UNIVERSITY OF SASKATCHEWAN
ME 323.3 – MECHANICS OF MATERIALS II
FINAL EXAM – APRIL 26, 2010
Professor A. Dolovich
A CLOSED BOOK EXAMINATION
TIME: 3 HOURS
For Marker’s
Use Only
LAST NAME (printed): ________________________________
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FIRST NAME (printed): ________________________________
2. __________
STUDENT NUMBER: _________________________________
3. __________
EXAMINATION ROOM: _______________________________
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SIGNATURE: ________________________________________
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Total:
INSTRUCTIONS
1. The examination consists of 7 questions.
Answer all seven questions.
The exam is out of a total of 100 marks.
The number of marks for each question is given in brackets.
PRINT YOUR NAME AT THE TOP OF EACH PAGE.
2. This is a closed book exam.
Calculators are permitted.
A list of formulas will be provided separately.
3. SHOW YOUR WORK CLEARLY.
Give final answers to 3 significant figures.
4. Your answers are to be given in the space below the question.
Continuation sheets have been provided within the exam paper.
In addition, the back of each page may be used as a continuation
sheet if required.
ME 323 Final 2010
Name: _____________________________
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T
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1. A torque T is applied to a solid tapered shaft BD of
circular cross section. The radius of the shaft at
section B is 2c, and the radius of the shaft at section
D is c, where c is a constant. The shaft is fixed at
section B, and is made of a nonlinear homogeneous,
isotropic material which has the stress-strain
D
c
1/5
relation τ = kγ , where k is a material constant.
L
(a)
(b)
Determine the relation between shearing
stress τ and applied torque T. Write your
answer for τ in terms of T, c, x, L, and radial
position r.
Find an expression for the relative angle of
twist φ of section D with respect to B. Give
your answer in terms of T, L, c, and k.
x
B
2c
ME 323 Final 2010
Name: _____________________________
Continuation Sheet – Problem 1
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ME 323 Final 2010
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Name: _____________________________
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2. A thin-walled beam has the cross section shown, with a uniform wall thickness
t = 6 mm.
74 mm
40 mm
50 mm
60 mm
50 mm
(a) Determine the location of the shear center. Clearly show this location on a
sketch of the cross section.
(b) Determine the maximum transverse shearing stress in the cross section due
to a vertical transverse shear force V = 15 kN applied through the shear
center.
ME 323 Final 2010
Name: _____________________________
Continuation Sheet – Problem 2
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ME 323 Final 2010
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Name: _____________________________
3. A cantilever beam of length L and rigidity EI is
subjected to a uniform load w and an endcouple MB as shown. Using energy methods,
determine the deflection of the beam at x = L/4.
Write your answer in a simple form in terms of
w, MB, L, E and I, as appropriate. Consider
bending only.
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w
MB
A
B
x
L
ME 323 Final 2010
Name: _____________________________
Continuation Sheet – Problem 3
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ME 323 Final 2010
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Name: _____________________________
4. A thin-walled, symmetrical tube is transmitting a
torsional moment of 2500 N·m. The tube is made
of aluminum for which E = 70 GPa and ν = 0.33.
The thickness of each wall is 5 mm. Determine
the average shear stress in each wall and the angle
of twist per unit length (in deg/m). In your
analysis, use a two-cell approach where
initially it is assumed that the shear flow q
around each cell is different. Then, by your
equations, show that in fact these two shear
flows are the same (and therefore consistent
with symmetry).
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25 mm
30 mm
30 mm
25 mm
ME 323 Final 2010
Name: _____________________________
Continuation Sheet – Problem 4
Page 9 of 15
ME 323 Final 2010
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Name: _____________________________
5. Regarding torsion of thin-walled closed sections, do the following:
(a) In detail, derive the equation τ =
T
showing all steps;
2 tA
(b) Show mathematically (without numerical values) that the equation
TL ds
agrees with circular torsion theory for a thin walled circular tube.
φ=
4GA 2 ∫ t
Neatness and clarity are especially important in this question to receive marks.
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ME 323 Final 2010
Name: _____________________________
Continuation Sheet – Problem 5
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ME 323 Final 2010
Name: _____________________________
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m
D
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6. A beam of length L, as shown, is
supported by a wire CD, and by a pin at
point O. The beam has a rectangular cross
section of b×2b, while the wire has a
circular cross section with radius r. The
wire has length d. A mass m is dropped
from height d and strikes the end of the
beam thereby subjecting it to a
concentrated force which acts through the
shear center. Using energy methods,
obtain a conservative estimate of the
resulting deflection of the beam at point C.
Use the numerical values given below the
sketch.
d
O
C
2b
B
b
3L
4
b = 2 in.
L = 60 in.
Ewire = Ebeam = 30×106 psi
L
4
d = 24 in.
r = 0.1 in.
Mass m has a weight of mg = 20 lb
ME 323 Final 2010
Name: _____________________________
Continuation Sheet – Problem 6
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ME 323 Final 2010
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Name: _____________________________
7. A beam has the cross section shown. Sketch the shear flow in this
section due to transverse loading (where the transverse force acts
through the shear center). You may draw a number of intermediate
sketches to help your thinking, but your final sketch must be large and
clear, with the relative size and direction of arrows clearly shown, and
with a neat box placed around this final answer. Marks will be based
only on the final sketch. On this sketch place a dark dot at every point
where the shear flow is zero.
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ME 323 Final 2010
Name: _____________________________
Continuation Sheet – Problem 7
End of Exam
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