Strain Energy

Strain Energy usually denoted by U is basically the energy stored in a body due to deformation.

It is a type of potential energy that is stored in a structure as a result of elastic deformation. For example, if a bar is bent by applying force on it then, then the bar is said to be deformed from its unstressed state, and the amount of strain energy stored in it is equal to the work done on the bar by that force.

It is calculated as the area under the stress-strain curve towards the point of deformation.

Strain Energy Formula

Strain energy is denoted by \(U\).

\(U = \frac{F \delta}{2}\)

Where,
\(\delta\) – Compression
\(F\) – Force Applied

If stress \(\sigma\) is proportional to strain \(\epsilon\), the formula can also be given by,

\(U = \frac{1}{2}V \sigma \epsilon\)

Where,
\(V\) – Volume of the body

Strain energy formula using Young’s Modulus,

\(U = \frac{\sigma^2}{2E}V\)

Solved Examples

Question 1. When a force of 100 N is applied to an object, it is compressed by 1.5 mm. Calculate the strain energy.

Solution. Given,

Force, \(F = 100 N\)
Compression, \(\delta = 1.5 mm\)

\(U = \frac{F \delta}{2} = \frac{100*1.5*10^{-3}}{2}\)

∴ \(U = 0.075 J\)

Question 2. A rod of area 100 mm² has a length of 5 m. Determine the strain energy if the stress of 500 MPa is applied when stretched. Young’s Modulus is taken to be 200 GPa.

Solution. Given,

Area, \(A = 100 {mm}^2\)
Length, \(l = 5m\)
Stress, \(\delta = 500 MPa = 500 * 10^6 Pa\)
Young’s Modulus, \(E = 200 GPa = 200 * 10^9 Pa\)

As we know, Volume = Area x Length

V = (100 x 10^-6) x 3

\(U = \frac{\sigma^2}{2E}V\)

\(U = \frac{(500 * 10^6 )^2}{2 * 200 * 10^9} (100 * 10^{-6}) * 3\)

∴ \(U = 18.75 J\)

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