NCEES Mechanics of Materials 5 — Questions and Answers
Question 1: A beam is made of two materials bonded together (composite beam). The transformed section method converts one material to an equivalent width using:
- The modular ratio n = E₁/E₂ (Correct answer)
- The yield strength ratio Sy₁/Sy₂
- The density ratio ρ₁/ρ₂
- The Poisson's ratio difference ν₁ − ν₂
Correct answer: The modular ratio n = E₁/E₂
The transformed section scales one material's width by the modular ratio n = E₁/E₂ so standard flexure formulas apply.
Question 2: In a biaxial stress state (σ_x and σ_y, no shear), which expression gives the principal stresses?
- σ₁,₂ = (σx + σy)/2 ± (σx − σy)/2 (Correct answer)
- σ₁,₂ = σx ± σy
- σ₁,₂ = (σx × σy)^(1/2)
- σ₁,₂ = σx/σy
Correct answer: σ₁,₂ = (σx + σy)/2 ± (σx − σy)/2
With no shear stress, Mohr's circle degenerates and principal stresses are simply σ_x and σ_y directly.
Question 3: A rod is loaded in fatigue. The endurance limit Se is 200 MPa and the mean stress is 100 MPa (Su = 500 MPa). Using the Goodman criterion, the allowable alternating stress amplitude is:
- 160 MPa (Correct answer)
- 200 MPa
- 100 MPa
- 140 MPa
Correct answer: 160 MPa
Goodman: σa/Se + σm/Su = 1 → σa/200 + 100/500 = 1 → σa = 200 × 0.8 = 160 MPa.
Question 4: Which boundary condition produces the LOWEST Euler critical buckling load for a column of given L and EI?
- Fixed-free (flagpole) with K = 2 (Correct answer)
- Pin-pin with K = 1
- Fixed-pin with K = 0.7
- Fixed-fixed with K = 0.5
Correct answer: Fixed-free (flagpole) with K = 2
P_cr = π²EI/(KL)²; the largest K = 2 (fixed-free) gives the smallest critical load.
Question 5: What does a stress concentration factor K_t represent in design?
- The ratio of maximum local stress to nominal (average) stress at a geometric discontinuity (Correct answer)
- The ratio of yield strength to ultimate strength
- The safety factor against fatigue failure
- The ratio of bending stress to shear stress
Correct answer: The ratio of maximum local stress to nominal (average) stress at a geometric discontinuity
K_t = σ_max/σ_nom; it quantifies the amplification of stress caused by holes, notches, fillets, or other geometric changes.
Question 6: A steel plate (E = 200 GPa, ν = 0.3) is under plane stress with σ_x = 100 MPa and σ_y = 60 MPa. What is the strain ε_x?
- 4.1 × 10⁻⁴ (Correct answer)
- 5.0 × 10⁻⁴
- 3.1 × 10⁻⁴
- 6.8 × 10⁻⁴
Correct answer: 4.1 × 10⁻⁴
ε_x = (σ_x − ν σ_y)/E = (100 − 0.3 × 60) / (200×10³) = 82/200000 = 4.1 × 10⁻⁴.
Question 7: The modulus of toughness of a material is best represented by:
- The total area under the engineering stress-strain curve from zero to fracture (Correct answer)
- The slope of the initial linear portion of the stress-strain curve
- The stress at the 0.2% offset yield point
- The area under the curve only in the elastic region
Correct answer: The total area under the engineering stress-strain curve from zero to fracture
Toughness is the total strain energy per unit volume a material can absorb before fracturing, equal to the entire area under the stress-strain curve.
NCEES Mechanics of Materials 5
A beam is made of two materials bonded together (composite beam).
The transformed section method converts one material to an equivalent width using: