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Fall 2006
Exam I
EM-3I9
M.E.Mear
Problem I: Five plates are joined together by two bolts of diameter d, and the assembly is
loaded as shown in the figure. Determine the maximum shear stress in each bolt.
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Problem 2: The first half of the bar shown has cross sectional area AJ ~ A and Young's
modulus EJ = E, while the second half has cross sectional area Az= A and Young's
modulus Ez = 2E. Twoloads are applied to the bar as indicated. Find: a) the maximum
axial stress in the bar and b) the displacement of the free end of the bar.
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Problem 3: The assembly shown in the figure is subjected to two loads of magnitude PI2
(applied to the rigid plate as shown). The bar labeled (1) has axial stiffness 3AE whereas
the bars labeled (2) and (3) have axial stiffness AE. Before the load is applied, there is a
small gap s between the rigid plate and the bar labeled (3). Assume that the magnitude of
the loads is such that the gap is fully closed after they are applied. Find: a) the force
induced in each bar by the applied loads and b) the displacement of the rigid plate.
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Problem 4: The beam shown in the figureis rigid whereas the two rods are linearly elastic
with axial stiffness AE, length h and coefficient of thermal expansion a. Determine the
force induced in each rod by a change in temperature AT.
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Problem 4:
1) Free-body diagram:
Ay
Ax
F1
F2
F1
F2
F1
F2
2) Equilibrium:
ΣM A = LF2 +
L
F1 = 0
2
2F2=-F1
3) Compatibility:
The rigid beam can rotate but not deform.
δ2
δ1
L/2
δ 2 = 2δ 1
L/2
4) Force-displacement relations:
F1 h
⎧
=
δ
⎪ 1 EA + α (∆T )h
⎨
Fh
⎪δ 2 = 2 + α (∆T )h
EA
⎩
Fh
F2 h
+ α (∆T )h = −4 2 + 2α (∆T )h
EA
EA
F2 =
α (∆T )EA
5
− 2α (∆T )EA
F1 =
5