Bending Moment Diagram - shape and curvature

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 and then draw bending moment diagrams.

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 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.