The bending moment M0 is represented in the figure by the vertical line BH where H is on the continuation of the side 4, the scale being given by
| BH | = | ½M0BC | ; |
| CE | 1⁄12wBC³ |
this appears from the diagrams of forces, fig. 22, in which the oblique lines are marked to correspond to the sides of the funiculars to which they are parallel.
In the application of the method to more complicated cases there are two systems of fixed points corresponding to F, by means of which the sides of the funiculars are drawn.
60. Finite Bending of Thin Rod.—The equation
curvature = bending moment ÷ flexural rigidity
may also be applied to the problem of the flexure in a principal plane of a very thin rod or wire, for which the curvature need not be small. When the forces that produce the flexure are applied at the ends only, the curve into which the central-line is bent is one of a definite family of curves, to which the name elastica has been given, and there is a division of the family into two species according as the external forces are applied directly to the ends or are applied to rigid arms attached to the ends; the curves of the former species are characterized by the presence of inflections at all the points at which they cut the line of action of the applied forces.
We select this case for consideration. The problem of determining the form of the curve (cf. fig. 23) is mathematically identical with the problem of determining the motion of a simple circular pendulum oscillating through a finite angle, as is seen by comparing the differential equation of the curve
| EI | d²φ | + W sin φ = 0 |
| ds² |
with the equation of motion of the pendulum