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