P. A. PETERSON, Vice-President, in the Chair. Paper, No. 27. CANTILEVER BRIDGES. BY C. F. FINDLAY, M.A., A.M. Inst. C.E. A structure which is supported at one end of its length only, so that it requires for equilibrium the application of other forces than those applied at the support, is commonly called a cantilever. If such a structure forms part of a bridge the bridge may be called a cantilever bridge. This description agrees sufficiently well with the ordinary usage of these terms, and any attempt at rigid definition would serve no good purpose. It is sometimes maintained that a cantilever bridge should be regarded and treated simply as a continuous-girder bridge, in which the points of contrary flexure are arbitrarily and permanently fixed by cutting the booms; but this point of view would exclude all cantilever bridges which are anchored at the abutments instead of being continuous with a neighboring span, and also it is both illogical and misleading to classify as a variety of continuous girders those in which the most prominent feature is the breach of continuity at certain points. The cantilever is in regard to the distribution of stress in it one of the simplest of framed structures, and no one who is familiar with the analysis of stresses in ordinary bridge and roof trusses, will find any difficulty in calculating the stresses in a cantilever bridge from given drawings and loads, difficulty, that is to say, inherent in the cantilever system. There are, however, several features peculiar to the cantilever which deserve attention in regard to design, and to these we shall confine our attention, taking it for granted that those points which the cantilever has in common with other bridges are familiar to our readers. The manner in which the cantilever is employed to form part of a bridge varies very much from one bridge to another. In some cases two similar cantilevers spring from neighbouring abutments in opposite directions, so that one forms a counterbalance to the other except as regards moving loads. In other cases, a cantilever stands alone and is kept in equilibrium by guys or backstays connected to an anchorage. Sometimes an opening is spanned by two cantilevers with a central span resting on their ends, and sometimes by one cantilever with an independent span from its end to the other abutment. Examples of these various types of cantilevers are shewn in the accompanying diagrams, Plate III of which Fig. 1 represents the bridge now in course of construction over the Forth near Edinburgh in Scotland; Fig. 2 represents a bridge being built over the Indus river in India; Fig. 3 represents the Jubilee bridge crossing the Hooghly river near Calcutta in India; Fig. 4 represents the bridge over the St. Lawrence below the Falls of Niagara. The great difference between the parts played by the cantilever in these different designs, is mainly accounted for by the character of the sites of the bridges. The reasons that may lead to the adoption of the cantilever-bridge in preference to any other type for certain situations are two :— First, where the span is very great, the cantilever bridge generally requires less material than any other rigid structure of equal strength, even though anchorage may have to be provided. Where two large spans have to be built, a double cantilever requiring no anchorage, may effect a very considerable saving in material, though it must not be forgotten that in this case, a double pier of sufficient width for stability under all conditions of loading is necessary. Secondly, where false works would be impossible or costly, the valuable property of the cantilever, that it can be made to suppport itself during erection, gives it an immense advantage. If the design of the cantilever be such that it can be built out rapidly and cheaply, it will often be the most economical form in the end, even where its total quantity of material is not so small as that required by some other type of bridge. In all engineering work quantity of material is only one of the elements of cost, and it is of the greatest importance to bear this in mind in designing a cantilever bridge, because a want of regard to the method of erection, may easily add to its cost an amount much greater than can be saved by economising material. The principal points which arise for consideration at the outset, in commencing to design a cantilever bridge are these: (1) The depth to be given to the cantilever at the abutment; (2) The outline the structure is to have in elevation, or, in other words, the rate at which the depth is to vary in passing from the abutment to the outer end of the cantilever; (3) The length of the panels into which the cantilever is to be divided by the bracing, and whether this length is to be uniform or not; (4) The length of the cantilever in relation to the entire span. The first three of these points are intimately connected with one another, and none of them can be settled by a mathematical investigation applicable to all circumstances. The theoretical investigations which have been made on questions of this character are all vitiated (as regards their application to cantilevers) by the assumption that the load on the bridge is uniform throughout its length. This assumption is nearly enough correct for ordinary truss bridges, because the web is heavier where the booms are lighter, and so the weight of the truss is distributed along its length with an approach to uniformity, and also because for all ordinary spans the weight of the girders is considerably less than that of the floor and the moving load together, so that a small variation in the load per foot due to the main girders is of less importance. With a cantilever, however, especially when used (as it generally is) for large spans, the case is entirely different. The web and booms are both heaviest at the abutment and diminished simultaneously on travelling away from the abutment. Also the weight per foot of the main framework of the cantilever, where greatest, will in a large span be generally much greater than that from all other causes. When it is also considered that the cantilever as a rule, has to carry a heavy concentrated load on its end, it is obvious that all general conclusions founded on an assumption of uniform weight must be fallacious. As an illustration, it may be mentioned that the cantilevers of the Indus Bridge (Fig. 2) weigh about 1 ton per foot at their outer ends, and about 6 tons per foot at the abutments (apart from the uniform load of the floor which is about ton per foot), and they are designed to carry a concentrated load of 300 tons at the end. In observing the important effect of this concentrated load on the cantilever we see how misleading it might be to regard the entire span as one continuous girder bridge with fixed points of contrary flexure, because that point of view masks the fact that when the continuity is destroyed the central part is just as much an independent span as if it rested on stone piers. Whatever proportions would lead to the greatest economy in an independent span of the same length should a fortiori be adopted for the central span of a cantilever bridge, because any extravagance in the central span has to be paid for over again in extra weight in the cantilevers. Half the benefit of destroying the continuity is lost, if we do not take advantage of it to design each part of the bridge in the way most suitable for that part. In a comparison of the weights of girder bridges with those of cantilever bridges, the concentration of weight near the abutments in the latter is an immense advantage on their side. The marked effect of taking into account the varying weight of the cantilever is well seen, very if the theoretical case be considered in which the web members consist only of vertical posts or suspenders. For a load uniformly distributed horizontally, we know that the curve of a suspension cable, of an arched rib or of the boom of a bow string girder, must be a parabola that no diagonal bracing may be necessary. These constructions are, each of them, as far as the stresses go, analogous to the cantilever without diagonal bracing, and we shall see that if the varying distribution of the load, owing to the varying depth of the structure, be taken into account, the parabola becomes approximately an ellipse. So long as the span is short, the ellipse and parabola would nearly coincide, but for a large span the difference would be considerable. In Figs. 5 and 6, let A be the abutment; "La be the length of a cantilever in feet; dent span resting on it. "L tons be the load at the end from an indepen Let O be the origin of coordinates, and one boom OX being horizontal, let x, y be the coordinates of any point P in the other boom. w be total weight per ft. in tons carried by the cantilever, at P (w varying with x) K be such a factor, that if T tons be the stress on any member of the cantilever and 1 ft. its length, its weight is KT1 tons, and assume K to be constant. (For steel with the stress usually allowed, and allowing for loss of section or increase of weight at connections K is about .0003.) .. d MS dx and d Swdx As it is assumed that there are no diagonal stresses, the stress in the horizontal boom is constant and = (say) H. Now w consists of two parts:-the uniform load per foot carried by the cantilever (including all parts of the bridge, such as flooring, etc., which are approximately uniform), and the variable weight of the web members and booms. Let the constant part of w = m. The weight of horizontal boom for horizontal distance dx = KH dx dx=KH (ds) (ds) * dx 2 of vertical suspenders or posts =Kwy dx [The last expression is not strictly correct, as the weight of the curved boom is not transmitted through the verticals, which do not therefore carry the whole of w. The error here made would, however, in prac tice, tend to counterbalance other incorrect assumptions, for it becomes more serious as the abutment is approached, and at the same time the secondary bracing and wind bracing, which have been supposed uniformly distributed, must in practice become heavier.] ds dx 2 + wy } We have then w = m + K { H + H m+ K = dx2 where C and D are arbitrary constants to be determined by the known = L, we have H or x2 (m + 2 KH) + y2 KH + 2 Lx-2 Hy=0 This is an equation to an ellipse, whose major axis is vertical, the m ratio of the squares of the axes being 2 + The vertical tangent KH. to the ellipse at the end of its minor axis touches the curve at the point depth of the longest cantilever that can be where y, so that the built with these loads, is H and L. K at the abutment, and is independent of m, |
