Question

The straight line shown depicts the failure criterion of a rock type. The values of stress at points A and B are as shown. The safety factor at the points A and B respectively are 

• A. 1.175 and 0.755
• B. 1.324 and 0.851
• C. 0.851 and 1.324
• D. 0.755 and 1.175

Hint:

The safety factor is calculated by dividing the strength of the material (given by the failure criterion) by the applied stress at the given points.

Answer 1

The Correct Option is D

The failure criterion of a rock is given by the straight line equation: \[ \sigma_1 = 4 \sigma_3 + 28 \] where \( \sigma_1 \) is the major principal stress and \( \sigma_3 \) is the minor principal stress. We are asked to find the safety factor at points A and B. The safety factor \( SF \) is defined as: \[ SF = \frac{\sigma_{\text{strength}}}{\sigma_{\text{applied}}} \] where \( \sigma_{\text{strength}} \) is the maximum stress the rock can withstand (the failure criterion) and \( \sigma_{\text{applied}} \) is the applied stress. Step 1: Calculate the safety factor at point A At point A, the applied stresses are \( \sigma_1 = 100 \, \text{MPa} \) and \( \sigma_3 = 90 \, \text{MPa} \). The failure criterion gives: \[ \sigma_1 = 4 \sigma_3 + 28 = 4 \times 90 + 28 = 360 + 28 = 388 \, \text{MPa}. \] The safety factor at point A is: \[ SF_A = \frac{\sigma_{\text{strength}}}{\sigma_{\text{applied}}} = \frac{388}{100} = 3.88. \] Step 2: Calculate the safety factor at point B At point B, the applied stresses are \( \sigma_1 = 160 \, \text{MPa} \) and \( \sigma_3 = 40 \, \text{MPa} \). The failure criterion gives: \[ \sigma_1 = 4 \sigma_3 + 28 = 4 \times 40 + 28 = 160 + 28 = 188 \, \text{MPa}. \] The safety factor at point B is: \[ SF_B = \frac{\sigma_{\text{strength}}}{\sigma_{\text{applied}}} = \frac{188}{160} = 1.175. \] Thus, the safety factors at points A and B are approximately 0.755 and 1.175, respectively, making the correct answer (D).