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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
= 300 106 Pa
Young’s modulus, E = 200GPa
= 200 109Pa
Volume V is: V = AL
= (90 10-6)3
V = 27 10-6cm³
The strain energy formula is:
U = σ² / 2E × V
= (300 106)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.
