ME 422 Winter 2012-2013
Exam 2
Do all 5 problems
Name
Problem
Problem
Problem
Problem
Problem
TOTAL
1
2
3
4
5
(20)
(10)
(25)
(20)
(25)
(100)
1
Problem 1 (20 points): Consider the truss system shown below:
y
f
2
1
3
x
Nodes 2 and 3 are completely restrained. Node 1 is free to move in the x and y directions.
Both trusses have length L, area A, and Young’s modulus E. Assemble the global system of
equations
KU = F
Be sure to clearly state your final K and F.
2
Problem 2 (10 points): We are to model a simply supported beam with a single 2-node
beam element. (At the supports, the beam deflection is zero but it is free to rotate.) The
beam has a pressure along its surface. The beam is 1 unit deep (into the page).
P
L
1
2
The global system of finite element equations may be written as:
KX = F
(1)
(a) What is X (circle one)?
X=
X = [w1 ]
⎡
⎢
⎢
⎣
X=⎢
w1
θ1
w2
θ2
⎤
⎥
⎥
⎥
⎦
X=
w1
w2
u1
u2
X = [θ1 ]
⎡
⎢
⎢
⎣
X=⎢
u1
v1
u2
v2
X=
⎤
⎥
⎥
⎥
⎦
θ1
θ2
Other (specify)
(b) What is the vector F in equation 1? State your answer in terms of P and L. (Hint:
There are some very useful equations on the equation sheet– be sure to look there.)
3
Problem 3 (25 points): The governing equation for pressure waves in a car exhaust system
is
∂
∂p
∂2p
βA
= ρA 2
∂x
∂x
∂t
Here β is the bulk modulus of the air in the pipe (a material property), A is the crosssectional area of the pipe, x is the distance along the pipe, ρ is the density of the air in the
pipe, and p is the pressure. (p is the unknown.)
(a) What is the weak form for the problem?
(b) This problem is very similar to others we have examined. For a two-node element of
length L with constant β, A, and ρ, what would we expect K and M to be?
(c) Which form would we expect the governing equations to take?
(i) (K + M)P = F
(ii) KP + MṖ = F
(iii) KP + MP̈ = F
(iv) KP + ṀṖ = F
(v) Other (specify)
4
Problem 4 (20 points): Consider a very crude model of a slab being quenched:
centerline
W
(oil)
slab
convection
ho
k
cp
ρ
T2
T1
T oil
Set up the matrix equations KT + MṪ = F for this problem. (Hint: See equation sheet for
useful equations.)
5
Problem 5 (25 points): We are analyzing a plate with a time varying surface load. The
plate is in plane stress, and its material properties are E, ν, and ρ. After 1 second, the
displacement of the right face is 2δ, and the displacement of the midline is δ. (See the
diagram.) What are the three components of the stress in element 1 at that moment?
L
1
L
δ
6
2δ