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

$\rho = \dfrac{[\,1 + (dy / dx)^2\,]^{3/2}}{\vert \, d^2y/dx^2 \, \vert}$

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

$\rho = \dfrac{EI}{M}$

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

$\rho = \dfrac{1}{d^2y/dx^2} = \dfrac{1}{y''}$

Thus, EI / M = 1 / y''

$y'' = \dfrac{M}{EI} = \dfrac{1}{EI}M$

If EI is constant, the equation may be written as:

where x and y are the coordinates shown in the figure of the elastic curve of the beam under load, y is the deflection of the beam at any distance x. E is the modulus of elasticity of the beam, I represent the moment of inertia about the neutral axis, and M represents the bending moment at a distance x from the end of the beam. The product EI is called the **flexural rigidity** of the beam.

The first integration y' yields the slope of the elastic curve and the second integration y gives the deflection of the beam at any distance x. The resulting solution must contain two constants of integration since EI y" = M is of second order. These two constants must be evaluated from known conditions concerning the slope deflection at certain points of the beam. For instance, in the case of a simply supported beam with rigid supports, at x = 0 and x = L, the deflection y = 0, and in locating the point of maximum deflection, we simply set the slope of the elastic curve y' to zero.

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