K. Bhaskar, Indian Institute of Technology, Madras, India
Professor K. Bhaskar is the Head of the Department of Aerospace Engineering at the Indian Institute of Technology, Madras. As part of the Structures Group, he teaches courses related to Solid Mechanics and Elasticity. His main research interest is theoretical modeling of thick laminated structures.

T. K. Varadan, formerly of Indian Institute of Technology, Madras, India (now retired)
T. K. Varadan worked in the Department of Aerospace Engineering at the Indian Institute of Technology, Madras for more than thirty-five years before retiring in 2001. Besides teaching a wide variety of courses related to Structural Mechanics and Aircraft Design, he has made significant research contributions in the areas of Non-linear Vibrations and Composite Structures, with more than one hundred refereed publications in international and national journals.

PART A

CLASSICAL THEORY AND STRAIGHTFORWARD APPLICATIONS

1 Definition of a Thin Plate 1

1.1 The Elasticity Approach 1

1.2 A Test Problem 4

1.3 The Case of a Thin Plate 7

2 Classical Plate Theory 11

2.1 Assumptions of Classical Plate Theory 12

2.2 Moment-Curvature Relations 16

2.3 Equilibrium Equations 18

2.4 Governing Biharmonic Equation 21

2.5 Boundary Conditions 21

2.6 Solution of a Problem 26

2.7 Inclusion of an Elastic Foundation / Thermal Effects 28

2.7.1 Elastic Foundation 28

2.7.2 Thermal Effects 29

2.8 Strain Energy of the Plate 30

3 A Critical Assessment of Classical Plate Theory 33

3.1 CPT Solution for the Test Problem of Section 1.2 33

3.2 Comparison with the Elasticity Solution 35

3.3 Why the Plane-Stress Constitutive Law? 37

4 Analysis of Rectangular Plates 40

4.1 Recapitulation of Fourier Series 40

4.2 Navier's Method 43

4.3 Levy's Method 50

4.4 Closed-form Solution for a Plate with Corner Supports 59

5 Analysis of Circular Plates 69

5.1 Equations of the Theory of Elasticity 69

5.2 Equations of CPT 71

5.3 Solution of Axisymmetric Problems 75

6 Free and Forced Vibrations 89

6.1 Equations of Motion 89

6.2 Free Vibration Analysis 91

6.3 Forced Vibration Analysis 99

7 Effect of In-plane Forces on Static Flexure, Dynamics and Stability 103

7.1 Governing Equations for Combined Bending and Stretching 103

7.2 Analysis for Stability 108

7.3 Static Flexure 115

7.4 Free Vibrations 117

8 Approximate Solutions 121

8.1 Analytical and Numerical Methods 121

8.2 Rayleigh-Ritz Method 122

8.2.1 Static Flexure 122

8.2.2 Buckling 130

8.2.3 Free Vibration Analysis 140

8.3 Galerkin’s Method 147

Appendix – Solutions for Problems 162

PART B

COMPLICATING EFFECTS AND CORRESPONDING THEORIES

9 Anisotropic, Laminated and Functionally-Graded Plates 191

9.1 CPT for Homogeneous Anisotropic Plates 191

9.1.1 The Anisotropic Constitutive Law 191

9.1.2 Plate Equations 199

9.2 Classical Laminated Plate Theory 203

9.3 CPT for Functionally-Graded Plates 215

10 Elasticity Solutions for Plates 220

10.1 Cylindrical Bending of a Cantilevered Plate Strip Under Tip Shear 220

10.1.1 Homogeneous Strip 221

10.1.2 A Laminated Strip 228

10.2 Flexure of Simply Supported Rectangular Plates/Laminates Due to Transverse Loading 233

10.3 Vibrations and Stability of Simply Supported Rectangular Plates and Laminates 238

10.4 Solutions for Rectangular Plates with Other Edge Conditions 240

10.5 Corner Reactions in Simply Supported Plates – Insight Obtained from Elasticity Solutions 242

10.6 Plates under Thermal Loads 246

11 Shear Deformation Theories 248

11.1 First-order Shear Deformation Theory 249

11.2 Higher-order Theories 260

12 Variable Thickness Plates 267

12.1 Stepped versus Smooth Thickness Variation 267

12.2 Rectangular Plates 268

12.3 Circular Plates 272

13 Plate Buckling due to Non-Uniform Compression 276

13.1 The In-plane Problem 276

13.2 Determination of the Critical Load 284

13.3 Some Other Approaches 287

14 Non-Linear Flexure and Vibrations 290

14.1 Cylindrical Bending of a Simply Supported Plate Strip 290

14.1.1 Case (a): Immovable Edges 290

14.1.2 Case (b): Freely Movable Edges 299

14.1.3 Observations from the Above Solutions 305

14.2 Moderately Large Deformation Theory 306

14.3 Flexure of a Simply Supported Rectangular Plate 312

14.4 Nonlinear Vibrations of a Rectangular Plate 318

15 Post-Buckling Behaviour 324

15.1 Post-Buckling of a Column 324

15.2 Post-Buckling of a Rectangular Plate 326

15.3 Effective Width 333

Index

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