FRAMES.] BRIDGES 321 call} the amount of metal required in this bridge and in a girder of the same span and 12 feet deep; the breadth given in the diagram to each member is proportional to the cross section required. The quantity of metal required for the girder exceeds that for the arch in the ratio of about 17 5 to 100 ; a similar calculation for a bowstring Fig. 52. lattice girder 17 feet deep at the centre gives the ratio between the weights of metal required as 100 for the arch and 155 for the bowstring. Even these ratios understate the great advantage to be derived from the braced arch or suspension bridge for large spans, since they assume that the loads to be carried are the same, whereas the permanent load is in large spans much less for the lighter con struction of bridge. The following table shows the probable weight in tons of several different types of trusses, assuming that the maxi mum intensity of stress on the metal in girders and arches is every where 4 tons per square inch, that the passing load to be carried is 1 "4 tons per foot run, and that the practical weight is 25 per cent, in excess of the minimum possible weight if no metal were wasted, vide Trans. R.S.S.A., vol. viii. p. 135. It must be re membered that the abutments for arches will in all cases be more expensive than those for girders ; in small spans this expense will often outweigh the saving which could be etfected in the super structure by employing the arch. TABLE XII. Weights of Trusses of differ Weight of Span. wrought iron Girder in tons. I. Teet. 2. Total. Per ft, run. 200 83 41: 300 222 74 400 475 1-19 500 940 1-88 600 2,220 37 700 3,900 5-57 800 1 2,800 16- 1000 1200 1400 1600 1800 2000 Weight of wrought iron P.o-.vstring, in tons. Weight of wrought iron Suspension Bridge, and wrought iron or cast iron Weight of wooden Arch, in tons. Weight of Suspension Bridge, in tons. Strain, 8 tons per Arch, in tons. square inch. 4. 5. G. 7. 8. 9. 10. 11. Total. Per foot Total. Per ft. run. Total. PCM- ft. run. Total. Per ft. run. run. 70 35 41 205 571 285 19 0-095 ISO 60 100 33 142 473 45 0-15 375 94 194 485 285 712 83 0-207 700 1-4 330 65 510 1-02 134 0-268 1260 2-1 530 883 870 1-45 200 0-333 2290 3-27 790 1-13 1440 2-06 280 0-400 4-150 5 56 1180 1-475 384 0-48 2490 2 49 655 0-655 5000 4-67 1060 0-885 1580 1-128 1 2360 1-475 3410 1-89
- 4950
2-475 The framed arch is a very suitable form for wooden bridges. The ties are few and subject to insignificant strains. In designing a series of arches supported on piers care must be taken to provide for the thrust from loaded to unloaded arches across the pier at the springing. 57. Stmif/th of Struts. When a strut or column is used as in framework to resist compression, i-, is usually so long in comparison with its cross section that it will bend and yield with a much less stress than would be required to crush the material. The strength of a strut of this kind can be approximately computed according to the following theory : Let a strut with a cross section S be pinned or hinged at both ends, or let it have round ends, so that if it yields under compression it bends as in fig. 83. Let the cross section have two axes of l; ->- symmetry, and let the section be such that the column will bend in the plane of one axis ; let the depth of cross section measured in this direction be called d, and the breadth measured at right angles to the depth d be called b. Let the maximum or breaking load be called J. , and the maximum deflection of the longitudinal axis of the strut from its unbent position be called v, this quantity being analogous to the deflection in a beam. The moment tending to produce flexure at the cross section where v is measured will be 1 y. This moment must, as in the case of a girder, be equal to the moment of the elastic forces, which we already know to be 2;>, I ~ where 2 and /have the same signification as for girders. (/ must be taken about the unstrained axis, or in other words about the axis running in the direction which has been called that of the breadth.) Hence we have 2 T P,-/7 TV r but we also know that for beams of uniform cross section under similar internal stresses v is proportional to y ; hence we have where a is a constant depending on the material only. Let^ = -* be the mean intensity of stress which would be produced if the load compressed all parts of the cross section equally, and ]ctf e as before be the ultimate strength of the material per unit of cross section, then, when the beam is on tJic point of yielding, we must have and calling r the radius of gyration from which / can be computed in terms of F and a, or P in terms offc and . The value of a does not change with a change in the design of the cross section ; the values for cast iron, wrought iron, and wood determined by experiment are given in the following table, S being measured in square inches, P in Ibs. The table also contains values offc. TAI;LE XIII. Constants for the Strength of Struts and Pillars. fc. . Cast-iron ................ 80,000 Ibs. r i T _ t Wrought iron ......... 36,000 Ibs. Dry timber ............ 7,200 Ibs. "When the direction of both ends of the column is fixed it must bend in the manner shown in fig. 84 ; if the curvature be uniform this would have the effect of reducing the deflection v to one-fourth of the amount caused by the same stresses when the ends arc hinged, giving and 3. Pi=4l al 2 7-0 ) . or , 1 + 7< ~ When one end is fixed in direction and the other hinged, the ulti mate load which the strut can bear may be taken as a mean between the strength of two pillars of the same length and cross section, one having both ends fixed in direction and the other having both ends hinged; for similar cross sections r" is proportional tod 2 , the square of the depth (measured in the direction of the shorter side or axis), I- l- thus the ratio 2 ^ . Let n = -J-^TJ , and let B = 3 -^.y . Then frofn equation 2 we have for hinged ends and from equation 3 for fixed ends fH 5. P = : 1 + ttlJ u has a constant value for all similar cross sections, and P> a constant
IV. 41