Question

Maximum shear stress in a solid shaft of diameter D and length L twisted through an angle θ is τ. A hollow shaft of the same material and length having outside and inside diameters of D and D/2  respectively is also twisted through the same angle of twist θ. The value of maximum shear stress in the hollow shaft will be

1. \(\frac{{16}}{{15}}\tau \)
2. \(\frac{{8}}{{7}}\tau \)
3. \(\frac{{4}}{{3}}\tau \)
4. \(\tau\)

Answer 1

Correct Answer - Option 4 : \(\tau\)

Concept:

\(\frac{T}{J} = \frac{τ }{R} = \frac{{G\;\theta }}{L}\)

where T is torque, J is the polar moment of inertia, τ is shear stress, R is the radius of the shaft, G is Modulus of rigidity, θ is angle of twist and L is length of shaft.

Calculation:

Given:

For solid shaft: diameter D, τ is maximum shear stress, θ is the angle of twist and L is the length of shaft.

For hollow shaft: outer diameter D and inner diameter D/2, τ is maximum shear stress, θ is the angle of twist and L is the length of the shaft.

\(\frac{T}{J} = \frac{τ }{R} = \frac{{G\theta }}{L}\)

\(\frac{τ }{R} = \frac{{G\theta }}{L}\)

For both the shaft G, θ, and L is the same.

so \(τ \propto R\)

\(τ \propto D/2\)

From the above equation, we can see that shear stress τ is dependent on outer diameter.

Since the outer diameter of the solid and hollow shaft are the same so shear stress is also the same.