Question

State of stress at a point of a loaded component is given by normal stresses \( \sigma_x = 30 \, \text{MPa} \), \( \sigma_y = 18 \, \text{MPa} \), and shear stress \( \tau_{xy} = 8 \, \text{MPa} \). What are the principal stresses?

• A. 38 MPa and 26 MPa
• B. 34 MPa and 14 MPa
• C. 24 MPa and 8 MPa
• D. 19 MPa and 13 MPa

Hint:

Use the principal stress formula and simplify the square root term to find the maximum and minimum stresses at a point.

Answer 1

The Correct Option is B

Step 1: Recall the formula for principal stresses.
The principal stresses \( \sigma_1 \) and \( \sigma_2 \) at a point are given by: \[ \sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}, \] where \( \sigma_x \) and \( \sigma_y \) are the normal stresses, and \( \tau_{xy} \) is the shear stress. Step 2: Substitute the given values.
Given: \[ \sigma_x = 30 \, \text{MPa}, \quad \sigma_y = 18 \, \text{MPa}, \quad \tau_{xy} = 8 \, \text{MPa}, \] \[ \frac{\sigma_x + \sigma_y}{2} = \frac{30 + 18}{2} = 24, \] \[ \frac{\sigma_x - \sigma_y}{2} = \frac{30 - 18}{2} = 6, \] \[ \left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2 = 6^2 + 8^2 = 36 + 64 = 100, \] \[ \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} = \sqrt{100} = 10. \] Step 3: Calculate the principal stresses.
\[ \sigma_1 = 24 + 10 = 34 \, \text{MPa}, \] \[ \sigma_2 = 24 - 10 = 14 \, \text{MPa}. \] Step 4: Verify the result.
The principal stresses are 34 MPa and 14 MPa, which match option (2). Step 5: Select the correct answer.
The principal stresses are 34 MPa and 14 MPa, matching option (2).