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.
For a beam with flexural rigidity EI constant, the differential equation is given by
Where y represents the vertical deflection; M is the bending moment at a distance x from the conveniently selected origin; E is the modulus of elasticity or Young's modulus; I is the moment of inertia of the beam section; and the product EI is called the flexural rigidity.
Integrating the above equation once will give us dy/dx which is the equation of the slope of elastic curve, usually denoted by ?. Integrating once more (thus, double integration) will result to y which is the equation of the deflection. We can write it into symbols as follows
and
The constants of integration C1 and C2 can be found by applying appropriate boundary conditions.
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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