Torsion of Shafts | Mechanical Engineering SSC JE (Technical) PDF Download

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 × 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 × 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
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 = angleof twi,st
Ip₁, Ip₂ = polar moments of inertia

(b) Shafts in parallel:

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

• 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
D_i = 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.

• 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