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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 crosssection of the shaft
I_{p}= Polar moment of Inertia of the section of the shaft
Assumptions in the theory of pure torsion:
Polar modulus:
Polar modulus =
The greatest twisting moment which a given shaft section can resist = Max. permissible shear stress × Polar Modulus
T = t_{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:
• Torque T will be same for both the shafts.
• 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:
q_{1} = q_{2} = q
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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