Application of lower bound limit analysis with second order cone programming for plane strain and axisymmetric geomechanics problems

APPLICATION OF LOWER BOUND LIMIT ANALYSIS WITH SECOND-ORDER CONE PROGRAMMING FOR PLANE STRAIN AND AXISYMMETRIC GEOMECHANICS PROBLEMS Tang Chong (Bachelor of Engineering, Southwest Jiao Tong University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NUS GRADUATE SCHOOL FOR DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014 i ii Acknowledgements This work would have never been possible without the support, dedication and help of many people towards who I feel deeply grateful. First of all, I want to express my most sincere thanks to my major supervisor, Prof. Phoon Kok-Kwang, for his successful guidance, excellent academic advice and for his outstanding intuitions and constant encouragement, especially for the tough time in my study. I truly appreciate his valuable friendship during all these years, becoming one of most influential people in my life, both professionally and personally. His guidance made me develop the personal skills needed to succeed in future work. I would like to thank my cosupervisor, Prof. Toh Kim-Chuan, whose deep knowledge about conic programming, iterative methods and Matlab enabled my codes to run successfully and much faster than expected. Working with Prof. Phoon and Prof. Toh has been a wonderful learning experience for me. Furthermore, I am also indebted to Dr. Goh Siang Huat for offering me the financial support to continue my research, when I was immersed in a great crisis during the study. I take this opportunity to thank Prof. Joseph Pastor (Savoie University, Polytech Annecy-Chambéry), Dr. C. M. Martin (University of Oxford), Prof. Alain Pecker (École Nationale des Ponts et Chaussées), Dr. Charles. Augarde (Durham University), Dr. Colin Smith (The University of Sheffield), Prof. Hai-Sui Yu (The University of Nottingham), Prof. Jean Salençon (École Polytechnique), Prof. Mosleh Al-Shamarani (King Saud University), Prof. D. V. Griffiths (Colorado School of Mines Golden), Prof. P. K. Basudhar (Indian Institute of Technology Kanpur), Prof. Muñoz (Universitat Politècnica de Catalunya), and Prof. Abdul–Hamid Soubra (Université de Nantes) for their suggestion during the course of the research work. I would like to express my gratitude to my friends, Miss Luo Ying, Miss Chen Zongrui, Miss Tran Huu Huyen Tran, Miss Ji Jiaming, Miss Yin Jing, Miss Liu Ziyi, Dr. Kong Fan, Dr. Sun Jie, Dr. Cheng Yonggang, Mr. Zhang Lei, Dr. Ye Feijian, Mr. Chen Jinbo, Dr. Wu Jun, iii Mr. Tang Xiaoxing, and Mr. Lu Yitan for their help during the course of study. I genuinely appreciate everyone’s help. I sincerely thank my parents (冷玉华 and 唐德华) and other family members (e.g. my uncle 初克波, 唐泽忠, 冷际国, and my auntie 刘朝晖, 刘慧平). I am more than grateful for their support and for encouraging me whenever I needed motivation. Although I was an ocean away, I always felt close to them. Finally, the work reported in this thesis was made possible by the financial support of the NUS research scholarship. iv Table of contents Acknowledgements . ii Table of contents v Abstract viii List of tables . x List of figures xi Notations xv Chapter Introduction . 1.1 General . 1.2 Motivation for the present study 1.3 Objectives and scope of the thesis . 1.4 Thesis organization Chapter Numerical lower bound limit analysis . 2.1 Literature review 2.2 Finite element lower bound limit analysis . 11 2.2.1 Formulation for plane strain problems . 11 2.2.2 Formulation for an axisymmetric analysis . 16 2.3 Concluding remarks . 21 Chapter Second-Order Cone Programming 23 3.1 General framework of SOCP . 23 3.2 Feasible primal-dual path-following interior point algorithms 24 3.3 SOCP solvers: MOSEK . 26 Chapter Application for axisymmetric lower bound limit analysis 28 4.1 Introduction 28 4.2 Numerical examples . 30 4.2.1 Circular footing 30 4.2.1.1 Problem definition and mesh details 30 4.2.1.2 Results and discussion . 32 4.2.2 Stability of circular anchor . 36 4.2.2.1 Problem definition and the mesh . 36 4.2.2.2 Results of anchors in purely cohesive Soil 36 4.2.2.3 Results of anchors in cohesionless Soil . 40 4.2.3 Multi-helical anchor . 43 4.2.3.1 Problem definition and review . 43 4.2.3.2 Results and discussion . 47 4.3 Conclusions 49 Chapter Stability analysis of geostructures under in the presence of soil inertia 52 5.1 Introduction 52 5.2 The ultimate lift capacity of anchors 52 5.2.1 General review . 52 5.2.2 Problem definition . 53 5.2.3 Static analysis . 54 5.2.3.1 Anchors in purely cohesive soil . 54 5.2.3.2 Anchors in cohesionless soil 57 5.2.3.3 Recommendation for practical design 60 5.2.4 Pseudo-static analysis 60 5.2.4.1 Results and discussion . 61 v 5.2.4.2 Comparison with the existing results . 65 5.3 Passive earth pressure 67 5.3.1 General review . 67 5.3.2 Problem definition 69 5.3.3 Results and discussion . 71 5.3.3.1 Static case . 71 5.3.3.2 Pseudo-static case 78 5.3.4 Summary 82 5.4 Conclusions 83 Chapter Bearing capacity of strip footings on slope under undrained combined loading 84 6.1 Introduction 84 6.2 Problem definition . 86 6.3 Results for horizontal ground surface 87 6.3.1 Ultimate uniaxial loads 87 6.3.1.1 Lateral load capacity 87 6.3.1.2 Vertical bearing capacity . 87 6.3.1.3 Moment capacity 89 6.3.2 The scaling concept 89 6.3.3 Soil with linearly increasing shear strength . 91 6.3.3.1 Vertical and eccentric loading 91 6.3.3.2 Vertical and horizontal loading 93 6.3.3.3 Vertical, horizontal and moment loading . 95 6.4 Results for sloping surface . 97 6.4.1 Slope and foundation failure 97 6.4.2 Vertical bearing capacity . 98 6.4.2.1 Comparison with the existing solution . 98 6.4.2.2 Soil with linearly increasing shear strength . 98 6.4.3 Vertical and horizontal loading 100 6.4.3.1 Effect of the ratio su0/γB . 100 6.4.3.2 Influence of normalized footing distance λ 100 6.4.3.3 Effect of slope inclination β . 101 6.4.3.4 Influence of normalized rate of shear strength increase with depth, κ=ρB/su0. 103 6.4.4 Combined loading 105 6.4.4.1 Uniform soil . 105 6.4.4.2 Soil with linearly increasing shear strength . 107 6.4.5 Suggested design procedure . 108 6.5 Conclusions 109 Chapter Effect of footing size on bearing capacity of surface footings 110 7.1 Introduction 110 7.2 Previous work 111 7.3 Problem definition . 112 7.3.1 Chosen domain and mesh details . 112 7.3.2 Mode of loading . 112 7.3.3 Soil properties 113 7.3.4 Remarks . 114 7.4 Results and discussion . 115 7.4.1 General observations for failure mechanism 115 vi 7.4.2 Case of constant friction angle . 117 7.4.2.1 Vertical loading 117 7.4.2.2 Effect of load eccentricity 118 7.4.2.3 Effect of load inclination 121 7.4.2.4 Combination of load eccentricity and inclination 122 7.4.3 Case of variable friction angle with stress level . 123 7.4.3.1 Vertical loading 123 7.4.3.2 Cross-section in the VM plane . 124 7.4.3.3 Cross-section in the HV plane 126 7.5 Conclusions 128 Chapter Conclusion and Future work . 129 8.1 Summary 129 8.2 Conclusions 130 8.3 Limitations and future work . 132 References 134 vii Abstract Geotechnical stability analysis is usually performed by a variety of approximate methods that are based on the theory of limit equilibrium. Although they are simple and appeal to engineering intuition, these techniques need to presuppose an appropriate failure mechanism in advance. This feature can lead to inaccurate predictions of the true collapse load, especially for problems involving heterogeneous soil profiles, complex loading, or three-dimensional deformation fields. A much more attractive approach for assessing the stability of geostructures is to use lower and upper limit analysis incorporated with finite elements and mathematical optimization developed in 1970s, which not require assumptions to be made about the mode of failure. These methods are very general and use only simple strength parameters that are familiar to geotechnical engineers. Since lower bound limit analysis can provide a safe design for engineers, the present thesis illustrates the application of this method to obtain the numerical solutions for various plane strain and axisymmetric stability problems. To ensure that the finite element formulation leads to a second-order cone programming (SOCP) problem, the yield criterion for plane strain and axisymmetric cases is formulated as a set of second-order cones. For solving different problems, computer programs are developed in MATLAB, and the toolbox MOSEK for conic programming is used. It is found that the present method in this thesis provides a computationally more efficient method for numerical lower bound limit analyses of plane strain and axisymmetric limit analysis. In the first part of this thesis, axisymmetric lower-bound limit analysis is applied to evaluate the bearing capacity of circular footings, the ultimate capacity of circular anchors and multi-plate helical anchors. It has been shown that the proposed axisymmetric formulation will be quite useful for solving various axisymmetric geotechnical problems in a rapid manner. However, it should be pointed out that for a circular footing or anchor under general loading which has been widely used in offshore foundation design, the axisymmetric assumption is invalid, and we have to resort to three-dimensional limit analysis, which is still a challenging problem. viii In a second part of this thesis, a set of rigorous investigations of geotechnical problems in plane strain condition such as the effect of soil inertia on the ultimate capacity of anchors and passive earth pressure on rigid walls, and the effect of footing width on the bearing capacity factor Nγ and failure envelopes of shallow foundations, are presented. Consideration is given to the wide range of parameters that influence the stability of geostructures. Based on the numerical results, some simple equations are proposed to approximate the ultimate capacity of geostructures. From the examples studied in this thesis, it is expected that the available plane strain formulation can yield quite satisfactory solutions even for complicated loading conditions. ix List of tables Table 4. 1. Table 4. 2. Table 4. 3. Table 4. 4. Table 4. 5. Table 4. 6. Table 4. 7. Table 5. 1. Table 5. 2. Table 5. 3. Table 5. 4. Table 5. 5. Table 6. 1. Table 6. 2. Table 7. 1. Table 7. 2. Table 7. 3. A comparison of obtained Nc values with published results from literature 33 A comparison of obtained Nγ values for a smooth footing with published results from literature .33 A comparison of obtained Nγ values for a rough footing with those published results 34 Iteration number and computational time for bearing capacity of circular foundations using LP and SOCP approach .35 Results for circular plate anchors in purely cohesive soil with or without selfweight γ .39 Results for Nγ for rough circular plate anchor in cohesionless soil 45 Ultimate capacity factor Nc for helical anchors embedded in purely cohesive soil 51 The seismic stability of inclined anchors embedded in frictional soils 62 Seismic passive earth pressure coefficient Kpγ (β=0, λ=0) 76 Seismic passive earth pressure coefficient Kpγ (β=0, λ≠0) for δ=ϕ . 77 Seismic passive earth pressure coefficient Kpγ (β≠0, λ=0) for δ=ϕ . 79 Seismic passive earth pressure coefficient Kpγ (β≠0, λ≠0) for δ=ϕ . 81 Coefficients and critical value h* for failure envelopes of combined vertical and horizontal loads 106 Coefficients and critical value h* for failure envelopes of combined vertical and horizontal loads 108 Vertical limit load values, where constant value of ϕ is used . 115 A comparison of Nγ values for a rough footing with available solutions from the literature 118 Comparison of Nγ for different footing widths with the method of stress characteristics . 125 x Chapter Conclusion and Future work 8.1 Summary Recent developments of upper and lower bound limit analyses in conjunction with finite elements and linear (Sloan 1988; Sloan and Kleeman 1995), nonlinear (Lyamin and Sloan 2002a, 2002b) programming, or SOCP (Makrodimopoulos and Martin 2006, 2007) offer an efficient approach to practical stability calculations in geotechnical engineering. In this thesis, the existing formulation for plane strain problems developed by Makrodimopoulos and Martin (2006) was used. On the other hand, following the terminology of Khatri and Kumar (2009a), a new formulation, which is computationally much more efficient and easier to implement, has been proposed to solve various axisymmetric problems. Compared with the conventional finite element method, the present lower bound formulation is much simpler to adopt since the complete description of load-deformation up to failure is not required. In relation to the method of stress characteristics or limit equilibrium technique, the present method is much easier to deal with problems involving complicated stress boundary conditions, layered soil media especially for spatially random soils, and any arbitrary problem geometries. Furthermore, although limit equilibrium techniques are simple and widely used in practice, there is a need to presuppose an appropriate failure mechanism in advance. This feature can lead to an inaccurate prediction of the ultimate collapse load for complex problems above. In the axisymmetric case, using the lower bound limit analysis incorporated with finite elements and SOCP as shown in Section 2.2.2, the bearing capacity of circular footings, stability of a single circular anchor embedded in clay and sand, and the ultimate capacity of multiplate helical anchors in clay have been investigated in detail. The obtained results are validated with the existing results. For plane strain case, the effect of soil inertia on the stability of inclined anchors (including two special cases, i.e. horizontal and vertical anchor), and the passive earth 129 pressure on rigid walls embedded in frictional soils is evaluated by using the existing formulation as described in Section 2.2.1. By combining the existing UB solutions, the actual collapse load can be bracketed. The third part of this thesis is related to the bearing capacity of shallow foundations under combined loading. The motivation for this work comes principally from the offshore industry, in which the foundation is usually subjected to combined loading due to the wind, wave or earthquake forces. In this section, the bearing capacity of strip footings on slope under undrained combined loading and the effect of footing width on the bearing capacity factor Nγ and the failure envelopes of shallow foundations are evaluated subsequently, in which an assumed linear relation between tan(ϕ) and σm on the log-log scale is used. 8.2 Conclusions Based on the various plane strain and axisymmetric lower bound limit analyses dealt in this thesis, the following major conclusions are drawn 1. For all the axisymmetric problems studied in the present thesis, a very good agreement is noted between the present lower bound solutions and the results reported in literature. It is also found that the proposed axisymmetric lower bound limit analysis in combination with SOCP is computationally much more efficient for solving various axisymmetric geotechnical problems accurately, compared with the existing linear programming approach. For circular footings, the variation of bearing capacity factors Nc and Nγ and failure mechanisms with soil friction angle has been obtained. For circular anchors in undrained clay, the variation of break-out factor Nc with embedment ratio and overburden ratio has also been obtained. It was observed that the ultimate uplift capacity increases up to constant as an increase in embedment depth or overburden pressure. This indicates the transition from a shallow to deep failure of the anchor. A similar observation can also be noted for multiplate helical anchors in clay. When an anchor is embedded in sand, the ultimate capacity increases with an increase in soil friction angle, and the failure mechanism always 130 extends up to the surface, regardless of the embedment depth. The numerical solutions obtained from this study is found match quite well with the existing solutions reported in literature. 2. For passive earth pressure, the present lower bound solutions are smaller than those of Caquot & Kérisel based on the method of characteristics and the existing upperbound solutions. In the presence of soil inertia, the design table for great values of ϕ, δ, β and negative values of λ provided by Caquot & Kérisel (1948) should be used with caution. By combing with the upper-bound solutions of Soubra (2000), the true collapse load can be bracketed within 18%. 3. For stability analysis of inclined anchors, it was also shown that the capacity of anchors increases with an increase in the soil internal friction angle, the embedment depth, and the anchor inclination angle from horizontal to vertical direction, and also decreases with increasing of soil inertia. In particular, because the anchor is embedded in soil to resist the pullout force, the effect of vertical soil inertia should be considered in practice for safety. The present results were also compared to the solutions reported earlier in literature and provide conservative predictions of the ultimate capacity of inclined anchors. 4. For bearing capacity of strip footings on slope under undrained combined loading, numerical lower bound limit analysis in conjunction with the scaling concept is used to derive a set of Green-type solutions to account for the effects of slope inclination, soil strength parameters. Therefore, these solutions can be viewed as a generalization of the Green solution, which is exact for the case of obliquely loaded strip footings on horizontal ground surface. Consequently, a practical procedure is suggested for practical design. 5. The magnitude of Nγ for a strip footing on sand is found to decrease significantly with an increase in the footing width B, load inclination α as well as load eccentricity e. For the range of footing width considered in the present thesis, the variation of Nγ/ 131  N  with B/B* has been found to be almost linear on a log-log scale under the vertical, inclined, or eccentric loading. For vertical case, this observation is quite similar to that reported earlier in literature. It has also been noted that the footing size has a significant influence on the failure envelopes. The obtained failure envelopes are also in a good agreement with finite element analyses and experimental results from literature. The velocity fields obtained from the present lower bound limit analysis suggest all the mechanisms were based on two different component mechanisms, namely, the “scoop” and “wedge” mechanism, which is in a good accordance with results from the finite element analysis. 8.3 Limitations and future work Although the lower bound limit analysis with finite elements and SOCP has been applied to various geotechnical problems successfully, it should be noted that the present method can only model the failure of soil, whereas the failure of structure cannot be considered. Ukritchon et al. (2003) described a modified version of numerical limit analyses combining the structural elements (beam and joint) with plane strain soil elements and established the ultimate resistance of laterally loaded piles and the stability of braced excavation. Later, Krabbenhoft et al. (2005) proposed a similar formulation including structural elements and applied it to ultimate capacity of sheet pile walls using nonlinear programming. More recently, Muñoz et al. (2013) also developed a method to consider the failure of sheet and rod. Therefore, introducing the structural elements with satisfying the equilibrium and yield criterion will make the present method to study a wider range of stability problems in practice. Secondly, the present work did not account for the effect of seepage force. In order to solve this problem, the element equilibrium, stress discontinuities between two adjacent elements, boundary conditions, and yield criterion should be formulated in terms of effective stresses. In this situation, the effective nodal stresses are treated as unknowns and also threenoded elements are used to carry out the analysis. In practice, the pore-water pressure should be determined in advance by using the traditional finite element method (e.g. Zienkiewicz et 132 al. 2005), and then substituted into lower bound limit analysis to obtain new stability solutions of geostructures subjected to seepage force as well as the pseudo-static case. Following this methodology, Sloan (2013) attempted to discuss the slope stability in the presence of seepage. Thirdly, the present work focuses on two-dimensional cases, which is a simplification of the real problems. The research for three-dimensional limit analysis is very limited. Lyamin and Sloan (2002a, b) developed an efficient algorithm of nonlinear programming for 3D limit analysis. Yang et al. (2003) studied 3D bearing capacity of rectangular surface footings using the sequential quadratic programming method. 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The finite element method: its basis and fundamentals. 6th edition, Elsevier. 144 [...]... attention of this thesis is focused on the application of a lower bound limit analysis to plane strain and axisymmetric stability problems in geotechnical engineering The lower bound limit analysis is implemented using finite elements and second- order cone programming (SOCP) The Mohr-Coulomb yield criterion is assumed to be applicable in all the cases The associated 2 computer programs for the different problems. .. conclusions, limitations of the present work and the recommendation for the future work 6 Chapter 2 Numerical lower bound limit analysis 2.1 Literature review The development of lower bound limit analysis incorporated with finite elements and mathematical optimization can be categorized into three types such as linear programming, nonlinear programming, and conic programming (e.g SOCP for 2-dimensional problems, ... tetrahedral elements) with encouraging results in terms of efficiency Therefore, it is possible to develop a new algorithm for larger and more complex problems of classical limit analysis in 3D, since SDP is a topic of great interest to researchers in mathematical programming 2.2 Finite element lower bound limit analysis 2.2.1 Formulation for plane strain problems The sign convention as shown in Figure 2.1(a)... numerical lower bound limit analysis is presented Then, the lower- bound limit analysis both 5 for plane strain and axisymmetric cases in conjunction with finite elements are then formulated as SOCP problems Chapter 3 briefly introduces the SOCP framework and presents, next, the main ideas about feasible primal-dual, path-following interior point methods Additionally, the canonical form required for general... the finite element analysis (Okamura et al 2002), and the finite element formulation of lower bound limit analysis with linear programming approach (Kumar and Khatri 2008a, b) However, the effects of load inclination and eccentricity on the bearing capacity of shallow foundations on 4 sand have not been investigated rigorously with considering the stress level, except for the work of Okamura et al (2002),... minimization of a sum of norms, convex quadratic programming, and convex quadratically constrained linear programming In spite of these advances in numerical lower bound limit analysis, research was mainly focused on two-dimensional problems Given that it is impractical to linearize the yield criterion in 3D, linear programming is usually not applicable to 3D problems in numerical lower bound limit analysis. .. the effects of load eccentricity and footing shape In the present thesis, the lower bound limit analysis incorporated with finite elements and SOCP is employed to study the variation of the bearing capacity factor Nγ and the failure envelopes lying in the H-V, V-M/B, or H-M/B load plane 1.3 Objectives and scope of the thesis As mentioned before, the scope of the thesis is limited to lower bounds only... nonlinear programming technique In order to perform 3-D stability analysis, most recently, semidefinite programming was implemented (e.g Martin and Makrodimopoulos et al 2008; Krabbenhøft et al 2008) in numerical limit analysis, however, for large-scale 3D case, the efficiency of the algorithm is still questionable Therefore, this section presents a new numerical formulation of lower bound limit analysis for. .. A bound ; Asoc    ;  A 2 N  M 1   0 N 2 M ; I M M    ; BN 1  f ; bstat ; bbound ; bsoc  The symbol I denotes the identity matrix Here, convex cone 1 1 is the linear cone and 2 is the Cartesian product of 3-dimensional second- order cones, respectively The dual cone is have 1   1 and 2   1 and  2 the linear and second- order cone are self-dual, we  2 The above problem is formulated... (2011) is adopted in this thesis, similar to the work of Makrodimopoulos & Martin (2006), because it is computationally efficient for very large optimization problems 2.2.2 Formulation for an axisymmetric analysis As described in section 2.2.1, finite element formulation for lower bound limit analysis has been applied extensively to deal with plane strain problems (Sloan 1988; Ukritchon et al 2003; Hjiaj . i APPLICATION OF LOWER BOUND LIMIT ANALYSIS WITH SECOND- ORDER CONE PROGRAMMING FOR PLANE STRAIN AND AXISYMMETRIC GEOMECHANICS PROBLEMS Tang Chong (Bachelor of Engineering,. lower bound limit analysis 11 2.2.1 Formulation for plane strain problems 11 2.2.2 Formulation for an axisymmetric analysis 16 2.3 Concluding remarks 21 Chapter 3 Second- Order Cone Programming. attention of this thesis is focused on the application of a lower bound limit analysis to plane strain and axisymmetric stability problems in geotechnical engineering. The lower bound limit analysis