Limit and shakedown analysis of structures using advanced discretisation methods and second order cone programming

Confirmation of Authorship I, Tran Trung Dung, confirm that this thesis entitled, ‘Limit and shakedown analysis of structures using advanced discretisation methods and second-order cone programming’ and the results presented in it are my own It has not been previously submitted for a degree to any other University or any other institution Ho Chi Minh City, March 2018 Tran Trung Dung i Acknowledgements The research presented in this thesis has been carried out in the framework of a doctorate program delivered at the Vietnam National University - Ho Chi Minh City, University of Science, Faculty of Mathematics and Computer Science This work would not have been possible without the help of many people to whom I feel deeply indebted I would first like to express my deep gratitude to my supervisors, Pham Duc Chinh and Le Van Canh, for their academic support and encouragement during the course of this work Their invaluable ideas, guidance and devotion helped me to overcome a number of difficulties arising in the process of conducting this research I would also like to acknowledge The Open University-Ho Chi Minh City and The National Foundation for Science and Technology Development (NAFOSTED, Vietnam) for their financial assistance throughout the research project Without their help this thesis would not have been completed on time I would like to say thanks to members of the Division of Computational Mechanics for their willingness to help me and for fruitful discussions about a range of topics Thanks are also extended to Assoc Prof Nguyen Xuan Hung, Assoc Prof Nguyen Thoi Trung and Dr Thai Hoang Chien for their discussions on computational aspects of XFEM and S-FEM Finally, my sincere thanks go to my family, especially to my wife Le Thi Ngoc Dung for their emotional support and encouragement throughout my study ii Abstract Limit and shakedown analyses provide efficient and powerful tools for structural design and safety assessment of many engineering components and structures, from simple metal forming problems to large-scale engineering structures and nuclear power plants In this thesis, a reduced shakedown kinematic formulation is developed to solve some typical plane stress problems, using finite element and smoothed finite element methods Whenever the comparisons are available, the present results agree with the available ones in the literature The advantage of the proposed approach lies on its simplicity, computational effectiveness, and the separation of collapse modes for possible different treatments Second-order cone programming developed for kinematic plastic limit analysis is effectively implemented to study the incremental plasticity collapse mode Moreover, the rotating plasticity collapse, which in the mathematical sense is a generalization of the alternating plasticity collapse, will also be derived analytically for general time-independent stress state, and yield criteria Various numerical examples of different complexities in terms of materials, structures, and loading combinations, are presented to show that an elastic–plastic body may fail by rotating plasticity collapse rather than the simpler alternating plasticity one among other possible modes Furthermore, the extended finite element method (XFEM) is extended to allow computation of the limit load of cracked structures It is demonstrated that the linear elastic tip enrichment basis with and without radial term r may be used in the framework of limit analysis, but the six-function enrichment basis based on the well-known Hutchinson-Rice-Rosengren (HRR) asymptotic fields appears to be the best The discrete kinematic formulation is cast in the form of a second-order cone problem, which can be solved using highly efficient interior-point solvers A solution strategy for a kinematic shakedown analysis formulation based on SFEM (ES-FEM and NS-FEM) has been described S-FEM is used in combination with second-order cone programming in the framework of the reduced shakedown kinematic formulation The comparative advantages of our approach are that the size of the optimization problem does not increase and accurate solutions can be obtained with minimal computational efforts iii Content Contents Confirmation of Authorship i Acknowledgements ii Abstract iii Contents iv List of Figures ix List of Tables xiv Nomenclature xv Chapter – Introduction 1.1 Overview 1.2 Historical reviews 1.2.1 Limit and shakedown theories 1.2.2 Mathematical programming 1.2.3 Discretisation techniques 1.3 Motivation, objectives and research methodology 1.3.1 The necessity of the research 1.3.2 Originality, relevance and scientific significance of the research 10 1.3.3 Research scope and content 11 1.3.4 Research methodology 11 1.4 Thesis outline 12 Chapter – Fundamentals 14 2.1 General relations in plasticity 14 2.1.1 Yield criterion 14 iv Content 2.1.2 Material model 15 2.1.3 Plastic dissipation function 18 2.1.4 Variational principles 18 2.2 Fundamental theorems of shakedown 20 2.2.1 Static shakedown theorem 21 2.2.2 Kinematic shakedown theorem 22 2.2.3 Shakedown analysis formulations 23 2.2.4 Nonshakedown modes 25 2.2.5 The shakedown theory for the limited kinematic hardening materials 27 2.3 Limit analysis 29 2.3.1 General theorems of limit analysis 30 2.3.2 Kinematic formulation of limit analysis 31 2.4 Second-order cone programming 32 2.5 Finite element method 34 2.5.1 Creation of shape function 35 2.5.2 Basic conditions for nodal shape functions 39 2.5.3 Strain evaluation 39 Chapter – Reduced shakedown kinematic formulation, separated collapse modes, and numerical implementation 41 3.1 Introduction 41 3.2 Shakedown kinematic formulations 43 3.3 Solution of discrete kinematic formulations 47 3.4 Numerical examples 51 3.4.1 Square plate with a circular hole 52 v Content 3.4.2 Grooved rectangular plate 55 3.4.3 Square plate with a circular hole under dynamic loads 57 3.5 Conclusion 60 Chapter – Extended finite element method for plastic limit load computation of cracked structures 62 4.1 Introduction 62 4.2 Kinematic limit analysis 64 4.3 Xfem-based limit analysis 66 4.3.1 The extended finite element method 66 4.3.2 XFEM discretization of kinematic formulation 72 4.4 Numerical examples 74 4.4.1 Simple-edge notched plate problem 75 4.4.2 Double-edge notched plate problem 80 4.4.3 Cylinder with longitudinal crack subjected to internal pressure 82 4.4.4 Inclined cracked under tension 84 4.5 Conclusion 85 Chapter – Rotating plasticity and nonshakedown collapse modes for elastic-plastic bodies under cyclic loads 87 5.1 Introduction 87 5.2 Shakedown theorems and collapse modes 90 5.2.1 Kinematic upper bound approach 90 5.2.2 Static lower bound approach 93 5.2.3 Research questions 98 5.3 Finite element discrete formulations 99 vi Content 5.3.1 Kinematic formulations 99 5.3.2 Static formulations 100 5.4 Numerical 105 5.4.1 Simple load program 106 5.4.2 Complicated load program 108 5.4.2.1 Homogeneous plate with no hardening 108 5.4.2.2 Homogeneous plate with hardening 109 5.4.2.3 Reinforced plate with no hardening 112 5.4.2.4 Reinforced plate with hardening 114 5.5 Conclusions 118 Chapter – Smoothed finite element method for shakedown analysis 119 6.1 Introduction 119 6.2 Brief of smoothed finite element method 120 6.2.1 Edge-Based Smoothed FEM 120 6.2.2 Node-Based Smoothed FEM 122 6.3 SFEM based on reduced shakedown kinematic formulation 124 6.4 Numerical examples 126 6.4.1 Square plate with a circular hole 126 6.4.2 Simple frame 130 6.4.3 A symmetric continuous beam 131 6.5 Conclusion 133 Chapter – DISCUSSION AND CONCLUSIONS 134 7.1 Discussion 134 7.1.1 The advantage of the reduced shakedown kinematic formulation 134 vii Content 7.1.2 Rotating plasticity mode for shakedown problem 136 7.1.3 Numerical methods for limit and shakedown analysis 138 7.1.3.1 Extended finite element method for limit analysis of cracked structures 138 7.1.3.2 Smoothed finite element method for shakedown analysis 140 7.2 Conclusions 140 7.3 Suggestions for future work 142 List of Publications 144 References 146 viii List of Figures List of Figures Figure 1.1 Bree-diagram (Bree, 1967) Figure 2.1 Material models: elastic-perfectly plastic (left) and rigid-perfectly plastic (right) 15 Figure 2.2 Stable (a) and unstable (b, c) materials (Le, 2009) 16 Figure 2.3 Normality rule 17 Figure 2.4 Structural model 19 Figure 2.5 Load domain D 21 Figure 2.6 Yield surfaces in the deviatoric stress coordinates 28 Figure 2.7 Nodal shape function NI for the node at xI for linear elements in a 1D domain (Liu and Nguyen, 2010b) 38 Figure 2.8 Nodal shape function NI for the node at xI for linear triangular elements in a 2D domain 39 Figure 3.1 The main flowchart of the solution strategy for reduced kinematic shakedown analysis 50 Figure 3.2 The upper-right quarter of the square plate with a circular hole subjected to quasi-static biaxial uniform loads, and a finite element mesh 52 Figure 3.3 The incremental plasticity collapse curve I = which coincides with the plastic limit curve, alternating plasticity collapse curve A = , proportional plastic limit curve, and the nonshakedown curve using FEM-DUAL method, for the square plate with a circular hole subjected to biaxial uniform loads ≤ p1′ ≤ p1 , ≤ p2′ ≤ p2 54 Figure 3.4 The incremental plasticity collapse curve I = , alternating plasticity collapse curve A = , proportional plastic limit curve, and the nonshakedown ix List of Figures curve using FEM-DUAL method, for the square plate with a circular hole subjected to biaxial uniform loads 0.4 ≤ p1′ ≤ p1 , 0.4 ≤ p2′ ≤ p2 55 Figure 3.5 A grooved rectangular plate subjected to varying tension and bending, and a finite element mesh 56 Figure 3.6 The incremental plasticity collapse curve I = , alternating plasticity collapse curve A = , proportional plastic limit curve (coincides with the plastic limit one), and the nonshakedown curve using FEM-DUAL method, for the grooved rectangular plate subjected to varying tension and bending 57 Figure 3.7 The incremental plasticity-static curve, incremental plasticity-dynamic curve, alternating plasticity collapse curve, for the square plate with a circular hole subjected to uniform dynamic load p1 = p0 (1 + 0.1sin ωt ), ϖ = ω DL ρ 58 h E Figure 3.8 The incremental plasticity-static curve, incremental plasticity-dynamic curve, alternating plasticity collapse curve, for the square plate with a circular hole subjected to uniform dynamic loads p1 = p0, p2 = 0.1sin ωt 59 Figure 4.1 Element and node categories in the classical XFEM (Belytschko and Black, 1999; Moës et al., 1999) 67 Figure 4.2 Element decomposition into subcells for integration 71 Figure 4.3 The main flowchart of the XFEM-based numerical procedure 74 Figure 4.4 Single-edge cracked plate under tension: (a) geometry and loading, (b) finite element mesh 75 Figure 4.5 Single-edge cracked plate under tension (plane strain): convergence behaviour 78 Figure 4.6 Limit load factor of single-edge cracked plate: (a) Plane stress, (b) Plane strain; Analytical solution is taken from Ewing and Richards (1974), LB–lower bound, UB–upper bound 79 x Chapter – Discussion and conclusion accurate limit loads and capturing localized plastic deformations at limit state A solution strategy for a kinematic shakedown analysis formulation based on S-FEM (ES-FEM and NS-FEM) has been described S-FEM associated with second-order cone programming and the reduced shakedown kinematic formulation has the advantages in dealing with shakedown analysis problems The size of optimization problems does not increase and accurate solutions can be obtained with minimal computational effort 7.3 Suggestions for future work Although the main aims of the thesis have been largely met, there are still a number of aspects of this work which require further investigation Future developments will address extension of the present work as follows Investigating the reduced shakedown kinematic formulation for bending plate problems, plane strain problems, and 3D problems which exhibit volumetric locking Developing a solution strategy for the incremental collapse mode, how to solve it numerically in the general case, and to prove ks = ksC The primal-dual interior-point method, which has been found to be efficient and robust, should be developed with the Koiter’s shakedown kinematic formulation for bending plate problems and 3D problems With an aim to bridge the gap between CAD and FEA, Hughes et al (2005) introducted the so-called isogeometric analysis By extending the isoparametric concept of the standard FEM to more general basis functions such as B-splines and Non-Uniform Rational B-splines (NURBS) that are common in CAD approaches, it is possible to fit exact geometries at the coarsest level of discretization and eliminate geometry errors from the very beginning It is, therefore, relevant to investigate the performance of NURBS 142 Chapter – Discussion and conclusion in combination with second-order cone programming for shakedown problems 143 List of Publications List of Publications Parts of this thesis have been published in international journals, national journals or presented in conferences These papers are: Articles in ISI-covered journal Tran, T.D., Le, C.V., Pham, D.C and Nguyen-Xuan, H (2014), Shakedown reduced kinematic formulation, separated collapse modes, and numerical implementation, International Journal of Solids and Structures, 51: 2893– 2899 T.D Tran and C V Le (2015), Extended finite element method for plastic limit load computation of cracked structures, International Journal For Numerical Methods In Engineering, 104: 2–17 Canh V.Le , T.D Tran, D.C Pham (2016), Rotating plasticity and nonshakedown collapse modes for elastic–plastic bodies under cyclic loads, International Journal of Mechanical Sciences, 111-112: 55–64 Articles in national scientific journal Tran Trung Dung, Le Van Canh, Nguyen Xuan Hung (2012), Computation of limit and shakedown using the NS-FEM and second-order cone programming, Journal of Science Ho Chi Minh City Open University, 2(5): 21–28 Tran Trung Dung, Le Van Canh, Lam Phat Thuan (2013), An XFEM based kinematic limit analysis formulation for plane strain cracked structures using SOCP, Journal of Science Ho Chi Minh City Open University, 3(8): 49-57 International Conference Tran Trung Dung, Le Van Canh, Lam Phat Thuan (2012), Limit analysis of cracked structures using XFEM and second – order cone programming, 144 List of Publications Proceedings of the International Conference on Advances in Computational Mechanics (ACOME) Vietnam, 191-202 National Conference Tran Trung Dung, Le Van Canh, Nguyen Xuan Hung, Pham Duc Chinh (2014), Limit analysis of 3-D structures using second – order cone programming, Proceedings of the National Conference on Mechanical Engineering HaNoi, 139-144 145 References References Anderheggen, E (1976), Finite element analysis assuming rigid-ideal-plastic material behavior, Limit 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Introduction The aim of limit and shakedown analysis (LSA) is to determine : i) the limit load multiplier to avoid collapse (limit analysis) , and ii) the shakedown load multiplier to avoid LCF and ratcheting... The shakedown theory for the limited kinematic hardening materials 27 2.3 Limit analysis 29 2.3.1 General theorems of limit analysis 30 2.3.2 Kinematic formulation of limit analysis