Engr Help

Calculation of Shear and Bending Moment Diagrams using relations relating
shear force to distributed load and change in bending moment to shear force

In a Nut Shell: The calculation of shear and bending moment diagrams using the

relations among the shear and bending moment involves the following steps:

Step 1

Construct a free body diagram (FBD) of the entire beam and use equilibrium

to determine the support reactions. These support reaction yield the shear

force and bending moment at each end of the beam.

Step 2

Use the value of shear at x = 0 for the starting value of shear at the left

end of the beam. Then dV/dx = w(x) establishes the slope of the shear

force across the beam and ʃ w(x) dx gives the change in shear along the

beam. Click here to recall the sign convention for positive shear force and

positive bending moment. Plot the shear diagram.

Step 3

Strategy: Use the shear diagram to construct the moment diagram.

The values of bending moment at the ends of the beam come from Step 1.

Start with the bending moment at x = 0 ( the left end of the beam).

The slope of the moment diagram (dM/dx) equals the shear since

dM/dx = V.

Step 4

Note: The change in the bending moment along the beam equals the area

under the shear diagram, M(x) = ∫ V(x) dx, provided there are no

concentrated moments. At a concentrated moment the bending moment

“jumps” by the amount of the concentrated moment. Keep in mind the

sign convention.

Strategy: The values of moment at the ends of the beam come from the

boundary conditions at each end of the beam. Use the area under the shear

diagram to establish the change in bending moment along the beam

and the value of shear to establish the slope of the moment diagram.

Step 5

Plot the bending moment diagram using the sign convention.

Step 6

Check that the values of shear and bending moment in both diagrams agree

with the values at the supports determined in step 1.