737. The use of concrete in the construction of arch bridges is becoming so extended and diversified that it would require a volume by itself to adequately cover the subject, and such a treatment of it is well merited. All that can be attempted here is to describe briefly one or two examples of well proportioned arches, and to give a few hints on methods of design and construction.
Concrete arches may be built with or without steel reinforcement, but for long spans concrete-steel is usually employed. The design of a concrete arch without steel is entirely similar to that of a stone masonry arch, except that planes of weakness corresponding to joints between voussoirs in a masonry arch, may be somewhat more arbitrarily arranged in the former.
In fixed concrete-steel arches, the arch ring is continuous, and is capable of resisting a bending moment. The computations are therefore somewhat more complicated, and until the action of concrete and steel in combination has been more carefully determined, it may be said, in the words of a prominent engineer, that "the development of the system of arches of concrete must necessarily be largely based upon empirical information coupled with sound judgment and work executed with great care." 1 Fortunately, the saving effected by this construction over a masonry arch is usually so great that it is possible to use a large factor of ignorance, and it is to be hoped that the use of concrete-steel for arches of long span will not be given a serious check by the failure, perhaps under unforeseen conditions, of some of the web-like structures that have been built of it.
739. Where the span and rise of the arch are not fixed by the local conditions, the comparative economy of different arrangements and the appearance of the completed structure must govern. Shortening the spans decreases the amount of concrete required in the arches, but increases the pier work, which is usually the most expensive part of the structure.
1 L. L. Buck, Trans. A. S. C. E., April, 1894.
1 Engineering News, Sept. 21, 1899.
These points having been decided, the form to be given the arch ring is next to be considered. While it is desirable that the neutral axis of the arch ring should nearly correspond with the line of pressures for a full load, there is still considerable choice allowed the designer as to the actual form to be given the intrados without serious changes in the amount of material required. As the semicircular arch can usually be adopted for very short spans only, the choice must lie between the segmental, the elliptical, and the polycenterecl arch approaching more or less closely the ellipse, the parabola, or the transformed catenary.
The segmental arch, the parabola and the catenary do not give a pleasing effect at the junction of the arch ring and the abutment, and the curve is sometimes departed from near the springing to make the intrados tangent to the face of the abutment. The final choice will thus usually lie between a true ellipse and the basket-handled arch..
Mr. Edwin A. Thacher, M. Am. Soc. C. E., designer of the Topeka bridge, considers that "arches with solid spandrel filling should be flat at the center and sharper at the ends, approaching an ellipse; while arches with open spandrel spaces should be sharp at the center and flatter at the ends approaching a parabola, or, which is better, sharp at the ends and center and flat at the haunches." 1
The form of the intrados having been fixed, the depth of keystone for an arch without reinforcement is derived, tentatively, from the rules of either Rankin or Trautwine, to be corrected later if necessary. The form of the extrados is then so chosen as to give the required depth of arch ring to confine the line of pressure within the middle third.
The computation of a concrete-steel arch is, as already stated, more involved. The graphical analysis is much the simplest method of deriving the bending moment, direct thrust and shear. The experience of Mr.
Thacher has led him to endeavor to have the line of pressure lie within the middle third of the arch ring, although this is not absolutely necessary in reinforced concrete. The same authority considers it good practice to design the steel work to be capable of taking the entire bending moment with a unit stress of about one-half the elastic limit of the steel.
The thrusts, bending moments and shears at successive sections of the arch ring having been determined, both for full and half span loads, by the graphical methods explained in Greene's "Arches" or Cain's "Elastic Arches," or by the analysis given in Howe's "Treatise on Arches," the dimensions of the arch ring and the steel reinforcement are to be derived by the aid of such formulas as are given by Mr. Thacher1 involving the allowable unit stresses in steel and concrete, and their respective moduli of elasticity.
In short spans, parallel spandrel walls with earth filling between, may be used, but for long spans the spandrels are usually open, that is, built of vertical piers or walls running parallel to the axis of the soffit, and arched over at the top to support the pavement or ballast. This treatment has the following advantages: only vertical forces are transmitted to the arch ring; decreased loads on arch and abutments; increased waterway in case of unusual floods; and better architectural effects.
The beauty of the structure is an important consideration, inasmuch as the decision in favor of a concrete arch as against a steel bridge is usually affected quite as much by considerations of aesthetic effect as of cheapness and durability. In this connection it may be said that in concrete-steel construction there may be little difference in the thickness of arch ring required at the crown and near the springing, but the appearance of the structure will usually be improved by accentuating a little, if necessary, the increased thickness at the springing, except in the case of the semicircular arch in which the eye is accustomed to a nearly uniform thickness of the voussoirs. The appearance is also frequently improved by molding the concrete at the crown to represent a keystone, projecting a little beyond the face of the rest of the arch ring.