Tensile Stress and Shearing Stress on a Bar

When a bar of cross-sectional area A is subjected to two equal and opposite forces at its ends, the bar will experience both tensile stress and shearing stress. The ratio of tensile stress to shearing stress on a plane BB' inclined at an angle θ with the length of the bar is equal to the tangent of θ.

Tensile Stress
Tensile stress is the ratio of the applied force to the cross-sectional area of the material. It represents the force per unit area acting parallel to the length of the bar. Mathematically, tensile stress (σ) is given by:

σ = F/A

where σ is the tensile stress, F is the applied force, and A is the cross-sectional area of the bar.

Shearing Stress
Shearing stress is the ratio of the shearing force to the cross-sectional area of the material. It represents the force per unit area acting parallel to the cross-section of the bar. Mathematically, shearing stress (τ) is given by:

τ = F/A

where τ is the shearing stress, F is the shearing force, and A is the cross-sectional area of the bar.

Analysis of Plane BB'
In the given scenario, the plane BB' is inclined at an angle θ with the length of the bar. This means that the applied forces are not parallel to the cross-section of the bar, but are at an angle θ to the plane BB'. To analyze the stresses on this plane, we need to resolve the forces into their components parallel and perpendicular to the plane.

Tensile Stress on Plane BB'
The component of the applied force parallel to the plane BB' will cause tensile stress on the plane. Let's denote this component as F'.

F' = F * cos(θ)

The tensile stress (σ') on plane BB' can be calculated using the formula for tensile stress:

σ' = F'/A = (F * cos(θ))/A = F * (cos(θ)/A)

Shearing Stress on Plane BB'
The component of the applied force perpendicular to the plane BB' will cause shearing stress on the plane. Let's denote this component as F''.

F'' = F * sin(θ)

The shearing stress (τ') on plane BB' can be calculated using the formula for shearing stress:

τ' = F''/A = (F * sin(θ))/A = F * (sin(θ)/A)

Ratio of Tensile Stress to Shearing Stress
To find the ratio of tensile stress to shearing stress on plane BB', we divide the tensile stress (σ') by the shearing stress (τ'):

(σ'/τ') = (F * (cos(θ)/A))/(F * (sin(θ)/A)) = (cos(θ)/sin(θ))

Using the trigonometric identity tan(θ) = sin(θ)/cos(θ), we can rewrite the ratio as:

(σ'/τ') = (cos(θ)/sin(θ)) = tan(θ)

Therefore, the ratio of tensile stress to shearing stress on plane BB' is equal to the tangent of the angle θ.