Consider a long steel bar under a tensile stress due to forces F, acting at the edges along the length of the bar

Consider a plane making an angle `theta` with the length. What are the tensile and shering stresses on this plane ? (a) For what angle is the tensile stress a maximum ? (b) For what angle is the shearing stress a maximum ?

Explanation

Let A be the area of cross -section of the bar. Consider the equilibrim of the plaen PR. A force F must be acting on this plane making an angle `(90^@ - theta)` with the normal ON. Refer

Resolving `vecF` into two rectangular components, we have Component of force F normal to plane, `F_N = F cos (90 - theta) = F sin theta`
Compont of force F along the plane, `F_P = F cos theta`
Let A' be the area of face PR, then `(A)/(A') sin theta A' =(A)/(sin theta)`
`:.` Tensile stress `= (Fsin theta)/(A') = (F sin theta)/((A//sin theta)) = (F)/(A) sin^2 theta` .... (i)
Shearing stress ` =(F cos theta)/(A') = (F cos theta)/((A//sin theta)) = (F)/(2 A) sin 2 theta` ..... (ii)
(a) Tensile stress will be maximum if `sin theta = 1` in equation (i) or `theta = 90^@` (b) Shering stress will be maximum if `sin 2 theta = 1` in equation (ii) or `2 theta = 90^@` or `theta = 45^@.`