## Strain Energy Stored Due To Bending

Let’s assume a beam that is subjected to a uniform moment M. Consider an elemental length **ds** of the beam between two sections *1-1* and **2-2.**

The elemental length of the beam may be assumed as consisting of an infinite number of element cylinders each of area **da** and length *ds. *Consider one such elemental cylinder located y units from the neutral layer between the section *1-1* and *2-2.*

Now, the intensity of stress in the element cylinder = [latex] f = \frac{M}{I}y [/latex]

Where* I* = Moment of inertia of the entire section of the beam about the neutral axis.

So, Energy stored by the element cylinder = (Energy stored per unit volume⨯Volume of the cylinder)

= [latex] \frac{f^{2}}{2E}\cdot da\cdot ds [/latex]

= [latex] \frac{1}{2E}\left (f^{2} \right )da\cdot ds [/latex]

= [latex] \frac{1}{2E}\left ( \frac{M}{I}y \right )^{2}da\cdot ds [/latex]

= [latex] \frac{M^{2}}{2EI^{2}} ds\cdot da\cdot y^{2} [/latex]

Energy stored by ds length of the beam = Sum of the energy stored by each elemental cylinder.

Between the two sections *1-1* and *2-2.*

= [latex] \sum \frac{M^{2}}{2EI^{2}} ds\cdot da\cdot y^{2} [/latex]

= [latex] \frac{M^{2}}{2EI^{2}} ds\cdot \sum da\cdot y^{2} [/latex]

But, ∑da.y^{2}^{ }= Moment of inertia of the beam section about the natural axis = **I**

So, The energy stored by the **‘***ds*‘ length of the beam

= [latex] \frac{M^{2}}{2EI^{2}} ds\cdot I [/latex]

= [latex] \frac{M^{2}}{2EI} ds [/latex]

And,

The total energy stored due to bending by the whole beam = [latex] \int \frac{M^{2}}{2EI} ds [/latex]