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
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