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.

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.

As we know that maximum strain energy stored in the body U will be provided by the following expression as mentioned here.
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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

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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.

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*According to 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

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**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google