# Torsional Shear Stress - Definition, Formula

By Aina Parasher|Updated : August 7th, 2022

Shear stress can be of two types: direct shear stress and torsional shear stress. Direct shear stress is produced due to the effect of direct shearing force. On the other hand, torsional shear stress is indirect shear stress produced inside the body subjected to torsion/twisting.

In an engineering context, the elements or members that are generally subjected to twisting have a circular cross-section, for example, shafts to transfer power from engines or motors, bolted connections, etc. For that reason in this article, we will discuss torsional shear stress acting on a circular cross-section.

## Torsional Shear Stress Definition

Before understanding torsional shear stress, we need to understand what causes it. As explained earlier, torsional shear stress is generated in a body that is subjected to a twisting moment. The figure below shows a circular bar subjected to MX, MY and MZ moments.

The moments MX and MY cause rotations about lateral axes; hence, they are bending moments. Whereas the moment MZ will cause rotation about the longitudinal axis (z-axis). Hence, MZ is a twisting moment. The twisting moment is also referred to as torque. Torque will tend to cause angular displacement in the body. Therefore, shear stress is generated in the body, referred to as torsional shear stress. Basically, torsional shear stress is the internal resistance of the body to resist twisting.

## Assumptions for Theory of Pure Torsion

When a circular shaft is subjected to torque, it will only generate torsional shear stress in the shaft (pure torsion). Our analysis makes certain assumptions to determine the torsional shear stress that will be produced. These assumptions are listed below -

• The material should be linearly elastic, isotropic and homogeneous.
• Hooke’s law is valid.
• The cross-section remains plane and circular before and after twisting.
• The shaft should be of uniform cross-section
• The shaft should be perfectly straight

## Torsional Shear Stress Equation

Consider a circular shaft fixed at one end and subjected to torque (T) on the other end. Due to this torque, shear stresses are produced in two mutually perpendicular planes: in the plane of the cross-section in the circumferential direction and normal to the cross-sectional plane in the longitudinal direction.

Due to the circular cross-section, torsional shear stress generated will be zero at the central axis and maximum at the outer surface as shown in the figure above. The torsional shear stressat a distance r from the centre of the cross-section is given by -

τ/r=τmax/R

where τmax = maximum torsional shear stress in the shaft

R = radius of the shaft

The torsional shear stress equation is given by-

T/IPmax/R=GΘ/L

where IP = polar moment of inertia

G = modulus of rigidity

θ = angle of twist in radian

L = length of the shaft

## Torsional Shear Stress in Different Circular Sections

The distribution of torsional shear stress will be different for a solid circular shaft, hollow circular shaft, and thin circular tube. The distribution and polar section modulus for each of these cases has been explained in this section.

### Solid Circular Shaft

Torsional stress distribution for a solid circular shaft subjected to a torque (T) is shown in the figure below -

IP=(π/32)D4

Polar section modulus,

ZP=IP/Rmax=π×D4/(32D/2)=πD3/16

Therefore, the maximum torsional shear stress will be

τmax=T/ZP=16T/πD3

### Hollow Circular Shaft

Torsional stress distribution for a hollow circular shaft subjected to a torque (T) is shown in the figure below -

IP=π/32(Do4-Di4)

Polar section modulus,

ZP=IP/Rmax=π(Do4-Di4)/32×Do/2=π(Do4-Di4)/16Do

Therefore, the maximum torsional shear stress can be calculated from -

τmax=T/ZP

### Thin Circular Tube With Mean Radius R

Torsional stress distribution for a thin circular tube of mean radius R and thickness t when subjected to a torque (τ) is shown in the figure below -

IP=2πR3t

Polar section modulus,

ZP=IP/Rmax=2πR3t/R=2πR2t

For a thin circular tube, the shear stress is assumed to be uniform across the whole thickness as shown in the figure above. Therefore, the torsional shear stress throughout the thickness will be

τ=T/ZP

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## FAQs on Torsional Shear Stress

• For a shaft of the circular cross-section of length L, radius R, subjected to torque T, torsional shear stress is given by the following equation -

T/IPmax/R=GΘ/L

where, IP = polar moment of inertia

G = modulus of rigidity

θ = angle of twist in radian

R = radius of the shaft

L = length of the shaft

• Torsional shear stress is indirect shear stress generated in a body subjected to a twisting moment (torque). Torsional shear stress is the body's internal resistance against angular deformation caused by twisting.

• Torsional rigidity of a shaft is defined as the minimum torque required to produce a twist of unit radian per unit length of the shaft. The torsional shear stress equation is given as:

T/IPmax/R=GΘ/L

From the above equation, torsional rigidity can be determined as,

Torsional rigidity = T/Θ/L=GIP

• The torsional shear stress in a solid shaft generated by the torque will be zero at the centroidal axis and it will be maximum at the outermost fibre. It is shown in the figure below. ### ESE & GATE CE

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