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Regarding the bending moments, it can be seen from the equi-iibrium of the element that they are not equal at two adjacent cross sections and that the increase dM in the bending moment equals the moment of the couple represented by the two equal and opposite forces V, i.e.,

and

dM = Vdx dx

(5°)

Thus, on all portions of a beam between loads the shearing force is the ratę of change of the bending moment with respect to x.

Let us now consider the case in which a distributed load of intensity q acts between the cross sections mn and mi»i, Fig. 60b. Then the total load acting on the element is qdx. If q is considered positive when the load acts downward, it may be concluded from the equilibrium of the element that the shearing. force at the cross section mtfi\ is different from that at mn by an amount

dV = — qdx, from which it follows that

d_V_

dx

~ 9-

(51)

Thus the ratę of change of the shearing force is equal to the intensity of the load with negative sign.

Taking the moment of all forces acting on the element we obtain

dx

dM = Vdx - qdx X — •

⁷    2

Neglecting the second term on the right side as a smali quantity of the second order, we again arrive at equation (50) and conclude that the ratę of change of the bending moment is equal to the shearing force in the case of a distributed load as well.