We were discussing Strain energy, Resilience, Proof resilience and Modulus of resilience in our recent post and also we have discussed
shear
force and bending moment diagrams for a simply supported beam with a point load
acting at midpoint of the loaded beam during our previous posts.

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

Let us consider a body which is subjected with tensile load which is increasing gradually up to its elastic limit from value 0 to value P and therefore deformation or extension of the body is also increasing from 0 to x and we can see it in following load extension diagram as displayed here.

We have
following information from above load extension diagram for body which is subjected
with tensile load up to its elastic limit.

σ= Stress
developed in the body

E = Young’s
Modulus of elasticity of the material of the body

A= Cross
sectional area of the body

P =
Gradually applied load which is increasing gradually up to its elastic limit from
value 0 to value P

P = Stress
developed in the body x Cross sectional area of the body

P = σ. A

x =
Deformation or extension of the body which is also increasing from 0 to x

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 = Area of the triangle ABC

U = (1/2)
x AB x BC

U = (1/2)
x. P

Let us use
the value of P = σ. A, which is determined above

U = (1/2)
x. σ. A

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*

*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 strain energy equation and we will have

U = (1/2)
(σ.
L/ E).σ. A

U = (1/2)
(σ

^{2}/E) L.A
U = (σ

^{2}/2E) V
U = (σ

^{2}/2E) V
Therefore strain
energy stored in a body, when load will be applied gradually, will be given by following
equation.

###
*Proof resilience*

*Proof resilience*

Proof
resilience is basically defined as the maximum strain energy stored in the
body. As we know that strain energy stored in the body will be maximum when
body will be loaded up to its elastic limit, therefore if σ could be taken at
the elastic limit then we will have following equation for Proof resilience.

###
*Modulus of resilience *

*Modulus of resilience*

Modulus of
resilience is a property of the material which is defined as the proof
resilience per unit volume of the body and we can express as

Modulus of
resilience = Proof resilience/Volume of the body

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google