

Figure 2. Contour diagram of joint poles by twodimensional sampling.

Step two: calculate the subsample of x, ζxn ζxn =

Figure 1.

t

Subsamples gained in this way form a point couple (ζ xi, ζ yi). The point couples gained from several sampling, i = 1, 2, . . ., n, are independent from each other

and their density distribution follows the distribution function, f(x, y). On the basis of the statistical data as shown in Figure 1, Figure 3 is the density distribution of pole points in accordance with the independent samples and Figure 2 the density distribution of pole points sampled by means of two-dimensional Montre-Carlo method. From comparison of these three diagraphs, it can be concluded that the general distribution by means of two-dimensional Montre-Carlo sampling method has much higher accuracy than that by means of independent sampling.

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structural plane as a plane or a polygonal plane and adjusts the mechanical parameters of the structural plane to compensate the impacts in form. Suppose that structural plane P is a plane disk with a radius of R, a normal vector of n(A, B, C) and a disk center of PC (xC , yC , zC ), then the disk can be expressed as:  M(x, y, z) ∈ P : Ax + By + Cz = D   (x − xC )2 + (y − yC )2 + (z − zC )2 = R 2 D = AxC + ByC + CzC  A2 + B2 + C2 = 1

Figure 3. Contour diagram of joint poles by independent sampling.

After the attitude of the structural plane is illustrated, there yet exist the following factors affecting the structural characteristics of the rock mass: a) density, b) dimensions, c) opening, d) characteristics of the filling and e) location of the structural plane. In the process of numerical analysis, characteristics of the opening and the filling of the fault can be expressed together in coefficients to show its characteristic parameters such as deformation, strength, seepage etc. The above mentioned five parameters have close correlation between each other. If they are sampled in an independent way, undoubtedly the results gained will have great difference from those of the actual cases. Therefore, multi-dimensional MontreCarlo method shall be adopted for sampling. The structural plane system simulated in this way is closer to the actual conditions in density, size, location and mechanical characteristics etc. The determined large faults or the discontinuities that has significant importance for an engineering analysis can be accurately defined in this method. 2.2

Expression of the structural plane

Barton and Long (4) think that the boundary of a structural plane can be deemed as a circle or an oval. When a structural plane develops to intersect with an earlier structural plane, it will not develop further more since the strain energy is be fully released. Therefore, the simulated structural plane is a circle at the very beginning but will not have a final boundary as a circle. A structural plane is three-dimensional in form, but its thickness is very small in comparison with its plane dimensions. In simulation, the author deems the

For convenience of calculation, the periphery of the said disk can be simplified in straight-line segments. For example, if a line segment is corresponding to a center angle of π/6, the start point of the segment is pr−1 (xr−1 , yr−1 , zr−1 ), and the terminal point is pr (xr , yr , zr ), r = 1, 2, . . ., 12, then the equation for this line can be written as x − xr−1 y − yr−1 z − zr−1 Lr : = = = tr xr − xr−1 yr − yr−1 zr − zr−1 (0 ≤ tr ≤ 1) The equations of 12 straight-line segments are calculated. In turn, the boundary of the disk can be approximately determined. After the random structural plane being calculated by the said method, the intersection with each other of the structural planes can be judged. 2.3 Judging the correlation of the structural planes Suppose that the coordinates of the center points of structural plane Pi and Pj are Pci (xci , yci , zci ) and Pcj (xcj , ycj , zcj ) respectively, the normal vectors Ni (Ai , Bi , Ci ) and Nj (Aj , Bj , Cj ), the radii Ri and Rj . The vector of the intersection line of the two planes is   i j k Nij = Ni × Nj = Ai Bi Ci  Aj Bj Cj = Aij i + Bij j + Cij k where Aij = Bi Cj − Bj Ci , Bij = Ci Aj − Cj Ai , Cij = Ai Bj − Aj Bi . On the basis of a point, Dij (xij , yij , zij ), on the intersection line of the two planes Pi and Pj , the equation of the intersection line of planes (or extended planes) Pi and Pj is gained as follows: x − xij y − yij z − zij = = = t0 Aij Bij Cij (−s∞ < t0 < ∞)

Lij :

Lij is an infinite long line. First use Lij to make the judge with the periphery straight line segments of plane Pi , then the segment Lij (of Lij ) within Pi can

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be gained. Consequently, make the judge with line Lij and the periphery straight lines segments of plane Pj . There can be three cases which follows: 1. With no intersection point: It illustrates that Lij can’t intersect the periphery line segments of Pj and there are two possibilities: the first is that Lij is completely out of Pj , planes Pi and Pj can’t intersect; the second is that Lij is completely in the domain of Pj and the intersected part of the planes Pi and Pj is Lij . 2. With two intersection points: it illustrates that the two planes must intersect with two possibilities: the first is that Lij is within the very scope of the periphery lines of Pj and the actual intersect line is Lij ; the second is that Lij has two ends beyond the domain of Pj and middle part is within the domain of Pj , and the actual intersected line of the two planes is the middle part. 3. With only one intersection point: It illustrates that the planes must intersect but the intersection line is only a part of Lij .

Figure 5.

Zone cut by one discontinuity.

Figure 6.

Zone cut by two discontinuities.

Figure 7.

Zone cut by three discontinuities.

Thus, it can be determined whether the two planes intersect or not. If they intersect, the specific location and scope of the intersection line can also be gained. Cap all the planes with one another in the system, for each structural plane, the following information can be gained: 1. Quantity, Nos. and the intersecting scope of the other planes intersecting with it. 2. Trace lines of all the planes which intersect with it, and the corresponding network relationship. 3. Mechanical characteristics and seepage characteristics of the structural planes intersecting with it. After these steps, the correlations of the structural plane system are clear and structural plane network system is formed, and, the slope outlook can be clearly expressed. The following figures (Figures 4–9) show the situation of a slope cut by structural planes.

Figure 4.

Model of analyzed zone.

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of information as follows: 1. Geometrical information such as space location, dimensions, direction etc. of each subset constituting the block. 2. Topological information expressing the relationship of the subsets.

Figure 8.

Zone cut by four discontinuities.

Figure 9.

Zone cut by five discontinuities.

3

Geometrical information and topological information are necessary for complete expression of a polyhedron. In the past, the method usually used for gaining the geometrical information was lower hemisphere equal area projection in combination with scaled solid project. For a simple polyhedron, this method is simple and effective. However, for any arbitrary polyhedron with over five boundary planes, the geometrical and topological information is pretty hard to gain. Furthermore, such an arbitrary polyhedron often occurs in an actual project. Therefore, the author initiates the searching method of a polyhedron as from the plane to the edges and then from the side to the planes. The method can make geometrical and topological information of any polyhedron be easily gained. Each plane has many trace lines intersected with other planes. Some of the trace lines can form closed loops but others can not. While the finite space plane enclosed by a closed loop must be a boundary plane of the polyhedron, an edge of the closed loop must be an edge of the polyhedron and nodes of the closed loop must be vertexes of the polyhedron. Therefore, after setting a closed loop of a structural plane Pi , and a point Q, in a given semi-space of this loop, the another plane Pj , which intersects Pi , can be sought out on the basis of edges of the loop, sand the Pj ’s semi-space in which the block is can be defined by point Q, so on and so forth, all the information of the block can be gained.

SEARCHING OF STRUCTURAL BODIES

A structural body in actual projects is a polyhedron with very complicated pattern. A polyhedron correctly defined shall not be self-intersecting and shall be directed and shall have no gap between each two intersected planes. Let H express finite number of three-dimensional structural planes, semi-space subdivision can be done by H. The set of the subdivided semi-spaces is a compound of the polyhedron, let’s say, A(H). Obviously, subsets ofA(H) are the points, the edges, planes and the polyhedron. A polyhedron is the sum of the directed boundary planes set and the inbody points set. The subsets of a polyhedron are points, sides and planes. Therefore, a polyhedron can be expressed by two parts

1. Coordinates of the point (on one side of the block) which all the polygonal planes in the block and constituting the block point to. 2. Array of the plane nos. 3. Number of the semi-space (of the plane) in which the block is. 4. Quantity of the edges of the plane. 5. Coordinates of each acnode. To avoid any omission in the block searching, a check matrix, ITEST(i, j, k), shall be established. The matrix expresses the searching case of the blocks in No. k semi-space of No. j closed loop of No. i plane (k = 1 expresses the upper semi-space of the loop and k = 2 the lower one). Before searching, let all array elements of ITEST be 0. If the blocks in No. k semispace of No. j closed loop of No. i plane have been sought out. Let ITEST(i, j, k) = 1. And do one check on ITEST after each block being sought out. If it is found that elements of an array are 0, using the plane,

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Figure 10. The main blocks in the structural system.

loop and semi-space expressed by this array element as the first data of the new block to conduct the searching. When all ITEST elements are equal to 1, it means that the block searching has been finished and the block system is formed. Figure 10 shows the main blocks in a slope cut by five structural planes. 4

CALCULATION OF INTERNAL FORCES OF THE STRUCTURAL PLANES NETWORK SYSTEM

After formation of the block system, circulating iteration is done on the system under external forces and self gravities so as to obtain the force-field of the whole system, and then force balancing analysis is made for individual blocks in the system. If any block can not be naturally balanced, a stability failure occurs. After the block falls, a new system will form, consequently

the force-field will redistribute to form a new balance, so on so forth, balance and failure of new system, rebalance and refailure, until a new fully-stable and balanced status forms. The force transmission calculation by TDNA method is greatly different from that by means of distinct element method. TDNA method deems that the system under study is transiently stable. The deformation of the system has been settled in the long geological history and the system is in stable conditions. Therefore, the contact relationship of all the blocks keeps unchangeable and the contact forces between in the blocks are static. The block is a rigid body that is unbreakable. Hence, the unbalanced forces acted on each block are balanced by the counter forces on the boundary planes. Magnitude and distribution of the counter forces can be gained by means of establishing three-axis force balancing and force couple balancing. Suppose that block A contacts elastically with blocks B, C, D, . . . surrounding it, block A must shift and rotate while it is acted by force F and force couple M. The shift value and the rotating angle can be gained by establishing balancing equations on the basis of the contact cases of block A with its surrounding blocks as well as the normal and tangential stiffness of the contact surfaces. On the basis of the gained shift value of the block and rotating angle of the block round its gravity center, the displacement increment of each acnode of the polyhedron, δ, can be solved. Superimpose δ on the locations of each acnode of block A before movement, the updated location of the form center, boundary planes and acnodes of block can be solved. Block A at the updated location will produce unbalancing forces and force couples on the surrounding blocks B, C, D, . . . By the same way for block A, the new locations after balancing of the block B, C, D, . . . can be solved. The calculation is done so on and so forth in such a pattern, until the locations of all the blocks in the system are in the balanced location and unbalanced forces will no longer exist. If the force system of one block can not be balanced, the block fails and consequently the systematic structure and form change and a new regulation process of unbalanced forces occurs. Calculation is done round and round in this way until new failure no longer occurs in the damaged system. The process of the block failure will be showed in three-dimensional blanking graphs by means of normal axonometric projection. The graphs have strong sense of three dimensions and clear dimensional scales. The drawings also show the location, shape, volume, supporting force to be provided and the failure pattern, which can be directly employed for the project site. The following figure (Figures 11–12) shows the slope shape after failure of two blocks.

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method, the probability distribution of the critical blocks in the slope can be sought out. Provided the statistical information is reliable, the method can predict and judge, in a microscopic view, the slope stability and the failure pattern as well as the chain reaction process of the collapse in a significant accuracy. This method can describe the form of any complicated block, including the concave blocks, as well as make accurate analysis on single dangerous block at the project site and provide corresponding supporting parameters.

REFERENCES Figure 11.

Figure 12. failure.

5

First falling block and zone outline after failure.

Second falling block and zone outline after

CONCLUSIONS

In the random three-dimensional structural plane network (of the rock slope) generated by statistical

Priest S.D. & Hudson J.A. Estimation of discontinuity spacing and trace length using scanline surveys. Int. J. Rock Mech. Mine Sci. and Geomech. Abstra. 18, 183–197 (1981). Sen Z. & Kazi A. Discontinuity Spacing and RQD Estimates from Finite Length Scanline. Int. J. Rock Mech. Mine Sci. and Geomech. Abstra. 21, 203–212 (1984). Pahl P.J. Estimation the Mean Length of Discontinuity Traces. Int. J. Rock Mech. Mine Sci. and Geomech. Abstra. 18, 221–228 (1981). Goodman R.E. & Shi, GenHua. Block Theory and its Application to Rock Engineering Prentice Hall INC 1985. Shi GenHua & Goodman R.E. The Key Blocks of Unroolled Joint Trace in Developed Maps of Tunnel Walls. Int. J. for Num. and And. Methods Geo. Vol. 13, No. 2, Mar–Apr. 1989. Hoek E. Methods for the Rapid Assessment of the Stability of Three-dimensional Rock Slopes. Q. JL Engeng. Geol. 6(3) (1973). Lin D. & Fairhurst C. Static Analysis of the Stability of Three-dimensional Blocky Systems Around Excavations in Rock. Int. J. Rock Mech. Mine Sci. and Geomech. Abstra. Vol. 25, No. 3, 139–147 (1988).

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