| |

| |

| |

Introduction | |

| |

| |

Brief History | |

| |

| |

Introduction to Matrix Notation | |

| |

| |

Role of the Computer | |

| |

| |

General Steps of the Finite Element Method | |

| |

| |

Applications of the Finite Element Method | |

| |

| |

Advantages of the Finite Element Method | |

| |

| |

Computer Programs for the Finite Element Method. | |

| |

| |

| |

Introduction To The Stiffness (Displacement) Method | |

| |

| |

Definition of the Stiffness Matrix | |

| |

| |

Derivation of the Stiffness Matrix for a Spring Element | |

| |

| |

Example of a Spring Assemblage | |

| |

| |

Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) | |

| |

| |

Boundary Conditions | |

| |

| |

Potential Energy Approach to Derive Spring Element Equations | |

| |

| |

| |

Development Of Truss Equations | |

| |

| |

Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates | |

| |

| |

Selecting Approximation Functions for Displacements | |

| |

| |

Transformation of Vectors in Two Dimensions | |

| |

| |

Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane | |

| |

| |

Computation of Stress for a Bar in the x-y Plane | |

| |

| |

Solution of a Plane Truss | |

| |

| |

Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space | |

| |

| |

Use of Symmetry in Structure | |

| |

| |

Inclined, or Skewed, Supports | |

| |

| |

Potential Energy Approach to Derive Bar Element Equations | |

| |

| |

Comparison of Finite Element Solution to Exact Solution for Bar | |

| |

| |

Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations | |

| |

| |

Other Residual Methods and Their Application to a One-Dimensional Bar Problem | |

| |

| |

Flowchart for Solutions of Three-Dimensional Truss Problems | |

| |

| |

Computer Program Assisted Step-by-Step Solution for Truss Problem | |

| |

| |

| |

Development Of Beam Equations | |

| |

| |

Beam Stiffness | |

| |

| |

Example of Assemblage of Beam Stiffness Matrices | |

| |

| |

Examples of Beam Analysis Using the Direct Stiffness Method | |

| |

| |

Distribution Loading | |

| |

| |

Comparison of the Finite Element Solution to the Exact Solution for a Beam | |

| |

| |

Beam Element with Nodal Hinge | |

| |

| |

Potential Energy Approach to Derive Beam Element Equations | |

| |

| |

Galerkin's Method for Deriving Beam Element Equations | |

| |

| |

| |

Frame And Grid Equations | |

| |

| |

Two-Dimensional Arbitrarily Oriented Beam Element | |

| |

| |

Rigid Plane Frame Examples | |

| |

| |

Inclined or Skewed Supports - Frame Element | |

| |

| |

Grid Equations | |

| |

| |

Beam Element Arbitrarily Oriented in Space | |

| |

| |

Concept of Substructure Analysis | |

| |

| |

| |

Development Of The Plane Stress And Strain Stiffness Equations | |

| |

| |

Basic Concepts of Plane Stress and Plane Strain | |

| |

| |

Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations | |

| |

| |

Treatment of Body and Surface Forces | |

| |

| |

Explicit Expression for the Constant-Strain Triangle Stiffness Matrix | |

| |

| |

Finite Element Solution of a Plane Stress Problem | |

| |

| |

Rectangular Plane Element (Bilinear Rectangle, Q4) | |

| |

| |

| |

Practical Considerations In Modeling: Interpreting Results And Exampels Of Plane Stress/Strain Analysis | |

| |

| |

Finite Element Modeling | |

| |

| |

Equilibrium and Compatibility of Finite Element Results | |

| |

| |

Convergence of Solution | |

| |

| |

Interpretation of Stresses | |

| |

| |

Static Condensation | |

| |

| |

Flowchart for the Solution of Plane Stress-Strain Problems | |

| |

| |

Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress-Strain Problems | |

| |

| |

| |

Development Of The Linear-Strain Traingle Equations | |

| |

| |

Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations | |

| |

| |

Example of LST Stiffness Determination | |

| |

| |

Comparison of Elements | |

| |

| |

| |

Axisymmetric Elements | |

| |

| |

Derivation of the Stiffness Matrix | |

| |

| |

Solution of an Axisymmetric Pressure Vessel | |

| |

| |

Applications of Axisymmetric Elements | |

| |

| |

| |

Isoparametric Formulation | |

| |

| |

Isoparametric Formulation of the Bar Element Stiffness Matrix | |

| |

| |

Isoparametric Formulation of the Okabe Quadrilateral Element Stiffness Matrix | |

| |

| |

Newton-Cotes and Gaussian Quadrature | |

| |

| |

Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature | |

| |

| |

Higher-Order Shape Functions | |

| |

| |

| |

Three-Dimensional Stress Analysis | |

| |

| |

Three-Dimensional Stress and Strain | |

| |

| |

Tetrahedral Element | |

| |

| |

Isoparametric Formulation | |

| |

| |

| |

Plate Bending Element | |

| |

| |

Basic Concepts of Plate Bending | |

| |

| |

Derivation of a Plate Bending Element Stiffness Matrix and Equations | |

| |

| |

Some Plate Element Numerical Comparisons | |

| |

| |

Computer Solutions for Plate Bending Problems | |

| |

| |

| |

Heat Transfer And Mass Transport | |

| |

| |

Derivation of the Basic Differential Equation | |

| |

| |

Heat Transfer with Convection | |

| |

| |

Typical Units; Thermal Conductivities K; and Heat-Transfer Coefficients, h | |

| |

| |

One-Dimensional Finite Element Formulation Using a Variational Method | |

| |

| |

Two-Dimensional Finite Element Formulation | |

| |

| |

Line or Point Sources | |

| |

| |

Three-Dimensional Heat Transfer by the Finite Element Method | |

| |

| |

One-Dimensional Heat Transfer with Mass Transport | |

| |

| |

Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method | |

| |

| |

Flowchart and Examples of a Heat-Transfer Program | |

| |

| |

| |

Fluid Flow In Porous Media And Through Hydraulic Networks; And Electrical Networks And Electrostatics | |

| |

| |

Derivation of the Basic Differential Equations | |

| |

| |

One-Dimensional Finite Element Formulation | |

| |

| |

Two-Dimensional Finite Element Formulation | |

| |

| |

Flowchart and Example of a Fluid-Flow Program | |

| |

| |

Electrical Networks | |

| |

| |

Electrostatics | |

| |

| |

| |

Thermal Stress | |

| |

| |

Formulation of the Thermal Stress Problem and Examples | |

| |

| |

| |

Structural Dynamics And Time-Dependent Heat Transfer | |

| |

| |

Dynamics of a Spring-Mass System | |

| |

| |

Direct Derivation of the Bar Element Equations | |

| |

| |

Numerical Integration in Time | |

| |

| |

Natural Frequencies of a One-Dimensional Bar | |

| |

| |

Time-Dependent One-Dimensional Bar Analysis | |

| |

| |

Beam Element Mass Matrices and Natural Frequencies | |

| |

| |

Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices | |

| |

| |

Time-Dependent Heat-Transfer | |

| |

| |

Computer Program Example Solutions for Structural Dynamics | |

| |

| |

| |

Matrix Algebra | |

| |

| |

Definition of a Matrix | |

| |

| |

Matrix Operations | |

| |

| |

Cofactor of Adjoint Method to Determine the Inverse of a Matrix | |

| |

| |

Inverse of a Matrix by Row Reduction | |

| |

| |

Properties of Stiffness Matrices | |

| |

| |

| |

Methods For Solution Of Simultaneous Linear Equations | |

| |

| |

Introduction | |

| |

| |

General Form of the Equations | |

| |

| |

Uniqueness, Nonuniqueness, and Nonexistence of Solution | |

| |

| |

Methods for Solving Linear Algebraic Equations | |

| |

| |

Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods | |

| |

| |

| |

Equations For Elasticity Theory | |

| |

| |

Introduction | |

| |

| |

Differential Equations of Equilibrium | |

| |

| |

Strain/Displacement and Compatibility Equations | |

| |

| |

Stress-Strain Relationships | |

| |

| |

| |

Equivalent Nodal Forces | |

| |

| |

| |

Principle Of Virtual Work | |

| |

| |

| |

Properties Of Structural Steel And Aluminum Shapes | |

| |

| |

Answers To Selected Problems | |

| |

| |

Index | |