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 = 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) . AB . 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

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) (σ2/E) L.A
U = (σ2/2E) V
U = (σ2/2E) V

Therefore strain energy stored in a body, when load will be applied gradually, will be given by following equation.

### Proof resilience

Proof resilience 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 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