# Strain Energy Stored Due To Bending

20th May 2019## Strain Energy Stored Due To Bending

Let’s assume a beam which is subjected to a uniform moment M. Consider an elemental length

*of the beam between two sections***ds****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

Now, the intensity of stress in the element cylinder =

Where

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

So,

The energy stored by ds length of the beam =

And,

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

*and length***da***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 =

**f = (M/I).y**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) = (

**f**^{2}**/***⨯***2E)****(da.ds) =***(1/2E)⨯(My/I)⨯(da.ds) = (M*^{2}*/2EI*^{2}^{)}*⨯ds.da.y*^{2}Energy stored by ds length of the beam = Sum of the energy stored by each elemental cylinder.

Between the two sections

**and***1-1**2-2.**= ∑*(M^{2}/2EI^{2}^{)}*⨯ds.da.y*^{2}*=***(M**^{2}/2EI^{2}^{)ds}*∑**da.y*^{2}**But**

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

The energy stored by ds length of the beam =

**(M**^{2}/2EI^{).ds}And,

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

*∫**(M*^{2}/2EI^{).ds}