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**Chapter 8**

**TORSION OF SHAFTS**

TORSION OF CIR CULAR SHAFTS

Theory of Pure Torsion

Torsional Moment of Resistance:

t_{s}= Shear intensity at the surface of the shaft

R = Radius of shaft

G = Modulus of rigidity of shaft material

l = Length of shaft

q = Angular movement due to strain in length of the shaft

T = total moment of resistance offered by the cross-section of the shaft

I|_{p}= Polar moment of Inertia of the section of the shaft

Assumptions in the theory of pure torsion:

- The material of the shaft is uniform throughout.
- Twist along the shaft is uniform.
- Shaft is of uniform circular section throughout, which may be hollow or solid.
- Cross section of the shaft, which are plane before twist remain plane after twist.
- All radii which are straight before twist remain straight after twist:

Polar modulus:

Polar modulus =

The greatest twisting moment which a given shaft section can resist = Max. permissible shear stress Ã— Polar Modulus

T = t_{s} Z_{p}

- For solid shaft,

for hollow shaft,

Where, d_{o}= outer diameter

d_{i} = inner diameter**Torsional rigidity:**

Where, G = rigidity modulus

I_{p} = Polar moment of Inertia

The quantity GI_{p} is called torsional rigidity. It is the torque required to produce a twist of 1 radian per unit length of the shaft.

**Power Transmitted by a shaft:**

= Torque Ã— angle turned per second

Where, P = Power transmitted (kW)

N = rotation per minute (rpm)

T = mean torque (kNm)

**SHAFTS IN SERIES AND SHAFTS IN PARALLEL****(a) shafts in series:**

â€¢ Torque T will be same for both the shafts.

â€¢ The twists 1_{q} and 2_{q} will be different for both the shafts.

Where, T = Torque

G_{1}, G_{2} = Modulus of rigidity for shafts 1& 2

l_{1}, l_{2} = length of shaft 1&2

if l1 = l2 G1 = G2 them

Where, q_{1},q_{2} = angleof twi,st

Ip_{1}, Ip_{2} = polar moments of inertia

**(b) Shafts in parallel:**

- In this case applied torque T is distributed to two shafts.

- T = T
_{1}+ T_{2} - The angle of twist will be same for each shaft,

q_{1} = q_{2} = q

- T = T
_{1}+ T_{2}=

**COMPARISON BETWEEN SOLID AND HOLLOW SHAFTS**

Let hollow shaft and solid shafts have same material and length.

D_{0} = external diameter of hollow shaft

D_{i} = nD_{0 }= Internal diameter of hollow shaft

D = Diameter of the solid shaft

Case (i): When the hollow and solid shafts have the same torsional strength.

- In this case polar modulus section of two shafts would be equal.
- USE: %Saving in weight can be calculated for same torsional strength.

Case (ii): When the hollow and solid shafts are of equal weights.

- In this case torsional strength is compared.

- USE: ratio of strength for same weight can be calculated.

Case (iii) : When the diameter of solid shaft is equal to the external diameter of the hollow shaft.

**SHEAR AND TORSIONAL RESILIENCE****Shear resilience:**

Let t = shear stress intensity at faces of a square block

- Strain energy stored per unit volume

= (uniform through the section)

Where G = rigidity modulus.

**Torsional resilience:**

- In this case shear stress due to torsion varies uniformly form zero at the axis to the maximum value t
_{s}at the surface. - Strain energy stored, per unit volume

- for hollow shaft,

Where, D = outer diameter of hollow shaft

d = internal diameter of hollow shaft

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