most beneficial UNDER EXAM CONDITIONS - carleton.ca · The purpose of this mock exam is to give you practice answering questions ... In axisymmetric problems, ... Which 3-D Mohr’s

Brandon Kerr MAAE 3202 AB

It is most beneficial to you to write this mock final UNDER EXAM CONDITIONS. This

means:

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After the time limit, go back over your work with a different colour or on a separate piece

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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.

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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.


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


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.


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


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


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


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)


(c) 𝐼𝑠𝑡

𝐼𝑠


(𝐼𝑦 , −𝐼𝑦𝑧)

(d)


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


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


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


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)


𝜏𝑠

𝜎𝑛

(c)


𝜏𝑠

𝜎𝑛

(d)

𝜎𝜃 𝜎𝑟 𝜎𝑧


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


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


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)


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

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most beneficial UNDER EXAM CONDITIONS - carleton.ca · The purpose of this mock exam is to give you practice answering questions ... In axisymmetric problems, ... Which 3-D Mohr’s