Equilibrium Problems by Graphical Methods

Solution. A free-body diagram of the truss with the forces identified by means of Bow’s Notation is shown in Fig. 3. The unknown forces , , and are shown with their correct lines of action; their magnitudes, of course, cannot as yet be drawn to scale. The notation should be so arranged that the known forces come first, so that the diagrams can be as completely constructed as possible. The diagram is then drawn to scale, and the force polygon and funicular polygon are constructed with the known forces.

In Fig. 4 we show as much of the force and funicular polygons as can be drawn with the known forces. The complete solution to scale is shown in Fig. 5.

It will be seen from the force polygon of Fig. 4 that it is possible to solve for one of the unknown forces, , directly from the force polygon, since it is known that the two forces and de must lie along a vertical line through , and that the force must lie along a horizontal line through , for closure of the force polygon. The intersection of these lines fixes the point , which determines the force and permits us to draw another string, , in the funicular polygon. The force polygon is now closed, but the funicular polygon is still open, since the string , which would join the points marked and in Fig. 5, is missing. For equilibrium, this string must be present, however, so we draw do, which closes the funicular polygon. We can now determine the point in the force polygon, because, working backwards, we can draw the ray do in the force polygon parallel to the string do in the funicular polygon. Having the point in the force polygon gives us directly the forces and and hence completes the solution.

Scaling off the values of the forces, we find:

To check our solution, let us determine the force analytically by taking moments about the left end of the truss: