Transcript Lecture 3. - DigitalAddis

FEA Course Lecture III – Outline
UCSD - 10/16/03
1. Intro to 2-Dimensional FEs [plates and shells].
1. Fluid Mechanics Problem
2. Heat Transfer (Thermal) Problem
3. Plane Elasticity
Finite Elements Principles and
Practices - Fall 03
2-Dimensional Finite Elements
1. Stiffness and Load Vector Formulations for mechanical, heat
transfer and fluid flow problems.
The system equation to be solved can be written in matrix form as:
[K] {D} = {q}
Where
[K] is traditional known as the ‘stiffness’ or ‘coefficient’ matrix
(conductance matrix for heat transfer, flow-resistance matrix for
fluid flow),
{D} is the displacement (or temperature, or velocity) vector and
{q} is the force (or thermal load, or pressure gradient) vector.
Finite Elements Principles and
Practices - Fall 03
C) For plane elasticity problem in 2-dimensional, we have:
1. Plane Problems – are also known as Plane Elasticity Problems
2. Plane – a flat element with constant thickness
3. There are two types of plane problems
• Plane Stress – thickness of the third dimension of an
elastic body is very small.
Sigma_XZ =
Sigma_YZ =
Sigma_ZZ = 0
• Plane Strain - thickness of the third dimension of an
elastic body is infinitely large.
t=infinity
Finite Elements Principles and
Practices - Fall 03
Governing Differential Equation:
s x s xy

 fx  0
x
y
s xy s xy

 fy  0
x
y
Stress-Strain Relationships
s=Ee
Strain
 s x  C11 C12 0   e x 
  
 

 s y   C12 C 22 0   e yx 
s   0 0 C 33 2e 
  xy 
 xy  
Plane Stress
C11 = C22 = E/(1-n2)
C12 = nE/(1-n2)
C66 = (1-n)E/2(1-n2)
Finite Elements Principles and
Practices - Fall 03
Plane
C11 = C22 = (1-n)E/(1+n)/(1-2n)
C12 = nE/(1+n)/(1-n)
C66 = (1-2n)E/2/(1+n)/(1-2n)
1.Types of Finite Elements used in Plane Elasticity
a) CST – Constant Strain Triangle
b) LST – Linear Strain Triangle (Quadratic Triangle)
c) Q4 – Bilinear Quadrilateral
Finite Elements Principles and
Practices - Fall 03
d) Q8 – Quadratic Quadrilateral
e) Q6 –Improved Bilinear Quadrilateral
f) Elements with Drilling DOF (Degree of Freedom)
When using beams and shell elements (say stiffeners)
Folded Plates and Shells
Finite Elements Principles and
Practices - Fall 03
A) For heat transfer problem in 2-dimensional, we have:
Finite Elements Principles and
Practices - Fall 03
B) For fluid flow problem in 2-dimensional, we have:
Finite Elements Principles and
Practices - Fall 03
Review of Analysis Results. E.g., Plane Elasticity. Exact Vs FE
solution. Error Estimation.
SOFTWARE-Specific Session:
Build 2D problem on ANSYS. Go through all steps.
Thermal problem on ANSYS
Plane-stress problem on ANSYS
2DFlow problem on ANSYS/FEMLAB.
Homework 2 and Reading Assignments.
Finite Elements Principles and
Practices - Fall 03