l d²φ+ g sin φ = 0.
dt²

The length L of the curve between two inflections corresponds to the time of oscillation of the pendulum from rest to rest, and we thus have

L √(W / EI) = 2K,

Fig. 24.

where K is the real quarter period of elliptic functions of modulus sin ½α, and α is the angle at which the curve cuts the line of action of the applied forces. Unless the length of the rod exceeds π√(EI / W) it will not bend under the force, but when the length is great enough there may be more than two points of inflection and more than one bay of the curve; for n bays (n + 1 inflections) the length must exceed nπ √(EI / W). Some of the forms of the curve are shown in fig. 24.

For the form d, in which two bays make a figure of eight, we have

L√(W / EI) = 4.6,   α = 130°

approximately. It is noteworthy that whenever the length and force admit of a sinuous form, such as α or b, with more than two inflections, there is also possible a crossed form, like e, with two inflections only; the latter form is stable and the former unstable.

Fig. 25.

61. The particular case of the above for which α is very small is a curve of sines of small amplitude, and the result in this case has been applied to the problem of the buckling of struts under thrust. When the strut, of length L′, is maintained upright at its lower end, and loaded at its upper end, it is simply contracted, unless L′²W > ¼π²EI, for the lower end corresponds to a point at which the tangent is vertical on an elastica for which the line of inflections is also vertical, and thus the length must be half of one bay (fig. 25, a). For greater lengths or loads the strut tends to bend or buckle under the load. For a very slight excess of L′²W above ¼π²EI, the theory on which the above discussion is founded, is not quite adequate, as it assumes the central-line of the strut to be free from extension or contraction, and it is probable that bending without extension does not take place when the length or the force exceeds the critical value but slightly. It should be noted also that the formula has no application to short struts, as the theory from which it is derived is founded on the assumption that the length is great compared with the diameter (cf. § 56).