Shear Stresses in Beams MCQ with answer and solution

Concept

In a rectangular beam, shear stress varies as a parabola. It is maximum at the neutral axis and zero at the top and bottom.

Formula

τ = (V/(2I)) × (d²/4 − y²), I = b d³/12

Step-by-step solution

1. b = 300 mm, d = 500 mm, V = 20 kN = 20000 N, y = 150 mm. 2. I = 300 × 500³/12 = 3.125 × 10⁹ mm⁴. 3. d²/4 − y² = 62500 − 22500 = 40000 mm². 4. τ = 20000 × 40000/(2 × 3.125 × 10⁹) = 0.128 MPa. 5. τ ≈ 0.13 MPa, so the answer is (D).

Explanation

Shear stress in a rectangular section has a parabolic shape. At 150 mm above the neutral axis, the stress is less than the maximum value. Putting the values into the formula gives about 0.13 MPa.

Why the other options are wrong

A: 0.30 MPa is larger than τ_max (0.20 MPa), so it is not possible. B: 0.60 MPa is far too large. C: 0.26 MPa is also larger than τ_max, so it is not possible.

Shortcut

τ_max = 1.5 V/(bd) = 0.2 MPa. Then τ = τ_max × (1 − (2y/d)²) = 0.2 × (1 − 0.36) = 0.128 MPa.

Common mistake

Using average shear stress (0.133 MPa) or maximum shear stress (0.2 MPa) instead of the stress at y = 150 mm.