CONTENTS Preface ix 1. Introduction 1-15 1.1 Typical Application Examples4 1.1.1Automotive Applications 4 1.1.2 Manufacturing Process Simulation7 1.1.3Electrical and Electronics Engineering Applications8 1.1.4 Aerospace Applications14 Summary14 2. Finite Element Formulation Starting from Governing Differential Equations 16-65 2.1 Weighted Residual MethodUse of a Single Continuous Trial Function 16 2.2 The General Weighted Residual (WR) Statement 28 2.3 Weak (Variational) Form of the Weighted Residual Statement 33 2.4 Comparison of Differential Equation, Weighted Residual and Weak Forms 36 2.5 Piece-wise Continuous Trial Function Solution of the Weak Form 41 2.6 One-dimensional Bar Finite Element 48 2.7 One-dimensional Heat Transfer Element 57 Summary 61 Problems61 3. Finite Element Formulation Based on Stationarity of a Functional 66-88 3.1 Introduction 66 3.2 Functional and Differential Equation Forms67 3.3 Principle of Stationary Total Potential (PSTP)73 3.3.1 Rayleigh-Ritz Method75 3.4 Piece-wise Continuous Trial FunctionsFinite Element Method81 3.4.1Bar Element Formulated from the Stationarity of a Functional81 3.4.2One-dimensional Heat Transfer Element Based on the Stationarity of a Functional83 3.5Meaning of Finite Element Equations84 Summary87 Problems 88 4. One-dimensional Finite Element Analysis 89-144 4.1General Form of the Total Potential for 1 -d89 4.2Generic Form of Finite Element Equations90 4.3The Linear Bar Finite Element93 4.4The Quadratic Bar Element101 4.4.1Determination of Shape Functions101 4.4.2Element Matrices102 4.5 Beam Element117 4.5.1Selection of Nodal d.o.f.117 4.5.2Determination of Shape Functions118 4.5.3Element Matrices119 4.6 Frame Element125 4.7One-dimensional Heat Transfer132 Summary138 Problems138 5. Two-dimensional Finite Element Analysis 145-231 5.1IntroductionDimensionality of a Problem145 5.2Approximation of Geometry and Field Variable148 5.2.1Simple Three-noded Triangular Element149 5.2.2Four-noded Rectangular Element152 5.2.3Six-noded Triangular Element153 5.3Natural Coordinates and Coordinate Transformation156 5.3.1Alternate Methods of Deriving Shape Functions157 5.3.2Natural CoordinatesQuadrilateral Elements159 5.3.3Natural CoordinatesTriangular Elements164 5.42-d Elements for Structural Mechanics167 5.4.1Generic Relations167 5.4.2Three-noded Triangular Element171 5.4.3Four-noded Rectangular Element179 5.4.4Compatibility of Displacements181 5.4.5Four-node Quadrilateral Element 183 5.4.6Eight-node Quadrilateral Element 188 5.4.7Nine-node Quadrilateral Element 190 5.4.8Six-node Triangular Element 192 5.5Numerical Integration194 5.5.1Trapezoidal Rule195 5.5.2Simpson's 1/3 Rule196 5.5.3Newton-Cotes Formula197 5.5.4 Gauss Quadrature Formula198 5.5.4Gauss Quadrature in Two Dimensions201 5.6Incorporation of Boundary Conditions205 5.7Solution of Static Equilibrium Equations206 5.82-d Fluid Flow220 Summary 225 Problems226 6. Dynamic Analysis Using Finite Elements 232-294 6.1Introduction232 6.2Vibration Problems232 6.3 Equations of Motion Based on Weak Form 235 6.3.1Axial Vibration of a Rod235 6.3.2Transverse Vibration of a Beam237 6.4 Equations of Motion Using Lagrange's Approach240 6.4.1Formulation of Finite Element Equations242 6.4.2Consistent Mass Matrices for Various Elements245 6.5Consistent and Lumped Mass Matrices246 6.5.1HRZ Lumping Scheme 247 6.6Form of Finite Element Equations for Vibration Problems253 6.7Some Properties of Eigenpairs255 6.8Solution of Eigenvalue Problems257 6.8.1Transformation Based Methods258 6.8.2Vector Iteration Methods264 6.9Transient Vibration Analysis 272 6.9.1Modelling of Damping272 6.9.2The Mode Superposition Scheme275 6.9.3Direct Integration Methods279 6.10 Thermal TransientsUnsteady Heat Transfer in a Pin-Fin289 Summary 293 Problems 293 7. Application Examples 295-307 7.1Finite Element Analysis of Crankshaft Torsional Vibrations 295 7.1.1Beam Element Model of Crankshaft Assembly296 7.1.2Results and Discussion299 7.1.3Dynamic Response Analysis301 7.2Axisymmetric Finite Element Analysis of a Pressure Vessel303 7.2.1Finite Element Formulation for Axisymmetric Loads304 7.2.2Stress Analysis of a Pressure Vessel305 Appendix ASuggested Mini-Project Topics 309-320 Project 1: Thermal Analysis of a Pressure Vessel309 Project 2: Structural Dynamic Analysis of a Pressure Vessel310 Project 3: Dynamics of a Scooter Frame312 Project 4: Automotive Chassis Dynamics313 Project 5: Analysis of a Turbine Disk 316 Project 6: Dynamic Analysis of a Building317 Project 7: Thermal Analysis of an 1C Engine Cylinder318 Project 8: Stress Concentration319 Project 9: Dynamics of a Hard Disk Drive Read/Write Head Assembly 319 Appendix BReview of Preliminaries 321-323 Bl.lMatrix Algebra 321 B1.2Interpolation 322 Appendix C-Typical Finite Element Program 324-328 Index329-330