Page 7 - Finite Element Modeling and Simulations with ANSYS Workbench

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