Relation between intensity bending moment and shear force:

What is the relation between load intensity bending moment and shear force

Consider the beam subjected to any type of transverse load of the general form shown in the figure given below. Isolate from the beam an element of length dx at a distance x from left end and draw its free body diagram as shown in the figure given below. As the element is of very small length, the loading over the beam is considered to be uniform and equal to w KN/m. The element is dependent on shear force F on the left hand side of it. Further bending moment M acts on left side of element and it changes to (M + dM) on right side.

Taking moment about the point C on right side, ΣM_C = 0

M - (M + dM) + F X dx - (W X dx) X dx/2 = 0

The UDL is considered to be acting at the Center of gravity
 dM = Fdx - [W(dx)²]/2 = 0

The last term comprises of the product of the two differentials and can be neglected

DM = Fdx, or

F = dM/dx

Therefore the Shear Force is equal to rate of change of bending moment with respect to x. Apply conditionΣV = 0 for equilibrium, we obtain

F - Wdx - (F + dF) = 0

Or                              W = dF/dx

That is intensity of loading is equal to rate of change of bending moment with respect to x.

F = dM/dx

and                           W = dF/dx = dM²/dx²