**ABC—XYZ**

BEAMS.] BRIDGES 301 the effect of increasing the curvature over it, and shorten ing the distance AA t ; analysis shows that in this case the point of inversion of flexure will be at a distance from the centre pier equal to 2G83L ; then the length of the part of the beam subject to a positive bending moment will be 7317L, instead of GGL as when the beam was of uniform strength ; the load on each of the end piers will be - 36585wL ; the load on the centre pier l 2G85u>L ; and vc hive 8 ..... M..= -1341wL s , ( J 10 In any actual bridge uniformly loaded the values of the moments and shearing stresses will be intermediate between those given for a beam of uniform strength and those for a beam of uniform cross section. It must be observed that the above theory assumes that the girder is unstrained when being built as a whole ; if, as is often the case, the separate spans are separately built and lifted into position, and then joined on the piers, special provision must be made to bring the desired bending moment over the piers into existence ; it is obvious that merely joining the two independent beams in the upper part of fig. 22 will not make these into a continuous girder, such as is shown in the lower figure, to do this, besides joining the upper and lower flanges, we must pull the top flanges together over the piers and put the lower flanges under compression. This may be done practically by tilting one end, or both ends, before the junction at the centre is made, and after wards allowing the ends to sink until the curve assumed by the girder shows that the required distribution of stress has been attained. The complete analysis of the problem of continuous girders of any number of spans equal or unequal with any number of loads has been given by Mr Heppel (Proc. R. S. t 1870-71). 27. Alloivance for Weiulit of Beam. Limiting Span. When the weight of the beam is a considerable and un certain part of the whole load, it can be allowed for as follows. Design a beam of the desired depth and span, fit to carry a total load equal to the external or passing load W 1 ; calculate the weight of this beam and call it Bj ; the beam so designed will really be fit to carry an external load Wj - B r Let b i be the area of any cross section of this beam ; let b be the area of cross section required at the same point for the beam of weight B actually necessary to carry a total load W. Then since the strength of the properly proportioned girder of constant depth and span is simply proportional to the quantity of metal em ployed, and therefore to the area of cross section, we have the proportion l> : b i = W x : W : - B 1? or 1 b- W 6 -W^B/ The weight B is given by the expression <> -a B I W , =W;--BV The whole load W is given by the expression W 2 3 For any given design of beam there is a limiting length which cannot be exceeded (the beams of different spans being assumed to be similar in the geometrical sense). Let L, be the limiting length of a beam of a given design which for the span L weighs B, and carries a gross load W, then the ratio of B to W can be shown to increase in direct proportion to the length of the span until this ratio reaches unity. Hence W or when 4. III. SUSPENSION BKIDGES. 28. Varieties of Suspension Bridyes. A very simple form of suspension bridge has long been used in Peru and Thibet. Two ropes are hung side by side across the gorge to be passed, a rude platform is laid on the ropes, and the dip of these is sufficiently small to allow the bridge to bo crossed by men or beasts passing down from the one side to the centre and up to the opposite bank. The modern suspension bridge consists of two or more chains, from which a level platform is hung by suspension rods. The chains may in some cases be secured directly to the sides of the chasm to be crossed, but the configuration of the ground seldom allows this to be done. The chains, therefore, usually pass over piers, aa in fig. 2, Plate XIX., and are led dowi on either side to an anchorage at a considerable distance from the piers. The chains between tin piers and the anchorage are generally used to support part of the platform. The chains where they pass over the piers rest on saddles f which are made of two different types. One construction, shown in fig. 27, allows the chain to slip backwards and Fig. 27. forwards over it with comparatively little friction, so that the stress on the rope may be taken as equal on both sides of the saddle. In the second type, as shown in fif. 28 V "^ J _L-- //. , the chain is secured to the saddle, which, however, is free to move horizontally on the top of the pier. With the first form of saddle the resultant pressure on the pier will not be vertical unless the chain leaves the pier with an equal inclination on each side, and even when the bridge is de signed with an equal slope of chain on both sides of the pier, a change in the distribution of weight due to any passing load will cause some departure from the equal slope of the chains, and therefore from the truly vertical pressure on the^ piers. This departure is easily allowed for in the

design of the pier. The friction on the saddle renders the