Finite Element Modeling and Simulation with ANSYS Workbench

408 Pages·2014·20.26 MB·English

Mechanical Engineering C
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Finite Element Modeling Simulation L
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“This book covers a wide range of engineering analyses using FEM with adequate i
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theoretical background, illustrative examples, and detailed tutorials on using ANSYS i
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Workbench for FEA. ... It covers both the theoretical and practical parts of FEM, which e

is very helpful to engineering students.” E
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—Dr. Anh-Vu Phan, University of South Alabama, Mobile, USA
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“The book is an excellent resource to introduce undergraduate and graduate engineer- e
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ing students to finite element analysis. A great deal of emphasis is [placed] on the t

finite element methods through simple application examples. …The main thing that I M
like about this book is that the authors have presented the material using ANSYS o
Workbench. For undergraduate students who are considered beginners with FEA, this d
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is an ideal software to use. The examples are simple and clear, and the ANSYS Work- l
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bench case studies are good applications of the techniques.” n
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—Dr. Mohammad H. Alimi, California State University, Fresno, USA
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“This book should be a great resource for engineers or senior undergraduate students d
who want to learn FEA and use ANSYS themselves and therefore can be adopted as a S
textbook for senior undergraduate students in engineering when introducing FEA.” i
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—Dr. Ernian Pan, University of Akron, Ohio, USA u
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Learn Basic Theory and Software Usage from a Single Volume ti
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Finite Element Modeling and Simulation with ANSYS Workbench combines finite
w
element theory with real-world practice. Providing an introduction to finite element
i
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modeling and analysis for those with no prior experience, and written by authors with h
a combined experience of 30 years teaching the subject, this text presents FEM
formulations integrated with relevant hands-on applications using ANSYS Workbench
for finite element analysis (FEA).
Incorporating the basic theories of FEA and the use of ANSYS Workbench in the
modeling and simulation of engineering problems, the book also establishes the
FEM method as a powerful numerical tool in engineering design and analysis.
This text benefits upper-level undergraduate students in all engineering disciplines,
as well as researchers and practicing engineers who use the finite element method
to analyze structures.
K13420
ISBN: 978-1-4398-7384-7
90000
Xiaolin Chen (cid:127) Yijun Liu
9 781439 873847

Xiaolin Chen (cid:127) Yijun Liu
Boca Raton London New York
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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 ......................................................................................................................52
2.8 Review of Learning Objectives .................................................................................52
Problems ..................................................................................................................................52
v
vi Contents
3. Beams and Frames ................................................................................................................57
3.1 Introduction .................................................................................................................57
3.2 Review of the Beam Theory ......................................................................................57
3.2.1 Euler–Bernoulli Beam and Timoshenko Beam .........................................57
3.2.2 Stress, Strain, Deflection, and Their Relations ..........................................59
3.3 Modeling of Beams and Frames ...............................................................................60
3.3.1 Cross Sections and Strong/Weak Axis .......................................................60
3.3.2 Support Conditions .......................................................................................61
3.3.3 Conversion of a Physical Model into a Line Model ..................................62
3.4 Formulation of the Beam Element ............................................................................62
3.4.1 Element Stiffness Equation: The Direct Approach ...................................63
3.4.2 Element Stiffness Equation: The Energy Approach .................................64
3.4.3 Treatment of Distributed Loads ...................................................................66
3.4.4 Stiffness Matrix for a General Beam Element ............................................67
3.5 Examples with Beam Elements .................................................................................68
3.6 Case Study with ANSYS Workbench .........................................................................77
3.7 Summary ......................................................................................................................96
3.8 Review of Learning Objectives .................................................................................96
Problems ..................................................................................................................................96
4. Two-Dimensional Elasticity .............................................................................................101
4.1 Introduction ...............................................................................................................101
4.2 Review of 2-D Elasticity Theory .............................................................................101
4.2.1 Plane Stress ...................................................................................................101
4.2.2 Plane Strain ...................................................................................................102
4.2.3 Stress–Strain (Constitutive) Equations .....................................................103
4.2.4 Strain and Displacement Relations ...........................................................104
4.2.5 Equilibrium Equations ................................................................................105
4.2.6 Boundary Conditions ..................................................................................105
4.2.7 Exact Elasticity Solution ..............................................................................105
4.3 Modeling of 2-D Elasticity Problems ......................................................................106
4.4 Formulation of the Plane Stress/Strain Element ..................................................107
4.4.1 A General Formula for the Stiffness Matrix ............................................108
4.4.2 Constant Strain Triangle (CST or T3) ........................................................108
4.4.3 Quadratic Triangular Element (LST or T6) ..............................................113
4.4.4 Linear Quadrilateral Element (Q4) ............................................................114
4.4.5 Quadratic Quadrilateral Element (Q8) ......................................................115
4.4.6 Transformation of Loads .............................................................................116
4.4.7 Stress Calculation ........................................................................................118
4.4.7.1 The von Mises Stress ...................................................................118
4.4.7.2 Averaged Stresses .........................................................................119
4.4.8 General Comments on the 2-D Elements .................................................120
4.5 Case Study with ANSYS Workbench .......................................................................121
4.6 Summary ....................................................................................................................138
4.7 Review of Learning Objectives ...............................................................................138
Problems ................................................................................................................................138
Contents vii
5. Modeling and Solution Techniques ...............................................................................143
5.1 Introduction ...............................................................................................................143
5.2 Symmetry ...................................................................................................................143
5.2.1 An Example ..................................................................................................144
5.3 Substructures (Superelements) ...............................................................................145
5.4 Equation Solving .......................................................................................................146
5.4.1 Direct Methods (Gauss Elimination) ........................................................146
5.4.2 Iterative Methods .........................................................................................146
5.4.3 An Example: Gauss Elimination ................................................................146
5.4.4 An Example: Iterative Method ...................................................................147
5.5 Nature of Finite Element Solutions ........................................................................148
5.6 Convergence of FEA Solutions ................................................................................149
5.7 Adaptivity (h-, p-, and hp-Methods) ........................................................................149
5.8 Case Study with ANSYS Workbench .......................................................................150
5.9 Summary ....................................................................................................................161
5.10 Review of Learning Objectives ...............................................................................162
Problems ................................................................................................................................162
6. Plate and Shell Analyses ...................................................................................................165
6.1 Introduction ...............................................................................................................165
6.2 Review of Plate Theory ............................................................................................165
6.2.1 Force and Stress Relations in Plates ..........................................................165
6.2.2 Thin Plate Theory (Kirchhoff Plate Theory) ............................................167
6.2.2.1 Example: A Thin Plate .................................................................169
6.2.3 Thick Plate Theory (Mindlin Plate Theory) .............................................170
6.2.4 Shell Theory ..................................................................................................171
6.2.4.1 Shell Example: A Cylindrical Container ...................................171
6.3 Modeling of Plates and Shells .................................................................................172
6.4 Formulation of the Plate and Shell Elements ........................................................173
6.4.1 Kirchhoff Plate Elements ............................................................................173
6.4.2 Mindlin Plate Elements ...............................................................................174
6.4.3 Discrete Kirchhoff Elements ......................................................................175
6.4.4 Flat Shell Elements .......................................................................................175
6.4.5 Curved Shell Elements ................................................................................176
6.5 Case Studies with ANSYS Workbench .....................................................................177
6.6 Summary ....................................................................................................................185
6.7 Review of Learning Objectives ...............................................................................185
Problems ................................................................................................................................185
7. Three-Dimensional Elasticity ..........................................................................................189
7.1 Introduction ...............................................................................................................189
7.2 Review of Theory of Elasticity ................................................................................189
7.2.1 Stress–Strain Relation .................................................................................190
7.2.2 Displacement ................................................................................................191
7.2.3 Strain–Displacement Relation ....................................................................191
7.2.4 Equilibrium Equations ................................................................................191
7.2.5 Boundary Conditions ..................................................................................192
viii Contents
7.2.6 Stress Analysis .............................................................................................192
7.3 Modeling of 3-D Elastic Structures ........................................................................192
7.3.1 Mesh Discretization .....................................................................................193
7.3.2 Boundary Conditions: Supports ................................................................193
7.3.3 Boundary Conditions: Loads .....................................................................194
7.3.4 Assembly Analysis: Contacts .....................................................................194
7.4 Formulation of Solid Elements ................................................................................195
7.4.1 General Formulation ...................................................................................195
7.4.2 Typical Solid Element Types ......................................................................196
7.4.3 Formulation of a Linear Hexahedral Element Type ...............................197
7.4.4 Treatment of Distributed Loads .................................................................200
7.5 Case Studies with ANSYS Workbench .....................................................................200
7.6 Summary ....................................................................................................................220
7.7 Review of Learning Objectives ...............................................................................220
Problems ................................................................................................................................220
8. Structural Vibration and Dynamics ...............................................................................225
8.1 Introduction ...............................................................................................................225
8.2 Review of Basic Equations .......................................................................................225
8.2.1 A Single DOF System ..................................................................................226
8.2.2 A Multi-DOF System ...................................................................................228
8.2.2.1 Mass Matrices ...............................................................................228
8.2.2.2 Damping ........................................................................................230
8.3 Formulation for Modal Analysis ............................................................................231
8.3.1 Modal Equations ..........................................................................................233
8.4 Formulation for Frequency Response Analysis ...................................................235
8.4.1 Modal Method ..............................................................................................235
8.4.2 Direct Method ..............................................................................................236
8.5 Formulation for Transient Response Analysis......................................................236
8.5.1 Direct Methods (Direct Integration Methods) .........................................237
8.5.2 Modal Method ..............................................................................................238
8.6 Modeling Examples ..................................................................................................239
8.6.1 Modal Analysis ............................................................................................239
8.6.2 Frequency Response Analysis ...................................................................240
8.6.3 Transient Response Analysis .....................................................................240
8.6.4 Cautions in Dynamic Analysis ..................................................................240
8.7 Case Studies with ANSYS Workbench .....................................................................241
8.8 Summary ....................................................................................................................260
8.9 Review of Learning Objectives ...............................................................................260
Problems ................................................................................................................................260
9. Thermal Analysis ...............................................................................................................267
9.1 Introduction ...............................................................................................................267
9.2 Review of Basic Equations .......................................................................................267
9.2.1 Thermal Analysis .........................................................................................267
9.2.1.1 Finite Element Formulation for Heat Conduction ...................269
9.2.2 Thermal Stress Analysis .............................................................................269
9.2.2.1 1-D Case .........................................................................................270
9.2.2.2 2-D Cases .......................................................................................271