EMPIRICAL FORMULAS FOR THE PROPORTIONS OF ARCM.
Numerous formulas derived from existing structures have been proposed for use in designing voussoir arches. Such formula3 are useful as guides in assuming proportions to be tested by theory, and also as indicating what actual practice is and thus affording data by which to check the results obtained by theory.
As proof of the reliability of such formulas, they are frequently accompanied by tables showing their agreement with actual struc tures. Ccneerning this method of proof, it is necessary to notice that (1) if the structures were selected because their dimensions agreed with the formula, nothing is proved; and (2) if the structures were designed according to the formula to be tested, nothing is proved except that the formula represents practice which is probably safe.
At best, a formula derived from existing structures only indicates safe construction, but gives no information as to the degree of safety. Such formulas usually state the relation between the principal dimen sions; but the stability of an arch can not be determined from the dime 3sions alone, for it depends upon various attendant circumstances —as the condition of the loading (if earth, upon whether loose or compact; and if masonry, upon the bonding, the mortar, etc.), the quality of the materials and of the workmanship, the manner of constructing and striking the centers, the spreading of the abutments, the settlement of the foundations, etc. The failure of an arch is a very instructive object lesson, and should be most carefully studied, since it indicates the least degree of stability consistent with safety.
Many masonry arches are excessively strong; and hence there are empirical formulas which agree with existing structures, but which differ from each other 300 or 400 per cent. All factors of the problem must be borne in mind in comparing empirical formulas either with each other or with theoretical results.
A number of the more important empirical formulas will now be given, but without any attempt at comparisons, owing to the lack of space and of the necessary data.
Let d = the depth at the crown, in feet; p = the radius of curvature of the intrados, in feet; r = the rise, in feet; s = the span, in feet.
American Practice. Trautwine's formula for the depth of the keystone for a first-class cut-stone arch, whether circular or elliptical, is "For second-class work, this depth may be increased about one eighth part; and for brick work or fair rubble, about one third." English Practice. Rankine's formula for the depth of keystone for a single arch is for an arch of a series, and for tunnel arches, where the ground is of the firmest and safest, and for soft and slipping materials twice the above.
The segmental arches of the Rennies and the Stephenson, which are generally regarded as models, "have a thickness at the crown of from - to s of the span, or of from - to iv of the radius of the intrados." French Practice.* Perronnet, a celebrated French engi neer, is frequently credited with the formula, as being applicable to arches of all forms—semicircular, segmental, elliptical, or basket-handled—and to railroad bridges or arches sustaining heavy surcharges of earth. "Perronnet does not seem, however, to have paid much attention to the rule; but has made his bridges much lighter than the rule would require." Other formulae of the above form, but having different constants, are also frequently credited to the same authority. Evidently Perronnet varied the proportions of his arches according to the strength and weight of the material, the closeness of the joints, the quality of mortar, etc.; and hence different examples of his work give different formulas. However, it is remarkable that according to all formulas credited to Perronnet the thickness at the crown is independent of the rise, and varies only with the span.