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# Strain Energy Formula

When a deformable structure, such as a spring, stretches, then it stores a type of energy known as strain energy. In many such cases, we can turn back this energy into kinetic energy relatively in an easy way. As the springs return to their original length, therefore its strain energy is transferred back to the form of kinetic energy. Strain energy is the key feature in such examples. This article will help students to understand the strain energy formula with examples. Let us start!

## Definition of Strain Energy

Strain energy is a type of potential energy that is stored in a structural member as a result of elastic deformation. The external work done on such a member when it is deformed from its unstressed state is transformed into (and considered equal to the strain energy stored in it. If, for instance, a beam that is supported at two ends is subjected to a bending moment by a load suspended in the canter, then the beam is said to be deflected from its unstressed state, and strain energy is stored in it.

The integration for strain energy can only be applied over a length of beam for which a continuous expression can be obtained. This generally will imply a separate integration for each section between two concentrated loads or reactions.

Source:en.wikipedia.org

Whenever we apply a force to an object of a deformable material, it will change its shape. Sometimes, it is a big change, like when we stretch out a rubber band. Also, it’s hard to see, like when a load is applied to a steel support beam. As we apply more and more force, the object will continue to stretch. Stress will be the amount of force applied divided by the cross-sectional area of the object.

## Formula for Strain Energy

Therefore, strain energy is the energy stored in a body due to its deformation.  So we refer to this strain energy per unit volume as strain energy density. Also, the area under the stress-strain curve towards the point of deformation. When the applied force is free then the whole system will get back to its original shape.

1] The strain energy formula is: $$U = \frac {F \delta } { 2}$$

 U Strain Energy $$\delta$$ Compression F Force applied

2] When stress $$\sigma$$ is proportional to strain \epsilon, the strain energy formula is: $$U = \frac {1}{2} V \sigma \epsilon$$

 U Strain Energy $$\sigma$$ Strain V Volume of body

3] Regarding young’s modulus E, the strain energy formula is : $$U = \frac {{\sigma}^2 }{2E} \times V.$$

 U Strain Energy $$\sigma$$ Strain V Volume of body E young’s modulus,

## Solved Examples for Strain Energy Formula

Q.1: A rod of area 90 square mm has a length of 3 m. Then find out the strain energy if the stress of 300 MPa is applied when stretched. Young’s modulus is 200 GPa.

Solution: Given parameters are,

Area A = 90 square mm,

Length, l = 3m,

Stress, $$\sigma = 300 MPa = 300 \times 10^{6} Pa$$

Young’s modulus, $$E = 200 GPa.= 200 \times 10^9 Pa$$

Volume V is: V = AL

$$= (90 × 10^{-6}) \times 3$$

$$V = 27 \times 10^{-6} cubic m$$

The strain energy formula is:

$$U = \frac {{\sigma}^2 }{2E} \times V.$$

$$= \frac {{ 300 \times 10^{6} }^2 }{2 \times 200 \times 10^9 } \times 27 \times 10^{-6}.$$

= 12.15 J

Therefore, the strain energy of the rod will be 12.15 J.

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