Elasto-Plastic and Damage Analysis of Plates and Shells

Professor Maciej Bieniek

Preface

Contents

Introduction

1.1 Shell Structures

1.2 Motivation and Scope

1.3 Basic Assumptions

References

Shell Constitutive Equations

2.1 Introduction

2.1.1 Thickness of the Shell

2.1.2 Initial Curvature and Radial (Transverse Normal) Stresses

2.2 Plate Constitutive Equations

2.2.1 Stresses and Stress Resultants in a Thin Plate

2.2.2 Equilibrium Equations and Governing Differential Equation of Plate

2.2.3 Transverse Shear and Transverse Normal Stresses in a Plate

2.3 Coordinate Transformation Ò Strains in Spherical Coordinates

2.4 Theoretical Formulation of the Shell Equations

2.4.1 Assumed Out-of-Plane Stress Components

2.4.2 Displacement Field

2.4.3 Stress Components

2.4.4 Stress Couples and Stress Resultants on the Middle Surface

2.4.5 Average Displacements

2.4.6 Equilibrium Equations and Boundary Conditions

2.4.7 The Non-Linear Nature of the Stress Distribution

2.4.8 The Equivalent Formulation for Thick Plates

2.5 Examples

2.5.1 Thick Sphere Subjected to Uniform Pressures

2.5.2 Thick Cylinder Subjected to Uniform Pressures

2.6 Summary

References

Shell Element Based on the Refined Theory of Thick Spherical Shells

3.1 Introduction

3.1.1 Shear Locking

3.1.2 Membrane Locking

3.1.3 Mesh Instabilities

3.2 Finite Element Formulation

3.2.1 Shell Constitutive Equations

3.2.2 Displacements and Boundary Conditions

3.2.3 Element Displacement and Strain Fields – Quasi- Conforming Method

3.2.4 Strain Energy and Stiffness Matrix

3.3 Numerical Examples

3.3.1 The Patch Test

3.3.2 Cantilevered Beam

3.3.3 Morley’s Hemispherical Shell (Morley and Moris, 1978)

3.3.4 Pinched Cylinder with Diaphragms

3.3.5 Scordellis-Lo Roof

3.3.6 Pinched Cylinder

3.4 Summary

References

Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells

4.1 Introduction

4.2 Updated Lagrangian Description

4.3 Shell Kinematics

4.3.1 Local Coordinates

4.3.2 Surface Coordinates

4.3.3 Base Coordinates

4.4 Explicit Tangent Stiffness Matrix

4.5 Numerical Example

4.6 Summary

References

Elasto-Plastic Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells

5.1 Introduction

5.2 Yield Criterion and Hardening Rule

5.2.1 Iliushin’s Yield Function (Iliushin, 1956)

5.2.2 Influence of the Shear Forces

5.2.3 Development of the Plastic Hinge

5.2.4 Bauschinger Effect and Kinematic Hardening Rule

5.3 Explicit Elasto-Plastic Tangent Stiffness Matrix with Large Displacements

5.4 Numerical Examples

5.4.1 Simply Supported Elasto-Plastic Beam

5.4.2 Simply Supported Plate

5.4.3 Cylindrical Shell Subjected to Ring of Pressure

5.4.4 Spherical Dome Subjected to Ring of Pressure

5.5 Summary

References

Elasto-Plastic Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells With Damage Due to Microvoids

6.1 Introduction

6.2 Yield and Damage Criterion

6.3 Explicit Tangent Stiffness Matrix

6.4 Numerical Examples

6.4.1 Clamped Square Plate Subjected to a Central Point Load

6.4.2 Spherical Dome Subjected to Ring of Pressure

6.5 Summary

References

Non-linear Post Buckling Finite Element Analysis of Plates and Shells

7.1 Introduction

7.2 Element Tangent Stiffness Matrix

7.2.1 Element Stiffness in Local Coordinates

7.2.2 Initial Surface Coordinates for Large Deformation Analysis

7.2.3 Transformation of Element Stiffness Matrix

7.3 Solution Algorithm

7.4 Numerical Examples

7.4.1 The Williams’ Toggle Frame

7.4.2 Simply Supported Circular Plate Subjected to Edge Pressure

7.4.3 Rectangular Plate Subjected to In-Plane Load

7.4.4 Cylindrical Shell Under a Central Load

7.4.5 Spherical Shell Subjected to Central Load

7.5 Summary

References

Determination of Transverse Shear Stresses and Delamination in Composite Laminates Using Finite Elements

8.1 Introduction

8.2 Kinematics of the Shell

8.3 Lamina Constitutive Equations

8.4 Failure Criteria for Composite Laminates

8.5 Implementation and Numerical Examples

8.5.1 Laminated Composite Strip under Three-Point Bending

8.5.2 Composite Cylinder under Internal Pressure

8.5.3 Cylindrical Shell Subjected to Ring of Pressure

8.6 Summary

References

Numerical Methods and Computational Algorithms

9.1 Introduction

9.2 Linear Elastic Analysis Ò System of Linear Algebraic Equations

9.3 Non-linear Analysis Ò System of Non-linear Algebraic Equations

9.3.1 Modified Newton-Raphson Method – Combined Incremental/ Iterative Solutions

9.3.2 The Arc-Length Technique

9.3.3 Integrating the Rate Equations – Return to the Yield Surface

References

Appendix

Interpolation Formulas for Displacement Field

Index