BME Advanced Mechanics 4 — Questions and Answers
Question 1: The stress intensity factor K_I for a through-crack of half-length a in an infinite plate under remote stress σ is:
- σ√(πa) (Correct answer)
- σ√(2πa)
- σπa
- σ/(√(πa))
Correct answer: σ√(πa)
For a central through-crack in an infinite plate, K_I = σ√(πa), the fundamental result of linear elastic fracture mechanics.
Question 2: Which of the following correctly defines the term 'plane strain' condition?
- All three principal strains are equal
- Strain in one direction (usually z) is constrained to zero (Correct answer)
- Only shear strains are non-zero
- The strain state is uniaxial
Correct answer: Strain in one direction (usually z) is constrained to zero
Plane strain occurs in thick components where strain in the thickness direction (ε_z = 0) is constrained by the surrounding material.
Question 3: In the finite element method, the stiffness matrix [K] relates nodal forces to nodal displacements via [K]{u} = {F}. The stiffness matrix is always:
- Asymmetric and non-singular
- Symmetric and positive semi-definite before boundary conditions are applied (Correct answer)
- Diagonal for any element type
- Singular after boundary conditions are applied
Correct answer: Symmetric and positive semi-definite before boundary conditions are applied
The global stiffness matrix is symmetric (from reciprocity) and positive semi-definite (singular until supports are applied to remove rigid body modes).
Question 4: A gear tooth experiences repeated bending stress that reverses each rotation. The appropriate design criterion is:
- Static yield strength
- Ultimate tensile strength
- Endurance limit under fully reversed bending (Correct answer)
- Fracture toughness K_Ic
Correct answer: Endurance limit under fully reversed bending
Gear teeth undergo fully reversed (R = −1) cyclic bending, so the endurance limit (or corrected fatigue strength) governs design life.
Question 5: The principle of virtual work states that a body is in equilibrium if and only if:
- The total potential energy is minimized
- Virtual work done by external forces equals virtual work done by internal forces for every admissible virtual displacement (Correct answer)
- All reaction forces are zero
- The kinetic energy equals the potential energy
Correct answer: Virtual work done by external forces equals virtual work done by internal forces for every admissible virtual displacement
The principle of virtual work (PVW) equates external and internal virtual work for any kinematically admissible virtual displacement, providing an equilibrium criterion.
Question 6: Which of the following materials exhibits the highest fracture toughness K_Ic?
- Soda-lime glass
- High-strength steel (4340)
- Aluminum alloy 7075-T6
- Low-carbon steel (A36) (Correct answer)
Correct answer: Low-carbon steel (A36)
Low-carbon structural steels like A36 have high fracture toughness (≈ 100–200 MPa√m) due to their high ductility, whereas high-strength steels trade toughness for yield strength.
Question 7: The strain energy density for a linear elastic body under uniaxial stress σ is:
- σε
- σ²/(2E) (Correct answer)
- Eε²
- σ/(2E)
Correct answer: σ²/(2E)
Strain energy density u = σ²/(2E) = Eε²/2, representing the area under the linear elastic stress-strain curve.
BME Advanced Mechanics 4
The stress intensity factor K_I for a through-crack of half-length a in an infinite plate under remote stress σ is: