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

Consider the adjacent diagram.
Let the cross-sectional area of the bar be A. Consider the equilibrium of the plane aa'. A force F must be acting on this plane making an angle `(pi)/(2)-theta` with the normal ON. Resolving F into components, along the plane (FP) and normal to the plane.
`" "F_(P)=F cos theta`
`" "F_(N) = F sin theta`
Let the area of the face aa' be A', then
`" "(A)/(A')=sin theta`
`therefore" "A' = (A)/(sin theta)`
The tensile stress = `("Normal force")/("Area")=(F sin theta)/(A')`
`" "=(F cos theta)/(A//sin theta)=(F)/(A) sin theta. cos theta`
Shearing stress = `("Parallel force")/("Area")`
`" "=(F cos theta)/(A//Sin theta)=(F)/(A) sin theta. cos theta`
`=(F)/(2A) (2 sin theta. cos theta) = (F)/(2A) sin 2 theta`
(a) For tensile stress to be maximum, `sin^(2)theta=1`
`implies " "sin theta = 1`
`implies " "theta=(pi)/(2)`
(b) For shearing stress to be maximum,
`" "sin 2theta = 1`
`implies " "2theta=(pi)/(2)`
`implies" "theta=(pi)/(4)`
Note We must not apply the formula for stress directly, forces must be resolved.