Suchen und Finden
Professor Maciej Bieniek
7
Preface
9
Contents
11
Introduction
15
1.1 Shell Structures
15
1.2 Motivation and Scope
18
1.3 Basic Assumptions
19
References
20
Shell Constitutive Equations
21
2.1 Introduction
21
2.1.1 Thickness of the Shell
22
2.1.2 Initial Curvature and Radial (Transverse Normal) Stresses
24
2.2 Plate Constitutive Equations
25
2.2.1 Stresses and Stress Resultants in a Thin Plate
25
2.2.2 Equilibrium Equations and Governing Differential Equation of Plate
27
2.2.3 Transverse Shear and Transverse Normal Stresses in a Plate
29
2.3 Coordinate Transformation Ò Strains in Spherical Coordinates
31
2.4 Theoretical Formulation of the Shell Equations
36
2.4.1 Assumed Out-of-Plane Stress Components
36
2.4.2 Displacement Field
39
2.4.3 Stress Components
42
2.4.4 Stress Couples and Stress Resultants on the Middle Surface
44
2.4.5 Average Displacements
48
2.4.6 Equilibrium Equations and Boundary Conditions
52
2.4.7 The Non-Linear Nature of the Stress Distribution
53
2.4.8 The Equivalent Formulation for Thick Plates
55
2.5 Examples
55
2.5.1 Thick Sphere Subjected to Uniform Pressures
56
2.5.2 Thick Cylinder Subjected to Uniform Pressures
58
2.6 Summary
59
References
60
Shell Element Based on the Refined Theory of Thick Spherical Shells
63
3.1 Introduction
63
3.1.1 Shear Locking
63
3.1.2 Membrane Locking
66
3.1.3 Mesh Instabilities
67
3.2 Finite Element Formulation
68
3.2.1 Shell Constitutive Equations
68
3.2.2 Displacements and Boundary Conditions
69
3.2.3 Element Displacement and Strain Fields – Quasi- Conforming Method
71
3.2.4 Strain Energy and Stiffness Matrix
76
3.3 Numerical Examples
78
3.3.1 The Patch Test
79
3.3.2 Cantilevered Beam
79
3.3.3 Morley’s Hemispherical Shell (Morley and Moris, 1978)
80
3.3.4 Pinched Cylinder with Diaphragms
83
3.3.5 Scordellis-Lo Roof
84
3.3.6 Pinched Cylinder
85
3.4 Summary
87
References
88
Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells
91
4.1 Introduction
91
4.2 Updated Lagrangian Description
92
4.3 Shell Kinematics
93
4.3.1 Local Coordinates
93
4.3.2 Surface Coordinates
94
4.3.3 Base Coordinates
95
4.4 Explicit Tangent Stiffness Matrix
96
4.5 Numerical Example
101
4.6 Summary
103
References
103
Elasto-Plastic Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells
105
5.1 Introduction
105
5.2 Yield Criterion and Hardening Rule
106
5.2.1 Iliushin’s Yield Function (Iliushin, 1956)
106
5.2.2 Influence of the Shear Forces
107
5.2.3 Development of the Plastic Hinge
108
5.2.4 Bauschinger Effect and Kinematic Hardening Rule
108
5.3 Explicit Elasto-Plastic Tangent Stiffness Matrix with Large Displacements
113
5.4 Numerical Examples
120
5.4.1 Simply Supported Elasto-Plastic Beam
121
5.4.2 Simply Supported Plate
122
5.4.3 Cylindrical Shell Subjected to Ring of Pressure
126
5.4.4 Spherical Dome Subjected to Ring of Pressure
128
5.5 Summary
129
References
130
Elasto-Plastic Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells With Damage Due to Microvoids
133
6.1 Introduction
133
6.2 Yield and Damage Criterion
135
6.3 Explicit Tangent Stiffness Matrix
142
6.4 Numerical Examples
149
6.4.1 Clamped Square Plate Subjected to a Central Point Load
150
6.4.2 Spherical Dome Subjected to Ring of Pressure
151
6.5 Summary
153
References
155
Non-linear Post Buckling Finite Element Analysis of Plates and Shells
159
7.1 Introduction
159
7.2 Element Tangent Stiffness Matrix
160
7.2.1 Element Stiffness in Local Coordinates
160
7.2.2 Initial Surface Coordinates for Large Deformation Analysis
164
7.2.3 Transformation of Element Stiffness Matrix
165
7.3 Solution Algorithm
166
7.4 Numerical Examples
167
7.4.1 The Williams’ Toggle Frame
167
7.4.2 Simply Supported Circular Plate Subjected to Edge Pressure
168
7.4.3 Rectangular Plate Subjected to In-Plane Load
169
7.4.4 Cylindrical Shell Under a Central Load
171
7.4.5 Spherical Shell Subjected to Central Load
173
7.5 Summary
174
References
174
Determination of Transverse Shear Stresses and Delamination in Composite Laminates Using Finite Elements
177
8.1 Introduction
177
8.2 Kinematics of the Shell
178
8.3 Lamina Constitutive Equations
180
8.4 Failure Criteria for Composite Laminates
185
8.5 Implementation and Numerical Examples
186
8.5.1 Laminated Composite Strip under Three-Point Bending
187
8.5.2 Composite Cylinder under Internal Pressure
192
8.5.3 Cylindrical Shell Subjected to Ring of Pressure
194
8.6 Summary
196
References
197
Numerical Methods and Computational Algorithms
199
9.1 Introduction
199
9.2 Linear Elastic Analysis Ò System of Linear Algebraic Equations
199
9.3 Non-linear Analysis Ò System of Non-linear Algebraic Equations
201
9.3.1 Modified Newton-Raphson Method – Combined Incremental/ Iterative Solutions
202
9.3.2 The Arc-Length Technique
204
9.3.3 Integrating the Rate Equations – Return to the Yield Surface
208
References
210
Appendix
213
Interpolation Formulas for Displacement Field
213
Index
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