《Plate and Shell Structures：Selected Analytical and Finite Element Soluti...
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板壳结构：解析和有限元解法
作者：
Maria Radwa´nska
Anna Stankiewicz
AdamWosatko
Jerzy Pamin
Cracow University of Technology, Poland
出版社：Wiley
出版时间：2017年
目录
Preface xvii
Notation xix
Part 1 Fundamentals: Theory andModelling 1
1 General Information 3
1.1 Introduction 3
1.2 Review ofTheories Describing Elastic Plates and Shells 6
1.3 Description of Geometry for 2D Formulation 9
1.3.1 Coordinate Systems, Middle Surface, Cross Section, Principal Coordinate
Lines 9
1.3.2 Geometry of Middle Surface 10
1.3.3 Geometry of Surface Equidistant from Middle Surface 12
1.3.4 Geometry of Selected Surfaces 13
1.3.4.1 Spherical Surface 13
1.3.4.2 Cylindrical Surface 14
1.3.4.3 Hyperbolic Paraboloid 15
1.4 Definitions and Assumptions for 2D Formulation 16
1.4.1 Generalized Displacements and Strains Consistent with the Kinematic
Hypothesis of Three-Parameter Kirchhoff–Love ShellTheory 16
1.4.2 Generalized Displacements and Strains Consistent with the Kinematic
Hypothesis of Five-Parameter Mindlin–Reissner ShellTheory 18
1.4.3 Force and Moment Resultants Related to Middle Surface 18
1.4.4 Generalized Strains in Middle Surface 20
1.5 Classification of Shell Structures 21
1.5.1 Curved, Shallow and Flat Shell Structures 22
1.5.2 Thin, ModeratelyThick, Thick Structures 22
1.5.3 Plates and Shells with Different Stress Distributions Along Thickness 23
1.5.4 Range of Validity of Geometrically Linear and Nonlinear Theories for Plates
and Shells 23
References 24
2 Equations for Theory of Elasticity for 3D Problems 26
Reference 30
viii Contents
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
3.2.1 Shells of Revolution in a Spherical Coordinate System 39
3.2.2 Shells of Revolution in a Cylindrical Coordinate System 40
3.2.3 Shallow Shells Represented by Rectangular Projection 41
3.2.4 Flat Membranes and Plates in Cartesian or Polar Coordinate System 41
3.3 Transition from General Shell Equations to Particular Cases of Plates and
Shells 42
3.3.1 Equations of Rectangular Flat Membranes 42
3.3.2 Equations of Rectangular Plates in Bending 43
3.3.3 Equations of Cylindrical Shells in an AxisymmetricMembrane-Bending
State 44
3.4 Displacement Equations for Multi-Parameter Plate and ShellTheories 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
4.1.1 Statics of aThin Plate in Bending 49
4.1.2 Buckling of a Plate 50
4.1.3 Transverse Free Vibrations of a Plate 51
4.2 Approximate Approach According to the Finite Difference Method 51
4.2.1 Set of Algebraic Equations for Statics of a Plate in Bending 52
4.2.2 Set of Homogeneous Algebraic Equations for Plate Buckling 53
4.2.3 Set of Homogeneous Algebraic Equations for Transverse Free Vibrations of a
Plate 53
4.3 Computational Analysis by Finite Element Method 54
4.4 ComputationalModels – 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
5.2.1 Flat Rectangular Four-Node Membrane FE 58
5.2.2 Conforming Rectangular Four-Node Plate Bending FE 60
5.2.3 Nonconforming Flat Three- and Four-Node FEs forThin Shells 63
5.2.4 Two-Dimensional Curved Shell FE based on the Kirchhoff–Love Thin Shell
Theory 64
5.2.5 Curved FE based on the Mindlin–ReissnerModerately Thick Shell
Theory 64
5.2.6 Degenerated Shell FE 65
5.2.7 Three-Dimensional Solid FE forThick Shells 65
5.2.8 Geometrically One-Dimensional FE for Thin Shell Structures 66
5.3 Remarks on Displacement-based FE Formulation 69
References 70
Contents ix
Part 2 Plates 73
6 Flat Rectangular Membranes 75
6.1 Introduction 75
6.2 Governing Equations 76
6.2.1 Local Formulation 76
6.2.2 Equilibrium Equations in Terms of In-Plane Displacements 78
6.2.3 PrincipalMembrane Forces and their Directions 78
6.2.4 Equations for a Flat Membrane Formulated using Airy’s Stress Function 79
6.2.5 Global Formulation 80
6.3 Square Membrane under Unidirectional Tension 81
6.3.1 Analytical Solution 81
6.3.2 Analytical Solution with Airy’s Stress Function 83
6.3.3 Numerical Solution 83
6.4 Square Membrane under Uniform Shear 83
6.4.1 Analytical Solution 83
6.4.2 FEM Results 84
6.5 Pure In-Plane Bending of a Square Membrane 85
6.6 Cantilever Beam with a Load on the Free Side 88
6.6.1 Analytical Solution 88
6.6.2 FEM Results 92
6.7 Rectangular Deep Beams 94
6.7.1 Beams and Deep Beams 94
6.7.2 Square Membrane with a Uniform Load on the Top Edge, Supported on Two
Parts of the Bottom Edge – FDM and FEM Results 94
6.8 Membrane with VariableThicknesses or Material Parameters 97
6.8.1 Introduction 97
6.8.2 Membrane with DifferentThicknesses inThree Subdomains – FEM
Solution 97
6.8.3 Membrane with Different Material Parameters inThree Subdomains – FEM
Solution 99
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 AnnularMembrane 105
7.3.1 Analytical Solution 107
7.3.2 FEM Solution 108
References 109
8 Rectangular Plates under Bending 110
8.1 Introduction 110
8.2 Equations for the Classical Kirchhoff–Love Thin Plate Theory 110
8.2.1 Assumptions and Basic Relations 110
8.2.2 Equilibrium Equation for a Plate Expressed by Moments 116
8.2.3 Displacement Differential Equation for a Thin Rectangular Plate According to
the Kirchhoff–Love Theory 116
x Contents
8.2.4 Global Formulation for a Kirchhoff–Love Thin Plate 117
8.3 Derivation of Displacement Equation for aThin Plate from the Principle of
Minimum Potential Energy 117
8.4 Equation for a Plate under Bending Resting on aWinkler Elastic
Foundation 118
8.5 Equations of Mindlin–ReissnerModerately Thick PlateTheory 119
8.5.1 Kinematics and Fundamental Relations for Mindlin–Reissner Plates 119
8.5.2 Global Formulation for ModeratelyThick Plates 120
8.5.3 Equations for Mindlin–ReissnerModerately Thick Plates Expressed by
Generalized Displacements 122
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
8.7.1 Application of Navier’s Method – Double Trigonometric Series 127
8.7.2 Idea of Levy’s Method – Single Trigonometric Series 130
8.8 Simply Supported or Clamped Square Plate with Uniform Load 131
8.8.1 Results Obtained using DTSM and FEM for a Simply Supported Plate 132
8.8.2 Results Obtained using STSM and FEM for a Clamped Plate 135
8.9 Rectangular Plate with a Uniform Load and Various Boundary
Conditions – Comparison of STSM and FEMResults 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
8.11.1 Idea of FDM 143
8.11.2 Application of FDM to the Solution of a Bending Problem for a Rectangular
Plate 144
8.11.3 Simply Supported Square Plate with a Uniform Load 147
8.11.4 Simply Supported Uniformly Loaded Square Plate Resting on a
One-Parameter Elastic Foundation 149
8.12 Approximate Solution to a Bending Plate Problem using the Ritz
Method 151
8.12.1 Idea of the Ritz Method 151
8.12.2 Simply Supported Rectangular Plate with a Uniform Load 152
8.13 Plate with VariableThickness 153
8.13.1 Description of Deformation 154
8.13.2 FEM Results 154
8.14 Analysis of Thin and Moderately Thick Plates in Bending 155
8.14.1 Preliminary Remarks 155
8.14.2 Simply Supported Square Plate with Uniform Load – Analytical and FEM
Results 156
8.14.3 Simply Supported Plate with a Concentrated Central Load – Analytical and
FEM Solutions 157
References 159
9 Circular and Annular Plates under Bending 160
9.1 General State 160
9.2 Axisymmetric State 162
Contents xi
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
9.6.1 Analytical Solution 172
9.6.2 FEM Solution 174
9.7 Uniformly Loaded Annular Plate with Static and Kinematic Boundary
Conditions 174
9.7.1 Analytical Solution 174
9.7.2 Numerical Solution using FEM 177
References 177
Part 3 Shells 179
10 Shells in theMembrane State 181
10.1 Introduction 181
10.2 General Membrane State in Shells of Revolution 182
10.3 Axisymmetric Membrane State 183
10.3.1 Membrane Forces for Shells Described in a Spherical Coordinate
System 184
10.3.2 Membrane Forces for Shells Described in a Cylindrical Coordinate
System 185
10.4 Hemispherical Shell 186
10.4.1 Shell under SelfWeight – Analytical Solution 186
10.4.2 Shell under Uniform Pressure – Analytical Solution 187
10.4.3 Suspended Tank under Hydrostatic Pressure – Analytical Solution 189
10.4.4 Supported Tank under Hydrostatic Pressure – Analytical and FEM
Solutions 191
10.4.4.1 Case I – upper shell part
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