Torsion of Shafts

By Aina Parasher|Updated : June 17th, 2022

Torsion of Shafts: When a torque or twisting moment is applied on a shaft, shear stress is developed in the shaft. Torsion of shafts is twisting of the shaft about its longitudinal axis. Torsion of shafts is a type of deformation in shafts. The moment which leads to a twisting moment is known as Torque.

In this article, we will try to see the Torsion of shafts, Torsional Shear Stress, Strength of these shafts, their design, and the power transmitted by these shafts. Torsion and Torque have many practical uses in real life.

Table of Content

What is Torsion of Shafts?

To get a better understanding of the Torsion of Shafts, let us suppose, a force ‘P’ which is acting at a distance ‘e’ from the centre generates a moment T, which acts along the longitudinal axis. The applied torque causes the bar to deform by twisting. In the case of an axis-symmetric section, the body will rotate relative to each other when the torque is applied but the body will not deform. A bar that is not axis-symmetric when subjected to torsion results in warping of bar cross-section.

Torsional Shear Stress 

It is the shear stress offered by a body to twisting moment or torque. In Torsion of Shafts, it is the shear stress resisted by the shafts when they are subjected to twisting moment or torque. Suppose a shaft of radius ‘r’ is subjected to a Torsional Moment ‘T’. 

Torsional Shear Stress will be given by:

τ/r= T/J

Where,

τ = Shear Stress

r = Radial Distance from Centre of Shaft

T = Torsional Moment

J = Polar Moment of Inertia

Problems on the torsion of shafts can be solved using the above formula.

τ= Tr/J

Shear Stress in a Shaft due to torsion moment is having a linear distribution. 

Shear Stress due to torsion on the shaft is maximum at extreme fiber where r=D/2.

Shear Stress due to torsion on the shaft is zero at the centre.

Maximum Shear Stress is given by τmax

τmax= Trmax/J

where rmax=D/2

τmax= T/ZP

Where ZP = Polar Section Modulus

ZP= J/rmax

ZP= πD3/16

τmax= 16T/πD3

Problems on torsion of shafts can be solved using the above formula.

Shear Stress is developed due to shear force (V) as well as torsion or Torque (T). Shear stress distribution due to shear force is parabolic with zero shear stress at extreme fiber and maximum shear stress at centre. Shear Stress due to shear force is given by 

τ= Vay ̅/Ib

Shear Stress Distribution due to torsion is linear with maximum shear stress at extreme fiber and zero at the centre.

τ= Tr/J

τmax for a hollow circular section is given by

τmax= T/ZP

ZP= D3(1-K4)/16

Where, k = d/D

d = Inner Diameter

D = Outer Diameter

After applying torque, the free end of the section rotates and changes position from B to B’. This creates an angle θ, which is known as angle of twist. The angle of twist varies linearly from 0 to π at the free end of the bar.

Θ= TL/GJ

Where, L = Length of the bar

T = Torsion/Torque applied

G = Shear Modulus

J = Polar Moment of Inertia

GJ – Torsional Rigidity

Strength of Shafts

Strength of Shafts is the torsional moment carrying capacity or torque carrying capacity of a shaft. We know that,

τmax= T/ZP

T= τmax/ZP

T- Torsional moment carrying capacity / Torsional strength

τmax≲ τP

Where τp is the permissible shear stress.

Strength of Solid Shaft

Ts= πds3τp/16

ds – Diameter of Solid Shaft

Torsion of Shafts 

d ≥ (16T/πZP)1/3

(For Solid Shafts)

d ≥ (16T/π(1-k4)ZP)1/3

(For Hollow Shafts)

d ≥ 32TL/πP

Where ΘP is the permissible angle of twist.

Power Transmitted by Shaft

P=Tw=2πNT/60

w=2πf

f= N/60

Where, f = Rotations per Second

N = Rotations per minute

Problems on Torsion of Shafts 

1. Find the maximum shear stress in a shaft subjected to a torque of 2kN-m having a diameter of 50mm.

Solution-

Given, T = 2kN-m, D = 50mm

τmax= 16T/πD3

τmax= 16×2×106503=81.48 MPa

2. Find the angle of twist in a shaft subject to a torque of 40kN-m and having a diameter of 200mm and length 4m.

Solution-

Given, G = 80GPa = 80 x 103 MPa

T = 40 kN-m

D = 200mm

L = 4m

θ= TL/GJ = [40 × 106 ×4000] / [80 × 103 × π/32 × 2004]=0.0127 radians or 0.73º

3. A circular shaft of 60mm diameter is requested to transmit torque from one shaft to another. Find the torque carrying capacity of the shaft if permissible shear stress is 40MPa. 

Solution-

Given, ds = 60mm, τp = 40MPa

Ts = τpZp

     = τp x π/16 x d3

     = 40 x π/16 x 603

     = 1.69 kN/m

4. Find the power transmitted by a circular shaft of diameter 100mm, number of rotations 150rpm and maximum shear stress 60MPa.

Solution-

Given, D = 100mm, N = 150rpm

τmax= 16T/πD3

60= 16 × T ×1000/ π×1003

T = 11780.97 N-m

P=2πNT/60= 2 ×150 ×11780.97/60000

P = 185 KW

5. Find the diameter of a shaft having a number of revolutions of 150 rpm. Maximum Shearing Stress is 80 N/mm2 and Power transmitted by the shaft is 400KW.

Solution-

P=2πNT/60

400=2π ×150 ×T/60000

T = 25464.8 N-m

τmax= 16T/πD3

d= (16 ×25464.8 ×1000/π ×80)1/3=117.5mm

Comments

write a comment

FAQs on Torsion of Shafts

  • When a shaft is subjected to two equal and opposite couples in a plane perpendicular to the longitudinal axis of the shaft (i.e. twisting couples), the amplitude of the twisting moment remains constant along the length of the shaft, it is said to be pure torsion of shafts.

  • We know that, 

    Ts= πds3τp/16

    where, 

    Ts - Strength of a Solid Shaft

    ds- Diameter of Solid Shaft

    τp-Permissible Shear Stress

  • After applying torque, the radial section of the object deviates from its original position. This creates an angle which is known as the angle of twist. The angle of twist varies linearly from 0 to π.

  • J is the Polar moment of inertia. The polar moment of inertia tells us about the resistance of the shaft with respect to the torsional shear deformation when torque is applied. The unit of polar moment of Inertia is m4.

  • Torsional Rigidity is the torque required for unit twist over unit length. 

    Torsional Rigidity = GJ

    Where, G - Shear Modulus

    J - Polar Moment of Inertia

  • Torsional Stiffness is the torsion required for unit twist over length.

    Torsional Stiffness = GJ/L

    Where,

    L- Length of bar

    J- Polar moment of inertia

    G- Shear Modulus

Follow us for latest updates