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LI. It may be proper in this place, to refer to still another form of trussing, which has enjoyed a degree of popular favor, and which differs somewhat from any we have hitherto considered. The plan is seen in outline, in Fig. 16. Each weight is sustained primarily by a pair of equally inclined tension members, and thereby transferred either to the king posts standing upon the abutments, or, to posts sustained by other pairs of equally inclined suspension rods of greater horizontal reach; which in turn, transfer a part to king posts, and another part to a post sustained by obliques of still greater reach, until finally, the whole remaining weight is brought to bear upon the abutments by a single pair of obliques, reaching from the centre to each abutment.
In Fig. 16, are represented three different lengths of obliques, in number, inversely as the respective horizontal reaches. The first set contains 8 pieces reaching horizontally across one panel, and sustaining each !". The next longer set, of four pieces, reach across two panels, and sustain each lw; one-half applied directly, and the other, through posts and short diagonals. The third and longest set, contains but two pieces, reach across four panels, and sustain together 4w; of which 1w is applied directly, lw through two short diagonals, and 2w through two intermediates.
Now, as each set sustains the same aggregate weight, namely 4w, the material in each set, will be represented by this weight multiplied by the square of the lengths respectively, and divided by v: and, making k = v= 1, the squares of respective lengths are 2, 5 and 17, which added together and multiplied by 4w, and w changed to m, gives 96m=amount of material in tension obliques, the only tension members iu the truss.
The upper chord sustains compression equal to the horizontal pull of one oblique member of each class, obviously equal to 107w, with length = 8. Hence, required material equals 84m. End posts sustain together, 7w, centre post 3w, and the two at the quarters, one w each, in all 12w, and the representative for material is 12M ; whence the total for thrust material is 96m, making a grand total of thrust and tension material= 192m. The 8 panels trapezoid with verticals, requires,... 135M Do “ “ without verticals,......... 130M
This comparison exbibits an amount of action in case of the first (Fig. 16), which, considering that it possesses no apparent advantage as to the efficient working of compression material, would seem to ex. clude it, practically, from the list of available plans of construction.
DISTINCTIVE CHARACTERISTICS OF THE ARCH. LII. We have seen that all heavy bodies near the earth's surface (except when falling by gravity or ascending by previous impulse), exert a pressure upon the earth equal to their respective weights. We have also seen that the object of a bridge, in general, is, to sustain bodies over void spaces, by transferring the pressure exerted by them upon the earth, from the
points immediately beneath them, to points at greater or less horizontal distances therefrom.
We have, moreover, seen that this horizontal transfer of pressure can only be effected by oblique forces (neither exactly horizontal nor exactly vertical), and have discussed and compared, in a general way, various combinations of members, capable of effecting this horizontal transfer of pressure.
But, without going into unnecessary recapitulation, we find two or three styles of trussing, possessing more or less distinctive features, which promise decidedly more economical and satisfactory results than any others; and, to make the properties and principles of action of the best and most promising plans as thoroughly understood as may be within the proposed limits of this work, will form a prominent object in the discussions of succeeding pages.
The distinctive feature of the arch, as a sustaining structure, consists in the fact that all the oblique action required to sustain a uniformly distributed load, is exerted by a single member of constantly varying obliquity from centre to ends; each section sustaining all the weight between itself and the centre, or crown of the arch, and none of the weight from the section to the end ; so that the weight sustained at any point, is as the horizontal distance of that point from the centre. Consequently (the arch being supposed in equilibrio under a uniform horizontal load), the hori. zontal thrust at all points must be the same, and the inclination of the tangent at any point should be such that the square of the sine, divided by the cosine of inclination (from the vertical), may give a constant quotient. For, regarding each indefinitely short section of the arch as a brace coinciding with the tangent at
the point of contact, its horizontal thrust equals the weight sustained, multiplied by the horizontal, and di. vided by the vertical reach of the brace. But the horizontal and vertical reaches are respectively as the sine and cosine of the angle made by the tangent with the vertical; that is, as ab and bd, Fig. 17, while the
weight is also as the sine ab, Fig 17.
of the angle adb. Hence, the weight by the horizontal reach, is as ab”, or as the square of the sine of adb; and the constant horizontal thrust of the arch at all points, is as
Now this condition is answered by the parabola, in which bc = cd = } bd, and * = =constant C, whence ab2 = ch x constant 2C, which is the equation of the parabola.
This quality of the arch truss, allowing nearly all of the compressive action to be concentrated upon almost the least possible length, and consequently, enabling the thrust material to work at better advantage than in plars where this action is more distributed, and acts upon a greater number and length of thrust members, enables it to maintain a more successful competition with other plans than we might be led to expect, in view of the greater amount of action upon materials in the arch truss, than what is shown in trusses with parallel chords. Hence, we should not too bastily come to a conclusion unfavorable to the arch truss, on account of the apparent disadvantage it labors under, as to amount of action upon material. These apparent disadvantages are frequently overbalanced by advantages of a practical character, which can not readily be reduced to measurement and calculation.
The preceding general comparisons are to be regarded only as approximations, and should not be taken as conclusive evidence of the superiority or otherwise, of any plan, except in case of very considerable difference in amount of action, with little or no probable advantage in regard to efficient action of material.
EFFECTS OF WEIGHT OF STRUCTURE.
LIII. In preceding analyses, and estimates of stresses upon the various members in bridge trusses, regard has only been had to the effects of movable load, which may be placed upou, or removed from the structure, producing more or less varying strains upon its several parts.
But the materials composing the structure, evidently act in a similar manner with the movable load, in producing stress upon its members; the only difference being, that the weight of structure is constant, always exerting or tending to exert the same influence upon the members, instead of a varying action, such as that produced by the movable load. In order, therefore, to know the absolute stress to which any member is liable, and thereby to be able to give it the required strength and proportions, we have to add the stresses due to constant and occasional loads together.
The weight of structure evidently acts upon the truss in the same manner as if it were concentrated at the nodes along the upper and lower chords, and of the arch, in case of the arch truss. And, since much the larger proportion of it acts at the points where the