engines, the whole bridge may be covered with the live load; this may be taken as uniform throughout. Case II. Three spans loaded with 1.3 ton per foot run. The equation of bending moments for the first and third spans becomes where x = the distance from the left and right abutment. If y = 0, x = 127·2 feet The equation of bending moments in the middle span is from which we find, by making y = 0, a 115 and 44 feet; hence the points of contra-flexure occur 44 feet from each pier. Shearing Stresses. Comparing Cases I. and II., we observe that the shearing stresses and bending moments over the piers is greater in Case II., while the bending moments between the left abutment and point of contra-flexure is greater in Case I. Case III.-First and second spans loaded with 1-3 ton per foot run; third span loaded with 0.6 ton per foot run. Here 8M2+2M2 = — w1l2 16M2+ 4M1 = − (w2ľ2 + w,F)}where w1 = w2 25281 = -3581.3 foot-tons 60 Equation to first span is y = 0.65(159x - x2) 22.52x therefore the point of contra-flexure is x = 124-35. Equation to second span is y=0.65(159x-2)- 2106-7 - 9.27x therefore the points of contra-flexure occur at 42.65 feet from the first pier, and 28.35 feet from the second pier. Case IV. First and third spans loaded with 0.6 ton per foot run; middle span loaded with 1.3 ton per foot run. Only the middle span need be considered for this case, the equation of which is therefore the points of contra-flexure occur at 28-27 feet from the piers. The shearing stress over the piers for the middle span is the same as in Case II. Case V.-First and third spans loaded with 1.3 ton per foot run; middle span loaded with 0.6 ton per foot run. Here M1 = M2, as in Case IV., viz. 2401.7 foot-tons. Equation of first and third spans is therefore the points of contra-flexure occur at x = 13576 feet from either abutment. The shearing stresses are less than in the cases already considered. Comparing the foregoing cases, we see that the maximum bending moment for the first and third spans between the point of contra-flexure and the abutments occurs for the loading considered under Case V., which gives the longest effective span. The maximum moment over the piers and in the cantilever portions occurs for the loading considered in Case III. The middle span is subjected to greatest bending moment about the centre in Case IV., and in the cantilever portions in Cases I., III., and V. The maximum shearing stresses in the side spans occur in Cases III. and V., and in the middle span in Cases III. and IV. The diagram of shearing stresses for a moving load will be semi-parabolas between the abutments and points of contraflexure in the first and third spans, and straight lines tangential to the parabolas for the cantilever portions. In the central span the shearing stresses in the portion between the two points of contra-flexure will be represented by parabolas and the two cantilevers on each side by straight lines. The diagram of maxima bending moments is shown in Fig. FIG. 214. 214. The moments of resistance of the flange plates provided to resist them may be plotted as in Fig. 173, Chapter VIII. The diagram of maxima shearing stresses is shown in Fig. 215. It will be observed that the points of contra-flexure are continually changing during the passage of the live load, hence the working stress must be taken much lower about the region of the points of contra-flexure, and the diagonal members of the web must be counterbraced for reversals of stress, or designed for compression as well as tension. Concentrated Loads.-The equation of three moments for concentrated loads, for the case when all the supports are on the same level or on the same uniform gradient, and the section of the girder is uniform throughout its length, may be expressed The use of this equation will be illustrated in the following example. A bressummer beam of two spans, continuous over one sup port, carries a shop-front and portions of the floors above. The whole of the load is applied to the beam through masonry piers built upon the upper flange. The magnitudes and points of application of these loads are illustrated in Figs. 216 and 217. In this case M, M = 0, and the equation becomes = M30, .. 7.7M2 + 502 5+ 62·1 + 51:03 + 74.94 + 223·1 = 0 The diagram of bending moments AbedB and BfgC, for the spans AB and BC considered as detached girders, should be plotted to a sufficiently large scale, as well as the diagram of upward bending moments APC, and their intersections noted, from which the points of contra-flexure x and x are obtained, 37 feet from B in the first span, and 1.8 foot from B in the second span. The shearing-stress diagram is drawn by considering Ax, and Ca2 as detached spans. The cantilever x,B is loaded at its extremity with the shearing stress at ; it is also loaded with 11.5 tons concentrated at a point 1-6 foot from B. The cantilever Be is loaded at its extremity with 65.4 tons. The diagrams of bending moments and shearing stresses are illustrated in Figs. 217 and 218. |