The Finite Element Analysis of Shells - Fundamentals (eBook)

Preface - Second Edition 6
Preface - First Edition 7
Contents 10
1. Introduction 13
1.1 Shells: from Nature to Engineering Designs 13
1.2 The Finite Element Analysis of Shells as Approached in this Book 16
2. Geometrical Preliminaries 20
2.1 Vectors and Tensors in Three-Dimensional Curvilinear Coordinates 20
2.1.1 Vectors and tensors 20
2.1.2 Covariant and contravariant bases. Metric tensor 22
2.1.3 Curvilinear coordinate systems 28
2.1.4 Covariant differentiation 31
2.2 The Shell Geometry 34
2.2.1 Geometric definition of a shell 34
2.2.2 Differential geometry on the midsurface 36
2.2.3 3D differential geometry for shells 48
3. Elements of Functional and Numerical Analysis 52
3.1 Sobolev Spaces and Associated Norms 52
3.1.1 General concepts in vector spaces 53
3.1.2 L2 and other Sobolev spaces 59
3.2 Variational Formulations and Finite Element Approximations 69
3.2.1 Basic error estimates for displacement-based and mixed formulations 70
3.2.2 Interpolation and a priori error estimates 99
3.2.3 Effect of numerical integration 104
4. Shell Mathematical Models 106
4.1 Shell Kinematics 106
4.2 Derivation of Shell Models 110
4.2.1 The "basic shell model" 111
4.2.2 The "shear-membrane-bending model" 114
4.2.3 The "membrane-bending model" 115
4.2.4 Plate models 118
4.2.5 Higher-order shell models, and the 3D-shell model 121
4.3 Mathematical Analysis of the Shell Models 125
4.3.1 Analysis of the s-m-b shell model 125
4.3.2 Analysis of the m-b shell model 134
4.3.3 Analysis of the basic shell model 136
4.3.4 Analysis of the 3D-shell model 141
5. Asymptotic Behaviors of Shell Models 146
5.1 General Asymptotic Analysis 147
5.1.1 Non-inhibited pure bending 154
5.1.2 Inhibited pure bending 157
5.1.3 Summary of asymptotic behaviors 161
5.1.4 Comparison of asymptotic behaviors for speci c shell models 163
5.2 Analysis of the Subspace of Pure Bending Displacements 167
5.2.1 Elliptic surfaces 168
5.2.2 Hyperbolic surfaces 169
5.2.3 Parabolic surfaces 170
5.3 Influence of the Loading 172
5.3.1 Effect of the loadings that do not activate the pure bending displacements 172
5.3.2 Effect of non-admissible membrane loadings 177
5.4 Asymptotic Analysis of the 3D-Based Shell Models 190
5.4.1 Asymptotic analysis of the basic shell model 191
5.4.2 Asymptotic analysis of the 3D-shell model 203
5.5 Asymptotic Considerations in Dynamic Analysis 219
5.5.1 Non-inhibited pure bending 219
5.5.2 Inhibited pure bending 220
5.5.3 Detailed numerical illustration for a clamped cylinder 221
6. Displacement-Based Shell Finite Elements 229
6.1 Discretizations of Shell Mathematical Models 229
6.2 Facet-Shell Elements 234
6.3 General Shell Elements 238
6.4 3D-Shell Elements 263
7. Influence of the Thickness in the Finite Element Approximation 269
7.1 Numerical Locking in Thin Structures 270
7.2 Treatments of Numerical Locking by Mixed Formulations 276
7.2.1 Basic principles: the Timoshenko beam example 277
7.2.2 Applications to the Reissner-Mindlin plate model 286
7.2.3 Basic principles of stabilized mixed formulations 301
7.2.4 MITC plate elements 305
7.3 Specific Difficulties Arising in the Analysis of Shells 314
8. Towards the Formulation of Effective General Shell Elements 325
8.1 Guidelines for Assessing and Improving the Reliability of Shell Finite Elements 325
8.1.1 Considerations on proper selection and use of test problems 325
8.1.2 Proposed set of test problems 333
8.2 Formulation of MITC Shell Elements 336
8.2.1 Formulation of quadrilateral MITC elements 336
8.2.2 Formulation of triangular MITC elements 337
8.2.3 Insight into MITC shell formulations 342
8.2.4 Considerations regarding 3D-shell elements 346
8.3 Assessment Results 348
8.3.1 Shell elements used in plate bending 348
8.3.2 Axisymmetric hyperboloid 354
9. On the Nonlinear Analysis of Shells 374
9.1 The Incremental Analysis to Obtain Nonlinear Response Solutions 374
9.2 The Finite Element Discretization of a Shell for General Nonlinear Analysis 376
9.3 The Fundamental Considerations of Linear Analysis Used in Nonlinear Analysis 379
9.4 Demonstrative Solutions 381
9.4.1 The "Myth of No-Locking" in nonlinear analysis of shells 381
9.4.2 Large deformation analysis of a simply-supported plate 384
9.4.3 Nonlinear analysis of thick cantilever beam 385
9.4.4 Contact analysis of Scordelis-Lo roof 386
9.4.5 Crash analysis of a tube 389
A. Tables of symbols 391
A.1 Latin Symbols 391
A.2 Greek Symbols 394
A.3 Special Symbols 395
B. Some Useful Mathematical Formulas 397
C. Distributions: Basic Definitions and Properties 399
Bibliography 403
Index 414