Question

derive an expression for the internal bending moment of a beam terms of radius of curvature

Answer 1

Explanation:

Let PQ be the chosen from the neutral axis. If R is the radius of curvature of the neutral axis and ᶿ is the angle subtended by it at its centre of curvature ’C’

Then we can write original length

PQ=Rᶿ ………………. 1

Let us consider a filament P’Q’ at a distance ‘X’ from the neutral axis.

We can write extended length

P’Q’ = (R + x)ᶿ   ………………2

From equations 1 and 2 we have,

Increase in length=P’Q’-PQ

On increase in its length = (R = x) θ - Rθ

Increase in length = xθ   ……………….3

We know linear strain=

Linear strain = ………………4

We know, the young's modulus of the material

Y =

Or              

stress = y × linear strain ………….5

Substituting 4 in 5, we have

Stress =

If δA is the area of cross section of the filament P’Q’, then,

The tensile force on the area δA = stress × Area

i.e. Tensile force = ( ).δa

We know the moment of force= force*Perpendicular distance

Moment of the tensile force about the neutral axis AB  

                                       or  

The moment of force acting on both the upper and lower halves of the neutral axis can be got by summing all the moments of tensile and compressive forces about the neutral axis