A ductile steel component has a yield strength σy = 300 MPa. At a critical point, the principal stresses are σ1 = 200 MPa (tensile), σ2 = 100 MPa (tensile) and σ3 = 0. What is the factor of safety against yielding according to the Maximum Shear Stress Theory?
- A1.25
- B1.5Correct
- C1.75
- D2
Solution
Concept
Factor of safety evaluation using Tresca theory
Formula
FOS = σy / (σ1 - σ3)
Step-by-step solution
1. The principal stresses are σ1 = 200 MPa, σ2 = 100 MPa, and σ3 = 0 MPa. 2. The maximum shear stress is τmax = (σ1 - σ3) / 2 = (200 - 0) / 2 = 100 MPa. 3. Under uniaxial yielding, yield shear strength is τy = σy / 2 = 300 / 2 = 150 MPa. 4. Factor of safety against yielding is FOS = τy / τmax = 150 / 100 = 1.5 (or FOS = σy / (σ1 - σ3) = 300 / 200 = 1.5). Answer: B
Explanation
Under plane stress with principal stresses σ1 = 200 MPa, σ2 = 100 MPa, and σ3 = 0, the maximum shear stress is τmax = (σ1 - σ3) / 2 = (200 - 0) / 2 = 100 MPa. According to Tresca theory, yielding occurs when 2·τmax = σy / FOS => FOS = σy / (σ1 - σ3) = 300 / 200 = 1.5.
Why the other options are wrong
A) 1.25 corresponds to using σ1 = 240 MPa. C) 1.75 is incorrect. D) 2.0 corresponds to using σ2 alone as the failure metric.
Shortcut
FOS = σy / (σ1 - σ3) = 300 / (200 - 0) = 1.5.
Common mistake
Considering in-plane stress difference (σ1 - σ2 = 100 MPa) and calculating FOS = 300 / 100 = 3.0.
Correct answer: B) 1.5