A Two-Step Perturbation Method in Nonlinear Analysis of Beams, Plates and Shells

The capability to predict the nonlinear response of beams, plates and shells when subjected to thermal and mechanical loads is of prime interest to structural analysis. In fact, many structures are subjected to high load levels that may result in nonlinear load-deflection relationships due to large deformations. One of the important problems deserving special attention is the study of their nonlinear response to large deflection, postbuckling and nonlinear vibration.

A two-step perturbation method is firstly proposed by Shen and Zhang (1988) for postbuckling analysis of isotropic plates. This approach gives parametrical analytical expressions of the variables in the postbuckling range and has been generalized to other plate postbuckling situations. This approach is then successfully used in solving many nonlinear bending, postbuckling, and nonlinear vibration problems of composite laminated plates and shells, in particular for some difficult tasks, for example, shear deformable plates with four free edges resting on elastic foundations, contact postbuckling of laminated plates and shells, nonlinear vibration of anisotropic cylindrical shells. This approach may be found its more extensive applications in nonlinear analysis of nano-scale structures. 

• Concentrates on three types of nonlinear analyses: vibration, bending and postbuckling
• Presents not only the theoretical aspect of the techniques, but also engineering applications of the method

ATwo-Step Perturbation Method in Nonlinear Analysis of Beams, Plates and Shells is an original and unique technique devoted entirely to solve geometrically nonlinear problems of beams, plates and shells. It is ideal for academics, researchers and postgraduates in mechanical engineering, civil engineering and aeronautical engineering.

Preface xi

List of Symbols xiii

1 Traditional Perturbation Method 1

1.1 Introduction 1

1.2 Load-type Perturbation Method 2

1.3 Deflection-type Perturbation Method 3

1.4 Multi-parameter Perturbation Method 4

1.5 Limitations of the Traditional Perturbation Method 5

References 6

2 Nonlinear Analysis of Beams 9

2.1 Introduction 9

2.2 Nonlinear Motion Equations of Euler–Bernoulli Beams 10

2.3 Postbuckling Analysis of Euler–Bernoulli Beams 13

2.4 Nonlinear Bending Analysis of Euler–Bernoulli Beams 17

2.5 Large Amplitude Vibration Analysis of Euler–Bernoulli Beams 21

References 25

3 Nonlinear Vibration Analysis of Plates 27

3.1 Introduction 27

3.2 Reddy’s Higher Order Shear Deformation Plate Theory 29

3.3 Generalized Karman-type Motion Equations 35

3.4 Nonlinear Vibration of Functionally Graded Fiber Reinforced Composite Plates 42

3.5 Hygrothermal Effects on the Nonlinear Vibration of Shear Deformable Laminated Plate 63

3.6 Nonlinear Vibration of Shear Deformable Laminated Plates with PFRC Actuators 69

References 74

4 Nonlinear Bending Analysis of Plates 79

4.1 Introduction 79

4.2 Nonlinear Bending of Rectangular Plates with Free Edges under Transverse and In-plane Loads and Resting on Two-parameter Elastic Foundations 80

4.3 Nonlinear Bending of Rectangular Plates with Free Edges under Transverse and Thermal Loading and Resting on Two-parameter Elastic Foundations 91

4.4 Nonlinear Bending of Rectangular Plates with Free Edges Resting on Tensionless Elastic Foundations 94

4.5 Nonlinear Bending of Shear Deformable Laminated Plates under Transverse and In-plane Loads 97

4.6 Nonlinear Bending of Shear Deformable Laminated Plates under Transverse and Thermal Loading 109

4.7 Nonlinear Bending of Functionally Graded Fiber Reinforced Composite Plates 116

Appendix 4.A 126

Appendix 4.B 131

Appendix 4.C 132

Appendix 4.D 133

Appendix 4.E 136

Appendix 4.F 137

References 141

5 Postbuckling Analysis of Plates 145

5.1 Introduction 145

5.2 Postbuckling of Thin Plates Resting on Tensionless Elastic Foundation 147

5.3 Postbuckling of Shear Deformable Laminated Plates under Compression and Resting on Tensionless Elastic Foundations 163

5.4 Thermal Postbuckling of Shear Deformable Laminated Plates Resting on Tensionless Elastic Foundations 171

5.5 Thermomechanical Postbuckling of Shear Deformable Laminated Plates Resting on Tensionless Elastic Foundations 178

5.6 Postbuckling of Functionally Graded Fiber Reinforced Composite Plates under Compression 185

5.7 Thermal Postbuckling of Functionally Graded Fiber Reinforced Composite Plates 194

5.8 Postbuckling of Shear Deformable Hybrid Laminated Plates with PFRC Actuators 200

References 211

6 Nonlinear Vibration Analysis of Cylindrical Shells 215

6.1 Introduction 215

6.2 Reddy’s Higher Order Shear Deformation Shell Theory and Generalized Karman-type Motion Equations 216

6.3 Nonlinear Vibration of Shear Deformable Cross-ply Laminated Cylindrical Shells 219

6.4 Nonlinear Vibration of Shear Deformable Anisotropic Cylindrical Shells 233

6.5 Hygrothermal Effects on the Nonlinear Vibration of Functionally Graded Fiber Reinforced Composite Cylindrical Shells 252

6.6 Nonlinear Vibration of Shear Deformable Laminated Cylindrical Shells with PFRC Actuators 257

Appendix 6.G 263

References 269

7 Postbuckling Analysis of Cylindrical Shells 273

7.1 Introduction 273

7.2 Postbuckling of Functionally Graded Fiber Reinforced Composite Cylindrical Shells under Axial Compression 274

7.3 Postbuckling of Functionally Graded Fiber Reinforced Composite Cylindrical Shells under External Pressure 295

7.4 Thermal Postbuckling of Functionally Graded Fiber Reinforced Composite Cylindrical Shells 312

7.5 Postbuckling of Axially Loaded Anisotropic Cylindrical Shells Surrounded by an Elastic Medium 320

7.6 Postbuckling of Internal Pressure Loaded Anisotropic Cylindrical Shells Surrounded by an Elastic Medium 325

Appendix 7.H 331

Appendix 7.I 339

Appendix 7.J 341

References 344

Index 349