Principal Stresses and Strains MCQ with answer and solution

Concept

Maximum principal stress calculation from stress components

Formula

σ_max = (σ_x + σ_y)/2 + √(((σ_x - σ_y)/2)² + τ_xy²)

Step-by-step solution

1. Given normal stresses σ₁ = 80 MPa, σ₂ = 20 MPa, and shear stress τ₁₂ = 30 MPa. 2. Maximum principal stress formula: σ_max = (σ₁ + σ₂)/2 + √(((σ₁ - σ₂)/2)² + τ₁₂²). 3. Center = (80 + 20) / 2 = 50 MPa. 4. Radius = √(((80 - 20)/2)² + 30²) = √(30² + 30²) = 30 × √(2) = 42.43 MPa. 5. σ_max = 50 + 42.43 = 92.43 MPa = 92 MPa. Answer: A

Explanation

The maximum principal stress for a 2D stress state is given by σ_max = (σ₁ + σ₂)/2 + √(((σ₁ - σ₂)/2)² + τ₁₂²). For normal stresses of 80 MPa and 20 MPa with shear stress 30 MPa: center = (80+20)/2 = 50 MPa, radius = √(30² + 30²) = 42.43 MPa; hence σ_max = 50 + 42.43 = 92.43 MPa = 92 MPa.

Why the other options are wrong

B) 102 MPa results from miscalculating the center as 60 MPa. C) 112 MPa adds un-squared terms. D) 122 MPa represents an incorrect combination of terms.

Shortcut

σ_max = 50 + √(30² + 30²) = 50 + 42.43 = 92 MPa.

Common mistake

Adding shear stress directly to the maximum normal stress (80 + 30 = 110 MPa).