## Strain Energy Due To Bending

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

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 =

*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

*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}
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