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| Introduction | |
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| Prologue | |
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| Brief History | |
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| Introduction to Matrix Notation | |
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| Role of the Computer | |
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| General Steps of the Finite Element Method | |
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| Applications of the Finite Element Method | |
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| Advantages of the Finite Element Method | |
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| Computer Programs for the Finite Element Method | |
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| References | |
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| Problems | |
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| Introduction to the Stiffness (Displacement) Method | |
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| Introduction | |
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| Definition of the Stiffness Matrix | |
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| Derivation of the Stiffness Matrix for a Spring Element | |
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| Example of a Spring Assemblage | |
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| Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) | |
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| Boundary Conditions | |
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| Potential Energy Approach to Derive Spring Element Equations | |
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| References | |
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| Problems | |
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| Development of Truss Equations | |
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| Introduction | |
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| Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates | |
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| Selecting Approximation Functions for Displacements | |
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| Transformation of Vectors in Two Dimensions | |
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| Global Stiffness Matrix | |
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| Computation of Stress for a Bar in the x-y Plane | |
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| Solution of a Plane Truss | |
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| Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space | |
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| Use of Symmetry in Structure | |
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| Inclined, or Skewed, Supports | |
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| Potential Energy Approach to Derive Bar Element Equations | |
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| Comparison of Finite Element Solution to Exact Solution for Bar | |
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| Galerkin's Residual Method and Its Application to a One-Dimensional Bar | |
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| References | |
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| Problems | |
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| Algor Program for Truss Analysis | |
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| Introduction | |
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| Overview of the Algor System and Flowcharts for the Solution of a Truss Problem Using Algor | |
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| Algor Example Solutions for Truss Analysis | |
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| References | |
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| Problems | |
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| Development of Beam Equations | |
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| Introduction | |
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| Beam Stiffness | |
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| Example of Assemblage of Beam Stiffness Matrices | |
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| Examples of Beam Analysis Using the Direct Stiffness Method | |
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| Distributed Loading | |
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| Comparison of the Finite Element Solution to the Exact Solution for a Beam | |
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| Beam Element with Nodal Hinge | |
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| Potential Energy Approach to Derive Beam Element Equations | |
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| Galerkin's Method for Deriving Beam Element Equations | |
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| Algor Example Solutions for Beam Analysis | |
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| References | |
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| Problems | |
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| Frame and Grid Equations | |
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| Introduction | |
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| Two-Dimensional Arbitrarily Oriented Beam Element | |
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| Rigid Plane Frame Examples | |
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| Inclined or Skewed Supports--Frame Element | |
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| Grid Equations | |
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| Beam Element Arbitrarily Oriented in Space | |
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| Concept of Substructure Analysis | |
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| Algor Example Solutions for Plane Frame, Grid, and Space Frame Analysis | |
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| References | |
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| Problems | |
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| Development of the Plane Stress and Plane Strain Stiffness Equations | |
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| Introduction | |
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| Basic Concepts of Plane Stress and Plane Strain | |
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| Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations | |
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| Treatment of Body and Surface Forces | |
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| Explicit Expression for the Constant-Strain Triangle Stiffness Matrix | |
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| Finite Element Solution of a Plane Stress Problem | |
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| References | |
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| Problems | |
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| Practical Considerations in Modeling; Interpreting Results; and Use of the Algor Program for Plane Stress/Strain Analysis | |
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| Introduction | |
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| Finite Element Modeling | |
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| Equilibrium and Compatibility of Finite Element Results | |
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| Convergence of Solution | |
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| Interpretation of Stresses | |
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| Static Condensation | |
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| Flowchart for the Solution of Plane Stress/Strain Problems and Typical Steps Using Algor | |
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| Algor Example Solutions for Plane Stress/Strain Analysis | |
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| References | |
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| Problems | |
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| Development of the Linear-Strain Triangle Equations | |
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| Introduction | |
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| Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations | |
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| Example LST Stiffness Determination | |
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| Comparison of Elements | |
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| References | |
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| Problems | |
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| Axisymmetric Elements | |
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| Introduction | |
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| Derivation of the Stiffness Matrix | |
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| Solution of an Axisymmetric Pressure Vessel | |
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| Applications of Axisymmetric Elements | |
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| Algor Example Solutions for Axisymmetric Problems | |
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| References | |
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| Problems | |
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| Isoparametric Formulation | |
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| Introduction | |
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| Isoparametric Formulation of the Bar Element Stiffness Matrix | |
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| Rectangular Plane Stress Element | |
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| Isoparametric Formulation of the Plane Element Stiffness Matrix | |
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| Gaussian Quadrature (Numerical Integration) | |
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| Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature | |
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| Higher-Order Shape Functions | |
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| References | |
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| Problems | |
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| Three-Dimensional Stress Analysis | |
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| Introduction | |
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| Three-Dimensional Stress and Strain | |
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| Tetrahedral Element | |
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| Isoparametric Formulation | |
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| Algor Example Solutions of Three-Dimensional Stress Analysis | |
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| References | |
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| Problems | |
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| Heat Transfer and Mass Transport | |
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| Introduction | |
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| Derivation of the Basic Differential Equation | |
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| Heat Transfer with Convection | |
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| Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, h | |
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| One-Dimensional Finite Element Formulation Using a Variational Method | |
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| Two-Dimensional Finite Element Formulation | |
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| Line or Point Sources | |
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| One-Dimensional Heat Transfer with Mass Transport | |
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| Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method | |
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| Flowchart of a Heat-Transfer Program | |
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| Algor Example Solutions for Heat-Transfer Problems | |
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| References | |
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| Problems | |
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| Fluid Flow | |
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| Introduction | |
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| Derivation of the Basic Differential Equations | |
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| One-Dimensional Finite Element Formulation | |
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| Two-Dimensional Finite Element Formulation | |
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| Flowchart of a Fluid-Flow Program | |
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| Algor Example Solutions for Two-Dimensional Steady-State Fluid Flow | |
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| References | |
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| Problems | |
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| Thermal Stress | |
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| Introduction | |
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| Formulation of the Thermal Stress Problem and Examples | |
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| Algor Example Solutions for Thermal Stress Problems | |
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| Reference | |
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| Problems | |
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| Structural Dynamics and Time-Dependent Heat Transfer | |
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| Introduction | |
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| Dynamics of a Spring-Mass System | |
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| Direct Derivation of the Bar Element Equations | |
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| Numerical Integration in Time | |
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| Natural Frequencies of a One-Dimensional Bar | |
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| Time-Dependent One-Dimensional Bar Analysis | |
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| Beam Element Mass Matrices and Natural Frequencies | |
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| Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matrices | |
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| Time-Dependent Heat Transfer | |
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| Algor Example Solutions for Structural Dynamics and Transient Heat Transfer | |
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| References | |
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| Problems | |
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| Plate Bending Element | |
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| Introduction | |
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| Basic Concepts of Plate Bending | |
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| Derivation of a Plate Bending Element Stiffness Matrix and Equations | |
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| Some Plate Element Numerical Comparisons | |
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| Algor Example Solution for Plate Bending Problems | |
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| References | |
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| Problems | |
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| Matrix Algebra | |
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| Introduction | |
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| Definition of a Matrix | |
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| Matrix Operations | |
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| Cofactor or Adjoint Method to Determine the Inverse of a Matrix | |
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| Inverse of a Matrix by Row Reduction | |
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| References | |
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| Problems | |
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| Methods for Solution of Simultaneous Linear Equations | |
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| Introduction | |
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| General Form of the Equations | |
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| Uniqueness, Nonuniqueness, and Nonexistence of Solution | |
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| Methods for Solving Linear Algebraic Equations | |
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| Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods | |
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| References | |
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| Problems | |
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| Equations from Elasticity Theory | |
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| Introduction | |
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| Differential Equations of Equilibrium | |
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| Strain/Displacement and Compatibility Equations | |
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| Stress/Strain Relationships | |
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| Reference | |
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| Equivalent Nodal Forces | |
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| Problems | |
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| Principle of Virtual Work | |
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| References | |
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| Basics of Algor | |
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| Introduction | |
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| Hardware Requirements for Windows Installation | |
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| Conventions | |
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| Getting Around the Menu System | |
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| Function Keys | |
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| Algor Processor Names | |
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| File Extensions Generated by the Algor System | |
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| Checking Model for Defects by Using Superview | |
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| Answers to Selected Problems | |
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| Index | |