Sarthaks Test
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Consider a long steel bar under a tensile stress due to forces F acting at the edges along the length of the bar (Fig. 9.5). Consider a plane making an angle θ with the length. What are the tensile and shearing stresses on this plane?

(a) For what angle is the tensile stress a maximum?

(b) For what angle is the shearing stress a maximum?

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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 (π/2)-θ with the normal ON. Resolving F into components, along the plane and normal to the plane

Fp = F cosθ

FN = Sinθ

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

(A/A') = sinθ

A' = A/Sinθ

The tensile stress T = (Fsinθ/A') = (F/A)sin2θ and the shearing stress Z = (F cosθ)/A' = (F/A)cosθ sinθ = (F sin2θ)/2A. Maximum tensile stress is when θ = π/2 and maximum shearing stress when 2θ = π/2 or θ = π/4.

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