Theories of Failure MCQ with answer and solution

Concept

Tresca theory safety check under biaxial tension compression

Formula

σ₁ - σ₂ ≤ σ_y

Step-by-step solution

1. The principal stresses in plane stress are: σ₁ = 180 MPa, σ₂ = -80 MPa, and σ₃ = 0 MPa. 2. Under Maximum Shear Stress Theory (Tresca), yielding occurs when max(|σ_i - σ_j|) ≥ σ_y. 3. The three stress differences are: |180 - (-80)| = 260 MPa; |-80 - 0| = 80 MPa; |0 - 180| = 180 MPa. 4. The maximum stress difference is 260 MPa, which is strictly less than the yield strength of 300 MPa. 5. Therefore, the component is Safe (FOS = 300 / 260 = 1.15). Answer: A

Explanation

Under plane stress with principal stresses σ₁ = 180 MPa, σ₂ = -80 MPa, and σ₃ = 0, the maximum stress difference is |σ₁ - σ₂| = 180 - (-80) = 260 MPa. According to the Maximum Shear Stress Theory, yielding occurs when this difference equals the yield stress (300 MPa). Because 260 MPa < 300 MPa, the component is safe.

Why the other options are wrong

B) Marginal is undefined in deterministic factor-of-safety criteria. C) Failing is incorrect because 260 MPa does not exceed the 300 MPa yield threshold. D) Principal stresses and yield limit provide complete determinacy.

Shortcut

Max stress difference = 180 - (-80) = 260 MPa < 300 MPa (Yield strength) => Safe.

Common mistake

Comparing only the maximum tensile stress (180 MPa) and ignoring the tensile-compressive stress span.