Cover
Contents
Preface to the SI Edition
Preface
Digital Resources
Notation
Chapter 1: Introduction
Chapter Objectives
Prologue
1.1 Brief History
1.2 Introduction to Matrix Notation
1.3 Role of the Computer
1.4 General Steps of the Finite Element Method
1.5 Applications of the Finite Element Method
1.6 Advantages of the Finite Element Method
1.7 Computer Programs for the Finite Element Method
References
Problems
Chapter 2: Introduction to the Stiffness (Displacement) Method
Chapter Objectives
Introduction
2.1 Definition of the Stiffness Matrix
2.2 Derivation of the Stiffness Matrix for a Spring Element
2.3 Example of a Spring Assemblage
2.4 Assembling the Total Stiffness Matrix byΒ Superposition (Direct Stiffness Method)
2.5 Boundary Conditions
2.6 Potential Energy Approach to Derive Spring ElementΒ Equations
Summary Equations
References
Problems
Chapter 3: Development of Truss Equations
Chapter Objectives
Introduction
3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates
3.2 Selecting a Displacement Function in Step 2 of the Derivation of Stiffness Matrix for the One-Dimensional Bar Element
3.3 Transformation of Vectors in Two Dimensions
3.4 Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane
3.5 Computation of Stress for a Bar in the x - y Plane
3.6 Solution of a Plane Truss
3.7 Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space
3.8 Use of Symmetry in Structures
3.9 Inclined, or Skewed, Supports
3.10 Potential Energy Approach to Derive Bar Element Equations
3.11 Comparison of Finite Element Solution to Exact Solution for Bar
3.12 Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations
3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem
3.14 Flowchart for Solution of Three-Dimensional TrussΒ Problems
3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem
Summary Equations
References
Problems
Chapter 4: Development of Beam Equations
Chapter Objectives
Introduction
4.1 Beam Stiffness
4.2 Example of Assemblage of Beam Stiffness Matrices
4.3 Examples of Beam Analysis Using the Direct Stiffness Method
4.4 Distributed Loading
4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam
4.6 Beam Element with Nodal Hinge
4.7 Potential Energy Approach to Derive Beam Element Equations
4.8 Galerkin's Method for Deriving Beam Element Equations
Summary Equations
References
Problems
Chapter 5: Frame and Grid Equations
Chapter Objectives
Introduction
5.1 Two-Dimensional Arbitrarily Oriented Beam Element
5.2 Rigid Plane Frame Examples
5.3 Inclined or Skewed Supports - Frame Element
5.4 Grid Equations
5.5 Beam Element Arbitrarily Oriented in Space
5.6 Concept of Substructure Analysis
Summary Equations
References
Problems
Chapter 6: Development of the Plane Stress and Plane Strain Stiffness Equations
Chapter Objectives
Introduction
6.1 Basic Concepts of Plane Stress and Plane Strain
6.2 Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations
6.3 Treatment of Body and Surface Forces
6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix
6.5 Finite Element Solution of a Plane Stress Problem
6.6 Rectangular Plane Element (Bilinear Rectangle, Q4)
Summary Equations
References
Problems
Chapter 7: Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis
Chapter Objectives
Introduction
7.1 Finite Element Modeling
7.2 Equilibrium and Compatibility of Finite Element Results
7.3 Convergence of Solution and Mesh Refinement
7.4 Interpretation of Stresses
7.5 Flowchart for the Solution of Plane Stress/Strain Problems
7.6 Computer Program-Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems
References
Problems
Chapter 8: Development of the Linear-Strain Triangle Equations
Chapter Objectives
Introduction
8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations
8.2 Example LST Stiffness Determination
8.3 Comparison of Elements
Summary Equations
References
Problems
Chapter 9: Axisymmetric Elements
Chapter Objectives
Introduction
9.1 Derivation of the Stiffness Matrix
9.2 Solution of an Axisymmetric Pressure Vessel
9.3 Applications of Axisymmetric Elements
Summary Equations
References
Problems
Chapter 10: Isoparametric Formulation
Chapter Objectives
Introduction
10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix
10.2 Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness Matrix
10.3 Newton-Cotes and Gaussian Quadrature
10.4 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature
10.5 Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements)
Summary Equations
References
Problems
Chapter 11: Three-Dimensional Stress Analysis
Chapter Objectives
Introduction
11.1 Three-Dimensional Stress and Strain
11.2 Tetrahedral Element
11.3 Isoparametric Formulation and Hexahedral Element
Summary Equations
References
Problems
Chapter 12: Plate Bending Element
Chapter Objectives
Introduction
12.1 Basic Concepts of Plate Bending
12.2 Derivation of a Plate Bending Element Stiffness Matrix and Equations
12.3 Some Plate Element Numerical Comparisons
12.4 Computer Solutions for Plate Bending Problems
Summary Equations
References
Problems
Chapter 13: Heat Transfer and Mass Transport
Chapter Objectives
Introduction
13.1 Derivation of the Basic Differential Equation
13.2 Heat Transfer with Convection
13.3 Typical Units; Thermal Conductivities, K; and Heat Transfer Coefficients, h
13.4 One-Dimensional Finite Element Formulation Using a Variational Method
13.5 Two-Dimensional Finite Element Formulation
13.6 Line or Point Sources
13.7 Three-Dimensional Heat Transfer by the Finite Element Method
13.8 One-Dimensional Heat Transfer with Mass Transport
13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method
13.10 Flowchart and Examples of a Heat Transfer Program
Summary Equations
References
Problems
Chapter 14: Fluid Flow in Porous Media and through Hydraulic Networks; and Electrical Networks and Electrostatics
Chapter Objectives
Introduction
14.1 Derivation of the Basic Differential Equations
14.2 One-Dimensional Finite Element Formulation
14.3 Two-Dimensional Finite Element Formulation
14.4 Flowchart and Example of a Fluid-Flow Program
14.5 Electrical Networks
14.6 Electrostatics
Summary Equations
References
Problems
Chapter 15: Thermal Stress
Chapter Objectives
Introduction
15.1 Formulation of the Thermal Stress Problem and Examples
Summary Equations
Reference
Problems
Chapter 16: Structural Dynamics and Time-Dependent Heat Transfer
Chapter Objectives
Introduction
16.1 Dynamics of a Spring-Mass System
16.2 Direct Derivation of the Bar Element Equations
16.3 Numerical Integration in Time
16.4 Natural Frequencies of a One-Dimensional Bar
16.5 Time-Dependent One-Dimensional Bar Analysis
16.6 Beam Element Mass Matrices and Natural Frequencies
16.7 Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices
16.8 Time-Dependent Heat Transfer
16.9 Computer Program Example Solutions for Structural Dynamics
Summary Equations
References
Problems
Appendix A: Matrix Algebra
Appendix B: Methods for Solution of Simultaneous Linear Equations
Appendix C: Equations from Elasticity Theory
Appendix D: Equivalent Nodal Forces
Appendix E: Principle of Virtual Work
Appendix F: Geometric Properties of Structural Steel Wide-Flange Sections (W Shapes)
Answers to Selected Problems
Index