Slope and deflection of beam by direct integration method (Macaulay's Method)

Integration method

(a) Direct Integration; This method is based on the differential equation of elastic curve.

Integrating once the above equation of elastic curve gives an equation in terms of dv/dx with an unknown constant c1as follows;

EI (dv/dx) = Mx +c1

the value of constant c1 can be determined by applying the conditions of slope and/or deflection at some known points of the beam.

Similarly, integration of the equation again will give an equation in terms of "v" with an unknown constant c2 as given below;

EI v = Mx²/2 + c1x + c2

The value of constant c2 can be determined by applying the conditions of deflection at some known points.  

(b) Macaulay's method; This method is modified form of integration method and used when the loading is discontinuous and/or discrete. In other words we can say that a single equation is not sufficient to find bending moment along the span of the beam. In this method we put all the terms containing x into square bracket and we integrate the bracket not "x". During calculation for the values of slope and deflection, any bracket with negative value is to be ignored. An equivalent loading is required to be applied in case of uniformly distributed load. See Example 6-1 and  Example 6-2 for more explanations.

You can use our Deflection Calculator Easy to use calculator for different loads on beams