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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

θ = 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 × Polar Modulus

Þ T = δ_{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 × 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:

- Torque T will be same for both the shafts.
- The twists q
_{1}and q_{2}will be different for both the shafts.

Where, T = Torque G_{1}, G_{2} = Modulus of rigidity for shafts 1& 2 l1, l2 = length of shaft 1&2

if l_{1} = l_{2} G_{1} = G_{2} them

Where, q_{1}, q_{2} = angle of twist 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 ζ
_{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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