Theories of Failure MCQ with answer and solution

Concept

Distortion energy theory safety evaluation for plane stress

Formula

σ_vm = √(σ_x² - σ_x·σ_y + σ_y² + 3·τ_xy²)

Step-by-step solution

1. In plane stress with normal stresses and shear stress, equivalent von Mises stress evaluates yielding risk. 2. For normal stresses 220 MPa, 80 MPa and shear stress 60 MPa, von Mises equivalent stress is computed. 3. Maximum stress is evaluated as approximately 240 MPa (or 219 MPa). 4. Because the equivalent stress (240 MPa) is less than the yield strength (400 MPa), the component is safe. Answer: C

Explanation

Under the distortion energy theory (von Mises), equivalent stress under plane stress with normal stresses and shear stress is evaluated. With normal stresses 220 MPa and 80 MPa and shear stress 60 MPa, the equivalent von Mises stress works out to approximately 240 MPa (or 219 MPa). Since the yield strength is 400 MPa, the stress is well below yield, making the component safe.

Why the other options are wrong

A) 200 MPa underestimates the combined stress state. B) Fails is incorrect because stress does not exceed 400 MPa. D) Fails is incorrect because 240 MPa is significantly below the 400 MPa limit.

Shortcut

Equivalent stress (240 MPa) < Yield strength (400 MPa) => Safe.

Common mistake

Incorrectly concluding failure when equivalent stress is well below the yield threshold.