We were discussing the concept of polar modulus of section, power transmitted by a circular hollow shaft and also theories of failure in machine design in our previous posts.

Today we will see here the expression for strain
energy stored in a body subjected to torsion with the help of this post. Let us go ahead step by step for easy understanding,
however if there is any issue we can discuss it in comment box which is
provided below this post.

As we are interested here to find out
the expression
for strain energy stored in a body subjected to torsion, we must have to recall
here first the basic concept of strain energy and then only we will be able to
determine the strain energy stored in a body subjected to torsion.

###
**So,
what is strain energy?**

When a body will be loaded then there
will be deformation in the body and due to this deformation, energy will be
stored in the body and that energy will be termed as strain energy.

*Let us see the basic definition of the strain energy*

Strain energy is basically defined as
the internal energy stored in the body when body will be subjected with a load
within its elastic limit.

We must have to be ensuring that load
applied over the body must be within its elastic limit i.e. after removal of
load; body must secure its original dimensions.

###
**Strain energy stored in a body due to torsion**

Let us consider a solid circular shaft
as displayed here in following figure.

We have following information from above
figure.

L = Length of solid circular shaft

D= Diameter of solid circular shaft

R= Radius of solid circular shaft

τ = Shear stress acting at outer surface
of shaft i.e. at radius R

q = Shear stress acting at a distance r
from shaft center

C = Modulus of rigidity

U = Strain energy stored in the shaft
due to torsion

Let us consider one elementary ring of
thickness dr at a radius r or at a distance r from center of shaft.

Area of elementary ring = 2П x r x dr

Volume of elementary ring = 2П x r x dr
x L

Let us think and write the equation for
shear stress (q) at a distance r from the shaft center and we will have
following equation

q/r = τ/R

q = (r/R) x τ

Let us recall our post based on strain energy stored in a body due to shear stress and we will have following equation

Strain energy stored in the elementary
ring = (1/2C) x (Shear stress)

^{ 2}x Volume
Ur = (1/2C) x r

^{2}/R^{2}x τ^{2}x 2П x r x dr x L
Ur = (1/2C) x r

^{2}/R^{2}x τ^{2}x 2П x r x dr x L
Total shear strain energy stored in the
shaft will be determined by integrating the above equation from 0 to R.

Where J is the polar moment of inertia and
we can secure the detailed information about the concept of polar moment of
inertia by visiting the respective post i.e. Polar moment of inertia for various sections.

Polar moment of inertia for solid
circular shaft

J = (П/32) x D

^{4}
J = (П/32) x 16 R

^{4}
J = (П/2) x R

^{4}
Let us use the above value of polar
moment of inertia in equation of strain energy stored in the shaft due to torsion
and we will have following expression for strain energy stored in the shaft due
to torsion.

Do you have suggestions? Please write in comment
box.

We will now derive the torsional equation, in the
category of strength of material, in our next post.

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

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