Chapter 8 Linear Strain Triangle Overview

1
Chapter 8 Linear Strain Triangle(Overview)

• Compare formulation of CST and LST
• Comparison of element performance

2
Comparison of CST and LST Formulations

• CST
• Three nodes per element
• 6 DOF per element

• LST
• 6 nodes per element
• 12 DOF per element

3
Displacement Interpolation

• CST
• LST

4
Strains within each element
Recall CST LST
? constant gt Constant Strain Triangle (CST)
? at most linear in x y gt Linear Strain
Triangle (LST)
5
Element Stiffness Matrix
CST LST Since Bij terms depend on x
y, numerical integration is required (Chap. 10)
6
CST vs. LST Performance Comparison

• Consider the following plane stress analysis of a
  cantilever beam

4 x 16 mesh
7
CST vs. LST Performance Comparison(cont.)

• Test Cases

8
CST vs. LST Performance Comparison(cont.)
9
Chapter 9 Axisymmetric Elements

• Examples of axisymmetric problems
• Pressure vessels
• Cylindrical shaft with notch or filleted step
• Stresses near a spherical void
• Hertzian contact between spheres
• Use of axisymmetric elements provides
  computational efficiency as compared to full 3-D
  analysis

10
Axisymmetric Example Pressure Vessel
11
Axisymmetric Example Soil Foundation
12
Axisymmetric Example Valve Stem
13
Axisymmetric Volume Element
14
Stresses Axisymmetric Problems

• Non-zero stresses - ?r, ??, ?z, ?rz
• ?r? ?z? 0 ( due to axisymmetry )

15
Strains Axisymmetric Problems
Non-zero strains - ?r, ??, ?z, ?rz ?r? ?z? 0
( due to axisymmetry )
Stress Strain Relations
4x4
16
Formulation of 3 Node Triangular Axisymmetric
Element
17
Axisymmetric Elements Displacement Fields

• Displacement fields

18
Axisymmetric strain components
19
Axisymmetric Stress-Strain Relations
20
Three node triangle(Note not constant strain)

• Assumed interpolation
• Nodal displacements

21
Interpolation functions Note same as CST
where
matrix form
22
Strain-Displacement Relations
23
Stresses
4 x 1
4 x 4
4 x 6
6 x 1
24
Element Stiffness Matrix