Relationship between intensity of load, Shear Force and Bending Moment

There is a relation among intensity of load 'w', Shear Force 'F' and Bending Moment 'M'. 

For the derivation of the relations among these three we consider a simply supported beam subjected to a uniformly distributed load w(x) throughout its length (L), as shown in Figure.

consider a small element of length dx at a distance of x form left support.

Let the shear force and bending moment at a section located at a distance of x from the left support be V and M, respectively, and at a section x + dx be V + dV and M + dM, respectively. 

The total load acting in the element is wdx, acts at the center of the element length dx.

Considering the element is in equilibrium ,take moments on right side of  element.

∑ Mx+dx = ∑ MR = 0 

                = -(M+ dM) + M + Vdx + wdx.dx/2 = 0 

(Neglecting the small term wdx.dx/2 )

- dM + Vdx =0

dM = V dx  

or 

 dM/dx  = V (x)

This relations implies that the first derivative of the bending moment with respect to the distance is equal to the shearing force.

Rate of change of Bending Moment is equal to Shear force. 

dM/dx = F(x) 

The equation also suggests that the slope of the moment diagram at a particular point is equal to the shear force at that same point.  suggests the following expression:

This  states that the change in moment equals the area under the shear diagram. 

Similarly, the vertical equilibrium in the element, 

∑ Fy = 0

 V – wdx - (V + dV) = 0

dV = - wdx  

or

dV/dx  = - w(x)

This relation implies that the first derivative of the shearing force with respect to the distance is equal to the intensity of the distributed load. 

Rate of change of Shear Force is equal to Intensity of loading. 

dF/dx = - w(x) 

The equation also suggests that the slope of the shear force diagram at a particular point is equal to the intensity of load at that same point

It also suggests the following expression:

This equation states that the change in the shear force is equal to the area under the load diagram.

By taking both relations 

dV/dx  = - w(x) and 

dM/dx  = V (x); we get one more relation

d/dx {dM/dx} = -w(x) 

The relation implies that the second derivative of the bending moment with respect to the distance is equal to the intensity of the distributed load.

Note:

a) Rate of change of Shear Force is equal to Intensity of loading
dF/dx = - w(x)
b) The change in the shear force in between any two points is equal to the area under the load diagram.
c) Rate of change of Bending Moment is equal to Shear force.
dM/dx = F(x)
d)The change in bending moment between any two points equals the area under the shear diagram.