Chapter 1 INTRODUCTION TO FINITE ELEMENT ANALYSIS 1.1Introduction, 1 1.2History of Finite Elements, 2 1.3Structural Mechanics, 4 1.4Concept, 7 1.5Basic Theory of the Finite Element Method, 8 1.6 Application of Finite Element Method, 8 1.7Advantages and Disadvantages of Finite Element Method, 11 1.8 Summary, 12 Chapter 2 TWO-DIMENSIONAL PIN JOINTED TRUSS 2.1Introduction, 14 2.2 Structure, 14 2.3 Displacement Coordinates, 14 2.4 Degrees of Freedom, 15 2.5Coordinate System, 15 2.6 Topology of the Structure, 15 2.7 Support Conditions, 16 2.8 Element Stiffness, 16 2.9 Element Assembly into Global Stiffness Matrix, 19 2.10 Introduction of Boundary conditions, 21 2.11 Skew Boundary Conditions, 28 2.12 Solution, 28 2.13 Calculation of stresses in Each Member, 33 2.14 Example, 33 2.15 Computer Program, 35 2.16 Summary, 39 Chapter 3 INTRODUCTION TO ELASTICITY 3.1 Introduction, 42 3.2Elasticity Theory, 42 3.3 Displacements and Strains, 43 3.4 Stress Transformation, 45 3.5 Strain Transformation, 46 3.6 Equilibrium, 47 3.7 Compatibility, 48 3.8 Constitutive Law, 48 3.9 Boundary Conditions, 49 3.10 Plane Stress, 49 3.11 Plane Strain, 49 3.12 Saint Venant's Principle, 50 3.13 Summary 51 Chapter 4. TWO-DIMENSIONAL STRESS ANALYSIS BY FINITE ELEMENT METHOD 52 4.1 Introduction, 52 4.2 Structural Idealization, 53 4.3 triangular Element, 54 4.4. Displacements in Terms of Nodal Displacements, 54 4.5 Strains in terms of Displacements, 55 4.6 Constitutive Relation, 56 4.7 Nodal Forces Using Equilibrium, 56 4.8 Assembly of Element Stiffness, 57 4.9 Introduction of Boundary Conditions, 58 4.10 Solution of Equations of Nodal Displacements, 59 4.11 Example, 59 4.12 Computer Program, 63 4.13 Summary, 71 Chapter 5 STORAGE SCHEMES AND SOLUTION OF LARGE SYSTEM OF LINEAR EQUATIONS 73 5.1 Introduction 73 5.2 Solution of Large Sets of Equations, 73 5.3 Band Form, 74 5.4 Skyline Storage, 74 5.5 Solution of Band Matrix in Core, 76 5.6 Bandwidth Reduction, 79 5.7 Frontal Solver, 82 5.8 Example, 83 5.9 Substructure Concept, 85 5.10 Summary, 88 ENERGY, VARIATIONAL PRINCIPLES AND RITZ TECHNIQUE 91 Introduction, 91 Work, Energy and Their Complementary Counterparts, 91 Principle of Virtual Work, 93 Derivation of Principle of Stationary Potential Energy, 96 Minimization of total Potential - Stability of Equilibrium 97 Reissner's Principle (Hybrid Models), 100 Variation Calculus - the Minima of Functionals, 100 Ritz Technique, 103 Method of Weighted Residual 105 Boundary Value Problem (by Method of Weighted Residual), 107 6.11 Initial Value Problem by Method of Weighted Residual 113 6.12 Axial Deformation of a Uniform Bar, 116 6.13 Galerkin's Method to Structured Elements, 122 6.14 Summary, 129 'hapter 7 PROPERTIES OF A SINGLE ELEMENT USING THE ASSUMED DISPLACEMENT FIELD 132 7.1 Various Element Shapes, 132 7.2 Displacement Model, 133 7.3 Generalized Coordinate form of Displacement Model, 133 7.4 Convergence and Compatibility Requirements, 135 7.5 Summary, 141 Chapter 8 ELEMENT STIFFNESS IN PLANE STRESS/STRAIN (C.S.T.) 142 8.1 Introduction, 142 8.2 Method 1 - K by Working direct from Polynomials, 143 8.3 Method 2 - Eliminate a Altogether by Writing Down Shape Functions Directly, 145 8.4 Method 3 - Use of Natural Coordinate System, 146 8.5 Summary, 153 Chapter 9 HIGHER ORDER TRIANGULAR ELEMENTS 154 9.1 Introduction 154 9.2 Linear Strain Triangle, 155 9.3 Derivation of Element Stiffness Using Natural Coordinate System, 157 9.4 Quadratic Strain Triangle or Cubic Displacement field (Type 1), 161 9.5 Quadratic Strain Triangle or Cubic Displacement Triangle (Type II), 164 9.6 Comparison of Type I and Type II Quadratic Strain Elements, 166 9.7 Comparison of Different Elements, 167 9.8 Numerical Examples, 170 9.9 Convenient Generation of Stiffness Matrices for the Family of Plane Triangular Elements, 173 9.10 Summary, 177 Chapter 10 RECTANGULAR AND QUADRILATERAL ELEMENTS 179 10.1 Introduction (Serendipity Family), 179 10.2 Derivation of Element Stiffness for Linear Serendipity Rectangle, 180 10.3 Higher Order Serendipity Elements 182 10.4 Comparison Studies, 185 10.5 Lagrangian Family, 186 10.6 Hermitian Family, 191 10.7 Quadrilateral Elements from CST 195 10.8 Numerical Examples, 198 10.9 Comparison of Results, 202 10.10 Summary, 203 Chapter 11 THREE-DIMENSIONAL ELEMENTS - AXISYMMETRIC STRESS ANALYSIS 207 11.1 Introduction, 207 11.2 Element Characteristics, 208 11.3 Special Cases, 211 11.4 Application of the Basic Axisymmetric Element, 212 11.5 Further Remarks on the Analysis of Axisymmetric Bodies, 216 11.6 Axisymmetric Solid Element under Non-axisymmetric Loads, 216 11.7 Numerical Examples, 218 11.8 Summary, 222 Chapter 12 THREE-DIMENSIONAL FINITE ELEMENTS - TETRAHEDRAL FAMILY 224 12.1 Introduction, 224 12.2 Tetrahedron Coordinates (Natural Coordinates), 225 12.3 Constant Strain Tetrahedron (T 12), 227 12.4 Linear Strain Tetrahedron (T 30), 229 12.5 Twenty Node 60 D.O.F. Quadratic Strain Tetrahedron, 231 12.6 Sixteen Node 48 D.O.F.. Tetrahedron, 232 12.7 Elements with Nodal Point Having 12 D.O.F. (T 48) 232 12.8 Parallelepiped Elements, 233 12.9. Numerical Results, 234 12.10. Numerical Examples, 235 12.11 Summary, 237 Chapter 13 STRESS ANALYSIS USING ISOPARAMETRIC ELEMENTS 239 13.1 Introduction, 239 13.2 Basic Principles of Shape Functions Mapping, 240 13.3 Uniqueness of Mapping, 244 13.4 Sub-Iso-Super Parametric Elements, 245 13.5 Isoparametric Two Noded Axial Element 246 13.6 Linear Isoparametric Quadrilateral, 248 13.7 Numerical Integration Using Gaussian Quadrature, 251 13.8 Numerical Examples, 255 13.9 Computer Program, 266 13.10 Examples (using the program given in Appendix E), 267 13.11 Hexahedron Element Family (ZIB 8 element) 282 13.12 Formulation of Stiffness Matrices for Isoparametric Element ZIB20 Twenty Nodel Hexahedron, 286 13.13 Examples, 289 13.14 Order of Quadrature Needed, 293 13.15 Incompatible Elements, 294 13.16 Lagrange and Hermitian Prism Elements, 298 13.17 Four Node Isoparametric Quadrilateral for Axisymmetric Problems 299 13.18 Examples, 301 13.19 Summary, 322 Chapter 14 FINITE ELEMENT ANALYSIS OF PLATES 326 14.1 Introduction, 326 14.2 Plate Theory, 326 14.3 Theory of Shells, 330 14.4 Plate Elements, 330 14.5 Assumed Displacement Formulation Based on Sub-domain Approach, 334 14.6 Rectangular Elements, 335 14.7 Sub-domain Formulation, 338 14.8 Computer Program, 338 14.9 Deflection Due to Transverse Shear, 339 14.10 Numerical Examples, 340 14.11 Summary, 342 Chapter 15 BUCKLING AND DYNAMIC ANALYSIS - THE EIGEN VALUE PROBLEMS 344 15.1 Introduction, 344 15.2 General Theory of Linear Stability Analysis, 345 15.3 Bifurcation Problem and Beam Column Problems, 364 15.4 Dynamic Analysis, 365 15.5 Concept of Masters and Slaves, 384 15.6 Summary, 386 Chapter 16 ANALYSIS OF LARGE SCALE STRUCTURES BY USING GENERAL PURPOSE FINITE ELEMENT PACKAGES 394 16.1 Introduction, 394 16.2 General Purpose Programs, 395 16.3 Case Histories, 398 16.4 General Purpose Programs on Personal Computers, 404 16.5 Summary, 404 Chapter 17 FINITE ELEMENT METHOD TO FIELD APPLICATIONS 405 17.1 Formulation for Heat Transfer and other Problems, 405 17.2 The Quasi. 405 17.3 Solution of Elliptic Equation Steady State Seepage Flow (Finite Element Method), 406 17.4 Finite Element Program for Seepage Flow in Matlab Laplaceseep, 420 17.5 Torsion of Prismatic Shafts, 429 17.6 Summary 445 Appendix A. Wilson's Simple Truss Program 446 Appendix B. Simple Two-Dimensional Finite Element Program Constant Strain Triangle 454 Appendix C. Program to Renumber the Nodes so that the Bandwidth in Minimum 476 Appendix D Axisymmetric Stress Analysis Using CST Element 482 Appendix E. Simple Two-Dimensional FE Program (Isoparametric Elements) Linear, Quadratic or Cubic Serendipity Elements 507 Appendix F. Program for Laminate 542 Appendix G. Program for Finding the Buckling Load of Plane Trusses 577 Appendix H. Program for Dynamic Analysis of Plane Rigid Frames 594 Appendix I. ANSYS - PC/ED 609 Index 628