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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 crosssection of the shaft
Ip= Polar moment of Inertia of the section of the shaft
Assumptions in the theory of pure torsion:
Polar modulus =
Þ T = δ_{s} Z_{p}
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:
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:
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.
Case (ii): When the hollow and solid shafts are of equal weights.
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
(uniform through the section)
Where G = rigidity modulus.
Torsional resilience:
Where, D = outer diameter of hollow shaft
d = internal diameter of hollow shaft
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