DISCUSSION. The writer has read Mr. Vautelet's paper, on Bridge Calculations, Mr. Shelford. with considerable interest; and on comparing his formula with a table, in accordance with which the writer has built a number of bridges in the last few years, finds that it agrees fairly well. This table is to be found in the Minutes of Proceedings of the Institution of Civil Engineers, Volume 82, Page 349, and probably represents the latest practice in England, where no larger work than the Hull & Barnsley Railway, for which the writer was Chief Engineer, has been constructed for many years. At the Birmingham Meeting of the British Association in 1886, the writer, jointly with Mr. A. H. Shield, Associate Member Inst. C.E., read a paper on the design of Girder Bridges, which the Committee did them the honor to print in extenso in their Proceedings. The author's first formulæ are of considerable interest, and afford a Mr. A. Shield. ready means of computing the stresses resulting from hypothetical distributed loads of different intensities, of which the extent of the load of greater intensity (and the intensities of the loads) is defined. It remains, however, to determine the values to be assigned to the intensity and extent of the distributed engine load P, in order that the formula may give results which will cover the stresses produced by the actual loads. In this respect a sign of weakness is apparent in the change made by the author in the values of P and V, when the span reaches 2 105 feet. The expression R=4,600-1360 (1) for spans from 21 to 105 feet takes into consideration the load due to one engine only (although it is usual to specify for two), and its limit of application is therefore about 55 feet. Beyond this length with a load specification of two engines of the usual type, there are two distributed loads of (say) 4,600 lbs. per foot run for lengths of (say) 21 feet, with a definite interval between them in which the load is (say) 3,000 lbs. per foot, and preceded and followed by a similar train load—a condition which is not covered by the formulæ. Having regard to the difficulties which attend the application of the author's formula, in the determination of the co-efficients and the limits of its applicability to spans, which either admit only one engine, or are Prof Burr. so great that the weight of two engines with their tenders may, without Referring to the second part of the author's first formula, viz., that relating to the equivalent for shearing stress, it is to be noted that the result obtained by the author, as an approximation to the case of a truss bridge, may be directly derived from the previous expressions for bending moments, and if so derived takes the form V 2 R = P — (P—p) (1—1—a) * This expression shews perhaps more clearly than the form R = P — (P—p) ( 1 − .1)' that in using an equivalent uniform load to estimate shearing stress, such equivalent must be determined separately for estimating the shear at each apex or panel, and is for each such apex equal to the equivalent. uniform load for a span equal to the length of that portion of the bridge, which must be loaded to give a maximum shear at that apex—i.e., nl or L-a. The wisdom of using arbitrary weight concentrations, more or less like those of locomotives and tenders, in the design of railway structures, is being widely questioned by engineers, and most properly. In the United States the matter has been receiving increasing and serious attention for several years past, and it seems probable that a more simple and equally accurate system will, at no remote date, be followed to a very considerable extent. The present system is certainly most wasteful of time, and the results are no more approximate in character than those reached by a much more simple and equally rational method. The constantly varying and increasing locomotive and tender wheel weights completely neutralize the accuracy of the best present methods, with weight concentrations, and render nugatory the extravagant expenditure of time in computations. Such a system cannot be too quickly changed; but it should not be supplanted by another equally objectionable. If the writer understands the first paragraph of Mr. Vautelet's paper, particularly the last sentence, he would propose to use an "equivalent distributed load," which "will vary with the length of span." In the first place, there is, in fact, no such thing as a uniform load equivalent to a given system of concentrations. The maxima stresses due to the latter in every chord panel and web member would, in the general case, each require a separate and independent "equivalent uniform load." In the second place, even if such an equivalent uniform load were a possibility, it would require a very considerable computation for each. different system of concentrations. In any case the method would effectually defeat its purpose. This matter of equivalent uniform load has been most thoroughly considered by many of the engineers of the United States within the past few years, and with rare exceptions the method has been rejected as entirely inadequate to accomplish the desired results. It has frequently been shown, however, particularly by Mr. Geo. H. Pegram, C.E., of St. Louis, that a simple uniform load with a single concentration at its head (or at some intermediate point, if the locomotives are so placed) will produce essentially the same stresses as the same uniform train load headed by two locomotives, or with the same locomotives placed at some intermediate point. Nothing could be simpler than such a system of loading, and the single concentration can be easily inferred from the locomotive weight. The method resulting from the use of such loading is now receiving very favorable attention from many quarters, and its early introduction is not unlikely. Regarding the particular equations found by Mr. Vautelet, it is to be observed that while their forms may be new, his method has long been used in the engineering office of the Phoenix Bridge Co., as well as in those of other American bridge companies; in fact the general method for parallel chords has for nearly four years been published in the writer's work on "Stresses in Bridge and Roof Trusses." Again, although Mr. Vautelet does not mention the fact, his results for shears do not give the maxima web stresses unless the two chords of the trusses are parallel. The writer believes that Mr. Henry W. Hodge, C.E. Student Institution of Civil Engineers, of the engineer. ing office of the Phoenix Bridge Co., was one of the first, if not the first, to point out the fact that different positions of loading from those for parallel chords are required for chords not parallel, when maxima web stresses are sought. The following investigation shows the effects of non-parallelism, and gives the positions of loading as well as the maxima web stresses themselves for the principal types of trusses. Two principal cases occur in connection with types of structures ordinarily used in engineering practices. That one to be tested first is the case of the intersection of the chord sections, in any panel, lying below the inclined web member in the same panel; the other is the case of the intersection lying above the inclined web member. Application of these principal cases to special features can easily be made after the general results are obtained. Case I. The intersection of chord sections below the inclined web member. Let 1 (Fig. 13.) be the length of span; d the distance from end of span to the point of intersection H of the chord sections in the panel in question; m the distance from the end of the span to the same panel whose length is p; S, the stress in the web member under consideration, and L its lever arm about H; a, b, c, etc., the distances separating W1 from W., W, from W3, etc., etc.; W1, W2, etc., the weights resting between G and D, and W,, W,, etc., the weights resting in the panel p, while W, distant x from E, is the last weight, resting on the span from C to toward E; b is the distance from D to the nearest weight W, The reaction at G is :— 2 Sh=Rd - (l+d—a−b - W1 (l + d — a − b − ... − x ) − W, (l + d− b − c − - (1 − ... - x) 1 By moving the entire load the distance Ax toward G, remembering that the change in the value of R will be 4 R = (W1 + W2 +............+Wn) 2 the change in Sh becomes: W, etc). Дх Δα AS h=(W1+W2+ + W1) d - (W1 + W2+ etc 1 1 For a maximum or minimum AS h = 0, hence :— Ah Τ (3) Bearing in mind that the first parenthesis in the second member of eq. (4) represents the load between the panel p and the left end of span, and that the second represents the load in panel p itself, it will be at once seen that, when the load extends from E to W, S is the maximum main stress, and that when it extends from G to W, (i.e., to the weight farthest towards E), S is the maximum counter stress. Eq. (4), therefore, as it stands, gives the conditions for maximum main or counter stresses. 4 Eq. (4) is perfectly general in character, and covers all systems of loading whatever, but it may be put in special forms for convenient application in special cases. Example (1) Uniform Load. If the load is continuous or only partially so, and w is its intensity (i.e., its amount per. lineal unit) at any point distant x from E, then its various concentrations W1, W2, W3, etc., will be separated by wdx. If, further, the load is uniform and continuous, w is constant, and W1 + W2+........+ W2 = w x1, x1 separating the length of uniform load on the bridge. In the same manner, if x, represent the length of uniform load from D towards G, 1 1 2 (W1 + W2+ etc.)= w X2; and (W 3 + W1 + etc. ) = wrp; r being the fractional part of the panel p covered by the uniform load m Equ. (4) then becomes : As with the general case so with equ. (5), it is so written as to give both maximum main and counter stresses. |
