Shear Stresses in Beams MCQ with answer and solution

Concept

In rectangular beams, the maximum shear stress is 1.5 times the average shear stress and acts at the neutral axis.

Formula

τ_max = (3 × V) / (2 × b × d), V = w × L

Step-by-step solution

1. Identify the given parameters: Span L = 1 m = 1000 mm, breadth b = 100 mm, depth d = 200 mm, limiting shear stress τ_max = 1.5 N/mm². 2. Recall the relation for maximum shear stress in a rectangular section: τ_max = (3 × V) / (2 × b × d). 3. Substitute the values to find the maximum shear force V: 1.5 = (3 × V) / (2 × 100 × 200). 4. V = (1.5 × 2 × 100 × 200) / 3 = 20,000 N = 20 kN. 5. For a cantilever beam with a UDL w over the entire span, maximum shear force V = w × L. 6. Therefore, w = V / L = 20 kN / 1 m = 20 kN/m.

Explanation

For a rectangular cross-section, the maximum shear stress occurs at the neutral axis and is given by τ_max = (3/2) × (V / (b × d)). For a cantilever beam carrying a uniformly distributed load w, the maximum shear force occurs at the fixed support and is equal to V = w × L. Equating τ_max to the limiting shear stress of 1.5 N/mm² allows us to solve for the maximum uniformly distributed load w.

Why the other options are wrong

A: Results from incorrect factor application. B: Arises from calculation errors. D: Arises from ignoring the 1.5 shape factor for rectangular sections.

Shortcut

For a rectangular section, remember that τ_max = 1.5 × τ_avg = 1.5 × (V / A).

Common mistake

Using the average shear stress formula V / (b·d) instead of multiplying by 1.5 for the maximum shear stress at the neutral axis.