Strain Energy Formula

Strain energy is the energy stored in a body as a result of deformation. It is represented by the symbol U. It’s unit of measurement is J. The dimensional formula of strain energy is given by [M¹L²T^-2]. The strain energy per unit volume strain energy density or the area under the stress-strain curve towards the site of deformation. The formula for strain energy is equal to half the product of the compression factor and force applied to the body.

Formula

U = 1/2 × F × δ

where,

δ is the compression factor,

F is the force applied on the body.

In terms of Young’s modulus, stress and volume of the body, the formula is given by,

U = σ²/2EV

where,

σ is the value of stress,

E is the Young’s modulus,

V is the volume of body.

When stress σ is proportional to strain ϵ, the strain energy formula is equal to half the product of stress, strain and volume of the body.

U = 1/2 × σ × ϵ × V

where,

σ is the stress,

ϵ is the strain,

V is the volume of body.

Sample Problems

Problem 1. Calculate the strain energy if a force of 1200 N compresses the body by 3 m.

Solution:

We have,

F = 1200

δ = 3

Using the formula we have,

U = 1/2 × F × δ

= 1/2 × 1200 × 3

= 1800 J

Problem 2. Calculate the strain energy if a force of 1000 N compresses the body by 4 mm.

Solution:

We have,

F = 1000

δ = 4 × 10^-3

Using the formula we have,

U = 1/2 × F × δ

= 1/2 × 1000 × 4 × 10^-3

= 2 J

Problem 3. Calculate the strain energy if the stress of 500 Pa is applied on a body of volume 270 cu. m. The value of Young’s modulus is given as 120 Pa.

Solution:

We have,

V = 270

σ = 500

E = 120

Using the formula we have,

U = σ²/2EV

= (500 × 500)/(2 × 120 × 270)

= 3.85 J

Problem 4. Calculate the strain energy if the stress of 160 Pa is applied on a body of volume 90 cu. m. The value of Young’s modulus is given as 50 Pa.

Solution:

We have,

V = 90

σ = 160

E = 50

Using the formula we have,

U = σ²/2EV

= (160 × 160)/(2 × 50 × 90)

= 2.84 J

Problem 5. Calculate the strain energy if the stress of 35 Pa is applied on a body of area of 12 sq. m and length of 4 m. The value of Young’s modulus is given as 25 Pa.

Solution:

We have,

A = 12

l = 4

σ = 35

E = 25

Calculate the volume V of the body.

V = Al

= 12 (4)

= 48 cu. m

Using the formula we have,

U = σ²/2EV

= (35 × 35)/(2 × 25 × 48)

= 0.51 J

Problem 6. Calculate the strain energy if the stress of 60 Pa and strain of 2 × 10^-6 are applied on a body of volume 100 cu. m such that the stress is proportional to strain.

Solution:

We have,

σ = 60

ϵ = 2 × 10^-6

V = 100

Using the formula we have,

U = 1/2 × σ × ϵ × V

= 1/2 × 60 × 2 × 10^-6 × 100

= 6 × 10^-3 J

Problem 7. Calculate the strain energy if the stress of 250 Pa and strain of 7 × 10^-3 are applied on a body of volume 400 cu. m such that the stress is proportional to strain.

Solution:

We have,

σ = 250

ϵ = 7 × 10^-3

V = 400

Using the formula we have,

U = 1/2 × σ × ϵ × V

= 1/2 × 250 × 7 × 10^-3 × 400

= 350 J