The propriety of leaving the dimensions as first assumed for ,trial figures, depends, therefore, on the following considerations: First—The loading assumed above for the uniformly distributed load is as great a loading as that produced by ordinary locomotives such as are used on the majority of railroads; while the locomotive requirements as assumed above are excessive, and are used on only a comparatively few railroads.
Second--If an equilibrium polygon had been started from a point nearer the intrados than the point in (using the same pole o.,'"), it would have passed a little below the point c, and likewise a little nearer the intrados than the point n. Although this would have brought the equilibrium polygon a little nearer to the intrados on the right-hand haunch of the arch, it would likewise have drawn it away from the extrados on the left-hand haunch. Although it is uncertain just which equilibrium polygon, among the infinite number which may mathematically be drawn, will actually represent the true equilib rium polygon, there is reason to believe that the true equilibrium polygon is the one of which the summation of the intensity of pres sures at the various joints is a minimum; and it is evident from mere inspection, that an equilibrium polygon drawn a little nearer the center (as described above) will have a slightly less summation of intensity of pressure, although the intensity of pressure on the joints on the right-hand haunch will rapidly increase as the polygon ap proaches the intrados. It is therefore quite possible that the true equilibrium polygon would have a less intensity of pressure at the joint between voussoirs Nos. 4 and 5.
If it is still desired to increase the thickness of the arch so that the line of pressure will pass further from the extrados, it may be clone approximately as indicated for a similar problem in Article 414. Evidently the keystone is sufficiently thick, and the voussoirs at the abutments alsc have ample thickness. The extrados must evidently be changed from an arc of a circle to some form of curve which shall pass through the same three points at the crown and the two abut ments. This may be either an ellipse or a three-centered or five centered curve. Although it will cause an extra loading on the haunches of the arch to increase the thickness of the arch on the haunches, and although this will cause the equilibrium polygon to rise somewhat, the rise of the equilibrium polygon will not be nearly so rapid as the increase in the thickness of the arch; and therefore the added thickness will add very nearly that same amount to the distance from the extrados to the equilibrium polygon. For example, by adding a little over three inches to the thickness of the arch at vous soirs Nos. 4 and 5, the distance from the equilibrium polygon to the extrados would be increased from three inches to six inches, and the maximum intensity of pressure on the joint would be approximately half of the previous figure. To be perfectly sure of the results, of course, the problem should be again worked out on the basis of the new dimensions for the arch.
The required radii for a multi-centered arch which should have this required extrados, or the axes of an arc of an ellipse which should pass through the required points, are best determined by trial. The effect of the added thickness on the load line for the right-hand side of the arch, will be to bring the load line nearer to the center of the voussoirs, and therefore will actually improve the conditions on that side of the arch. Of course, when the concentrated load is over the right-hand side of the arch instead of the left, the form of the equilib rium polygon will be exactly reversed. It is quite probable that, for mere considerations of architectural effect, the revised extrados would be made the same kind of a curve as the intrados. This would practically be done by selecting a radius which would leave the same thickness at the crown, allow the required thickness on the haunches, and let the thickness come what it will at the abutments, even though it is needlessly thick.
420. Stability of the Pier between the Arches. The stability of the pier on the right-hand side of the arch in Fig. 227, is deter mined on the assumption of the concentrated locomotive loading on the left-hand end of the next arch which is at the right of the given arch, and the uniform loading over the right-hand end of the given arch. We therefore draw through the point a line of force parallel to 'Ink, and also produce the line In until it intersects the other line of force in the point s. A line from s parallel to R„ therefore, gives the line of action of the resultant of the forces passing down the pier, for this system of loading. Since this system of loading will give the most unfavorable condition, or the condition which will give a resultant with the greatest variation from the perpendicular, we shall consider this as the criterion for the stability of the pier. The piers were drawn with a batter of 1 in 12, and it should be noted that the resultant R., is practically parallel to the batter line. If the slope of I?, were greater than it is, the batter should then be increased. The value of I?, is scaled from the force diagram as 55,650 pounds. The force I?, is about 14 inches from the face of the pier, and this would indicate a maximum intensity of pressure of 221 pounds per square inch. This is a safe pressure for a good class of masonry work. The actual pressure on the top of the pier is somewhat in excess of this, on account of the weight of that portion of the arch between the virtual abutment at n and the top of the pier; and the total pressure at any lower horizontal section, of course, gradually increases; but on the other hand, the weight of the pier combines with the resultant thrust of the two arches to form a resultant which is more nearly vertical than and the center of pressure therefore approaches more nearly to the axis of the pier. The effect of this is tc reduce the intensity of pressure on the outer edge of the pier; and since the numerical result obtained above is a safe value, the actual maximum intensity of pressure is certainly safe.