Page:EB1911 - Volume 09.djvu/169

Fig. 19.

58. Let l be the length of a span of a loaded beam (fig. 19), M₁ and M₂ the bending moments at the ends, M the bending moment at a section distant x from the end (M₁), M′ the bending moment at the same section when the same span with the same load is simply supported; then M is given by the formula

M = M′ + M1	l − x	+ M2	x	,
	l		l

Fig. 20.

and thus a fictitious load statically equivalent to M/EI can be easily found when M′ has been found. If we draw a curve (fig. 20) to pass through the ends of the span, so that its ordinate represents the value of M′/EI, the corresponding fictitious loads are statically equivalent to a single load, of amount represented by the area of the curve, placed at the point of the span vertically above the centre of gravity of this area. If PN is the ordinate of this curve, and if at the ends of the span we erect ordinates in the proper sense to represent M₁/EI and M₂/EI, the bending moment at any point is represented by the length PQ.^[1] For a uniformly distributed load the curve of M′ is a parabola M′ = 1/2wx (l − x), where w is the load per unit of length; and the statically equivalent fictitious load is 1/12wl³ / EI placed at the middle point G of the span; also the loads statically equivalent to the fictitious loads M₁ (l − x) / lEI and M₂x / lEI are 1/2M₁l / EI and 1/2M₂l / EI placed at the points g, g′ of trisection of the span. The funicular polygon for the fictitious loads can thus be drawn, and the direction of the central-line at the supports is determined when the bending moments at the supports are known.

Fig. 21.

Fig. 22.

Fig. 23.

59. When there is more than one span the funiculars in question may be drawn for each of the spans, and, if the bending moments at the ends of the extreme spans are known, the intermediate ones can be determined. This determination depends on two considerations: (1) the fictitious loads corresponding to the bending moment at any support are proportional to the lengths of the spans which abut on that support; (2) the sides of two funiculars that end at any support coincide in direction. Fig. 21 illustrates the method for the case of a uniform beam on three supports A, B, C, the ends A and C being freely supported. There will be an unknown bending moment M₀ at B, and the system^[2] of fictitious loads is 1/12wAB³/EI at G the middle point of AB, 1/12wBC³ / EI at G′ the middle point of BC, −1/2M₀AB / EI at g and −1/2M₀BC / EI at g′, where g and g′ are the points of trisection nearer to B of the spans AB, BC. The centre of gravity of the two latter is a fixed point independent of M₀, and the line VK of the figure is the vertical through this point. We draw AD and CE to represent the loads at G and G′ in magnitude; then D and E are fixed points. We construct any triangle UVW whose sides UV, UW pass through D, B, and whose vertices lie on the verticals gU, VK, g′W; the point F where VW meets DB is a fixed point, and the lines EF, DK are the two sides (2, 4) of the required funiculars which do not pass through A, B or C. The remaining sides (1, 3, 5) can then be drawn, and the side 3 necessarily passes through B; for the triangle UVW and the triangle whose sides are 2, 3, 4 are in perspective.

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	=	1/2M0BC	;
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

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 1/2α, 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

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1. The figure is drawn for a case where the bending moment has the same sign throughout.
2. M₀ is taken to have, as it obviously has, the opposite sense to that shown in fig. 19.