THE ARCH TRUSS. CXXXVII. A parabolic Arch Truss of the same length, depth and load as allowed in the five preceding cases, and having 9 panels, will compare, as to representative of amount of material, as follows: Let w, represent the variable, and w,,, fw,, the permanent panel-load. Then, taking the greatest depth of truss (15f.), as the unit of length, as before, the length of chord will be 6.666, and the verticals respectively 1, 0.9, 0.7, and 0.4. .74074 The length of panel (11.111 f.), being divided by 15f. (the unit), gives 0.74074. Hence, tension of chord = 4 (w, +w,,) × = 13 × 7.4074w,, which, multiplied by length of chord (= 6.666), and w,, changed to м,, gives representative of material 9.8765 × 63м, = M, 65.843M,; in which M, is the unit of material, proportional to the unit of length (15',) x unit of stress, w,. = The maximum tension of diagonals, as determined instrumentally by process explained [XXVII, &c.,] varies from 1.11w,, to 13w,; and, taking the highest, multiplying by the aggregate length (15.4), and changing w, to M,, we obtain material = 20.52M,. The verticals sustain tension, each, aggregate length of 6, giving material a total of tension material = 94.376м,. = = 14w,, with an 8м,; making The horizontal thrust of the arch, must be in all parts the same as the tension of the chord (at the maximum under full load), and it is manifest that the material for each segment, must be to that of the middle segment, as the squares of respective lengths to unity; that is, equal to material in said middle segment, multiplied by squares of respective lengths. = = But the representative for the middle piece equals th that of the lower chord, 7.316м,. Hence, this amount multiplied by the sum of squares of all the others, +1 for the middle segment, found to be 9.058+1, 10.058, gives, to represent material for the whole arch, 73.584м,. Then, the vertical members are liable to be exposed to compressive action, represented by the small amount of 2.058M,, which added to the above, gives a total of compression material, equal to 75.642м,. = Now, the factor M,, here used, is to the factor M used in the preceding cases, manifestly, as ×}, to }, as, whence, 12м, 8M; and we reduce the coefficients of M,, by, and change м, to M, to bring the last results to the same standard measure as in the preceding. Effecting these changes, we have, for tension material, Chord 43.895M, + Diagonals 13.689M + Verticals 5.333M, equal to a total of 62.917 M. For compression, Arch, 49.056+Verticals, 1.372, 50.428M. = SYNOPSIS OF PRECEDING DEDUCTIONS. The following tabulated statement may promote the convenience of comparison: 95.755 50.428 113.345 Actual, but not a determinate quantity. Thrust Diagonals + Arch. CXXXVIII. The figures in this table are to be understood in all cases as prefixed to the quantity M, which, as far as relates to tension material, represents a determinate amount of wrought iron; while, as it relates to compression material, M represents an amount of cast or wrought iron, varying as the forms and proportions of parts vary. But, in the present discussion M may be assumed to have a uniform value in ex pressions relating to material under the heading of chords; and of ends, whether oblique or vertical. The quantities under the head posts, require in general, probable twice as high a value for M, as that required for the other classes of thrust members, as it regards all but the first named truss, while the first is not represented in that column at all, although the parts there referred to are as indispensible, practically, and require nearly as much material as corresponding parts in the other plans. With regard to plan No 2 (the Finck). 6 posts actually required (two of which, at the quarterings, sustain determinate weight equal to W each), are also omitted in the table, to place this plan upon an equal footing with the preceding one. There is also a consideration with regard to the ef fects of load upon these two trusses, especially the first, which render it partially necessary to use diagonal ties, or "panel rods" in the several panels; and such have usually been introduced wherever such bridges have been constructed. As any one pair of suspension rods in the Bollman truss may be under full load, while the others are without load, the loaded node would, in such case, be depressed, while that on either side would retain nearly its normal position. Thus would result an obliquity in panels adjacent to the loaded point, and consequently, a tendency to kink in the upper chord, by opening the joint above the loaded point upon the under side, and the next joint either way, upon the upper side. Hence the compression of certain chord segments would be thrown upon the extreme upper side at one end, and the lower side at the other end. This would be decidedly an unfavorable condition, which the panel rods are used to obviate by distributing the load of loaded points over adjacent, and more remote parts of the truss. Otherwise, the bridge would act under a passing load, somewhat in the manner of a pontoon bridge. By estimating a reasonable amount of material for posts and panel ties, the figures in the table, opposite the first two trusses would be materially increased. Hence, it must be obvious that the necessary material for the two above named trusses, is not so fully represented in the table, as in the case of the other four; with regard to which assigning proper values to M in the different columns of the table, and assuming the members to adhere to one another as firmly as the different portions of each cohere among themselves, a complete truss would be formed in either case (of dimensions as above assumed), sufficient to be used in a bridge required to bear a gross load equal to 4 times the weight of superstructure; provided the proper ratio of safe variable load to weight of structure be as 3 to 1; as is nearly the case with regard to a 100 foot bridge.* *M, in the preceding table, represents a piece of iron, 15' long sus ficient to sustain with safety, a weight W, equal to of the gross maximum load for one truss of a 100ft. bridge. Allowing 1,000lbs. to the linea! foot for movable, and 3331bs. for permanent load, W, represents 133,333lbs. 16,6661bs. Then, reckoning the safe stress of In such case, the results already obtained, would show the relative cost of the several trusses (excepting the first two), with almost absolute exactness. But, as the parts of a truss can not be so connected and welded into a single piece, without enlargements at the joinings, by any skill or process now in use, we have to include as an item of cost, in all plans, a considerable amount of material above and beyond the net lengths and cross-sections, as here before determined with regard to the trusses under discussion, required for the lapping of parts, screws and nuts, eyes and pins, &c., to form the connections of the different members with one another. With regard to the trusses under comparison, no obvious reason presents itself, why any one should require a percentage of allowance for connections materially greater than another. Leaving out the two first, as perhaps already sufficiently discussed, the others consist of about the same number of necessary members, and with the exception of the arch truss, admit of nearly the same forms and connections of parts. The Isometric, or Trapezoid without verticals, presents the fewest lines in the diagram; but some six of those lines represent both tension and thrust members, either separate or combined, which probably complicates the iron (thrust or tension), at- say 10,000lbs. to the inch of cross-section, it takes 1 square inches to sustain the weight W; being about 5lbs. to the foot, or 84glbs. for 15'. This, increased by-say, 20 per c. for extra material in connections, gives the practical value of M; which, multiplied by the co-efficient of M in the table, produces approximately, the respective weights of trusses. = Now, 1 x 84.37 1011lbs. which multiplied by 113.345, the coefficient for the Arch truss, gives for the weight of that truss, 11.476lbs. Add for 10 feet width of platform (with wooden beams), say 5,000ft. b. m. of timber and plank, equal to about 20,000 lbs., and we have 31.476lbs. to represent the permanent load of the truss. But we have assumed a truss proportioned to sustain with safety 133,3331b., which is a little more than 4 times the weight of structure here above esti mated as supported by the truss. |