Principal Stresses and Strains MCQ with answer and solution

Concept

Normal stress on an inclined plane

Formula

σ_n = (σ_x + σ_y) / 2 + ((σ_x - σ_y) / 2) × cos(2 × θ)

Step-by-step solution

1. Given principal stresses σ1 = 500 N/mm² and σ2 = 500 N/mm². 2. Normal stress on an inclined plane at angle θ is given by σn = (σ1 + σ2)/2 + ((σ1 - σ2)/2) × cos(2θ). 3. Substituting values with θ = 45°: σn = (500 + 500)/2 + ((500 - 500)/2) × cos(90°) = 500 N/mm². Answer: B

Explanation

When equal principal stresses of the same sign act on a plane element, the state of stress is hydrostatic or isotropic. Under this condition, the normal stress acting on any plane at any angle of inclination remains constant and equal to the principal stress value. Therefore, the normal stress on a plane inclined at 45° remains equal to 500 N/mm².

Why the other options are wrong

A) Normal stress is equal to 500 N/mm² on all planes in equal biaxial stress states, not zero. C) 1000 N/mm² is the sum of the principal stresses rather than the normal stress. D) 707 N/mm² incorrectly applies a trigonometry factor to equal principal stresses.

Shortcut

When σₓ = σ_y the normal stress on any inclined plane remains equal to σ_x (500 N/mm²).

Common mistake

Assuming normal stress varies with angle in a state of equal bi-axial principal stresses.