
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 = Mx^{2}/2 + c1x + c2
The value of constant c2 can be determined by applying the conditions of
deflection at some known points.

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(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
61
and
62
for more explanations.
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