| |
| |
| |
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 | |