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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 | |