Question

The state of stress at the critical location in a structure is \[ \sigma_{xx} = 420 \, MPa, \sigma_{yy} = 100 \, MPa, \sigma_{zz} = 0, \tau_{xy} = \tau_{yz} = \tau_{zx} = 0 \] The yield stress of the material in uniaxial tension is $400 \, MPa$. Select the correct statement among the following:

• A. The structure is safe by both Tresca (maximum shear stress) theory and von-Mises (distortion energy) theory.
• B. The structure is safe by Tresca (maximum shear stress) theory and unsafe by von-Mises (distortion energy) theory.
• C. The structure is unsafe by Tresca (maximum shear stress) theory and safe by von-Mises (distortion energy) theory.
• D. The structure is unsafe by both Tresca (maximum shear stress) theory and von-Mises (distortion energy) theory.

Hint:

Tresca criterion is more conservative than von-Mises. If a material fails Tresca but passes von-Mises, the design is generally still acceptable but with reduced safety margin.

Answer 1

The Correct Option is C

Step 1: Principal stresses.
From the given state: \[ \sigma_1 = 420 \, MPa, \sigma_2 = 100 \, MPa, \sigma_3 = 0 \]
Step 2: Tresca criterion (Maximum shear stress).
Maximum shear stress: \[ \tau_{max} = \frac{\sigma_{max} - \sigma_{min}}{2} = \frac{420 - 0}{2} = 210 \, MPa \] Tresca yield limit: \[ \tau_{max} \leq \frac{\sigma_y}{2} = \frac{400}{2} = 200 \, MPa \] Here: \[ 210>200 \Rightarrow \text{Unsafe by Tresca} \]
Step 3: von-Mises criterion (Distortion energy).
Von-Mises equivalent stress: \[ \sigma_{vm} = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}} \] Substitute: \[ \sigma_{vm} = \sqrt{\frac{(420 - 100)^2 + (100 - 0)^2 + (0 - 420)^2}{2}} \] \[ = \sqrt{\frac{(320)^2 + (100)^2 + (420)^2}{2}} = \sqrt{\frac{102400 + 10000 + 176400}{2}} \] \[ = \sqrt{\frac{288800}{2}} = \sqrt{144400} \approx 379.8 \, MPa \] Compare with yield stress $400 \, MPa$: \[ \sigma_{vm} = 379.8<400 \Rightarrow \text{Safe by von-Mises} \] Final Answer: \[ \boxed{\text{Unsafe by Tresca, Safe by von-Mises (Option C)}} \]