length of joists. Hence, if we could find the point where the cost of connections (consisting of extra material in the lappings of parts, connecting pins, screws and nuts, and enlarged sections at the ends of members, together with the extra labor in forming the connections), becomes equal to the whole cost of material in joists or stringers; that would seem to indicate the proper width of panel, or value of h, as far as depends upon these elements.

But aside from the fact that our data upon this question are so few and so imperfect, that it would be mere charlatanism to attempt to reduce the matter to a mathematical formula, the occasions would be so rare which would admit of the application of such formula, without incurring disadvantages in other respects, such as improper inclination of diagonals, unsuitable ratio of length to depth of truss, &c., that no attempt will be here made to give any thing more definite upon this point, than to refer to the best precedents and practice of the times; which seem to confine the range of width of panel mostly within the limits of 8 and 15 feet.

Within these limits, and seldom reaching either extreme, plans may be adapted to any of the ordinary lengths of span, by adopting the single or double cancelated trusses, Figs. 12 and 13, or 18 and 19, or the arch truss Fig. 11, (which unquestionably contain the essential principles and combinations of the best trusses in use), according to length of span, the purposes of the bridges respectively, or the taste and judgment of engineers and builders.

ARCH BRIDGES.

LXIX. An arch bridge may be distinguished from an Arch Truss Bridge, by the fact that in the former, the bridge and its load are sustained by one or more arches without chords; and, consequently, requiring external means to withstand the horizontal thrust or action of the arches at either end; which means are afforded by heavy abutments and piers, in case of erect arches, and by towers and anchorage in the earth, in case of inverted, or suspension arches.

It is not the purpose of this work to treat elaborately of either of these forms of bridging, as the author's experience and investigations have been mostly confined to truss bridge construction. But as some of the largest bridge enterprises and achievements of the age are designed upon the principles here referred to, a brief notice of the subject, and some of the conditions affecting the use of these classes of bridges, may be regarded as desirable in a work of this kind.

Suspension, or inverted arch bridges of very great spans, have long been in use, both in this and foreign countries; and the capabilities of that system have been pretty thoroughly tested experimentally and practically.

But bridges supported by erect metallic arches, have hitherto been confined to structures of moderate span. Within a few years, however, the magnificent enterprise of spanning the Mississippi at St. Louis by three noble stretches of about 500 feet each, supported each by four arched ribs of cast steel, has been undertaken and is understood to be in rapid process of execution. The interest naturally felt in the progress and final result of this grand enterprise, by students and practi

tioners in the engineering profession, will perhaps aid in rendering the following brief, and somewhat superficial discussion acceptable.

LXX. An erect arch subjected to the action of weight, or vertical pressure, is in a condition of unstable equilibrium; and can only stand while the weight is so distributed that all the forces acting at each point of its length, are in equilibrio. To illustrate this, we may assume the arch to be composed of short straight segments meeting and forming certain angles with one another, and the weights applied at the angular points.

A weight at c, Fig. 22, for instance, acts vertically, and, if de be produced till it meet the vertical drawn

through bin m, then the triangle bem has its sides. respectively parallel with the directions of three forces acting at the point c; namely, the weight at the point c, the thrust of the segment bc, and that of de. Hence, if these three forces be to one another as the sides of said triangle,—that is, if the weight (w): thrust of bc: thrust of de:: bm: bc: cm, then they are in equilibrio. If w be greater than is indicated by this proportion, the point c will be depressed, bcd approaching nearer and nearer to a straight line, and becoming less and

less able to support the weight, and a collapse must result.

If w be less than the above proportion indicates, it will be unable to withstand the upward tendency of the point c, due to the thrust of be and de (or, to the preponderance of the vertical thrust of bc, over that of de), the point c will rise, the upward tendency becom ing greater and greater, and the result will be a collapse, as before. The same reasoning, and the same inference, apply to any other angular point, as at c. It is, therefore, only in theory that such a thing as an equilibrated erect arch, can exist. The arch is here considered as a geometrical line without breadth or thickness.

It is this property of instability, in the Erect Arch, that the diagonals in the Arch Truss, [Figs. 5 and 11] are designed to obviate, and to enable the arch to retain its form and stability under a variable load.

LXXI. Still, in theory, an arch may be in equilibrio with any given distribution of load, whenever the points a, b, c, etc., are so situated that the sides of the triangle bem, for instance, formed by a vertical with lines respectively coinciding or parallel with the two segments meeting at c, are proportional to the 3 forces acting at c, as above stated, and so at the other angular points of the figure.

To construct an equilibrated arch adapted to a given distribution of load, consisting of determinate weights at given horizontal intervals between the extremities of the arch, we may proceed as follows:

Draw a horizontal line representing the chord ak, and upon the vertical Cft, erected from its centre, take Cf equal to the required versed sine, or depth of the

arch at the centre. Also, take ft=Cf, and erect verticals upon the chord, at all the points at which the load is applied, and join a and t.

Then, if the load be uniformly distributed (horizontally) upon the arch, we have seen, [LII], the arch should be a parabola, to which of course, at is tangent at the extremity, a. But, regarding ab as tangent to the curve at r, half way between a and b (horizontally), we seek the abscissa fs, which is to Cf:: rs2: aC2. Then, taking the distance of fu=fs, au is tangent to the curve at r, and coincides with the first segment (ab) of the arch. (These segments are supposed to be so short, that the tangent and curve may be regarded as essentially coinciding, for the length of a single segment).

Now, the thrust of ab, is to the whole weight bearing at a, as ru to us; and, erecting the vertical al, such that al: ab:: weight at b: thrust of ab, and drawing the straight line lbc, cutting the second vertical in c, we have be for the second segment of the arch, being in the line of lb, which represents the resultant of the two forces acting at b; namely the weight at b, and the thrust of ab.

In like manner, take bm, representing the weight at c, and the straight line mcd, meeting the third vertical in d, gives cd as the third segment of the arch.

Repeat the same operation for each of the succeeding segments de, ef, &c., till the arch is completed, and it is obvious that the forces acting at each of the several angular points b, c, d, &c., are in equilibrio; and that the arch throughout is, theoretically, in a state of equilibrium.

We may vary this process so as to secure greater accuracy of construction, in the following manner :