Question

Consider a long steel bar under a tensile stress due to forces F acting at the edges along the length of the bar (Figure). Consider a plane making an angle θ with the length. What are the tensile and shearing stresses on this plane

1. For what angle is the tensile stress a maximum?
2. For what angle is the shearing stress a maximum?

Answer 1

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 θ`

`F_N = F sin θ`

Let the area of the face aa' be A, then

`A/A^' = sin θ`

∴ `A^' = A/sin θ`

The tensile stress = `"Normal force"/"Area" = (F sin θ)/A^'`

= `(F sin θ)/(A/sin θ)`

= `F/A sin^2 θ`

Shearing stress = `"Parallel force"/"Area"`

= `(F sin θ)/(A/sin θ)`

= `F/A sin θ. cos θ`

= `F/(2A) (2 sin θ. cos θ)`

= `F/(2A) sin 2θ`

a. For tensile stress to be maximum, `sin^2theta` = 1

⇒ `sin θ` = 1

⇒ θ = `pi/2`

b. For shearing stress to be maximum,

sin 2θ = 1

⇒ 2θ = `pi/2`

⇒ θ = `pi/4`