Summary of Shear force and Bending moments

1. The Shear Force at a section may be defined as "the algebraic sum of the total vertical forces on either side of the cross section".

The unit of shear force is : N,or kN

2. The Bending moment at a section may be defined as "the algebraic sum of the moments about that section of all external forces acting to one side of that section".

The unit of bending moment is : N-mm,or kN-m

3. A Shear Force diagram (SFD) is one which shows the variation of the shear force along the length of the beam.

A bending moment diagram (BMD) is one which shows the variation of the bending moment along the length of the beam.

4.Draw the shearing force, and bending moment diagram for the structure, noting the sign conventions.

Sign convention for Shear and Bending moments

To draw SFD or BMD the negative values are plotted to scale below a horizontal reference line and positive values are plotted above the line.

5.Key points in Shear force and Bending moments Diagrams.

a) Bending Moment is maximum at the point where the Shear Force is zero or where it changes its direction from +ve to -ve or vice versa.

b) The point of contraflexure or point of inflexion is the point where the Bending moment changes its direction from +ve to -ve or vice versa. At this point the magnitude of B.M. is zero.

c) The maximum shear force usually occur at the support or at the point under concentrated load.

d) Shear force is constant in unloaded sections, linearly varies in uniformly loaded sections and parabolic over uniformly varying sections.

e)Bending moment varies linearly over unloaded sections, parabolic over uniformly loaded sections and cubic parabolic over uniformly varying sections.

These variations are shown below -

6. The relation among the intensity of load (w), Shear force (V) and Bending moment (M)

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.

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