The proper depth of keystone for an arch should theoretically depend on the.total pressure on the keystone of the arch as developed from the force diagram; and the depth should be such that the unit-pressure shall not be greater than a safe working load on that stone. But since we cannot compute the stresses in the arch, until we know, at least approximately, the dimensions of the arch and its thickness, from which we may compute the dead weight of the arch, it is necessary to make at least a trial determination of the thickness. The mechanics of such an arch may then be computed, and a correction may subsequently be made, if necessary. Usually the only correction which would be made would be to increase the thickness of the arch, in case it was found that the unit-pressure on any voussoir would become dangerously high. Trautwine's Handbook quotes a rule which he declares to be based on a very large number of cases that were actually worked out by himself, the cases including a very large range of spans and of ratios of span to rise. The rule is easily applied, and is sufficiently accurate to obtain a trial depth of the keystone. It will probably be seldom, if ever, that the depth of the keystone, as determined by this rule, would need to be altered. The rule is as follows:

Depth of Keystone, in feet =

For architectural reasons, the actual keystone of an arch is usually made considerably deeper than the voussoirs on each side of it, as illustrated in Fig. 218. When computing the maximum permissible pressure at the crown, the actual depth of the voussoirs on each side of the keystone is used as the depth of the keystone; or perhaps it would be more accurate to say that the extrados is drawn as a regular curve over the keystone (as illustrated in Fig. 223), and then any extra depth which may subsequently be given to the keystone should be considered as mere ornamentation and as not affecting the mechanics of the problem.

408. Numerical Illustration

The above principles will be applied to the case of an arch having a span of 20 feet and a rise of

3 feet (see Fig. 223). If this arch is to be a circular or segmental arch, the radius which will fulfil these conditions may be computed as illustrated in Fig. 222. We may draw a horizontal line, at some scale, which will represent the span of 20 feet. At the center of this line we may erect a perpendicular which shall be 3 feet long (at the same scale). Joining the points a and c, and bisecting ac at d, we may draw a line from the bisecting point, which is perpendicular to ac, and this must pass through the center of the required arc. A vertical line through c will also pass through the center of the required arc, and their intersection will give the point o. As a graphical check on the work, a circle drawn about o as a center, and with oc as a radius, should also pass through the points a and b. Since some prefer a numerical solution to determine the radius for a given span and rise, the radius for this case may be computed as follows: The line ac equals the square root of the sum of the squares of the half-span and the rise, which equals but the angle cae = angle aod, and, from similar triangles, we may write the proportion:

Fig. 223. Resultant vertical pressure.

This equals numerically in the above case, 109 ÷ 6 = 18.17.

Applying the above rule for the depth of the keystone, we would find for this case that the depth should be:

=5.31/4+ 0.2

=1.33+0.2 = 1.53 feet.

Since the total pressure on the voussoirs is always greater at the abutment than at the crown, the depth of the stones near the end of the arch should be somewhat greater than the depth of the keystone. We shall therefore adopt, in this case, the dimensions of 18 inches for the depth of the keystone, and 2 feet for the depth at the skewback.