Matrix and Finite Element Analyses of Structures

Preface
About This Book
Contents
About the Authors
1 Basic Concepts of Structural Analysis
	1.1 Types of Structures
	1.2 Objective of Structural Analysis
	1.3 Materials and Basic Assumptions
	1.4 Loads
	1.5 General Methods of Analysis
		1.5.1 Equilibrium Conditions
		1.5.2 Compatibility Conditions
	1.6 Force–Displacement Relationship
	1.7 Statical Indeterminacy
		1.7.1 Plane Structure
		1.7.2 Space Structures
	1.8 Kinematic Indeterminacy
	1.9 Two Approaches of Structural Analysis
2 Energy Principles
	2.1 Introduction
	2.2 Principle of Virtual Work
	2.3 Principle of Complementary Virtual Work
	2.4 Principle of Minimum Potential Energy
	2.5 Principle of Minimum Complementary Energy
	2.6 Castigliano’s Theorems
	2.7 Determination of Displacements
	References and Suggested Readings
3 Introduction to the Flexibility and Stiffness Matrix Methods
	3.1 Introduction
	3.2 The Flexibility Matrix Method
	3.3 The Stiffness Method
	3.4 Incorporation of Different Loading Conditions
	3.5 Other Types of Loadings
		3.5.1 Treatment by the Flexibility Matrix Method
		3.5.2 Treatment by the Stiffness Method
	3.6 Incorporation of Shear Deformation
	3.7 Relation Between Flexibility and Stiffness Matrices
	3.8 Equivalent Joint Loads
	3.9 Choice of the Method of Analysis
	References and Suggested Readings
4 Direct Stiffness Method
	4.1 Introduction
	4.2 Local and Global Coordinate System
	4.3 Transformation of Variables
		4.3.1 Transformation of Member Coordinate Axes
		4.3.2 Transformation of Member Displacement Matrix
		4.3.3 Transformation of the Member Force Matrix
		4.3.4 Transformation of the Member Stiffness Matrix
	4.4 Transformation of the Stiffness Matrix of the Member of a Truss
	4.5 Transformation of the Stiffness Matrix of the Member of a Rigid Frame
	4.6 Transformation of the Stiffness Matrix of the Member of a Grillage
	4.7 Transformation of the Stiffness Matrix of the Member of a Space Frame
	4.8 Horizontally Circular Curved Beam Element
	4.9 Overall Stiffness Matrix
	4.10 Boundary Conditions
		4.10.1 Boundary Conditions Corresponding to Skewed Supports
	4.11 Computation of Internal Forces
	4.12 Computer Program for the Truss Analysis by the Direct Stiffness Method
	4.13 Computer Program for the Frame Analysis by Direct Stiffness Method
	4.14 Computer Program for the Grillage Analysis by the Direct Stiffness Method
	References and Suggested Readings
5 Substructure Technique for the Analysis of Structural Systems
	5.1 Introduction
	5.2 Basic Concepts
	5.3 Direct Stiffness Method Restated
	5.4 The Substructure Technique
	5.5 An Illustrative Example
	5.6 Computer Program for the Truss Analysis by the Substructure Technique
	References and Suggested Readings
6 The Flexibility Matrix Method
	6.1 Introduction
	6.2 Element Flexibility Matrix
	6.3 Principle of Contragredience
	6.4 The Equilibrium Matrix
	6.5 Construction of the Flexibility Matrix of the Structure
	6.6 Matrix Determination of the Displacement Vector
	6.7 Determination of Member Forces
	6.8 Procedure of the Analysis of Statically Indeterminate Structures
	6.9 Illustrated Examples
	6.10 Choice of the Released Structure
	References and Suggested Readings
7 Elements of Elasticity
	7.1 Introduction
	7.2 Some Notations and Relations in the Theory of Elasticity
		7.2.1 Surface and Body Forces
		7.2.2 Components of Stresses
		7.2.3 Components of Strain
		7.2.4 Stress–Strain Relationship
	7.3 Two-Dimensional Problems
		7.3.1 Plane Stress
		7.3.2 Plane Strain
		7.3.3 Differential Equations of Equilibrium
	7.4 Bending of Thin Plates
		7.4.1 Basic Assumptions
		7.4.2 Deformation of the Plate
		7.4.3 Strain–Displacement Relationship
		7.4.4 Stress–Strain Relationship
		7.4.5 Equilibrium Equations
		7.4.6 Differential Equation for Deflection
		7.4.7 Shearing Forces
	7.5 Boundary Conditions
		7.5.1 Simply Supported Edge
		7.5.2 Clamped Edge
		7.5.3 Free Edge
		7.5.4 Elastically Supported Edge
		7.5.5 Edge Having Elastic Rotational Restraint
	7.6 Concluding Remarks
	References and Suggested Readings
8 Introduction to the Finite Element Method
	8.1 Introduction
	8.2 The Finite Element Method
	8.3 Brief History of the Development of the Finite Element Method
	8.4 Basic Steps in the Finite Element Method for the Solution of Static Problems
	8.5 Advantages and Disadvantages of the Finite Element Method
	References and Suggested Readings
9 Finite Element Analysis of Plane Elasticity Problems
	9.1 Introduction
	9.2 Three-Noded Triangular Element
		9.2.1 Displacement Function
		9.2.2 Displacement Function Expressed in Terms of Nodal Displacements
		9.2.3 Strain–Nodal Parameter Relationship
		9.2.4 Stress–Strain Relationship
		9.2.5 Derivation of the Element Stiffness Matrix
		9.2.6 Determination of Element Stresses
	9.3 Criteria for the Choice of the Displacement Function
	9.4 Polynomial Displacement Functions
	9.5 Verification of the Convergence Criteria of the Displacement Function of 3-Noded Triangular Element
	9.6 Number of Terms in a Polynomial
	9.7 Four-Noded Rectangular Element
		9.7.1 Displacement Function
		9.7.2 Displacement Function in Terms of Nodal Displacements
		9.7.3 Strain-Nodal Displacement Relationship
		9.7.4 Stress–Strain Relationship
		9.7.5 Derivation of the Element Stiffness Matrix
		9.7.6 Evaluation of Element Stresses
	9.8 A Note on the Rectangular Element
	9.9 A Note on Element Stresses
	9.10 Computer Program for the Plane Stress Analysis Using Three–Noded Triangular Element
	Bibliography
10 Isoparametric and Other Element Representations and Numerical Integrations
	10.1 Introduction
	10.2 Shape Function or Interpolation Function
	10.3 Determination of Shape Functions
		10.3.1 Linear 2-D Element
		10.3.2 Quadratic 2-D Element
	10.4 Plane Stress Isoparametric Linear Element
		10.4.1 Displacement Function in Terms of Nodal Parameters
		10.4.2 Strain-Nodal Parameter Relationship
		10.4.3 Evaluation of [B] Matrix
		10.4.4 Element Stiffness Matrix
		10.4.5 Convergence of Isoparametric Elements
		10.4.6 Concept of Isoparametric Element
	10.5 Numerical Integration
		10.5.1 Gaussian Quadrature Formula
		10.5.2 Gaussian Integration of Two Variables
	10.6 Lagrangian Interpolation
	10.7 Natural Coordinates and Higher Order Triangular Elements
		10.7.1 One-Dimensional Element
		10.7.2 Higher Order Triangular Elements
	10.8 The Quadratic Triangle for the Plane Stress Problem
	10.9 Numerical Integration of Area Coordinates
	10.10 Triangular Isoparametric Elements for the Analysis of Plane Stress Problems
	10.11 Allman’s Triangular Plane Stress Element
	10.12 Computer Program for the Solution of Plane Stress Problem Using Isoparametric Element
	References and Suggested Readings
11 Finite Element Analysis of Plate Bending Problems
	11.1 Introduction
	11.2 Beam Element
		11.2.1 Displacement Function
		11.2.2 Displacement Function in Terms of Nodal Displacements
		11.2.3 Strain (Curvature)–Nodal Parameter Relationship
		11.2.4 Stress (Moment)–Strain (Curvature) Relationship
		11.2.5 Derivation of the Element Stiffness Matrix
		11.2.6 Determination of Equivalent Loading on the Beam
	11.3 Rectangular Plate Bending Element
		11.3.1 Displacement Function
		11.3.2 Displacement Function Expressed in Terms of Nodal Displacements
		11.3.3 Strain–Nodal Parameter Relationship
		11.3.4 Stress (Moment)–Strain (Curvature) Relationship
		11.3.5 Derivation of the Element Stiffness Matrix
	11.4 Parallelogram Element of Plate Bending
		11.4.1 Displacement Function
		11.4.2 Displacement Function in Terms of Nodal Displacements
		11.4.3 Curvature–Nodal Parameter Relationship
		11.4.4 Moment–Curvature Relationship
		11.4.5 Element Stiffness Matrix
	11.5 Hermitian Polynomial Interpolation
	11.6 A Conforming Plate Bending Element
	11.7 Isoparametric Plate Bending Element
		11.7.1 Displacement Function
		11.7.2 Strain–Nodal Displacement Relationship
		11.7.3 Stress–Strain Relationship
		11.7.4 Element Stiffness Matrix
		11.7.5 Reduced Integration Technique
	11.8 Smoothed Stresses
	11.9 Triangular Plate Bending Elements
	11.10 DKT Element
		11.10.1 Constraint Equations
		11.10.2 Transformation Matrix
		11.10.3 Element Stiffness Matrix
	11.11 The Patch Test
		11.11.1 The Patch Test for the Plane Stress Element
		11.11.2 The Patch Test for Plate Bending Elements
	11.12 Horizontally Curved Isoparametric Beam
		11.12.1 Displacement Function in Terms of Nodal Parameters
		11.12.2 Stress–Strain Relations
		11.12.3 Strain–Displacement Relationship
		11.12.4 Element Stiffness Matrix
	11.13 Nonuniform Straight Beam Element
	11.14 Computer Program for Isoparametric Quadratic Bending Element
	References and Suggested Readings
12 Finite Element Analysis of Shells
	12.1 Introduction
	12.2 Flat Shell Element
		12.2.1 Transformation of the Stiffness Matrix and Assembly
	12.3 Shell of Revolution
	12.4 General Shell Finite Element of Triangular Shape
		12.4.1 Derivation of the Stiffness Matrix
		12.4.2 Consistent Load Vector
		12.4.3 Condensation of Stiffness Matrix
	12.5 Isoparametric General Shell Element
		12.5.1 Geometry of the Shell Element
		12.5.2 Displacement Field
		12.5.3 Strains Inside the Element
		12.5.4 Stress–Strain Relationship
		12.5.5 Stiffness Matrix of the Shell Element
	12.6 Vertically Curved Beam Element
	12.7 Computer Program for the Finite Element Analysis of Shallow Shells of General Shape Using Triangular Element
	References and Suggested Readings
13 Semi-analytical and Spline Finite Strip Method of Analysis of Plate Bending
	13.1 Introduction
	13.2 Beam Function
	13.3 Model of the Plate
	13.4 The Displacement Function
	13.5 Curvature-Nodal Parameter Relationship
	13.6 Moment—Curvature Relationship
	13.7 Strip Stiffness Matrix
	13.8 Loading Matrix
	13.9 Force Displacement Relationship
	13.10 Spline Finite Strip Method of Analysis of Plate Bending
		13.10.1 The Spline Function
		13.10.2 Displacement Functions
		13.10.3 Strain–Displacement Relationship
		13.10.4 Stiffness Matrix
		13.10.5 The Loading Matrix
	13.11 Computer Program for the Spline Finite Strip Method of Analysis of Plates in Bending
	References and Suggested Readings
14 Dynamic and Instability Analyses by the Finite Element Method
	14.1 Introduction
	14.2 Dynamic Analysis
		14.2.1 Torsional Vibration of Shafts
		14.2.2 An Example
		14.2.3 Flexural Vibration of Beams
		14.2.4 In-Plane Vibration of Plates
		14.2.5 Flexural Vibration of Plates
	14.3 Elastic Instability Analysis
		14.3.1 Column Instability Analysis
		14.3.2 Plate Instability Analysis
	References and Suggested Readings
15 The Finite Difference Method for the Analysis of Beams and Plates
	15.1 Introduction
	15.2 Finite Difference Representation of Derivatives
		15.2.1 First Derivative
		15.2.2 Second Derivative
		15.2.3 Third Derivative
		15.2.4 Fourth Derivative
	15.3 Errors in the Finite Difference Expressions
	15.4 Equivalent Concentrated Load
	15.5 Boundary Conditions for Beam Bending
		15.5.1 Simple Support
		15.5.2 Fixed End
		15.5.3 Free End
	15.6 A Statically Determinate Static Problem
	15.7 A Statically Indeterminate Static Problem
	15.8 Free Vibration of Beams
	15.9 Buckling of Columns
	15.10 Finite Difference Representation of the Plate Equation
		15.10.1 A Plate Example
	References and Suggested Readings
16 Adaptive Finite Element Analysis
	16.1 Introduction
	16.2 The Adaptive Finite Element Technique
	16.3 Superconvergent Patch Recovery Technique
	16.4 Example of Verification of SPR
	16.5 Error Estimation
	16.6 ZZ Error Estimator
	16.7 ZZ—Refinement Framework
	16.8 Adaptive Mesh Generation
		16.8.1 Mesh Generation Based on Mapping
		16.8.2 Delaunay Triangulation Method
		16.8.3 Domain Decomposition Method (Quadtree)
		16.8.4 Advancing Front Technique
	References and Suggested Readings
17 Geometrical Nonlinear Finite Element Analysis
	17.1 Introduction
	17.2 Nonlinear Equation Solving Procedures
		17.2.1 Direct Iteration Method
		17.2.2 Newton–Raphson Method
		17.2.3 Modified Newton–Raphson Method
		17.2.4 Incremental Techniques
	17.3 Formulation of the Geometric Nonlinear Problem
		17.3.1 Equilibrium Equations
		17.3.2 Incremental Equilibrium Equation
	17.4 Large Deflection Analysis of Plates in b-notation
	17.5 Large Deflection Analysis of Plates in n-notation
	17.6 Example of a Pin-Jointed bar
	17.7 Computer Program for Geometrically Nonlinear Analysis of Plates
	References and Suggested Reading
18 Finite Element Method of Analysis of Stiffened Plates
	18.1 Introduction
	18.2 Modeling the Plate and the Stiffener
	18.3 Rectangular Stiffened Plate Bending Element
		18.3.1 Stiffness Matrix of the Stiffener Element
	18.4 Isoparametric Stiffened Plate Bending Element
		18.4.1 Stiffness Matrix of Arbitrarily-Oriented Eccentric Stiffener
	References and Suggested Readings
19 Selected Topics
	19.1 Rayleigh–Ritz Method
	19.2 An Example
	19.3 Rayleigh–Ritz Finite Element Method
	19.4 Weighted Residual Methods
	19.5 Galerkin Method
		19.5.1 An Example of Galerkin Method
	19.6 Galerkin Finite Element Method
	19.7 Torsional Stiffness of Prismatic Beam Element
	19.8 Torsion of Noncircular Sections
	19.9 Axi-symmetrical Element
	19.10 Three-Dimensional Elements
		19.10.1 Linear Element (8 Nodes) (Fig. 19.8a)
		19.10.2 Quadratic Element (20 Nodes) (Fig. 19.8b)
		19.10.3 Cubic Element (32 Nodes) (Fig. 19.8c)
		19.10.4 A16 Noded Solid (Fig. 19.9a)
		19.10.5 A24 Noded Solid (Fig. 19.9b)
	References and Suggested Readings
Appendix A Fixed-End Forces
Appendix B
Index