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

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.. 39 Chapter

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

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

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 FEM (ES-FEM and NS-FEM) has been described S-FEM is used in combination

S-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

Contents

Confirmation of Authorship i

Acknowledgements ii

Abstract iii

Contents iv

List of Figures ix

List of Tables xiv

Nomenclature xv

Chapter 1 – Introduction 1

1.1 Overview 1

1.2 Historical reviews 3

1.2.1 Limit and shakedown theories 3

1.2.2 Mathematical programming 5

1.2.3 Discretisation techniques 6

1.3 Motivation, objectives and research methodology 9

1.3.1 The necessity of the research 9

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 2 – Fundamentals 14

2.1 General relations in plasticity 14

2.1.1 Yield criterion 14

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 3 – 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

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 4 – 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 5 – 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

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 6 – 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 7 – DISCUSSION AND CONCLUSIONS 134

7.1 Discussion 134

7.1.1 The advantage of the reduced shakedown kinematic formulation 134

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

List of Figures

Figure 1.1 Bree-diagram (Bree, 1967) 1

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 N I for the node at x I for linear elements in a 1D domain (Liu and Nguyen, 2010b) 38

Figure 2.8 Nodal shape function N Ifor 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 = 1 which coincides with the plastic limit curve, alternating plasticity collapse curve A = 1, 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 ′ ′ ≤ 1≤ 1 ≤ 2 ≤ 2 0 p p , 0 p p 54 Figure 3.4 The incremental plasticity collapse curve I = 1, alternating plasticity collapse curve A = 1, proportional plastic limit curve, and the nonshakedown

curve using FEM-DUAL method, for the square plate with a circular hole subjected

0.4 p p , 0.4 p p 55Figure 3.5 A grooved rectangular plate subjected to varying tension and bending, and a finite element mesh 56Figure 3.6 The incremental plasticity collapse curve I = 1, alternating plasticity collapse curve A = 1, 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 57Figure 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 p 1 p 0(1 0.1sin t), DL .

Figure 4.7 Double-edge cracked tensile specimen: geometry and loading (L = b =

1) 80Figure 4.8 Limit load factor of double-edge cracked tensile specimen: PS–Plane stress, PD–Plane strain 80Figure 4.9 Cylinder with longitudinal crack under internal pressure: (a) geometry and loading, (b) finite element mesh 83Figure 4.10 Limit load factor of cylinder with longitudinal crack under internal pressure 83Figure 4.11 Cylinder with longitudinal crack under internal pressure: plastic dissipation distribution (a h =0.2) 84Figure 4.12 Inclined cracked under tension: geometry and loading (a), finite element mesh 84Figure 4.13 Limit load factor of inclined cracked under tension (reference solutions were taken from Miller (1988), α = 150) 85Figure 5.1 Geometrical illustration of the bounded cyclic mode 95Figure 5.2 The main flowchart of the solution strategy for rotating plasticity and nonshakedown collapse 104Figure 5.3 The upper-right quarter of the square plate with a circular hole subjected

to quasi-static biaxial uniform loads 105Figure 5.4 Homogeneous square plate with a circular hole: finite element mesh 106Figure 5.5 The square plate with a circular hole ≤ ′ ≤ ≤ ′ ≤

Figure 5.7 Results of rotating plasticity and non-shakedown modes: homogeneous plate with no hardening 109Figure 5.8 Results of rotating plasticity and non-shakedown modes: homogeneous plate with hardening σI =σ , σU =1.5σ

Y Y Y Y 110Figure 5.9 Location of rotating plasticity and alternating plasticity failure point

(the element with red dot, [x y] = [1.0060, 0.0110]) 110

Figure 5.10 Results of rotating plasticity and non-shakedown modes: homogeneous plate with hardening (σI =σ , σU =1.5σ

Y Y Y Y, load vertices {[0.5 0.5;

p 1 –0.5; 0.5 –p 2]}) 111Figure 5.11 Square plate with a reinforced circular hole 112Figure 5.12 Square plate with a reinforced circular hole: finite element mesh with

588 elements in the reinforced area 113Figure 5.13 Results of rotating plasticity and non-shakedown modes: reinforced plate with no hardening 114Figure 5.14 Reinforced plate with hardening: elements having C <A (the elements with magenta dot) 115Figure 5.15 Results of rotating plasticity and non-shakedown modes: reinforced plate with hardening 116

Figure 5.16 Location of rotating plasticity failure point (the element with red dot, [x

y ] = [1.1497, 1.0443 ]) 117 Figure 6.1 Smoothing cell associated with interior edge k (Le, 2013) 121

Figure 6.2 Three-node triangular mesh and smoothing domains (Liu, 2010) 123Figure 6.3 The upper-right quarter of the square plate with a circular hole subjected

to quasi-static biaxial uniform loads, and a finite element mesh 127

Figure 6.5 Symmetric continuous beam: (a) the geometry and (b) finite element discretization using 1200 three-node linear triangular elements 132

List of Tables

Table 4.1 Single-edge cracked plate under tension (plane strain): influence of the asymptotic enrichment functions on collapse multiplier (a b =0.4) 78Table 4.2 Double-edge cracked tensile specimen (plane strain): comparison with literature solutions 81Table 5.1 Reinforced plate with hardening: computed stresses 116Table 5.2 Reinforced plate with hardening: alternating and rotating plasticity limits 117Table 6.1 The shakedown load factor of the S-FEM in comparison with those of other methods for the square plate with a central circular hole 128

Table 6.2 The shakedown load factor for different values of D/L of the square plate

with a central circular hole subjected to biaxial uniform loads

0 p Y, 0 p Y 128

Table 6.3 The shakedown load factor for D/L = 0.2 of the square plate with a

central circular hole subjected to biaxial uniform loads 0.4σY ≤p1 ≤0.4σY,

0.4 Y p Y 129Table 6.4 The shakedown load factor of the S-FEM in comparison with those of other methods for simple frame 131Table 6.5 The shakedown load factor of the S-FEM in comparison with those of other methods with various load domains for symmetric continuous beam 132

ξ integral weight at the ith Gaussian integral point

ρ residual stress field



B smoothed strain – displacement matrix

I incremental collapse mode

NG total number of integration points

Abbreviation

CS-FEM Cell-based smoothed finite element method

ES-FEM Edge-based smoothed finite element method

FS-FEM Face-based smoothed finite element method

NS-FEM Node-based finite element method

NURBS Non-uniform rational B-splines

S-FEM Smoothed finite element method

Chapter 1 – Introduction

1.1 Overview

The elasto-plastic behaviour of a structure subjected to variable loads of different intensities can vary as shown in Figure 1.1 (Bree, 1967): from purely elastic behaviour, purely elastic behaviour after initial plastic flow (shakedown), low cycle fatigue (LCF) by alternating plasticity, to incremental collapse by accumulation of plastic deformations over subsequent load cycles (ratcheting) and instantaneous collapse by unrestricted plastic flow at limit load

Figure 1.1 Bree-diagram (Bree, 1967)

ratcheting

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 It should be noted that shakedown analysis is an extension of plastic limit analysis to the case of variable repeated loads It allows for effects such as accumulation of plastic strain increments over subsequent load cycles as well as the low-cycle fatigue to be accounted for

To solve this problem, some analytical tools are available but they are limited to solving only simple cases Numerical methods, from very simple examples to very complicated applications, have shown their great competence In general, there are two methods, namely the “step by step” method and the “direct” method (LSA) The “direct” method is reportedly considered a faster and more efficient method in estimating the load multiplier at the collapsed state of structures than the “step by step” method By applying the fundamental theorems of plasticity and discretisation techniques, LSA becomes a problem of optimisation involving either linear or nonlinear programming, and lower and upper bounds of the load multiplier can be determined directly without intermediate steps Various iterative optimization algorithms have been developed to provide solutions to such non-linear programming (Zouain et al., 2002; Vu et al., 2004b; Garcea et al., 2005; Li and Yu, 2006) However, these methods can tackle problems with a moderate number of variables, and hence it is still desirable to develop an alternative solution procedure that can solve large-scale shakedown analysis problems in engineering practices Besides that, finite element method (FEM), one of the most robust and popular discretisation methods, is still in need of improvement, for instance locking problems, mesh distortion and highly sensitive to the geometry of the original mesh, particularly in the region of stress or displacement singularities In the following, a brief overview of the historical developments of related fields will be presented

1.2 Historical reviews

1.2.1 Limit and shakedown theories

The theorems of limit analysis, which provide upper and lower bounds of the true collapse load, were first presented by Gvozdev (1938) and independently proved by Hill (1951) for the rigid-perfectly plastic materials, and by Drucker et a1 (1952) for

the elastic-perfectly plastic materials Prager (1972) and Martin (1975) also made

landmark contributions and since then there has been a continuing interest in the development and application of limit analysis Significant contributions to the application of limit analysis in structural engineering were made by Hodge (1959,

1961)

The concept and method of shakedown analysis were first brought up in the 1930s and widely explored in the 1950s The most significant achievement in shakedown theory of elasto-plastic structures are the pioneering works by Bleich (1932), Melan (1936), and Koiter (1953; 1960) They brought up two crucial shakedown theorems, namely static shakedown theorem (also called the Melan’s theorem, the first shakedown theorem, or the lower bound shakedown theorem), and kinematic shakedown theorem (also called the Koiter’s theorem, the second shakedown theorem or the upper bound shakedown theorem) The later developments of shakedown analysis can be categorized into the static and the kinematic shakedown analysis methods Based on their work, a lot of researchers have intensively studied shakedown and formed it into a very active independent research topic In parallel, the first separate criterion of shakedown (the incremental collapse criterion) was found by Gokhfeld (1966) and Sawczuk (1967) Konig completed the theory in

1979 with his work on the alternative criterion The separated shakedown theory is based on the fact that two different types of failure modes cause the inadaptation of structures It suggests the use of different formulations in dealing with the two corresponding load factors, see for example König (1987) However, in the literature the unified kinematic theorem derived using Koiter’s and two convex-

cycle theorems (König and Kleiber, 1978; König, 1978) is commonly used to determine the upper bound shakedown limit, which is the smaller one of the incremental limit and the alternative limit

Extensions of the classical theorems to more realistic structures have attracted much interest with the relaxation of one or more of the restrictions of the classical theorems such as geometrically linear structures; elastic perfectly-plastic material models; quasi-static mechanical and thermal loadings; temperature-independent mechanical properties; negligible time-dependent effects Among them, hardening and non-associative flow rules have been studied e.g by Maier (1973), Pycko and Maier (1995), Heitzer et al (2000) Studies on the shakedown problem under geometric nonlinearity can be found in Polizzotto et al (1996), Weichert (1986) Another important case concerns the effects of temperature on the yield surface (Borino, 2000; Kleiber and König, 1984)

Recently, the separated shakedown method was further developed by Pham (2003b) and named as the reduced kinematic shakedown theorem Moreover, Pham (2003b) proposed a modified shakedown kinematic theorem using a fictitious material that can yield in bulk tension and compression has been constructed for subsequent treatment of real engineering materials, which cannot yield but fail under high hydrostatic stresses The kinematic theorem should have vanishing hydrostatic plastic strain rate solution for the safety of the body against hydrostatic fracture In this way, the modified kinematic formulation including the limits on hydrostatic stresses is suggested for application

Shakedown theorems for certain elastic-plastic limited kinematic hardening materials have been formulated by Pham and Weichert (2001), Pham (2007, 2008)

An upper bound reduced shakedown kinematic formulation with separated collapse modes has been developed and implemented for a number of typical simple structures, which yields semianalytical solutions in Pham and Stumpf (1994), Pham (2000a, 2000a, 2014) It still awaits developments of appropriate numerical

methods to solve the shakedown problem for more complicated practical structures and to check certain theoretical predictions

or the more recently developed family of interior point methods can be applied to solve linear optimsation problems

By means of a Newton type scheme, nonlinear yield functions based on von Mises

or other yield criteria can be applied directly in nonlinear programming formulations In most algorithms developed to solve non-linear problems, Lagrangian multipliers are introduced to eliminate linear or non-linear constraints The problem then changes into an unconstrained functional, and some iterative methods have been suggested to treat such a problem (Gaudrat, 1991; Andersen, 1996)

Recently, a primal-dual interior-point method proposed in Andersen et al (2001, 2003) has been proved to be one of the most robust and efficient algorithms in treating large-scale nonlinear optimization problems (Ciria et al., 2008; Munoz et al., 2009; Le et al., 2009, 2010c, 2012) The algorithm has been extended to both static and kinematic shakedown analysis problems (Le et al., 2010c; Bisbos et al., 2005; Makrodimopoulos, 2006; Weichert and Simon, 2012) In the algorithm, LSA problems are transformed into a second-order cone programming (SOCP) problem with linear objective function and feasible region defined by some cones (Andersen

et al, 2001; Ben-Tal & Nemirovski, 2001, chapter 2; Renegar, 2001) This problem can be solved effectively by commercial codes such as the MOSEK software package (Mosek, 2010)

1.2.3 Discretisation techniques

FEM is one of the most popular numerical analysis technique used for problems of engineering and mathematical physics Therefore, it is comprehensible that finite elements have been the subject of numerous publications not only in the field of LSA but also in other areas of interest including heat transfer, fluid flow and mass transport There are two main basic types of finite element models, namely, equilibrium-based model and displacement-based model

A lower-bound solution for limit analysis using equilibrium-based finite elements has been developed by several researchers such as Hodge & Belytschko (1968), Nguyen-Dang (1976), Krabbenhoft & Damkilde (2002), Lyamin & Sloan (2002) In the equilibrium finite element, the moment or assumed stress fields of each element are usually described in terms of spatial coordinates and parameters that are combinated with nodal stress/moment values The approximated moment or assumed stress fields are needed to satisfy a priori the boundary equilibrium conditions and equilibrium at interfaces between continuous finite elements For this reason, a set of linear constraints on the stress/moment parameters have to be established in order to satisfy static admissibility This often leads to a more difficult construction of such fields

In displacement-based finite elements, the velocity fields are often approximated by

a continuous function described in terms of spatial coordinates and nodal velocities The displacement formulation is more popular when compared with equilibrium models because (i) the internal compatibility condition can be satisfied directly in the assembly scheme, and (ii) boundary conditions can be imposed directly Displacement finite elements have been applied to solve limit analysis problems by

investigators such as Hodge & Belytschko (1968), Anderheggen (1976), Capsoni & Corradi (1997) and Krabben-hoft et al (2005)

Recently, some advanced discretisation techniques such as the meshfree methods, the smoothed finite element methods (SFEM), the Isogeometric Analysis (IGA) have been successfully developed to solve the LSA problem Among them, meshfree or meshless methods use sets of nodes distributed across the problem domain, and also along domain boundaries These methods have gained increasing attention due to its rapid convergence characteristics and its ability to obtain highly accurate solutions Moreover, the naturally conforming property of meshfree approximations (with no nodal connectivity required) facilitates the implementation

of h-adaptivity This method, combined with SOCP, has been used very effectively

in limit analysis and shakedown (Le et al., 2009; Le et al., 2012)

The strain smoothing technique was recently extended to the standard finite element method by Liu et al (2007), who named it the smoothed finite element method (S-FEM) The smoothing cell can be obtained from divisions of elements or based on edges or nodes It was shown that the S-FEM retains most properties of the strain smoothing technique and also advantages of the FEM, and hence yields solutions that are accurate and computationally inexpensive (Liu et al, 2007; Nguyen-Xuan et al., 2008) S-FEM, combined with second-order programming or primal-dual algorithm respectively has been successfully applied to limit and shakedown analysis in 2D and 3D structures (Ho et al., 2017 ; Nguyen-Xuan et al., 2012)

Based on the idea of bridging the gap between the computer aided design (CAD)

and the finite element analysis (FEA), Hughes et al (2005) have introduced 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 The NURBS-based finite element method is thus

very promising because it can directly use CAD data to describe an exact geometry and field variables Only one deflection variable (without rotational degrees of freedom) is used for each control point This technique is applied for the assessment

of collapse limit loads of plastic thin plates in bending and the numerical results in very good agreement with analytical solutions (Nguyen-Xuan et al., 2014a) It

isintegrated into an associated primal-dual algorithm and show the better accuracy and convergence than several finite element solutions (Hien, 2017)

In addition, various aspects of discretisation techniques have been further

investigated in efforts to provide more robust and efficient procedures and overcome drawbacks when applying it to LSA Volumetric locking, as was first pointed out by Nagtegaal et al (1974), is the use of traditional low-order displacement finite elements that imposes excessive kinematic constraints in plasticity This may lead to unboundedness in a limit analysis model in plane strain and 3D structures Various techniques have been proposed to overcome this problem These include reduced or selective integration, and augmenting the strain field (Simo and Rifai, 1990) The most robust and effective method to overcome locking is the use of higher order elements (Sloan and Randolph, 1982) such as the six-noded triangle For interelement discontinuities problem, the slip-line methods, which allow for discontinuities of tangential displacement, can provide exact limit analysis solution for some geometrically simple plane strain cases (see e.g Martin, 1975) LSA that allow for interelement discontinuities in both displacement and stress fields are of interest for current computational issues (Yan, A M et al., 1999) Recently, a numerical procedure based on extended isogeometric elements in combination with SOCP for assessing collapse limit loads of cracked structures have beeninvestigated by Nguyen-Xuan et al (2014b)

1.3 Motivation, objectives and research methodology

1.3.1 The necessity of the research

As mentioned above, the unified shakedown kinematic is commonly used to determine the upper bound shakedown limit However, this method can only tackle problems with a moderate number of variables, and hence challenges still remain in large-scale shakedown analysis problems in engineering pratice Because of the complexity of the shakedown theorems, which are the nonlinear optimization problems, many methods have been developed to study the separated nonshakedown collapse modes such as the incremental plasticity, and alternating plasticity ones (Gokhfeld, 1966; Gokhfeld and Cherniavski, 1980; König, 1987; Bree, 1989; Pham, 1992, 2000, 2003a, 2014; Pham and Stumpf, 1994; Pham and Weichert, 2001; Zhang and Rad, 2002) Separation of collapse modes not only simplifies the shakedown analysis, but is also useful and necessary for the treatment

of different collapse modes at limit state (Pham, 2000), and they may be determined

by different material plastic constants (Pham, 2007, 2008, 2010, 2013) However, this research approach has not performed numerically for the real problems in engineering practice In the context of limit analysis, a primal-dual interior-point method proposed in Andersen et al (2001, 2003) has been proved to be one of the most robust and efficient algorithms in treating large-scale nonlinear optimization problems (Ciria et al., 2008; Munoz et al., 2009; Le et al., 2009, 2010c, 2012)

Therefor, the combination of the primal-dual interior-point algorithm with the reduced shakedown kinematic formulation can lead to more advantages for application of shakedown analysis in engineering

Besides, limit analysis has been proved to be an effective tool for the direct estimation of fracture toughness and for safety assessment of fracture failure of ductile cracked structures (Yan and Nguyen-Dang, 1999) The slip-line method is a long established and extremely effective means of estimating the plastic collapse load of plane cracked structures For certain special cases of simple geometries, it

has been possible to calculate provably exact failure load factors (Hill, 1952; Ewing, 1967; Ewing and Richards, 1974) However, for most real-world geometrical configurations in engineering practice exact limit load factors are not available Consequently, various numerical limit analysis procedures, based on finite element method and mathematical programming, have been developed over the past few decades (Andersen at el., 2001, 2003; Le at el., 2010b) However, these procedures require the finite element edges to coincide with the cracks, complicating mesh generation since both the regular geometric features and the cracks must be considered Recently, Belytschko & Black (1999) and Moes et al (1999) proposed the so-called extended finite element method (XFEM), which was originally used to treat crack problems In this method, discontinuities are permitted

to cross elements and are often realised by the level-set method Therefore, it may also be appropriate to apply the XFEM to limit analysis problems for cracked structures to make the problem more efficient and convenient Additionally, in order

to improve computational efficiency of FEM, the strain smoothing technique of the reduced shakedown numerical formulation will be used to increase efficiency

1.3.2 Originality, relevance and scientific significance of the research

The reduced limit and shakedown analysis formulations proposed by Pham (2003b) has not performed numerically for the real problems in engineering practice, and hence in this research the reduced shakedown formulations will be discretized using various advanced numerical methods associated with conic programming The efficiency of the method will be investigated by comparing with the traditional one Furthermore, this research is to extend various numerical approaches which have developed in recent years to the field of limit and shakedown: a class of smoothed finite element methods (SFEM), extended finite element method (XFEM) Second-order cone programming will also be used in these numerical procedures, showing that the proposed method have many advantages in computational aspects than those reported in the literature

1.3.3 Research scope and content

The following tasks will be undertaken to achieve these overall aims :

The primal-dual interior-point algorithm will be used with the reduced shakedown kinematic formulation for plane stress problems, making the use

of the optimization method in direct manner

The exact shakedown solutions of rotating plasticity, which in the mathematical sense is a generalization of alternating plasticity collapse (Pham, 2007, 2008, 2013), will be proposed

The performance of the XFEM for plastic limit analysis of 2D cracked structures will be investigated Several tip enrichment strategies will be compared to provide accurate limit load and capturing localized plastic deformations at limit state

Numerical reduced shakedown kinematic formulations will be developed based on SFEM

1.3.4 Research methodology

Study theoretical aspects: theories of the reduced kinematic shakedown

formulation will be studied Various numerical methods will also be reviewed

Numerical simulation: Computational tools for LSA problems will be developed.To test the performance of the proposed numerical procedures, it will be applied to a number of examples of interest in engineering practice

Analyze results: The efficacy of the proposed procedures will be demonstrated by comparing its solutions with analytical solutions, or with those numerically obtained in the literature The advantages and disadvantages of each procedure will be withdrawn

1.4 Thesis outline

The thesis consists of seven chapters Four of these (Chapters 3, 4, 5 and 6) are presented as self-contained manuscripts which have been published or submitted for publication Due to the use of these ‘four-paper’ format, some minor overlaps in the content may occur The contents of each chapter will now be briefly described Chapter 1 consists of two main sections which provides a historical review of limit and shakedown theories and research motivations

In Chapter 2, a brief description of plasticity relations and theory of limit and shakedown analysis is first provided The next part of this chapter provides a brief description of the general framework of conic programming The chapter is closed with a discussion of computational aspects of FEM

Chapter 3 describes a solution strategy for a kinematic shakedown analysis formulation based on the finite element method Three main ingredients are presented: shakedown kinematic formulation, solution of discrete kinematic formulations and second-order cone programming Various numerical examples are examined to test the performance of the proposed procedure The present solutions are in good agreement with the results from the literature

Chapter 4 describes an extended finite element method (XFEM)-based numerical procedure for limit load computation of cracked structures Firstly, a brief overview

on the XFEM will be recalled, and its extension to plastic limit analysis problems will be described Then some numerical examples are presented in which several tip enrichment strategies are presented to demonstrate that the six-function enrichment basis is capable of providing accurate limit loads and capturing localized plastic deformations at limit state

In Chapter 5, finite element methods are implemented for both kinematic and static shakedown theorems to study the nonshakedown modes In the next part some theoretical results from both kinematic and static shakedown theorems are resumed,

with some particular questions raised Then the discrete formulations of the shakedown problems and respective solution methods are presented following by particular numerical implementations Finally, various numerical examples 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

In Chapter 6, numerical reduced shakedown formulations based on SFEMs is proposed Firstly, the chapter briefly describes the smoothed finite element method The kinematic shakedown analysis formulation is then recalled and SFEM based discretization problems are then formulated as a second-order cone programming in the next part Finally, numerical examples are provided to illustrate the performance

of the proposed procedure

In Chapter 7, some of the broad issues which were first discovered in the course of the research are discussed Finally, key conclusions are drawn and suggestions for future work are presented Equation Section 2 (từ đây bắt đầu đánh số phương trình

là 2.1)

( ) ( 1, ,2 3) 0,

where J1 is the first invariant of the stress tensor, while J2 and J3 are the second and the third invariants of the deviatoric stress tensor

In many problems, the yield criterion may not depend on hydrostatic pressure, and

therefore the yield criterion is independent of J1 The most commonly used yield

criterion which is independent of hydrostatic pressure is the von Mises one The von Mises yield criterion states that yielding will begin when the octahedral shearing stress reaches the critical value kv such that

2.1.2 Material model

The elastic-perfectly plastic model are used to simulate the mechanical behaviour of rate-independent plastic and non-hardening solids (Figure 2.1)

Figure 2.1 Material models: elastic-perfectly plastic (left) and rigid-perfectly plastic (right)

In this model, if the stress intensity is below the yield stress, the material behaves elastically However, once the yield stress is reached, plastic deformation appears and it cannot be recovered even after the stress is removed In fact, the plastic strains are significantly larger than the elastic ones, and therefore the elastic characteristics may be ignored Consequently, throughout the thesis, limit and shakedown analysis solutions of rigid perfectly plastic bodies are theoretically also valid for elastic-plastic bodies

In the framework of a limit and shakedown analysis problem, only plastic strain rates are considered and can be determined by the following formula, which is called the associated flow law:

where µ
is a non-negative scalar multiplier called the plastic multiplier and f( )σ

is the yield function which defines a time independent yield surface

The elastic region: f( )σ <0

The plastic: f( )σ =0

The inaccessible region: f( )σ >0.

Besides, the material model must also obey the maximum plastic work principle, which is a mathematical statement of the two important consequences, namely convexity and the normality rule The notations outlined earlier are just mathematical ideas To implement a missing link between these mathematical ideas and material behavior, a fundamental stability postulate is introduced called the

Drucker’s stability postulate

The Drucker’s stability postulate

For a rigid-perfectly plastic model, a stable plastic material is defined to be one that satisfies the following conditions (Chen, 1988, chapter 3):

“During the application of the added set of forces, the work done by the external agency on the changes in displacements it produces is positive Over the cycle of application and removal of the added set of forces, the new work performed by the external agency on the changes in the displacements

it produces is non-negative”

Figure 2.2 Stable (a) and unstable (b, c) materials (Le, 2009)

The postulate is illustrated graphically in Figure 2.2 The Drucker’s stability postulate may be represented by:

( 0)

0,dε

where ∫ is the integral taken over a cycle of applying and removing the added stress set, σ is the stress tensor on the yield surface satisfying the yield condition

Figure 2.3 Normality rule

When n differentiable surfaces intersect at a singular point, the relation (2.3) is replaced by

1

n

i i

fµ

Convexity of yield surface

It is clear from Figure 2.3 that the yield surface must be strictly convex to satisfy inequality (2.4) This is one of the most important consequences in plasticity theory

Y ield su rfa c e

(Lubliner, 1990, chapter 3) so that limit and shakedown analysis problems can be solved by using convex programming tools

2.1.3 Plastic dissipation function

The plastic dissipation function is defined by

D eare not available in general terms and must be constructed on the basis of the specific form of yield function used

For the Mises criterion and associated flow rule we have

Figure 2.4 Structural model

A stress field and a force field can be called statically admissible if they satisfy the equilibrium equation and the boundary condition

,,

t

in Von

∇ denotes the differential operator and n is the outward normal vector to V

f is the volume force and g is the surface load acting on Γt

Similarly, the kinematically admissible strain rate and velocity fields are the fields which satisfy the kinematic equation

Principle of virtual work

The principle of virtual work expresses the equilibrium condition between the internal stress field and an external force field acting on the body It states that the necessary and sufficient condition to make a possible statically admissible stress

field σ on any virtual strain field equilibrium is to satisfy the following equation

Principle of complementary virtual work

The principle of complementary virtual work expresses that the internal complementary virtual work done by any random virtual stresses on the kinematically admissible strains is equal to the external complementary virtual work done by the associated virtual surface forces on the prescribed displacements

2.2 Fundamental theorems of shakedown

Shakedown analysis is to determine the load-carrying capacity of a structure

subjected to n time-dependent loads 0( )

k

P t , each of which can vary independently

within a given range of itself

These loads form a convex polyhedral domain D of n dimensions with M = 2 n

vertices in load space as shown in Figure 2.5 for two variable loads

Figure 2.5 Load domain D

In the following, the classical shakedown theorems are restated followed by the formulation of a shakedown analysis problem

2.2.1 Static shakedown theorem

Let σe( )x,t denote the fictitious elastic stress response of the body V to external

agencies over a period of time (x∈V t, ∈ 0,T) under the assumption of perfectly elastic behavior, called a loading process (history) If the body has already shaken down, there must exist a constant residual stress field ρ( )x such that the actual stress field σ x( ),t :

Note that, the obtained shakedown load multiplier ks− is generally a lower bound

2.2.2 Kinematic shakedown theorem

According to Koiter, an admissible cycle of plastic strain field ∆εp

is

recommended corresponding to a cycle of displacement field ∆ui Note that at

each instant during the time cycle t,the plasticstrain rate ep may not be compatible,

but the plastic strain accumulated over the cycle ∆ε must be compatible p

and

1

,2

( )

inon

0 0

0 0

min

1,1

V T

e p V T

i j j i T

2.2.3 Shakedown analysis formulations

In practical computations, in most cases, it is impossible to apply the above theorems to directly find the shakedown limit load multiplier The difficulty here is the presence of the time-dependent stress field e( ),

t

σ x in (2.19) or the time

integration in the formulation (2.26) The time integration must be removed because the evaluation of plastic strains over a loading cycle would be difficult These obstacles can be overcome based on the two convex-cycle theorems, presented by König (1978):

to the static theorem (2.27) and the kinematic theorem (2.28):

The lower bound

t e