Elasto-plastic and damage analysis of plates and shells

Shells and plates are critical structures in numerous engineering applications. Analysis and design of these structures is of continuing interest to the scienti c and engineering communities. Accurate and conservative assessments of the maximum load carried by a structure, as well as the equilibrium path in both the elastic and inelastic range, are of paramount importance to the engineer. The elastic behavior of shells has been closely investigated, mostly by means of the nite element method. Inelastic analysis however, especially accounting for damage effects, has received much less attention from researchers. In this book, we present a computational model for nite element, elasto-plastic, and damage analysis of thin and thick shells. Formulation of the model proceeds in several stages. First, we develop a theory for thick spherical shells, providing a set of shell constitutive equations. These equations incorporate the effects of transverse shear deformation, initial curvature, and radial stresses. The proposed shell equations are conveniently used in nite element analysis. 0 AsimpleC quadrilateral, doubly curved shell element is developed. By means of a quasi-conforming technique, shear and membrane locking are prevented. The element stiffness matrix is given explicitly, making the formulation computationally ef cient. We represent the elasto-plastic behavior of thick shells and plates by means of the non-layered model, using an Updated Lagrangian method to describe a small-strain geometric non-linearity. For the treatment of material non-linearities, we adopt an Iliushin's yield function expressed in terms of stress resultants, with isotropic and kinematic hardening rules.

1 Introduction	p. 1
1.1 Shell Structures	p. 1
1.2 Motivation and Scope	p. 4
1.3 Basic Assumptions	p. 5
References	p. 6
2 Shell Constitutive Equations	p. 7
2.1 Introduction	p. 7
2.1.1 Thickness of the Shell	p. 8
2.1.2 Initial Curvature and Radial (Transverse Normal) Stresses	p. 10
2.2 Plate Constitutive Equations	p. 11
2.2.1 Stresses and Stress Resultants in a Thin Plate	p. 11
2.2.2 Equilibrium Equations and Governing Differential Equation of Plate	p. 13
2.2.3 Transverse Shear and Transverse Normal Stresses in a Plate	p. 15
2.3 Coordinate Transformation - Strains in Spherical Coordinates	p. 17
2.4 Theoretical Formulation of the Shell Equations	p. 22
2.4.1 Assumed Out-of-Plane Stress Components	p. 22
2.4.2 Displacement Field	p. 25
2.4.3 Stress Components	p. 28
2.4.4 Stress Couples and Stress Resultants on the Middle Surface	p. 30
2.4.5 Average Displacements &ubar;, &vbar;, &wbar; and Rotations ¿¿, ¿¿	p. 34
2.4.6 Equilibrium Equations and Boundary Conditions	p. 38
2.4.7 The Non-Linear Nature of the Stress Distribution	p. 39
2.4.8 The Equivalent Formulation for Thick Plates	p. 41
2.5 Examples	p. 41
2.5.1 Thick Sphere Subjected to Uniform Pressures	p. 42
2.5.2 Thick Cylinder Subjected to Uniform Pressures	p. 44
2.6 Summary	p. 45
References	p. 46
3 Shell Element Based on the Refined Theory of Thick Spherical Shells	p. 49
3.1 Introduction	p. 49
3.1.1 Shear Locking	p. 49
3.1.2 Membrane Locking	p. 52
3.1.3 Mesh Instabilities	p. 53
3.2 Finite Element Formulation	p. 54
3.2.1 Shell Constitutive Equations	p. 54
3.2.2 Displacements and Boundary Conditions	p. 55
3.2.3 Element Displacement and Strain Fields - Quasi-Conforming Method	p. 57
3.2.4 Strain Energy and Stiffness Matrix	p. 62
3.3 Numerical Examples	p. 64
3.3.1 The Patch Test	p. 65
3.3.2 Cantilevered Beam	p. 65
3.3.3 Morley's Hemispherical Shell (Morley and Moris, 1978)	p. 66
3.3.4 Pinched Cylinder with Diaphragms	p. 69
3.3.5 Scordellis-Lo Roof	p. 70
3.3.6 Pinched Cylinder	p. 71
3.4 Summary	p. 73
References	p. 74
4 Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells	p. 77
4.1 Introduction	p. 77
4.2 Updated Lagrangian Description	p. 78
4.3 Shell Kinematics	p. 79
4.3.1 Local Coordinates	p. 79
4.3.2 Surface Coordinates	p. 80
4.3.3 Base Coordinates	p. 81
4.4 Explicit Tangent Stiffness Matrix	p. 82
4.5 Numerical Example	p. 87
4.6 Summary	p. 89
References	p. 89
5 Elasto-Plastic Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells	p. 91
5.1 Introduction	p. 91
5.2 Yield Criterion and Hardening Rule	p. 92
5.2.1 Iliushin's Yield Function (Iliushin, 1956)	p. 92
5.2.2 Influence of the Shear Forces	p. 93
5.2.3 Development of the Plastic Hinge	p. 94
5.2.4 Bauschinger Effect and Kinematic Hardening Rule	p. 94
5.3 Explicit Elasto-Plastic Tangent Stiffness Matrix with Large Displacements	p. 99
5.4 Numerical Examples	p. 106
5.4.1 Simply Supported Elasto-Plastic Beam	p. 107
5.4.2 Simply Supported Plate	p. 108
5.4.3 Cylindrical Shell Subjected to Ring of Pressure	p. 112
5.4.4 Spherical Dome Subjected to Ring of Pressure	p. 114
5.5 Summary	p. 115
References	p. 116
6 Elasto-Plastic Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells With Damage Due to Microvoids	p. 119
6.1 Introduction	p. 119
6.2 Yield and Damage Criterion	p. 121
6.3 Explicit Tangent Stiffness Matrix	p. 128
6.4 Numerical Examples	p. 135
6.4.1 Clamped Square Plate Subjected to a Central Point Load	p. 136
6.4.2 Spherical Dome Subjected to Ring of Pressure	p. 137
6.5 Summary	p. 139
References	p. 141
7 Non-linear Post Buckling Finite Element Analysis of Plates and Shells	p. 145
7.1 Introduction	p. 145
7.2 Element Tangent Stiffness Matrix	p. 146
7.2.1 Element Stiffness in Local Coordinates	p. 146
7.2.2 Initial Surface Coordinates for Large Deformation Analysis	p. 150
7.2.3 Transformation of Element Stiffness Matrix	p. 151
7.3 Solution Algorithm	p. 152
7.4 Numerical Examples	p. 153
7.4.1 The Williams' Toggle Frame	p. 153
7.4.2 Simply Supported Circular Plate Subjected to Edge Pressure	p. 154
7.4.3 Rectangular Plate Subjected to In-Plane Load	p. 155
7.4.4 Cylindrical Shell Under a Central Load	p. 157
7.4.5 Spherical Shell Subjected to Central Load	p. 159
7.5 Summary	p. 160
References	p. 160
8 Determination of Transverse Shear Stresses and Delamination in Composite Laminates Using Finite Elements	p. 163
8.1 Introduction	p. 163
8.2 Kinematics of the Shell	p. 164
8.3 Lamina Constitutive Equations	p. 166
8.4 Failure Criteria for Composite Laminates	p. 171
8.5 Implementation and Numerical Examples	p. 172
8.5.1 Laminated Composite Strip under Three-Point Bending	p. 173
8.5.2 Composite Cylinder under Internal Pressure	p. 178
8.5.3 Cylindrical Shell Subjected to Ring of Pressure	p. 180
8.6 Summary	p. 182
References	p. 183
9 Numerical Methods and Computational Algorithms	p. 185
9.1 Introduction	p. 185
9.2 Linear Elastic Analysis - System of Linear Algebraic Equations	p. 185
9.3 Non-linear Analysis - System of Non-linear Algebraic Equations	p. 187
9.3.1 Modified Newton-Raphson Method - Combined Incremental/Iterative Solutions	p. 188
9.3.2 The Arc-Length Technique	p. 190
9.3.3 Integrating the Rate Equations - Return to the Yield Surface	p. 194
References	p. 196
Appendix	p. 199
Index	p. 201

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