Thursday, 28 September 2017

DERIVATION OF STRAIN ENERGY DUE TO TORSION

DERIVATION OF STRAIN ENERGY DUE TO TORSION

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 r2/R2 x τ2 x 2П x r x dr x L
Ur = (1/2C) x r2/R2 x τ2 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 D4
J = (П/32) x 16 R4
J = (П/2) x R4

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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