FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS With Mathematica and MATLAB Computations M. ASGHAR BHATTI WILEY JOHN WILEY & SONS, INC.
CONTENTS OF THE BOOK WEB SITE PREFACE xi xiii 1 FINITE ELEMENT METHOD: THE BIG PICTURE 1 1.1 Discretization and Element Equations / 2 1.1.1 Plane Trass Element / 4 1.1.2 Triangular Element for Two-Dimensional Heat Flow / 7 1.1.3 General Remarks on Finite Element Discretization / 14 1.1.4 Triangular Element for Two-Dimensional Stress Analysis / 16 1.2 Assembly of Element Equations / 21 1.3 Boundary Conditions and Nodal Solution / 36 1.3.1 Essential Boundary Conditions by Rearranging Equations / 37 1.3.2 Essential Boundary Conditions by Modifying Equations / 39 1.3.3 Approximate Treatment of Essential Boundary Conditions / 40 1.3.4 Computation of Reactions to Verify Overall Equilibrium / 41 1.4 Element Solutions and Model Validity / 49 1.4.1 Plane Truss Element / 49 1.4.2 Triangular Element for Two-Dimensional Heat Flow / 51 1.4.3 Triangular Element for Two-Dimensional Stress Analysis / 54 1.5 Solution of Linear Equations / 58 1.5.1 Solution Using Choleski Decomposition / 58 1.5.2 Conjugate Gradient Method / 62
Vi CONTENTS 1.6 Multipoint Constraints / 72 1.6.1 Solution Using Lagrange Multipliers / 75 1.6.2 Solution Using Penalty Function / 79 1.7 Units / 83 MATHEMATICAL FOUNDATION OF THE FINITE ELEMENT METHOD 98 2.1 Axial Deformation of Bars / 99 2.1.1 Differential Equation for Axial Deformations / 99 2.1.2 Exact Solutions of Some Axial Deformation Problems / 101 2.2 Axial Deformation of Bars Using Galerkin Method / 104 2.2.1 Weak Form for Axial Deformations / 105 2.2.2 Uniform Bar Subjected to Linearly Varying Axial Load / 109 2.2.3 Tapered Bar Subjected to Linearly Varying Axial Load / 113 2.3 One-Dimensional BVP Using Galerkin Method / 115 2.3.1 Overall Solution Procedure Using Galerkin Method / 115 2.3.2 Higher Order Boundary Value Problems / 119 2.4 Rayleigh-Ritz Method / 128 2.4.1 Potential Energy for Axial Deformation of Bars / 129 2.4.2 Overall Solution Procedure Using the Rayleigh-Ritz Method / 130 2.4.3 Uniform Bar Subjected to Linearly Varying Axial Load / 131 2.4.4 Tapered Bar Subjected to Linearly Varying Axial Load / 133 2.5 Comments on Galerkin and Rayleigh-Ritz Methods / 135 2.5.1 Admissible Assumed Solution / 135 2.5.2 Solution Convergence the Completeness Requirement / 136 2.5.3 Galerkin versus Rayleigh-Ritz / 138 2.6 Finite Element Form of Assumed Solutions / 138 2.6.1 Linear Interpolation Functions for Second-Order Problems / 139 2.6.2 Lagrange Interpolation / 142 2.6.3 Galerkin Weighting Functions in Finite Element Form / 143 2.6.4 Hermite Interpolation for Fourth-Order Problems / 144 2.7 Finite Element Solution of Axial Deformation Problems / 150 2.7.1 Two-Node Uniform Bar Element for Axial Deformations / 150 2.7.2 Numerical Examples / 155 ONE-DIMENSIONAL BOUNDARY VALUE PROBLEM 173 3.1,-Selected Applications of ID BVP / 174 3.1.1 Steady-State Heat Conduction / 174 3.1.2 Heat How through Thin Fins / 175
Vii 3.1.3 Viscous Fluid Flow between Parallel Plates Lubrication Problem / 176 3.1.4 Slider Bearing / 177 3.1.5 Axial Deformation of Bars / 178 3.1.6 Elastic Buckling of Long Slender Bars / 178 3.2 Finite Element Formulation for Second-Order ID BVP / 180 3.2.1 Complete Solution Procedure / 186 3.3 Steady-State Heat Conduction / 188 3.4 Steady-State Heat Conduction and Convection / 190 3.5 Viscous Fluid Flow Between Parallel Plates / 198 3.6 Elastic Buckling of Bars / 202 3.7 Solution of Second-Order ID BVP / 208 3.8 A Closer Look at the Interelement Derivative Terms / 214 4 TRUSSES, BEAMS, AND FRAMES 222 4.1 Plane Trasses / 223 4.2 Space Trasses / 227 4.3 Temperature Changes and Initial Strains in Trasses / 231 4.4 Spring Elements / 233 4.5 Transverse Deformation of Beams / 236 4.5.1 Differential Equation for Beam Bending / 236 4.5.2 Boundary Conditions for Beams / 238 4.5.3 Shear Stresses in Beams / 240 4.5.4 Potential Energy for Beam Bending / 240 4.5.5 Transverse Deformation of a Uniform Beam / 241 4.5.6 Transverse Deformation of a Tapered Beam Fixed at Both Ends / 242 4.6 Two-Node Beam Element / 244 4.6.1 Cubic Assumed Solution / 245 4.6.2 Element Equations Using Rayleigh-Ritz Method / 246 4.7 Uniform Beams Subjected to Distributed Loads / 259 4.8 Plane Frames / 266 4.9 Space Frames / 279 4.9.1 Element Equations in Local Coordinate System / 281 4.9.2 Local-to-Global Transformation / 285 4.9.3 Element Solution / 289 4.10 Frames in Multistory Buildings / 293
Viii CONTENTS TWO-DIMENSIONAL ELEMENTS 311 5.1 Selected Applications of the 2D BVP / 313 5.1.1 Two-Dimensional Potential Flow / 313 5.1.2 Steady-State Heat Flow / 316 5.1.3 Bars Subjected to Torsion / 317 5.1.4 Waveguides in Electromagnetics / 319 5.2 Integration by Parts in Higher Dimensions / 320 5.3 Finite Element Equations Using the Galerkin Method / 325 5.4 Rectangular Finite Elements / 329 5.4.1 Four-Node Rectangular Element / 329 5.4.2 Eight-Node Rectangular Element / 346 5.4.3 Lagrange Interpolation for Rectangular Elements / 350 5.5 Triangular Finite Elements / 357 5.5.1 Three-Node Triangular Element / 358 5.5.2 Higher Order Triangular Elements / 371 MAPPED ELEMENTS 381 6.1 Integration Using Change of Variables / 382 6.1.1 One-Dimensional Integrals / 382 6.1.2 Two-Dimensional Area Integrals / 383 6.1.3 Three-Dimensional Volume Integrals / 386 6.2 Mapping Quadrilaterals Using Interpolation Functions / 387 6.2.1 Mapping Lines / 387 6.2.2 Mapping Quadrilateral Areas / 392 6.2.3 Mapped Mesh Generation / 405 6.3 Numerical Integration Using Gauss Quadrature / 408 6.3.1 Gauss Quadrature for One-Dimensional Integrals / 409 6.3.2 Gauss Quadrature for Area Integrals / 414 6.3.3 Gauss Quadrature for Volume Integrals / 417 6.4 Finite Element Computations Involving Mapped Elements / 420 6.4.1 Assumed Solution / 421 6.4.2 Derivatives of the Assumed Solution / 422 6.4.3 Evaluation of Area Integrals / 428 6.4.4 Evaluation of Boundary Integrals / 436 6.5 Complete Mathematica and MATLAB Solutions of 2D BVP Involving Mapped Elements / 441 6.6 Triangular Elements by Collapsing Quadrilaterals / 451 6.7 Infinite Elements / 452 6.7.1 One-Dimensional BVP / 452 6.7.2 Two-Dimensional BVP / 458
7 ANALYSIS OF ELASTIC SOLIDS 467 7.1 Fundamental Concepts in Elasticity / 467 7.1.1 Stresses / 467 7.1.2 Stress Failure Criteria / 472 7.1.3 Strains / 475 7.1.4 Constitutive Equations / 478 7.1.5 Temperature Effects and Initial Strains / 480 7.2 Governing Differential Equations / 480 7.2.1 Stress Equilibrium Equations / 481 7.2.2 Governing Differential Equations in Terms of Displacements / 482 7.3 General Form of Finite Element Equations / 484 7.3.1 Potential Energy Functional / 484 7.3.2 Weak Form / 485 7.3.3 Finite Element Equations / 486 7.3.4 Finite Element Equations in the Presence of Initial Strains / 489 7.4 Plane Stress and Plane Strain / 490 7.4.1 Plane Stress Problem / 492 7.4.2 Plane Strain Problem / 493 7.4.3 Finite Element Equations / 495 7.4.4 Three-Node Triangular Element / 497 7.4.5 Mapped Quadrilateral Elements / 508 7.5 Planar Finite Element Models / 517 7.5.1 Pressure Vessels / 517 7.5.2 Rotating Disks and Flywheels / 524 7.5.3 Residual Stresses Due to Welding / 530 7.5.4 Crack Tip Singularity / 531 8 TRANSIENT PROBLEMS 545 8.1 Transient Field Problems / 545 8.1.1 Finite Element Equations / 5^46 8.1.2 Triangular Element / 549 8.1.3 Transient Heat Flow / 551 8.2 Elastic Solids Subjected to Dynamic Loads / 557 8.2.1 Finite Element Equations / 559 8.2.2 Mass Matrices for Common Structural Elements / 561 8.2.3 Free-Vibration Analysis / 567 8.2.4 Transient Response Examples / 573
9 p-formulation 586 9.1 p-formulation for Second-Order ID BVP / 586 9.1.1 Assumed Solution Using Legendre Polynomials / 587 9.1.2 Element Equations / 591 9.1.3 Numerical Examples / 593 9.2 ^-Formulation for Second-Order 2D BVP / 604 9.2.1 p-mode Assumed Solution / 605 9.2.2 Finite Element Equations / 608 9.2.3 Assembly of Element Equations / 617 9.2.4 Incorporating Essential Boundary Conditions / 620 9.2.5 Applications / 624 A USE OF COMMERCIAL FEA SOFTWARE 641 A.I ANSYS Applications / 642 A. 1.1 General Steps / 643 A. 1.2 Trass Analysis / 648 A. 1.3 Steady-State Heat Flow / 651 A. 1.4 Plane Stress Analysis / 655 A.2 Optimizing Design Using ANSYS / 659 A.2.1 General Steps / 659 A.2.2 Heat Flow Example / 660 A.3 ABAQUS Applications / 663 A.3.1 Execution Procedure / 663 A.3.2 Trass Analysis / 665 A.3.3 Steady-State Heat Flow / 666 A.3.4 Plane Stress Analysis / 671 B VARIATIONAL FORM FOR BOUNDARY VALUE PROBLEMS 676 B.I Basic Concept of Variation of a Function / 676 B.2 Derivation of Equivalent Variational Form / 679 B.3 Boundary Value Problem Corresponding to a Given Functional / 683 BIBLIOGRAPHY 687 INDEX 695