The scheme below represents a simply supported beam loaded by a single concentrated load (yellow vector P). By dragging around the red points, you can change magnitude and direction of the load and the distance (blue vector a) at which it is applied, computed from node x.
The red vectors represent the reactions on nodes x (pinned point) and y (simple support):
A = -P.(y-x)/(y-x)
B = -P^(y-x-a)/(y-x)
C = -P^a/(y-x).
The green vectors show that equilibrium is respected, as the force polygon is closed. The (light green and violet) highlighted areas represent the bivector momenta of load P and reaction C with respect to node x:
P^a = - C^(y-x),
they are always of equal magnitude but opposite sign, and ensure that momenta equilibrium is respected as well.
The second scheme below represents the magnitude of the maximum bending moment, i.e. computed at location a, where the load is applied. Note that the value of the moment changes accordingly as the load vector P is changed in magnitude or position. When the load is located at point L (midpoint of the beam) the bending moment is maximum for that particular load vector, while it vanishes as vector a approaches either tip of the beam.
Mmax = (1-a/l)(P^a) with l=y-x.
[ GA with Cinderella | structural mechanics ]
Created with Cinderella by Luca Redaelli