Torsion of Shafts | Civil Engineering SSC JE (Technical) - Civil Engineering (CE) PDF Download

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 q₁ and q ₂ 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 = angle of twist 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 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