TORSION OF SHAFTS

TORSION OF SHAFTS | THEORY OF PURE TORSION | ASSUMPTION | DERIVATION OF TORSION EQUATION | POLAR MODULUS | TORSIONAL RIGIDITY | POWER TRANSMITTED BY SHAFT | HOLLOW SHAFT | SOLID SHAFT |

TORSION OF SHAFTS

When a shaft is fixed at one end and is subjected to a torque at the other end, then every cross section of the shaft will be subjected to share stresses.

THEORY OF PURE TORSION: -

Fig. 1 | Theory of Torsion

f_s=  G Φ [ G = Share stress/ Share strain = f_s/ Φ]

f_s = (BC/ l) G [∵ Φ = BC/l]

f_s/R = G θ/l [∵BC = Rθ]

If share stress intensity ‘q’ on an elements distance from centre is ‘r’ then –

f_s/R = q/r = G θ/l

TORSIONAL MOMENT OF RESISTANCE: -

T/I_P= f_s/R = G θ/l

Where,

•    T = Total moment of resistance offered by the cross section of the shaft.
•    I_P = Polar Moment Inertia.
•    f_s = Shear Stress
•    R = Radius of the cross section of the shaft.
•    G = Modulus Of Rigidity
•    θ = Angle of twist of the shaft.
•    l = Length of the shaft.

ASSUMPTION: -

1.   The material of the shaft is uniform throughout the length.

2.   The twist along the shaft is uniform.

3.   The shaft is uniform circular section throughout.

4.   All radiuses which are straight before twist remain straight after twist.

5.   Cross sections of the shaft which are straight before twist remain straight after twist.

PURE TORSION: -

If a torque is applied on a shaft causes to induce only torsional stress, then the shaft is said to be in Pure Torsion.

POLAR MODULUS: -

T = f_s (I_P /R)

T = f_S. Z_P [Z_P = I_P /R]

The ratio of Polar Moment of Inertia of the shaft section to the maximum radius of the shaft is called is called the Polar Modulus of the section.

The greatest twisting moment which a given shaft section can resist is equal to the multiplication of the maximum permissible share stress and the Polar Modulus.

T = f_S. Z_P

Maximum permissible shear stress is inversely proportional to the Polar modulus of the section.

f_S ∝ (1/Z_P)

The magnitude of the polar modulus is a measure of its strength in resisting torsion.

For same length and material the shaft which has greatest polar modulus can resist greatest twisting moment.

 

1.   POLAR MODULUS FOR A SOLID SHAFT: -

I_P = 𝝅 D⁴ /32

R = D/2

Z_P= 𝝅 D³ /16

Moment of resistance = f_{S .} (𝝅 D³ /16)

2.   POLAR MODULUS FOR A HOLLOW SHAFT: -

I_P = 𝝅 (D₁⁴ – D₂⁴) /32

R = D₁/2

Z_P= {𝝅 (D₁⁴ – D₂⁴)} /(16 D₁)

 

Moment of resistance = f_{S .} [{𝝅 (D₁⁴– D₂⁴)} /(16 D₁)]

TORSIONAL RIGIDITY: -

T/I_P= G θ/l

G. I_P = T.l /θ

When l =1 & θ = 1 radian,

Then, G. I_P = T [or C. I_P = T]

It is defined as the torque required to produce a twist of one radian per unit length of the shaft.

POWER TRANSMITTED BY A SHAFT:-

P = 2𝝅NT/60 = T. ω

THE STRENGTH OF SHAFT / MAX. TORQUE (OR POWER) TRANSMITTED BY A SHAFT: -

T = f_s (I_P /R)

1.   FOR A SOLID SHAFT: -

T = (𝝅f_{S .} D³/16)

 

2.   FOR A HOLLOW SHAFT: -

T = [𝝅 f_S (D₁⁴– D₂⁴)] /(16 D₁)