Question

The three-dimensional state of stress at a point is given by \[ \sigma = \begin{pmatrix} 10 & 0 & 0
0 & 40 & 0
0 & 0 & 0 \end{pmatrix} \text{ MPa.} \] The maximum shear stress at the point is

• A. 20 MPa
• B. 15 MPa
• C. 5 MPa
• D. 25 MPa

Hint:

For principal stress calculations in a stress matrix, the diagonal elements represent normal stresses, and the maximum shear stress is determined by taking half of the difference between the maximum and minimum principal stresses.

Answer 1

The Correct Option is A

Step 1: The principal stresses are the diagonal elements of the stress matrix: \[ \sigma_1 = 40 \text{ MPa}, \quad \sigma_2 = 10 \text{ MPa}, \quad \sigma_3 = 0 \text{ MPa} \] Step 2: The formula to calculate the maximum shear stress is: \[ \text{Maximum Shear Stress} = \frac{\sigma_{\max} - \sigma_{\min}}{2} \] Substituting the values: \[ \text{Maximum Shear Stress} = \frac{40 - 0}{2} = 20 \text{ MPa} \] Conclusion: The maximum shear stress at the point is \( \mathbf{20} \) MPa, which corresponds to option (A).