details of connection, quite as much as the extra three

members in truss No. 4. The Post truss presents the larger number of acting members, even omitting six counter ties seen in the diagram, with apparently no advantage as to modes of connection. Both the Post and the Isometric have 10 members represented in the 4th column of the table, whereas the Whipple truss has only 7, and these the shortest of all; and, as the material in these parts manifestly acts at a disadvantage, they being comparatively long and slim, and sustaining slight action, any excess in their number, would seem to be unfavorable to economy.

It is believed, however, that the Post truss would be improved in economy by reducing it to a trapezoidal contour, as, for instance, by removing the parts outside of bx and kn (Fig. 49), and changing the tension pieces av and ol for others connecting b with v, and o with k; thus converting the figure to a trapezoid very similar to that of Fig. 50; and, by striking out one panel from the latter, and arranging parts as in Fig. 20, except as to inclination, the relative merits of inclined and vertical posts, as represented in these two plans may be fairly tested.

Analysis of trusses modified as just indicated, show tension material slightly in preponderance with the vertical, and thrust material a little the greater with inclined posts; the average being about one per cent greater in the case of vertical posts.

This balance, though trifling in amount, is upon the side where it was to be lookod for, in view of the result of investigations had with reference to Figures 12 and 13 [XXXIX-XLVI], as well as the case of the Isometric. Both the Post truss and the Isometric, as to principle of action, may be classed with Fig. 13, where

weight is transferred from oblique to oblique, and not from oblique to vertical, and the contrary. The same may be said of truss Fig. 15, sometimes called the Triangular, in which verticals are used merely to transfer the action of weight from the point of application to the connections of the obliques; after which, the weight has no action upon verticals.

Now finally, we see by table of results, that if the Post truss be changed to the trapezoidal form, as above suggested, it will occupy a position, as to amount of material, or more strictly speaking, the amount of ac. tion upon material, between Fig. 50 and Fig. 51; which latter differ from one another less than 2 per cent; a difference, which would undoubtedly be increased somewhat, under different general proportions of trusses. For instance, while Fig. 50, shows an inclination of diagonals used in connection with verticals, probably nearly approaching the optimum, Fig. 51, though superior to the true Isometric (with angles of 60°), in the greater inclination of its obliques, would give still better results with an inclination of about 40°.

CXXXIX. On the whole, we must look to other quarters than the amount of action upon material, for plausible ground upon which to found a decided preference for either of the three plans in question. A difference of two or three per C., and even more, may easily result from greater or less facility of constructing and erecting the structure, while a regard for appearance may also be worthy of consideration. Hence, Engineers and builders will adopt one or another plan, according to individual taste and judgment, and the one who carries out the principles of either system with

the greatest skill, and the best materials and workmanship, will probably produce the best bridge.

Judging from the preceding tabulated statement, the arch truss seems, prima facie, to labor under a somewhat formidable disadvantage in the fact that it shows an amount of action upon material 10 or 15 per cent. greater than the three preceding plans just especially referred to. But for the light of experience, we might be led to discard the plan without a trial.

But, having chanced to be the first plan of iron Truss successfully put in use, aud having had its capabilities fully tried and demonstrated, before any formidable competitor appeared in the field, it could not be dislodged from its position, until a rival plan could not only theoretically, but also practically demonstrate its superior claim to public favor.

The result has been such as to show that even a very considerable excess of action upon material, may be overbalanced by more advantageous action of thrust material, and greater simplicity and facility of constructtion; insomuch that the Whipple Patent Arch Truss, with trifling modifications from the original pattern, has competed successfully with all other plans, for the class of structures it was originally designed and recommended for (common bridges of 50 to 100 feet), during more than a quarter of a century, which has been fruitful in efforts at improvement in iron bridge construction.

COUNTER BRACING.

The elasticity of solid materials, is manifested in bridge trusses, by their downward deflection under.

load, and the recovery of their previous form and po sition on the removal of the load.

This arises principally, from the temporary elongation of parts exposed to tension, and the contraction of those exposed to compression, according to lawa and principles supposed to be understood.

The deflection of trusses within the usual limits, when properly proportioned, is not essentially detrimental to their safety and durability; but rather enables them the better to resist sudden impulses,— except in case of a regular succession of impulses, at intervals corresponding with those of the natural vibrations of the structure, or with some multiple or even division thereof; a result frequently noticeable, and sometimes, to a degree somewhat unpleasant to the eye, as well as suggestive of danger. Hence, great emphasis is often employed, in expressing the supposed advantages of "counter bracing," as a means of stiffening trusses, and preventing, or diminishing their vibration.

What is technically called "counter-bracing," as applied to bridge trusses, is the introduction of a set of diagonal, or oblique pieces or members, to act in antagonism to the main diagonals, whether acting by tension or thrust, which contribute toward sustaining the weight of structure and load; the object being, to retain in the truss when unloaded, more or less of the deflection produced by the load, when the truss is loaded.

My object at the present time is, to exhibit the process and results of my investigations as to the theory and effects of this counter-bracing, as usually practiced in bridge building, and to state the conclusions arrived

at, as to the value of counter-braces, towards effecting the object proposed.

FIG. 52.

a b

d

e

h i

8 r q po n m

I assume a truss (see Fig. 52) composed of horizontal chords (of equal lengths), at top and bottom, vertical posts, and diagonal tension rods, inclined at 45°, or at any other given inclination,-the truss being uniformly loaded from end to end, and so proportioned that all of the above named parts, in that condition of the load, shall undergo an amount of extension or compression, proportional to the respective lengths of parts, multiplied by a constant factor (E), equal to the elastic change effected in a length equal to that of the uprights between centres of chords, which is assumed as the unit of length for the occasion. Then, let L represent the length of truss, P, the number of panels, H, equal to LP, the horizontal reach of diagonals, and D (equal to 2LE), the difference in length, occasioned by extension of lower, and compression of upper chord.

Now, assuming no change in lengths of diagonals and verticals, it is manifest that the chords assume, in these circumstances, the forms of two similar and concentric arcs of circles, of which the difference in length is to the mean length, as the difference of radii is to the mean radius, R.

But the difference of radii manifestly equals the distance between chords, equal to 1. Using, then, the representative signs before adopted, we have D: L: 1: R; whence ...... R L + D.

=