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Clearly, the distribution of the load by the rail girder considerably alters the distribution of shear due to a load in the bay in which the section considered lies. The total shear due to a series of loads Pj, P.,, ... at distances mj, m2, • • • from the left abutment, yx, y2, . . . being the ordinates of the influence curve under the loads, is S = Pi2/i + I>22/2+ • • • Generally, the greatest shear S at C will occur when the longer of the segments into which C x — divides the girder is fully loaded and the other is unloaded, Fig- 16. the leading load being at C. If the loads are very unequal or unequally spaced, a trial or two will determine which position gives the greatest value of S. The greatest shear at C' of the opposite sign to that due to the loading of the longer segment occurs with the shorter segment loaded. For a uniformly-distributed load w per foot run the shear at C is iv x the area of the influence curve under the segment covered by the load, attention being paid to the sign of the area of the curve. If the load rests directly on the main girder, the greatest + and - shears at C will be x AGO and -wx CHB. But if the load is distributed to the bracing intersectionsr by rail and cross girders, then the shear at C' will be greatest w hen the load extends to N, and will have the values w x AD1ST and -wx NEB. Influence lines were described by Frankel, Der Civilingenieur, 1876. See also Handbuch der Ingcnieur-ioissenschaften, vol. ii. ch. x., 1882, and Levy, La statique graphiqiw, 1886. There is a useful paper by Prof. G. F. Swain (Trans. Am. Soc. G. E. xvii., 1887), and another by L. M. Hoskins (Pros. Am. Soc. C. E. xxv., 1899). Another method of investigating the maximum shear at a section due to any distribution of a travelling load has been given by Prof. H. T. Eddy (Trans. Am. Soc. C. K xxii., 1890). Let hk (Fig. 17) represent in magnitude and position a load W, at a; from the left abutment, on a girder AB of span l. Lay off kf hg, horizontal and equal to l. Join f and g to h and k. Draw verticals at A, B, and join no. Obviously no is horizontal and equal to l. Also mn/mf=hklkf or mn = 'W(l-x)/l, which is the reaction at A due to the load at C, and is the shear at any point of AC. Similarly, po is the reaction at B and shear at any point of CB. The shaded rectangles represent the distribution of shear due to the load at C, while no may be termed the datum line of s
shear. Let the load move to D, so that its distance from the left abutment is x+a. Draw a vertical at D, intersecting fh, kg, in s and q. Thon qrlro=hkjhg or ro = W(l-x-a)/l, which is the reaction at A and shear at any point of AD, for the new position of the load. Similarly, rs = W(x + a)/l is the shear on DB. The distribution of shear is given by the partially-shaded rectangles. For the application of this method to a series of loads Prof. Eddy’s paper must be referred to. In the case of a bridge of many spans, there is a length of span which makes the cost of the bridge least. The cost of abutments and bridge flooring is practically independent of the length of span adopted. Let P be the cost of one pier *, G the cost of the main girders for one span, erected ; n the number of spans ; l the length
of one span, and L the length of the bridge between abutments. Then, n = L/l nearly. Cost of piers (?i - 1) P. Cost of main girders nG. The cost of a pier will not vary materially with the span adopted. It depends mainly on the character of the foundations and height at which the bridge is carried. The cost of the main girders for one span will vary nearly as the square of the span for any given type of girder and intensity of live load. That is, G=aP, where a is a constant. Hence the total cost of that part of the bridge which varies with the span adopted is — C — (n- VjV + naP = LP//-P + LaZ. Differentiating and equating to zero, the cost is least when dC LP T „ “77 — - 72 +La —0, P=a^=G ; that is, when the cost of one pier is equal to the cost erected of the main girders of one span. Sir Guildford Molesworth puts this in a convenient but less exact form. Let G be the cost of superstructure of a 100-feet span erected, and P the cost of one pier with its protection. Then the economic span is 7 = 100 ^/P/ ^/G. All the earlier arched bridges were of masonry or brickwork, so built that they could be treated as blockwork structures composed of rigid voussoirs. The stability of such structures depends on the position of the line of pressure in relation to the extrados and intrados of the arch ring. Generally the line of pressure lies within the middle half of the depth of the arch ring. In finding the line of pressures some principle such as the principle of least action must be used in de- 'bridges termining the reactions at the crown and springings. For an elastic arch of metal there is a complete theory, but it is difficult, and there is always some doubt as to the validity of some of the assumptions which must be made. If hinges are introduced at crown and springings, the calculation of the stresses in the arch ring becomes simple, as the line of pressures must pass through the hinges. Such hinges have been used not only for metal arches, but in a modified form for masonry and concrete arches. Three cases therefore arise: (a) The arch is rigid at crown and springings; (6) the arch is twohinged (hinges at springings) ; (c) the arch is three-hinged (hinges at crown and springings). For an elementary account of the theory of arches, hinged or not, reference may be made to a paper by Mr H. M. Martin (Proc. Inst. G. E. vol. xciii. p. 462) ; and for that of the elastic arch, to a paper by Mr A. E. Young {Proc. Inst. G. E. vol. cxxxi. p. 323). In Germany three hinged arches of masonry and concrete have been built, up to 150 feet span, with much economy, and the calculations being simple, an engineer can venture to work closely to the dimensions required by theory. For hinges, Mr Liebbrand, of Stuttgart, uses sheets of lead about one inch thick extending over the middle third of the depth of the voussoir joints, the rest of the joints being left open. As the lead is plastic this construction is virtually an articulation. If the pressure on the lead is uniformly varying, the centre of pressure must be within the middle third of the width of the lead ; that is, it cannot deviate from the centre of the voussoir joint by more than one-eighteenth of its depth. In any case the position of the line of pressures is confined at the lead articulations within very narrow limits, and ambiguity as to the stresses is greatly diminished. The restricted area on which the pressure acts at the lead joints involves greater intensity of stress than has been usual in arched bridges. In the Wiirtemberg hinged arches a limit of stress of 110 tons per square foot was allowed, while in the unhinged arches at Cologne and Coblentz the limit was 50 to 60 tons per square foot (Annales des Fonts et Chaussies, 1891). At Rechtenstein a bridge of two concrete arches has been constructed, span 75| feet, with lead articulations: width of arch 11 feet; depth of arch at crown and springing 21 and 2‘96 feet respectively. The stresses were calculated to be 15, 17, and 12 tons per square foot at crown, joint of rupture, and springing respectively. At Cincinnati, a concrete arch of 70 feet span has been built, with a rise of 10 feet. The concrete is reinforced by eleven 9-inch steel-rolled joists, spaced 3 feet apart and supported by a cross channel joist at each springing. The arch is 15 inches thick at the crown and 4 feet at the abutments. The concrete consisted of 1 cement, 2 sand, and 3 to 4 broken stone. An important series of experiments on the strength of masonry,
