Torsion of Shafts
TORSION OF CIR CULAR SHAFTS
Theory of Pure Torsion
Torsional Moment of Resistance:
ts= 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:
Polar modulus =
Þ T = δs Zp
for hollow shaft,
Where, do = outer diameter
di = inner diameter
Where, G = rigidity modulus
Ip = Polar moment of Inertia
The quantity GIp 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 G1, G2 = Modulus of rigidity for shafts 1& 2 l1, l2 = length of shaft 1&2
if l1 = l2 G1 = G2 them
Where, q1, q2 = angle of twist Ip1, Ip2 = polar moments of inertia
(b) Shafts in parallel:
T = T1 + T2
The angle of twist will be same for each shaft,
q1 = q2=q
T = T1 + T2 =
COMPARISON BETWEEN SOLID AND HOLLOW SHAFTS
Let hollow shaft and solid shafts have same material and length.
D0 = external diameter of hollow shaft
Di = nD0 = 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.
Where, D = outer diameter of hollow shaft
d = internal diameter of hollow shaft