ME: 5117 Finite Element Analysis
Midterm Exam
1) Consider the following advection-diffusion equation:
d ( X , t)
U ( X , t) ( X , t) f( X )
2
dt
Where is the gradient operator and 2 is the Laplacian operator. U, >0 and f are given function
of X= (x, y, z
FEM of single-variable problems in 2-D
FEM Model
Interpolation Function and Elements
Requirement of Interpolation Function:
X
X
Interpolation Function: Triangular
Interpolation Function: Rectangle
Evaluation of Boundary Integrals
s=0
Local
coordinate
Equi
Lecture 11 Euler-Bernoulli Beam
Governing Equation
Transverse deflection w governed by 4th-order DE:
Distributed loading density
d 2 d 2w
EI 2 c f w q (x) for 0 x L (1)
2
dx
dx
Standard Procedures for FEA of Eq. (1)
1) Discretization of the Domain
2)
CHAPTER
3
Development
of Truss Equations
Introduction
Having set forth the foundation on which the direct stiffness method is based, we will
now derive the stiffness matrix for a linear-elastic bar (or truss) element using the general steps outlined in Ch
Lecture 2: Basic FEA of Truss
Structures
Three Principle Steps of FEA
FEA: Preprocessing- Example from ABAQUS
1
3
2
4
FEA: Analysis Example from ABAQUS
FEA: Postprocessing Example from ABAQUS
Matrix Analysis of Trusses
K u f
Kij u j fi
u and f are know
Transverse deflection of membranes (Poissons equation)
The equation governing the transverse deflection w of the clamped unit rhombus membrane is given by
the Poissons equation
2w 2w
a
f x, y
2
2
y
x
in 0 ( x , y ) 1
with the boundary condition
w
Dimensional Analysis: FEA Analysis
Calculate the deflection of a cantilever beam:
f ( E , I , P, L )
Q: Calculate the function f numerically
Dimensionless: variables with no units
Fundamental unit :
Length : m
Mass : kg
Time : s
m
Lm
I m4
P kg m / s 2
E
Gauss quadrature
for Numerical Integration
Axially loaded elastic bar
y
A(x) = cross section at x
b(x) = body force distribution
(force per unit length)
x E(x) = Youngs modulus
F
x
x=L
x=0
x1
1
For each element
x2
2
Element stiffness matrix
x2
k B EB Adx
Short Introduction about
the FEM
Jie Yin
What is FEM?
FEM: Method for numerical solution of field problems
Number of degrees of freedoms (DOF)
Continuum: Infinite
FEM: Finite
Descriptions:
http:/nylander.wordpress.com/category/engineering/femfea/
Dividin
A guide to using Finite Element Software
Finite element method: to solve partial differential equations
Solids:
predict deformation and stress fields within solid bodies subjected to external
forces
Fluid flow
heat transfer
electromagnetic fields
Dif