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 |
Table of Contents provided by Syndetics. All Rights Reserved. |