BME Advanced Mechanics 3 — Questions and Answers
Question 1: The critical buckling load for a pin-pin column of length L, modulus E, and moment of inertia I is given by Euler's formula as:
- π²EI/L² (Correct answer)
- 2π²EI/L²
- π²EI/(4L²)
- 4π²EI/L²
Correct answer: π²EI/L²
Euler's critical load for a pin-pin column is P_cr = π²EI/L², corresponding to an effective length of L.
Question 2: In a Mohr's circle construction, the center of the circle lies at:
- (σ_x, τ_xy)
- ((σ_x + σ_y)/2, 0) (Correct answer)
- ((σ_x − σ_y)/2, τ_xy)
- ((σ_x + σ_y)/2, τ_xy)
Correct answer: ((σ_x + σ_y)/2, 0)
The center of Mohr's circle is at the average normal stress ((σ_x + σ_y)/2, 0) on the σ–τ plane.
Question 3: Which statement correctly describes Saint-Venant's principle?
- Stress concentrations are permanent near any boundary
- The effects of localized loads become negligible at distances large compared to the load region (Correct answer)
- All distributed loads can be replaced by a single point force anywhere
- Stress is always uniform in statically determinate structures
Correct answer: The effects of localized loads become negligible at distances large compared to the load region
Saint-Venant's principle states that the stress distribution away from the point of load application is independent of how the load is distributed, provided the resultant is the same.
Question 4: For a linearly elastic material, the relationship between shear modulus G, Young's modulus E, and Poisson's ratio ν is:
- G = E/(2(1+ν)) (Correct answer)
- G = E/(1+2ν)
- G = 2E/(1+ν)
- G = E/(2(1−ν))
Correct answer: G = E/(2(1+ν))
The elastic constants are related by G = E / [2(1 + ν)], valid for isotropic materials.
Question 5: A stepped shaft transmits torque T. At a step where the radius changes sharply, the stress concentration factor K_t is used because:
- The material becomes anisotropic at the step
- Local geometry causes stress to exceed the nominal value (Correct answer)
- Torque is not conserved through the step
- Shear modulus changes at the step
Correct answer: Local geometry causes stress to exceed the nominal value
Geometric discontinuities like fillets and steps create local stress concentrations where actual stress = K_t × nominal stress.
Question 6: The compatibility equations in elasticity theory ensure that:
- Equilibrium is satisfied at every point
- The displacement field is single-valued and continuous (Correct answer)
- Material remains in the elastic regime
- Boundary conditions are met on all surfaces
Correct answer: The displacement field is single-valued and continuous
Compatibility equations guarantee that the strain field derived from stresses corresponds to a continuous, single-valued displacement field with no gaps or overlaps.
Question 7: Creep in metals at elevated temperatures is best described as:
- Instantaneous plastic deformation upon loading
- Time-dependent permanent deformation under constant stress (Correct answer)
- Elastic deformation recovered upon unloading
- Fatigue damage accumulated over cyclic loading
Correct answer: Time-dependent permanent deformation under constant stress
Creep is the slow, time-dependent plastic deformation that occurs under sustained stress, especially above ~0.4 T_m (homologous temperature).
BME Advanced Mechanics 3
The critical buckling load for a pin-pin column of length L, modulus E, and moment of inertia I is given by Euler's formula as: