Finite Element Analysis

اسم المؤلف

S. S. BHAVIKATTI

التاريخ

10 أغسطس 2024

التصنيف

هندسة الإنتاج والتصميم الميكانيكى

المشاهدات

114

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Finite Element Analysis
S. S. BHAVIKATTI
Professor and Head Department of Civil Engineering and Dean (Academic)
Sh Dharmaslhal Manjunalhcshwara College of Engineering ami
Technology. Dhavalagin. Dharwad • 580 002
(Formerly). Professor and Head Department of Civil Engineering
Karnataka Regional Engineering College. Surathkal
Contents
Preface v
Acknowledgments vi

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 32viii Contents
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 127Contents ix
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 224x Contents
    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 325Contents xi
    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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