Then, by taking moments about b, we have 1. The value of T depends upon 2 h k—the sum of the moments of the horizontal component of the external forces. In discussing and applying this principle, the term 2 h k is usually neglected. Ordinarily this gives an increased degree of stability; but this is not necessarily the case. The omission of the effect of the horizontal component makes the computed value of T less than it really is, and causes the line of resistance found on this assumption to approach the intrados at the haunches nearer than it does in fact; and hence the conditions may be such that the actual line of resistance will be unduly near the extrados at the haunches, and consequently endanger the arch in a new direction.
2. From equation 6 it appears that, other things remaining the same, the larger y the smaller T; and hence, for a minimum value of T, a should be as near C as is possible without crushing the stone. Usually it is assumed that aC is equal to one third of the thickness of the arch at the crown, and hence the average pressure per unit of area is to be equal to one half of the assumed unit working pressure; i.e., twice the thrust T divided by the thickness of the crown is to be equal to the maximum unit compressive stress at the crown.
3. To determine y, it is necessary that the direction of T should be known. It is usually assumed that T is horizontal. If the arch is symmetrical and is loaded uniformly over the entire span, this assumption is reasonable; but if the arch is subject to a moving load which is heavy in comparison with the weight of the arch and the spandrel filling, the thrust at the crown is not horizontal and hence can not be determined directly (see § 1237).
4. If the joint AB is horizontal, then b is to be taken as near A as is consistent with the crushing strength of the stone, or at, say, one third of the length of the joint AB from A. Notice that if the joint AB is inclined, as in general it will be (see the last para graph of § 1217 and of § 1222), moving b toward A decreases x and at the same time increases y and k. Hence the position of b corre sponding to a minimum value of T can be found only by trial. It is usual, however, to assume that Ab is one third of AB, whatever' the inclination of the joint.
small, some one of the joints 1, 2, 3, etc., Fig. 194, may open at the extrados; and on the other hand, if T is too large, some one of the joints may open at the intrados. Neither of these conditions would be consistent with the condition of equilibrium assumed, i.e., with the assumption that the center of pressure is to remain within the middle third of any joint. if the center of pressure remains within the middle third, every part of all joints will be under compression, and hence no joint can open at either the extrados or the intrados. It remains then to find a value of T that shall keep the center of pressure within the middle third of every joint.
If b, the origin of moments for equation 5, page 621, be taken successively at the inner or lower end of the middle third of each joint, the corresponding value of T will be the crown thrust for which that particular joint is on the point of opening at the extrados; and if under this condition the greatest value of T that will prevent any joint 1•rom opening on the intrados be found, then that value is the crown thrust required by the hypothesis, for a less value will permit one or more joints to open at the extrados and a greater value will cause one or more joints to open at the intrados.
The joint for which the tendency to open at the extrados is the greatest is called the joint of rupture. The joint of rupture of an arch is analogous to the dangerous section of a beam. Practically, the joint of rupture is the springing line of the arch, the arch masonry below that joint being virtually only a part of the abutment. There fore the first step in testing the stability of a given arch is to find the joint bf rupture.
Example of the Method of Determining the Joint of Rupture. Assume that it is required to determine the joint of rupture of the 16-foot arch shown in Fig. 195 which is the arch of the standard voussoir-arch culvert formerly employed on the Chicago, Rock Island and Pacific Railroad. Assume that the arch supports an embankment of earth extending 10 feet above the crown, and that the earth weighs 100 pounds per cubic foot and the masonry 160. For simplicity, consider a section of the arch only a foot thick per pendicular to the plane of the paper. The half-arch ring and the earth embankment above it are divided into eight sections, which for a more accurate determination of the joint of rupture are made smaller near the supposed position of that joint.