Sunday, 16 April 2017

STRAIN ENERGY STORED IN A BODY DUE TO SUDDENLY APPLIED LOAD

STRAIN ENERGY STORED IN A BODY DUE TO SUDDENLY APPLIED LOAD

We were discussing Strain energy, Resilience, Proof resilience and Modulus of resilience in our recent post and also we have discussed strain energy stored in a body when load will be applied gradually during our previous posts.

Today we will discuss strain energy stored in a body when load will be applied suddenly with the help of this post. 

 
Earlier we were discussing a body which is subjected with tensile load which is increasing gradually up to its elastic limit from value 0 to value P, but in this case we have considered that load is applied suddenly over the body and therefore sudden applied load P will be constant throughout the deformation process of the body.
Let us see the load extension diagram as displayed here for this case where body will be subjected with sudden load and we will find out here the stress induced in the body due to sudden applied load and simultaneously we will also secure the expression for strain energy for this situation.
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.

We have following information from above load extension diagram for body which is subjected with sudden applied load.

σ = Stress developed in the body due to sudden applied load
E = Young’s Modulus of elasticity of the material of the body
A= Cross sectional area of the body
P = Sudden applied load which will be constant throughout the deformation process of the body
x = Deformation or extension of the body
L = Length of the body
V= Volume of the body = L.A
U = Strain energy stored in the body

As we have already discussed that when a body will be loaded within its elastic limit, the work done by the load in deforming the body will be equal to the strain energy stored in the body.

Strain energy stored in the body = Work done by the load in deforming the body
Strain energy stored in the body = Area of the load extension curve
Strain energy stored in the body = P. x
U = P. x

 
As we know that maximum strain energy stored in the body U will be provided by the following expression as mentioned here.
Now we will secure the value of extension x in terms of Stress, Length of the body and Young’s modulus of the body by using the concept of Hook’s Law.

According to Hook’s Law

Within elastic limit, stress applied over an elastic material will be directionally proportional to the strain produced due to external loading and mathematically we can write above law as mentioned here.

Stress = E. Strain
Where E is Young’s Modulus of elasticity of the material

σ = E. ε
σ = E. (x/L)
x = σ. L/ E

Let use the value of the extension or deformation “x” in above equation and we will have 
 
Therefore, we can say here that maximum stress induced in the body due to sudden applied load will be twice the stress induced in the body with same value of load applied gradually.


Once we will have value of the stress (σ) induced in the body due to sudden applied load, we will easily determine the value of strain energy stored in the body due to sudden applied load

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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