Year: 2005 Language: english Author: S.S. Bhavikatti Genre: Textbook Publisher: New Age International Publishers ISBN: 978-81-224-2524-6 Format: PDF Quality: eBook Pages count: 347 Description: Finite Element Analysis was developed as a numerical method of stress analysis, but now it has been extended as a general method of solution to many complex engineering and physical science problems. As it involves lot of calculations, its growth is closely linked with the developments in computer technology. Now-a-days a number of finite element analysis packages are available commercially and number of users is increasing. A user without a basic course on finite element analysis may produce dangerous results. Hence now-a-days in many M.Tech. programmes finite element analysis is a core subject and in undergraduate programmes many universities offer it as an elective subject. The experience of the author in teaching this course to M.Tech (Geotechnical Engineering) and M.Tech. (Industrial Structures) students at National Institute of Technology, Karnataka, Surathkal (formerly, K.R.E.C. Surathkal) and to undergraduate students at SDM College of Eingineering and Technology, Dharwad inspired him to write this book. This is intended as a text book to students and as an introductory course to all users of finite element packages.
Contents
1. Introduction 1 1.1 General 1 1.2 General Description of the Method 1 1.3 Brief Explanation of FEA for a Stress Analysis Problem 2 1.4 Finite Element Method vs Classical Method 4 1.5 FEM vs FDM 5 1.6 A Brief History of FEM 6 1.7 Need for Studying FEM 6 1.8 Warning to FEA Package Users 7 Questions 7 References 7 2. Basic Equations in Elasticity 9 2.1 Introduction 9 2.2 Stresses in a Typical Element 9 2.3 Equations of Equilibrium 12 2.4 Strains 14 2.5 Strain Displacement Equations 14 2.6 Linear Constitutive Law 15 Questions 20 3. Matrix Displacement Formulation 21 3.1 Introduction 21 3.2 Matrix Displacement Equations 21 3.3 Solution of Matrix Displacement Equations 28 3.4 Techniques of Saving Computer Memory Requirements 30 Questions 32 4. Element Shapes, Nodes, Nodal Unknowns and Coordinate Systems 33 4.1 Introduction 33 4.2 Element Shapes 33 4.3 Nodes 38 4.4 Nodal Unknowns 39 4.5 Coordinate Systems 40 Questions 53 5. Shape Functions 55 5.1 Introduction 55 5.2 Polynomial Shape Functions 56 5.3 Convergence Requirements of Shape Functions 59 5.4 Derivation of Shape Functions Using Polynomials 61 5.5 Finding Shape Functions Using Lagrange Polynomials 82 5.6 Shape Functions for Serendipity Family Elements 89 5.7 Hermite Polynomials as Shape Functions 95 5.8 Construction of Shape Functions by Degrading Technique 98 Questions 102 6. Strain Displacement Matrix 104 6.1 Introduction 104 6.2 Strain—Displacement Matrix for Bar Element 104 6.3 Strain Displacement Matrix for CST Element 105 6.4 Strain Displacement Relation for Beam Element 107 Questions 108 7. Assembling Stiffness Equation—Direct Approach 110 7.1 Introduction 110 7.2 Element Stiffness Matrix for CST Element by Direct Approach 110 7.3 Nodal Loads by Direct Approach 114 Questions 117 8. Assembling Stiffness Equation—Galerkin’s Method, 118 Virtual Work Method 8.1 Introduction 118 8.2 Galerkin’s Method 118 8.3 Galerkin’s Method Applied to Elasticity Problems 119 Questions 127 9. Assembling Stiffness Equation—Variational Method 128 9.1 Introduction 128 9.2 General Variational Method in Elasticity Problems 128 9.3 Potential Energy in Elastic Bodies 134 9.4 Principles of Minimum Potential Energy 136 9.5 Rayleigh—Ritz Method 140 9.6 Variational Formulation in Finite Element Analysis 150 Questions 153 10. Discritization of a Structure 154 10.1 Introduction 154 10.2 Nodes as Discontinuities 154 10.3 Refining Mesh 156 10.4 Use of Symmetry 157 10.5 Finite Representation of Infinite Bodies 157 10.6 Element Aspect Ratio 158 10.7 Higher Order Element vs Mesh Refinement 159 10.8 Numbering System to Reduce Band Width 159 Questions 160 11. Finite Element Analysis—Bars and Trusses 161 11.1 Introduction 161 11.2 Tension Bars/Columns 161 11.3 Two Dimensional Trusses (Plane Trusses) 180 11.4 Three Dimensional Trusses (Space Trusses) 197 Questions 201 12. Finite Element Analysis—Plane Stress and Plane Strain Problems 204 12.1 Introduction 204 12.2 General Procedure when CST Elements are Used 204 12.3 Use of Higher Order Elements 216 Questions 217 13. Isoparametric Formulation 219 13.1 Introduction 219 13.2 Coordinate Transformation 221 13.3 Basic Theorems of Isoparametric Concept 222 13.4 Uniqueness of Mapping 223 13.5 Isoparametric, Superparametric and Subparametric Elements 224 13.6 Assembling Stiffness Matrix 225 13.7 Numerical Integration 230 13.8 Numerical Examples 232 Questions 240 References 241 14. Analysis of Beams and Rigid Frames 242 14.1 Introduction 242 14.2 Beam Analysis Using two Noded Elements 242 14.3 Analysis of Rigid Plane Frame Using 2 Noded Beam Elements 259 14.4 A Three Dimensional Rigid Frame Element 266 14.5 Timoshenko Beam Element 269 Questions 278 References 279 15. Bending of Thin Plates 280 15.1 Introduction 280 15.2 Basic Relations in Thin Plate Theory 281 15.3 Displacement Models for Plate Analysis 282 15.4 Rectangular Plate Element with 12 Degrees of Freedom 284 15.5 Rectangular Plate Element with 16 Degrees of Freedom 289 15.6 Mindlin’s Plate Element 292 Questions 299 References 299 16. Analysis of Shells 301 16.1 Introduction 301 16.2 Force on Shell Element 301 16.3 Finite Element for Shell Analysis 302 16.4 Finite Element Formulation Using Four Noded Degenerated Quadrilateral Shell Element 307 Questions 317 References 317 17. Nonlinear Analysis 318 17.1 Introduction 318 17.2 Nonlinear Problems 318 17.3 Analysis of Material Nonlinear Problems 320 17.4 Analysis of Geometric Nonlinear Problems 325 17.5 Analysis of Both Material and Geometric Nonlinear Problems 328 Questions 328 References 328 18. Standard Packages and Their Features 329 18.1 Introduction 329 18.2 Commercially Available Standard Packages 329 18.3 Structure of a Finite Element Analysis Program 330 18.4 Pre and Post Processors 331 18.5 Desirable Features of FEA Packages 333 Questions 333
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Finite Element Analysis
Language: english
Author: S.S. Bhavikatti
Genre: Textbook
Publisher: New Age International Publishers
ISBN: 978-81-224-2524-6
Format: PDF
Quality: eBook
Pages count: 347
Description: Finite Element Analysis was developed as a numerical method of stress analysis, but now it has been extended
as a general method of solution to many complex engineering and physical science problems. As it involves
lot of calculations, its growth is closely linked with the developments in computer technology. Now-a-days a
number of finite element analysis packages are available commercially and number of users is increasing. A
user without a basic course on finite element analysis may produce dangerous results. Hence now-a-days in
many M.Tech. programmes finite element analysis is a core subject and in undergraduate programmes many
universities offer it as an elective subject. The experience of the author in teaching this course to M.Tech
(Geotechnical Engineering) and M.Tech. (Industrial Structures) students at National Institute of Technology,
Karnataka, Surathkal (formerly, K.R.E.C. Surathkal) and to undergraduate students at SDM College of
Eingineering and Technology, Dharwad inspired him to write this book. This is intended as a text book to
students and as an introductory course to all users of finite element packages.
Contents
1. Introduction 11.1 General 1
1.2 General Description of the Method 1
1.3 Brief Explanation of FEA for a Stress Analysis Problem 2
1.4 Finite Element Method vs Classical Method 4
1.5 FEM vs FDM 5
1.6 A Brief History of FEM 6
1.7 Need for Studying FEM 6
1.8 Warning to FEA Package Users 7
Questions 7
References 7
2. Basic Equations in Elasticity 9
2.1 Introduction 9
2.2 Stresses in a Typical Element 9
2.3 Equations of Equilibrium 12
2.4 Strains 14
2.5 Strain Displacement Equations 14
2.6 Linear Constitutive Law 15
Questions 20
3. Matrix Displacement Formulation 21
3.1 Introduction 21
3.2 Matrix Displacement Equations 21
3.3 Solution of Matrix Displacement Equations 28
3.4 Techniques of Saving Computer Memory Requirements 30
Questions 32
4. Element Shapes, Nodes, Nodal Unknowns and Coordinate Systems 33
4.1 Introduction 33
4.2 Element Shapes 33
4.3 Nodes 38
4.4 Nodal Unknowns 39
4.5 Coordinate Systems 40
Questions 53
5. Shape Functions 55
5.1 Introduction 55
5.2 Polynomial Shape Functions 56
5.3 Convergence Requirements of Shape Functions 59
5.4 Derivation of Shape Functions Using Polynomials 61
5.5 Finding Shape Functions Using Lagrange Polynomials 82
5.6 Shape Functions for Serendipity Family Elements 89
5.7 Hermite Polynomials as Shape Functions 95
5.8 Construction of Shape Functions by Degrading Technique 98
Questions 102
6. Strain Displacement Matrix 104
6.1 Introduction 104
6.2 Strain—Displacement Matrix for Bar Element 104
6.3 Strain Displacement Matrix for CST Element 105
6.4 Strain Displacement Relation for Beam Element 107
Questions 108
7. Assembling Stiffness Equation—Direct Approach 110
7.1 Introduction 110
7.2 Element Stiffness Matrix for CST Element by Direct Approach 110
7.3 Nodal Loads by Direct Approach 114
Questions 117
8. Assembling Stiffness Equation—Galerkin’s Method, 118
Virtual Work Method
8.1 Introduction 118
8.2 Galerkin’s Method 118
8.3 Galerkin’s Method Applied to Elasticity Problems 119
Questions 127
9. Assembling Stiffness Equation—Variational Method 128
9.1 Introduction 128
9.2 General Variational Method in Elasticity Problems 128
9.3 Potential Energy in Elastic Bodies 134
9.4 Principles of Minimum Potential Energy 136
9.5 Rayleigh—Ritz Method 140
9.6 Variational Formulation in Finite Element Analysis 150
Questions 153
10. Discritization of a Structure 154
10.1 Introduction 154
10.2 Nodes as Discontinuities 154
10.3 Refining Mesh 156
10.4 Use of Symmetry 157
10.5 Finite Representation of Infinite Bodies 157
10.6 Element Aspect Ratio 158
10.7 Higher Order Element vs Mesh Refinement 159
10.8 Numbering System to Reduce Band Width 159
Questions 160
11. Finite Element Analysis—Bars and Trusses 161
11.1 Introduction 161
11.2 Tension Bars/Columns 161
11.3 Two Dimensional Trusses (Plane Trusses) 180
11.4 Three Dimensional Trusses (Space Trusses) 197
Questions 201
12. Finite Element Analysis—Plane Stress and Plane Strain Problems 204
12.1 Introduction 204
12.2 General Procedure when CST Elements are Used 204
12.3 Use of Higher Order Elements 216
Questions 217
13. Isoparametric Formulation 219
13.1 Introduction 219
13.2 Coordinate Transformation 221
13.3 Basic Theorems of Isoparametric Concept 222
13.4 Uniqueness of Mapping 223
13.5 Isoparametric, Superparametric and Subparametric Elements 224
13.6 Assembling Stiffness Matrix 225
13.7 Numerical Integration 230
13.8 Numerical Examples 232
Questions 240
References 241
14. Analysis of Beams and Rigid Frames 242
14.1 Introduction 242
14.2 Beam Analysis Using two Noded Elements 242
14.3 Analysis of Rigid Plane Frame Using 2 Noded Beam Elements 259
14.4 A Three Dimensional Rigid Frame Element 266
14.5 Timoshenko Beam Element 269
Questions 278
References 279
15. Bending of Thin Plates 280
15.1 Introduction 280
15.2 Basic Relations in Thin Plate Theory 281
15.3 Displacement Models for Plate Analysis 282
15.4 Rectangular Plate Element with 12 Degrees of Freedom 284
15.5 Rectangular Plate Element with 16 Degrees of Freedom 289
15.6 Mindlin’s Plate Element 292
Questions 299
References 299
16. Analysis of Shells 301
16.1 Introduction 301
16.2 Force on Shell Element 301
16.3 Finite Element for Shell Analysis 302
16.4 Finite Element Formulation Using Four Noded
Degenerated Quadrilateral Shell Element 307
Questions 317
References 317
17. Nonlinear Analysis 318
17.1 Introduction 318
17.2 Nonlinear Problems 318
17.3 Analysis of Material Nonlinear Problems 320
17.4 Analysis of Geometric Nonlinear Problems 325
17.5 Analysis of Both Material and Geometric Nonlinear Problems 328
Questions 328
References 328
18. Standard Packages and Their Features 329
18.1 Introduction 329
18.2 Commercially Available Standard Packages 329
18.3 Structure of a Finite Element Analysis Program 330
18.4 Pre and Post Processors 331
18.5 Desirable Features of FEA Packages 333
Questions 333
Screenshots
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