Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/350

Rh 330 A E C H . w strain at P is, while the corresponding strain at Q is , and if ~Pq were horizontal these would be alike ; but the obliquity of Tq causes the load Sw which is placed on it to generate a horizontal pressure 68w, wherefore w w+lw M , ^ = 6(&amp;gt;iv, whence t t+6t . w %t 1 n w Si u = or fjt = 1 . t 2 8 to t 6w t Now, when the forms of the intrados and extrados are both given, the values of w, t, 8w, 8t, are thence deducible, so that the value of may always be computed by help of differentiations only; excepting, indeed, that integrations may be needed for determining the value of w, which is the area included between the two curves. In this very simple investigation we have the com plete solution of the principal problem in bridge-building. The data needed for determining the shape of the inner end of the arch-stone are already in the hands of the architect, who must know, from his plans, the weight of each part and the inclination of each joint ; so that, with a very small addition to the labour of his calculations, he is enabled to put the structure completely in equilibrium, even on the supposition of there being no cohesion and no friction ; that is to say, he is enabled to obtain the greatest stability of which a structure having the prescribed outlines is susceptible. Even although he may not care to have the stones actually cut to the computed shape, and may regard their usual roughness and the cement as enough, he may judge, by help of the above formula, of the practi cability of his design ; for if at any place the value of 68t come out with the wrong sign, that is, if u&amp;gt;.8t be less than t.Sw, the building is unstable, whereas if w.ot be greater than t.Sw everywhere, the design, as far as these details go, is a safe one. In every possible arrangement of the details, the hori zontal thrust at the crown of the arch is transmitted to and resisted by the ultimate abutments. The only effect, in this respect, of varieties in the form of con struction is to vary the manner of the distribution of that strain among the horizontal courses. Hence one great and essential element of security, the first thing, indeed, to be seen to, is that the ground at the ends of the pro posed bridge be able to resist this out-thrust. Another, and not less important one is, that the arch-stones be able to withstand the strains upon them. In this respect much depends on the workmanship ; it is all important that the stones touch throughout their whole surfaces : if these surfaces be uneven the stones must necessarily be subjected to transverse strains, and so be liable to fracture. The practice, too common among house-masons, of cheaply obtaining an external appearance of exactitude, by confining their attention to a chisel-breadth around the outside, is not permissible here, nor should any reliance be placed on the layer of mortar for making up the inequalities. The limit to the span of an arch depends primarily on the quality of the material of the arch-stones. At the crown of the arch the horizontal thrust is the weight of as much of the masonry as fills a rectangle whose length is equal to R, the radius of curvature, and whose breadth is A, the effective thickness there ; now this strain has to be borne by the arch-stones, whose depth we shall denote by d, and therefore these stones must be subjected, as it were, to the EA direct pressure of a vertical column whose height is 7-. This column must be much shorter than that which the stone is actually able to bear. The ability of a substance to resist a crushing pressure is generally measured by the length of the column which it is able to support, without reference to the horizontal section ; but it may be questioned whether this mode of estimation be a sound one, for it does seem natural to suppose that a block three inches square should bear a greater load than nine separate blocks each one inch square, seeing that the centre block in the entire stone is protected on all sides ; and thus it is possible that we under-estimate the greatest practicable span of a stone arch. This difficult subject belongs to the doctrine of &quot; Strength of Materials.&quot; ARCH, SKEWED. In the earlier days of bridge-building the road was led so as to cross the river or ravine perpen dicularly, but in modern engineering we cannot always afford to make the detour necessary for this purpose, and must have recourse to the skeived or oblique arch, having its plan rhomboidal, not rectangular. If AB, CD, figure 8, represent the roadway, and EF, GH, the boundaries of the abutment walls placed obliquely, we easily perceive that the thrust cannot be perpendicular to the abutments, for then it would go out on the side walls which have no means of resistance ; the thrust can only be resisted in the direction of the road. Hence if the structure be divided into a multitude of slices by vertical planes parallel to the parapet, the strains belonging to each slice must be resisted within that slice, and each should form an arch capable of standing by itself. The abutment, therefore, cannot have a continuous surface as in the common or right arch, but must be cut in steps to resist the oblique pressure ; wherefore also the ultimate founda tion stones must present surfaces perpendicular to the road. Attending for the moment to one only of these divisions, say to a thin slice contiguous to the side wall EG, let us study the manner in which the arch-stones in it must be shaped. At the crown I the pressure is horizontal in the plane EIG, and therefore the joint of the stones there must be perpendicular to AB, and so also must be its projection on the horizontal plane. Proceeding along the line of the curve to the point R, we observe that the pressure there must be in the direction of a tangent to the curve, wherefore the surface of a joint at R must be perpendicular to that tangent, and the