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**Uniform Torsion**

**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**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**Moment of Inertia About polar Axis****(i) For solid circular shaft:**

**(ii) For hollow circular shaft:****Power Transmitted in the Shaft****(i)****Power transmitted by shaft:**

P = (2πNT / 60000)kW

Where, N = Rotation per minute.**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 = θ_{1}+ θ_{2}

T = T_{1 }+ T_{2}

Therefore,

θ = TL_{1}/ G_{1}J_{1}+ TL_{2}/ G_{2}J_{2}

Where,

θ_{1 }= Angular deformation of 1^{st}shaft

θ_{2}= 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.

θ_{1 }= θ_{2}

T = T_{1}+ T_{2}

Therefore,

T_{1}L / G_{1}J_{1}= T_{2}L / G_{2}J_{2}**Strain energy (U) stored in shaft due to torsion**

**(i)**G = Shear modulus**(ii)**T = Torque**(iii)**J = Moment of inertia about polar axis**Effect of Pure Bending on Shaft**

The effect of pure bending on shaft can be defined by the relation for the shaft,

σ = 32M / πD^{3}

Where, σ = Principal stress

D = Diameter of shaft

M = Bending moment**Effect of Pure Torsion on Shaft**

It can be calculated by the formula, which are given below

τ_{max}= 16T/πD^{3}

Where, τ = Torsion

D = Diameter of shaft**Combined effect of bending and torsion**

**(i) Principal stress****(ii) Maximum shear stress****(iii) Equivalent bending moment****(iv) Equivalent torque****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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