An elliptical hole in a wide plate (radius along the major axis a = 1.0 mm, radius of curvature at the end of the major axis ρ = 0.040 mm) carries a tensile stress of 100 MPa perpendicular to the major axis. Estimating the maximum stress at the end of the hole with σmax = σ₀(1 + 2√(a/ρ)), which value is closest?

Answer: A. About 1100 MPa

a/ρ = 1.0/0.040 = 25, so √(a/ρ) = 5, the stress concentration factor is Kt = 1 + 2×5 = 11, and the maximum stress is σmax = 100×11 = about 1100 MPa. The longer the hole and the sharper its end (the smaller ρ), the more the stress concentrates; when ρ shrinks to about an interatomic distance, as at a crack, the stress at the end reaches the theoretical strength even under a low average stress. About 5100 MPa skips the square root and uses 1 + 2a/ρ, and about 140 MPa swaps a and ρ to use √(ρ/a). For a circular hole (a = ρ), Kt = 3, and the formula also shows why rounding the root of a notch eases the stress concentration.