This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
An inspection of the equilibrium polygon for the first condition of loading, shows that it passes everywhere within the middle third. The maximum total pressure on a joint, of course, occurs at the abutment, where the pressure equals 24,750 pounds. Since the joint is here about 42 inches thick, and a section one foot wide has an area of 504 square inches, the pressure on the joint is at the rate of.49 pounds per square inch. At the keystone, the actual pressure is 19,750 pounds; and since the keystone has an area of 228 square inches, the pressure is at the rate of 87 pounds per square inch.
At the joint between forces Nos. 13 and 14, the line of force passes just inside the edge of the middle third. The ray from the pole o1' to the joint between voussoirs Nos. 13 and 14 of the force diagram, has a scaled length of 20,250 pounds. The joint has a total thickness of about 24 inches, and therefore an area of 288 square inches. This gives an average pressure of 70 pounds per square inch; but since the line of pressure passes near the edge of the middle third, we may double it, and say that the maximum pressure at the upper edge of the joint is 140 pounds per square inch. All of these pressures for the first condition of loading are so small a proportion of the crushing strength of any stone such as would be used for an arch, or even of the good quality of mortar which would of course be used in such a structure, that we may consider the arch as designed, to be perfectly safe for the first condition of loading.
The special equilibrium polygon for the second condition of loading shows that the stability of the arch is far more questionable - under this condition, since the special equilibrium polygon passes outside the middle third, especially on the left-hand haunch of the arch. The critical joint appears to be between voussoirs Nos. 4 and 5. The pressure at this joint, as determined by scaling the distance from the point o2" to the load line between forces Nos. 4 and 5, is approximately 24,500 pounds. The section of the equilibrium polygon parallel to this ray passes through the joint at a distance of a little over three inches from the edge. On the basis of the distribution of pressure at a joint, the compression at this joint would be confined to a width of 9 inches from the upper edge, the pressure being zero at a distance of 9 inches from the edge. This gives an area of pressure of 108 square inches, and an average pressure of 227 pounds per square inch. At the upper edge of the joint, there would therefore be a pressure of double this, or 454 pounds per square inch. This pressure approaches the extreme limit of intensity of pressure which should be used in arch work; and even this should not be used unless the voussoirs were cut and dressed in a strictly first-class manner, and the joints were laid with a first-class quality of mortar.
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 m (using the same pole o2'"), 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 equilibrium polygon, there is reason to believe that the true equilibrium polygon is the one of which the summation of the intensity of pressures 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 approaches 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 done approximately as indicated for a similar problem in Article 414. Evidently the keystone is sufficiently thick, and the voussoirs at the abutments also 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 abutments. 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, wall 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 equilibrium 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.