Plates
Theories and Applications
Published on 11. July 2014
Book
Hardback
352 pages
978-1-118-89387-6 (ISBN)
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Article is exhausted; no reprint
Description
Plates: Theories and Applications provides a comprehensive introduction to plate structures, covering classical theory and applications. It considers plate structures in several forms, starting from the simple uniform, thin, homogeneous metallic structure to more efficient and durable alternatives involving features such as variable-thickness, lamination, sandwich construction, fiber reinforcement, functional gradation, and moderately-thick to very-thick geometry. Different theoretical models are then discussed for analysis and design purposes starting from the classical thin plate theory to alternatives obtained by incorporation of appropriate complicating effects or by using fundamentally different assumptions. Plates: Theories and Applications alsocovers the latest developments on the topic.
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Persons
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.
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.
Content
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