427. Value of n. Still another simplification may be made, on the assamptiun that the moment of inertia varies as the cube of the depth, and also that we may increase the depth of the rib as desired. Assume that the depth of the rib is increased so that at any point n= ds d_x (sec Fig. 230); ds is always greater than day, and n is a ratio varying from one upward. Then, on the assumption that: we may compute a series of values for It in terms of the height at the center h, which will correspond to various angles a. For each angle, we find the ratio between it and that will correspond to the value which n has for that particular angle on the basis that . If we dx substitute this value of in Equations 52, 53, and 54, we shall have Therefore, when all sections have the same moment of inertia, and n is uniformly 1, use Equations 52, 53, and 54, ignoring the n. When an increase in depth of section, as indicated above, will fulfil the ultimate requirements, there is an advantage of simplicity in inn-king the sections accordingly, and using the Equations 56, 57, and 58. When it proves necessary to vary the sections according to some different law, U must be determined at frequent intervals, spaced by a uniform ds, and the summations of Equations 52, 53, and 54 determined. The remainder of this method follows out the assumption that n varies as ds — or that ds d.r 42S. Position of yin. We may locate via by satisfying Equa 51, which may be written z' thrz =0. But this integral is represented by the shaded area (Fig. 231), which is the equivalent of saying that the segment OCB =-- the rectangle OK X OB. If OCT were a parabola, OK would exactly equal 3 CD. Even with circular arcs, the ratio is approximately correct if the angle is small. There fore, for flat circular arcs, draw via at a the height of the arc. If neces sary, increase the height according to the figures given in the accom panying tabular form: Of course, for full-centered arches in which 2 a = LSO°, the error of the s rule is very great, but the tabular values are correct.
Since an elliptic arc may be considered as a circle in which the vertical ordinates have all been shortened by some constant ratio, the same law and same percentage of error will hold true.
For any other curve, particularly multi-centered curves, the position of via may be found by determining by trial a position such that the summation of equally spaced ordinates is zero.
429. Weight and Thickness of Arch. Theoretically, this should be known before any calculations are made; but since the weight of filling and pavement are always large, and their unit-weight is but little less than that of the concrete, it is possible to estimate from experience on the required crown thickness, and to make the thickness at other points in the required ratio. If this should prove too thin (or too thick), all sections can be changed in the same ratio. If the outline of the intrados is determined (as in the case of an arch spanning railroad tracks), and the upper surface line (of earthwork or pavement) is also known, the change in the arch ring will mean only a change in weight dub to the difference of uilit-weight of con crete and earth filling. If the original assumption is even reasonably close, this difference will hardly exceed the uncertainties in the loading.
430. Intrados. The span and rise are frequently predetermined. Fortunately this method is applicable to almost any form of curve, if the change in curvature is not too extreme. Even if the arch is very flat and the curvature very sharp near the abutments, it only means that the virtual abutment is some where on the haunches. There fore, draw the intrados; assume and lay off a reasonable crown thickness ; 'multiply this thickness by the factors given in the tabular form in Article 427 for the angles with vertical lines made by the various normals to the curve. These thicknesses can be laid off, and the extrados can be drawn through the points.
But since the curve OCB of Fig. 22S does not represent either the intrados or extrados, but the center line of the rib, we should draw a line midway between the intrados and extrados which will represent the center line of the rib, and which corresponds to the line OCB in the figures which refer to the theoretical demonstrations. This also means that the span of the rib, measured between the centers of the skewbacks, will be slightly greater than the nominal clear span. The rise of the center of the rib above the line joining the abutment points 0 and B will in general be slightly different from the nominal rise of the arch.