Torsion of Circular Shafts | Solid Mechanics - Mechanical Engineering PDF Download

Uniform Torsion

1. Torsion of Shaft and Combined Stresses
   Torsion means twisting a structural Member when it is loaded by a couple that Produces rotation about the longitudinal axis.
   If  τ be the intensity of shear stress, on any layer at a distance r from the centre of shaft, then
   τ_l / r = T / J = Gθ / l
   
2. Sign Convention
   (i) Sign convention of torque can be explained by right hand thumb rule.
   (ii) A positive torque is that in which there is tightening effect of nut on the bolt. From either side of the cross-section. If torque is applied in the direction of right hand fingers than right hand thumbs direction represents movement of the nut.
   TMD = Torsion moment diagram
   T = Torque
   Total angle of twist :
   θ = Tl / GJ
   Where, T = Torque,
   J = Polar moment of inertia
   G = Modulus of rigidity,
   θ = Angle of twist
   L = Length of shaft,
   GJ = Torsional rigidity
   GJ / l → Torsional stiffness;
   l / GJ → Torsional flexibility
   EA / l → Axial stiffness
   l / EA → Axial flexibility
3. Moment of Inertia About polar Axis
   (i) For solid circular shaft:
    
   (ii) For hollow circular shaft:
   
4. Power Transmitted in the Shaft
   (i) Power transmitted by shaft:
   P = (2πNT / 60000)kW
   Where, N = Rotation per minute.
5. Compound Shaft
   An improved type of compound coupling for connecting in series and parallel are given below
   (i) Series connection: Series connection of compound shaft as shown in figure. Due to series connection the torque on shaft 1 will be equal to shaft 2 and the total angular deformation will be equal to the sum of deformation of 1^st shaft and 2^nd shaft.
   θ = θ₁ + θ₂
   T = T₁ + T₂
   Therefore,
   θ = TL₁ / G₁J₁ + TL₂ / G₂J₂ 
   Where,
   θ₁ = Angular deformation of 1^st shaft
   θ₂ = Angular deformation of 2^nd shaft
   (ii) Parallel connection: Parallel connection of compound shaft as shown in figure. Due to parallel connection of compound shaft the total torque will be equal to the sum of torque of shaft 1 and torque of shaft 2 and the deflection will be same in both the shafts.
    θ₁ = θ₂
   T = T₁ + T₂
   Therefore,
   T₁L / G₁J₁ = T₂L / G₂J₂
6. Strain energy (U) stored in shaft due to torsion
    
   (i) G = Shear modulus
   (ii) T = Torque
   (iii) J = Moment of inertia about polar axis
7. Effect of Pure Bending on Shaft
   The effect of pure bending on shaft can be defined by the relation for the shaft,
    
   σ = 32M / πD³
   Where, σ = Principal stress
   D = Diameter of shaft
   M = Bending moment
8. Effect of Pure Torsion on Shaft
   It can be calculated by the formula, which are given below
   τ_max = 16T/πD³
   Where, τ = Torsion
   D = Diameter of shaft
9. Combined effect of bending and torsion
   (i) Principal stress
   
   (ii) Maximum shear stress
   
   (iii) Equivalent bending moment
   
   (iv) Equivalent torque
   
10. Shear Stress Distribution
    (i) Solid Circulation Section:
    (ii) Hollow Circulation Section
    (iii) Composite Circular Section
    (iv) Thin Tubular section: In view of small thickness-shear stress is assumed to be uniform.