Question

how to find deflection in fixed beam with udl by double integration

Answer 1

The double integration method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve.

In calculus, the radius of curvature of a curve y = f(x) is given by

ρ=[1+(dy/dx)2]3/2|d2y/dx2|ρ=[1+(dy/dx)2]3/2|d2y/dx2|

In the derivation of flexure formula, the radius of curvature of a beam is given as

ρ=EIMρ=EIM

Deflection of beams is so small, such that the slope of the elastic curve dy/dx is very small, and squaring this expression the value becomes practically negligible, hence

ρ=1d2y/dx2=1y′′ρ=1d2y/dx2=1y″

Thus, EI / M = 1 / y''

y′′=MEI=1EIM