At a point in a stressed body the state of stress on two planes $$45$$⁰ apart is as shown below. Determine the two principal stresses

The three - dimensional state of stress at a point is given by: $$$\left[ \sigma \right] = \left[ {\matrix{ {30} & {10} & { - 10} \cr {10} & 0 & {20} \cr { - 10} & {20} & 0 \cr } } \right]MN/{m^2}$$$

The shear stress in the $$x-y$$ plane at the same point is then equal to