52 example problems solved and il
Table of Contents
Preface xvii
Notation xix
Part 1 Fundamentals: Theory and Modelling 1
1 General Information 3
1.1 Introduction 3
1.2 Review of Theories Describing Elastic Plates and Shells 6
1.3 Description of Geometry for 2D Formulation 9
1.4 Definitions and Assumptions for 2D Formulation 16
1.5 Classification of Shell Structures 21
References 24
2 Equations for Theory of Elasticity for 3D Problems 26
Reference 30
3 Equations of Thin Shells According to the Three-Parameter Kirchhoff–Love Theory 31
3.1 General Equations for Thin Shells 31
3.2 Specification of Lame Parameters and Principal Curvature Radii for Typical Surfaces 38
3.3 Transition from General Shell Equations to Particular Cases of Plates and Shells 42
3.4 Displacement Equations for Multi-Parameter Plate and Shell Theories 45
3.5 Remarks 47
References 47
4 General Information about Models and Computational Aspects 48
4.1 Analytical Approach to Statics, Buckling and Free Vibrations 49
4.2 Approximate Approach According to the Finite Difference Method 51
4.3 Computational Analysis by Finite Element Method 54
4.4 Computational Models – Summary 55
Reference 55
5 Description of Finite Elements for Analysis of Plates and Shells 56
5.1 General Information on Finite Elements 56
5.2 Description of Selected FEs 58
5.3 Remarks on Displacement-based FE Formulation 69
References 70
Part 2 Plates 73
6 Flat Rectangular Membranes 75
6.1 Introduction 75
6.2 Governing Equations 76
6.3 Square Membrane under Unidirectional Tension 81
6.4 Square Membrane under Uniform Shear 83
6.5 Pure In-Plane Bending of a Square Membrane 85
6.6 Cantilever Beam with a Load on the Free Side 88
6.7 Rectangular Deep Beams 94
6.8 Membrane with Variable Thicknesses or Material Parameters 97
References 101
7 Circular and Annular Membranes 102
7.1 Equations of Membranes – Local and Global Formulation 102
7.2 Equations for the Axisymmetric Membrane State 104
7.3 Annular Membrane 105
References 109
8 Rectangular Plates under Bending 110
8.1 Introduction 110
8.2 Equations for the Classical Kirchhoff–Love Thin Plate Theory 110
8.3 Derivation of Displacement Equation for a Thin Plate from the Principle of Minimum Potential Energy 117
8.4 Equation for a Plate under Bending Resting on a Winkler Elastic Foundation 118
8.5 Equations of Mindlin–Reissner Moderately Thick Plate Theory 119
8.6 Analytical Solution of a Sinusoidally Loaded Rectangular Plate 122
8.7 Analysis of Plates under Bending Using Expansions in Double or Single Trigonometric Series 127
8.8 Simply Supported or Clamped Square Plate with Uniform Load 131
8.9 Rectangular Plate with a Uniform Load and Various Boundary Conditions – Comparison of STSM and FEM Results 135
8.10 Uniformly Loaded Rectangular Plate with Clamped and Free Boundary Lines – Comparison of STSM and FEM Results 139
8.11 Approximate Solution to a Plate Bending Problem using FDM 143
8.12 Approximate Solution to a Bending Plate Problem using the Ritz Method 151
8.13 Plate with Variable Thickness 153
8.14 Analysis of Thin and Moderately Thick Plates in Bending 155
References 159
9 Circular and Annular Plates under Bending 160
9.1 General State 160
9.2 Axisymmetric State 162
9.3 Analytical Solution using a Trigonometric Series Expansion 164
9.4 Clamped Circular Plate with a Uniformly Distributed Load 166
9.5 Simply Supported Circular Plate with a Concentrated Central Force 169
9.6 Simply Supported Circular Plate with an Asymmetric Distributed Load 171
9.7 Uniformly Loaded Annular Plate with Static and Kinematic Boundary Conditions 174
References 177
Part 3 Shells 179
10 Shells in the Membrane State 181
10.1 Introduction 181
10.2 General Membrane State in Shells of Revolution 182
10.3 Axisymmetric Membrane State 183
10.4 Hemispherical Shell 186
10.5 Open Conical Shell under Self Weight 193
10.6 Cylindrical Shell 195
10.7 Hemispherical Shell with an Asymmetric Wind Action 199
References 204
11 Shells in the Membrane-Bending State 205
11.1 Cylindrical Shells 205
11.2 Spherical Shells 221
11.3 Cylindrical and Spherical Shells Loaded by a Uniformly Distributed Boundary Moment and Horizontal Force 229
11.4 Cylindrical Shell with a Spherical Cap – Analytical and Numerical Solution 232
11.5 General Case of Deformation of Cylindrical Shells 237
11.6 Cylindrical Shell with a Semicircular Cross Section under Self Weight – Analytical Solution of Membrane State 238
11.7 Cylindrical Scordelis-Lo Roof in the Membrane-Bending State – Analytical and Numerical Solution 242
11.8 Single-Span Clamped Horizontal Cylindrical Shell under Self Weight 246
References 254
12 Shallow Shells 256
12.1 Equations for Shallow Shells 256
12.2 Pucher’s Equations for Shallow Shells in the Membrane State 260
12.3 Hyperbolic Paraboloid with Rectangular Projection 262
12.4 Remarks on Engineering Applications 266
References 267
13 Thermal Loading of Selected Membranes, Plates and Shells 268
13.1 Introduction 268
13.2 Uniform Temperature Change along the Thickness 270
13.3 Linear Temperature Change along the Thickness – Analytical Solutions 275
References 286
Part 4 Stability and Free Vibrations 287
14 Stability of Plates and Shells 289
14.1 Overview of Plate and Shell Stability Problems 289
14.2 Basis of Linear Buckling Theory, Assumptions and Computational Models 291
14.3 Description of Physical Phenomena and Nonlinear Simulations in Stability Analysis 298
14.4 Analytical and Numerical Buckling Analysis for Selected Plates and Shells 301
14.5 Snap-Through and Snap-Back Phenomena Observed for Elastic Shallow Cylindrical Shells in Geometrically Nonlinear Analysis 319
References 321
15 Free Vibrations of Plates and Shells 323
15.1 Introduction 323
15.2 Natural Transverse Vibrations of a Thin Rectangular Plate 325
15.3 Parametric Analysis of Free Vibrations of Rectangular Plates 328
15.4 Natural Vibrations of Cylindrical Shells 333
15.5 Remarks 337
References 338
Part 5 Aspects of FE Analysis 339
16 Modelling Process 341
16.1 Advantages of Numerical Simulations 341
16.2 Complexity of Shell Structures Affecting FEM 342
16.3 Particular Requirements for FEs in Plate and Shell Discretization 343
References 346
17 Quality of FEs and Accuracy of Solutions in Linear Analysis 347
17.1 Order of Approximation Function versus Order of Numerical Integration Quadrature 347
17.2 Assessment of Element Quality via Spectral Analysis 347
17.3 Numerical Effects of Shear Locking and Membrane Locking 350
17.4 Examination of Element Quality – One-Element and Patch Tests 354
17.5 Benchmarks for Membranes and Plates 357
17.6 Benchmarks for Shells 359
17.7 Comparison of Analytical and Numerical Solutions, Application of Various FE Formulations 361
References 362
18 Advanced FE Formulations 365
18.1 Introduction 365
18.2 Link between Variational Formulations and FE Models 366
18.3 Advanced FEs 373
References 383
A List of Boxes with Equations 387
B List of Boxes with Data and Results for Examples 389
Index 391
