Development of Stability Evaluation Methods for Soil- Title Masonry Structure Interactive Problems and Application to Historic Structures( Dissertation_全文 )
Author(s) Hashimoto, Ryota
Citation 京都大学
Issue Date 2017-03-23
URL https://doi.org/10.14989/doctor.k20327
Right 許諾条件により本文は2017-10-01に公開
Type Thesis or Dissertation
Textversion ETD
Kyoto University Development of Stability Evaluation Methods for Soil-Masonry Structure Interactive Problems and Application to Historic Structures
2017
Ryota HASHIMOTO
Abstract
Many historic masonry structures exist that have collapsed due to deformation and failure of their foundation ground. In order to achieve rational restoration of th ese structures, stability evaluation methods considering the mechanical interaction between soils and masonry structures must be employed. The main purpose of this thesis is to develop a numerical method that can simulate the deformation of composite structures constructed from both soil and masonry stones. Additionally, the mechanical behaviors of masonry structure foundations have been studied in the proposed method, and a simplified design scheme based on the numerical simulation results has been developed for application in the practical restoration process.
A numerical method to solve the interaction problems between soil foundation and masonry structures has been newly developed via the introduction of elasto-plastic constitutive models to coupled Numerical Manifold Method and Discontinuous Deformation Analysis (NMM-DDA), which is a discontinuum-based numerical method. Furthermore, to increase the calculation accuracy and avoid volumetric locking, node-based uniform strain element has also been introduced to the NMM. To check the verification of the newly developed elasto-plastic NMM-DDA, cantilever bending problem and bearing capacity problem of strip footing were solved numerically and compared with theoretical solutions.
Subsequently, the elasto-plastic NMM-DDA was applied to deformation analysis of an actual masonry building of the Angkor ruins in Cambodia deteriorated due to deformation of the foundation soil mound. The mechanical behaviors of in-situ soils were modeled using an elasto-plastic constitutive model based on corresponding laboratory test results; additionally, analysis reflecting detailed structural conditions was performed. To discuss the applicability of the numerical method, the simulation results were compared with the field observation. A series of parametric studies on the initial density of the man-made mound was also conducted to investigate the influence of the compaction quality of the foundation ground on the stability of the masonry structures.
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Finally, a series of numerical experiments to investigate the bearing capacity of the masonry platform, which is commonly used in the construction of Angkor monuments, was performed using elasto-plastic NMM-DDA. The stability analyses of the platforms with different structural features were executed, and the ultimate bearing capacity and failure modes/mechanisms were investigated from the simulated results. A simplified method to estimate the ultimate bearing capacity of the masonry platform was also formulated based on the limit equilibrium considering the failure mechanisms revealed by the numerical simulations. The ultimate load estimated with the simplified analytical method was compared with the numerical simulation results, and the validity of the simplified analytical method was confirmed. In addition, the design scheme of the masonry structure foundation utilizing the proposed method in a practical restoration process has also been suggested.
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Acknowledgements
The present study was conducted at the Geofront System Engineering Laboratory in the Department of Urban Management of Kyoto University from 2011 to 2017. The work could not have been achieved without the support and assistance from professors, friends, and my family. I would like to express my gratitude to them.
Primarily, I would like to express my deepest gratitude to my supervisor, Professor Mamoru Mimura, for his valuable advice and encouragement throughout this research study. His suggestions based on deep insights and wide experiences have greatly helped me to complete the present study. I am very grateful to Professor Mimura for giving me the opportunity to study under his direction, and proud of having been his student.
I would like to express my sincere appreciation to the members of my thesis committee, Professor Hiroyasu Ohtsu and Associate Professor Yosuke Higo, for their thorough and helpful reviewing. Their constructive comments and discussions definitely helped me to enhance the quality of this thesis.
It is also a pleasure to express my deep gratitude to Associate Professor Tomofumi Koyama of Kansai University, who was my original supervisor, and Associate Professor Mamoru Kikumoto of Yokohama National University. They have given me continuous and enormous support since I began this research in 2011. The research life with them has become the origin of my aspiration to become a researcher, and I could never have completed the present study without their direction and advice.
I would next like to express my appreciation for Emeritus Professor Yuzo Ohnishi of Kyoto University, Professor Satoshi Nishiyama of Okayama University (former Associate Professor in the Geofront System Engineering Laboratory), Dr. Takeshi Sasaki at Suncoh Consultants Co., Ltd., and Dr. Shigeru Miki at Kiso-Jiban Consultants Co., Ltd.. They have offered me much advice and the opportunity to join the research community for analysis of discontinuous deformation. The discussions with them have helped immensely in the development of the present simulation code.
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I want to express my gratitude to Professor Kiyoshi Kishida of Kyoto University for his continuous encouragement. He has given me many opportunities to learn the forefront of the rock mechanics, and I was thus able to broaden my knowledge.
I would like to show my appreciation to Dr. Takao Yano, the technical officer, for his persisting considerate guidance on soil testing in spite of my clumsy works.
I am grateful to Dr. Mai Sawada, who is now Assistant Professor of the Geofront System Engineering Laboratory, for her advice during the compilation step of the thesis.
I am continually grateful to the members of the geotechnical and structural team of JAPAN-APSARA Safeguarding Angkor (JASA), Dr. Yoshinori Iwasaki, Dr. Mitsuharu Fukuda, Emeritus Professor Koichi Nakagawa and Associate Professor Tsuyoshi Haraguchi of Osaka City University, Emeritus Professor Masato Araya of Waseda University, and Assistant Professor Shunsuke Yamada of Yasuda Women’s University for their valuable advice regarding proper restoration of the Angkor monuments. I also deeply appreciate Emeritus Professor Takeshi Nakagawa of Waseda University, a co-director of JASA, Dr. Ichita Shimoda, and Mr. Mitsumasa Ishizuka, the technical assistant of JASA, for their tremendous support in organizing the field survey of the monuments. I have always been impressed with their passion and dedication for the Angkor ruins.
I want to show my gratitude to Mr. Toru Saito, a former student at Yokohama National University, who is now at Central Nippon Expressway Company Ltd., for his cooperation in soil testing and constitutive modeling.
I would also like to show my appreciation to my friends in the doctoral program, Mr. Toshifumi Akaki, Mr. Yuma Daito, Dr. Risa Matsumoto, Dr. Tomohiro Tanaka, and Mr. Kyohei Noguchi. I spent joyful time with them and got much motivation for the research work.
I would like to express my special thanks to Ms. Reiko Tomiyama for her continuous support. The time spent with her was a great encouragement for me to complete this thesis.
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Special acknowledgement is extended to the members of the Geofront System Engineering Laboratory. Particularly, I wish to show my gratitude to Mr. Yuki Ohta, a former student who is now at Tokyo Gas Co., Ltd., for his guidance on the numerical analysis during the initial phase of this research. I also express my appreciation to Mr. Kohei Kawakami, a Master course student, for his great contribution in extending the numerical method. Of course, I also want to acknowledge all the past and present members of the laboratory. They have made my research life supremely meaningful and pleasurable. My six years in the laboratory will be an unforgettable memory.
This work has been supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for JSPS Fellows [Subject No. 14J00077]. I am deeply grateful for their financial support.
Finally, I want to thank my parents, Sumito and Yoko, my sister, Tomomi, and all my family for their love and support throughout my life, which has allowed me to advance to the doctoral course.
January 2017 Ryota HASHIMOTO
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Table of Contents
Page
List of Tables, Figures, and Photos xiii
Chapters 1. Introduction 1
References ······································································································ 4
2. Literature Review on Stability Evaluation Methods of Masonry Structures 5
2.1 Restoration projects and geotechnical treatments in the Angkor ruins ··························· 5
2.1.1 Prasat Suor Prat N1 tower ·········································································· 6
2.1.2 Southern library of Bayon ·········································································· 9
2.1.3 Problems with previous restoration designs ···················································· 10
2.2 Numerical methods for stability evaluation of masonry structures ····························· 11
2.3 Scopes and structure ··················································································· 14
References ···································································································· 16
3. Method of Deformation Analysis for Composite Structures of Soils and Masonry Stones 19
3.1 Governing equations of continuum kinematics and mutual contact problem ·················· 20
3.2 Formulation of the elasto-plastic NMM-DDA ····················································· 23
3.2.1 Displacement field approximation in DDA and NMM ········································ 24
3.2.2 Contact detection and discretization of contact penalty term ······························· 29
3.2.3 Discretized form of the governing equations ·················································· 31
3.3 Volumetric locking and implementation of node-based uniform strain element in NMM ··· 32
3.3.1 Outlines of NB element ··········································································· 33
3.3.2 Modifications for NMM··········································································· 36
3.4 Numerical examples ··················································································· 38
3.4.1 Cantilever beam ···················································································· 38
3.4.2 Bearing capacity of strip footing under vertical load ········································· 41
3.4.3 Bearing capacity of strip footing under eccentric vertical load ····························· 46
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3.5 Summary ································································································ 50
References ···································································································· 51
4. Deformation Analyses of an Existing Masonry Structure in the Angkor Ruins 53
4.1 Overview of Prasat Suor Prat N1 tower ···························································· 54
4.2 Constitutive modeling of foundation soils ························································· 57
4.2.1 Man-made compacted mound soil ······························································· 57
4.2.2 Natural layer soil ·················································································· 60
4.2.3 Constitutive modeling with subloading Cam-clay model ···································· 61
4.3 Deformation analysis of Prasat Suor Prat N1 tower ·············································· 64
4.3.1 Analytical conditions ·············································································· 64
4.3.2 Simulation results ·················································································· 68
4.4 Parametric study on the construction quality of the platform mound ·························· 76
4.4.1 Analytical conditions ·············································································· 76
4.4.2 Simulation results ·················································································· 77
4.5 Summary ································································································ 85
References ···································································································· 86
5. Numerical Experiment on Bearing Capacity Characteristics of Masonry Platform Structures 87
5.1 Structural parameters of masonry platform ························································ 88
5.2 Analytical conditions ·················································································· 88
5.3 Simulation results ······················································································ 91
5.3.1 Influence of the number of steps ································································ 93
5.3.2 Influence of the overlapping width ······························································ 96
5.3.3 Influence of the stone thickness ································································ 101
5.4 Summary ······························································································· 105
References ··································································································· 106
6. Simplified Estimation Method of Ultimate Bearing Capacity for Masonry Platform Structures 107
6.1 Simplified estimation method of bearing capacity for masonry platforms: Proposal ······· 108
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6.1.1 Assumptions of failure mechanisms ···························································· 108
6.1.2 Derivation of the bearing capacity equation ·················································· 109
6.1.3 Implementation of the proposed method ······················································· 112
6.2 Validation of the proposed method ································································· 115
6.3 Implementation in the restoration process························································· 118
6.4 Summary ······························································································· 121
References ··································································································· 121
7. Conclusions and Future Studies 123
Appendix
A. Spatial Discretization by NMM-DDA 129
B. Temporal Discretization in NMM-DDA 131
References ··································································································· 132
C. Outlines of subloading Cam-clay model 133
C.1 Introduction ···························································································· 133
C.2 Modified Cam-clay model ··········································································· 134
C.3 Introduction of the subloading surface concept ·················································· 139
References ··································································································· 142
D. Treatment of material boundaries in node-based NMM 145
References ··································································································· 147
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List of Tables, Figures, and Photos
Tables
Table 3.1 Mesh information. Table 3.2 Material properties of the ground and footing. Table 3.3 Material surface properties of the ground and footing. Table 3.4 Ultimate bearing capacity of strip footing under vertical eccentric load for varying eccentricities. Table 4.1 Initial void ratio of mound soil specimens for oedometer test. Table 4.2 Initial void ratio of natural layer soil specimens for oedometer test. Table 4.3 Material parameters of each soil used in the subloading Cam-clay model. Table 4.4 Material properties of the laterite block. Table 4.5 Properties of the material interface. Table 4.6 Estimated uneven settlement and inclination for each case. Table 5.1 Material properties for platform bearing capacity analyses. Table 5.2 Material surface parameters for platform bearing capacity analyses. Table 5.3 List of simulated structural conditions. Table 5.4 Ultimate bearing capacity for varying number of steps (h/B = 0.40, l/B = 0.50). Table 5.5 Ultimate bearing capacity for varying l/B (h/B = 0.40, n = 6). Table 5.6 Ultimate bearing capacity for varying h/B (l/B = 0.50, n = 6). Table 6.1 Minimum value of P and corresponding d for each layer.
Figures
Figure 1.1 Schematic figure of the platform structure of Angkor monuments. Figure 2.1 Bearing capacity failure in the platform (added description to reference 2-1)). Figure 2.2 Failure mechanisms considered in the restoration of the PSPN1 (after 2-1)): (a) circular slip through the natural layer, (b) linear slip line in the man-made layer. Figure 2.3 North-south cross section of the southern library 2-2): (a) mode of the measured settlement, (b) inner structure of the platform. Figure 2.4 Joint element. Figure 2.5 Combined FEM-DEM. Figure 2.6 Contact with spring-dashpot model. Figure 2.7 Discretization in NMM-DDA. Figure 2.8 Structure of the thesis and corresponding subjects. Figure 3.1 Deformation problem for a single continuum. Figure 3.2 Contact problem for two continua. Figure 3.3 Modeling in NMM-DDA. Figure 3.4 Polygonal approximation and variables of displacement and deformation in DDA. Figure 3.5 Approximation scheme in NMM: (a) physical mesh of the continuum body, (b) mathematical cover, and (c) manifold element.
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Figure 3.6 Polygonal physical mesh and 3-node triangular mathematical mesh produced with the hexagonal covers. Figure 3.7 Weight assigned by the hexagonal cover: (a)center of the hexagonal cover and (b)weighting function of the cover. Figure 3.8 Contact determination: (a) before contact and (b) after contact. Figure 3.9 Node-based uniform strain element. Figure 3.10 Equal division of CST element. Figure 3.11 Element with irregularly shaped physical domain in NMM. Figure 3.12 Methods of determining the element area rate: (a) conventional method and (b) using the weighting functions of the mathematical covers. Figure 3.13 Cantilever beam bending problem. Figure 3.14 Numerical meshes considered in the simulation: (a) nodes are located on the physical boundaries of the beam and (b) nodes are not located on the physical boundaries of the beam. Figure 3.15 Distribution of the deflection (DOF=4402). Figure 3.16 Error in tip deflection. Figure 3.17 Analytical domain, boundary conditions, and the finest mesh applied to the bearing capacity problem of a footing: (a) entire model and (b) locations of loading point. Figure 3.18 Load-displacement relationship for each DOF: (a) CST element and (b) NB element. Figure 3.19 Error of the ultimate bearing capacity. Figure 3.20 Collapse mechanism represented with the deviator strain contours for the finest mesh of 9968 DOFs: (a) CST element and (b) NB element. Figure 3.21 Bearing capacity problem of the strip footing under eccentric vertical load. Figure 3.22 Load-displacement relationship for each eccentricity. Figure 3.23 Final distribution of deviator strain for each eccentricity: (a) e = 0.2, (b) e = 0.4, (c) e = 0.6, and (d) e = 0.8. Figure 3.24 Relationship between normalized eccentricity e/B and the ultimate vertical load. Figure 4.1 Location of Prasat Suor Prat N1 tower4-2). Figure 4.2 Northwest view of Prasat Suor Prat N1 tower after restoration 4-1). Figure 4.3 Schematic figure of inner structure of the platform. Figure 4.4 Observed damages of Prasat Suor Prat N1 tower before restoration (west side view) (added explanation to reference 4-1)). Figure 4.5 Observed deformation of the platform (north-south cross section) 4-1),4-3).
Figure 4.6 e-logσv relationship of the mound soil. Figure 4.7 Stress-strain relationship of the soil for compacted mound obtained from the drained triaxial compression test. Figure 4.8 Sampling location of the natural layer soil (Map data: Google, CNES/Astrium).
Figure 4.9 e-logσv relationship of the natural layer soil. Figure 4.10 Simulated results of oedometer test on man-made mound soil. Figure 4.11 Simulated results of triaxial test on man-made mound soil. Figure 4.12 Simulated results of oedometer test on the natural layer soil.
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Figure 4.13 Analytical domain of the Prasat Suor Prat N1 tower. Figure 4.14 Construction process assumed to estimate the initial stress: (a) Step 1, (b) Step 2, (c) Step 3, (d) Step 4, (e) Step 5, (f) Step 6. Figure 4.15 Initial stress conditions: (a) mean stress, (b) deviator stress. Figure 4.16 Final distribution of the settlement. Figure 4.17 Settlement at the monitoring points. Figure 4.18 Inclination of the tower. Figure 4.19 Enlarged view of Figure 4.17 around the window. Figure 4.20 Final distribution of the deviator strain. Figure 4.21 Mechanical behaviors around the northern platform: (a) final distribution of the settlement, (b) final distribution of the deviator strain, (c) final distribution of the deviator stress. Figure 4.22 Mechanical behaviors of element 5016: (a) stress path, (b) e-logp relationship. Figure 4.23 Mechanical behaviors of element 5246: (a) stress path, (b) e-logp relationship. Figure 4.24 Mechanical behaviors of element 5619: (a) stress path, (b) e-logp relationship. Figure 4.25 Deformation of element 3443: (a) element location, (b) evolution of the strains. Figure 4.26 Mechanical behaviors of element 3443: (a) stress path, (b) e-logp relationship. Figure 4.27 Final distribution of the volumetric strain (compression is positive) . Figure 4.28 Comparison of the e-logp relationship between elements 2997 and 3043. Figure 4.29 Final distribution of the vertical stress. Figure 4.30 Initial void ratio distribution: (a) Case 1, (b) Case 2 and (c) Case 3. Figure 4.31 Final distribution of the settlement for varying initial void ratios of the mound: (a) Case 1, (b) Case 2, (c) Case 3. Figure 4.32 Joint openings around the window: (a) Case 1, (b) Case 2, (c) Case 3. Figure 4.33 Final distribution of the volumetric strain (compression is positive) for varying initial void ratios of the mound: (a) Case 1, (b) Case 2, (c) Case 3. Figure 4.34 Comparison of simulated e-logp curves: (a) element 2997, (b) element 3043. Figure 4.35 Final distribution of the deviator strain around the northern platform for varying initial void ratios of the mound: (a) Case 1, (b) Case 2, (c) Case 3. Figure 4.36 Comparison of mechanical behaviors of element 5016 for varying initial void ratios :
(a) stress path, (b) εd-q relationship, (c) e-logp relationship, (d) εd-εv relationship. Figure 4.37 Comparison of mechanical behaviors of element 5246 for varying initial void ratios:
(a) stress path, (b) εd-q relationship, (c) e-logp relationship, (d) εd-εv relationship. Figure 4.38 Comparison of mechanical behaviors of element 5619 for varying initial void ratios:
(a) stress path, (b) εd-q relationship, (c) e-logp relationship, (d) εd-εv relationship. Figure 5.1 Structural parameters of masonry platform. Figure 5.2 Example of analytical model for platform bearing capacity analyses (h/B = 0.40, l/B = 0.50, n = 6). Figure 5.3 Load-displacement curve (h/B = 0.40, l/B = 0.50, n = 6). Figure 5.4 Final distribution of the deviator strain (h/B = 0.40, l/B = 0.50, n = 6). Figure 5.5 Final distribution of the horizontal displacement (h/B = 0.40, l/B = 0.50, n = 6).
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Figure 5.6 Vertical load applied on each stone at the final state (h/B = 0.40, l/B = 0.50, n = 6). Figure 5.7 Load-displacement curves for varying number of steps (h/B = 0.40, l/B = 0.50). Figure 5.8 Final distribution of deviator strain for varying number of steps (h/B = 0.40, l/B = 0.50): (a) n = 6, (b) n = 5, (c) n = 4. Figure 5.9 Final distribution of horizontal displacement for varying number of steps (h/B = 0.40, l/B = 0.50): (a) n = 6, (b) n = 5, (c) n = 4. Figure 5.10 Vertical load applied on each stone at the final state (h/B = 0.40, l/B = 0.50): (a) n = 6, (b) n = 5, (c) n = 4. Figure 5.11 Load-displacement curves for varying l/B (h/B = 0.40, n = 6). Figure 5.12 Relationship between l/B and the ultimate bearing capacity (h/B = 0.40, n = 6). Figure 5.13 Final distribution of deviator strain for varying l/B (h/B = 0.40, n = 6): (a) l/B = 0.50, (b) l/B = 0.60, (c) l/B = 0.65, (d) l/B = 0.75, (e) l/B = 0.90. Figure 5.14 Final distribution of horizontal displacement for varying l/B (h/B = 0.40, n = 6): (a) l/B = 0.50, (b) l/B = 0.60, (c) l/B = 0.65, (d) l/B = 0.75, (e) l/B = 0.90. Figure 5.15 Vertical load and load distribution rate of each stone for varying l/B (h/B = 0.40, n = 6): (a) l/B = 0.50, (b) l/B = 0.60, (c) l/B = 0.65, (d) l/B = 0.75, (e) l/B = 0.90. Figure 5.16 Load-displacement curves for varying h/B (l/B = 0.50, n = 6). Figure 5.17 Relationship between h/B and the ultimate bearing capacity (l/B = 0.50, n = 6). Figure 5.18 Final distribution of deviator strain for varying h/B (l/B = 0.50, n = 6): (a) h/B = 0.30, (b) h/B = 0.40, (c) h/B = 0.50, (d) h/B = 0.60. Figure 5.19 Final distribution of horizontal displacement for varying h/B (l/B = 0.50, n = 6): (a) h/B = 0.30, (b) h/B = 0.40, (c) h/B = 0.50, (d) h/B = 0.60. Figure 5.20 Vertical load and load distribution rate of each stone for varying h/B (l/B = 0.50, n = 6): (a) h/B = 0.30, (b) h/B = 0.40, (c) h/B = 0.50, (d) h/B = 0.60. Figure 5.21 Location of the center of the slip circle for varying h/B (l/B = 0.50, n = 6): (a) h/B = 0.30, (b) h/B = 0.40, (c) h/B = 0.50, (d) h/B = 0.60. Figure 6.1 Assumed failure mechanisms of the platform. Figure 6.2 Load distribution rate of the jth stone from the top. Figure 6.3 Parameter of the center coordinates of the slip circle. Figure 6.4 Relationship between d and bearing capacity for each layer (B = 1.0m, h/B = 0.4， l/B = 0.5，n = 6): (a) i = 2, (b) i = 3, (c) i = 4, (d) i = 5, (e) i = 6. Figure 6.5 Constraint conditions on the location of the center of the slip circle: (a) condition for y coordinate, (b) condition for x coordinate. Figure 6.6 Comparison of the ultimate load estimated using the simplified method and NMM-DDA: (a) h/B = 0.20, (b) h/B = 0.30, (c) h/B = 0.40, (d) h/B = 0.50, (e) h/B = 0.60. Figure 6.7 Error map of the ultimate bearing capacity. Figure 6.8 Comparison of failure mechanisms of the simplified method and NMM-DDA: (a) (h/B, l/B) = (0.20, 0.60), (b) (h/B, l/B) = (0.40, 0.65), (c) (h/B, l/B) = (0.50, 0.75), (d) (h/B, l/B) = (0.60, 0.90). Figure 6.9 Flowchart of the restoration process in the Angkor ruins.
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Figure 6.10 Example of the relationship between the mound soil cohesion, ultimate bearing
capacity Pult, and safety factor Fs. Figure C.1 Water content ratio contours (after Henkel C-8)). Figure C.2 Critical state of normally consolidated soil: (a) p’-q plane, (b) p’-v plane. Figure C.3 Critical state line on lnp’-v plane. Figure C.4 Schematic figure of Eqs. (C.9) and (C.10). Figure C.5 Schematic figure of e-lnp’ relationship of the overconsolidated soil: (a)actual soil, (b)modified Cam-clay model. Figure C.6 Compression of overconsolidated soil. Figure C.7 Schematic figure of subloading surface. Figure D.1 Material boundary treatment in node-based FEM (after Kurumatani et al. D-2)). Figure D.2 Material boundary treatment of NMM.
Photos
Photo 1.1 South sanctuary of Western Prasat Top 1-3). Photo 2.1 Prasat Suor Prat N1 tower 2-1). Photo 2.2 Southeast view of the southern library of Bayon (before restoration) 2-2). Photo 4.1 Test construction of the compacted soil mound: (a) soil compaction, (b) after completion4-4).
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Chapter 1 Introduction
Today, there are many historic remains or cultural properties as represented by UNESCO World Heritage, and they include a large number of structures that retain important values in civil and/or architectural engineering history. Among them, the masonry structure, which uses stones or rocks as the main material, is one of the most popular architectural styles. Masonry structures, such as the pyramids in Egypt and Machu Picchu in Peru, have been constructed throughout all ages and all over the world. Even in Japan, there are numerous masonry structures such as stone huts in the tumulus, and masonry castle walls. Stone material was often used in these ancient construction fields because of its durability, fire resistance, and scenic beauty. However, many structures have deteriorated due to natural and/or artificially derived factors persisting over hundreds or thousands of years, so they require restoration and conservation. Because a number of them are facing geotechnical problems of their foundations and/or backfill soils, investigations into suitable geotechnically-sourced restoration methods are currently in high demand.
An example suffered by geotechnical problems is the Angkor ruins 1-1),1-2), a world cultural heritage in the Kingdom of Cambodia. The Angkor ruins is the ancient capital of the Khmer Empire (9–15 Century), which consists of numerous masonry buildings, including the famous tourist site, Angkor Wat. The typical structural feature of Angkor monuments is illustrated in Figure 1.1. The structures usually comprise a platform and an upper masonry structure. The platform was constructed from man-made compacted
1 soil mound with stones as the foundation, and the upper structure was constructed by stacking stones on the platform without joint bonding, a method termed “dry masonry”. Since the weight of the stones is sustained by the soil mound and natural ground, the deformation and failure of the foundation ground are direct consequences of inclination of the upper structure and the collapse of the masonry stones and joint openings (see Photo 1.1). As an example, during restoration construction of the Prasat Suor Prat N1 tower performed by the Japanese government team for Safeguarding Angkor (JSA) 1-1), dismantling and reconstruction of the entire building, including the platform, were performed because the tower inclined due to large, uneven settlement of the foundation. Hence, the geotechnical problem resulting in the instability of the structures is one of the main topics in the restoration of the Angkor ruins.
Weight of Upper structure upper structure
Platform Man-made mound Natural ground
Figure 1.1 Schematic figure of the platform structure of Angkor monuments.
Photo 1.1 South sanctuary of Western Prasat Top 1-3).
2
In current restoration works, the design of the platform mound (compaction density, necessity of soil improvement) is based on the allowable bearing capacity concept with consideration for the safety factor1-1),1-2). However, the failure mechanisms of the platform structure, which yield complex stress transmission paths among the stones and the mound, have not been revealed systematically and therefore the ultimate load is estimated using an empirical method. In addition, although the deformation process of the upper structure due to ground deformation should be considered because the joint openings between the stones may cause the stress concentration on the ground, it has not been accomplished currently.
The main reason for the difficulties in the stability evaluation of the masonry structures is the strong nonlinearity of the problems due to the material nonlinearity of the soils and the discontinuity among the stones and foundation ground. In order to solve such initial-boundary value problems with strong nonlinearities, numerical methods would be a very powerful tool. Numerical methods enable us to investigate the internal conditions of the foundation ground, which are difficult to observe in experimental studies, and parametric studies on the difference in behavior of the structures under various conditions can be performed easily. Additionally, the selection of a suitable restoration method, which should be able to secure the stability of the structure while preserving the cultural values of the original construction methods, can be achieved by simulations considering the various structural countermeasures. In consideration of the previously mentioned reasons to adopt numerical methods in the restoration of the heritages, many researches on analysis methods for masonry structures have been done. However, methods to incorporate all of the previously mentioned nonlinearities have not yet been established.
Based on the background described above, in the present study, a numerical method for the deformation analysis of the composite structure of soil and the masonry stones is newly proposed and applied to the studies on failure modes/mechanisms and the bearing capacity characteristics of masonry structure foundations. More specifically, the coupled Numerical Manifold Method and Discontinuous Deformation Analysis (NMM-DDA)1-4), which is one of the discontinuum-based numerical methods, is further
3 extended by introducing elasto-plastic constitutive models and an analytical precision improvement technique. The verification of the method is discussed through the simulations of the fundamental boundary value problems of the applied mechanics and deformation analysis of an existing structure in the Angkor ruins. Subsequently, the bearing capacity characteristics of the platform structure has been studied by a series of numerical experiments employing NMM-DDA, and finally, a simple analytical method to estimate the bearing capacity of masonry platforms that is applicable to practical designs is proposed.
References 1-1) Japanese Government Team for Safeguarding Angkor (JSA): Report on the conservation and
restoration work of the Prasat Suor Prat Tower, OGAWAINSATSU Co., Ltd., 2005.
1-2) JAPAN APSARA Safeguarding Angkor (JASA): Report on the conservation research
of the Bayon, Angkor Thom, Kingdom of Cambodia , 2 011.
1-3) Nara National Research Institute for Cultural Properties: Annual Report on the Research and
Restoration Work of the Western Prasat Top Dismantling Process of the Southern Sanctuary II ,
2015.
1-4) Miki, S., Sasaki, T., Koyama, T., Nishiyama, S. and Ohnishi, Y.: Development of coupled
discontinuous deformation analysis and numerical manifold method (NMM-DDA), Int. J.
Comput. Methods, 7(1), pp.1-20, 2010.
4
Chapter 2 Literature Review on Stability Evaluation
Methods of Masonry Structures
In this chapter, to clarify the scope of this thesis, comprehensive literature reviews on stability evaluation methods of masonry structures are provided. Firstly, in Section 2.1, the outlines of past restoration projects and the geotechnical treatments in the Angkor ruins are introduced to illustrate previous restoration design methods and their limitations. Secondly, in Section 2.2, previous studies on numerical methods to analyze the mechanical behaviors of masonry structures are reviewed, and the features, including advantages and the disadvantages of each method, are explained. Finally, based on the reviews in Sections 2.1 and 2.2, the subjects dealt in this thesis are summarized, and the chapter structures of this thesis are explained in Section 2.3.
2.1 Restoration projects and geotechnical treatments
in the Angkor ruins
The restoration and conservation works of the monuments in the Angkor ruins were started by Ecole Francaise d’Extreme Orient (EFEO) from the beginning of the 20th century. However, the restoration projects were once temporarily halted in 1972 due to the intensification of the civil war in Cambodia, and the destruction of the ruins progressed again. As a result, UNESCO registered the Angkor ruins in the “List of World Heritage in Danger” at the same time as the ruins were selected as a World
5
Cultural Heritage in 1992, leading to the recommence of restoration by many countries, including Japan. Although the Angkor ruins was removed from the List of World Heritage in Danger in 2004 due to international cooperation, the restoration works have been continued until now. From Japan, the Japanese Government Team for Safeguarding Angkor (JSA) 2-1),2-2), Sophia University' Angkor International Mission2-3), and the Nara National Research Institute for Cultural Properties 2-4) are working on the restoration projects.
Through the restoration researches for more than 20 years, it has been recognized that the stability of the Angkor monuments is significantly related to the foundation ground conditions. As previously mentioned in Chapter 1, the masonry structures of the Angkor ruins are usually composed of a platform and an upper structure (Figure 1.1). The upper structure is constructed by stacking the stones on the platform, which is a composite structure comprising stones and the man-made compacted soil mound. Since the stones are directly stacked on the mound, any deformation or failure of the platform or the mound will affect the instability of the upper structure immediately. Considering this structural feature, JSA has performed intensive geotechnical surveys of the damaged monuments and reflected the results in the restoration designs. Among many of the past restoration projects undertaken by JSA, two main projects are introduced below.
2.1.1 Prasat Suor Prat N1 tower Prasat Suor Prat N1 tower2-1) is the structure restored by JSA from 2002 to 2005. This structure consists of a platform with a height of 2.5 m, and an upper structure with the height of approximately 19 m. Since the tower had inclined due to uneven settlement of the platform (Photo 2.1), dismantling and reconstruction of the entire building, including the platform, was performed. In the trench survey of the platform during the dismantling process, shear deformation was observed in the mound (Figure 2.1), and this failure of the foundation was regarded as a cause of the uneven settlement. Since this is a typical example of structures damaged due to ground deformation, in Chapter 4, the applicability of the new numerical method proposed in Chapter 3 is discussed through the deformation analysis of this structure. Therefore, more detailed damage information is explained again in Chapter 4.
6
Photo 2.1 Prasat Suor Prat N1 tower 2-1).
Sheared area
Slip lines
Figure 2.1 Bearing capacity failure in the platform (added description to reference 2-1)).
Next, the restoration method of the platform is briefly introduced. When the platform was reconstructed, the mound soil was improved with slaked lime to increase the bearing capacity of the platform. The mixing rate of the slaked lime was designed so that the allowable bearing capacity becomes larger than the expected structural load on the top stone of the platform. The allowable bearing capacity Pa was estimated by the following equation:
7
qu Pa , (2.1) Fs
where qu is the unconfined compression strength of the test specimen of the improved soil, and Fs is the safety factor, which was set to 3.0 for this structure. It should be noted that this equation contains two bold assumptions: 1) that the platform mound is not confined in the lateral directions, and 2) that the uniaxial compression behavior will be predominant; these were assumed to simplify the problem. After determining the design of the improved soil, the additional stability check for the failure mechanisms shown in Figure 2.2 was performed, but the failure mechanisms are assumed empirically.
Man-made layer
Natural layer Circular slip line
(a)
Linear slip line
Man-made layer
Natural layer
(b)
Figure 2.2 Failure mechanisms considered in the restoration of the PSPN1 (after 2-1)):
(a) circular slip through the natural layer, (b) linear slip line in the man-made layer.
8
2.1.2 Southern library of Bayon The southern library of Bayon2-2) (Photo 2.2) was restored by JSA from 2006 to 2011. This structure consists of a relatively high platform with the height of approximately 5.3 m, and an upper structure with the height of approximately 4.3 m. Regarding this structure, uneven settlement appeared on the top surface of the platform, and the columns and walls of the upper structure inclined inward due to the settlement (Figure 2.3(a)). Thus, dismantling and reconstruction of the largely damaged part of the platform were performed.
Photo 2.2 Southeast view of the southern library of Bayon (before restoration) 2-2).
Uneven settlement
(a) (b)
Figure 2.3 North-south cross section of the southern library 2-2): (a) mode
of the measured settlement, (b) inner structure of the platform.
9
Figure 2.3(b) shows the archaeological sketch of the north-south cross section of the inner structure of the platform, drawn during the dismantling. The platform stones were more vertically stacked as compared with the Prasat Suor Prat N1 tower. Local failure in the platform mound was not found in this structure, and the horizontal spreading of the mound driven by the structural weight caused the settlement. Based on this insight, soil improvement with slaked lime was also applied in the reconstruction of this structure in order to increase the stiffness of the mound, as well as the bearing capacity. The mixing rate of the improved soil was determined using the same methods as with the Prasat Suor Prat N1 tower, which were based on the allowable bearing capacity of the top stone of the platform. In addition, a construction manual considering the stiffness control of the mound was created and implemented in the reconstruction.
2.1.3 Problems with previous restoration designs The above-mentioned examples detail that deformation of the platform mound caused the destabilization of the upper structure in the Angkor ruins. Additionally, masonry platforms with different structural features show different deformation mo des, such as the local deformation of the mound and the deformation of the entire building including the platform. Since the restoration projects introduced above are the rare cases that the dismantling was performed as far as the platform, their works would be important references for future restorations. However, previous restoration method s still contain the following issues to be improved.
(1) The bearing capacity of the platform is estimated by an overly simplified method (e.g. Eq. (2.1)) without considering the structural conditions.
(2) The changes in load conditions due to the deformation of the masonry structure are never considered.
In regards to the first point, the bearing capacity characteristics of the masonry platform should vary and depend on the structural features because of the complex mechanical conditions that transmit the load from the upper layer to both the stone and soil of the lower layers. To establish a rational method to estimate the bearing capacity of masonry structures that is able to consider the different mechanical features, the failure mechanisms of the masonry platform should be investigated in detail. Regarding
10 the second point, the deformation of masonry structures containing many discontinuous planes will cause changes in stress transmission paths and result in an unexpected stress concentration. In the previous design process, the possibility of load distribution changes was considered in the safety factor, but the validity of the value of the safety factor is not clear. Investigation into deterioration/failure mechanisms and adequate restoration methods considering the deformation process of the entire structure is desirable for important monuments. The difficulties of the stability evaluation of masonry structures mainly arise from the material nonlinearity of the soils and the discontinuity of the masonry stones. In order to solve such strongly nonlinear problems, numerical methods would be useful. The past studies on numerical methods for masonry structures are reviewed in the next section.
2.2 Numerical methods for stability evaluation of masonry structures
Because the stability of masonry structures is governed by the mechanical interaction between the masonry stones and the foundation ground, the analysis method s should incorporate both the discrete bodies of the masonry stones and the continuum bodies of the soils.
Many numerical methods have been developed to address discontinuity in the context of continuum analysis. The most popular of these is the Finite Element Method (FEM) with zero-thickness joint (interface) elements 2-5)–2-7) (Figure 2.4). FEM is the most widely used numerical method in geotechnical and structural engineering, where various constitutive models are introduced for different materials, and the joint element techniques enable the simple modeling of the discontinuous behavior between the continuum bodies. Therefore, this technique is employed in many studies on structural analyses of masonry buildings. For example, Lourenço and Rots2-8), and Senthivel and Lourenço2-9) performed stability analysis of a masonry shear wall, and Parajuli et al.2-10) performed dynamic analysis of a historic masonry building in Nepal. However, the applicability of the method to the behavior of masonry structures is limited because many interfacial surfaces exist among the stones and the ground, and introducing an
11
Finite elements Joint element (zero thickness)
Figure 2.4 Joint element.
Triangular finite element
Discrete element
Figure 2.5 Combined FEM-DEM. Figure 2.6 Contact with spring-dashpot model. excessive number of joint elements would result in ill-conditioning of the global stiffness matrix. Additionally, it is difficult to model perfect separation, rotation, and generation of new contact surfaces because the node connectivity should be maintained throughout the analyses. The interface element approach was also applied to the Finite Difference Method (FDM)2-11), however, application of FDM results in similar complications. Although it may be possible to apply this method to the soil-masonry structure interactive problems if the displacement is sufficiently small, to the best of the author’s knowledge there is no current literature that has attempted to solve such a problem.
In order to overcome these limitations, several novel approaches have been developed and applied. One of the popular methods that incorporates both continua and discontinua is a method combining FEM and the discrete element method (DEM) 2-12). Munjiza2-13) proposed a combined FEM-DEM approach that discretizes each discrete element (block) into finite elements (see Figure 2.5) and considers contacts between discrete blocks by the spring-dashpot model of the DEM (see Figure 2.6); this method was applied to the analyses of dry stone masonry structures2-14),2-15). Michael et al.2-16) developed another combined FEM-DEM scheme that incorporates the contacts between the finite elements and the particle discrete elements, and applied this method to a tire-terrain interaction problem. However, the applicability of the FEM-DEM approach
12 to quasi-static and long-term problems is rather limited because it treats contacts based on DEM algorithms that explicitly integrate the equation of motion. Since DEM requires smaller time intervals to satisfy the governing equation, it is difficult for the c ombined FEM-DEM method to obtain an accurate solution that satisfies the equilibrium of the entire soil and masonry system. In addition, there exists no example employing the FEM-DEM approach to simulate soil-masonry structure interaction.
Identifying the problems with the current approaches at the time, Miki et al.2-17) proposed coupling Discontinuous Deformation Analysis (DDA) 2-18) with Numerical Manifold Method (NMM)2-19), which are numerical methods for the dynamic and quasi-static analyses of contact between discontinuous elastic bodies, respectively. This coupled NMM-DDA method affords strong coupling between the behaviors of continua and discontinua. The difference between the two methods is the discretization of the displacement field in the materials. DDA employs rigid body displacement, rotation, and strain at the centroid of the block as the unknown variables, and can adequately simulate rockfall2-20), dynamic landslide of a rock slope 2-21),2-22) and dynamic stability analyses of masonry stones2-23)-2-26). In contrast, NMM takes the nodal displacements of a mesh composed of finite mathematical covers over the analytical domain as the unknown variables, and is able to generate detailed deformation of the continuum as well as the contacts. In the coupled NMM-DDA, an analytical domain consisting of multiple discrete continua is divided into continua modeled by DDA and by NMM (see Figure 2.7), and the simultaneous analysis is accomplished by solving the deformation of both domains and the contacts among them. Since the original DDA and NMM utilize the same contact treatment algorithms and the “implicit” time integration of the equation of motion, the strong coupling analysis, which satisfies the equilibrium of the entire analytical domain consisting of both continua and discontinua, would be properly achieved. Considering this advantage of the NMM-DDA, Koyama et al.2-27) performed a deformation analysis of a masonry structure in the Angkor ruins. However, it is noted that the original NMM-DDA was formulated based on the principle of potential energy minimization, and is only applicable to linear elastic material. In order to incorporate the elasto-plastic behavior of soils, further extension of this method is required.
13
Material modeled by DDA (e.g. masonry stones)
Material modeled by NMM (e.g. foundation ground)
Figure 2.7 Discretization in NMM-DDA.
2.3 Scopes and structure
This chapter initially introduced the conventional geotechnical restoration processes and their problems in reference to the Angkor ruins; additionally, previous research studies on applying numerical methods to simulate masonry structures were also described. Based on the reviews, the following subjects should be studied to establish a more suitable restoration design process of masonry structures that is founded in geotechnical engineering.
(1) Development of a numerical method for soil-masonry structure interaction problems
(2) Verification and validation of the developed numerical method
(3) Investigation into the deformation and failure mechanisms of masonry structure foundation
(4) Proposal of a rational design method of the foundation based on the revealed failure mechanisms
In order to resolve the above subjects, a series of research studies, shown in Figure 2.8, is provided in this thesis. Outlines of Chapter 3 to Chapter 7 are summarized below.
14
In Chapter 3, a new numerical method for solving the interaction problem between the soil and masonry structure is developed by extending the NMM-DDA to include elasto-plastic constitutive models, and further improved by applying a method to prevent volumetric locking2-28), which results in an unrealistically stiff solution when an incompressible material is used. Several numerical examples are provided using the newly developed method, and the verification of the method is discussed in comparison with the theoretical solutions.
In Chapter 4, the laboratory test results of the in-situ soil samples of the Angkor ruins, and the constitutive modeling with an elasto-plastic model are presented, followed by the deformation analysis of an actual damaged masonry structure in the Angkor ruins that was performed using the newly developed elasto-plastic NMM-DDA. The applicability of the NMM-DDA to the stability analysis of an actual building is discussed by comparing the simulated results with the corresponding field observation data. The possible causes of the masonry structure deterioration are also discussed.
In Chapter 5, the numerical experiments on the bearing capacity of the masonry platform using the newly developed elasto-plastic NMM-DDA are conducted. The stability analyses of the masonry platforms with different structural features are shown, and the changes in the ultimate bearing capacity and failure modes/mechanisms of the platform structures are explicated.
In Chapter 6, a simplified method to estimate the bearing capacity of the masonry platform structure is proposed. The method is formulated based on the limit equilibrium assuming the failure mechanisms revealed in Chapter 5. The ultimate load of the platforms is predicted by the proposed method and the applicability is discussed by comparison with the results obtained from the NMM-DDA simulations shown in Chapter 5. Application scheme of the proposed method to a practical restoration project is also discussed.
In Chapter 7, the conclusions of this thesis and recommendations for future works are provided.
15
Chapter 3 Method of Deformation Analysis for Composite Structures Subject (1) and (2) of Soils and Masonry Stones
Chapter 4 Deformation Analysis of an Existing Masonry Structure in the Angkor Ruins Subject (3) Chapter 5 Numerical Experiment on Bearing Capacity Characteristics of Masonry Platform Structure
Chapter 6 Simplified Estimation Method Subject (4) of Ultimate Bearing Capacity for Masonry Platform Structure
Chapter 7 Conclusions and Future Studies
Figure 2.8 Structure of the thesis and corresponding subjects.
References 2-1) Japanese Government Team for Safeguarding Angkor (JSA): Report on the conservation and
restoration work of the Prasat Suor Prat Tower, OGAWAINSATSU Co., Ltd., 2005.
2-2) JAPAN APSARA Safeguarding Angkor (JASA): Report on the conservation research
of the Bayon, Angkor Thom, Kingdom of Cambodia , 2 011.
2-3) Sophia University Angkor International Mission : Report of the conservation and
restoration of the Angkor Wat western causeway, 2011.
2-4) Nara National Research Institute for Cultural Properties: Annual Report on the Research and
Restoration Work of the Western Prasat Top Dismantling Process of the Southern Sanctuary II ,
2015.
16
2-5) Goodman, R.E., Taylor, R. and Brekke R.L., A model for the mechanics of jointed rock, J. Soil
Mech. Found. Eng. Div., ASCE, 94(SM3), pp. 637– 65 9 , 1968.
2-6) Zienkiewicz, O.C., Best, B., Dullage, C. and Stagg, K.C.: Analysis of non -linear problems in
rock mechanics with particular reference to jointed rock systems, Proc. 2nd Cong. Int. Soc.
Rock Mech., pp. 8– 14, 1970.
2-7) Ghaboussi, J., Wilson, E.L. and Isenberg, V.: Finite element for rock joints and interfaces, J.
Soil Mech. Found. Div., ASCE, 99, pp. 833–848, 1973.
2-8) Lourenço, P.B. and Rots J.G.: Multisurface interface model for analysis of masonry structures,
J. Eng. Mech., 123(7), pp. 660-668, 1997.
2-9) Senthivel, R. and Lourenço, P.B.: Finite element modelling of deformation characteristics of
historical stone masonry shear walls, Eng. Struct., 31, pp. 1930–1943, 2009.
2-10) P ar aju l i , H .R ., K i yo no , J ., Taniguchi , H ., To ki , K ., F u ku kaw a , A. and M as ke y, P.N.:
Parametric study and dynamic analysis of a historical masonry building of
K at h man du , Proc. of Disaster Mitigation of Cu ltural Heritage and Historic Cities , 4,
pp. 14 9– 156, 2010.
2-11) Itasca. FLAC: fast lagrangian analysis of continua , Version 7.0 [computer program] ,
Minneapolis, Itasca Consulting Group Inc , 20 11 .
2-12) Cu nd al l , P.A.: Computer model for simulating progressive large scale movements in
blocky systems , P ro c. of ISRM Symp ., II- 8, pp. 12 9 –13 6, 1971.
2-13) Mu nj i za , A.: The finite/discrete element method , Chichester, NJ, USA: John Wiley
and S on s , 2 00 4.
2-14) Smoljanović , H ., Ži val j i ć , N . and N i kol i ć , Ž.: A combined finite - discrete element
analysis of dry stone masonry structures , E ng . S t ru ct ., 52, pp. 89– 100, 2013.
2-15) Smoljanović , H ., N i ko li ć , Ž. and Ž i val j i ć , N.: A fi n i t e - discrete element model for dry
stone masonry structures strengthened with steel clamps and bolts , E ng . S t r u ct ., 90,
pp.117 –1 29 , 20 15 .
2-16) Mi ch ael , M ., Vo gel , F. and P et er s B . : DEM- FEM coupling simulations of the
interactions between a tire tread and granular terrain , Co m pu t . Methods Appl . Mech .
En g., 2 89 , pp. 2 27 –2 48 , 2016.
2-17) Mi ki , S ., S as aki , T., K o ya ma , T., Nishiyama , S . and O hn i sh i , Y. : Development of
coupled discontinuous deformation analysis and numerical manifold method
(NMM - DDA) , In t . J. Co m pu t. Met ho d s , 7 (1), pp. 1 – 20, 2010.
17
2-18) Sh i , G.H . and Goo d man , R .E.: Generalization of two - dimensional discontinuous
deformation analysis for forward modelling , I n t. J. N u m er. A na l . Methods Geomech .,
13, pp. 3 59– 38 0 , 1989.
2-19) Shi, G.H.: Manifold method of material analysis, Trans. 9th Army Conference on Applied
Mathematics and Computing, Report No. 92-1, U.S. Army Research Office, 1991.
2-20) Shimauchi, T., Nakamura, K., Sakai, I., Hagiwara, Y., Ohnishi, Y. and Nishiyama, S.: Studies
on the property of impact velocity ratio and application to the rockfall simulation by DDA ball,
Proc. Int. Mini-Symp. Numerical Discontinuous Analysis, pp. 53–62, 2008.
2-21) Wu, J.H.: Seismic landslide simulations in discontinuous deformation analysis ,
Co m pu t . G eo t ech ., 37(5 ) , pp. 59 4– 601, 2010.
2-22) Wu, J .H . and Ch en , C .H.: Application of DDA to simulate characteristics of the
Tsaoling landslide , C o mp ut . G eo t ech ., 38(5 ) , pp. 74 1–7 50 , 2011.
2-23) K a mai , R . and H at zo r, Y.H.: Numerical analysis of block stone displacements in
ancient masonry structures: a new met hod to estimate historic ground motions , I nt . J.
Nu m er. A na l . Methods Geo mech ., 32, pp. 13 21 –13 40 , 2008.
2-24) Thavalingam, A., B i can i c , N ., Ro bi n so n , J .I ., P on ni ah , D.A.: Computational
framework for discontinuous modeling of masonry arch bridges , C o mp u t . S t r u ct ., 79,
pp. 18 21 –18 30 , 2001.
2-25) J i an g , H ., Wan g , L., Li , L. and Gu o , Z.: Safety evaluation of an ancient masonry
seawall structure with modified DDA method , Co m pu t. G eot ech ., 55, pp. 2 77– 28 9,
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2-26) Yago d a - B i r an , G. and H at zo r, Y.H .: Benchmarking the numerical discontinuous
deformation analysis method , Co m pu t . G eo t ech ., 71, pp. 30–46, 2016.
2-27) Koyama, T., Yasuda, Y., Yamada, S., Araya, M., Iwasaki, Y. and Ohnishi, Y.: Development of
NMM-DDA and stability analysis for Prasat Suor Prat, Angkor, Proc. 14th Asian Regional
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Englewood Cliffs, NJ, USA: Prentice-Hall; 1987.
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Chapter 3 Method of Deformation Analysis for Composite Structures of Soils and Masonry Stones
This chapter provides the basic theory and the performance evaluation study of the elasto-plastic NMM-DDA (coupled Numerical Manifold Method and Discontinuous Deformation Analysis) 3-1), which is the newly developed numerical method for the mechanical interaction problem between the soils and the masonry structures. Firstly, the governing equations of the contact problem of continua and their weak forms are described in Section 3.1, and the formulation of the elasto-plastic NMM-DDA is shown in Section 3.2. Additionally, a method to avoid the volumetric locking in NMM using a node-based uniform strain element is explained in Section 3.3. In Section 3.4, several numerical examples are presented and solved, such as the elastic cantilever bending problem typically affected by volumetric locking, and the bearing capacity problems of strip footing under central or eccentric loading. Then, based on the simulation results, the verification of the proposed method for the soil-masonry structure interaction problems is discussed. Finally, the achievements in this chapter are summarized in Section 3.5.
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3.1 Governing equations of continuum kinematics and mutual contact problem
NMM-DDA analyzes the mechanical behaviors of systems that consist of multiple continua by considering their mutual contacts. This problem is governed by the equation of motion for each continuum and additional constraint conditions that must be satisfied wherever contacts occur among the objects. A system that includes n independent continua is hereafter assumed. The domains occupied by each continuum are named as
i (i = 1, 2, , n) and the boundary surfaces of i are named as i . The kinematic problem of a single material , shown in Figure 3.1, is described with the equation of motion:
iui σi bi 0 ini (3.1) the strain compatibility condition:
1 T εi ui ui (3.2) 2 and the incremental form of the constitutive equation:
σi Di : εi ; (3.3)
In Eqs. (3.1)–(3.3), the subscript i indicates the physical quantity of , the dot ( ) means the material derivative, i is the density, ui is the displacement vector, σi is the Cauchy stress tensor, bi is the known body force vector, εi is the infinitesimal strain tensor, and Di is the constitutive relations tensor. In addition, the displacement field must satisfy the displacement boundary conditions:
σ i ni t i
it
bi
r i y
iu x u u i i Figure 3.1 Deformation problem for a single continuum.
20
ui ui oniu ; (3.4) and the stress boundary conditions:
ti σi ni ti oni . (3.5)
In Eqs. (3.4) and (3.5), the displacement boundary iu and the stress boundary i of
i satisfy i iu i and iu i ∅. In addition, ui is the known displacement on , ni is the unit vector normal to i , and ti is the known traction vector on .
By integrating Eq. (3.1) within the domain after multiplying by a trial function ui , which is an arbitrary vector function that satisfies ui 0 on , the weak form of the equation of motion is derived as
u σ b u d 0 i i i i i . (3.6) i
The trial function is usually called virtual displacement. After applying Gauss’s divergence theorem, the second term on the left side of Eq. (3.6) is transformed as
σ u d σ n u d σ : u d i i i i i i i . (3.7) i i i
Considering on and the stress boundary condition given in Eq. (3.5), Eq. (3.7) becomes
σ u d t u d σ :ε d i i i i i i . (3.8) i i i
T where εi is defined as εi ui ui 2 and the symmetry of the Cauchy stress tensor is also considered. Substituting Eq. (3.8) into Eq. (3.6), we obtain
u u d ε :σ d u b d u t d 0 i i i i i i i i i . (3.9) i i i i
In order to incorporate the incremental form of the constitutive equation given by Eq.
(3.3), we assume that Eq. (3.9) represents the equilibrium condition at the time t t , and we rewrite ui , εi , σi , bi , and as the summations of the known values at time t and the incremental values from t to t t :
ui ui|tt ui|t ui ; (3.10)
εi εi|tt εi|t εi ; (3.11)
σi σi|tt σi|t σi ; (3.12)
21
bi bi|tt bi|t bi ; and (3.13)
ti ti|tt ti|t ti ; (3.14) where the subscript | t means the physical quantity at the time t. Substituting Eqs. (3.10)–(3.14) and Eq. (3.3) into Eq. (3.9), the following equation can be obtained: ui iui|tt d εi :Di : εi d ui bi d ui ti d i i i i (3.15) ui bi|t d ui ti|t d εi :σi|t d 0, i i i where the third and the fourth terms on the left side are the incremental external forces, and the fifth, sixth, and seventh terms represent the equilibrium condition at time t, which are the residual terms in the numerical simulation. Eq. (3.15) is the weak form of the governing equations of a continuum that does not contact with any other bodies. The displacement field has to satisfy both Eq. (3.15) and the displacement boundary condition given by Eq. (3.4).
Next, the contact between two continua j and k (j k) along the boundary
jk is assumed, as shown in Figure 3.2. In this situation, the relative normal distance
dn between two objects must be zero on jk , and the tractions acting against each other on , called the contact force, must balance. Therefore, the solution has to satisfy,
σ k nk t k
kt σ j n j t j jk bk jt ku u u b j k k k r j y u u ju x j j Figure 3.2 Contact problem for two continua.
22
dn 0 onjk , and (3.16)
t j tk 0 onjk , (3.17) in addition to Eqs. (3.15) and (3.4). Furthermore, when there is friction at the contact surface, and the tangential traction force is less than the frictional strength criterion (Coulomb’s criterion), the constraint condition of the tangential (shear) displacement between j and k ,
ds 0 onjk , (3.18) must be satisfied. In NMM-DDA, Eq. (3.16)–(3.18) are approximated by the penalty method. The penalty term of the contact is added to the weak form of the continua j and k that are in contact with each other, and we consequently obtain ul l ul|tt d εl :Dl : εl d ul bl d ul tl d l l l l ul bl|t d ul tl|t d εl :σl|t d (3.19) l l l d p d d d p d d 0, n n n s s s jk jk where l = j or k, m = j or k, and l m; and pn and ps are the penalty coefficients for the contact condition in the normal and tangential directions, respectively, both of which are sufficiently large real numbers to allow us to ignore the other terms on jk in Eq. (3.19). When the contacts are established at multiple points and/or with multiple objects, penalty terms corresponding to each contact point need to be added to Eq. (3.19).
3.2 Formulation of the elasto-plastic NMM-DDA
This section describes the formulation of the NMM-DDA that incorporates an elasto-plastic constitutive model, which is needed for proper analysis of soil deformation. In NMM-DDA3-1), the n independent continua assumed in Section 3.1 are divided into those continua modeled by DDA and those modeled by NMM, as shown in Figure 3.3. The weak forms of the governing equations for each object, given by Eq. (3.19), will be discretized over the space domain by DDA and NMM, respectively, and solved by incremental time intervals. In this section, the procedures for approximating
23
i Material modeled by NMM
i Mathematical mesh of NMM
k
Material modeled by DDA j k
j
Contacting surface jk Figure 3.3 Modeling in NMM-DDA.
v0 r 0 u 0 i x0 , y0
y x y xy
x
Figure 3.4 Polygonal approximation and variables of displacement and deformation in DDA. the displacement fields in continua using DDA and NMM are introduced, and the characteristics of each method are described in terms of the approximation scheme. In addition, the contact treatment commonly applied to both DDA and NMM is explained.
3.2.1 Displacement field approximation in DDA and NMM In DDA3-2), the continua are modeled as polygons, called DDA blocks, to apply the contact detection algorithm described in Section 3.2.2. Assuming that the stress and the strain are uniform in one DDA block, the displacement field can be expressed with three displacement variables and three strain variables, as shown in Figure 3.4. For a two-dimensional problem,
D T di u0 v0 r0 x y xy . (3.20)
Here, the superscript D means a variable relating to an object modeled by DDA, and the
D superscript T means the transposition of the relevant vectors and matrices. di
24 represents the unknown variables of i ; u0 and v0 are the displacement of the block centroid in the x and y directions, respectively; r0 is the rotation around the centroid;
x and y are the normal strain in the x and y directions, respectively; and xy is the shear strain in the x-y plane. Adopting these variables, the displacement vector uD x, y at an arbitrary point P(x, y) in a DDA block can be expressed as i D T D D ui x, y ux, y vx, y Ti di ; (3.21)
D where Ti is the so-called displacement matrix of and is expressed as
y y0 1 0 y y0 x x0 0 T D T D x, y 2 ; i i x x (3.22) 0 1 x x 0 y y 0 0 0 2 in which (x0, y0) are the coordinates of the centroid of . The strain component vector
i is expressed as
T D D (3.23) i x y xy Bi di , where
0 0 0 1 0 0 BD 0 0 0 0 1 0 i . (3.24) 0 0 0 0 0 1
Because DDA explicitly treats the rotation of the object as an unknown variable, as described in Eq. (3.20), the rolling of the stones can be appropriately considered in analyses of the collapse behavior of masonry structures.
On the other hand, in NMM3-3), three other concepts are introduced to approximate the complex displacement and strain fields in the continua: the physical mesh (Figure 3.5(a)), mathematical cover (Figure 3.5(b)), and manifold element (Figure 3.5(c)). The physical mesh is the physical domain that must satisfy the governing equations. The physical mesh will be fully covered by the mathematical cover, which is the mathematical domain, and the overlapping area of several covers e in
Figure 3.5(c) is called the manifold element (ME). For each cover Cl, we define the local displacement approximation function, called the cover displacement function, ~ T ul x, y ul x, y vl x, y , (3.25) and the weighting function
wl x, y. (3.26)
25
Cover Cn
Manifold element
i e
Cover Cl Cover Cm
(a) (b) (c)
Figure 3.5 Approximation scheme in NMM: (a) physical mesh of the continuum body,
(b) mathematical cover, and (c) manifold element.
For each ME, the weighting function satisfies
wl x, y 0 x, yCl , (3.27)
wl x, y 0 x, yCl , (3.28) and
nc wl x, y1 x, ye . (3.29) l1
Here, nc is the total number of mathematical covers over the physical domain. Based on the above characteristics of the weighting function, the cover displacement functions are connected together as a weighted summation, and the global displacement approximation function of i can be constructed:
nc M ~ ui x, y wl x, yul x, y, (3.30) l1 where the superscript M means a variable relating to an object modeled by NMM.
Although any arbitrary shapes can be employed for the physical meshes and the mathematical covers, this study employs hexagonal covers, as shown in Figure 3.6, because these can generate triangular meshes, simplifying the treatments of the contacts and the systematic element quadrature. As shown in this figure, three hexagonal covers produce a triangle ME, and the vertices of the covers correspond to the nodes of the triangle mesh. As the weighting function for the hexagonal cover, a linear polynomial function weights the cover from 1 at its center (xl, yl) to 0 at its edge, as shown in Figure 3.7. The uniform function
26
~ ~ ~ ~ T ul x, y dl ul vl (3.31) ~ is used as the cover displacement function. Here, dl is the displacement vector at the center of the cover, representing the nodal displacement of the triangular mesh.
Utilizing these weighting and cover displacement functions, the displacement at an arbitrary point P(x, y) in a ME e can be described as 3 ~ ˆ ˆ ˆ ue x, y wel x, ydel Te de , (3.32) l1 where w 0 w 0 w 0 ˆ ˆ e1 e2 e3 Te Te x, y (3.33) 0 we1 0 we2 0 we3 and ˆ T de ue1 ve1 ue2 ve2 ue3 ve3 , (3.34)
Hexagonal covers
Triangular NMM element
Figure 3.6 Polygonal physical mesh and 3-node triangular mathematical mesh
produced with the hexagonal covers.
wl(x, y) 1 0 0 (x , y ) l l 0
(xl, yl) 0
0 0 ( a) (b ) Figure 3.7 Weight assigned by the hexagonal cover: (a)center of the hexagonal cover
and (b)weighting function of the cover.
27
In Eqs. (3.32)–(3.34), el (l = 1, 2, 3) is the global node number of the node of the ˆ ˆ manifold element e, Te is the displacement matrix of e, and d e is the vector consisting of the displacement components of each node of e. The strain component vector ˆe of the element e is expressed as T ˆ ˆ (3.35) ˆe x y xy Be de, ˆ where Be is the displacement-strain relationship matrix, expressed as we1 we2 we3 0 0 0 x x x w w w ˆ e1 e2 e3 Be 0 0 0 . (3.36) y y y w w w w w w e1 e1 e2 e2 e3 e3 y x y x y x
As the weighting function wel employed here is spatially linear, as shown in Figure 3.7, the strain component is spatially constant in each triangular ME. The displacement and the strain field of the entire domain of i are described by M M M (3.37) ui x, y Ti di ;
M M M (3.38) i x, y Bi di ;
w1 0 w2 0 wn 0 T M c ; (3.39) i 0 w 0 w 0 w 1 2 nc
w w w 1 0 2 0 nc 0 x x x w w wn BM 0 1 0 2 0 c ; and (3.40) i y y y w w w w w w 1 1 2 2 nc nc y x y x y x
d M u v u v u v T ; (3.41) i 1 1 2 2 nc nc
M M where Ti and Bi are the displacement matrix and the displacement-strain matrix M of the entire domain of i , and di is the vector consisting of all the nodal displacement components of the mathematical mesh covering . In NMM, the mathematical mesh can be generated independently from the physical mesh because the weighting functions of the covers satisfy Eq. (3.29), the so-called Partition of Unity (PU) condition3-4),3-5). Therefore, unlike other mesh-based numerical methods such as FEM, NMM easily generates the mathematical mesh no matter how complicated the
28 shape of the physical domain. In addition, because the boundary and contact conditions are defined on the physical mesh, the same algorithms can be used with DDA, and the two methods can be closely coupled.
3.2.2 Contact detection and discretization of contact penalty term The contact problem can be properly solved in the NMM-DDA by detecting the onset of the contact and applying the penalty terms in the governing equation of the contact problem with the equation of motion. Because the contact mathematically occurs when the normal distance to the material surfaces dn becomes zero, the onset of the contact can be judged by calculating dn for all contact pairs in the analysis. In both DDA and NMM, based on the polygonal approximation of the material surface, contact is determined by calculating dn from the coordinates of the vertices. Figure 3.8(a) shows an example of two closely located objects j and k (j k), where the distance between the two objects can be evaluated from the distance between the vertex
P1 of and the edge P2P3 of , given as follows:
1 x1 y1 1 S d 1 x y 0 , (3.42) n l 2 2 l 1 x3 y3 where (xi, yi) (i = 1, 2, 3) are the coordinates of the vertex Pi, S0 is twice the area of the triangle P1P2P3, and l is the length of the edge . Using Eq. (3.42), contact is mathematically determined to occur when dn becomes zero. In the proposed numerical analysis, which is implemented by incremental time intervals, the vertex P1 usually intersects the edge , and the two objects slightly overlap, as shown in Figure
3.8(b). When P1 penetrates into , the value dn changes from positive to negative.
Therefore, contact is considered to occur when the sign of dn reverses.
Whenever contact is detected, the penalty terms are applied for objects in contact with other objects to satisfy the contact constraint condition described in Eq. (3.19). The contact term in Eq. (3.19) has to be discretized over the spatial domain in a manner similar to that applied to the other terms. In Eq. (3.19), dn represents the normal distance after the contact; it can be expressed as the normal penetration length of the vertex P1 to the edge :
29
Ωj
Ωj P (x , y ) P 1 1 1 2 P0 P P (x , y ) 3 2 2 2 dn dn P3 (x3, y3) Ω Ω k k P1
d 0 not contacted d 0 contacted (a) n (b) n
Figure 3.8 Contact determination: (a) before contact and (b) after contact.
1 x1 u1 y1 v1 1 d 1 x u y v n l 2 2 2 2 1 x3 u3 y3 v3
S 1 u1 (3.43) 0 y y x x 2 3 3 2 l l v1 u u y y x x 2 y y x x 3 . 3 1 1 3 1 2 2 1 v2 v3
Here, (Δui, Δvi) (i = 1, 2, 3) is the incremental displacement of the vertex Pi. By discretizing (Δu1, Δv1) with the displacement matrix of j , and discretizing (Δu2, Δv2) and (Δu3, Δv3) with the displacement matrices of k , Eq. (3.43) is expressed as follows:
S0 I j I j Ik Ik dn E j d j Gk dk (3.44) l where
I j 1 I j E j y2 y3 x3 x2Tj x1, y1, and (3.45) l
1 1 GIk y y x x T Ik x , y y y x x T Ik x , y . (3.46) k l 3 1 1 3 k 2 2 l 1 2 2 1 k 3 3
Meanwhile, the tangential displacement ds is defined as the relative displacement along P P between P and P , which is the displacement on of the base of the 2 3 1 0 perpendicular line extending from P1, and is expressed as follows:
30
1 d P P P P s l 0 1 2 3 T 1x1 u1 x0 u0 x3 u3 x2 u2 l y1 v1 y0 v0 y3 v3 y2 v2 (3.47) S 1 u u s x x y y 1 x x y y 0 3 2 3 2 3 2 3 2 l l v1 v0
Ss I j I j Ik Ik H j d j Lk dk . l where
I j 1 I j H j x3 x2 y3 y2Tj x1, y1 , (3.48) l
1 LIk x x y y T Ik x , y , and (3.49) k l 3 2 3 2 k 0 0
x3 x2 Ss x1 x0 y1 y0 . (3.50) y3 y2
Here, the superscript Ii (i = j, k) is either D or M indicating that i is modeled by DDA or NMM, respectively. It should be noted that the tangential contact force (shear force) can be estimated as psds, and the constraint condition on ds is detached if the shear force violates the friction strength criterion (Coulomb’s criterion). Finally, the penalty term in Eq. (3.19) can be spatially discretized using Eqs. (3.44)–(3.50).
3.2.3 Discretized form of the governing equations
Here, we assume n discrete continua 1 , 2 , , m modeled by DDA and
m1 , m2 , , n modeled by NMM, and a contact between j modeled by DDA and k (j k) modeled by NMM. Based on the spatial discretization scheme explained in Section 3.2.1 and 3.2.2, and the temporal discretization scheme shown in Appendix B, we finally obtain the discretized equation for the entire system of n continua as a single simultaneous linear equation:
~ D D ~D K1 0 0 0 d1 F1 ~ D D ~D 0 K2 0 0 d2 F2 0 0 ~ D DM DM D ~D D K j Kcjj Kcjk d j Fj Fcj ~ ~ D D D . Km dm Fm (3.51) ~ ~ K M d M F M m1 m1 m1 DM ~ M DM M ~M M Kckj Kk Kckk dk Fk Fck 0 ~ M M ~M 0 0 0 Kn dn Fn
31
~ Ii Here, Ki (i = 1, 2, , n) is the stiffness matrix of the continua i including the
I jIk I jIk I jIk I jIk inertia term, and Kcjj , Kcjk , Kckj , Kckk are the contact penalty stiffness
Ii matrices between j and k , di is the incremental displacement vector of i ;
~Ii Ii and Fi is the external force vector of including the inertia force, and Fci is the contact force vector of . The discretized equations of j and are coupled through the contact stiffness matrix; thus, NMM-DDA can satisfy the governing equation of entire system, implementing simultaneous DDA and NMM analyses. Although we assumed only the contact between a material modeled by DDA and a material modeled by NMM, NMM-DDA analysis can also be applied to the contact of other arbitrary pairs such as DDA-DDA contact or NMM-NMM contact. The details of the spatial and temporal discretization procedures for the generalized contact pair are shown in Appendix A and Appendix B, respectively.
3.3 Volumetric locking and implementation of node-based uniform strain element in NMM
The elasto-plastic NMM-DDA formulated in Section 3.1 and 3.2 suitably incorporates essential features of the analysis of the interaction between the ground and a masonry structure, such as the material nonlinearity of soils and the discontinuities between the materials. However, the NMM-DDA can exhibit an unrealistically stiff behavior when modeling domains containing incompressible materials (some typical examples are shown in Section 3.4). This phenomenon is also significant when the simulation of an elasto-plastic soil model predicts constant volume shearing at a critical state, and it is rather difficult to obtain theoretically accurate solutions without an appropriate countermeasure. Similar difficulties have often been reported in the finite element simulation, in which the phenomenon is usually known as volumetric locking3-6). This volumetric locking appears when a finite element mesh uses low-order full integration elements, such as constant strain triangular (CST) elements and/or 4-node isoparametric quadrilateral elements, to model incompressible materials, and several countermeasures have been proposed to avoid volumetric locking in the FEM3-7),3-8). Now, recalling Eqs. (3.32)–(3.36), which approximate the manifold
32 elements (MEs), these equations have a similar structure to that of the finite eleme nt approximation when their weighting functions are replaced by the shape functions applied in FEM. In this study, focusing on this theoretical similarity, we consequently introduce into NMM a method based on the so-called node-based uniform strain element (NB element), which was originally proposed by Dohrmann et al. 3-9) to avoid the volumetric locking of a 3-node triangular FEM mesh. This section provides the outline of the NB element and the modification required to implement the NB element in NMM.
3.3.1 Outlines of NB element In FEM analyses, incompressible materials usually exhibit volumetric locking when there are too many integration points at which strain is evaluated compared to the degree of freedom of the nodes, and when there are too many constraint conditions on volumetric strain3-6). Considering these aspects, several countermeasures were proposed to avoid this locking with regard to isoparametric elements, based on applying a reduced integration technique during the computation of the stiffness matrix 3-7),3-8). However, it is impossible to reduce the number of integration points in case of the CST element, which has only one integration point. The same argument also holds in NMM because it has a theoretical structure similar to that of the FEM.
In order to overcome this problem, Dorhmann et al. 3-9) proposed the node-based uniform strain element (NB element), which consists of the partial domain of the CS T elements around a node, as shown in Figure 3.9. In this method, strain components, which are usually evaluated at the center of the element, are evaluated at each node by averaging the strain components of the triangle elements sharing the node. For a 3 -node triangular mesh, the number of CST elements, which is equal to the number of volumetric constraints, is nearly the same as the nodal degrees of freedom3-9). On the other hand, if the NB element, in which the integration points correspond with the nodes, is used in a 2D problem, the number of integration points will be reduced by half, and the volumetric locking can be avoided. The strain components are evaluated at each node using the strain components of the CST elements containing the given node as follows:
33
Node-based
uniform strain element N N
: 3-node triangular mesh : Edges of the node-based uniform strain elements
Figure 3.9 Node-based uniform strain element.
Node e(3)
CST element e 1 Ae 3 Centroid of element 1 A 3 e Node e(1) Node e(2)
Figure 3.10 Equal division of CST element.
1 mN N ˆ N e Ae e. (3.52) AN e1
Here, N is the strain components vector at the node N, ˆe is the strain components vector of a CST element e, mN is the number of CST elements containing N the node N, and Ae is the area of a CST element e. e is the partition rate of Ae to the node N , and AN is the area of the NB element corresponding with the node N. N e and satisfy the following conditions:
3 el e 1, and (3.53) l1
mN N AN e Ae , (3.54) e1 where el (l = 1, 2, 3) is the global node number of the CST element e. Eq. (3.53) ensures that Ae and of all elements are distributed to the nodes without any loss or surplus. The value of is usually determined by a geometrical division procedure applied to the triangular mesh, such as a Voronoi division or a simple trisection, as
el shown in Figure 3.10. For example, when applying a trisection, e is set as
34
1 e1 e2 e3 . (3.55) e e e 3
Because the strain vector of a CST element ˆe is defined as Eq. (3.35), Eq. (3.52) can be expressed as
mN 1 N ˆ ˆ N e Ae Be de BN d N , (3.56) AN e1 where BN is the displacement-strain relationship matrix for the NB element at node
N, which consists of Be (e = 1, 2, , mN); and d N is the vector of the displacement components of nodes consisting of the CST elements with respect to the node N . Because the strains are evaluated at each node, the stress components will also be updated at each node with following equation,
σ N DN N . (3.57)
where σ N is the stress vector of the node N, and DN is the stress-strain relationship matrix at the node N. When an elasto-plastic constitutive model is used,
DN is calculated with the stress components of the node N. Discretizing Eq. (3.15) or Eq. (3.19) using Eq. (3.56) and Eq. (3.57), the stiffness matrix for the NB element is derived as
M Ki K N , (3.58) N where
T T K B D B d A B D B , N N N N N N N N (3.59) N
Here, N is the domain over which the NB element of the node N exists. Applying Eq. M (3.58) instead of Ki in Eq. (A.6) of Appendix A, we can evaluate the stresses and strains at the nodes. It should be noted that the formulations of the matrices except and the implementation of the boundary conditions and the contact conditions are the same because this method is based on the conventional displacement field approximation with the nodes and the shape functions of the CST elements, which correspond to the mathematical covers and weighting functions in NMM.
35
3.3.2 Modification for NMM The NB element formulation needs to be further modified for its application in
N NMM. As mentioned in Section 3.3.1, in FEM, the area rate of the CST elements e is usually determined by Voronoi division or the trisection of the elements. In NMM, however, because the mathematical mesh can be generated independently of the physical domain, irregularly shaped elements may exist in which nodes are not located on a physical boundary, as shown in Figure 3.11. In this case, only the area of the effective physical domain Ae in the CST element should be distributed to the nodes. If Ae is divided by the conventional trisection method, as shown in Figure 3.12(a), the partition
el rate e for each node el (l = 1, 2, 3) will be, e1 e2 e3 e1 Ae e2 Ae e3 Ae e 0 , e 0 and e 0, (3.60) Ae Ae Ae
e1 e2 e3 respectively, because Ae 0 , Ae 0 and Ae 0 . In this case, recalling Eqs.
(3.57) and (3.59), all values of Be1 and K e1 are zero, and consequently, the M computation will fail because the components of the stiffness matrix Ki corresponding to the node e1 becomes zero, and the simultaneous linear equation cannot be solved. In this study, therefore, we suppose a method to determine using weighting functions of the mathematical covers as follows, and as shown in Figure 3.12(b):
el e welxg , yg l 1,2,3. (3.61)
Here, (xg, yg) are the coordinates at the centroid of the effective physical domain, and
wel xg , yg is the weighting function of the node el described in Section 3.1. always remains positive in this formulation because (xg, yg) is not on the element edge if
Ae > 0, and because the weighting functions satisfy the partition of unity condition set in Eq. (3.29) in arbitrary CST elements, Eq. (3.61) automatically satisfies Eq. (3.53). It should be noted that if the entire area of an element is filled with the physical domain,
(xg, yg) corresponds with the centroid of the mathematical CST element and every becomes 1/3, which is consistent with the conventional method given by Eq. (3.55). The influence of the irregular elements on the computational accuracy is shown in the next section.
36
Node e(3) Effective physical domain
CST element e Ae
Node e(1) Node e(2)
Figure 3.11 Element with irregularly shaped physical domain in NMM.
Node e(3)
Centroid of CST element
e3 Ae
e2 Ae e1 Ae 0 Node e(1) Node e(2)
e1 e2 e2 e3 e3 e 0, e Ae Ae 0, e Ae Ae 0 (a)
Node e(3) Centroid of physical domain
in the CST element x g , y g
Node e(1) Node e(2)
el e wel x g , y g 0, l 1,2,3 (b) Figure 3.12 Methods of determining the element area rate: (a) conventional method
and (b) using the weighting functions of the mathematical covers.
37
3.4 Numerical examples
3.4.1 Cantilever beam The bending of an incompressible elastic beam is a typical problem in which significant volumetric locking is exhibited 3-7),3-8). This problem is first simulated under a plane strain condition to verify the proposed NMM with the NB element. The cantilever beam considered in this simulation is shown in Figure 3.13 (length L = 480 mm and height H = 60 mm), and its material properties are set to have a Young’s modulus E = 3.0 107 kPa, and Poisson’s ratio ν = 0.4999. The Cartesian coordinates (x, y) are set along the parallel and normal directions to the beam axis, respectively. A distributed load P of 1000 kPa was applied on the tip (right side edge) of the beam in
3-10) the –y direction. The theoretical solution of the vertical deflection vtheoretical is expressed as
P 2 1 2 3 vtheoretical 3y L x 4 5 H x 3L xx , (3.62) 6EI 4
1 I H 3 , (3.63) 12
E E , and (3.64) 1 2
. (3.65) 1
Here, the deflection is positive in the y direction, and I is the second moment of area when the unit depth is assumed.
y L=480 mm
H/2 x H=60 mm E=3.0×107 kPa, ν=0.4999 A P=1000 kPa
Figure 3.13 Cantilever beam bending problem.
38
(a)
70mm
(b)
Figure 3.14 Numerical meshes considered in the simulation: (a) nodes are located on the physical
boundaries of the beam and (b) nodes are not located on the physical boundaries of the beam.
0.5 0 -0.5-0.5 -1-1 Theoretical -1.5-1.5 NB - mesh(a) -2-2 NB - mesh(b) -2.5
Deflection [mm] -2.5 CST - mesh(a) Deflection [mm] -3-3 CST - mesh(b) -3.5-3.5 0 100 200 300 400 500500 x [mm] x [mm] Figure 3.15 Distribution of the deflection (DOF=4402).
Examples of NMM meshes used in the simulations are shown in Figure 3.14. Two types of mesh having the same number of nodes and elements are used: (a) one in which nodes are located on the physical boundaries of the beam; and (b) one in which nodes are not located on the physical boundaries of the beam. The simulations are performed using the CST element and the NB element, respectively. The simulation results are compared with the theoretical solution from Eq. (3.62), and the influences of the degrees of freedom and the existence of irregularly shaped elements are discussed. All cases are performed under static condition in which an inertia term is not considered because contact does not occur in these cases.
Figure 3.15 shows the computed deflection distribution along the horizontal axis of the beam (y=0) using each element and in the finest mesh types, in which there are 4402 degrees of freedom (DOFs). The theoretical solution obtained by Eq. (3.62) is also shown in the figure by the solid line. The results using the CST element clearly show a
39 smaller deflection than the theoretical solution, regardless of the mesh type, as was predicted in the description of Eq. (3.62) above. On the other hand, the results using the NB element agreed quite well with the theoretical deflection for both mesh types.
Figure 3.16 shows the relationship between the DOFs and the error in the vertical deflection of the tip (point A in Figure 3.13). Here, the error from the theoretical solution vtheoretical is defined as
v vtheoretical Error 100%. (3.66) vtheoretical
First we discuss the results of mesh (a), in which the node locations match with the physical boundary. The results using the CST element (indicated by the closed triangles) show 80–90% smaller deflections than the theoretical values, regardless of the number of DOFs, which is a typically stiff response resulting from volumetric locking. On the other hand, the results using the NB element (indicated by the closed squares) show significant improvement, especially when the DOF is sufficiently increased. The deflections computed with the CST element show smaller (stiffer) results than the theoretical value and become softer as the DOFs increase. In contrast, the deflections computed with the NB element show larger (softer) values than the theoretical values with small DOFs and converge to the theoretical solution as the DOFs increase, which is mainly caused by the underestimation of the strain energy during the deformation due to the nodal strain averaging process described by Eq. (3.56). A similar tendency has been reported in the FEM simulations using the NB element 3-9).
120 100 NB - mesh (a) 80 NB - mesh (b) 60 CST - mesh (a) 40 CST - mesh (b) 20 0 -20
Error [%] Error -40 -60 -80 -100 -120 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Number of DOF Figure 3.16 Error in tip deflection.
40
For mesh (b), the deflections computed with the CST element (open triangle) show that the error is significant regardless of the DOF. In contrast, the deflections computed with the NB element (open square) show an error that converges to zero a s the DOF increases. These results demonstrate that the NB element improves the performance of NMM in an elastic problem. In addition, it should be noted that although both types of the element showed larger error than the results of mesh (a), the increase of the error as the DOF increases can only be disregarded when the NB element is used. In a past literature3-11), the accuracy degradation in the finite cover method (FCM), which is another name for NMM, was revealed to result from distortion of the physical domain due to node discordance with physical boundary. On the other hand, Kurumatani et al.3-12) examined the performance of the NB element in distorted FE mesh, and the NB element showed little accuracy degradation even when the mesh was strongly distorted. Therefore, the NB element is considered capable of improving NMM performance when nodes are not located on the physical boundaries, which should be investigated in more detail in future research.
3.4.2 Bearing capacity of strip footing under vertical load In this section, the performance of the proposed method applied to elasto -plastic analyses of soils will be examined with the bearing capacity problem of the vertically loaded strip footing on cohesive soils 3-13). This problem usually suffers from volumetric locking because the volumetric strain of soil during failure is governed by plastic flow regardless of the setting of the elastic parameters as Poisson’s ratio. Figure 3.17 shows an example of the analytical model in a 2D plane strain condition. The model consists of two discontinuous materials, a strip footing modeled by DDA and a ground modeled by NMM. The analytical domain is set to be sufficiently large so that the boundary conditions do not affect the simulation results. The lateral side of the ground is fixed in the horizontal direction, and the bottom of the ground is fixed in all directions. As in the cantilever problem of Section 3.4.1, meshes with different fineness having the same aspect ratio of the elements are considered. The numbers of DOFs and triangular elements beneath the footing are summarized in Table 3.1; Figure 3.17(a) shows the finest of the meshes: mesh No. 5 with 9968 DOFs.
41
B=2.0m H=0.1m Strip footing (DDA)
4.0m Ground (NMM)
12.0m (a)
CL Displace 10mm
(b) Figure 3.17 Analytical domain, boundary conditions, and the finest mesh applied to the bearing capacity problem of a footing: (a) entire model and (b) locations of loading point.
Table 3.1 Mesh information.
Mesh No.
1 2 3 4 5
Footing (DDA) 6 6 6 6 6
DOF Ground (NMM) 458 1682 3674 6434 9962
Total 464 1688 3680 6440 9968 Number of triangular elements 4 8 12 16 20 beneath the footing
Table 3.2 Material properties of the ground and footing.
Ground Footing
(von Mises) (Elastic)
Young’s modulus [kPa] 1.0×105 1.0×107
Poisson’s ratio 0.3 0.0
Cohesion [kPa] 10.0 N/A
Table 3.3 Material surface properties of the ground and footing.
Normal direction [kN/m] 1.0×106 Contact penalty coefficient Tangential direction [kN/m] 1.0×106
Friction angle [°] 89.5
Cohesion [kPa] 0.0
42
The simulations were performed for each mesh with the CST element and the NB element, respectively, to investigate the relationship between the number of DOFs and the solution accuracy. The material properties of the footing and ground used in the simulations are summarized in Table 3.2. The footing is modeled as a linear elastic material with large stiffness, in which deformation can be almost disregarded. The ground is modeled as elasto-perfectly plastic material using von Mises’s failure criterion, assuming the associated flow rule. It should be noted that the weights of the materials are ignored to compare the simulation results with Prandtl’s theoretical solution3-14) of the bearing capacity of a strip footing on a weightless cohesive ground. Table 3.3 shows the parameters of the material surface. Here, the bottom of the footing is assumed to be rough, and a sufficiently large friction angle is set so that the relative displacements between the footing and ground are zero along the contacted surface. In the NMM-DDA, the penalty coefficients for the contact treatment have to be set for both the normal and the tangential directions to the contact surface. Akao et al. 3-15) showed that an accurate contact solution could be obtained if the normal penalty coefficient is set to be approximately 1.0 104 times larger than the maximum contact force (kN) that occurs in the simulations. Therefore, we performed some preliminary analyses to
4 estimate the order of a contact force with various penalty coefficients (pn = 1.0 10 , 1.0 105, 1.0 106 and 1.0 107 kN/m), and the contact force at each contact pair (vertex and edge) was estimated by multiplying the penalty coefficient by the penetration distance dn. From the preliminary analysis, considering the relationship between the order of the maximum contact force and the penalty coefficient, we selected
6 pn = 1.0 10 kN/m. The tangential penalty coefficient corresponds to the shear stiffness of the discontinuous surface. Because the bottom face of the footing is assumed to be rough in this study, the tangential penalty is also set to be 1.0 106 kN/m to prevent the relative tangential displacement between the footing and the ground. The simulations were performed in quasi-static condition for the reasons explained in Appendix B. A downward vertical displacement of 10 mm was applied at the center point on the top surface of the footing (see Figure 3.17(b)) in 10,000 steps. The displacement of the footing, which is the displacement boundary condition, was applied by the penalty method with a very stiff spring in the same manner as in a reference3-2). In the series of
43 analyses, the bearing capacity was defined as the reaction force at the loading point, and the reaction force was estimated by multiplying the penalty coefficient by the residual of the applied displacement.
The computed relationships between the footing displacement and the applied force for each DOF are shown in Figure 3.18. Figure 3.18(a) shows the results using the CST element, and Figure 3.18(b) shows the results using the NB element. In the figures, Prandtl’s solution3-14),
Vtheoretical 2 c B , (3.67) is also indicated by the gray line. Here, c is the cohesion of the soil, B is the footing width, and Vtheoretical 102.8 kN in the case of the present problem, where c = 10.0 kPa and B = 2.0 m. When using the NB element, the vertical load converged to an almost constant value, and the ultimate load could be easily determined. However, when using the CST element, the vertical load continued to increase even after yielding. Therefore, we compared the final value of the computed vertical load with the theoretical solution.
200 180 160
[kN] 140
V 120 100 Prandtl’s solution 80 60 DOF=464 DOF=1688
40 DOF=3680 DOF=6440 Vertical load Vertical 20 DOF=9968 0 0 2 4 6 8 10 12 Displacement of the footing [mm] (a) 200 180 160
[kN] 140
V 120 100 Prandtl’s solution 80 60 DOF=464 DOF=1688
40 DOF=3680 DOF=6440 Vertical load Vertical 20 DOF=9968 0 0 2 4 6 8 10 12 Displacement of the footing [mm] (b) Figure 3.18 Load-displacement relationship for each DOF: (a) CST element and (b) NB element.
44
100 90 80 NB 70 CST 60 50 40
Error [%] Error 30 20 10 0 -10 0 2000 4000 6000 8000 10000 Number of DOF
Figure 3.19 Error of the ultimate bearing capacity.
Figure 3.19 shows the relationship between the number of DOFs and the error of the computed ultimate load for each element type. The error of the ultimate load is defined as
V Vtheoretical Error 100%. (3.68) Vtheoretical
In the case of the CST element, the error slightly decreases as the DOFs increase, but remains larger than 20% with 9968 DOFs. In contrast, in the case of the NB element, the error is relatively small even with few DOFs. Furthermore, the results adequately converge to the theoretical solution as the DOFs increase.
Figure 3.20 shows the final distribution of the deviator strain in the case of the finest mesh (DOF = 9968). In the figure, the slip line assumed in Prandtl’s solution is also indicated by the dotted line. In the case of the CST element shown in Figure 3.20(a), the shear bands are predicted to form from the tip of the footing along the arrangement of the elements, but the bands do not correspond to those determined in Prandtl’s solution. In addition, the slip lines that reach the ground surface are not clearly seen in the surrounding ground, which is why the ultimate load is not clear in Figure 3.18(a). In the case of the NB element shown in Figure 3.20(b), clear slip lines are predicted that nearly agree with the theoretical ones. From these results, the NMM-DDA with the NB element is demonstrated to simulate the bearing capacity problem in elastoplastic ground with sufficiently high precision.
45
2.10 1.96 1.82 4 1.68 1.54
1.40 Strain [%] 1.26 Plandtl’s solution 1.12 0.98 0.84 0.70 0.56 0.42 0.28 0.14 0.0 (a) 2.10 1.96 1.82 4 1.68 1.54
1.40 Strain [%] 1.26 Plandtl’s solution 1.12 0.98 0.84 0.70 0.56 0.42 0.28 0.14 0.0 (b) Figure 3.20 Collapse mechanism represented with the deviator strain contours for the finest mesh of 9968 DOFs: (a) CST element and (b) NB element.
3.4.3 Bearing capacity of strip footing under eccentric vertical load Because each of the masonry structure’s stones will move and/or rotate individually, load eccentricity will often occur at the interface between stones and the ground. Therefore, the ability of the proposed method to simulate the change in th e state of the contact between stone and ground and to predict the ultimate bearing capacity under varied load eccentricities is discussed here. Toward this objective, we solved the bearing capacity problem of the strip footing on the weightless cohesive g round under an eccentric vertical load. In Figure 3.21, e is the distance from the center of the footing to the loading point, the so-called eccentricity. Although the theoretical solution of the bearing capacity for this problem has not been obtained, an empirical equation proposed by Meyerhof has been widely adopted in practice3-16):
Ve 2 c B , (3.69) where B B1 2e B, (3.70)
Here, Ve is the ultimate load under an eccentric load, c is the cohesion of the soil, B is the effective footing width, and e/B is the normalized eccentricity.
46
CL e Displace 10mm
Figure 3.21 Bearing capacity problem of the strip footing under eccentric vertical load.
The NMM-DDA simulation using the NB element was conducted under a quasi-static condition assuming the same analytical domain shown in Figure 3.17. The mesh employed for the simulation was the mesh No. 5 used in Section 3.4.2, which was the finest one, with 9968 DOFs. The boundary conditions except loading were also the same as those in Section 3.4.2: the bottom of the ground was fixed in all directions; and the lateral sides were fixed in the horizontal direction. The material properties and the parameters for the material surfaces were also the same as those listed in Table 3.2 and Table 3.3. The ground was thus assumed to be a weightless von Mises elasto-perfectly plastic material, and the surface condition of the bottom of the footing was assumed to be rough. The downward vertical displacement of 10 mm was applied at the loading point in 10,000 steps. It should be noted that the loading point was not constrained in the horizontal direction during the loading process. In order to obtain the relationship between the eccentricity and the ultimate load, five cases with different eccentricities were considered: e = 0.0, 0.2, 0.4, 0.6 and 0.8. Here, e = 0.0 represents the case of the central loading that was already demonstrated in Section 3.4.2.
The load-displacement relationships obtained from each case are shown in Figure 3.22. It can be seen that the bearing capacity for every case finally reaches its ultimate value, and the ultimate bearing capacity tends to decrease as the eccentricity increases. Figure 3.23 shows the final distribution of the deviator strain for each case, which is already shown in Figure 3.20 for e = 0.0. As the eccentricity increases, the footing clearly rotates in the anti-clockwise direction, the bottom of the footing partially separates from the ground surface, and the contacting area of the footing reduces. Consequently, the slip line in the ground has diminished, and the ultimate bearing capacity has reduced. The relationship between the normalized eccentricity e/B and the
47 computed ultimate load is summarized in Table 3.4 and depicted in Figure 3.24. As a reference, the values predicted by Eq. (3.69), proposed by Meyerhof, are also indicated. The computed ultimate loads are slightly larger than Meyerhof’s hypothesis, but show a quite similar tendency in the ultimate load reduction due to load eccentricity. These results demonstrate that the proposed method can reproduce the changes due to load eccentricity in the contact between the footing and ground. Moreover, the computed ultimate bearing capacity almost agrees with Meyerhof’s equation.
120 e=0.0 100 e=0.2 [kN] 80 V e=0.4 60 e=0.6 40 e=0.8
Vertical load Vertical 20
0 0 2 4 6 8 10 12 Displacement of the footing [mm]
Figure 3.22 Load-displacement relationship for each eccentricity.
3.0 2.8 2.6 2.4 2.2
2.0 Strain [%] 1.8 1.6 1.4 1.2 1.0 0.80 0.60 0.40 0.20 (a) 0.0 3.0 2.8 2.6 2.4 2.2
2.0 Strain [%] 1.8 1.6 1.4 1.2 1.0 0.80 0.60 0.40 0.20 (b) 0.0 Figure 3.23 Final distribution of deviator strain for each eccentricity: (a) e = 0.2, (b) e = 0.4, (c) e = 0.6, and (d) e = 0.8 (continues to next page).
48
3.0 2.8 2.6 2.4 2.2
2.0 Strain [%] 1.8 1.6 1.4 1.2 1.0 0.80 0.60 0.40 0.20 (c) 0.0 3.0 2.8 2.6 2.4 2.2
2.0 Strain [%] 1.8 1.6 1.4 1.2 1.0 0.80 0.60 0.40 0.20 (d) 0.0 Figure 3.23 Final distribution of deviator strain for each eccentricity: (a) e = 0.2, (b) e = 0.4, (c) e = 0.6, and (d) e = 0.8.
Table 3.4 Ultimate bearing capacity of strip footing under vertical eccentric load
for varying eccentricities. Normalized eccentricity e/B
0.0 0.1 0.2 0.3 0.4
Meyerhof’s hypothesis 102.8 82.24 61.68 41.12 20.56 Ve [kN] NMM-DDA 104.8 86.64 65.61 44.56 24.12
120
[kN] 100 NMM-DDA
Ve Meryerhof's hypothesis 80
60
40
20
Ultimate vertical loadverticalUltimate 0 0 0.1 0.2 0.3 0.4 0.5 Normalized eccentricity e/B
Figure 3.24 Relationship between normalized eccentricity e/B and the ultimate vertical load.
49
3.5 Summary
A numerical method that can fully consider both nonlinear characteristics of materials and interactions between continua has been developed in order to analyze the mechanical behaviors of composite soil and masonry structures. The elasto -plastic NMM-DDA was formulated and its advantages over other numerical methods in the strong coupling analyses of continua and discontinua are described. A drawback of the method is identified that is caused by volumetric locking of the conventional CST element in NMM, which has not been reported in the previous literature. In order to avoid volumetric locking, a NB element is newly introduced to NMM-DDA, and the modification needed to implement the NB element in NMM is proposed as well. The proposed method is applied to three fundamental boundary value problems and the method’s performance is examined. First, the bending of an incompressible elastic cantilever beam was simulated by the proposed method, and the calculated deflection was compared with both the theoretical solution and the results obtained by the NMM with the conventional CST elements. Although the CST element showed extremely stiff results, the NMM with the NB element predicted the deflection much more precisely. In addition, the error increase due to the irregularly shaped elements was proven to be insignificant as long as the mesh used for the simulation is sufficiently fine. Second, the bearing capacity problem of the strip footing under vertical load was solved as an example of a boundary value problem concerning the material nonlinearity of the soil. The precision of the computed ultimate load is improved with the NB element, and the collapse mechanism obtained from the results almost agreed with the theoretical results. Third, the bearing capacity of the strip footing under eccentric vertical load was simulated, and the proposed method reproduced the reduction of the ultimate load due to the footing rotation and its separation with the ground. The simulation results shown in this chapter proved the verification of the proposed elasto-plastic NMM-DDA for the mechanical interaction problems between the soil and the blocky structures.
50
References 3- 1) Mi ki , S ., S as aki , T., K o ya ma , T., Nishiyama , S ., Oh ni s h i, Y. : Development of coupled
discontinuous deformation anal ysis and numerical manifold method (NMM - DDA) ,
In t . J. C o mp ut . Met hod s , 7, pp. 1– 20, 2010.
3- 2) Sh i , G.H . and Go od man , R .E.: Generalization of two - dimensional discontinuous
deformation analysis for forward modelling , I n t. J. N u m er. A na l . Methods Geomech .,
13, 35 9– 380, 1989.
3- 3) Shi, G.H.: Manifold method of material analysis, Trans. 9th Army Conference on Applied
Mathematics and Computing, U.S. Army Research Office, Report No. 92-1, 1991.
3- 4) M el en k , J .M ., B ab u š ka , I.: The partition of unity finite element method: bas i c th eo r y
and applications , Co m pu t . Method Appl . Mech . E ng ., 13 9, pp. 28 9– 31 4 , 1996.
3- 5) B abu š ka , I ., M el en k , J .M.: The partition of unity method , I nt . J. N u m er. Met ho d En g .,
40, pp. 7 27– 75 8 , 1997.
3- 6) Hughes, T.J.R.: The finite element method: linear static and dynamic finite element analysis,
Englewood Cliffs, NJ, USA, Prentice-Hall, 1987.
3- 7) Fl an agan , D .P. and B el yt s c h ko , T.: A uniform strain hexahedron and quadrilateral with
orthogonal hourglass control , I nt . J. N u m er. Met h o d En g ., 17, 67 9– 706, 1981.
3- 8) B el yt s ch ko , T. and B in d e man , L.P.: Assumed strain stabilization of the 4 - no d e
quadrilateral with 1 -point quadrature for nonlinear problems , Co m pu t . Method Appl .
Mech . E ng ., 88, 3 11 –3 40 , 1991.
3- 9) D oh r man n , C .R ., Heinstein , M .W., J un g , J ., K e y, S .W. and Wi t ko ws ki , W.R.:
N od e - based uniform strain elements for three - node triangular and four -n od e
tetrahedral meshes , I n t. J. N u m er. Met h od . E ng ., 47, pp. 1 54 9–1 56 8 , 2000.
3- 10 ) Timoshenko SP, Gere JM. Theory of elastic stability. New York, NY, USA: McGraw -Hill;
1961.
3- 11 ) Terada, K. and Kurumatani, M.: Performance assessment of generalized elements in the finite
cover method, Finite Elem. Anal. Des., 41, 111–132, 2004.
3- 12 ) Kurumatani, M., Kojima, T. and Terada, K.: Performance assessment of nodal -integration
finite element method, Trans. JSCES, Paper No. 20070015, 2007 (in Japanese).
3- 13 ) Sloan, S. and Randolph, M.: Numerical prediction of collapse loads using finite element
methods, Int. J. Numer. Anal. Method Geomech., 6, pp. 47–76, 1982.
3- 14 ) Prandtl, L.: Über die härte plastischer körper, Nachr. Ges. Wiss Göttingen Math. Phys., Kl, pp.
74–85, 1920.
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3- 15 ) Akao, S., Ohnishi, Y., Nishiyama, S. and Nishimura, T.: Comprehending DDA for a block
behavior under dynamic condition, Proc. 8th Int. Conf. on the Anal. of Discontinuous
Deformation, Beijing, China: China University of Mining and Technology, August 14–19,
2007.
3- 16 ) Meyerhof G.G.: The bearing capacity of foundations under eccentric and inclined loads, Proc.
3rd Int. Conf. Soil Mech. Found. Eng., 1, pp. 440–445, 1953.
52
Chapter 4 Deformation Analyses of an Existing Masonry Structure in the Angkor Ruins
In this chapter, elasto-plastic NMM-DDA enhanced with a node-based uniform strain element formulated in Chapter 3 is applied to deformation analyses of an existing masonry structure in the Angkor ruins, and the applicability of the method is discussed in comparison with field observation results obtained by JSA (Japanese government team for Safeguarding Angkor)4-1). Firstly, the conditions before and after restoration works of the Prasat Suor Prat N1 tower (PSPN1) will be overviewed in Section 4.1. In 4.2, the outline of the laboratory mechanical tests for in-situ soil samples obtained from the PSPN1, and the methodology to determine the parameters for the elasto-plastic constitutive model are explained. In Section 4.3, deformation analysis of the PSPN1 under self-weight load is carried out. The applicability of the NMM-DDA is discussed with respect to the behaviors of both the upper masonry structure and the foundation ground, and furthermore, the deterioration/failure mechanisms of the PSPN1 caused by soil deformation (uneven settlement) are reinvestigated based on the simulated results. In Section 4.4, parametric study on the initial density of the foundation soil is performed to investigate the influence of the compaction quality of the man -made mound on the stability of the entire masonry building. Finally, the findings obtained in this chapter are summarized in Section 4.5.
53
4.1 Overview of Prasat Suor Prat N1 tower
This chapter focuses on the PSPN1 in the Angkor ruins as a target for simulations. The PSPN1 is a member of the twelve Prasat Suor Prat towers, which are arranged in a row at the east end of the Royal Plaza, and face the Terrace of the Elephants (see Figure 4.14-2)). The structure of all twelve towers is nearly identical and all towers show signs of deterioration such as the inclination and/or the collapse of the stones, which require future restorations. Since PSPN1 had the largest inclination and highest risk of collapse among the towers, the restoration of the entire tower was performed from 2002 to 2005 by JSA as the pilot restoration project of the twelve towers. The structural features and damages prior to the restoration reported by JSA 4-1) are described below.
N6 N Angkor Thom
N5
N4 Angkor Wat
N3 North Pond
N2
N1
S1
S2
South Pond Prasat Suor Prat S3
S4 Bayon S5
S6
Figure 4.1 Location of Prasat Suor Prat N1 tower4-2).
54
The PSPN1 consists of a main tower and an antechamber (see Figure 4.2), similar to other eleven towers. The antechamber faces south and the bathing pond is located to the north of the tower. The main tower is composed of the platform, three stories (1 st, 2nd, and 3rd), and the roof. The main building material of the tower is laterite blocks and the sand stones are partially used, and the stones are stacked without mortar ( i.e., “dry masonry”).
Figure 4.3 provides a schematic figure of the foundation. The foundation comprises natural and man-made soil layers. The natural ground is excavated to a depth of 2.0 m, and filled with compacted soil for reinforcement. Then, the platform is constructed using the compacted soil mound covered by laterite blocks. The soil types of the natural ground and compacted soil layers are clayey and sandy soil, respec tively. This is the typical foundation structure of the Angkor ruins.
G.L.+21
Roof
G.L.+17 3rd Story
G.L.+13.5
2nd Story
G.L.+9.5
Antechamber
1st Story
G.L.+1.6 Platform G.L.+0
Pond
Figure 4.2 Northwest view of Prasat Suor Prat N1 tower after restoration 4-1).
North South Man-made Approx. 2.5m mound
Pond Man-made filling Approx. 2.0m
Natural ground
Figure 4.3 Schematic figure of inner structure of the platform.
55
According to the survey by JSA 4-1), the structural damages before the restoration are summarized as follows (also see Figure 4.4).
· The tower inclined 4.60%/2.63° toward north.
· Fallen stones, due to the inclination of the tower, were observed.
· A number of critical joint openings were observed along the wall of the tower (particularly around windows).
With regard to the foundation, uneven settlement of the ground surface was observed (see Figure 4.4). The north side settled approximately 40 cm larger than the southern side, and this uneven settlement caused the inclination of the tower and other structural damages. In order to determine the cause of the uneven settlement, an archaeological trench survey to investigate the interior conditions of the platform was performed during the dismantling process4-1),4-3). Figure 4.5 illustrates the sketch of the north-south cross section of the platform drawn during the archaeological survey4-1),4-3).
Vertical axis
Northward inclination Approx. 4.60%
Apparent joint openings between the stones
Horizontal axis Uneven settlement of the foundation ground
Figure 4.4 Observed damages of Prasat Suor Prat N1 tower before restoration (west side view) (added explanation to reference 4-1)).
North South Sheared area
Slip lines
Figure 4.5 Observed deformation of the platform (north-south cross section) 4-1),4-3).
56
A sheared zone was observed just behind the northern platform stones (gray colored area), in addition to several slip lines that initiated at the corner of the stones. The geotechnical group of the JSA pointed out that the original mound had barely enough strength in the unsaturated state. However, strength reduction due to wetting of the mound during rainfalls may have resulted in the failures observed at stress concentrated points4-3). As a fact that further support this hypothesis, sudden increase of inclination after a strong rainfall was observed among the field monitoring data before the restoration4-1).
As described above, PSPN1 is a typical example of a building deteriorated due to ground deformation, and therefore would be a suitable example to examine the applicability of the NMM-DDA to the soil-masonry structure interaction problem. Additionally, since the other eleven Prasat Suor Prat towers remain untreated, a detailed investigation into the foundation behavior observed through simulated results may provide useful information for future restoration projects.
4.2 Constitutive modeling of foundation soils
To predict soil deformations of actual structures, appropriate constitutive modeling is essential. However, the mechanical characteristics (stress-strain relationship) of the soils in the Angkor area have not been studied to date. Hence, this section provides results of the newly conducted mechanical testing for in-situ soil samples and their constitutive modelings. As mentioned in Section 4.1, the foundation ground of PSPN1 contains two types of soil, the man-made layer and the natural layer. Hence, this section begins by providing the mechanical test results of the man-made and the natural layers, respectively, followed by a description of the constitutive modeling of each soil type.
4.2.1 Man-made compacted mound soil In order to understand the fundamental mechanical characteristics of the compacted mound, oedometer test and drained triaxial compression test were performed. The sample used in the laboratory tests was soil used by Fukuda et al.4-4) in test construction of the compacted soil mound shown in Photo 4.1.
57
(a) (b) Photo 4.1 Test construction of the compacted soil mound:
(a) soil compaction, (b) after completion 4-4).
Table 4.1 Initial void ratio of mound soil specimens for oedometer test.
Reconstituted specimen Compacted
A B Specimen
Soil particle density [g/cm3] 2.664 2.664 2.664
Dry density [g/cm3] 1.414 1.525 1.904
Void ratio 0.884 0.747 0.399
This soil is composed of the original excavated soil and some fine grains, blended to improve the compaction characteristics. More specifically, this sample is not the original soil; however, its mechanical characteristics would be similar to those of the original soil. The optimal water content and maximum dry density of the sample soil obtained from the compaction test were 14% and 1.9 g/cm3, respectively4-4). Based on these testing data, the test mound was constructed controlling the water content at 14%.
An oedometer test was performed for three specimens; one specimen being the specimen obtained from the test mound without disturbance, while the remaining two specimens were the reconstituted samples, with density lesser than that of the compacted (undisturbed) specimen. Table 4.1 provides the initial void ratio of each specimen. Dry density of the compacted specimen was found to be 1.904 g/cm3, which was found to be in accordance with the maximum dry density obtained from the compaction test4-4). Considering this, the compacted specimen was expected to yield
58 nearly identical behavior to that of the man-made mound when the compaction quality was properly controlled. Figure 4.6 demonstrates the e-logσv relationship obtained from oedometer tests. Reconstituted specimens, which are relatively loose, were shown to converge to one line, i.e., the normal consolidation line, as the stress increases. The compacted specimen showed small and almost reversible compressibility, which is typical behavior of overconsolidated soil.
Triaxial compression test was performed for the two reconstituted specimens. One specimen was loose (e0 = 0.651), while the remaining specimen was relatively dense (e0 = 0.466). In both cases, the initial confining pressure was 196 kPa, and the axial strain was applied under drained and constant mean effective stress conditions. Figure 4.7 shows the stress-strain relationship for each specimen. The loose specimen shows monotonous hardening and compression behavior. Contrastively, the dense specimen shows strain softening and dilation behavior.
Vertical stress σv [kPa] 1 10 100 1000 10000 1 Reconstituted specimen A Reconstituted specimen B
0.8 Compacted specimen e 0.6
0.4 Void ratio Void
0.2
0
Figure 4.6 e-logσv relationship of the mound soil. 2 -10
Volumetric strain
1 -5
'
p
/ q Axial strain εa [%] 0 0 0 5 10 15 20
ε
v Stress rate Stress
[%] -1 e0=0.651 5 e0=0.466
-2 10
Figure 4.7 Stress-strain relationship of the soil for compacted mound
obtained from the drained triaxial compression test.
59
These testing results indicate that the soil samples obtained from the man-made compacted mound shows different behavior depending on the initial void ratio, which is the typical behavior of the soil. Since the compacted mound is usually dense and overconsolidated, a suitable elasto-plastic constitutive model that can reproduce the influence of the soil density should be used to model the soils described in this section.
4.2.2 Natural layer soil The sample of the natural layer soil was obtained at a location near the Prasat Suor Prat towers, as indicated in Figure 4.8. The soil was sampled from approximately 1.1 m depth without disturbing the sample (by block sampling). Since it was difficult to obtain a sufficiently large in-situ soil sample from the heritage area, only one case of oedometer test could be performed.
Sampling point N
Prasat Suor Prat towers N1
100m
Figure 4.8 Sampling location of the natural layer soil (Map data: Google, CNES/Astrium).
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Table 4.2 provides the initial void ratio of each specimen. Since the specimen was undisturbed, this information could be used in the stability analyses of the PSPN1 in
Section 4.3 and Section 4.4. Figure 4.9 shows the e-logσv relationship obtained from the laboratory tests. According to the consolidation curve, the specimen was initially slightly overconsolidated; however, typical yielding and compression behavior of normally consolidated soil was observed as the load increased. During the unloading and reloading processes, nearly reversible behavior may be observed. In consideration of these testing results, the natural layer also should be modeled using an elasto-plastic model that can represent mechanical behaviors of overconsolidated soil.
4.2.3 Constitutive modeling with subloading Cam-clay model As described in Section 4.2.1 and 4.2.2, both the man-made mound soil and the natural layer soil yield typical mechanical behaviors of overconsolidated soil. Therefore, in this section, the modified Cam-clay model4-5) extended with a subloading surface concept4-6) (hereafter referred to as ‘subloading Cam-clay model’), an elasto-plastic constitutive model for normally consolidated and overconsolidated soil, is introduced to NMM-DDA, and is applied to model the soils in the Angkor area. A detailed explanation of the subloading Cam-clay model is provided in Appendix C.
Table 4.2 Initial void ratio of natural layer soil specimens for oedometer test. Soil particle density [g/cm3] 2.664
Dry density [g/cm3] 1.662
Void ratio 0.603
Vertical stress σv [kPa] 1 10 100 1000 10000 1
0.8 e 0.6
0.4 Void ratio Void
0.2
0
Figure 4.9 e-logσv relationship of the natural layer soil.
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The material parameters of the man-made mound and natural layer are summarized in Table 4.3. These values were determined by simulating the test results shown in Section 4.2.1 and 4.2.2, where only the critical state stress ratio of the natural layer soil has been set, assuming the internal friction angle was 30 because triaxial compression test had not been performed.
Figure 4.10 shows the simulated e-logσv relationship of the man-made mound soil. The symbols represent the results of the laboratory tests, and the solid lines represent the simulated results. The subloading Cam-clay model adequately reproduced the laboratory test results regardless of the initial void ratio. Figure 4.11 illustrates the simulated results of drained triaxial compression with constant mean effective stress. Although the simulated results yielded a moderately higher stress ratio for dense specimen, the model successfully reproduced hardening and compression behavior of the loose specimen, and the strain softening and dilatancy behavior of dense specimen with same material parameter set.
Figure 4.12 illustrates the simulated e-logσv relationship for the natural layer soil. The symbols and solid lines are the results obtained from the laboratory tests and simulation, respectively. The simulated results were found to be in accordance with those obtained from laboratory testing. Additional experiments such as triaxial compression test are necessary to discuss the validity of the parameter set of Table 4.3 for shear deformations, but gradual change to a normally consolidated state during the compression process was reproduced successfully.
Table 4.3 Material parameters of each soil used in the subloading Cam-clay model.
Parameters Man-made mound Natural layer
Compression index: λ 0.0580 0.0719
Swelling index: κ 0.00484 0.00711
Void ratio on normal consolidation line 0.700 0.478 at p’=98.0 kPa: eNC Critical state stress ratio: M 1.37 1.20
Poisson’s ratio: ν 0.3 0.3
Parameter for evolution rule of 25 100 reducing overconsolidation: a
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The above-mentioned results confirm that the fundamental mechanical behaviors of the soils in the Angkor area may be adequately modeled using the subloading Cam-clay model. Considering this, this model has been employed in the deformation analysis of the PSPN1, which is provided in the next section, using the material parameters determined in this section.
Vertical stress σv [kPa] 1 10 100 1000 10000 1 Reconstituted specimen A Reconstituted specimen B
0.8 Compacted specimen e 0.6
0.4 Void ratio Void
0.2
0
Figure 4.10 Simulated results of oedometer test on man-made mound soil.
2 -10
Volumetric strain
1 -5
'
p
/ q Axial strain εa [%] 0 0 0 5 10 15 20
ε
v Stress rate Stress
[%] -1 e0=0.651 5 e0=0.466
-2 10
Figure 4.11 Simulated results of triaxial test on man-made mound soil.
Vertical stress σv [kPa] 1 10 100 1000 10000 1
0.8 e 0.6
0.4 Void ratio Void
0.2
0
Figure 4.12 Simulated results of oedometer test on the natural layer soil.
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4.3 Deformation analysis of Prasat Suor Prat N1 tower
4.3.1 Analytical conditions Figure 4.13 illustrates the numerical model of the Prasat Suor Prat N1 tower (in 2D, Plane strain condition), which utilized the sketch of the west side view after restoration drawn by JSA architectonics group. The masonry blocks were modeled using DDA, and the foundation ground consisting of the natural layer and the man-made layer was modeled using NMM with a node-based uniform strain element. This modeling enables to focus on the large displacement of the blocks (including joint openings) and the local stress/strain fields in the foundation ground simultaneously. Regarding the boundary conditions, the displacements in all directions, and only horizontal direction were fixed along the bottom and lateral side edges of the analytical domain, respectively. It should be noted that the natural layer and the man-made compacted mound were modeled separately as independent blocks. The reason for this is that the constitutive relationship matrix is defined at each node when the node-based element is used, and thus different materials cannot share the nodes. Therefore, equal displacement conditions should be applied between each of the two layers along material boundaries to maintain continuity of the displacement fields. A detailed explanation for this specific treatment is provided in Appendix D.
North South
3rd story
2nd story Masonry stones Approx. 18m Approx. Window (no stone) 1st story Approx. 8m
Man-made layer Platform
Embankment
4.4m Approx.
Natural layer 6.5m
34m
Figure 4.13 Analytical domain of the Prasat Suor Prat N1 tower.
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Table 4.4 Material properties of the laterite block. Density [g/cm3] 3.06 Young’s modulus: E [kPa] 1.0×106 Poisson’s ratio: ν 0.2
Table 4.5 Properties of the material interface. Normal direction [kN/m] 5.0×106 Contact penalty coefficient Tangential direction [kN/m] 5.0×103 Friction angle [°] 36.0 Cohesion [kPa] 0.0
The soils were modeled using the subloading Cam-clay model4-6), and the material parameters determined in Section 4.2 (see Table 4.3) were applied to each soil layer. The masonry stones (laterite blocks) were modeled as linear elastic material, and the material properties are summarized in Table 4.4. All values of the stones were determined from laboratory tests. Table 4.5 provides the properties of the contacted material interfaces determined using results from the direct shear test for laterite blocks.
The initial stress and initial void ratio of the ground, and the initial stress of the masonry stones, were determined from self-weight analyses assuming the following steps of construction (also see Figure 4.14):
Step 1: Self-weight analysis of flat natural ground Step 2: Excavation analysis for the pond embankment Step 3: Excavation analysis for the man-made filling Step 4: Construction analysis of the man-made filling Step 5: Construction analysis of the Embankment stones Step 6: Construction analyses of the platform
For Step 1, the initial void ratio of the natural layer was set to 0.603 based on the data of the undisturbed sample shown in Table 4.2, and a unit weight of 16.2 kN/m3 was applied assuming the dry density that corresponds to the void ratio. For Steps 3–6, the initial void ratio of the newly constructed man-made layer was set to 0.399 based on the data of the undisturbed sample of the test mound shown in Section 4.2.1. This condition implies that the man-made mound was constructed with optimal compaction control. A unit weight of 18.6 kN/m3 was applied assuming dry density. For the laterite blocks, a unit weight of 30.0 kN/m3 was applied as determined by the density shown in Table 4.4.
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Natural layer
(a)
Natural layer
(b)
Natural layer
(c)
Man-made filling
Natural layer
(d)
Embankment Man-made filling
Natural layer
(e)
Platform Embankment Man-made filling
Natural layer
(f)
Figure 4.14 Construction process assumed to estimate the initial stress:
(a) Step 1, (b) Step 2, (c) Step 3, (d) Step 4, (e) Step 5, (f) Step 6.
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The initial mean stress and the initial deviator stress conditions are shown in Figure 4.15. The following features may be seen from the figure.
· Stress concentration appeared beneath the platform stones of the bottom layer.
· The mean stress distributed symmetrically around the tower.
· The deviator stress was relatively high in the northern (left) part of the ground due to the northern side not being confined in the lateral direction due to the existence of the pond.
From the above-mentioned initial conditions, the weight of all masonry stones above the platform was applied gradually over 100,000 steps in quasi-static condition, and the load was kept for another 100,000 steps because the inertia term of NMM-DDA may result in the delay of load transmission between the materials, even in the quasi-static condition. It should be noted that this analysis does not consider the stiffness and strength increase of soils due to suction in the unsaturated state because the material properties of soils applied here were determined from laboratory testing of the saturated specimen. Therefore, the final deformation after the deterioration of soils has been computed.
165.0 154.0 143.0 132.0 121.0 110.0 99.0 88.0 77.0 66.0 55.0 44.0 33.0 22.0 11.0 0.0 Mean stress [kPa] (a) 75.0 70.0 65.0 60.0 55.0 50.0 45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 Deviator stress [kPa] (b) Figure 4.15 Initial stress conditions: (a) mean stress, (b) deviator stress.
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4.3.2 Simulation results Figure 4.16 illustrates the final distribution of the settlement. This figure clearly shows that the northern part of the tower settled larger than the southern part. In order to estimate the uneven settlement, evolution of the settlement at the monitoring points (centroid of Stones A and B located at the bottom of the 1st story shown in Figure 4.16) is shown in Figure 4.17. This figure shows that settlement of both stones almost converges at the final state, and the northern stone A settled approximately 11 cm larger than the southern stone B.
0.660 0.630 0.600 0.570 0.540 0.510 0.480 0.450 Stone A Stone B 0.420 0.390 0.360 0.330 0.300 0.270 0.240
Settlement [m]
Figure 4.16 Final distribution of the settlement.
Step Number 0 50000 100000 150000 200000 0 0.1 0.2 Stone A Stone B 0.3 0.4 0.5
Settlement [m] Settlement 0.6 0.7 0.8 Figure 4.17 Settlement at the monitoring points.
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Figure 4.18 shows the inclination of the tower at each calculation step estimated by the following equation: y y B A 100%, (4.1) xB xA where (xI, yI) are the centroid coordinates of stone I (= A, B). The inclination of the tower nearly converges at the final state, similar to the settlement, and the final calculated inclination was determined to be 1.63% toward the north. Figure 4.19 shows the enlarged view of the wall around the window at the final state. From this figure, many joint openings between the stones on the wall caused by the uneven settlement and the inclination of the tower were observed. Joint openings mainly appeared in areas where a joint was vertically connected, such as around the window, as observed from the survey prior to the restoration. Although the estimated inclination and uneven settlement smaller than the actual damages before the restoration (uneven settlement: approx. 40 cm, inclination: approx. 4.60%), the deterioration behaviors of the upper structure could be qualitatively reproduced.
2 1.8 1.6 1.4 1.2 1 0.8
0.6 Inclination [%] 0.4 0.2 0 0 50000 100000 150000 200000 Step Number Figure 4.18 Inclination of the tower.
Figure 4.19 Enlarged view of Figure 4.17 around the window.
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The next topic focuses on the mechanical behaviors of the foundation ground, which is a significant factor in uneven settlements and deformation of upper structures.
Figure 4.20 shows the final distribution of the deviator strain. The behavior of the man-made layer and the platform are discussed and compared with that obtained from the trench survey, as shown in Section 4.1. This figure illustrates that the strain concentration area initiated in the man-made mound just behind the northern platform stones where the uneven settlement occurred. A comparable strain concentration may be observed in the results of the trench survey of the actual building before restoration (see Figure 4.5). An enlarged figure illustrating the final distribution of the settlement, the deviator strain, and the deviator stress near the northern platform are shown in Figure 4.21(a), (b) and (c), respectively. Figure 4.21 illustrates that the deviator stress was concentrated at the corners of the platform stones. In addition, the strain concentration area is formed from the same corners where the strain concentration initiated. This implies that the shear deformation in the man-made mound was caused by the stress concentration due to the platform structure, which consists of soil and masonry stones. These simulated results were found to be in accordance with those obtained in the pre-restoration survey, as described in Section 4.1.
0.180 0.168 0.156 0.144 0.132 0.120 0.108 0.0960 0.0840 0.0720 0.0600 0.0480 0.0360 0.0240 0.0120 0.00 Deviator strain
Figure 4.20 Final distribution of the deviator strain.
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0.660 0.630 0.600 0.570 0.540 0.510 0.480 0.450 0.420 0.390 0.360 0.330 0.300 0.270 0.240
Settlement [m]
(a)
0.180 0.168 0.156 0.144 0.132 0.120 0.108 0.0960 0.0840 0.0720 0.0600 0.0480 0.0360 0.0240 0.0120 0.00 Deviator strain
(b)
900.0 840.0 780.0 720.0 Element 5619 660.0 600.0 540.0 Element 5246 480.0 Element 5016 420.0 360.0 300.0 240.0 180.0 120.0 60.0 0.0 Deviator stress [kPa]
(c)
Figure 4.21 Mechanical behaviors around the northern platform: (a) final distribution of the settlement, (b) final distribution of the deviator strain, (c) final distribution of the deviator stress.
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Figure 4.22, Figure 4.23, and Figure 4.24 illustrate the stress paths and e-logp relationships at the elements where stress concentration, as shown in Figure 4.21(c) (element number 5016, 5246, and 5619), was observed. Although no element reached the critical state line (C.S.L.), the deviator stress increased for all elements as the structural load increased. Deviator stress q [kPa] C.S.L. Deviator stress q [kPa] e 700 C.S.L. e q=Mp 0.9 N.C.L. 600
0.8 ]
500 0.7
kPa [
q 0.6 400 e
0.5 e 300 0.4 Initial state
Void ratio Void 0.3 Deviator stress q [kPa] 200
Deviator stress stress Deviator 0.2 100 0.1 Initial state 0.0 0 10 100 1000 0 100 200 300 400 500 600 700 MeanMean stress stress p [kPa]p [kPa] MeanMean stressstress pp [kPa][kPa] (a) (b) Deviator stress q [kPa] C.S.L. Figure 4.22 Mechanical behaviors of Deviator element stress q [kPa] 5016: (a) stress path, (b) e-logp relationship. e 700 C.S.L. e q=Mp 0.9 N.C.L. 600
0.8 ]
500 0.7
kPa [
q 0.6 400 e
0.5 e
300 0.4 Initial state Void ratio Void
Deviator stress q [kPa] 0.3 200 Deviator stress stress Deviator 0.2 100 0.1 Initial state 0.0 0 0 100 200 300 400 500 600 700 10 100 1000 MeanMean stress stress p [kPa]p [kPa] MeanMean stressstress pp [kPa][kPa] (a) (b) Deviator stress q [kPa] C.S.L. Figure 4.23 Mechanical behaviors ofDeviator element stress q [kPa] 5246: (a) stress path, (b) e-logp relationship. e 700 C.S.L. e q=Mp 0.9 N.C.L. 600
0.8 ]
500 0.7
kPa [
q 0.6 400 e
0.5 e
300 0.4 Initial state Void ratio Void
Deviator stress q [kPa] 0.3 200 Deviator stress stress Deviator 0.2 100 0.1 Initial state 0.0 0 0 100 200 300 400 500 600 700 10 100 1000 MeanMean stress stress p [kPa]p [kPa] MeanMean stressstress pp [kPa][kPa] (a) (b) Figure 4.24 Mechanical behaviors of element 5619: (a) stress path, (b) e-logp relationship.
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The behavior of the natural layer, which was not directly observed in the past field survey will now be discussed. From Figure 4.20, significantly large strain was observed in the natural soil layer located to the rear of the pond embankment. Figure 4.25 shows the normal strain in x (horizontal) and y (vertical) directions for each calculation step for element 3443, with the black circled area indicating where the deviator strain was the largest (Figure 4.25(a)). Figure 4.25(b) demonstrates that element 3443 was compressed vertically and extended horizontally. This phenomenon was a consequence of the northern side not being confined in the lateral direction due to the existence of the pond, thus resulting in larger deviator strain occurring in the natural layer behind the embankment. However, regardless of the large strain, the settlement of this area was not very large (see Figure 4.16). Therefore, shear deformation of this portion was not determined as a contributing factor to the uneven settlement. Finally, the stress path and e-logp curve of element 3443 (see Figure 4.25(a)) are shown in Figure 4.26. Since the natural layer is relatively loose, the element remains near to the failure state, but does not reach the C.S.L. G8 G9 0.3 0.2 x y
0.1
G8 0.0
Element 3443 Strain -0.1
-0.2
-0.3 0 50000 100000 150000 200000 Step numberG7 (a) (b)
Figure 4.25 Deformation of element 3443:Deviator stress (a) q [kPa] element location, (b) evolution of the strains. C.S.L. Deviator stress q [kPa]
200 e C.S.L. e q=Mp 0.7 N.C.L.
] 150 0.6 kPa
[ 0.5 q q
e Initial state
100 0.4 e
0.3 Void ratio Void Deviator stress q [kPa] 0.2
Deviator stress stress Deviator 50 0.1 Initial state 0.0 0 0 50 100 150 200 10 100 1000 MeanMean stress stress p [kPa]p [kPa] MeanMean stress p [kPa][kPa] (a) (b) Figure 4.26 Mechanical behaviors of element 3443: (a) stress path, (b) e-logp relationship.
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Figure 4.27 shows the final distribution of the volumetric strain. As shown in Figure 4.23, the man-made mound does not exhibit a large change in volume. Conversely, uneven compression was observed in the natural layer, which resulted in uneven settlement. Figure 4.28 compares e-logp curves of elements 2997 and 3043 as shown in Figure 4.27, which are located at the northern and southern region of the upper natural layer, respectively. The initial conditions and inclinations of the curves of the two elements were found to be comparable. However, the final mean stress of element 2997 was found to be larger than that of element 3043, and consequently, element 2997 was largely compressed. In order to investigate the cause of the uneven load, the distribution of the vertical stress in the foundation ground is shown in Figure 4.29. The larger vertical load was transmitted from the corner of the northern platform stones. Hence, the uneven load on the natural layer may have been generated by the inclination of the tower due to shear deformation of the platform mound.
0.100 0.0867 0.0733 0.0600 0.0467 0.0333 0.0200 0.00667 -0.00667 -0.0200 -0.0333 -0.0467 Element 2997 Element 3043 -0.0600 -0.0733 -0.0867
Volumetric strain
Figure 4.27 Final distribution of the volumetric strain (compression isNorth positive) . C.S.L. North N.C.L. 0.7 South South Element 2997 0.6 E Element 3043 0.5
e Initial state
0.4 e
0.3 Void ratio Void 0.2
0.1
0.0 10 100 1000 Mean stress p [kPa] Mean stress p [kPa] Figure 4.28 Comparison of the e-logp relationship between elements 2997 and 3043.
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450.0 420.0 390.0 360.0 330.0 300.0 270.0 240.0 210.0 180.0 150.0 120.0 90.0 60.0 30.0 0.0 Vertical stress [kPa]
Figure 4.29 Final distribution of the vertical stress.
From these simulated results, the deformation process of the PSPN1 may be summarized as follows.
Phase 1: Shear deformation in the man-made mound occurred behind the stones of the northern platform due to stress concentration at the stone corners. Phase 2: Uneven settlement occurred due to the shear deformation in the ground , and the tower inclined northward. Phase 3: The inclination of the tower instigated progression of the load concentration at the northern platform. Phase 4: The load concentration at the northern platform produced uneven compression of the natural layer, prompting further shear deformation in the platform mound. Phase 5: Progression in uneven settlement and tower inclination occurred.
In this section, deformation analysis of PSPN1 was performed with newly developed elasto-plastic NMM-DDA. While the estimated inclination of the tower was found to be smaller than that of the actual building, the pre-restoration damages were qualitatively reproduced using this method. Additionally, the deterioration mechanisms of the tower were evaluated through the simulated results. Considering the mechanisms shown above, future restoration project works for the Prasat Suor Prat towers, which have structures similar to PSPN1, should take into account the compressibility of the natural layer and the bearing capacity of the platform structure. However, in the case of the actual construction, since the compression (consolidation) of the natural layer had already finished during past hundreds of years, the bearing capacity evaluation of the platform and the construction control of the mound pose significant problems that require consideration.
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4.4 Parametric study on the construction quality of the platform mound
The simulated results provided in Section 4.3 demonstrate that the degradation of the platform due to shear deformation of the mound is an important factor in the restoration. The mechanical behavior of the man-made mound soil was significantly affected by its density (or void ratio), as described in Section 4.2. Therefore, the compaction degree during the construction process of the mound ma y have significant impact on the stability of the structures. This section describes the series of parametric studies performed on the initial density of the man-made mound, a state parameter that determines its stiffness and strength. Additionally, the influences of construction quality of the mound on the structural stability are also discussed.
4.4.1 Analytical conditions The simulated analytical domain and boundary conditions were identical to those observed, as presented in Section 4.3 (see Figure 4.13). In addition, the material parameters provided in Table 4.3 and Table 4.4 were applied to the soils and masonry stones, respectively, and the contact parameters of Table 4.5 were used. The initial void ratio in the initial stress analysis was altered to study its influences on the construction quality of the platform mound. In Section 4.3, the initial void ratio was set as 0.399 (ρd = 1.89 g/cm3), assuming a well-compacted state. The result of in-situ density test of the
4-1) 3 mound soil prior to restoration was calculated as ρd = 1.67–1.73 g/cm . Therefore, an
3 3 additional two cases assuming e0 = 0.50 (ρd = 1.77 g/cm ) and e0 = 0.60 (ρd = 1.66 g/cm ) were considered for the loosely compacted cases. Hereafter, the cases with e0 = 0.399, 0.50, 0.60 will be termed Case 1, 2, and 3, respectively. The void ratio distributions obtained from initial stress analyses are shown in Figure 4.30. The distributions of the void ratio in the natural layer are almost identical for all cases, and variations in the man-made filling and mound may be observed. From these initial conditions, the weight of masonry stones above the platform was gradually applied in steps of 100,000 under quasi-static condition, and the load is kept for another 100,000 steps, as same as the analysis in Section 4.3.
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(a) 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 (b) 0.40 0.38 0.36 0.34 0.32 0.30 Void ratio
(c)
Figure 4.30 Initial void ratio distribution: (a) Case 1, (b) Case 2 and (c) Case 3.
4.4.2 Simulation results Figure 4.31 illustrates the final distribution of the settlement for each case. The simulated results for e0 = 0.399, which has previously been shown in Section 4.3, has also been included for comparison (the range of the contour has been modified). The northern side of the tower settled more than the southern side in all cases, and the settlement of the tower increased as the density of the mound soil reduced. Regarding the foundation ground, the settlement distribution in the natural layer was not found to differ among the cases; however, local settlements in the platform mound progressed as the mound density decreased. The amount of uneven settlement between stones A and B has been compared in Figure 4.31, and the inclination estimated using Eq. (4.1) is summarized in Table 4.6. Figure 4.31 demonstrates that the uneven settlement and inclination increases as the mound soil density decreases, and that the overall deformation increases, particularly in Cases 2 and 3. Figure 4.32 provides an enlarged view of the wall around the window at the final state. The joint openings among the stones was also found to increase when the mound soil was loosely compacted.
77
0.810 0.770 0.730 0.690 0.650 0.610 0.570 0.530 Stone A Stone B 0.490 0.440 0.410 0.370 0.330 0.290 0.250
Settlement [m] (a)
0.810 0.770 0.730 0.690 0.650 0.610 0.570 0.530 Stone A Stone B 0.490 0.440 0.410 0.370 0.330 0.290 0.250
Settlement [m] (b)
0.810 0.770 0.730 0.690 0.650 0.610 0.570 0.530 Stone A Stone B 0.490 0.440 0.410 0.370 0.330 0.290 0.250
Settlement [m] (c) Figure 4.31 Final distribution of the settlement for varying initial void ratios of the mound:
(a) Case 1, (b) Case 2, (c) Case 3.
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Table 4.6 Estimated uneven settlement and inclination for each case.
Simulation case Observed value 1 2 3 before restoration
Uneven settlement [cm] 11.3 12.1 14.6 Approx. 40.0
Inclination [%] 1.63 1.73 2.05 Approx. 4.6
(a) (b)
(c)
Figure 4.32 Joint openings around the window: (a) Case 1, (b) Case 2, (c) Case 3.
As described in the previous section, the main cause of tower instability was the uneven compression of the natural layer and the shear deformation of the mound behind the stones of the northern platform. Therefore, the results of each case were compared, focusing on the above-mentioned behaviors. Figure 4.33 illustrates the final distribution of the volumetric strain for each case, and Figure 4.34 compares the simulated e-logp curves at the elements (elements 2997 and 3943) within natural layer (see Figure 4.33). These figures show the difference of the compression behavior in the natural layer among the cases to be relatively small. Therefore, increase of the uneven settlement and inclination shown in Table 4.6 was determined as not having been caused by compression in the natural ground layer.
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0.100 0.0867 0.0733 0.0600 0.0467 0.0333 0.0200 0.00667 -0.00667 -0.0200 -0.0333 -0.0467 Element 2997 Element 3043 -0.0600 -0.0733 -0.0867
Volumetric strain (a)
0.100 0.0867 0.0733 0.0600 0.0467 0.0333 0.0200 0.00667 -0.00667 -0.0200 -0.0333 -0.0467 Element 2997 Element 3043 -0.0600 -0.0733 -0.0867
Volumetric strain (b)
0.100 0.0867 0.0733 0.0600 0.0467 0.0333 0.0200 0.00667 -0.00667 -0.0200 -0.0333 -0.0467 Element 2997 Element 3043 -0.0600 -0.0733 -0.0867
Volumetric strain (c) Figure 4.33 Final distribution of the volumetric strain (compression is positive)
for varying initial void ratios of the mound: (a) Case 1, (b) Case 2, (c) Case 3.
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e0=0.399 C.S.L. e0=0.399 0.6 N.C.L. e0=0.50 e0=0.50 e0=0.60 e0=0.60 0.5
e Initial state
e 0.4
Void ratio Void Case 1 0.3 ECase 2 Case 3
0.2 10 100 1000 MeanMean stress pp [kPa][kPa] (a) e0=0.399 C.S.L. 0.6 e0=0.399 N.C.L. e0=0.50 e0=0.50 e0=0.60 e0=0.60 0.5
e Initial state
e 0.4
Void ratio Void Case 1 0.3 ECase 2 Case 3
0.2 10 100 1000 MeanMean stress pp [kPa][kPa] (b) Figure 4.34 Comparison of simulated e-logp curves: (a) element 2997, (b) element 3043.
Figure 4.35 illustrates the final distribution of the deviator strain for each simulated case. The strain in the man-made mound obviously increased as the initial void ratio of the mound increased. A comparison of the mechanical behaviors in the simulated cases at elements 5016, 5246, and 5619 are shown in Figure 4.36, Figure 4.37 and Figure 4.38, respectively. These elements are located just behind the stones in the northern platform. Figure 4.36(a) demonstrates that element 5016 yields comparable stress paths throughout all cases. Monotonic hardening appears for all cases with regard to the d q relationship (see Figure 4.36(b)); however, the inclinations of the curves vary significantly among the cases (the inclination of the curves is steepest and most gradual in Cases 1 and 3, respectively). This may be due to the stiffness of the soil decreasing as its void ratio increases, which consequently resulted in the largest strain being observed in Case 3. It should be noted that these elements are compressed monotonically as shown in Figure 4.36(c) and (d), and do not reach the critical state. The mechanical behaviors of element 5246 shown in Figure 4.37(a)-(d) imply similar tendencies to those of element 5016.
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0.180 Element 5619 0.168 0.156 Element 5246 0.144 Element 5016 0.132 0.120 0.108 0.0960 0.0840 0.0720 0.0600 0.0480 0.0360 0.0240 0.0120 0.00 Deviator strain (a)
0.180 Element 5619 0.168 0.156 Element 5246 0.144 Element 5016 0.132 0.120 0.108 0.0960 0.0840 0.0720 0.0600 0.0480 0.0360 0.0240 0.0120 0.00 Deviator strain (b)
0.180 Element 5619 0.168 0.156 Element 5246 0.144 Element 5016 0.132 0.120 0.108 0.0960 0.0840 0.0720 0.0600 0.0480 0.0360 0.0240 0.0120 0.00 Deviator strain (c) Figure 4.35 Final distribution of the deviator strain around the northern platform
for varying initial void ratios of the mound: (a) Case 1, (b) Case 2, (c) Case 3.
82
e0=0.399 e0=0.50 e0=0.60 700 C.S.L. e0=0.399 q=Mp e0=0.50 e0=0.60 500
600
]
] kPa
500 [ 400
kPa
[
q q 400 300
300 200
Deviator stress q [kPa] Case 1 200 e0=0.399
Case 1 stressDeviator q [kPa] e0=0.50 Deviator stress stress Deviator Deviator stress Deviator ECase 2 E 100 e0=0.60 Case 2 Case 3 100 Case 3 Initial state 0 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0 100 200 300 400 500 600 700
MeanMean stress stress p [kPa]p [kPa] e0=0.399 DeviatorDeviator strain strain ƒ εd Ãd C.S.L. (a) (b) e0=0.399 0.7 N.C.L. e0=0.50 Case 1 e0=0.50 e0=0.60 ECase 2 e0=0.60-0.02 Case 1
0.6 Case 3 ECase 2
v ε
e Case 3 Initial state -0.01
e 0.5 Deviator strain εd Initial state 0.00 0.02 0.04 0.06 0.08 0.10 0.12
Void ratio Void 0.00 0.4 e0=0.399 Initial state e0=0.50
0.01 e0=0.60 Volumetric strain Volumetric 0.3 strainVolumetric ƒ Ãv 10 100 1000 0.02 MeanMean stress p [kPa][kPa] (c) (d) Figure 4.36 Comparison of mechanical behaviors of element 5016 for varying initial void ratios :
(a) stress path, (b) εd-q relationship, e0=0.399 (c) e-logp relationship, (d) εd-εv relationship. e0=0.50 e0=0.60 700 C.S.L. 700 e0=0.399 e0=0.50 q=Mp e0=0.60
600 ] 600
] kPa
500 [ 500
q
kPa [
q 400 400 300 300 200 Case 1 Deviator stress q [kPa] e0=0.399
200 Case 1 stressDeviator q [kPa] e0=0.50 Deviator stress Deviator ECase 2 Deviator stress stress Deviator e0=0.60 ECase 2 100 Case 3 100 Case 3 Initial state 0 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0 100 200 300 400 500 600 700 MeanMean stress stress p [kPa]p [kPa] DeviatorDeviator strain strain ƒ εd Ãd (a) (b) e0=0.399 C.S.L. 0.7 e0=0.399 N.C.L.-0.03 Case 1 e0=0.50 e0=0.50 Case 1 ECase 2 e0=0.60 Case 3 e0=0.60-0.02 ECase 2
0.6 v ε Case 3 e Initial state -0.01 Deviator strain ε e 0.5 d 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Initial state 0.00 Void ratio Void 0.4 e0=0.399 Initial state 0.01 e0=0.50
e0=0.60
Volumetric strain Volumetric Volumetric strainVolumetric ƒ Ãv 0.3 0.02 10 100 1000 MeanMean stress p [kPa][kPa] (c) (d) 0.03 Figure 4.37 Comparison of mechanical behaviors of element 5246 for varying initial void ratios:
(a) stress path, (b) εd-q relationship, (c) e-logp relationship, (d) εd-εv relationship.
83
e0=0.399 e0=0.50 e0=0.60 700 C.S.L. e0=0.399 500 e0=0.50 q=Mp e0=0.60
600 ]
400
] kPa
500 [ e0=0.399
kPa q [ e0=0.50 q 300 e0=0.60 400
300 200
Deviator stress q [kPa] Case 1 200
Case 1 stressDeviator q [kPa] Deviator stress stress Deviator Deviator stress Deviator 100 ECase 2 ECase 2 Case 3 100 Case 3 Initial state 0 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0 100 200 300 400 500 600 700 MeanMean stress stress p [kPa]p [kPa] DeviatorDeviator strain strain ƒ εd Ãd 500 (a) (b) 400 e0=0.399 e0=0.50 0.7 -0.02 300 e0=0.60 200
Case 1 Deviator stress q [kPa] Case 1 100 ECase 2 0 Case 3 ECase 2 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
0.6 v Deviator strain ƒ Ãd
ε -0.01 Case 3 e Initial state Deviator strain ε e 0.5 d 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
Initial state 0.00 Void ratio Void 0.4
Initial state e0=0.399 Volumetric strain Volumetric Volumetric strainVolumetric ƒ Ãv 0.01 e0=0.50 0.3 e0=0.60 10 100 1000 MeanMean stress p [kPa][kPa] (c) (d) 0.02 Figure 4.38 Comparison of mechanical behaviors of element 5619 for varying initial void ratios:
(a) stress path, (b) εd-q relationship, (c) e-logp relationship, (d) εd-εv relationship.
The trend of the mechanical behaviors of element 5619 was found to be different from that of the other two elements. Figure 4.38(a) shows that the stress paths were comparable among all cases; however, the stress level at the final state was found to decrease significantly as the mound soil density decreased. From Figure 4.38(b): the hardening behavior may be observed in Cases 1 and 2 as the deviator strain increases until the final state. The deviator stress of Case 3 was observed to nearly plateau regardless of the deviator strain continuing to increase. From Figure 4.38(c) and (d): the monotonic compression similar with elements 5016 and 5246 may be observed in Case 1. In Cases 2 and 3, compression was initially observed, but was changed to dilation in spite of the mean stress increase. In general, overconsolidated soil shows failure with strain softening after changing to dilation during drained shear deformation, as indicated in Figure 4.7. Hence, element 5619 in Case 2 and 3 is comparatively close to failure, and the difference in the final stress levels between the simulated cases was determined to be the result of differences in peak shear strength that are dependent on soil density.
84
These simulated results confirm that the loosely compacted mound induces changes in mound stiffness and strength, and furthermore destabilizes the entire masonry building. However, the calculated uneven settlements and tower inclinations were still found to be smaller than the observation values even for the loosest mound. One of the possible reasons for larger actual deformation could be reduction of the effective stress and shear strength of the soils that resulted from rainwater infiltration. For further study, in order to account for such a phenomenon, NMM-DDA should be extended to hydro-mechanical coupled analysis for unsaturated soil.
4.5 Summary
This chapter provided constitutive modeling of in-situ soils in the Angkor area, and the deformation analysis of the Prasat Suor Prat N1 tower and a se ries of parametric studies on the construction quality of the man-made mound were performed using a novel elasto-plastic NMM-DDA method. The findings obtained from constitutive modeling of in-situ soils are as follows.
(1) From mechanical test of the man-made mound soil: the well-compacted specimens showed typical behaviors of overconsolidated soil.
(2) From oedometer test of the natural layer soil: the undisturbed specimen exhibited a moderately overconsolidated state.
(3) Mechanical test results of in-situ soils were reproduced successfully using the subloading Cam-clay model, and material parameters were uniquely determined from laboratory tests.
In addition, findings obtained from deformation analysis of PSPN1 and parametric studies on the construction quality of the man-made mound are as follows.
(4) Computed mechanical behaviors of the upper masonry structure and the man -made mound were found to be in accordance with pre-restoration observations, and the applicability of the NMM-DDA was demonstrated.
(5) Simulated results indicated that the shear deformation that occurred to the rear of the northern platform caused the tower inclination and directly influenced the load
85
concentration, and consequently, instigated the uneven compression of the natural layer and further inclination of the tower.
(6) Bearing capacity evaluation of the platform structure will be important in future restoration projects of the remaining Prasat Suor Prat towers.
(7) Loosely compacted mound induces change in mound stiffness and strength, and furthermore destabilizes the entire masonry building.
(8) The computed uneven settlement and tower inclination remained smaller than that of the observation even in the case of the loosest mound.
For further study, an extension of the NMM-DDA to include coupled stress-flow simulations for unsaturated soil, and the investigation into the deterioration process considering the influence of rainwater infiltration using the method, are recommended.
References 4- 1) Japanese Government Team for Safeguarding Angkor (JSA) : Report on the
conservation and restoration work of the Prasat Suor Prat Tower , OGAWAINSATSU
Co . Lt d. , 2 00 5.
4- 2) Uchida, E., Suda, C., Ueno, A., Shimoda, I. and Nakagawa, T.: Estimation of the construction
period of Prasat Suor Prat in the Angkor monuments, Cambodia, based on the characteristics of
it stone materials and the radioactive carbon age of charcoal fragments, J. Archaeol. Sci., 32,
pp. 1339–1345, 2005.
4- 3) Iwasaki, Y., Fukuda, M., Nakagawa, K., Akazawa, Y., Shimoda, I. and Naka gawa, T.:
Geotechnical aspects of the N1 Tower, Prasat Suor Prat, Angkor Thom, Cambodia, Advanced
Material Research, 133– 134, pp. 113– 118, 2010.
4- 4) Fukuda, M., Iwasaki, Y., Nakagawa, T., Araya, M., Yamada, S. and Shimoda, I.: Bearing
capacity of foundation of Angkor ruin and reconstruction, Proc. 58th Japanese Geotechnical
Symposium, pp. 241– 248, 2013 (in Japanese).
4- 5) Roscoe, K.H. and Burland, J.B.: On the generalized stress-strain behaviour of wet clay,
Engineering Plasticity, eds. Hyeyman, J. and Leckie, F.A., Cambridge England. Cambridge
University Press, pp. 535–609, 1968.
4- 6) Hashiguchi, K. and Ueno, M.: Elastoplastic constitutive laws of granular material. Constitutive
Equations of Soils, Proc. 9th Int. Conf. Soil Mech. Found. Engrg., eds. Murayama, S.,
Schofield. A.N., Tokyo, JSSMFE, pp. 73– 82, 1977.
86
Chapter 5 Numerical Experiment on Bearing Capacity Characteristics of Masonry Platform Structures
In this chapter, a series of numerical experiments using the elasto -plastic NMM -DDA formulated in Chapter 3 to reveal the characteristics of the bearing capacity for masonry platform structures will be discussed. As described in Chapter 2 and Chapter 4, the most significant geotechnical consideration in the restoration of Angkor monuments is the estimation of the bearing capacity of the platform sustaining the weight of the upper structure. The bearing capacity of shallow foundations may often be evaluated using the limit equilibrium method that assumes the slip lines explicitly. However, no research studies have to date systematically studied failure mechanisms of the masonry platform. This is due to the structure of masonry platforms being exceedingly complex, as it contains material nonlinearity of soils and discontinuous material interfaces among the stones and ground, which may indicate slip and/or separation. This chapter begins with an introduction of the parameters expressing the structural features of the platform in Section 5.1. Then, the analytical conditions in the numerical experiments are explained in Section 5.2. In Section 5.3, results from parametric studies on the structural parameters of the platform are provided, and the failure mechanisms and bearing capacity characteristics are discussed in detail. To conclude, the findings obtained in this chapter are summarized in Section 5.4.
87
5.1 Structural parameters of masonry platform
This chapter describes the series of numerical experiments performed for masonry platforms constructed by stacking stones of equal size with a constant overlapping width, as shown in Figure 5.1. In this figure, B is the stone width, h is the stone thickness, l is the overlapping width between the stacked stones, and n is the number of steps. When the stone width B is fixed, the structural features may be expressed using the following three parameters: the aspect ratio of the stones h/B, the ratio of the overlapping width to the stone width l/B, and the number of steps n. In this study, B was fixed at 1.0 m, and the loading analyses of the platform under various parameter sets of h/B, l/B, and n were conducted, and the influence of each parameter on the bearing capacity and failure mechanisms are discussed based on the numerical results.
l
h Number of steps n
B
Figure 5.1 Structural parameters of masonry platform.
5.2 Analytical conditions
The analytical conditions of the numerical experiments, which are common to all simulated cases performed in Section 5.3, are explained in this section. An example of the analytical domain (h/B = 0.40, l/B = 0.50, n = 6) is shown in Figure 5.2. The model consists of the ground and the n masonry stones. The foundation ground was modeled using NMM with a node-based uniform strain element, and the masonry stones were modeled using DDA blocks. All the stones were equal size and stacked with uniform overlapping width. Regarding the boundary conditions, the bottom and lateral sides of the ground were fixed in all directions, and only in the horizontal direction, respectively. Table 5.1 and Table 5.2 provide the material properties and surface parameters in all
88 simulated cases. Each masonry stone was modeled as a linear elastic body. The ground was assumed an elasto-perfectly plastic material in compliance with von Mises’ failure criterion and associated flow rule. This assumption is taking into account the fact that the current restoration design of the platform mound in the Angkor ruins by JSA is dependent on the unconfined compression strength of the soil improved by slaked lime5-1). It should be noted that the influence of the weight of the materials is not considered in the analyses due to the assumption of the stress condition of the ground being isotropic at the initial state. In addition, the friction strength parameter between the materials, i.e., the surface friction angle s = 30 , is applied to both the masonry stones and the ground. Based on these conditions, a vertical displacement of 4 mm was applied downward at the center point of the upper surface of the top stone (point A in Figure 5.2) in 40,000 steps with the penalty method under the quasi-static condition. In this study, we define the reaction force at the loading point, which was estimated using the penalty coefficient and residual of the applied displacement, as the bearing capacity of the masonry platform. At each calculation step, the stress conditions of the foundation ground modeled by NMM elements with an elasto-plastic model were updated using the return mapping algorithm, and the equilibrium equation was thus solved using Newton-Raphson iteration5-2).
1.5m 1.0m 0.5m 0.5m 0.5m 0.5m 0.5m 1.5m A 0.4m Masonry stones (DDA) 0.4m 0.4m 0.4m 0.4m Ground (NMM) 0.4m
1.4m
Figure 5.2 Example of analytical model for platform bearing capacity analyses
(h/B = 0.40, l/B = 0.50, n = 6).
89
Table 5.1 Material properties for platform bearing capacity analyses. Ground Stones
(von Mises) (Elastic)
Young’s modulus [kPa] 1.0×105 1.0×107
Poisson’s ratio 0.3 0.2
Cohesion [kPa] 10.0 N/A
Table 5.2 Material surface parameters for platform bearing capacity analyses . Normal direction [kN/m] 1.0×105 Contact penalty coefficient Shear direction [kN/m] 1.0×103
Friction angle [°] 30.0
Cohesion [kPa] 0.0
Table 5.3 List of simulated structural conditions. Investigated Number Section h/B l/B n parameter of cases
Reference case 0.40 0.50 6 1
5 5.3.1 n 0.40 0.50 2 4
0.60
0.65 5.3.2 l/B 0.40 6 4 0.75
0.90
0.30
5.3.3 h/B 0.50 0.50 6 3
0.60
The simulated cases are summarized in Table 5.3. The simulation model with (h/B, l/B, n) = (0.40, 0.50, 6) illustrated in Figure 5.2 is defined as the reference case, and two cases for varying number of steps n, four cases for varying l/B, and three cases for varying h/B were simulated.
90
5.3 Simulation results
The results for the structure under conditions of (h/B, l/B, n) = (0.40, 0.50, 6), as illustrated in Figure 5.2, are provided as an example of the simulations. Figure 5.3 shows the simulated load-displacement curve at the loading point. In this figure, the curve yields near the displacement of 0.6 mm, and the bearing capacity converges to almost constant value which is termed the ultimate state. The bearing capacity at the final state was found to be approximately 30.9 kN. The distribution of the deviator strain in the foundation ground at the final state is shown in Figure 5.4. Two circular-shaped slip surfaces were generated from the lower left corner of the top stone, and they reached to the lower left corner of the third and fourth stones from the top, respectively.
40 h B 0.40,l B 0.50,n 6
30
20
Bearing capacity [kN] capacity Bearing 10
0 0 1 2 3 4 5 Displacement [mm] Figure 5.3 Load-displacement curve (h/B = 0.40, l/B = 0.50, n = 6).
0.045 0.042 0.039 0.036
0.033 Deviator strain 0.030 0.027 0.024 0.021 0.018 0.015 0.012 0.0090 0.0060 0.0030 0.0
Figure 5.4 Final distribution of the deviator strain (h/B = 0.40, l/B = 0.50, n = 6).
91
Figure 5.5 illustrates the distribution of the horizontal displacement at the final state. The obvious discontinuous displacement was observed between the third and fourth, and fourth and fifth, stones from the top, respectively. This implies that the shear stress has reached the friction strength and induced slippage along the interface between these stones. From comparative analysis of the results shown in Figure 5.4 and Figure 5.5, the shear bands formed in the foundation ground were connected to the slipped interfaces. Since the horizontal displacement is largest at the third stone from the top, progressive failure was found to have mainly occurred along the interface between the third and fourth stones. These results demonstrate that failure of bearing capacity in a masonry platform shows the complex failure mechanisms, including shear failure in the foundation ground and sliding between the stones. In addition, the ultimate bearing capacity of the masonry platform was found to be governed by both the shear resistance of the soil along the shear band and the friction along the sliding plane. The vertical loads, a factor influencing the friction strength between the stones, transmitted from the upper to the lower stones, as indicated in Figure 5.6. The vertical load that acts on a stone will decrease as it is transferred to lower layers because the total load acting on each stone is partially supported by the foundation ground. This means that the friction
7.5 7.0 6.5 Horizontal [mm]displacement 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.50 0.0
Figure 5.5 Final distribution of the horizontal displacement (h/B = 0.40, l/B = 0.50, n = 6).
30.9kN
15.1kN
7.35kN
1.23kN
0.405kN
0.0472kN
Figure 5.6 Vertical load applied on each stone at the final state (h/B = 0.40, l/B = 0.50, n = 6).
92 strength between the stones weakens in the lower layer of the stones. Contrastively, since the shear resistance of von Mises materials is purely generated from cohesion and is proportional to the length of the slip plane, the bearing capacity increases as the slip plane passes into a deeper domain. Hence, regarding evaluation of the failure mode in a platform, it would be essential to find the layer where the summation of the shear resistance force of the ground and the frictional resistance along the surfaces becomes minimal. Failure mechanisms of the platform are further discussed below by comparing results of the varying structural conditions.
5.3.1 Influence of the number of steps The influence of the number of steps on the bearing capacity of the platform is described in this section. Setting h/B = 0.40 and l/B = 0.50, the number of steps n was reduced from six (reference case) to five, and then four. The load-displacement curves for these three cases are shown in Figure 5.7. All cases show similar curves, and the ultimate bearing capacity is nearly identical in these three cases. Table 5.4 provides the ultimate bearing capacity for each case as estimated by averaging the bearing capacity following 2 mm displacement. The ultimate bearing capacity was not significantly altered, and thus determined to be independent of the number of steps.
40 h B 0.40,l B 0.50
30
20
n 6 Bearing capacity [kN] capacity Bearing 10 n=6 n=5n 5 n=4n 4
0 0 1 2 3 4 5 Displacement [mm] Figure 5.7 Load-displacement curves for varying number of steps (h/B = 0.40, l/B = 0.50).
Table 5.4 Ultimate bearing capacity for varying number of steps (h/B = 0.40, l/B = 0.50).
Number of steps n
6 5 4
Ultimate bearing capacity [kN] 30.70 30.59 30.73
93
Figure 5.8 and Figure 5.9 illustrate the distribution of the deviator strain and the horizontal displacement at the final state for each case, respectively, with a varying number of steps. These figures confirm failure along the interface between the third and fourth stones for all cases; additionally, the failure mode remains nearly constant and independent of the number of steps.
0.045 0.042 0.039 0.036
0.033 Deviator strain 0.030 0.027 0.024 0.021 0.018 0.015 0.012 0.0090 0.0060 0.0030 0.0 (a)
0.045 0.042 0.039 0.036
0.033 Deviator strain 0.030 0.027 0.024 0.021 0.018 0.015 0.012 0.0090 0.0060 0.0030 0.0 (b)
0.045 0.042 0.039 0.036
0.033 Deviator strain 0.030 0.027 0.024 0.021 0.018 0.015 0.012 0.0090 0.0060 0.0030 0.0 (c) Figure 5.8 Final distribution of deviator strain for varying number of steps (h/B = 0.40, l/B = 0.50): (a) n = 6, (b) n = 5, (c) n = 4.
94
7.5 7.0 6.5 Horizontal [mm]displacement 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.50 0.0 (a)
7.5 7.0 6.5 Horizontal [mm]displacement 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.50 0.0 (b)
7.5 7.0 6.5 Horizontal [mm]displacement 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.50 0.0 (c) Figure 5.9 Final distribution of horizontal displacement for varying number of steps (h/B = 0.40, l/B = 0.50): (a) n = 6, (b) n = 5, (c) n = 4.
95
30.9kN
15.1kN
7.35kN
1.23kN
0.405kN
0.0472kN (a)
31.1kN
15.2kN
7.19kN
1.51kN
0.424kN (b)
30.5kN
15.0kN
7.17kN
1.13kN (c) Figure 5.10 Vertical load applied on each stone at the final state (h/B = 0.40, l/B = 0.50): (a) n = 6, (b) n = 5, (c) n = 4.
Lastly, a comparison of the vertical load acting on each stone at the final state is provided in Figure 5.10. This figure illustrates load transmission conditions that are nearly identical throughout all cases, in addition to the failure mechanisms and bearing capacity. From these results, it was determined that platform structures with constant h/B and l/B yield identical failure mechanisms, and therefore, yields an ultimate bearing capacity that is independent of the number of steps n.
5.3.2 Influence of the overlapping width The influence of the overlapping width is discussed in this section. Setting h/B = 0.40 and n = 6, the ratio of the overlapping width to the stone width l/B was changed from 0.50 (reference case) to 0.60, 0.65, 0.75, and 0.90. The load-displacement curves for five cases are provided for comparison in Figure 5.11. This figure confirms that all curves reach the ultimate state. The ultimate bearing capacity for each case is summarized in Table 5.5, and the relationship between l/B and the ultimate bearing
96 capacity is illustrated in Figure 5.12. In these results, the ultimate bearing capacities of l/B = 0.50 and 0.60 were found to be almost the same, however, the ultimate load was found to increase with increasing l/B. Although the size of all the stones was equivalent and same strength parameters for the foundation ground were applied, the ultimate bearing capacity of the platform was significantly different; additionally, the ultimate load for l/B = 0.90 was observed to be twice as large as that for l/B = 0.50.
70 h B 0.40,n 6 60
50
40
30 l/B=0.50l B 0.50 20
Bearing capacity [kN] capacity Bearing l/B=0.60l B 0.60 l/B=0.65l B 0.65 10 l/B=0.75l B 0.75 l/B=0.90l B 0.90 0 0 1 2 3 4 5 Displacement [mm]
Figure 5.11 Load-displacement curves for varying l/B (h/B = 0.40, n = 6).
Table 5.5 Ultimate bearing capacity for varying l/B (h/B = 0.40, n = 6). l/B
0.50 0.60 0.65 0.75 0.90
Ultimate bearing capacity [kN] 30.70 30.84 33.21 41.43 61.28
h/B=0.40
80 h B 0.40,n 6
60
40
20 Ultimate bearing capacity [kN] capacity bearing Ultimate
0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 l/Bl/B
Figure 5.12 Relationship between l/B and the ultimate bearing capacity (h/B = 0.40, n = 6).
97
Figure 5.13 and Figure 5.14 illustrate the distribution of the deviator strain and horizontal displacement at the final state, respectively. Figure 5.13(a) and Figure 5.14(a) show the results of the reference case (h/B = 0.40, l/B = 0.50, n = 6), as previously shown (Figure 5.4 and Figure 5.5, respectively). Results illustrated in Figure 5.13 confirm that the sheared zone in the ground extends deeper as l/B is increased. Additionally, results shown in Figure 5.14 indicate that the layer where discontinuity of the displacement field originates becomes lower with increasing l/B. In the case where l/B = 0.90, the failure mode was observed to be significantly different from that of the other cases, where slide between the stones did not occur and ground beneath the bottom stone showed failure of the bearing capacity. As shown in the above-mentioned results, the relocation of the slip was determined to caused the increase in the ultimate bearing capacity with increasing l/B.
0.045 0.075 0.042 0.070 0.039 0.065 0.036 0.060
0.033 Deviator strain 0.055 Deviator strain 0.030 0.050 0.027 0.045 0.024 0.040 0.021 0.035 0.018 0.030 0.015 0.025 0.012 0.020 0.0090 0.015 0.0060 0.010 0.0030 0.0050 0.0 0.0 (a) (b)
0.075 0.075 0.070 0.070 0.065 0.065 0.060 0.060
0.055 Deviator strain 0.055 Deviator strain 0.050 0.050 0.045 0.045 0.040 0.040 0.035 0.035 0.030 0.030 0.025 0.025 0.020 0.020 0.015 0.015 0.010 0.010 0.0050 0.0050 0.0 0.0 (c) (d)
0.060 0.056 0.052 0.048
0.044 Deviator 0.040 0.036 0.032
strain 0.028 0.024 0.020 0.016 0.012 0.0080 0.0040 0.0 (e) Figure 5.13 Final distribution of deviator strain for varying l/B (h/B = 0.40, n = 6): (a) l/B = 0.50, (b) l/B = 0.60, (c) l/B = 0.65, (d) l/B = 0.75, (e) l/B = 0.90.
98
7.5 6.0
7.0 5.6 Horizontal Horizontal displacement [mm] 6.5 Horizontal [mm]displacement 5.2 6.0 4.8 5.5 4.4 5.0 4.0 4.5 3.6 4.0 3.2 3.5 2.8 3.0 2.4 2.5 2.0 2.0 1.6 1.5 1.2 1.0 0.80 0.50 0.40 0.0 0.0 (a) (b)
7.5 7.5 7.0 7.0 6.5 Horizontal [mm]displacement 6.5 Horizontal [mm]displacement 6.0 6.0 5.5 5.5 5.0 5.0 4.5 4.5 4.0 4.0 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.50 0.50 0.0 0.0 (c) (d)
6.0 5.6 5.2 Horizontal displacement [mm] 4.8 4.4 4.0 3.6 3.2 2.8 2.4 2.0 1.6 1.2 0.80 0.40 0.0 (e) Figure 5.14 Final distribution of horizontal displacement for varying l/B (h/B = 0.40, n = 6):
(a) l/B = 0.50, (b) l/B = 0.60, (c) l/B = 0.65, (d) l/B = 0.75, (e) l/B = 0.90.
The vertical load acting on each stone at the final state is shown in Figure 5.15. For all cases, the acting load was found to decrease in the lower layers of stones; however, it was difficult to perform a direct comparison due to the stress level being different among the simulated cases. Hence, “load distribution rate ri” was developed and defined as follows: Vertical load on i 1th stone from the top r . (5.1) i Vertical load on ith stone from the top
Calculated results of the load distribution rate are shown as bolded numbers in Figure 5.15. As shown in the figure, the values of the load distribution rate varied according to l/B. More specifically, the load distribution rate was observed to increase with increasing l/B. This may be a result of the increasing contacting (overlapping) width
99 between the stones. Furthermore, increased load distribution rates were confirmed to result in an increase of the vertical load acting on lower layers, while the friction strength between the stones throughout the platform remained largely unchanged. As a result, failure of the platform occurred at lower layers, and the ultimate bearing capacity increased as l/B increased. In addition, the number of steps n was not found to affect the failure mechanisms or bearing capacity (see Section 5.3.1), because both l/B and the load distribution rate are independent of n. The above results suggest that the load distribution rate is a significant factor in the evaluation of the stability of masonry platform structures.
30.9kN 0.491 31.5kN 0.592
15.1kN 0.486 18.6kN 0.603
7.35kN 0.160 11.2kN 0.538
1.23kN 0.324 6.05kN 0.211
0.405kN 0.0850 1.27kN 0.202
0.0472kN 0.257kN
(a) (b) 32.5kN 0.636 42.7kN 0.768 20.7kN 0.663 32.8kN 0.800
13.7kN 0.699 26.3kN 0.845
9.60kN 0.594 22.2kN 0.831
5.70kN 0.325 18.5kN 0.780
1.85kN 14.4kN
(c) (d) 61.3kN 0.919
56.4kN 0.917
51.7kN 0.934
48.3kN 0.937
45.2kN 0.936 42.3kN
(e) Figure 5.15 Vertical load and load distribution rate of each stone for varying l/B (h/B = 0.40, n = 6):
(a) l/B = 0.50, (b) l/B = 0.60, (c) l/B = 0.65, (d) l/B = 0.75, (e) l/B = 0.90.
100
5.3.3 Influence of the stone thickness In this section, influence of stone thickness on the bearing capacity and failure mechanisms of the platform structure is discussed. Setting l/B = 0.50 and n = 6, the aspect ratio of the stones h/B was changed from 0.40 (reference case) to 0.30, 0.50, and lastly, 0.60. Figure 5.16 illustrates the load-displacement curves of four different simulation cases. This figure shows that all curves reach the ultimate state. The ultimate load in each case was estimated by averaging the bearing capacity after 2 mm of displacement; the results are summarized in Table 5.6. In addition, the relationship between h/B and the ultimate bearing capacity is illustrated in Figure 5.17.
40 l B 0.50,n 6
30
20
h/B=0.30h B 0.30 h B 0.40 Bearing capacity [kN] capacity Bearing 10 h/B=0.40 h/B=0.50h B 0.50 h/B=0.60h B 0.60
0 0 1 2 3 4 5 Displacement [mm] Figure 5.16 Load-displacement curves for varying h/B (l/B = 0.50, n = 6). Table 5.6 Ultimate bearing capacity for varying h/B (l/B = 0.50, n = 6). h/B
0.30 0.40 0.50 0.60
Ultimate bearing capacity [kN] 32.50 30.70 31.13 32.97 l/B=0.50
40 l B 0.50,n 6
30 Ultimate bearing capacity [kN] capacity bearing Ultimate
20 0.2 0.3 0.4 0.5 0.6 0.7 h/Bh/B
Figure 5.17 Relationship between h/B and the ultimate bearing capacity (l/B = 0.50, n = 6).
101
The simulated results show that h/B also affects the bearing capacity, although its influence is not as significant as that of l/B. Additionally, the relationship between h/B and the ultimate bearing capacity was not observed to increase or decrease monotonically, but exhibit a downward convex curve with a minimal value that can be observed at h/B = 0.40. The reason for this phenomenon is further discussed below.
Figure 5.18 and Figure 5.19 are the distribution of the deviator strain and the horizontal displacement at the final state for each case, respectively. Figure 5.18 illustrates shear band extension to the third and fourth stones from the top when h/B is 0.30 and 0.40. However, the shear band could only be found at the third stone from the top when h/B is 0.50 and 0.60. Regarding Figure 5.19, displacement is shown to be its largest at the third stone from the top in all cases. Therefore, failure of the platform structures was determined to mainly progress along the interface between the third and fourth stones throughout all cases. When this occurs, the shear resistance force along the shear band in the foundation ground will increase with increasing h/B because the length of the shear band increases.
0.060 0.045 0.056 0.042 0.052 0.039 0.048 0.036
Deviator Deviator strain 0.044 0.033 Deviator strain 0.040 0.030 0.036 0.027 0.032 0.024 0.028 0.021 0.024 0.018 0.020 0.015 0.016 0.012 0.012 0.0090 0.0080 0.0060 0.0040 0.0030 0.0 0.0 (a) (b)
0.075 0.045 0.070 0.042 0.065 0.039 0.060 0.036
Deviator Deviator strain 0.055 0.033 Deviator strain 0.050 0.030 0.045 0.027 0.040 0.024 0.035 0.021 0.030 0.018 0.025 0.015 0.020 0.012 0.015 0.0090 0.010 0.0060 0.0050 0.0030 0.0 0.0 (c) (d) Figure 5.18 Final distribution of deviator strain for varying h/B (l/B = 0.50, n = 6):
(a) h/B = 0.30, (b) h/B = 0.40, (c) h/B = 0.50, (d) h/B = 0.60.
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9.0 7.5 8.4 7.0
Horizontal Horizontal [mm]displacement 7.8 6.5 Horizontal [mm]displacement 7.2 6.0 6.6 5.5 6.0 5.0 5.4 4.5 4.8 4.0 4.2 3.5 3.6 3.0 3.0 2.5 2.4 2.0 1.8 1.5 1.2 1.0 0.60 0.50 0.0 0.0 (a) (b)
9.0 7.5 8.4 7.0
Horizontal Horizontal displacement [mm] 7.8 Horizontal displacement [mm] 6.5 7.2 6.0 6.6 5.5 6.0 5.0 5.4 4.5 4.8 4.0 4.2 3.5 3.6 3.0 3.0 2.5 2.4 2.0 1.8 1.5 1.2 1.0 0.60 0.50 0.0 0.0 (c) (d) Figure 5.19 Final distribution of horizontal displacement for varying h/B (l/B = 0.50, n = 6): (a) h/B = 0.30, (b) h/B = 0.40, (c) h/B = 0.50, (d) h/B = 0.60.
The vertical load acting on each stone at the final state and the load distribution rate between the stones (bold numbers) are shown in Figure 5.20. Comparison of these results with those shown in Figure 5.15 confirm that differences in load distribution rates resulting from h/B dependency were found to be less significant than those resulting from dependency on l/B. Under the assumption of the shear band in the ground being an arc shape, the center of the slip circle for each case was drawn in Figure 5.21. The center of the slip circle was observed to shift leftward as h/B decreased; additionally, the horizontal distance between the loading point and the center of the circle was found to decrease. This may be a result of the gradient of the platform being small when h/B is small. Considering the limit equilibrium condition, the ultimate load of the platform would be governed by the balance of the moment around the center of the slip circle of the applied load, the shear resistance in the ground, the reaction force, and the friction force along the stone interface. Therefore, the ultimate bearing capacity may be large even if h/B is small when the loading point is close to the center of the slip circle in the ground; this would be the reason for the masonry platform with h/B = 0.30 yielding a larger bearing capacity than that with h/B = 0.40. Based on these results, it has been concluded that the ultimate bearing capacity is affected by the complex effect of changes of the length and center position of the slip circle due to the change of h/B.
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30.9kN 0.491
15.1kN 0.486 32.6kN 0.483 15.7kN 0.348 7.35kN 0.160 0.00 5.48kN 1.23kN 0.324 0.00kN N/A 0.405kN 0.0850 0.00kN N/A 0.00kN 0.0472kN (a) (b)
32.3kN 0.530
30.2kN 0.493 17.1kN 0.603
0.557 14.9kN 10.34kN 0.364
8.30kN 0.371 3.76kN 0.310 3.08kN 0.340 1.17kN 0.250 1.05kN 0.136
0.292kN 0.142kN (c) (d) Figure 5.20 Vertical load and load distribution rate of each stone for varying h/B (l/B = 0.50, n = 6):
(a) h/B = 0.30, (b) h/B = 0.40, (c) h/B = 0.50, (d) h/B = 0.60.
0.99m 0.81m
(a) (b) 1.89m
1.17m
(c) (d) Figure 5.21 Location of the center of the slip circle for varying h/B (l/B = 0.50, n = 6):
(a) h/B = 0.30, (b) h/B = 0.40, (c) h/B = 0.50, (d) h/B = 0.60.
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5.4 Summary
In this chapter, failure mechanisms and bearing capacity characteristics of masonry platform structures were investigated through a series of numerical experiments using the elasto-plastic NMM-DDA. The structural features of the platform were first characterized by three parameters: the aspect ratio of the stones h/B, the ratio of the overlapping width to the stone width l/B, and the number of steps n. Next, loading analyses of the platform structures under various structural conditions were performed, and the influences of each structural parameter on the failure mechanisms and bearing capacity were described. The findings obtained in this chapter are summarized as follows.
(1) The masonry platform structures show complex failure mechanisms, including shear failure in the ground and sliding between the stones.
(2) The ultimate bearing capacity of the masonry platform is dependent on both the shear resistance of the soil along the shear band and the friction force between the slipped masonry stones.
(3) The number of steps n of the platform yields no effects on failure mechanisms or the ultimate bearing capacity.
(4) As l/B increases, the failed layer shifts downward to lower layers and the bearing capacity increases. This is because the friction strength between the stones varies according to the load distribution rate from the upper to the lower stone.
(5) h/B affects the ultimate bearing capacity, as variation in h/B was shown to result in variations in length and the center position of the slip circle in the ground.
As described above, failure mechanisms and bearing capacity characteristics of masonry platforms were explained through the numerical experiments. The most remarkable feature of masonry platform structures as the foundation is described above in (4). The ultimate bearing capacity of the masonry platform increases with increasing l/B, which implies that the platform becomes steeper; however, it is well-known that a strip footing on slope ground exhibits a decrease in ultimate bearing capacity with increasing slope angle5-3),5-4). This difference is due to the structural characteristic that transmits the
105 structural load from upper stones to lower stones. Therefore, this load transmission feature, with inclusion of the load distribution rate, is a critical consideration in the stability evaluation of platforms. Based on the results obtained in this chapter, a simplified method to estimate the bearing capacity of the platform has been developed and will be proposed in Chapter 6.
References 5- 1) Japanese Government Team for Safeguarding Angkor (JSA) : Report on the
conservation and restoration work of the Prasat Suor Prat Tower , OGAWAINSATSU
Co . Lt d. , 2 00 5.
5- 2) Simo, J.C. and Taylor R.L.: Consistent tangent operators for rate -independent elastoplasticity,
Comput. Methods Appl. Mech. Eng., 48, pp. 101– 118, 1985.
5- 3) Akai, K. and Sano, I.: The bearing capacity of foundation on the inclined ground, Disaster
Prevention Research Institute Annuals, Kyoto University Disaster Prevention Research
Institute, 23(B-2), pp. 57– 71, 1980.
5- 4) Georgiadis, K.: Undrained bearing capacity of strip footings on slopes, J. Geotech. Geoenviron.
Eng., 136(5), pp. 677– 685, 2010.
106
Chapter 6 Simplified Estimation Method of Ultimate Bearing Capacity for Masonry Platform Structures
In Chapter 5, deformation analyses on a masonry platform were conducted with a numerical method that can precisely simulate the mechanical interaction problem between discontinua and continua. However, if there were a simpler stability evaluation method that properly considers failure mechanisms of platform structures, it would be highly beneficial for use in actual restoration projects of monuments. Therefore, a simplified estimation method of the ultimate bearing capacity of a masonry platform, which applies the limit equilibrium method and further incorporates the failure mechanisms revealed from results of numerical experiment conducted in Chapter 5, is described in this chapter. In Section 6.1, failure mechanisms of the platform structure are reviewed, and a simplified estimation method of the ultimate bearing capacity, which is based on the limit equilibrium of the moment of force, is subsequently proposed. In Section 6.2, the ultimate bearing capacity estimated by the proposed estimation method is compared with that estimated from the simulated results obtained with the elasto-plastic NMM-DDA, and validation of the proposed method is described. In Section 6.3, the utilization scheme of the proposed method in the restoration design process of an actual masonry structure is explained. Lastly, the resulting outcomes are summarized in Section 6.4.
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6.1 Simplified estimation method of bearing capacity for masonry platforms: Proposal
6.1.1 Assumptions of failure mechanisms At the beginning, the following conditions, which are equivalent to those applied in the numerical experiments using NMM-DDA in Chapter 5, were established for application in the development of a simplified estimation method of the ultimate bearing capacity for a platform. The conditions are provided again below for reference.
1) The initial stress of the platform mound is in isotropic condition, and shear stress is zero. In addition, the failure of the ground is determined according to von Mises’ criterion.
2) The platform structures constructed by stacking masonry stones of equal size with uniform overlapping width are assumed.
Then, the following assumptions were made based on results of the numerical experiments.
3) Masonry platform structures yield failure mechanisms that include circular slippage in the ground and sliding between the stones connected with the slip circle.
4) The vertical load acting on each stone is represented by the load distribution rate defined in Eq. (5.1).
Additionally, in the proposed method, the ground is assumed as a rigid-plastic material, and the stones are assumed to be rigid bodies.
From conditions 1 and 2, and assumption 3, the failure mechanisms shown in Figure 6.1 were assumed for a platform with n rectangular stones and ground with cohesion c under downward loading conditions. Thus, the circular slip passing the lower left corner of top stone A(0,0), the lower left corner of the ith stone from the top B
i 1 B l, i 1 h, and the sliding along the bottom surface of the ith stone are assumed.
108
OxO , yO P
y A0,0 1 x R : Ultimate bearing capacity 2 : Shear resistance of soil : Reaction force along the bottom of the stone : Friction force along the bottom of the stone
Cohesion c [kPa] i f s m i fi l Bi 1B l, i 1h F s F m i i n h B
Figure 6.1 Assumed failure mechanisms of the platform.
Fj
j
Fj 1 rj Fj rj
Figure 6.2 Load distribution rate of the jth stone from the top.
6.1.2 Derivation of the bearing capacity equation The forces acting on the sliding domain, including the stones, are defined as the ultimate load P, the shear resistance force of the soil along the slip circle, the normal
s s reaction force Fi and the friction force fi from the ground beneath the ith stone, and
m m the normal reaction force Fi and the friction force fi from the masonry stone beneath the ith stone. Considering the equilibrium of the rotation moment by these forces around the center of the slip circle OxO, yO , the following equation can be derived:
B 2 s B l P xO cR Fi xO i 1B l 2 2 (6.1) m l s m Fi xO iB l fi yO i 1h fi yO i 1h 2 where R is the radius of the slip circle, and is the central angle of the arc of the circle. Now, as determined from assumption 4, the load distribution rate at the bottom of each stone is defined as r j j 1,2,n (Figure 6.2), and the reaction force against the ground and the stone beneath the ith stone can be expressed as follows:
109
i1 s , and Fi P rj 1 ri (6.2) j1
i m , Fi P rj (6.3) j1 respectively. The friction force from the ground and the stone beneath the ith stone at the failure state are described as follows as based on Coulomb’s friction law.
i1 s s , and fi Fi tanms P rj 1 ri tanms (6.4) j1
i m m . fi Fi tanmm P rj tanmm (6.5) j1
Here, ms is the friction angle between the masonry stone and the ground, and mm is the friction angle between the stones. By incorporating Eqs. (6.2)–(6.5), Eq. (6.1) can be rewritten as follows:
i1 B 2 B l P xO cR P rj 1 ri xO i 1B l 2 j1 2 i l
P rj xO iB l (6.6) j1 2 i1 i P rj 1 ri tanms yO i 1h P rj tanmm yO i 1h. j1 j1
From Eq. (6.6), the ultimate load P can be derived as follows:
cR2 P (6.7) g
B g xO 2 i1 B l rj 1 ri xO i 1B l tanms yO i 1h (6.8) j1 2 i l rj xO iB l tanmm yO i 1h j1 2
Incidentally, in the bearing capacity analyses for platform structures with different l/B (Section 5.3.2), the load distribution rate of the stones above the slip line (Figure 5.15) were found to yield similar values with l/B. Hence, the load distribution rate r j is subsequently defined as follows:
l r r , j 1,2,n. (6.9) j B
Eq. (6.8) can be rewritten using Eq. (6.9) as follows:
110
i1 B g 1 r xO 2 (6.10) i1 i1 r i 1B l r 1 rtanms r tanmmyO i 1h.
In Eqs. (6.7) and (6.10), B, h, l, and r = l/B are the structural parameters, and c, ms , and
mm are the material properties. Furthermore, the radius of the slip circle R and the center angle of the arc are the variables determined by the center position of the slip circle O xO, yO . If the layer to where the slip circle reaches is fixed as i, the ultimate load P is affected only by the center coordinates of the slip circle xO, yO . Therefore, the ultimate bearing capacity for failure mechanisms assumed at the ith stone can be estimated by setting xO, yO so that the left-hand side of Eq. (6.7) gives the minimum value.
In order to simplify Eqs. (6.7) and (6.10), xO,yO, R, and are parameterized with a single parameter below. As shown in Figure 6.1, Eqs. (6.7) and (6.10) assume the circular slip that passes the lower left corner of the top stone A and the lower left corner of the ith stone from the top B. In this condition, since the center of the slip circle is located on the perpendicular bisector of AB, xO and yO can be defined using the distance d between point C (midpoint of ) and point O (see Figure 6.3), as expressed by the following equations. i 1 B l h xO d (6.11) 2 B l2 h2 i 1 h B l yO d (6.12) 2 B l2 h2
OxO , yO
y A0,0 1 d x R 2
C
i Bi 1B l, i 1h l n h B
Figure 6.3 Parameter of the center coordinates of the slip circle.
111
Additionally, R and can be parameterized with d as follows:
2 2 2 i 1 2 R AC OC B l h2 d 2 , and (6.13) 4
2 2 AC i 1 B l h 2arctan 2arctan , (6.14) OC 2d respectively. Hence, Eqs. (6.7) and (6.10) can thus be parameterized with d as follows: cRi,d2i,d Pi,d , and (6.15) gi,d
i1 B i1 gi,d 1 r xO r i 1B l 2 (6.16) i1 r 1 rtanms r tanmmyO i 1h, respectively. Lastly, the ultimate bearing capacity of the structure can be obtained by finding the parameter set of i and d that minimizes Eq. (6.15).
6.1.3 Implementation of the proposed method An example of estimating the bearing capacity using the proposed method is introduced below, and applies the following conditions: B = 1.0 m, h/B = 0.40, l/B =
0.50, n = 6, the cohesion c = 10 kPa, and surface friction angle ms mm 30 .
Firstly, the relationship between d and P represented by Eq. (6.15) was calculated for i = 2, 3, 4, 5, 6 (without the top stone), as shown in Figure 6.4. The cross marks in the figures represent the minimum values of P for each i. The minimum values of P for each i and corresponding values of d are summarized in Table 6.1. This table shows that the ultimate load estimated by Eq. (6.15) is minimized when i = 3 and d = 1.106 m. Therefore, the platform assumed here was predicted to fail at the third layer from the top, and the ultimate bearing capacity was estimated to be 29.26 kN. As described above, the proposed method is able to perform simple and definite predictions of the ultimate bearing capacity and failure mechanisms of masonry platform structures. It should be noted that P estimated using Eq. (6.15) may yield a negative value for some parameter sets (i, d). However, failure mechanisms corresponding to such parameters do not occur under downward loading conditions assumed here. Therefore, minimum value of P in the positive range is sought in this method.
112
80 h B 0.40,l B 0.50,i 2 80 h B 0.40,l B 0.50,i 3
60 60
] ]
kN kN
[ 40 [ 40
P P P [kN]
P [kN]
20 20
0 0 0 2 4 6 0 2 4 6 8 10 i=2 i=3 d [m] i=2 d [m] i=3 % (3) (a) (b) % (4)
100 h B 0.40,l B 0.50,i 4 100 h B 0.40,l B 0.50,i 5
80 80
] 60 ] 60
kN kN
[ [
P P
P [kN] P [kN] 40 40
20 20
0 0 0 2 4 6 8 10 0 2 4 6 8 10 d [m] d [m] d [m] d [m] (c) (d)
120 h B 0.40,l B 0.50,i 6
100
80
] kN
[ 60
P P [kN]
40
20
0 0 2 4 6 8 10 d [m] (e) Figure 6.4 Relationship between d and bearing capacity for each layer (B = 1.0m, h/B = 0.4，l/B = 0.5，n = 6): (a) i = 2, (b) i = 3, (c) i = 4, (d) i = 5, (e) i = 6.
Table 6.1 Minimum value of P and corresponding d for each layer. i d [m] Minimum value of P [kN]
2 0.265 33.64
3 1.106 29.26
4 1.794 32.77
5 2.338 38.12
6 2.801 44.25
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Additionally, the failure modes, which comprise a slip circle passing through the domain to the left of the left edge of the top stone (Figure 6.5(a)) or a domain below the bottom of the ith stone (Figure 6.5(b)), are unacceptable as failure mechanisms of platforms that shows the sliding between the stones. Indeed, such failure modes were not observed in the numerical simulations described in Chapter 5, and have thus been omitted from all estimations. The modes of Figure 6.5(a) and (b) may be removed by applying the constraint conditions to d as follows:
i 1 h B l2 h2 y 0 d , and (6.17) o 2B l
i 1B l B l2 h2 x i 1B l d , (6.18) o 2h respectively. In the estimations, the effective range of d is defined so as to satisfy both Eqs. (6.17) and (6.18).
y y Ox , y A0,0 x A0,0 x O O
OxO , yO
y 0 Unacceptable y 0 Acceptable O O (a)
OxO , yO
OxO , yO
y y A0,0 x A0,0 x
x i 1 B l Unacceptable x i 1 B l Acceptable O O (b) Figure 6.5 Constraint conditions on the location of the center of the slip circle: (a) condition for y coordinate, (b) condition for x coordinate.
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6.2 Validation of the proposed method
In this section, validation of the simplified estimation method for masonry platform ultimate bearing capacity proposed in Section 6.1 is demonstrated via a comparison with the ultimate bearing capacity estimated using the calculated results of the elasto-plastic NMM-DDA. Figure 6.6 illustrates the comparison of ultimate bearing capacity of platforms with n = 6 and h/B = 0.20, 0.30, 0.40, 0.50, and 0.60 between the simplified method and the NMM-DDA for varying l/B (0.50, 0.60, 0.65, 0.75, and 0.90).
100 100
80 NMM-DDANMM-DDA 80 NMMNMM-DDA-DDA Simplified method Simplified method
60 60
40 40
20 20 Ultimate bearing capacity [kN] capacity bearing Ultimate Ultimate bearing capacity [kN] capacity bearing Ultimate h B 0.20 h B 0.30 0 0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 l/B l/B (a) l/B (b) l/B
100 100
80 NMMNMM-DDA-DDA 80 NMMNMM-DDA-DDA Simplified method Simplified method
60 60
40 40
20 20
Ultimate bearing capacity [kN] capacity bearing Ultimate h B 0.40 [kN] capacity bearing Ultimate h B 0.50 0 0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 l/B l/B (c) l/B (d) l/B 100
80 NMMNMM-DDA-DDA Simplified method
60
40
20
Ultimate bearing capacity [kN] capacity bearing Ultimate h B 0.60 0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 l/B (e) l/B Figure 6.6 Comparison of the ultimate load estimated using the simplified method and NMM-DDA: (a) h/B = 0.20, (b) h/B = 0.30, (c) h/B = 0.40, (d) h/B = 0.50, (e) h/B = 0.60.
115
From these figures, the simplified estimation method is proved to be able to predict the ultimate bearing capacity quantitatively when h/B ranges from 0.20 to 0.40. Moreover, when h/B = 0.50 and h/B = 0.60, the ultimate load is accurately predicted when l/B ranges from 0.50 to 0.65. Figure 6.7 shows the error map of the ultimate load by the simplified method to that by the NMM-DDA drawn on h/B-l/B plane. Here, the error was estimated using the following equation: PSimplified PNMM-DDA ult ult (6.19) Error NMM-DDA 100%, Pult
Simplified NMM-DDA where Pult and Pult are the ultimate bearing capacity estimated using the simplified method and NMM-DDA, respectively. As mentioned above, the error tended to increase as h/B and l/B increased.
Figure 6.8 illustrates the comparison of failure mechanisms determined with the NMM-DDA and that predicted with the simplified method for the structural conditions indicated with cross marks in Figure 6.7. Regarding Figure 6.8(a) and (b), which yield relatively small error, the slip circle in the ground determined by the simplified method was found to be not significantly different to that determined by NMM-DDA. On the other hand, regarding the conditions applied to results shown in Figure 6.8(c) and (d), which show large error, significantly different mechanisms were indicated for each of the two methods. Hence, the simplified method was found to accurately predict the ultimate load and failed layer under the conditions of platform failure mechanisms assumed in Section 6.1.1, and from the results shown in Figure 6.7, the applicable ranges were found to be as follows: h/B 0.40 or l/B 0.65.
0.2 0.3 0.4 0.5 0.6 0.9 0.9 30.0 (d)
20.0 10.0 0.8 0.8 (c)
15.0 B
/ 0.7 0.7
l l/B
(b)
0.6 0.6 (a) 5.00
0.5 0.5 0.2 0.3 0.4 0.5 0.6 h/Bh/B Figure 6.7 Error map of the ultimate bearing capacity.
116
Predicted arc with the simplified method
(a)
Predicted arc with the simplified method
(b)
Predicted arc with the simplified method
(c)
Predicted arc with the simplified method
(d) Figure 6.8 Comparison of failure mechanisms of the simplified method and NMM-DDA:
(a) (h/B, l/B) = (0.20, 0.60), (b) (h/B, l/B) = (0.40, 0.65), (c) (h/B, l/B) = (0.50, 0.75),
(d) (h/B, l/B) = (0.60, 0.90).
117
6.3 Implementation in the restoration process
The results in Section 6.2 demonstrated that the proposed simplified method is able to perform accurate predictions of the ultimate bearing capacity and failure mechanisms under proper structural conditions. This section suggests an implementation scheme of the proposed method in the restoration process.
A flowchart of the restoration of the Angkor monuments is shown in Figure 6.9. The general restoration process is explained here. The presence/absence of foundational damage is judged based on results of an external investigation on structural damage. If detailed research on the platform is needed, dismantling and a survey of the platform should be conducted after upper structure dismantling. During the platform survey, the plate loading test and the cone penetration test on the original platform mound are performed to estimate the strength, and when possible, sampling of the soil and physical
Investigation on the structural damages
Is the investigation of No Dismantling of the upper structure the platform necessary? and repair of the stones
Yes
Dismantling including the platform
Testing the physical and the strength characteristics of the platform mound
Does the original Yes Determination of the compaction platform have sufficient control criteria of the platform mound bearing capacity?
No
Determination of the best reinforcement method
Reconstruction of the platform
Reconstruction of the upper structure
Figure 6.9 Flowchart of the restoration process in the Angkor ruins.
118 and mechanical tests are conducted. From the test results, the ultimate bearing capacity and the allowable bearing capacity considering the safety factor are estimated, and it is decided whether the original platform can sustain the structural weight, which is presumed from the stone density and height of the structure. If the original platform possesses a sufficient bearing capacity, the compaction control criteria for the platform mound are determined and the reconstruction is performed according to the criteria. However, if the original platform is unable to bear the structural weight, a reinforcement method that is able to satisfy requirements for both stability and conservation of the cultural value of the structure should be investigated and performed. In the present restoration process of Angkor monuments, improvement of the mound soil with slaked lime is mainly adopted 6-1),6-2),6-3). For the above process, the following questions need to be answered:
1) Does the original platform have sufficient bearing capacity?
2) Where and to what extent should the structure be reinforced?
3) What is the most effective method to secure the stability of the structure?
Methods to obtain practical answers to these questions via application of the proposed simplified method are introduced below.
The first question may be answered by estimating the ultimate bearing capacity using the procedure explained in Section 6.1.3, employing the observed structural parameters (B, h/B, l/B, n) and the unconfined compression strength (UCS) of the mound estimated from results of the cone penetration test during the dismantling process. It should be noted that if the ultimate bearing capacity is sufficiently high, there may be other reasons for deterioration, allowing for elimination of platform failure as the cause. In this situation, application of detailed deformation analysis using the elasto-plastic NMM-DDA is a viable option.
In regards to question two, it is possible to check the extent of reinforcement necessary to satisfy a safety factor by creating a chart of the ultimate bearing capacity for the relating parameters (e.g., the cohesion of the mound soil c, the surface friction angle ms and mm , or the cohesion along the stone interfaces if the mortar is injected).
119
160 80-4 -4
120 ] 60-3 -3 kN 80 [ 40-2 -2
ult Original state P 40 20-1 Allowable state -1 Cohesion
of original soil Design cohesion [kN]
0 [kN] 00 0 ult
0 10 20 30 40 50ult 0 5 10 15 20 25 P
-40 c [kPa] P -201 c [kPa] 1 s
-80 F -402 2
-120 -603 3 Design safety factor -160 -804 4 Figure 6.10 Example of the relationship between the mound soil cohesion,
ultimate bearing capacity Pult, and safety factor Fs.
For example, the situation with a structural load of 20 kN applied to a platform, as
provided in Section 6.1.3 (B = 1.0 m, h/B = 0.4, l/B = 0.5, n = 6, c = 10 kPa, ms = mm = 30), is used as reference here. If the design safety factor is set to 3.0, this platform posseses insufficient strength because its ultimate bearing capacity is 29.26 kN and allowable bearing capacity is 9.75 kN. In the case of mound reinforcement being planned, the cohesion necessary to satisfy the safety factor may be determined from the relationship chart for soil cohesion, ultimate bearing capacity, and safety factor, as shown in Figure 6.10. Selection of appropriate soil material for reconstruction, compaction control criteria, and necessity of soil improvement may then be determined based on the design cohesion. It should be noted that the proposed bearing capacity estimation method is capable of incorporating considerations for multiple related factors such as friction strength between stones, whereas the conventional method only considers the UCS of the mound soil. This enables us to suggest multiple reinforcement plans.
Lastly, regarding question three, a reasonable restoration method would be selectable through trial application of the construction methods suggested through the question two. Comparing the cost and construction convenience to fulfil the required reinforcement performance, the most effective method or most effective combination of several methods could be selected.
As described above, a more rational restoration could be achieved by adopting the simplified estimation method throughout the entire restoration process.
120
6.4 Summary
In this chapter, a simplified estimation method of the ultimate bearing capacity for the masonry platform was proposed to establish a simple design method that could be applicable in actual restoration projects. The proposed method is a completely novel approach that considers the complex mechanisms of ground failure and sliding between stones. The ultimate bearing capacity and failure mechanism predicted with the proposed method was found to be in agreement with those calculated through detailed numerical simulations, regardless of its simple implementation procedures. Additionally, the author suggested a design scheme using the proposed simplified method in a restoration process. Because the proposed method incorporates several essential factors related to the ultimate bearing capacity, while conventional methods only consider the soil strength, multiple reinforcement methods may be suggested and quantitatively compared. Considering these aspects, the proposed method can contribute to a rational restoration of the masonry structures. In future studies, it is important to investigate the applicability of the method to actual structures through model experiments.
References 6- 1) Japanese Government Team for Safeguarding Angkor (JSA): Report on the
conservation and restoration work of the Prasat Suor Prat Tower , OGAWAINSATSU
Co., Ltd., 2005.
6- 2) JAPAN APSARA Safeguarding Angkor (JASA): Report on the conservation research
of the Bayon, Angkor Thom, Kingdom of Cambodia , 2 011.
6- 3) Sophia University Angkor International Mission: Report of the conservation and
restoration of the Angkor Wat western causeway , 2 011.
121
122
Chapter 7 Conclusions and Future Studies
In the current study, a numerical method that can simulate the deformation of composite structures constructed from both soil and masonry stones was newly developed, and the applicability of the method was confirmed through simulations of fundamental problems. Additionally, mechanical behaviors of a masonry structure foundation were investigated using the proposed method, and a simplified design method that reflected the results of the numerical methods was subsequently developed. The conclusions obtained in each chapter are described below.
In Chapter 2, the current geotechnical restoration process and the problems in the Angkor ruins were introduced, and previous research studies on numerical methods for masonry structures were also described. Based on the reviews, topics to be addressed to establish a rational restoration design method of masonry structures are summarized as follows.
(1) Development of a numerical method to solve interaction problems for soil and masonry structures
(2) Verification and validation of the developed numerical method
(3) Investigation into the deformation and failure mechanisms of masonry structure foundation
(4) Proposal of a rational design method for the foundation based on the revealed failure mechanisms Considering these topics, a series of studies were performed as described below.
123
In Chapter 3, a numerical method yielding effective incorporation of both the nonlinear characteristics of materials and interactions between continua was developed in order to analyze the mechanical behaviors of composite soil and masonry structures. The elasto-plastic NMM-DDA was formulated and its advantages over other numerical methods as determined via strong coupling analyses of continua and discontinua were described. A disadvantage of the method was identified, and was found to be the result of volumetric locking of the conventional CST element in NMM, which, to date, has not been reported in previous literature. In order to avoid volumetric locking, an NB element was introduced to NMM-DDA, in addition to the modification needed to implement the NB element in NMM. The proposed method was applied to three fundamental boundary value problems and the performance was evaluated. First, the bending of an incompressible elastic cantilever beam was simulated with the proposed method, and the calculated deflection was then compared with both the theoretical solution and the results obtained by the NMM incorporating conventional CST elements. Although the calculated performance using CST elements significantly overestimated the stiffness, the NMM using an NB element was found to predict the deflection much more precisely. In addition, the error increase due to irregularly shaped elements was proven to be insignificant as long as the mesh used for the simulation is sufficiently fine. Second, the bearing capacity problem of the strip footing under vertical load was solved as an example of a boundary value problem concerning the material nonlinearity of soils. The accuracy of the calculated ultimate load yielded improvement with the NB element, and the collapse mechanism obtained from the results were nearly equivalent to that obtained from theoretical solutions. Third, the bearing capacity of the strip footing under eccentric vertical load was simulated, and the proposed method reproduced the reduction of the ultimate load due to footing rotation and its separation with the ground. The simulated results shown in this chapter provided verification of the proposed elasto-plastic NMM-DDA for mechanical interaction problems between soil and blocky structures.
In Chapter 4, the constitutive modeling of in-situ soils in the Angkor area, deformation analysis of the Prasat Suor Prat N1 tower, and parametric studies on the construction quality of man-made mound were conducted with elasto-plastic
124
NMM-DDA. From the mechanical tests of the platform mound soil with varying initial density, well-compacted specimen were shown to exhibit typical behaviors of overconsolidated soil. Oedometer test on the undisturbed specimen of the natural layer soil yielded moderately overconsolidated behavior. Considering these test results, constitutive modeling using the modified Cam-clay model extended with the subloading surface concept, an elasto-plastic model for normally and overconsolidated soil, was performed. The model was able to reproduce laboratory test results, and the material parameters were determined. Then, deformation analysis of the Prasat Suor Prat N1 tower was conducted using the above constitutive modeling. Computed behaviors of the upper masonry structure and the man-made mound were shown to be in accordance with the observed damages before the restoration, thus confirming the applicability of the NMM-DDA. In the simulated results, shear deformation behind the northern platform was found to cause the inclination of the tower and the load concentration, and consequently, this resulted in uneven compression of the natural layer and further inclination of the tower. Lastly, in order to study the influence of the compaction quality of the platform mound, which was found to yield significant effects on foundation strength, parametric studies on the initial density of the man-made mound soil were performed. As a result, loosely compacted mound was found to result in changes in stiffness and strength of the mound, and consequently destabilized the entire building. The above simulated results indicated that the failure of the platform and the strength of its mound control the stability of the upper structure. Hence, in the future restoration of other Prasat Suor Prat towers, precise bearing capacity evaluation of the platform structure is essential.
In Chapter 5, failure mechanisms and bearing capacity characteristics of masonry platform structures were shown to be investigated through numerical analyses using the elasto-plastic NMM-DDA. The structural features of the platform were parameterized with the aspect ratio of the stones h/B, the ratio of the overlapping width to the stone width l/B, and the number of steps n. Then, loading analyses of the platform structures under various structural conditions were performed, and the influences of each structural parameter were studied. In the simulated results of the reference case, the masonry platform structures exhibited complex failure mechanisms, including shear
125 failure in the ground and the sliding between the stones. Based on these failure mechanisms, it was concluded that the ultimate bearing capacity of a masonry platform is governed by both the shear resistance of the soil along the shear band and the friction strength along the discontinuous surface. Additionally, the following characteristics were observed for each structural parameter.
(1) The number of steps n does not affect failure mechanisms or the ultimate bearing capacity.
(2) As the ratio l/B increases, the failed layer shifts to lower layers, and the bearing capacity increases. This is because the friction strength between the stones varies according to the load distribution rate from the upper to the lower stone.
(3) The ratio h/B affects the ultimate bearing capacity as a result of the change in h/B altering the length and the center position of the slip circle in the ground.
As described above, the failure mechanisms and bearing capacity characteristics of the masonry platform were demonstrated and explained through numerical analyses.
In Chapter 6, a simplified estimation method of the ultimate bearing capacity for the masonry platform was proposed to establish a simple design method that could be applicable to actual restoration projects. The proposed method was constructed by entirely novel approach that considers the complex mechanisms of failures in the ground and sliding between the stones. The ultimate bearing capacity and failure mechanisms predicted by the proposed method were in good agreement, quantitatively, with those obtained via detailed numerical simulations, regardless of its simple implementation procedure. Additionally, a design scheme with the proposed simplified method in a restoration process was suggested. Because the proposed method incorporates the effects of structural characteristics and friction between the stones, in addition to soil strength, multiple reinforcement methods may be suggested and compared quantitatively. Considering these aspects, the proposed method is suggested to have the potential to contribute positively in the restoration of masonry structures.
126
Finally, recommendations for future studies are summarized below.
(1) The elasto-plastic NMM-DDA developed in this study was adapted for single-phase materials. However, the stability of the masonry structures is often threatened by strength reduction of the foundation ground due to rainwater infiltration. To consider such destabilization factors in the analys es and restoration designs, further enhancement of the method to include coupled seepage-deformation analysis of unsaturated soil is required.
(2) Numerical analyses of the bearing capacity of the platform structure performed in Chapter 5 treated only cohesive soils. This is because only unconfined compression strength or the cone penetration testing results are usually attainable in the restoration of heritage structures. However, simulation using frictional soil should also be conducted to detail mechanical behaviors of the platform.
(3) The simplified estimation method of the ultimate bearing capacity of the platform proposed in Chapter 6 was validated through comparison with the simulated results of NMM-DDA. Further validation including comparison with the experimental results is required in order to apply the proposed method to actual in-situ problems.
127
128
Appendix A Spatial Discretization by NMM-DDA
Here, we assume a system consists of n discrete continua 1 , 2 , , n and a contact between j and k (j k) at a single contact point as shown in Figure 3.8. Although only the DDA-NMM contact is assumed in Section 3.2, a generalized contact pair is considered here. With the approximation equations of displacement and strain by DDA and NMM (Eqs. (3.21), (3.23), (3.37), and (3.38)), the displacement and the strain field of i (i = 1, 2, , n) can be described by the following generalized form:
Ii Ii Ii ui x, y Ti di and (A.1)
Ii Ii Ii i Bi di , (A.2)
respectively. Here, the superscript Ii is either D or M, indicating that the variables of the object are modeled by DDA or NMM, respectively. The spatially discretized form of the governing equations for is derived by substituting Eqs. (A.1), (A.2), (3.44), and (3.47) into Eq. (3.19) as
I I I I I I I I I I I j j j j k j j k Ik j j j M j d j|tt K j Kcjj d j Kcjk dk Fj R j|t Fcj ; (A.3)
and the spatially discretized form of the governing equations for k is derived by substituting Eqs. (A.1), (A.2), (3.44), and (3.47) into Eq. (3.19) as
I I I I I j k j Ik Ik Ik j k Ik Ik Ik Ik Kckj d j M k dk|tt Kk Kckk dk Fk Rk|t Fck , (A.4)
Ii Ii Ii Ii where M i , Ki , Fi , and Ri|t are the mass matrix, the stiffness matrix, the force increment vector, and the residual force vector at time t, respectively, expressed as follows:
T M Ii T Ii T Ii d i i i i , (A.5) i
T K Ii BIi D BIi d i i i i , (A.6) i
129
T T F Ii T Ii b d T Ii t d i i i i i , and (A.7) i i
T T T RIi T Ii b d T Ii t d BIi d i|t i i|t i i|t i i|t . (A.8) i i i
IjIk IjIk I jIk Here, Di represents the stress-strain relationship of i . Kcjj , Kcjk , Kckj and
I jIk I j Ik Kckk are the contact penalty stiffness matrices, and Fcj and Fck are the contact force vectors:
T T I jIk I j I j I j I j Kcjj pn E j E j ps H j H j , (A.9)
T T I jIk I j Ik I j Ik Kcjk pn E j Gk ps H j Lk , (A.10)
T T I jIk Ik I j Ik I j Kckj pn Gk E j ps Lk H j , (A.11)
T T I jIk Ik Ik Ik Ik Kckk pn Gk Gk ps Lk Lk , (A.12)
T T I j pnS0 I j ps Ss I j Fcj E j H j , (A.13) l l
p S T p S T F Ik n 0 GIk s s LIk . (A.14) ck l k l k
The discretized equations above are the governing equations of the contact objects j and k . The discretized equations of the other objects can be derived by deleting the contact term from Eq. (A.3) as follows.
Ii Ii Ii Ii Ii Ii M i di|tt Ki di Fi Ri|t . (A.15)
Because Eqs. (A.3), (A.4), and (A.15) include the acceleration term, a temporal discretization is needed to solve the governing equations. The details of the temporal discretization are shown in Appendix B.
130
Appendix B Temporal discretization in NMM-DDA
The NMM-DDA method adopts an implicit time discretization method based on Newmark’s method, following the original DDA B-1) and NMMB-2). The acceleration
Ii Ii di|tt at the time t t is expressed with the displacement increment di between
Ii the time t and t t and the velocity di|t at the time t:
2 2 dIi d Ii d Ii . (B.1) i|tt t 2 i t i|t
Here, Ii (i = j, k) denotes D or M; Eq. (B.1) is commonly applied to both DDA and NMM. Using Eq. (B.1), Eqs. (A.3) and (A.4) are rewritten as Eqs. (B.2) and (B.3), respectively.
2 I j I j I jIk I j I jIk Ik M j K j Kcjj d j Kcjk dk t 2 (B.2) 2 I j I j I j I j I j M j d j|t Fj R j|t Fcj t
I jIk I j 2 Ik Ik I jIk Ik Kckj d j M k Kk Kckk dk t 2 (B.3) 2 M Ik d Ik F Ik RIk F Ik t k k|t k k|t ck
If we define
~ Ii 2 Ii Ii Ki M i Ki , and (B.4) t 2
~ 2 F Ii M Ii d Ii F Ii RIi (B.5) i t i i|t i i|t
Eqs. (B.2) and (B.3) become
~ I j I jIk I j I jIk Ik ~I j I j K j Kcjj d j Kcjk dk Fj Fcj , and (B.6)
I jIk I j ~ Ik I jIk Ik ~Ik Ik Kckj d j Kk Kckk dk Fk Fck , respectively. (B.7)
131
Combining these two equations, we finally obtain the discretized equation for the entire system of n continua as a single simultaneous linear equation:
~ I1 I1 ~ I1 K1 0 0 0 d1 F1 ~ I2 I2 ~ I2 0 K2 0 0 d2 F2 0 0 ~ I j I jIk I jIk I j ~ I j I j K j Kcjj Kcjk d j Fj Fcj (B.8) I I ~ I I ~ j k Ik j k Ik Ik Ik Kckj Kk Kckk dk Fk Fck 0 ~ I ~ In n In 0 0 0 Kn dn Fn
This is the matrix form of the equations to be solved in the proposed NMM-DDA. Eq. (B.8) fully couples the contacting objects with the contact penalty terms, and implements simultaneous DDA and NMM analyses.
Although both dynamic and quasi-static problems are analyzed in NMM-DDA with the same formulation, including the inertial term given by Eq. (B.8), the infinite displacement of unconfined objects must be carefully treated, especially in quasi -static analysis. A so-called dynamic damping procedure is implemented for this purpose, in which the velocities of the objects are set to zero at the beginning of each time step to make the objects converge to a static equilibrium state immediately, in the same manner as that of previous studiesB-1),B-2). In the quasi-static analysis, the time increment t is a virtual quantity that does not necessarily correspond to an actual time increment.
References B-1) Sh i , G.H . and Goo d man, R .E.: Generalization of two - dimensional discontinuous
deformation analysis for forward modelling , I n t. J. N u m er. A na l . Methods Geomech .,
13, pp. 3 59– 38 0 , 1989.
B-2) Shi, G.H.: Manifold method of material analysis, Trans. 9th Army Conference on Applied
Mathematics and Computing, Report No. 92-1, U.S. Army Research Office, 1991.
132
Appendix C Outlines of subloading Cam-clay model
C.1 Introduction
As described in Section 4.2.1 and Section 4.2.2, the soils of the Angkor area exhibited different mechanical behaviors that varied depending on the density, such as strain hardening/softening behaviors and positive/negative dilatancy. Thus, in Chapter 4, as a constitutive model able to explain the above characteristics, the modified Cam-clay modelC-1) extended with subloading surface concept C-2) (thus termed the ‘subloading Cam-clay model’) was employed to model the soils in the Angkor area. In this appendix, outlines of the modified Cam-clay model are first described, and subsequently, an introduction to the subloading surface is provided.
Variables used in the model formulation will first be introduced. For the stress variable, which govern deformations of the soil skeleton, effective stress tensor,
σ σ pwI (C.1) was used. Here, σ is the Cauchy total stress tensor (compression is positive), σ is the Cauchy effective stress tensor, pw is the pore water pressure (compression is positive), and I is the second order unit tensor. The mean effective stress p and the deviator stress q are respectively provided as follows: 1 p trσ, and (C.2) 3
3 q s : s . (C.3) 2
Here, s is the deviatoric stress tensor defined as follows: s σ pI , (C.4) and the stress ratio is defined as follows: q . (C.5) p
133
Additionally, as the invariant of the strain tensor, the volumetric strain v and the deviatoric strain d are respectively defined as follows: 1 trε, and (C.6) v 3
2 e : e , (C.7) d 3 where ε is the infinitesimal strain tensor (compression is positive), and e is the deviatoric strain tensor defined as follows: 1 e ε I . (C.8) 3 v
C.2 Modified Cam-clay model
The original Cam-clay model and the modified Cam-clay model, which were proposed by the soil mechanics group of Cambridge University in United Kingdom, together with the critical state theoryC-3), are the first elasto-plastic constitutive models able to describe the shear and consolidation behaviors of normally consolidated soil in a uniform manner. These models are commonly derived by assuming the internal energy dissipation concept developed by TaylorC-4) as the plastic work and applying the associated flow rule. However, Akai and TamuraC-5), and Asaoka et al.C-6) derived the yield function of the Cam-clay models based on the experimental results, demonstrating that the void ratio of normally consolidated soil is uniquely determined for each set of p and q . This section provides an explanation of the modified Cam-clay model as according to Akai and TamuraC-5), and Asaoka et al.C-6).
The basis of the Cam-clay models can be determined from the results of consolidated drained/undrained triaxial compression tests for remolded normally consolidated clay under several confining pressures, as described by Bishop and HenkelC-7),C-8) at Imperial College, London. Henkel arranged the series of test results as water content ratio contours on the effective stress plane, as shown in Figure C.1. This figure indicates that the water content ratio of normally consolidated soil may be uniquely determined for each set of mean effective stress and deviator stress. Since only saturated soil is relevant in this figure, this logic may also be applied to the void ratio.
134
Figure C.1 Water content ratio contours (after Henkel C-8)).
From the same test results obtained by HenkelC-8), the relationships among the mean effective stress p , the deviator stress q , and the specific volume v = 1 + e at the critical state, at which point the soil exhibits shear deformation without changes in the stress ratio and void ratio, may be drawn as illustrated in Figure C.2. Figure C.2(a) and (b) are p q and p v relationships, respectively. As shown in these figures, the critical state of normally consolidated soil may be represented as a single line on p q and p v planes (or in p q v space), regardless of drained or undrained conditions, and this line is termed the “critical state line (C.S.L.)”. Figure C.3 is the rearranged figure of Figure C.2(b) as applied on the ln p v plane; this figure demonstrates that the C.S.L. is parallel with the normal consolidation line (N.C.L.) in the plane. Considering the N.C.L. yields a constant gradient in the plane, the experimental results obtained by HenkelC-8) may be summarized as follows: p v 1 e ln , 0 and (C.9) Pa
p v 1 e ln , M , (C.10) Pa where the material parameters and are the specific volumes under atmospheric pressure (Pa = 98 kPa) on the N.C.L. and the C.S.L., respectively; λ is the compression index, and M is the stress ratio at the critical state.
135
(a) (b)
Figure C.2 Critical state of normally consolidated soil: (a) p’-q plane, (b) p’-v plane.
Figure C.3 Critical state line on lnp’-v plane.
v=1+e N
Γ
1 λ 1 λ
Pa ln p'
Figure C.4 Schematic figure of Eqs. (C.9) and (C.10).
Figure C.4 is the schematic representation of Eqs. (C.9) and (C.10). As shown in this figure, Eqs. (C.9) and (C.10) illustrate the state of normally consolidated soil when 0 and M , respectively. Therefore, the p q v relationship for normally consolidated soil in an arbitrary stress state may be represented by introducing an
136 interpolation function between and that is parameterized with the stress ratio into Eq. (C.9) as follows: p v 1 e ln . (C.11) Pa
Here, is a function of , which satisfies 0 0 . (C.12) 1 M
In addition, because normally consolidated soil exhibits monotonic compression behavior during the undrained shear process, a monotonically increasing function should be employed for . In the case of the modified Cam-clay model, the following interpolation function is used:
2 ln1 (C.13) M . ln2
The yield function of the modified Cam-clay model may be derived from Eqs. (C.11) and (C.13). Normally consolidated soil in the state of p,,e p0 ,0,e0 is assumed. When the soil is loaded and the soil state changed to p,,e p,,e, the specific volume at each state can be described as follows: p 0 v0 1 e0 ln (initial state), and (C.14) Pa
(post-loading). (C.15)
Therefore, the void ratio change e and the volumetric strain v during this process may be represented, respectively, as follows: p e 1 e0 1 e ln , and (C.16) p0
e p v ln . (C.17) 1 e0 1 e0 p0 1 e0
e If the swelling index is defined as , the elastic volumetric strain v is p e v ln , (C.18) 1 e0 p0
p and the plastic volumetric strain v is
137
p p e v v v ln . (C.19) 1 e0 p0 1 e0
Since Eq. (C.19) expresses the yield condition, the yield function of the modified Cam-clay model may be derived from Eqs. (C.19) and (C.13) to form the following equation: 2 p p 1 p f p,q, v ln ln1 v . (C.20) 1 e0 p0 ln2 1 e0 M
Since the plastic volumetric strain becomes zero at the critical state ( M ), if the associated flow rule dεp f σ is assumed, the following relation will be obtained: f p,M , p d p v 0 v p f p,M , p 1 1 (C.21) v 0 . p 1 e0 p ln2 1 e0 p ln2
Therefore, Eq. (C.20) may be rewritten as follows: 2 p p p f p,q, v ln ln1 v . (C.22) 1 e0 p0 1 e0 M As mentioned above, the yield function of the modified Cam-clay model was derived from experimental results on normally consolidated soil. Therefore, this model is not capable of accurately describing all behaviors of overconsolidated soil. For example, it is impossible to represent plastic deformation in the overconsolidated state, because the model assumes the elastic condition when f is negative, which means that the soil is overconsolidated. In order to overcome this problem, the subloading surface concept was thus proposed by Hashiguchi and Ueno C-2).
138
C.3 Introduction of the subloading surface concept
Before introducing the subloading surface to the modified Cam-clay model, the general behavior of overconsolidated soil should be reviewed. Figure C.5(a) is a schematic representation of the e ln p relationship of overconsolidated soil during isotropic consolidation, and Figure C.5(b) illustrates the e ln p relationship modeled with the modified Cam-clay model. The curve of the actual overconsolidated soil asymptotically approaches the N.C.L. as consolidation progresses. On the other hand, the curve reproduced using the modified Cam-clay model exhibits a linear relationship with the gradient of during the overconsolidated state, immediately yields when the consolidation line reaches the N.C.L., and then subsequently continues along the N.C.L.. This difference arises from the assumption applied in the conventional elasto-plastic theory including the modified Cam-clay model that states that the material exhibits purely elastic behavior in the yield surface. However, actual soil exhibits an elasto-plastic deformation with hardening even when in the overconsolidated state. Therefore, the objective of constitutive modeling of overconsolidated soil is to properly associate the changes in density (or overconsolidation ratio) of soil with the corresponding plastic strain generation.
In order to model the relationship between density change and plastic strain, a variable to represent the degree of overconsolidation should be defined. As mentioned above, overconsolidation of soil will decrease gradually as the consolidation
1 1
λ λ
e e ρ
1 κ
Void ratio Void Void ratio ratio Void
ln p' ln p'
(a) (b)
Figure C.5 Schematic figure of e-lnp’ relationship of the overconsolidated soil:
(a) actual soil, (b) modified Cam-clay model.
139 progresses and its e ln p curve approaches the N.C.L. asymptotically. During this process, the difference between the current void ratio e and the void ratio on the N.C.L. at same stress eN is defined as follows:
eN e , (C.23) which is illustrated in Figure C.5(a), showing a monotonic decrease to zero. Therefore, may be used as the state variable to represent the degree of overconsolidation.
Next, plastic deformation of overconsolidated soil will be described using defined above. The compression process of the overconsolidated soil shown in Figure C.6 (arrow from point A to point B) is assumed here. Using , the void ratio change
e during the process can be expressed as follows: p e e0 e eN0 0 eN ln 0 , (C.24) p0
where eN0 and eN are the void ratio on the N.C.L. at the stress of the initial and current states, respectively. Eq. (C.24) may be rewritten with the volumetric strain v as follows: p 0 v ln . (C.25) 1 e0 p0 1 e0
e Assuming the elastic volumetric strain v may be expressed using Eq. (C.18) in a manner identical to that in the modified Cam-clay model, the plastic volumetric strain
p v may be derived as follows: p p 0 v ln . (C.26) 1 e0 p0 1 e0
e 1 λ
eN0
ρ0 A e0 eN ρ e B
p'0 p' ln p'
Figure C.6 Compression of overconsolidated soil.
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q=Mp' q
Normal yield surface
O p' Subloading surface
Figure C.7 Schematic figure of subloading surface.
If the volume change due to the stress ratio change is considered as identical to that of the modified Cam-clay model, Eq. (C.26) may be extended as follows: 2 p p 0 v ln ln1 . (C.27) 1 e0 p0 1 e0 M 1 e0 Rearranging Eq. (C.27) to obtain the yield function of overconsolidated soil may be performed as follows: 2 p p 0 p f p,q, v , ln ln1 v . (C.28) 1 e0 p0 1 e0 M 1 e0 Since this yield function results in an yield surface that is analogous to, and within range of the normal yield surface provided by the modified Cam-clay model, as shown in Figure C.7, the yield surface of Eq. (C.28) is termed the “subloading surface”. When is zero, which means that the soil is normally consolidated, the subloading surface is equivalent to the normal yield surface.
From the yield function of Eq. (C.28), the elasto-plastic constitutive relations tensor Dep may be derived using the associated flow rule:
f dε p , (C.29) σ and Prager’s compatibility condition:
f f p f v p df dσ p p dε d 0 , (C.30) σ v ε in the same manner as is derived in conventional elasto-plastic models. Since Eq. (C.30) includes d , an evolution rule of d able to yield a proper representation of the
141 monotonic decrease of with plastic deformation should be defined. An arbitrary function that satisfies the above requirement may be applied to the evolution rule, and in this study, the following equations were employed: d Gdε p (C.31) 1 e0 G sgn a 2 . (C.32)
Here, a is the material parameter that represents the reduction rate of overconsolidation; this is the only parameter added due to the introduction of the subloading surface. From Eqs. (C.28)–(C.32), Dep of the subloading Cam-clay model is defined as follows: dσ Dep :dε (C.33) f f De : : De ep e σ σ D D , (C.34) f f f f G : De : σ p σ σ where De is the elastic constitutive relations tensor. When is large, which implies that the soil is strongly overconsolidated, G also increases, and the second term of the right hand side of Eq. (C.34) will be small. Consequently, the soil exhibits nearly elastic behavior. Conversely, if decreases to zero with plastic deformation, the terms associated with will vanish from Eq. (C.34) because G 0 , and will thus correspond to that of normally consolidated soil. Based on this theoretical framework, the subloading Cam-clay model is able to reproduce the smooth transition from an overconsolidated state to a normally consolidated state in the uniform manner. It should be noted that since the stress condition always locates on the subloading surface in this model, the generation of plastic strain is determined by only the loading or unloading condition.
References C -1) Roscoe, K.H. and Burland, J.B.: On the generalized stress-strain behaviour of wet clay,
Engineering Plasticity, Cambridge University Press, pp. 535–609, 1968.
C -2) Hashiguchi, K. and Ueno, M.: Elastoplastic constitutive laws of granular material. Constitutive
Equations of Soils, Proc. 9th Int. Conf. Soil Mech. Found. Engrg., eds. Murayama, S. and
Schofield, A.N., Tokyo, JSSMFE, pp. 73– 82, 1977.
C -3) Schofield, A.N. and Wroth, C.P.: Critical state soil mechanics, McGraw-Hill, London, 1968.
C -4) Taylor, D.W.: Fundamentals of soil mechanics, Wiley, New York, 1948.
142
C -5) Akai, K. and Tamura, T.: Numerical analysis of multi-dimensional consolidation accompanied
with elasto-plastic constitutive equation, J. JSCE, 269, pp. 95– 104, 1978.
C -6) Asaoka, A., Noda, T., Yamada, E., Kaneda, K., Nakano, M.: An elasto -plastic description of
two distinct volume change mechanisms of soils, Soils. Found., 42(5), pp. 47–57, 2002.
C -7) Bishop, A.W. and Henkel D.J.: The measurement of soil properties in the triaxial Test, Edward
Arnold, London. 1962.
C -8) Henkel, D.J.: The shear strength of saturated remolded clay, Proc. Res. Conf. Shear Strength
Cohesive Soils at Boulder, Colorado, pp. 533–540, 1960.
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Appendix D Treatment of material boundaries in node-based NMM
This appendix provides the treatment of material boundaries when the node-based uniform strain element is applied for material modeled by NMM. As described in Chapter 3, node-based uniform strain elementD-1) is a method employed to prevent volumetric locking of 3-node triangular mesh, which reduces the number of integration points by evaluating the strain and stress variables at the nodes instead of at each triangular element. This method possesses the capacity to eliminate defects caused by volumetric locking, but different materials may not share the nodes because the constitutive relationship is defined at the nodes, as with the strain. This implies that this method is unable to handle an object composed of multiple materials with a single mesh effectively. To overcome this problem, using node-based FEM, Kurumatani et al.D-2) set two nodes at identical coordinates along material boundaries, and the constraint conditions to generate continuous displacement and traction fields were applied to the node sets using a penalty method (see Figure D.1).
F ig ure D. 1 Material boundary treatment in node -based FEM (after Kurumatani et al. D - 2)) .
145
The node-based NMM proposed in this thesis adopted a countermeasure as according to Kurumatani et al.D-2). As mentioned above, in the method developed by Kurumatani et al. D-2), double nodes are set along material boundaries. However, in the case of NMM, the nodes are not always located just on the material surface or along the material boundaries, as explained in Section 3.2; thus, the double nodes scheme may not be applied. Therefore, in NMM or node-based NMM, different materials were modeled as independent polygonal physical domains (or blocks), as shown in Figure D.2, and the continuity conditions of the displacement and tractions along material boundaries, provided as follows:
u1 u2 on12 , and (D.1)
t1 t2 on12, (D.2) were applied to the vertices of polygons with identical coordinates. Here, ui and ti (i
= 1, 2) are the displacement and traction of the material i , respectively, and 12 represents material boundaries between 1 and 2 . The constraint conditions of Eqs. (D.1) and (D.2) were incorporated into the weak form of the equation of motion for each material (Eq. (3.9)) using the penalty method. The following equation was subsequently derived:
ui i ui d εi :σi d ui bi d ui ti d i i i i (D.3) u p u u d 0, i m i j 12 where i = 1, 2, j = 1, 2, and i j, and pm is the penalty coefficient for material boundaries. By applying spatial and temporal discretization as explained in Chapter 3, analysis considering material boundaries may be implemented independent of node-based elements.
12 2 pm
1
Figure D.2 Material boundary treatment o f N M M .
146
References
D-1) Dohrmann, C.R., Heinstein, M.W., Jung, J., Key, S.W. and Witkowski, W.R.: Node -based
uniform strain elements for three-node triangular and four-node tetrahedral meshes, Int. J.
Numer. Meth. Engng., 47, pp. 1549– 1568, 2000.
D-2) Kurumatani, M., Kojima, T. and Terada, K.: Performance assessment of nodal -integration
finite element method, Trans. JSCES, Paper No. 20070015, 2007 (in Japanese).
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