Question

The infinitesimal element shown in the figure (not to scale) represents the state of stress at a point in a body. What is the magnitude of the maximum principal stress (in N/mm², in integer) at the point? 

Hint:

To calculate the principal stresses, use the formula for \( \sigma_1 \) and \( \sigma_2 \). The maximum principal stress is the higher value of the two.

Answer 1

Given the stress components from the figure: - \( \sigma_x = 6 \, \text{N/mm}^2 \) - \( \sigma_y = 3 \, \text{N/mm}^2 \) - \( \tau_{xy} = 4 \, \text{N/mm}^2 \) The principal stresses are calculated using the following formula: \[ \sigma_1, \sigma_2 = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2} \] Substitute the given values into the formula: \[ \sigma_1 = \frac{6 + 3}{2} + \sqrt{\left( \frac{6 - 3}{2} \right)^2 + 4^2} \] \[ \sigma_1 = 4.5 + \sqrt{(1.5)^2 + 16} \] \[ \sigma_1 = 4.5 + \sqrt{2.25 + 16} \] \[ \sigma_1 = 4.5 + \sqrt{18.25} \] \[ \sigma_1 = 4.5 + 4.27 = 8.77 \, \text{N/mm}^2 \] Thus, the magnitude of the maximum principal stress is approximately 7 N/mm² when rounded to the nearest integer. \[ \boxed{\text{The maximum principal stress is } 7 \, \text{N/mm}^2.} \]