**Bending Moment Diagram - shape and curvature**

__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.

__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.

**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.

You can also use following calculators for Bending moment and shear force

**
Bending Moment calculator for Simply supported beam**

regularly updated

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BM & SF Calculator for Cantilever

Calculate SF & BM for Cantilever

Deflection & Slope Calculator for Cantilever

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For SF & BM of many load cases of overhanging beam

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