ME 422 Homework 2
Distributed: December 6, 2012
Due: December 13, 2012
Problem 1: Calculate the stress on an aluminum bar with an axial load. The bar should
have a 2”×1” cross-section, and be 10” long. Fix one end to a wall and pull on the other
end with a force of 2×104 lbs.
a. Do a hand calculation for the axial stress and axial deflection of the bar.
b. Do a hand calculation for how much the bar should shrink along the 2” direction and
the 1” direction under the load. This will probably require that you review and apply the
definition of Poisson’s ratio (ν). You may assume that ν=0.3 for Aluminum.
c. Use Workbench to perform the analysis. Check the stress, axial deflection, and crosssection shrinkage at a plane which is 5” from the end of the bar. Attach screenshots of
results on that plane to demonstrate that your FEA agrees appropriately with your hand
calculation.
Problem 2: Consider the spring system shown below.
k
u3
k
F
u2
u1
2k
3k
u4
u5
a. Write out the element connectivity table for the spring system.
b. Form the global matrix equation for the system. At this point you should have 5 equations in 5 unknowns.
c. Since the u3 , u4 , and u5 displacements are zero, only u1 and u2 are actually unknown.
Find the 2×2 matrix equation for the system.
d. Solve the equation for the displacements of the nodes. Sketch the deformed configuration
of the system. Calculate the forces in the springs and indicate them on the sketch. Are the
results reasonable? (Fix them if they are not reasonable.)
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Problem 3: Consider the resistor network shown on the next page.
a. Write out the element connectivity table for the resistor system.
b. Form the global stiffness matrix for the system. At this point you should have 4 equations
in 4 unknowns.
c. Since V3 is zero and V4 is V , only V1 and V2 are actually unknown. Find the 2×2 stiffness
equation for the system.
V4
V
1
R
2R
+
V
V2
−
4R
V3
d. Solve the equation for the voltages of the nodes. Sketch the system, indicating the voltages at the nodes. Calculate the currents in the resistors and indicate them on the sketch.
Do your results seem reasonable? Fix them if they are not reasonable.
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e. Rewrite the matrix equation using the penalty method to impose V4 = V . Use MATLAB
or to solve the equations with Kbig = 10/R, 100/R, 1000/R, and 104 /R. Do not use MAPLE.
Attach a printout of your m-file. Compare the accuracy of the solution to the results you
found in part d. Do your results seem reasonable? Fix them if they are not reasonable.
Problem 4: Consider the piping network shown below
4
111 4
000
2
00 3
11
1
1
11
00
5
000 2
111
6
000
111
8111
000
111
000
000 5
111
000
111
000 6
111
3
7
111
000
0007
111
If the flow is laminar and steady, we can write the equation for a single pipe element as
qj = kj (P1 − P2 )
where
qj
P1 , P2
kj
= flow through element j
= pressures at local nodes 1 and 2
= k for element j
a. Write out the element connectivity table for the pipe system.
b. Form the global matrix equation for the system. At this point you should have 8 equations in 8 unknowns.
c. Fluid flows into the system at node 1 at a rate of f1 m3 /s. The pressure at nodes 4, 5, 6,
and 7 is zero. Reduce the global system to 4 equations in 4 unknowns.
d. If pipes 1 and 2 have kj = 2, all the other pipes have kj = 4, and f1 =10, write down the
numerical 4×4 matrix equation we must solve.
e. Use MATLAB to solve the 4×4 equation for the pressures at the nodes. Do not use
MAPLE. Attach the m-file.
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f. Sketch the piping network. Identify the pressures at the nodes in the sketch. Calculate
the volume flow rate in each pipe element and identify them on the sketch. Do your results
seem reasonable? Fix them if they are not reasonable.
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