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 Mx 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 Mx at a distance "x" from the free end = 10 x (x) x (x/2)= 0.5 x2
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.
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