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Contents
Preface ..............................................................................................................................................xi
Authors .........................................................................................................................................xiii
1. Introduction .............................................................................................................................1
1.1 Some Basic Concepts ....................................................................................................1
1.1.1 Why FEA? .........................................................................................................1
1.1.2 Finite Element Applications in Engineering ................................................1
1.1.3 FEA with ANSYS Workbench ..........................................................................3
1.1.4 A Brief History of FEA ....................................................................................3
1.1.5 A General Procedure for FEA ........................................................................4
1.2 An Example in FEA: Spring System ...........................................................................4
1.2.1 One Spring Element .........................................................................................5
1.2.2 A Spring System ...............................................................................................6
1.2.2.1 Assembly of Element Equations: Direct Approach .....................6
1.2.2.2 Assembly of Element Equations: Energy Approach ...................8
1.2.3 Boundary and Load Conditions ....................................................................9
1.2.4 Solution Verification ......................................................................................10
1.2.5 Example Problems .........................................................................................10
1.3 Overview of ANSYS Workbench ................................................................................13
1.3.1 The User Interface ..........................................................................................13
1.3.2 The Toolbox .....................................................................................................14
1.3.3 The Project Schematic ...................................................................................14
1.3.4 Working with Cells ........................................................................................16
1.3.5 The Menu Bar .................................................................................................16
1.4 Summary ......................................................................................................................17
Problems ..................................................................................................................................18
2. Bars and Trusses ...................................................................................................................21
2.1 Introduction .................................................................................................................21
2.2 Review of the 1-D Elasticity Theory .........................................................................21
2.3 Modeling of Trusses ...................................................................................................22
2.4 Formulation of the Bar Element ................................................................................23
2.4.1 Stiffness Matrix: Direct Method ..................................................................23
2.4.2 Stiffness Matrix: Energy Approach .............................................................25
2.4.3 Treatment of Distributed Load ....................................................................27
2.4.4 Bar Element in 2-D and 3-D ..........................................................................28
2.4.4.1 2-D Case ...........................................................................................28
2.4.4.2 3-D Case ...........................................................................................31
2.4.5 Element Stress ................................................................................................31
2.5 Examples with Bar Elements .....................................................................................32
2.6 Case Study with ANSYS Workbench .........................................................................40
2.7 Summary ......................................................................................................................59
2.8 Review of Learning Objectives .................................................................................59
Problems ..................................................................................................................................59
v
vi Contents
3. Beams and Frames ................................................................................................................65
3.1 Introduction .................................................................................................................65
3.2 Review of the Beam Theory ......................................................................................65
3.2.1 Euler–Bernoulli Beam and Timoshenko Beam .........................................65
3.2.2 Stress, Strain, Deflection, and Their Relations ..........................................67
3.3 Modeling of Beams and Frames ...............................................................................68
3.3.1 Cross Sections and Strong/Weak Axis .......................................................68
3.3.2 Support Conditions .......................................................................................69
3.3.3 Conversion of a Physical Model into a Line Model ..................................70
3.4 Formulation of the Beam Element ............................................................................70
3.4.1 Element Stiffness Equation: The Direct Approach ...................................71
3.4.2 Element Stiffness Equation: The Energy Approach .................................72
3.4.3 Treatment of Distributed Loads ...................................................................74
3.4.4 Stiffness Matrix for a General Beam Element ............................................75
3.5 Examples with Beam Elements .................................................................................76
3.6 Case Study with ANSYS Workbench .........................................................................85
3.7 Summary ....................................................................................................................112
3.8 Review of Learning Objectives ...............................................................................112
Problems ................................................................................................................................112
4. Two-Dimensional Elasticity .............................................................................................117
4.1 Introduction ...............................................................................................................117
4.2 Review of 2-D Elasticity Theory .............................................................................117
4.2.1 Plane Stress ...................................................................................................117
4.2.2 Plane Strain ...................................................................................................118
4.2.3 Stress–Strain (Constitutive) Equations .....................................................119
4.2.4 Strain and Displacement Relations ...........................................................120
4.2.5 Equilibrium Equations ................................................................................121
4.2.6 Boundary Conditions ..................................................................................121
4.2.7 Exact Elasticity Solution ..............................................................................121
4.3 Modeling of 2-D Elasticity Problems ......................................................................122
4.4 Formulation of the Plane Stress/Strain Element ..................................................123
4.4.1 A General Formula for the Stiffness Matrix ............................................124
4.4.2 Constant Strain Triangle (CST or T3) ........................................................124
4.4.3 Quadratic Triangular Element (LST or T6) ..............................................129
4.4.4 Linear Quadrilateral Element (Q4) ............................................................130
4.4.5 Quadratic Quadrilateral Element (Q8) ......................................................131
4.4.6 Transformation of Loads .............................................................................132
4.4.7 Stress Calculation ........................................................................................134
4.4.7.1 The von Mises Stress ...................................................................134
4.4.7.2 Averaged Stresses .........................................................................135
4.4.8 General Comments on the 2-D Elements .................................................136
4.5 Case Study with ANSYS Workbench .......................................................................137
4.6 Summary ....................................................................................................................155
4.7 Review of Learning Objectives ...............................................................................155
Problems ................................................................................................................................156
5. Modeling and Solution Techniques ...............................................................................161
5.1 Introduction ...............................................................................................................161
Contents vii
5.2 Symmetry ...................................................................................................................161
5.2.1 An Example ..................................................................................................162
5.3 Substructures (Superelements) ...............................................................................163
5.4 Equation Solving .......................................................................................................164
5.4.1 Direct Methods (Gauss Elimination) ........................................................164
5.4.2 Iterative Methods .........................................................................................164
5.4.3 An Example: Gauss Elimination ................................................................164
5.4.4 An Example: Iterative Method ...................................................................165
5.5 Nature of Finite Element Solutions ........................................................................166
5.6 Convergence of FEA Solutions ................................................................................167
5.7 Adaptivity (h-, p-, and hp-Methods) ........................................................................167
5.8 Case Study with ANSYS Workbench .......................................................................168
5.9 Summary ....................................................................................................................182
5.10 Review of Learning Objectives ...............................................................................183
Problems ................................................................................................................................183
6. Plate and Shell Analyses ...................................................................................................187
6.1 Introduction ...............................................................................................................187
6.2 Review of Plate Theory ............................................................................................187
6.2.1 Force and Stress Relations in Plates ..........................................................187
6.2.2 Thin Plate Theory (Kirchhoff Plate Theory) ............................................189
6.2.2.1 Example: A Thin Plate .................................................................191
6.2.3 Thick Plate Theory (Mindlin Plate Theory) .............................................192
6.2.4 Shell Theory ..................................................................................................193
6.2.4.1 Shell Example: A Cylindrical Container ...................................193
6.3 Modeling of Plates and Shells .................................................................................194
6.4 Formulation of the Plate and Shell Elements ........................................................195
6.4.1 Kirchhoff Plate Elements ............................................................................195
6.4.2 Mindlin Plate Elements ...............................................................................196
6.4.3 Discrete Kirchhoff Elements ......................................................................197
6.4.4 Flat Shell Elements .......................................................................................197
6.4.5 Curved Shell Elements ................................................................................198
6.5 Case Studies with ANSYS Workbench .....................................................................199
6.6 Summary ....................................................................................................................214
6.7 Review of Learning Objectives ...............................................................................214
Problems ................................................................................................................................214
7. Three-Dimensional Elasticity ..........................................................................................219
7.1 Introduction ...............................................................................................................219
7.2 Review of Theory of Elasticity ................................................................................219
7.2.1 Stress–Strain Relation .................................................................................220
7.2.2 Displacement ................................................................................................221
7.2.3 Strain–Displacement Relation ....................................................................221
7.2.4 Equilibrium Equations ................................................................................221
7.2.5 Boundary Conditions ..................................................................................222
7.2.6 Stress Analysis .............................................................................................222
7.3 Modeling of 3-D Elastic Structures ........................................................................222
7.3.1 Mesh Discretization .....................................................................................223
7.3.2 Boundary Conditions: Supports ................................................................223
viii Contents
7.3.3 Boundary Conditions: Loads .....................................................................224
7.3.4 Assembly Analysis: Contacts .....................................................................224
7.4 Formulation of Solid Elements ................................................................................225
7.4.1 General Formulation ...................................................................................225
7.4.2 Typical Solid Element Types ......................................................................226
7.4.3 Formulation of a Linear Hexahedral Element Type ...............................227
7.4.4 Treatment of Distributed Loads .................................................................230
7.5 Case Studies with ANSYS Workbench .....................................................................230
7.6 Summary ....................................................................................................................255
7.7 Review of Learning Objectives ...............................................................................255
Problems ................................................................................................................................255
8. Structural Vibration and Dynamics ...............................................................................261
8.1 Introduction ...............................................................................................................261
8.2 Review of Basic Equations .......................................................................................261
8.2.1 A Single DOF System ..................................................................................262
8.2.2 A Multi-DOF System ...................................................................................264
8.2.2.1 Mass Matrices ...............................................................................264
8.2.2.2 Damping ........................................................................................266
8.3 Formulation for Modal Analysis ............................................................................267
8.3.1 Modal Equations ..........................................................................................269
8.4 Formulation for Frequency Response Analysis ...................................................271
8.4.1 Modal Method ..............................................................................................271
8.4.2 Direct Method ..............................................................................................272
8.5 Formulation for Transient Response Analysis......................................................272
8.5.1 Direct Methods (Direct Integration Methods) .........................................273
8.5.2 Modal Method ..............................................................................................274
8.6 Modeling Examples ..................................................................................................275
8.6.1 Modal Analysis ............................................................................................275
8.6.2 Frequency Response Analysis ...................................................................276
8.6.3 Transient Response Analysis .....................................................................276
8.6.4 Cautions in Dynamic Analysis ..................................................................276
8.7 Case Studies with ANSYS Workbench .....................................................................277
8.8 Summary ....................................................................................................................293
8.9 Review of Learning Objectives ...............................................................................294
Problems ................................................................................................................................294
9. Thermal Analysis ...............................................................................................................301
9.1 Introduction ...............................................................................................................301
9.2 Review of Basic Equations .......................................................................................301
9.2.1 Thermal Analysis .........................................................................................301
9.2.1.1 Finite Element Formulation for Heat Conduction ...................303
9.2.2 Thermal Stress Analysis .............................................................................303
9.2.2.1 1-D Case .........................................................................................304
9.2.2.2 2-D Cases .......................................................................................305
9.2.2.3 3-D Case .........................................................................................305
9.2.2.4 Notes on FEA for Thermal Stress Analysis ..............................305
9.3 Modeling of Thermal Problems ..............................................................................306
9.3.1 Thermal Analysis .........................................................................................306
Contents ix
9.3.2 Thermal Stress Analysis .............................................................................306
9.4 Case Studies with ANSYS Workbench .....................................................................308
9.5 Summary ....................................................................................................................330
9.6 Review of Learning Objectives ...............................................................................330
Problems ................................................................................................................................330
10. Introduction to Fluid Analysis ........................................................................................337
10.1 Introduction ...............................................................................................................337
10.2 Review of Basic Equations .......................................................................................337
10.2.1 Describing Fluid Motion .............................................................................337
10.2.2 Types of Fluid Flow .....................................................................................337
10.2.3 Navier–Stokes Equations ............................................................................338
10.3 Modeling of Fluid Flow ............................................................................................339
10.3.1 Fluid Domain ...............................................................................................339
10.3.2 Meshing .........................................................................................................339
10.3.3 Boundary Conditions ..................................................................................339
10.3.4 Solution Visualization .................................................................................340
10.4 Case Studies with ANSYS Workbench .....................................................................341
10.5 Summary ....................................................................................................................368
10.6 Review of Learning Objectives ...............................................................................368
Problems ................................................................................................................................368
11. Design Optimization .........................................................................................................373
11.1 Introduction ...............................................................................................................373
11.2 Topology Optimization ............................................................................................373
11.3 Parametric Optimization .........................................................................................374
11.4 Design Space Exploration for Parametric Optimization .....................................374
11.4.1 Design of Experiments ................................................................................375
11.4.2 Response Surface Optimization ................................................................377
11.5 Case Studies with ANSYS Workbench .....................................................................377
11.6 Summary ....................................................................................................................415
11.7 Review of Learning Objectives ...............................................................................415
Problems ................................................................................................................................415
12. Failure Analysis ..................................................................................................................419
12.1 Introduction ...............................................................................................................419
12.2 Static Failure ..............................................................................................................419
12.2.1 Ductile Failure ..............................................................................................419
12.2.1.1 Maximum Shear Stress Theory (Tresca Criterion) ..................419
12.2.1.2 Distortion Energy Theory (von Mises Criterion) .....................420
12.2.2 Brittle Failure ................................................................................................420
12.2.2.1 Maximum Normal Stress Theory ..............................................420
12.2.2.2 Mohr–Coulomb Theory ..............................................................420
12.3 Fatigue Failure ...........................................................................................................421
12.3.1 Soderberg Failure Criterion ........................................................................422
12.3.2 Goodman Failure Criterion ........................................................................422
12.3.3 Gerber Failure Criterion ..............................................................................423
12.4 Buckling Failure ........................................................................................................424
12.5 Case Studies with ANSYS Workbench .....................................................................425
