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Brandon Kerr MAAE 3202 AB
It is most beneficial to you to write this mock final UNDER EXAM CONDITIONS. This
means:
• Complete the exam in 3 hour(s). • Work on your own. • You may use your notes, but not the textbook. • Attempt every question.
After the time limit, go back over your work with a different colour or on a separate piece
of paper and try to do the questions you are unsure of. Record your ideas in the
margins to remind yourself of what you were thinking when you take it up at PASS.
The purpose of this mock exam is to give you practice answering questions in a timed
setting and to help you to gauge which aspects of the course content you know well and
which are in need of further development and review. Use this mock exam as a
learning tool in preparing for the actual exam.
Please note:
Come to the PASS workshop with your mock exam complete. During the
workshop you can work with other students to review your work.
Often, there is not enough time to review the entire exam in the PASS workshop.
Decide which questions you most want to review – the Facilitator may ask
students to vote on which questions they want to discuss in detail.
Facilitators do not bring copies of the mock exam to the session. Please print out
and complete the exam before you attend.
Facilitators do not produce or distribute an answer key for mock exams.
Facilitators help students to work together to compare and assess the answers
they have. If you are not able to attend the PASS workshop, you can work alone
or with others in the class.
Good Luck writing the Mock Exam!!
Dates and locations of mock exam take-up: Thursday, December 14th, 6:00-9:00 pm
in ME 3275.
Brandon Kerr MAAE 3202 AB
Question 1
1. An element of a link in a structure is subjected to the stresses shown below. Which Mohr’s circle represents the stress state?
𝜏𝑠
𝜎𝑛
(𝜎𝑥, − 𝜏𝑥𝑦)
(𝜎𝑦, 𝜏𝑥𝑦)
(a) 𝜏𝑠
𝜎𝑛
(𝜎𝑥, − 𝜏𝑥𝑦)
(𝜎𝑦, 𝜏𝑥𝑦)
(b)
𝜏𝑠
𝜎𝑛
(𝜎𝑥, 𝜏𝑥𝑦)
(𝜎𝑦, − 𝜏𝑥𝑦)
(c) 𝜏𝑠
𝜎𝑛
(𝜎𝑥, 𝜏𝑥𝑦)
(𝜎𝑦, − 𝜏𝑥𝑦)
(d)
𝜎𝑦
𝜏𝑥𝑦
𝜎𝑥
Brandon Kerr MAAE 3202 AB
2. For concrete, steel, and cast iron, which choice of failure criteria is most
appropriate?
a) Concrete: Maximum principle stress Steel: Mohr’s criterion Cast Iron: Maximum shear stress
b) Concrete: Mohr’s criterion Steel: Maximum shear stress Cast Iron: Maximum principle stress
c) Concrete: Mohr’s criterion Steel: Maximum principle stress Cast Iron: Maximum shear stress
d) Concrete: Maximum principle stress Steel: Maximum shear stress Cast Iron: Mohr’s criterion
3. Which one of the following statements is true?
a) Any stress field that satisfies the compatibility equation is a valid solution to
an elasticity problem.
b) The neutral axis is always perpendicular to the bending load.
c) If no shear stress is applied to an element, then 𝜎𝑥 and 𝜎𝑦 are the principle
stresses.
d) The maximum shear stress is always given by |𝜎1−𝜎2
2|.
4. (Again!) Which one of the following statements is true?
a) In axisymmetric problems, in-plane shear stress 𝜏𝑟𝜃 = 0.
b) The maximum shear stress in a plane is oriented perpendicular to the
principle stress.
c) Body forces can be neglected in axisymmetric problems.
d) The principle strains are oriented at 90° to the principle stresses.
Brandon Kerr MAAE 3202 AB
5. An element in a machine with a yield strength of 200 MPa is subjected to the
following stress state:
𝜎1 = 150 MPa
𝜎2 = 0
𝜎3 = −100 MPa
a) �̂� = 75 MPa; yielding occurs
b) �̂� = 75 MPa; yielding does not occur
c) �̂� = 125 MPa; yielding occurs
d) �̂� = 125 MPa; yielding does not occur
6. For the following stress state, the maximum shear stress is in which plane?
𝜎1 = 200 MPa
𝜎2 = 0
𝜎3 = −200 MPa
a) The “1-2” plane
b) The “2-3” plane
c) The “1-3” plane
d) �̂� = 0 throughout the element
7. The ratio of the change in radius over the original radius is the:
a) axial strain
b) hoop strain
c) radial strain
d) normal strain
Brandon Kerr MAAE 3202 AB
8. What is the value of the radial stress at
the outer radius, ro, for a thick-walled
cylinder loaded with an internal pressure
pi?
a) 𝜎𝑟(𝑟𝑜) = −𝑝𝑖
b) 𝜎𝑟(𝑟𝑜) = 𝑝𝑖
c) 𝜎𝑟(𝑟𝑜) =2𝑝𝑖
𝑘2−1
d) 𝜎𝑟(𝑟𝑜) = 0
9. If the internal pressure increases until
the thick-walled cylinder is loaded beyond
its yield limit, where will yielding first
occur?
a) At 𝑟 = √𝑟𝑖𝑟𝑜
b) At 𝑟 = 𝑟𝑜
c) At 𝑟 = 𝑟𝑖
d) Thick walled cylinders are magical;
and therefore, they do not yield
10. A cylindrical vessel is closed on one end and compressed by a moveable
piston on the other end. Which of the following is correct, regarding the axial
stress/strain of the
vessel?
a) 𝜎𝑧 = 0, 휀𝑧 < 0
b) 𝜎𝑧 > 0, 휀𝑧 < 0
c) 𝜎𝑧 > 0, 휀𝑧 < 0
d) 𝜎𝑧 = 0, 휀𝑧 = 0
𝑧 𝑃
𝑷 +
𝒓𝒊 𝒓𝒐
Brandon Kerr MAAE 3202 AB
11. Which section would fail first under pure torsional loading?
12. Where will the maximum shear stress occur when the segment below is
subjected to a torque?
a) Along the top flange
b) Along the vertical web
c) Along the bottom flange
d) Specifically at the shear centre, S
13. Which statement is correct for the cantilever elbow beam loaded through its centroid as shown? (The neutral axis is denoted N.A.)
a) The maximum tensile stress is at pt. B
b) The beam does not twist; it only bends
c) The moment vector points in the negative z-direction
d) The maximum compressive stress is at pt. B
(a) (b) (c) (d)
𝑏
𝑏
5𝑏
3
𝑡
2𝑡
𝑡
× 𝑆
+
𝐵
𝐴
Brandon Kerr MAAE 3202 AB
14. An asymmetric beam cross-section has the following second and product
moments of area:
𝐼𝑧 = +100 × 106 mm4
𝐼𝑦 = + 50 × 106 mm4
𝐼𝑦𝑧 = − 40 × 106 mm4
This corresponds with which of the following Mohr’s circles of area moments?
𝐼𝑠𝑡
𝐼𝑠
(𝐼𝑧 , 𝐼𝑦𝑧)
(𝐼𝑦 , − 𝐼𝑦𝑧)
(a) 𝐼𝑠𝑡
𝐼𝑠
(𝐼𝑦 , 𝐼𝑦𝑧)
(𝐼𝑧 , − 𝐼𝑦𝑧)
(b)
𝐼𝑠𝑡
𝐼𝑠
(𝐼𝑦 , −𝐼𝑦𝑧)
(𝐼𝑧 , 𝐼𝑦𝑧)
(c) 𝐼𝑠𝑡
𝐼𝑠
(𝐼𝑧 , 𝐼𝑦𝑧)
(𝐼𝑦 , −𝐼𝑦𝑧)
(d)
Brandon Kerr MAAE 3202 AB
15. Which statement is false for the beam section shown below when it is subjected
to a vertical load through its shear centre?
a) The shear centre is located on the z-axis,
left of pt. C
b) The maximum shear stress occurs at pt. C
c) The first moment of area is largest at points
A and E
d) The location of the shear centre depends on the magnitude of the bending
load.
16. Which of the following shows the correct shear flow directions in the action plane of a cantilever T-beam loaded at the free end by a downward point load?
(a) (b)
(c) (d)
𝑧
A B
C
D E
Brandon Kerr MAAE 3202 AB
17. Castigliano’s 2nd theorem is being used to find what deflection of the beam shown below?
𝐷 = 𝜕𝑈
𝜕𝐹
a) The vertical deflection at A, (𝛿𝑉)𝐴
b) The horizontal deflection at A, (𝛿𝐻)𝐴
c) The vertical deflection at B, (𝛿𝑉)𝐵
d) The horizontal deflection at B, (𝛿𝐻)𝐵
18. Which of the following is an admissible configuration for the system shown below?
(a)
(b)
(c)
(d)
3L L 2L
A
B C D
𝑃 𝐹
𝐻
𝑤
Brandon Kerr MAAE 3202 AB
19. A brass ring is fitted onto a copper ring via interference fit. The brass ring is
heated uniformly, then placed onto the copper ring, and the assembly is left to
cool. Which of the following Lamé plots correctly represents this situation?
(d) None of the above
20. Which one of these is false for torsion of a thin, closed, single-celled section?
a) The shear stress is assumed uniform through the thickness
b) The shear flow is constant along the perimeter
c) Closed sections twist much less under torsion loads than open sections
d) The shear stress is constant along the perimeter of the section
𝟏
𝒓𝟐
𝝈
1
𝑟𝑖2
1
𝑟𝑚2
1
𝑟𝑜2
1
𝑟𝑜2
1
𝑟𝑚2
1
𝑟𝑖2
𝑃𝑜 𝑃𝑚 (𝜎
𝜃)
𝑟𝑜
(𝜎𝜃
)𝑚
1 (𝜎
𝜃)
𝑚2
(𝜎𝜃
)𝑟
𝑖
(a)
𝟏
𝒓𝟐
𝝈
1
𝑟𝑖2
1
𝑟𝑚2
1
𝑟𝑜2
1
𝑟𝑜2
1
𝑟𝑚2
1
𝑟𝑖2
𝑃𝑚
(𝜎𝜃
)𝑟𝑜
(𝜎𝜃
)𝑚
1 (𝜎
𝜃)
𝑚2
(𝜎𝜃
)𝑟
𝑖
(b)
𝟏
𝒓𝟐
𝝈
1
𝑟𝑖2
1
𝑟𝑚2
1
𝑟𝑜2
1
𝑟𝑜2
1
𝑟𝑚2
1
𝑟𝑖2 𝑃𝑖
𝑃𝑚
(𝜎𝜃
)𝑚
2 (𝜎
𝜃)
𝑚1
(𝜎𝜃
)𝑟
𝑖
(c)
(𝜎𝜃
)𝑟𝑜
Brandon Kerr MAAE 3202 AB
21. Which statement is true regarding beam bending?
a) The principle axes are the axes with respect to which the product moment of area is zero
b) For a beam with an axis of symmetry in the y-direction, the moment vector always lies along the neutral axis.
c) If a beam has an axis of symmetry, then the centroid is also the shear center
d) If the moment vector is perpendicular to one of the principle centroidal axes, then symmetric bending occurs
22. Which 3-D Mohr’s circle represents the stress state of an element at the bore of a thick-walled cylindrical pressure vessel subjected to external pressure and built in between rigid supports?
𝜏𝑠
𝜎𝑛
(a)
𝜎𝑧 𝜎𝑟 𝜎𝜃
𝜏𝑠
𝜎𝑛
(b)
𝜎𝑧 𝜎𝑟 𝜎𝜃
𝜏𝑠
𝜎𝑛
(c)
𝜎𝑧 𝜎𝑟 𝜎𝜃
𝜏𝑠
𝜎𝑛
(d)
𝜎𝜃 𝜎𝑟 𝜎𝑧
Brandon Kerr MAAE 3202 AB
23. A thin, solid circular disc with blades attached to its outer edge spins at high speed. Which of the following is true?
a) The maximum hoop stress occurs at the outer edge
b) The radial stress at the outer edge is zero
c) The maximum radial stress occurs at the center
d) The radial stress is independent of the number of blades attached
24. Which of the following is the correct set up for the curved beam shown using Castigliano’s 2nd theorem?
25. Which of these is correct? The stress field is…
a) valid because it satisfies equilibrium
b) valid because it satisfies both equilibrium
and compatibility
c) invalid because it does not satisfy
equilibrium
d) invalid because it does not satisfy
compatibility
a) (𝛿𝐻)𝐴 =
𝜕𝑈
𝜕𝐹= ∫
𝑀𝐴𝐵
𝐸𝐼
𝜕𝑀𝐴𝐵
𝜕𝐹
𝜋/2
0
𝑅𝑑𝜃
b) (𝛿𝑉)𝐴 =
𝜕𝑈
𝜕𝐹= ∫
𝑀𝐴𝐵
𝐸𝐼
𝜕𝑀𝐴𝐵
𝜕𝐹
𝜋/2
0
𝑅𝑑𝜃
c) Both (a) and (b)
d) None of the above
𝑭
A
B
+
R
𝑑𝑠 = 𝑅𝑑𝜃
𝜃
𝑭
𝜎𝑥 = 𝑥3𝑦 − 2𝑎𝑥𝑦 + 𝑏𝑦
𝜎𝑥 = 𝑥𝑦3 − 2𝑥3𝑦
𝜏𝑥𝑦 =𝑥4
2−
3
2𝑥2𝑦2 + 𝑎𝑦2
𝑎, 𝑏 constants
Brandon Kerr MAAE 3202 AB
Question 2
The structure shown is of constant flexural rigidity EI throughout. It carries a
load P at the free end, where horizontal movement is restricted, but it is free to
slide in the vertical direction. Determine expressions for the horizontal reaction at
C, and the horizontal deflection at B.
𝑷 2𝐿
𝑅 = 𝐿
A
B C
Brandon Kerr MAAE 3202 AB
Question 3
A super awesome new turbine stage design consists of a thin nickel-chromium disc
of outer radius ro = 500 mm and thickness 50-mm carrying rigid, uniformly spaced
blades at its outer periphery. The total mass of the blades is 55 kg, with the centre
of mass of each blade located on a pitch circle of 575-mm radius. A gap of 1-mm
exists between the tips of the blades are the engine casing. The disc is shrink-fitted
onto a steel shaft. Material properties are given in the table below. Assume plane
stress conditions apply throughout.
a) Given that the mating pressure between the disc and the shaft upon assembly
(i.e. when 𝜔 = 0 rad/s) is to be Pm = 300 MPa, what is the required radial
interference, 𝛿?
b) Once the turbine engine is in service, what is the maximum rotational speed, 𝜔,
in RPM, that would cause the blades to hit the casing?
Material 𝑬 (GPa) 𝝂 𝝆 (𝐤𝐠/𝐦𝟑)
Ni-Cr 600 0.25 8400
Steel 200 0.33 7850
Ni-Cr alloy disc, ro = 500 mm
solid steel shaft,
rm = 62.5 mm
(ri = 0)
casing
𝝎
blade pitch radius (575 mm)
Brandon Kerr MAAE 3202 AB
Question 4
A uniform beam, 1 m long, is simply supported at its ends and subjected to a
transverse force P at mid-span. It has a cross-section comprised of two solid
rectangular sections as shown below. Determine the magnitude of the force P such
that the bending stress at the points A do not exceed 150 MPa in magnitude and
points B are stress-free. What must be the orientation of the force P in the plane
of the cross-section? Indicate this clearly in a figure.
20 mm
20 mm
20 mm
30 mm 30 mm
𝐵
𝐵
𝐴
𝐴