**ABC—XYZ**

302 B 11 I D G E S [SUSPENSION BRIDGES. assumption of equal stresses on each side slightly incorrect, and with this type of saddle care must be taken to provide against the wear produced by the motion of the chain. With the second type the use of rollers under the solid saddle leaves the motion of the saddle very free ; the resultant pressure on the pier is always sensibly vertical, and the chains may leave the pier at any angle, equal or unequal. The chain must in no case be rigidly attached to the pier unless the support itself is free to rock on its base, as in fig. 29, where Fig. 29. the place of the pier is taken by cast-iron struts, working on a horizontal axis. Suspension bridges are chiefly used for very large spans, because, as we shall find, they can be constructed to carry the same load with a less weight of material than a beam or girder, subject, however, to the disadvantage of flexi bility or deformation under a passing load, a disadvantage which is very serious where, as in small bridges, the passing load is a large proportion of the whole load, but which is of less importance where the chief load to be carried is the weight of the structure itself. We will first consider the usual or simple suspension bridge, as shown in fig. 2, Plate XIX., and will then pass to the various modifications introduced to remedy its defects. 29. Form of Chain with given Load. Let the platform be hung from the chain by equidistant vertical rods ; then the load may be treated as hanging from each joint where the rods are attached, and will consist at each joint of the weight of one subdivision of the chain and of one sub division of the platform and its load. If the position of the vertical tie-rods be assumed as definite relatively to the points of suspension, so that the assumed loads on the joints act at known distances from the points of support, a form of chain, which will remain in equilibrium (or undis turbed) under these loads, can easily be found by the following graphic me thod : C Let the vertical line QN (fig. 30) represent the whole load to be carried, and the subdivisions QA, AB, BC, CD, &c., the loads referred to each joint of the chain (QA and NK will be the portion of the load referred directly to the saddle or point of support, and will be simply half the weight of the piece of chain between the saddle and the joint pin). Let QP and NP be the weights carried by each pier, equal if the distribution of load is symmetrical, otherwise to be determined as for a girder. Let a horizontal line, PO, represent the horizontal component of the tension to be allowed on the Fie. 30. chain, or the whole tension on that part of the chain the tangent to which is horizontal ; join with Q, with A, B, &c. ; then the lines OA, OB, &c., give the slopes of each successive link as shown in fig. 31, where the line parallel to OA in fig. 30 lies between the two spaces containing the letters and A in fig. 31, similarly the line parallel to OB in fig. 30 is represented by the link between the two spaces lettered and B in fig. 31, and so forth. The line in fig. 31 lying between and Q parallel to OQ in fig. 30, represents the direction of the force on the point of support a, being equal and opposite to the resultant of the tension on the first link and the weight carried directly by the support. The triangle QOA (fig. 30) is the polygon of forces in equilibrium at the point of support a (fig. 31). The triangle OAB is the polygon of forces in equilibrium at the first joint, and similarly each component triangle of fig. 30 represents the equilibrated forces at one joint of the chain in fig. 31. This theorem is one example of the general theory of reciprocal figures, which will be treated hereafter under the general head of " Frames," 53. Fig. 32. When the maximum dip is given instead of the horizontal com ponent of the stress, it is easy to find the latter from the former by the method of moments when the point is known where the chain will be horizontal; for then, let the link cd, fig. 32, be horizontal ; let the dip be called y, and let the distances of weights w v w s , w 3 , &c., from the point of support be called x lt a? 2 , x 3 , &c., and let the horizontal tension represented by PO in fig. 30 be called H. Then taking moments round the point a, we have 1 H.y S.iux, from which H can always be found. When the length is given of each link (or portion of the chain between the joints where the platform is suspended), and conse quently the length of the whole chain, the problem of determining the form assumed under any distribution of load is difficult, for the proportion of the load carried by each pier and the position of each load relatively to the piers vary when the form of the chain varies. The problem may be solved tentatively, but it is seldom attempted. The converse problem of finding the load which will keep a chain in equilibrium when the dimensions and curve are given is per fectly easy. From a point 0, fig. 30, draw a series of lines parallel to the given links. At any convenient distance, OP, draw a vertical line cutting the lines diverging from at the points Q, A, B, C, D, &c. The vertical loads required to keep the chain in equilibrium are proportional to the lengths QA, AB, BC, &c. 30. Relation between the Curve of Bending Moments and the Curve assumed by a Loaded Chain. The vertical ordinates at the joints of an equilibrated chain, measured from a horizontal axis passing through the two points of support (these being at the same level), are proportional to the bending moments for similarly chosen sections of a girder similarly loaded. Let us consider any joint, say that at which w 3 is hanging in fig.

33. Let V be the vertical component of the resultant pull on the left