In the case of a beam bent by transverse loads acting in a plane of symmetry, the bending moment M varies along the length of the beam and we represent the variation of bending moment in B.M diagram. Futher, it is assumed that the simple bending theory equation holds good.
If we look at the elastic line or the deflection curve, this is obvious that the curvature at every point is different; hence the slope is different at different points.
To express the deflected shape of the beam in rectangular co-ordinates let us take two axes x and y, x-axis coincide with the original straight axis of the beam and the y – axis shows the deflection.
Futher let us consider element ds of the deflected beam. At the ends of this element let us construct the normal which intersect at point O denoting the angle between these two normal be di
But for the deflected shape of the beam the slope i at any point C is defined,
This is the differential equation of the elastic line for a beam subjected to bending in the plane of symmetry.
Its solution y = f(x) defines the shape of the elastic line or the deflection curve as it is frequently called.