
Bending moment is required for design of beam and
also for the calculation of
slope and
deflection of beam. The following examples will
illustrate how to write bending moment equation for different types
of load.
Case I Bending moment due to
point load
Bending moment due to a point
load is the product of the load and its perpendicular distance from
the point of moment. as shown below.
Consider a cantilever subjected to a
point load at its free end.
Bending moment at the fixed end = W
x L =
WL
Bending moment M_{x }at a
distance x from free end = W x
x = Wx
This is equation of a straight line and
the plotted bending moment diagram in the above figure shows that the variation of
bending moment along the span of a cantilever is a straight line.

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Case II Bending moment due to
a uniformly distributed load
Bending moment due to a uniformly
distributed load (udl) is
equal to the intensity of the load
x length of load
x
distance of its center from the point of moment as shown in the
following examples.
Bending moment at the fixed end =
10 x 2
x 1= 20 kNm
Bending moment M_{x} at a
distance "x" from the free end = 10 x
(x) x
(x/2)= 0.5 x^{2}
which is a second degree function of
"x" and therefore parabolic. 

You can also use our Free online calculators given below
(i)
bending moment & shear force calculator for
Cantilever
(ii)
Bending moment & shear force calculator
for Simple supported beam
(iii)
Bending moment & shear force calculator
for Overhanging beam
(iv)
Bending moment & shear force calculator for Fixed beam
(v)
Calculator for Continuous beam

Case III Bending moment due to
uniformly varying load
Bending moment due to a varying load is
equal to the area of load diagram
x distance of its centroid
from the point of moment.
The shape of bending moment diagram due
to a uniformly varying load is a cubic parabola.
Case IV Bending moment due to a
couple
Bending moment at a section due to a
couple is equal to the magnitude of the couple and in the same sense
as the couple.
Example 51
Example 52
Example 53
Example 54 will be more helpful in explaining how to
write the equations for shear force
and bending moment calculations and to draw the diagrams for
cantilever, simply supported and overhanging beams.
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