ME 422 Winter 2012-2013
Do all 4 problems
Name
Problem
Problem
Problem
Problem
TOTAL
1
2
3
4
(25)
(25)
(25)
(25)
(100)
1
Problem 1 (25 points): Consider the tapered bar with internal heat generation shown
below. The area of the bar is A = x/2.
x1
x2
x=0
The portion of the weak form for internal heat generation for the element that goes from x1
to x2 is
x2
T gAdx
x1
Find the contribution to F due to internal heat generation for this element. Set up all
integrals as completely as possible but do not integrate.
2
Problem 2 (25 points): Consider the CYLINDER of radius R shown below. We will use
two 2-node elements to discretize this cylinder.
R
x2
x1
The stiffness matrix for a typical 2-node finite element which has node one at x = x1 and
node two at x = x2 is (this is given, you don’t need to show this):
πk
(x2 + x1 )
K=
Le
1 −1
−1
1
Assemble the contributions to the global stiffness matrix from these terms. Write your
answer in terms of k and R.
3
Problem 3 (25 points): We have a straight bar with internal heat generation which is
held at zero temperature at the two ends. The bar is discretized with a single 3-node finite
element. (See equation sheet for matrices.)
L
T2
T1 =0
T3 =0
g, A, k = constant
Create the finite element equation for T2 and solve for T2 in terms of g, A, k, and L.
DO NOT USE TWO 2-NODE ELEMENTS– USE ONE 3-NODE ELEMENT! (Recall that
the middle node is number 3 in the master 3-node element.)
4
Problem 4 (25 points): Consider the free vibration of a tapered bar:
x=0
x
X=L
For this problem, the governing differential equation is
dδ
d
(EA ) = −ρAω 2 δ
dx
dx
and the boundary conditions are
at x = 0
δ=0
dδ
=0
at x = L EA
dx
What is the weak form for this specific problem? (Note: don’t worry that it is “vibration”–
ω is just another constant here.)
5