The first condition of loading is given by duplicating about the center, in the force diagram, the system of forces from No. 17 to No. 9 inclusive. Since this system of forces is symmetrical about the center, we know that its resultant R, passes through the center of the arch, and that it must be a vertical force. We may draw from the middle of force No. 9 a horizontal line, and also draw a vertical from the lower end of the load line. Their intersection is evidently at the center of the resultant R„ which is therefore carried above this horizontal line for an equal amount. Joining the upper end of with the upper end of force No. 9, we have the direction and amount of the force 14". The intersection of ny with the force at the point j, gives a point which, when joined with the point in, gives one line of a trial equilibrium polygon passing through the required points in and II, but which does not pass through the required point c. The inter section of fin with the force R," at the point p, gives us the line pg, which is the same kind of line for this trial polygon as the line kg was for the other.
By a similar method to that used before and as described in detail in Article 401, we obtain the line qr passing through c, which gives us also the section of our true equilibrium polygon between forces Nos. S and 9. The line rn also gives us that portion of the true equilibrium polygon for this system of loading, from the point n up to the force No. 17.
By drawing a line from the lower end of the load line, parallel to nr, until it intersects the horizontal line through the middle of force No. 0 at the point we have the pole of the special equilibrium polygon for this system of loading, which is the first condition of load ing. The rays are drawn from o,' only to the forces from No. 9 to No. 17 inclusive, and the special equilibrium polygon is completed between n and c by drawing them parallel to these rays.
On account of the symmetry of loading, we know that the equilib rium polygon would be exactly similar on the left-hand side of the arch. In discussing these equilibrium polygons, we must therefore remember that of the two equilibrium polygons lying between the extrados and intrados on the right-hand side of the arch, the upper line represents the line of pressure for a uniform loading over the whole arch (the first condition of loading), while the loler line on the right-hand side, and also the one equilibrium polygon which is shown on the left-hand side of the arch, represent the special equilibrium polygon for the second condition of loading.
419. Intensity of Pressures on the Voussoirs of the Arch. An inspection of the equilibrium polygon for the first condition of loading, shows that it passes everywhere within the middle third. The maxi
mum 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 o,' 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 2SS square inches. This gives an average pressure of 70 pounds per square inch; 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 he 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 out side 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 sealing the distance from the point o," to the load line between forces Nos. 4 and 5, is approxi mately 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 10S 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.