Introduction | p. 1 |

Prologue | p. 1 |

Brief History | p. 2 |

Introduction to Matrix Notation | p. 3 |

Role of the Computer | p. 6 |

General Steps of the Finite Element Method | p. 6 |

Applications of the Finite Element Method | p. 13 |

Advantages of the Finite Element Method | p. 18 |

Computer Programs for the Finite Element Method | p. 19 |

References | p. 22 |

Problems | p. 25 |

Introduction to the Stiffness (Displacement) Method | p. 26 |

Introduction | p. 26 |

Definition of the Stiffness Matrix | p. 26 |

Derivation of the Stiffness Matrix for a Spring Element | p. 27 |

Example of a Spring Assemblage | p. 32 |

Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) | p. 35 |

Boundary Conditions | p. 37 |

Potential Energy Approach to Derive Spring Element Equations | p. 50 |

References | p. 58 |

Problems | p. 59 |

Development of Truss Equations | p. 63 |

Introduction | p. 63 |

Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates | p. 63 |

Selecting Approximation Functions for Displacements | p. 69 |

Transformation of Vectors in Two Dimensions | p. 71 |

Global Stiffness Matrix | p. 74 |

Computation of Stress for a Bar in the x-y Plane | p. 78 |

Solution of a Plane Truss | p. 80 |

Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space | p. 87 |

Use of Symmetry in Structure | p. 92 |

Inclined, or Skewed, Supports | p. 95 |

Potential Energy Approach to Derive Bar Element Equations | p. 101 |

Comparison of Finite Element Solution to Exact Solution for Bar | p. 112 |

Galerkin's Residual Method and Its Application to a One-Dimensional Bar | p. 116 |

References | p. 119 |

Problems | p. 120 |

Development of Beam Equations | p. 137 |

Introduction | p. 137 |

Beam Stiffness | p. 138 |

Example of Assemblage of Beam Stiffness Matrices | p. 143 |

Examples of Beam Analysis Using the Direct Stiffness Method | p. 145 |

Distributed Loading | p. 154 |

Comparison of the Finite Element Solution to the Exact Solution for a Beam | p. 165 |

Beam Element with Nodal Hinge | p. 171 |

Potential Energy Approach to Derive Beam Element Equations | p. 176 |

Galerkin's Method for Deriving Beam Element Equations | p. 179 |

References | p. 181 |

Problems | p. 181 |

Frame and Grid Equations | p. 188 |

Introduction | p. 188 |

Two-Dimensional Arbitrarily Oriented Beam Element | p. 188 |

Rigid Plane Frame Examples | p. 192 |

Inclined or Skewed Supports--Frame Element | p. 211 |

Grid Equations | p. 212 |

Beam Element Arbitrarily Oriented in Space | p. 229 |

Concept of Substructure Analysis | p. 234 |

References | p. 240 |

Problems | p. 240 |

Development of the Plane Stress and Plane Strain Stiffness Equations | p. 264 |

Introduction | p. 264 |

Basic Concepts of Plane Stress and Plane Strain | p. 265 |

Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations | p. 270 |

Treatment of Body and Surface Forces | p. 284 |

Explicit Expression for the Constant-Strain Triangle Stiffness Matrix | p. 289 |

Finite Element Solution of a Plane Stress Problem | p. 291 |

References | p. 301 |

Problems | p. 301 |

Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis | p. 307 |

Introduction | p. 307 |

Finite Element Modeling | p. 308 |

Equilibrium and Compatibility of Finite Element Results | p. 318 |

Convergence of Solution | p. 320 |

Interpretation of Stresses | p. 321 |

Static Condensation | p. 323 |

Flowchart for the Solution of Plane Stress/Strain Problems | p. 327 |

Computer Program Results for Some Plane Stress/Strain Problems | p. 328 |

References | p. 331 |

Problems | p. 332 |

Development of the Linear-Strain Triangle Equations | p. 344 |

Introduction | p. 344 |

Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations | p. 344 |

Example LST Stiffness Determination | p. 349 |

Comparison of Elements | p. 352 |

References | p. 354 |

Problems | p. 355 |

Axisymmetric Elements | p. 358 |

Introduction | p. 358 |

Derivation of the Stiffness Matrix | p. 358 |

Solution of an Axisymmetric Pressure Vessel | p. 368 |

Applications of Axisymmetric Elements | p. 376 |

References | p. 380 |

Problems | p. 381 |

Isoparametric Formulation | p. 386 |

Introduction | p. 386 |

Isoparametric Formulation of the Bar Element Stiffness Matrix | p. 386 |

Rectangular Plane Stress Element | p. 392 |

Isoparametric Formulation of the Plane Element Stiffness Matrix | p. 395 |

Gaussian Quadrature (Numerical Integration) | p. 404 |

Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature | p. 407 |

Higher-Order Shape Functions | p. 413 |

References | p. 417 |

Problems | p. 417 |

Three-Dimensional Stress Analysis | p. 421 |

Introduction | p. 421 |

Three-Dimensional Stress and Strain | p. 421 |

Tetrahedral Element | p. 423 |

Isoparametric Formulation | p. 430 |

References | p. 436 |

Problems | p. 436 |

Plate Bending Element | p. 441 |

Introduction | p. 441 |

Basic Concepts of Plate Bending | p. 441 |

Derivation of a Plate Bending Element Stiffness Matrix and Equations | p. 445 |

Some Plate Element Numerical Comparisons | p. 450 |

Computer Solution for a Plate Bending Problem | p. 452 |

References | p. 454 |

Problems | p. 455 |

Heat Transfer and Mass Transport | p. 458 |

Introduction | p. 458 |

Derivation of the Basic Differential Equation | p. 459 |

Heat Transfer with Convection | p. 462 |

Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, h | p. 463 |

One-Dimensional Finite Element Formulation Using a Variational Method | p. 464 |

Two-Dimensional Finite Element Formulation | p. 478 |

Line or Point Sources | p. 487 |

One-Dimensional Heat Transfer with Mass Transport | p. 490 |

Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method | p. 491 |

Flowchart and Examples of a Heat-Transfer Program | p. 495 |

References | p. 499 |

Problems | p. 499 |

Fluid Flow | p. 508 |

Introduction | p. 508 |

Derivation of the Basic Differential Equations | p. 508 |

One-Dimensional Finite Element Formulation | p. 513 |

Two-Dimensional Finite Element Formulation | p. 521 |

Flowchart and Example of a Fluid-Flow Program | p. 526 |

References | p. 527 |

Problems | p. 528 |

Thermal Stress | p. 532 |

Introduction | p. 532 |

Formulation of the Thermal Stress Problem and Examples | p. 532 |

Reference | p. 553 |

Problems | p. 554 |

Structural Dynamics and Time-Dependent Heat Transfer | p. 559 |

Introduction | p. 559 |

Dynamics of a Spring-Mass System | p. 559 |

Direct Derivation of the Bar Element Equations | p. 561 |

Numerical Integration in Time | p. 565 |

Natural Frequencies of a One-Dimensional Bar | p. 577 |

Time-Dependent One-Dimensional Bar Analysis | p. 581 |

Beam Element Mass Matrices and Natural Frequencies | p. 586 |

Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matrices | p. 591 |

Time-Dependent Heat Transfer | p. 595 |

Computer Program Example Solutions for Structural Dynamics | p. 602 |

References | p. 609 |

Problems | p. 610 |

Matrix Algebra | p. 616 |

Introduction | p. 616 |

Definition of a Matrix | p. 616 |

Matrix Operations | p. 617 |

Cofactor or Adjoint Method to Determine the Inverse of a Matrix | p. 624 |

Inverse of a Matrix by Row Reduction | p. 626 |

References | p. 628 |

Problems | p. 628 |

Methods for Solution of Simultaneous Linear Equations | p. 630 |

Introduction | p. 630 |

General Form of the Equations | p. 630 |

Uniqueness, Nonuniqueness, and Nonexistence of Solution | p. 631 |

Methods for Solving Linear Algebraic Equations | p. 632 |

Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods | p. 643 |

References | p. 649 |

Problems | p. 650 |

Equations from Elasticity Theory | p. 652 |

Introduction | p. 652 |

Differential Equations of Equilibrium | p. 652 |

Strain/Displacement and Compatibility Equations | p. 654 |

Stress/Strain Relationships | p. 656 |

Reference | p. 659 |

Equivalent Nodal Forces | p. 660 |

Problems | p. 660 |

Principle of Virtual Work | p. 663 |

References | p. 666 |

Answers to Selected Problems | p. 667 |

Index | p. 689 |

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