Notice that the distance which the center of pressure may vary from the center of the joint without the masonry's being crushed depends upon the ratio between the ultimate crushing strength and the mean pressure on the joint. In other words, if the mean pressure is very nearly equal to the ultimate crushing strength, then a slight departure of the center of pressure from the center of the joint may crush the voussoir; but, on the other hand, if the mean pressure is small, the center of pressure may depart considerably from the center of the joint without the stone's .being crushed. This can be shown by equation 2, page 611. If both P and W _ 1 are large, .d must be small; but if P is large and W = 1 small, then d may be large. Essentially the same result can be de duced from equation 3, page 611.
Even though the line of resistance approaches so near the edge of the joint that the stone is crushed, the stability of the arch is not necessarily endangered. For example, conceive a block of stone resting upon an incompressible plane, AB, Fig. 191; and assume that the center of pressure is at N. Then the pressure is applied over an area projected in AV, such that AN = AV. The pressure at A is represented by AK, and the area of the triangle AKV represents the total pressure on the joint. Assume that AK is the ultimate crushing strength of the stone, and that the center of pressure is moved to N'. The pressure is borne on an area projected in AV'. The pressure in the vicinity of A is uni form and equal to the crushing strength AK; and the total pressure on the joint is represented by the area of the figure AKGV', which has its center of gravity in the vertical through N'. Eventually, when the center of pressure approaches so near A that the area in which the stone is crushed becomes too great, the whole block will give way, and the arch will fall.* Open Joints. It is frequently prescribed that the line of resistance shall pass through the middle third of each joint, "so that the joint may not open on the side most remote from the line of resistance." If the line of resistance departs from the middle third, the remote edge of the joint will be in tension; but since cement mortar is now quite generally employed, if the masonry is laid with ordinary care the joint will be able to bear considerable tension; and hence it does not necessarily follow that the joint will open.
If the line of pressure departs from the middle third and the mortar is incapable of resisting tension, the joint will open on the side farthest from the line of resistance. For example, if the center of pressure is at N, Fig. 191, then a portion of the joint AV ( = 3 AN) is in compression, while the portion VB has no force acting upon it; and hence the yielding of the portion AV will cause the joint to open a little at B.. This opening will increase as the center
of pressure approaches A, and when the material at that point begins to crush the increase will become comparatively rapid.
Notice that if there are open joints in an arch, it is certain that the actual line of resistance does not lie within the middle third of such joints. Notice, however, that the opening of a joint does not indicate that the stability of the arch is in danger. In most cases, an open joint is no serious matter, particularly if it is in the soffit. If in the extrados, it is a little more serious, since water might get into it and freeze. To guard against this danger, it is customary to cover the extrados with a layer of puddle or some coating imper vious to water.
However, there is no probability that an arch will receive its full load before the mortar has begun to set; and hence the angle of friction is virtually much greater than 27°. It is customary to arrange the joints of the arch at least nearly perpendicular to the line of resistance, in which case little or no reliance is placed on the re sistance of friction or the adhesion of the mortar.