**Strain Energy Formula**

Strain energy is a sort of potential energy stored in a structural element due to elastic deformation. When a part is deformed from its unstressed state, the external work done on it is changed into (and deemed equal to) the strain energy stored in it.

For example, When a beam supported at both ends is subjected to a bending moment caused by a load suspended in the centre, the beam is said to be deflected from its unstressed state and strain energy is stored.

When a force is given to an object formed of deformable material, it will change shape. In the case of a rubber band when we stretch it. It’s also difficult to see when a load is placed on a steel support beam. The object will continue to stretch as we apply greater force. Stress is equal to the amount of force exerted divided by the object’s cross-sectional area.

Formula for Strain energy

U = (Fδ)/2

Where,

U = Strain Energy

δ = Compression

F = Force applied

When stress and strain are proportional, the formula is:

U = (1/2) × Vσε

Where,

U = Strain Energy

σ = Strain

V = Volume of body

The strain energy formula for Young’s modulus E is

U = σ² /2E × V

Where,

U = Strain Energy

σ = Strain

V = Volume of body

E = young’s modulus

**Solved Examples**

**Q.1: A rod with a surface area of 90 square mm is 3 metres long. If tension of 300 MPa is applied when stretched, calculate the strain energy.**

Given parameters are,

Area A = 90 mm2,

Length, l = 3m,

Stress, σ = 300MPa

= 300 10^{6} Pa

Young’s modulus, E = 200GPa

= 200 10^{9}Pa

Volume V is: V = AL

= (90 10^{-6})3

V = 27 10^{-6}cm³

The strain energy formula is:

U = σ² / 2E × V

= (300 10^{6})²/ 2 × 200 × 109 × 27 × 10^{-6}

= 12.51 J

Thus, the strain energy of the rod is 12.15 J.

**Q.2: An object is crushed by 1.5 mm when a force of 100 N is applied to it. Determine the strain energy.**

Given,

Force, F = 100N

Compression, δ = 1.5 mm

U = Fδ / 2

= 100 × 1.5 × 10-³/ 2

∴ U = 0.075 J

Thus, the strain energy of the object is 0.075 J.