Question A2.9: A point is subjected to a tensile stress of 60 N/mm² and a c...
A point is subjected to a tensile stress of 60 N/mm² and a compressive stress of 40 N/mm² acting on two mutually perpendicular planes and a shear stress of 10 N/mm² on these planes. Determine the principal stresses as well as maximum shear stress. [KU, June 2008]
Given : \sigma_x=60 \mathrm{~N} / \mathrm{mm}^2, \sigma_y=-40 \mathrm{~N} / \mathrm{mm}^2, \tau_{x y}=10 \mathrm{~N} / \mathrm{mm}^2
Principal stresses are given by,
\begin{aligned}\sigma_{1,2} &=\frac{1}{2}\left(\sigma_x+\sigma_y\right) \pm \frac{1}{2} \sqrt{\left(\sigma_x-\sigma_y\right)^2+4 \tau_{x y}^2} \\&=\frac{1}{2}(60-40) \pm \frac{1}{2} \sqrt{(60+40)^2+4 \times 10^2} \\&=10 \pm 50.99 \\\sigma_1 &=60.99 \mathrm{~N} / \mathrm{mm}^2 \text { and } \sigma_2=-40.99 \mathrm{~N} / \mathrm{mm}^2\end{aligned}
Maximum shear stress =\frac{1}{2}\left(\sigma_1-\sigma_2\right)
=\frac{1}{2}(60.99+40.99)=50.99 \mathrm{~N} / \mathrm{mm}^2