# Strength of Materials : Torsion Notes

By Deepanshu Rastogi|Updated : July 30th, 2021

## 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  Sign Convention

• Sign convention of torque can be explained by right hand thumb rule.
• 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 : Where, T = Torque,

J = Polar moment of inertia

G = Modulus of rigidity,

θ = Angle of twist

L = Length of shaft,

GJ = Torsional rigidity Torsional stiffness; Torsional flexibility Axial stiffness Axial flexibility

Moment of Inertia About polar Axis:

• For solid circular shaft,: • For hollow circular shaft: Power Transmitted in the Shaft

• Power transmitted by shaft: Where, N = Rotation per minute.

Compound Shaft

An improved type of compound coupling for connecting in series and parallel are given below

1. 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.  Therefore, Where,

θ1 = Angular deformation of 1st shaft

θ2 = Angular deformation of 2nd shaft

1. 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.  Therefore, Strain energy (U) stored in shaft due to torsion:

• G = Shear modulus
• T = Torque
• 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,  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  Where, τ = Torsion

D = Diameter of shaft

Combined effect of bending and torsion • Principal stress • Maximum shear stress • Equivalent bending moment • Equivalent torque Shear Stress Distribution:

• Solid Circulation Section: • Hollow Circulation Section • Composite Circular Section • Thin Tubular section: In view of small thickness-shear stress is assumed to be uniform

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