2.092/2.093
FINITE ELEMENT ANALYSIS OF SOLIDS AND FLUIDS I
FALL 2009
Quiz #1
Instructor:
TA:
Prof. K. J. Bathe
Seounghyun Ham
Problem 1 (10 points):
Consider the solution of the problem shown below. A rod is spinning in steady-state at ω rad/sec.
The rod is idealized as two one-dimensional elements. Assume linear analysis.
ω rad/sec
100
60
Cross­
sectional
area=1
Young’s modulus E
η
Mass density ρ
2
η ⎞
⎛
A(η)= ⎜ 4 ⎟
20 ⎠
⎝
U1=0
x for element 1
U2
U3
x for element 2
a) Establish the finite element displacement interpolation functions, for each element.
b) Give the element stiffness matrices and load vectors, but do not perform any integration.
1 of 2
Problem 2 (10 points):
Consider the two-dimensional plane stress element shown below.
y,v
2
v1
1 u1
fyB
x,u
4
fxB
4 node element
Thickness t
Young’s modulus E
3
6
4
Poisson’s ratio ν
f xB = ( 4+x )
f yB =0
⎡
⎤
⎢ 1 ν 0 ⎥ ⎡ ε xx ⎤
⎡ τ xx ⎤
⎥⎢ ⎥
⎢ ⎥ E ⎢
=
ν
1
0
τ
yy
⎢
⎥
2
⎢ ⎥ 1-ν
⎢ ε yy ⎥ ; ε xx = ∂u ; ε yy = ∂v ; γ xy = ⎛⎜ ∂u + ∂v ⎟⎞
∂x
∂y
⎢
⎢ τ xy ⎥
⎝ ∂y ∂x ⎠
1-ν ⎥ ⎣⎢γ xy ⎥⎦
⎣ ⎦
⎢0 0
⎥
⎣
2 ⎦
a) Establish all displacement interpolation functions.
b) Give the K matrix and the load vector RB of the element. Give all required expressions
but do not perform any integration.
Give all answers but write as little as possible !
2 of 2
MIT OpenCourseWare
http://ocw.mit.edu
2.092 / 2.093 Finite Element Analysis of Solids and Fluids I
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.