## 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

**and**

*1-1*

**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

**and**

*1-1*

*2-2.*Now, the intensity of stress in the element cylinder =

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)

=

=

=

=

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.*=

=

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

=

=

And,

The total energy stored due to bending by the whole beam =