Theory of Plates and Shells

Preface
Contents
Part I Fundamentals
1 Basics of Elasticity Theory
	1.1 Introduction
	1.2 Stress State
		1.2.1 Stress Vector and Stress Tensor
		1.2.2 Transformation Rules
		1.2.3 Principal Stresses, Invariants, Mohr\'s Circles
		1.2.4 Equilibrium Conditions
	1.3 Deformations and Strains
		1.3.1 Introduction
		1.3.2 Green-Lagrangian Strain Tensor
		1.3.3 Von-Kármán Strains
		1.3.4 Infinitesimal Strain Tensor
		1.3.5 Compatibility Equations
	1.4 Constitutive Law
		1.4.1 Introduction
		1.4.2 The Generalized Hooke\'s Law
		1.4.3 Strain Energy
	1.5 Boundary Value Problems
	1.6 Material Symmetries
		1.6.1 Full Anisotropy
		1.6.2 Monotropic Material
		1.6.3 Orthogonal Anisotropy/Orthotropy
		1.6.4 Transversal Isotropy
		1.6.5 Isotropy
		1.6.6 Representation in Engineering Constants
	1.7 Transformation Rules
	1.8 Representation of the Basic Equations in Cylindrical Coordinates
	1.9 Plane Problems
		1.9.1 Plane Strain State
		1.9.2 Plane Stress State
		1.9.3 Stress Transformation
		1.9.4 Formulation for Orthotropic Materials
		1.9.5 Formulation in Polar Coordinates
2 Energy Methods of Elastostatics
	2.1 Work and Energy
		2.1.1 Introduction
		2.1.2 Inner and Outer Work
		2.1.3 Principle of Work and Energy and the Law of Conservation of Energy
		2.1.4 Strain Energy and Complementary Strain Energy
		2.1.5 General Principle of Work and Energy of Elastostatics
	2.2 The Principle of Virtual Displacements
		2.2.1 Virtual Displacements and Virtual Work
		2.2.2 The Principle of Virtual Displacements
		2.2.3 Analysis Rules for the Variational Operator δ
		2.2.4 Formulation for the Continuum
		2.2.5 Application to the Rod
		2.2.6 Application to the Euler-Bernoulli Beam
	2.3 Principle of the Stationary Value of the Total Elastic Potential
		2.3.1 Introduction
		2.3.2 Application to the Rod
		2.3.3 Application to the Euler-Bernoulli Beam
	2.4 Approximation Methods of Elastostatics
		2.4.1 The Ritz Method
		2.4.2 The Galerkin Method
Part II Disks
3 Isotropic Disks in Cartesian Coordinates
	3.1 Introduction
	3.2 Fundamentals
		3.2.1 Basic Equations
		3.2.2 The Displacement Method
		3.2.3 The Force Method
		3.2.4 Boundary Conditions
	3.3 Energetic Consideration
		3.3.1 Strain Energy
		3.3.2 Energetic Derivation of the Basic Equations
		3.3.3 Disks with Arbitrary Boundaries
	3.4 Elementary Solutions
		3.4.1 Solutions of the Disk Equation
		3.4.2 Elementary Cases
	3.5 Beam-type Disks
	3.6 St. Venant\'s Principle
	3.7 The Isotropic Half-Plane
		3.7.1 Decay Behaviour of Boundary Perturbations
		3.7.2 The Half-Plane Under Periodic Boundary Load
		3.7.3 The Half-Plane Under Non-periodic Load
	3.8 The Effective Width
		3.8.1 Effective Width of Flanges of Beams Under Bending
		3.8.2 Effective Width for Load Introductions
4 Isotropic Disks in Polar Coordinates
	4.1 Fundamentals
		4.1.1 Basic Equations
		4.1.2 The Displacement Method
		4.1.3 The Force Method
	4.2 Energetic Consideration
		4.2.1 Strain Energy
		4.2.2 Energetic Derivation of the Basic Equations
	4.3 Elementary Cases
	4.4 Rotationally Symmetric Disks
	4.5 Non-rotationally Symmetric Circular Disks
	4.6 Wedge-shaped Disks
	4.7 Disks with Circular Holes
5 Approximation Methods for Isotropic Disks
	5.1 The Displacement-Based Ritz Method
	5.2 The Force-Based Ritz Method
	5.3 Finite Elements for Disks
6 Anisotropic Disks
	6.1 Basic Equations
		6.1.1 Cartesian Coordinates
		6.1.2 Polar Coordinates
	6.2 Elementary Cases
	6.3 Beam-type Disks
	6.4 Decay Behaviour of Edge Perturbations
	6.5 Orthotropic Circular Ring Disks
	6.6 Orthotropic Circular Arc Disks
	6.7 Layered Circular Ring Disks
	6.8 Layered Circular Arc Disks
Part III Plates
7 Kirchhoff Plate Theory in Cartesian Coordinates
	7.1 Introduction
	7.2 The Kirchhoff Plate Theory
		7.2.1 Assumptions, Kinematics and Displacement Field
		7.2.2 Strain and Stress Field
		7.2.3 Force and Moment Flows, Constitutive Law
		7.2.4 Transformation Rules
	7.3 Effective Stiffnesses for Selected Plate Structures
		7.3.1 Homogeneous Plate of Orthotropic Material
		7.3.2 Homogeneous Plate of Isotropic Material
		7.3.3 Reinforced Concrete Plate
		7.3.4 Isotropic Plate Reinforced by Equidistant Stiffeners
		7.3.5 Isotropic Plate Reinforced by Equidistant Ribs
		7.3.6 Corrugated Metal Sheet
		7.3.7 Symmetrical Cross-Ply Composite Laminate
	7.4 Basic Equations of Plate Bending in Cartesian Coordinates
		7.4.1 Displacement Differential Equation
		7.4.2 Equivalent Transverse Shear Forces
		7.4.3 Boundary Conditions
	7.5 Elementary Solutions of the Plate Equation
	7.6 Bending of Plate Strips
	7.7 Navier Solution for Static Plate Bending Problems
		7.7.1 Determination of the Plate Deflection
		7.7.2 Moments, Forces and Stresses of the Plate
		7.7.3 Special Load Cases
	7.8 Lévy-type solutions for static plate bending problems
		7.8.1 Introduction
		7.8.2 Orthotropic Plates
		7.8.3 Isotropic Plates
	7.9 Energetic Consideration of Plate Bending
		7.9.1 Principle of the Minimum of the Total Elastic Potential
		7.9.2 Principle of Virtual Displacements
		7.9.3 Plate with Arbitrary Boundary
	7.10 Plate on Elastic Foundation
	7.11 The Membrane
8 Approximation Methods for the Kirchhoff Plate
	8.1 The Ritz Method
	8.2 The Galerkin Method
	8.3 The Finite Element Method
9 Kirchhoff Plate Theory in Polar Coordinates
	9.1 Transition to Polar Coordinates
	9.2 Basic Equations
	9.3 Rotationally Symmetric Bending of Circular Plates
		9.3.1 Basic Equations
		9.3.2 Plates Under Constant Surface Load
		9.3.3 Plates Under Centric Point Force
		9.3.4 Plate Under Edge Moments
		9.3.5 Plate Under Partial Load
		9.3.6 Circular Ring Plates
	9.4 Asymmetric Bending of Circular Plates
	9.5 Strain Energy
10 Higher-order Plate Theories
	10.1 First-Order Shear Deformation Theory
		10.1.1 Kinematics and Constitutive Equations
		10.1.2 Determination of the Shear Correction Factor K
		10.1.3 Equilibrium and Boundary Conditions
		10.1.4 Strain Energy
		10.1.5 Bending of Plate Strips
		10.1.6 Navier Solution
		10.1.7 Lévy-type solutions
		10.1.8 The Ritz Method
	10.2 Third-Order Shear Deformation Theory According to Reddy
		10.2.1 Kinematics
		10.2.2 Strains and Constitutive Equations
		10.2.3 Equilibrium Conditions
		10.2.4 Navier Solution
		10.2.5 The Ritz Method
11 Plate Buckling
	11.1 Basic Equations
	11.2 Navier Solution
		11.2.1 Biaxial Load
	11.3 Energy Methods for the Solution of Plate Buckling Problems
		11.3.1 Introduction
		11.3.2 The Rayleigh Quotient
		11.3.3 The Ritz Method
12 Geometrically Nonlinear Analysis
	12.1 Kirchhoff Plate Theory
		12.1.1 Energetic Consideration
		12.1.2 Th. V. Kármán equations
		12.1.3 Discussion of the Boundary Terms
		12.1.4 Inner and External Potential
		12.1.5 Special Cases
	12.2 Bending of Plates with Large Deflections
		12.2.1 Solution by Series Expansion
		12.2.2 The Galerkin Method
		12.2.3 The Ritz Method
	12.3 First-Order Shear Deformation Theory
13 Laminated Plates
	13.1 Introduction
	13.2 Classical Laminated Plate Theory
		13.2.1 Introduction
		13.2.2 Assumptions and Kinematics
		13.2.3 Strains and Stresses
	13.3 Constitutive Law
	13.4 Coupling Effects
		13.4.1 Shear Coupling
		13.4.2 Bending-Twisting Coupling
		13.4.3 Bending-extension Coupling
	13.5 Special Laminates
		13.5.1 Isotropic Single Layer
		13.5.2 Orthotropic Single Layer
		13.5.3 Anisotropic Single Layer/Off-axis Layer
		13.5.4 Symmetric Laminates
		13.5.5 Cross-ply Laminates
		13.5.6 Angle-ply Laminates
		13.5.7 Quasi-isotropic Laminates
	13.6 Basic Equations and Boundary Conditions
		13.6.1 Equilibrium Conditions
		13.6.2 Displacement Differential Equations
		13.6.3 Boundary Conditions
	13.7 Navier Solutions
		13.7.1 Bending of a Symmetric Cross-Ply Laminate
		13.7.2 Bending of an Unsymmetric Cross-Ply Laminate [(0°/90°)N]
		13.7.3 Bending of an Unsymmetric Angle-ply Laminate [(pmθ)N]
Part IV Shells
14 Introduction to Shell Structures
	14.1 Introduction
	14.2 Shells of Revolution
	14.3 Load Cases
	14.4 Classical Shell Theory
		14.4.1 Assumptions
		14.4.2 Stresses; Force and Moment Quantities
		14.4.3 Strains and Displacements
15 Membrane Theory of Shells of Revolution
	15.1 Assumptions
	15.2 Equilibrium Conditions for Shells of Revolution
		15.2.1 Equilibrium Conditions
		15.2.2 Rotational Symmetric Load
	15.3 Selected Solutions for Shells of Revolution
		15.3.1 Circular Cylindrical Shells
		15.3.2 Spherical Shells
		15.3.3 Conical Shells
	15.4 Kinematics of Shells of Revolution
	15.5 Constitutive Equations
	15.6 Displacement Solutions for Rotationally Symmetric Loads
	15.7 Energetic Derivation of the Basic Equations
16 Bending Theory of Shells of Revolution
	16.1 Basic Equations
		16.1.1 Equilibrium Conditions
		16.1.2 Kinematic Equations
		16.1.3 Constitutive Equations
		16.1.4 Displacement Differential Equations for the Circular Cylindrical Shell
		16.1.5 Boundary Conditions under Rotationally Symmetric Load
	16.2 Container Theory of the Circular Cylindrical Shell
		16.2.1 Basic Equations
		16.2.2 The Container Equation
		16.2.3 Solutions for the Container Equation
	16.3 The Force Method
	16.4 Edge Perturbations of the Spherical Shell
	16.5 Edge Perturbations of Arbitrary Shells of Revolution
	16.6 Circular Cylindrical Shell under Arbitrary Load
		16.6.1 Basic Equations
		16.6.2 Approximation According to Donnell
		16.6.3 Solution of the Basic Equations
		16.6.4 Boundary Conditions
	16.7 Laminated Shells
		16.7.1 Basic Equations
		16.7.2 Cross-ply Laminated Cylindrical Shells under Rotationally Symmetric Load
Index
Index