GATE ME 2023 Set 1 — Question 26

Given principal stresses: $\sigma_1 = 70$ MPa, $\sigma_2 = -70$ MPa, $\sigma_3 = 0$ MPa.
Yield stress of the material: $S_y = 100$ MPa.We need to check the predictions of material failure using two common theories:1. Maximum Normal Stress Theory (Rankine's Theory):
According to this theory, failure occurs if the maximum principal stress (in magnitude) exceeds the yield strength of the material.$|\sigma_1| = |70| = 70$ MPa
$|\sigma_2| = |-70| = 70$ MPa
$|\sigma_3| = |0| = 0$ MPaThe maximum absolute principal stress is $70$ MPa.
Since $70 \text{ MPa} < S_y = 100 \text{ MPa}$, the material does not fail according to the Maximum Normal Stress Theory.Therefore, option (A) is incorrect, and option (C) is correct.2. Maximum Shear Stress Theory (Tresca's Theory):
According to this theory, failure occurs if the maximum shear stress in the material exceeds half the yield strength in uniaxial tension (which is the yield shear strength, $S_{ys} = S_y/2$).The maximum shear stress $\tau_{max}$ is given by:
$\tau_{max} = \max\left(\frac{|\sigma_1 - \sigma_2|}{2}, \frac{|\sigma_2 - \sigma_3|}{2}, \frac{|\sigma_3 - \sigma_1|}{2}\right)$Let's calculate the three possible shear stresses:
$\frac{|\sigma_1 - \sigma_2|}{2} = \frac{|70 - (-70)|}{2} = \frac{|140|}{2} = 70$ MPa
$\frac{|\sigma_2 - \sigma_3|}{2} = \frac{|-70 - 0|}{2} = \frac{|-70|}{2} = 35$ MPa
$\frac{|\sigma_3 - \sigma_1|}{2} = \frac{|0 - 70|}{2} = \frac{|-70|}{2} = 35$ MPaSo, $\tau_{max} = 70$ MPa.Now, calculate the yield shear strength:
$S_{ys} = \frac{S_y}{2} = \frac{100}{2} = 50$ MPa.Since $\tau_{max} = 70 \text{ MPa} > S_{ys} = 50 \text{ MPa}$, the material fails according to the Maximum Shear Stress Theory.Therefore, option (B) is correct, and option (D) is incorrect.Combining the results, the correct predictions are that the material fails by Maximum Shear Stress Theory and does not fail by Maximum Normal Stress Theory.The final answer is $\boxed{B, C}$