Torsional shear stress

Torsional shear stress definition:

Torsional shear stress is the shear stress offered by the body against torsional load or twisting load. it is denoted by the symbol β€˜πœβ€™.

The value of torsional shear stress varies within the cross-section of the object. The value for shear stress is minimum at the neutral axis of the cross-section while it is maximum at the outermost surface of the cross-section of the object.

The units of torsional shear stress are N/mΒ² in the SI system while lb/ftΒ² in the FPS system.

Torsional shear stress equation:

Torsional shear stress can be found by using the torsional equation.

Therefore,

`\tau= \frac{T}{J}\times r`

Where,
T = Applied torque
J = Polar moment of inertia
r = Distance between neutral axis and the point where shear stress is to be calculated

Torsional shear stress formula for circular shaft:

A] For solid shaft:-

Torsional shear stress for circular solid shaft

The above diagram shows the torsional shear stress distribution in a hollow circular shaft. In that figure, the value for 𝜏 is minimum at the neutral axis while it is maximum at r = d/2

For the solid circular shaft, the shear stress at any point in the shaft is given by,

`\tau= \frac{T}{J}\times r`

But for solid shaft,

`J= \frac{\pi }{32}\times d^{4}`

Therefore the torsional shear stress for the circular shaft is given by,

`\tau= \frac{32T}{\pi\times d^{4} }\times r`

B] For hollow shaft:-

Torsional shear stress for circular hollow shaft

The above diagram shows the torsional shear stress distribution in a hollow circular shaft. In that figure, the value for 𝜏 is minimum at r = di/2 while it is maximum at r = do/2

For hollow circular shaft with outer diameter (do) and inner diameter (di), the polar moment of inertia is given by,

`J= (\frac{\pi }{32})\times [do^{4}-di^{4}]`

Therefore the torsional shear stress for hollow shaft is given by,

`\tau= \frac{32T}{\pi\times (do^{4}-di^{4}) }\times r`

Maximum torsional shear stress for circular shaft:

A] For solid shaft:

The maximum shear stress in solid circular shaft is observed at the outermost surface where r = d/2

`\tau_{max}= \frac{32T}{\pi\times d^{4} }\times \frac{d}{2}`

`\tau_{max}= \frac{16T}{\pi\times d^{3} }`

B] For hollow shaft:

The maximum shear stress in hollow circular shaft is observed at outer diameter (do).
∴ r = do/2

`\tau_{max}= \frac{32T}{\pi\times (do^{4}-di^{4}) }\times\frac{do}{2}`

`\tau_{max}= \frac{16T\times do}{\pi\times (do^{4}-di^{4}) }`


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