Mechanics of Materials Flashcards

A beam is made of two materials bonded together (composite beam). The transformed section method converts one material to an equivalent width using:

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.

In a biaxial stress state (σ_x and σ_y, no shear), which expression gives the principal stresses?

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.

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:

Answer: 160 MPa

Goodman: σa/Se + σm/Su = 1 → σa/200 + 100/500 = 1 → σa = 200 × 0.8 = 160 MPa.

Which boundary condition produces the LOWEST Euler critical buckling load for a column of given L and EI?

Answer: Fixed-free (flagpole) with K = 2

P_cr = π²EI/(KL)²; the largest K = 2 (fixed-free) gives the smallest critical load.

What does a stress concentration factor K_t represent in design?

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.

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?

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.

The modulus of toughness of a material is best represented by:

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.