NEW CONSTRUCTIONS

IN

GRAPHICAL STATICS.

ordinates

in which E is the modulus of elasticity surface is the polygon or curve, above of the material, and I is the moment of described, is considered to have the inertia of the girder; and as is well same effect as a series of concentrated known, the summation is to be extended loads proportional to the from the point O to a free end of the yp acting at the assumed points of girder, or, if not to a free end, the sum- division. If the points of division be mation expresses the effect only of the assumed sufficiently near to each other, quantities included in the summation. the assumption is sufficiently accurate.

Let a number of points be taken at equal distances along the girder, and let the values of P, S, M, B, D be computed for these points by taking O at these points successively, and also erect ordinates at these points whose lengths are proportional to the quantities computed. First, suppose I is the same at each of the points chosen, then the values of these ordinates may be expressed as follows, if a, b, c, etc., are any

real constants whatever :

The ordinates ym and ym' are not equal, but can be obtained one from the other when we know the ratio of the moments of inertia at the different cross sections.

If a polygon be drawn in a similar manner by joining the extremities of the ordinates ym computed from equation (3), it is known that this polygon is an equilibrium polygon for the applied weights P, and it can also be constructed directly without computation by the help of a force polygon having some as

sumed horizontal stress.

Now, it is seen by inspection that equations (3) and (5), or (3') and (5'), have the same relationship to each other that equations (1) and (3) have. The relationship may be stated thus:—If the ordinates ym (or ym') be regarded as the depth of some species of loading, so that the polygonal part of the equilibrium polygon is the surface of such load, then a second equilibrium polygon constructed for this loading will have for its ordinates proportional to yd. But deflections of the girder. these last are proportional to the actual

Hence a second equilibrium polygon, so constructed, might be called the deflection polygon, as it shows on an exaggerated scale the shape of the neutral axis of the deflected girder.

The first equilibrium polygon having the ordinates ym may be called the moment polygon.

It may be useful to consider the physical significance of equations (3), (4), (5), or (3), (4′), (5′).

According to the accepted theory of perfectly elastic material, the sharpness of the curvature of a uniform girder is Equation (1) expresses the loading, directly proportional to the moment of and yp may be considered to be the the applied forces, and for different depth of some uniform material as earth, shot or masonry constituting the girder, it is inversely proportional to the girders or different portions of the same load. Lines joining the extremities of these ordinates will form a polygon, or approximately a curve which is the upper surface of such a load. When the

load is uniform the surface is a horizontal line.

Now this resistance varies directly as I resistance which the girder can afford. varies, hence curvature varies as M÷I, which is equation (3) or (3').

Now curvature, or bending at a point, is expressed by the acute angle between For the purposes of our investiga- two tangents to the curve at the distance tion, a distributed load whose upper of a unit from each other; and the total

bending, i.e. the angle between the tan- due to the forces applied to the arch will gent at O, and that at some distant point be sustained at those points which are A is the sum of all such angles between not flexible, partly in virtue of its being O and the point A. Hence the total

bending is proportional to ≥(M÷I), approximately an equilibrium polygon, the summation being extended from O and partly in virtue of its resistance as a to the point A, which is equation (4) or girder. (4').

Again, if bending occurs at a point distant from 0, as A, and the tangent at A be considered as fixed, then O is deflected from this tangent, and the amount of such deflection depends both upon the amount of the bending at A, and upon its distance from 0. Hence the deflection from the tangent at A is proportional to (Mx÷I) which is equation (5) or (5').

It is evident from the nature of the equilibrium polygon that it is possible with any given system of loading to make an arch of such form (viz., that of an equilibrium polygon) as to require no bracing whatever, since in that case there will be no tendency to bend at any point. Also it is evident that any deviation of part of the arch from this equilibrium polygon would need to be braced. As, for example, in case two distant points It will be useful to state explicitly be joined by a straight girder, it must several propositions, some of which are be braced to take the place of part of implied in the foregoing equations. The importance and applicability of some of them has not, perhaps, been sufficiently recognized in this connection.

Prop. I. Any girder (straight or otherwise) to which vertical forces alone are applied (i. e., there is no horizontal thrust) sustains at any cross-section the stress due to the load, solely by develop

forces.

the arch. Furthermore, the greater the deviation the greater the bending moment to be sustained in this manner. Hence appears the general truth stated in the proposition.

called into action, at any point of a straight girder, depends not only on the applied forces which furnish the polygonal part of the equilibrium polygon, but also on the resistance which the girder is capa

It will be noticed that the moment

ing one internal resistance equal and opble of sustaining at joints or supports, or posed to the shearing, and another equal the like. For example, if the girder and opposed to the moment of the applied rests freely on its end-supports, the moment of resistance vanishes at the ends, Prop. II. But any flexible cable or and the "closing line" of the polygon arch with hinge joints can offer no re-joins the extremities of the polygonal part. If however the ends are fixed sistance at these joints to the moment horizontally and there are two free of the applied forces, and their moment (hinge) joints at other points of the giris sustained by the horizontal thrust de- der, the polygonal part will be as before, veloped at the supports and by the ten- but the closing line would be drawn so that the moments at those two points sion or compression directly along the vanish. Similarly in every case (though cable or arch. the conditions may be more complicated than in the examples used for illustration) the position of the closing line is fixed by the joints or manner of support of the girders, for these furnish the conditions which the moments (i. e., the ordinates of the equilibrium polygon) must fulfill. For example, in a straight uniform girder without joints and fixed Prop. III. If an arch not entirely flexi- horizontally at the ends, the conditions ble is supported by abutments against vanishes when taken from end to end, are evidently these; the total bending which it can exert a thrust having a and the deflection of one end below the horizontal component, then the moment tangent at the other end also vanishes.

It is well known that the equilibrium polygon receives its name from its being the shape which such a flexible cable, or equilibrated arch, assumes under the action of the forces. In this case we may say for brevity, that the forces are sustained by the cable or arch in virtue of its being an equilibrium polygon.

Prop. IV. If in any arch that equilibrium various, and so cannot be considered in polygon (due to the weights) be construct- a general demonstration. The obscurity, ed which has the same horizontal thrust however, will disappear after the treatment of some particular cases, where we as the arch actually exerts; and if its shall take pains to render the truth of closing line be drawn from consideration the proposition evident. We may, howof the conditions imposed by the supports, ever, make a statement which will posetc.; and if furthermore the curve of the sibly put the matter in a clearer light by arch itself be regarded as another equilib- saying that A" is a figure easily found, rium polygon due to some system of load- the determination of A' which is unand we, therefore, employ it to assist in ing not given, and its closing line be also known, and of A which is partially unfound from the same considerations re- known. And we arrive at the peculiar specting supports, etc., then, when these property of A", that its closing line is found two polygons are placed so that these in the same manner as that of A, by noclosing lines coincide and their areas partially cover each other, the ordinates intercepted between these two polygons are proportional to the real bending moments acting in the arch.

Suppose that an equilibrium polygon due to the weights be drawn having the same horizontal thrust as the arch. We

ticing that the positions of the closing lines of A and A' are both determined in the same manner by the supports, etc.; for the same law would hold when the rise of the arch is nothing as when it has any other value. But A" is the difference of A and A'. Hence what is true of A and A' separately is true of their difference A", the law spoken of being a mere matter of summation. are in fact unable to do this at the outFrom this proposition it is also seen that set as the horizontal thrust is unknown. the curve of the arch itself may be reWe only suppose it drawn for the pur- garded as the curved closing line of the pose of discussing its properties. Let polygon whose ordinates are the actual also the closing line be drawn, which bending moments, and the polygon itmay be done, as will be seen hereafter. self is the polygonal part of the equiliCall the area between the closing line brium polygon due to the weights. and the polygon, A. Draw the closing line of the curve of the arch itself (regarded as an equilibrium polygon) according to the same law, and call the area between this closing line and its curve A". Further let A' be the area of a polygon whose ordinates represent the actual moments bending the arch, and drawn on the same scale as A and A". Since the supports etc., must influence the position of the closing line of this polygon in the same manner as that of A, we have by Prop. III not only

A=A'+A"

It is believed that Prop. IV contains an important addition to our previous knowledge as to the bending moments in an arch, and that it supplies the basis for the heretofore missing method of obtaining graphically the true equilibrium polygon for the various kinds of arches.

Prop. V. If bending moments M act on a uniform inclined girder at horizontal distances x from O, the amount of the vertical deflection yd will be the same as that of a horizontal girder of the same cross section, and having the same

which applies to the entire areas, but horizontal span, upon which the same also

y=y'+y"

moments M act at the same horizontal distances from 0. Also, if bending as the relation between the ordinates of moments M act as before, the amount of these polygons at any of the points of the horizontal deflection, say a, will be division before mentioned, from which the same as that of a vertical girder of the truth of the proposition appears.

This demonstration in its general form the same cross section, and having the may seem obscure since the conditions same height, upon which the same moimposed by the supports, etc., are quite ments M act at the same heights.

specting deflections, which the reader can easily enunciate for himself.

Before entering upon the particular discussions and constructions we have in view, a word or two on the general question as to the manner in which the problem of the arch presents itself, will perhaps render apparent the relations between this and certain previous investigations. The problem proposed by Rankine, Yvon-Villarceaux, and other analytic investigators of the arch, has been this:-Given the vertical loading, what must be the form of an arch, and

what must be the resistances of the spandrils and abutments, when the weights produce no bending moments whatever? By the solution of this question they obtain the equation and properties of the particular equilibrium polygon which would sustain the given weights. Our graphical process completely solves this question by at once constructing this equilibrium polygon.

For the small deflections occurring in a It may be remarked in this connection, girder or arch, AOC=90°

The same may be proved of any other moments at other points; hence a similar result is true of their sum; which proves the proposition.

It may be thought that the demonstration is deficient in rigor by reason of the assumption that AOC=90°.

that the analytic process is of too complicated a nature to be effected in any, except a few, of the more simple cases, while the graphical process treats all cases with equal ease.

But the kind of solution just noticed, is a very incomplete solution of the problem presented in actual practice; for, any moving load disturbs the distribution of load for which the arch is the equilibrium polygon, and introduces bending moments. For similar reasons it is necessary to stiffen a suspension bridge. The arch must then be propor tioned to resist these moments. Since this is the case, it is of no particular consequence that the form adopted for the arch in any given case, should be such as to entirely avoid bending moments when not under the action of the moving load.

Such, however, is not the fact as ap- So far as is known to us, it is the pears from the analytic investigation of universal practice of engineers to asthis question by Wm. Bell in his at-sume the form and dimensions, as tempted graphical discussion of the arch well as the loading of any arch proin Vol. VIII of this Magazine, in which jected, and next to determine whether the only approximation employed is that the assumed dimensions are consistent admitted by all authors in assuming that the curvature is exactly proportional to the bending moment.

We might in this proposition substitute f. M÷I for e. M, and prove a similar but more general proposition re

with the needful strength and stability. If the assumption is unsuited to the case in hand, the fact will appear by the introduction of excessive bending moments at certain points. The considerations set forth furnish a guide to a new