# Plate and Shell Structures: Selected Analytical and Finite Element Solutions

ISBN: 978-1-118-93453-1

Dec 2016

424 pages

Select type: O-Book

## Description

Plate and Shell Structures:

Selected Analytical and Finite Element Solutions

Maria Radwañska, Anna Stankiewicz, Adam Wosatko, Jerzy Pamin

Cracow University of Technology, Poland

Comprehensively covers the fundamental theory and analytical and numerical solutions for different types of plate and shell structures

Plate and Shell Structures: Selected Analytical and Finite Element Solutions not only provides the theoretical formulation of fundamental problems of mechanics of plates and shells, but also several examples of analytical and numerical solutions for different types of shell structures. The book contains advanced aspects related to stability analysis and a brief description of modern finite element formulations for plates and shells, including the discussion of mixed/hybrid models and locking phenomena.

Key features:

• 52 example problems solved and illustrated by more than 200 figures, including 30 plots of finite element simulation results.
• Contents based on many years of research and teaching the mechanics of plates and shells to students of civil engineering and professional engineers.
• Provides the basis of an intermediate-level course on computational mechanics of shell structures.

The book is essential reading for engineering students, university teachers, practitioners and researchers interested in the mechanics of plates and shells, as well as developers testing new simulation software.

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