Question

Derive relation between load shear force and bending moment

Answer 1

Explanation:

Shear and moment diagrams by shear and moment equationsThe vertical shear at C in the figure shown in previous section (also shown to the right) is taken as

VC=(ΣFv)L=R1−wx

where R1 = R2 = wL/2

Vc=wL2−wx

The moment at C is

MC=(ΣMC)=wL2x−wx(x2)

MC=wLx2−wx22

If we differentiate M with respect to x:

dMdx=wL2⋅dxdx−w2(2x⋅dxdx)

dMdx=wL2−wx=shear

thus,

dMdx=V

Thus, the rate of change of the bending moment with respect to x is equal to the shearing force, or the slope of the moment diagram at the given point is the shear at that point.

Differentiate V with respect to x gives

dVdx=0−w

thus,

dVdx=Load

Answer 2

Thus the relation between shear force and bending momentum is F = dM / dx

Explanation:

Consider a beam of light subjected to bending and transverse shear. In order to maintain equilibrium, the forces and moments are added together. Equilibrium of forces and moments exist together at all points so it is convenient to look at the equilibrium of moment and force at the bottom corner of the beam.

Now

F σ x + M = M + σM

F = σx /σM

σx = 0  

so we have  

F = dM / dx

Thus the relation between shear force and bending momentum is F = dM / dx