Torsion of Circular Shafts Notes | EduRev

Solid Mechanics

Civil Engineering (CE) : Torsion of Circular Shafts Notes | EduRev

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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/ 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 1st shaft and 2nd shaft.
    θ = θ1 + θ2
    T = T+ T2
    Therefore,
    θ = TL1 / G1J1 + TL2 / G2J2 
    Where,
    θ= Angular deformation of 1st shaft
    θ2 = Angular deformation of 2nd 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.
     θ= 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
    τmax = 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.
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