Question

An element, as shown in the figure, is subjected to stresses \(\sigma_x = 500\ \text{N/m}^2\), \(\sigma_y = 300\ \text{N/m}^2\) and \(\tau = 120\ \text{N/m}^2\). If \(\sigma_1\) and \(\sigma_2\) are the principal stresses, then the absolute value of the angle \(\varphi\) is ________________ degree (rounded off to one decimal place).

Hint:

Principal plane angle is always computed using \(\tan(2\varphi)\), not \(\tan(\varphi)\).

Answer 1

Correct Answer: 24.5

The angle of principal planes is given by: \[ \tan(2\varphi) = \frac{2\tau}{\sigma_x - \sigma_y} \] Substituting values: \[ \tan(2\varphi) = \frac{2 \times 120}{500 - 300} = \frac{240}{200} = 1.2 \] \[ 2\varphi = \tan^{-1}(1.2) \approx 50.2^\circ \] \[ \varphi = \frac{50.2^\circ}{2} \approx 25.1^\circ \] Thus, the required principal angle lies between: \[ \boxed{24.5^\circ\ \text{to}\ 25.5^\circ} \]
Final Answer: 24.5–25.5 degree