The direct integration method may become more involved if the expression for entire beam is not valid for the entire beam. Let us consider a deflection of a simply supported beam which is subjected to a concentrated load W acting at a distance 'a' from the left end.
Let R1 & R2 be the reactions then,
These two equations can be integrated in the usual way to find ‘y' but this will result in four constants of integration two for each equation. To evaluate the four constants of integration, four independent boundary conditions will be needed since the deflection of each support must be zero, hence the boundary conditions (a) and (b) can be realized.
Further, since the deflection curve is smooth, the deflection equations for the same slope and deflection at the point of application of load i.e. at x = a. Therefore four conditions required to evaluate these constants may be defined as follows:
(a) at x = 0; y = 0 in the portion AB i.e. 0 ≤ x ≤ a
(b) at x = l; y = 0 in the portion BC i.e. a ≤ x ≤ l
(c) at x = a; dy/dx, the slope is same for both portion
(d) at x = a; y, the deflection is same for both portion
By symmetry, the reaction R1 is obtained as