BME Advanced Mechanics 2 — Questions and Answers
Question 1: A thin-walled pressure vessel with radius r and wall thickness t is subjected to internal pressure p. What is the hoop stress?
- pr/(2t)
- pr/t (Correct answer)
- 2pr/t
- pr/(4t)
Correct answer: pr/t
Hoop (circumferential) stress in a thin-walled cylinder is σ_h = pr/t, twice the longitudinal stress.
Question 2: In Euler-Bernoulli beam theory, which assumption is NOT made?
- Plane sections remain plane after bending
- The beam material is linearly elastic
- Shear deformation is negligible
- Large deflections are permitted (Correct answer)
Correct answer: Large deflections are permitted
Euler-Bernoulli beam theory assumes small deflections; large-deflection problems require nonlinear analysis.
Question 3: Which failure criterion is most appropriate for ductile metals under multiaxial loading?
- Maximum normal stress criterion
- von Mises (distortion energy) criterion (Correct answer)
- Coulomb-Mohr criterion
- Maximum principal strain criterion
Correct answer: von Mises (distortion energy) criterion
The von Mises criterion accurately predicts yielding in ductile metals by considering all stress components through the distortion energy.
Question 4: A simply supported beam of length L carries a uniform distributed load w. The maximum bending moment occurs at:
- x = L/4
- x = L/3
- x = L/2 (Correct answer)
- x = 2L/3
Correct answer: x = L/2
For a uniformly loaded simply supported beam, the bending moment is parabolic and peaks at mid-span with M_max = wL²/8.
Question 5: The polar moment of inertia J for a solid circular shaft of diameter d is:
- πd³/16
- πd⁴/32 (Correct answer)
- πd⁴/64
- πd²/4
Correct answer: πd⁴/32
J = πd⁴/32 for a solid circle; this quantity is used in the torsion formula τ = Tr/J.
Question 6: Which of the following best describes the shear center of a cross-section?
- The point where normal stress is zero
- The point through which a transverse load must act to produce no twisting (Correct answer)
- The centroid of the cross-section
- The point of maximum shear stress
Correct answer: The point through which a transverse load must act to produce no twisting
The shear center is the unique point where a transverse load causes pure bending without twisting of the beam.
Question 7: Castigliano's second theorem states that the partial derivative of strain energy U with respect to a force P equals:
- The stress at the point of load application
- The deflection at the point and in the direction of P (Correct answer)
- The reaction force at the support
- The bending moment due to P
Correct answer: The deflection at the point and in the direction of P
Castigliano's second theorem: δ = ∂U/∂P, giving the displacement at the load's point of application in the direction of the load.
BME Advanced Mechanics 2
A thin-walled pressure vessel with radius r and wall thickness t is subjected to internal pressure p.
What is the hoop stress?