Integration Method

This technique provides a means of obtaining not only the V and M diagrams but the formulas for any or all parts of the diagrams.  The concepts will be used to study beam deflections later in the semester.

The internal forces and moments are governed by simple differential equations:

dN/dx = 0 slope of normal-force diagram equals zero
dV/dx = -w slope of shear-force diagram equals negative of distributed load intensity
dM/dx = V slope of bending-moment diagram equals shear force

Later we will also use the following relationships.

EI d2y/dx2 = M
dy/dx = tan θ

where

y = deflection of beam at x
 θ = slope of beam at x

Steps

  1. Sketch the V and M diagrams using the area method.
  2. For unloaded or distributed load portions of the beam, integrate dV/dx = -w to determine V = V(x).
  3. Use points on the V diagram to solve for the integration constant(s).
  4. Once you have determined V = V(x) exactly, integrate dM/dx = V to determine M = M(x).
  5. Use points on the M diagram to solve for the integration constant(s).