ME 422 Exam 2
Do all 8 problems
Name
Problem
Problem
Problem
Problem
Problem
Problem
Problem
Problem
TOTAL
1
2
3
4
5
6
7
8
(5
(5
(5
(5
( 25
( 25
( 10
( 20
)
)
)
)
)
)
)
)
(100)
1
Problem 1 (5 points): Consider a thin slab which is fixed to a wall and has a pressure,
P , on its free edge:
w
L
pressure,
P
E ν
2L
Using standard Mechanics of Materials and elasticity equations and concepts, and assuming
that the wall doesn’t have any effect except to keep the slab from moving sideways, approximately what is the τxx stress in the slab?
Problem 2 (5 points) For a transient stress analysis with uniform temperature throughout
the body, using isoparametric stress analysis elements, which properties need to be specified?
(Circle all that are necessary.)
(a) E (Young’s modulus)
(b) I (moment of inertia)
(c) ρ (density)
(d) ν (Poisson’s ratio)
(e) A (cross-sectional area)
(f) α (coefficient of thermal expansion)
(g) k (conductivity)
(h) Other (specify):
2
Problem 3 (5 points) In static heat transfer analysis, we write
q = −kBT
How do we modify this calculation in the case of transient heat transfer analysis?
(a) We need to add the inertia term
q = −kBT − ρcp MṪ
(b) We don’t add the inertia term, that’s all there is:
q = −ρcp MṪ
(c) It’s the same for static and transient
q = −kBT
(d) In transient analysis, we just use the positive gradient of the temperature:
q = BT
(e) None of the above. Explain:
Problem 4 (5 points) What is the default boundary condition in heat transfer if we don’t
specify anything on a boundary?
(a) Zero temperature
(b) Zero heat flux
(c) The program just solves using the boundary conditions it does know– it doesn’t make
any assumption.
(d) None of the above. Explain:
3
Problem 5 (25 points) Consider the bar shown below.
2 two-node
bar elements
E,A
ρ
F
F
L
(a) If the problem is static, form the global system of equations for the problem. (Apply all
of the boundary conditions.)
(b) If the problem is now switched to a transient analysis, what would the global system of
equations look like? (Again, apply all of the boundary conditions.)
4
Problem 6 (25 points) A static thermal analysis has been completed for the object shown
below. The temperatures at three of the corners of the mesh (which contains 2 three-node
triangular elements) are zero, and the temperature at the last node is Tc . What is the heat
flux vector
"
#
q̇x′′
q̇y′′
in the center of element 1? (Express your answer in terms of L, H, w, k, and Tc .)
y
W
Tc
0
2
conductivity
k
H
x
1
0
L
5
0
Problem 7 (10 points): The Helmholtz equation is often used to find the vibration modes
of cavities, including exhaust systems. In 1-D the governing differential equation for the
problem is
d2 p
+ k2p = 0
dx2
with k 2 a constant, and p the pressure in the cavity. Suppose that p(0) = 0 and p(L) = 0.
What is the weak form for the problem? (Hint: the weighting function would be called p
since the unknown is p.)
6
Problem 8 (20 points) Find K13 (first row, third column) for the elasticity element shown
below. Set up all integrals completely but do not integrate. Assume plane stress with nonzero ν.
y
W
2
1
E, ν
H
x
3
L
7
4