The student should carefully check over all these calcula tions, drawing the arch at the scale of one-half inch to the foot, and the load line of the force diagram at the scale of 1,500 pounds per inch; then the rays of the true equilibrium polygon will represent at that scale the pressure at the joints. Dividing the total depth of any joint by the pressure found at that joint, gives the average pressure. In the case of the joint at the crown, the total pressure at the joint is 13,900 pounds. The depth of the joint is 1.5 feet, and the area of the joint is 216 square inches; therefore the average unit-pressure is 64 pounds per square inch; if it is assumed that the line of pressure passes through either edge of the middle third, then the pressure at the edge of the joint is twice the average, or is 12S pounds per square inch. This is a very low pressure for any good quality of building stone.
Similarly, the maximum pressure at the skewback is scaled from the force diagram as 16,350 pounds; but since the arch is here two feet thick, and the square inches, it gives an average pressure of 57 pounds per square inch. Since this equilibrium polygon is supposed to start from the center of this joint, this repre sents the actual pressure.
Usually it is only a matter of form to make the test for uniform full loading. Eccentric loading nearly always tests an arch more severely than uniform loading. The ability to carry a full uniform load is no indication of ability to carry a partial eccentric loading, except that if the arch appeared to be only just able to carry the uni form load, it might be predicted that it would probably fail under the eccentric load. On the other hand, if an arch will safely carry a heavy eccentric load, it will certainly carry a load of the same in tensity uniformly distributed over it.
411. Test for the Second Condition, or Loading of Maximum Load over One=Half of the Arch. Since the arch has a dead load over the entire arch, and a live load over only one-half of the arch, the load line for the entire arch must be drawn. The load line for the loaded half of the arch will be identical with that already drawn for the previous case. The load line for the remainder of the arch may be similarly drawn. This case is worked out by precisely the same general method as that already employed in the similar case given in detail in Article 410. As in that article, we select a trial pole which in general will give an oblique closing line for the equilibrium polygon. This closing line must be brought down to the horizontal by the method already explained in Article 400; then a second trial must be made in order to shift the polygon so that it shall pass through the middle third at the crown joint. This line should pass through the middle of the crown joint; then the real test is to determine how it passes through the haunches of the arch. As in the previous case, the total pressure at any joint will be determined by the corresponding lines in the force diagram, and the unit-pressure at the joint may be determined from the area of the joint and the position of the line of force with respect to the center of the joint. Even though a line of
force passed slightly outside of the middle third, it would not neces sarily mean that the arch will fail, provided that the maximum in tensity of pressure, determined according to the principles enunciated in Article 405, does not exceed the safe unit-pressure for the kind of stone used.
An inspection of the force diagram with the pole at o,', shows that the rays are all shorter than those of the force diagram for the first condition of loading—with pole at o,'. This means that the actual pressure at any joint is less than for the first case; but since the true equilibrium polygon for this case does not pass so near the center of the joints as it does for the first condition of loading, the intensity of pressure at the edges of the joints may be higher than in the first case. However, since the equilibrium polygon for this second case is always well within the middle third at every joint, and since even twice the average joint pressure for the first case is well within the safe allowable pressure on any good building stone, we may know that the second condition of loading will be safe, even without exactly measuring and computing the maximum intensity of pressure produced by this loading.
412. Test for the Third Condition, Involving Concentrated Load. The method of making this test is exactly similar to that previously given; but on account of a load eccentrically placed, the force diagram will be more distorted than in either of the cases previ ously given, and there is greater danger that the arch will prove to be unstable on such a test. An inspection of the equilibrium polygon for this case shows that the critical point is the joint between vous soirs Nos. 3 and 4. This is what might be expected, since it is the joint under the heavy concentrated load. The ray in the force dia gram which is parallel to the section of the equilibrium polygon passing through this joint, is the ray which reaches the load line between loads 3 and 4. This ray, measured at the scale of 1,500 pounds per square inch, indicates a pressure of 15,625 pounds on the joint. The line of pressure is 44 inches from the upper edge of the joint; it is outside of the middle third; and therefore the joint will probably open somewhere under this loading. According to the theory of the distribution of pressure over a stone joint, the pressure will be maximum on the upper edge of this joint, aria will be zero at three times 4 inches, or 14.25 inches, from the upper edge. The area of pressure for a joint 12 inches wide will be 14.25 X 12 = 171 square inches. Dividing 171 into 15,625, we have an average pres sure of 91 pounds, or a maximum pressure of twice this, or 182 pounds, per square inch at the edge of the joint. But this is such a safe working pressure for such a class of masonry as cut-stone you,. soirs, that the arch certainly would not fail, even though the elasticit, of the stone caused the joint to open slightly at the intrados during the passage of the steam roller.