406. External Forces Acting on an Arch. There is always some uncertainty regarding the actual external forces acting on ordinary arches. The ordinary stone arch consists of a series of voussoirs, which are overlaid usually with a mass of earth or cinders having a depth of perhaps several feet, on top of which may be the pavement of a roadway. The spandrel walls over the ends of the arch, especially when made of squared stone masonry, also develop an arch action of their own which materially modifies the loading on the arch rings. As this, however, invariably assists the arch, rather than weakens it, no modification of plan is on this account. The actual pressure of the earth filling, together with that caused by the live load passing over the arch, on any one stone, is uncertain in very much the same way as the pressure on a retaining wall is uncer tain, as previously explained.
The simplest plan is to consider that each voussoir is carrying a load of earth equal to that indicated by lines from the joints in the voussoir vertically upward to the surface. The development of the graphical method makes it more convenient to draw what is called a reduced load line on top of the arch, in which the depth of earth above the arch is reduced in the ratio of the relative weights per cubic foot of the earth filling and of the stone of which the arch is made (sec Fig. 220). Even the live load on the arch is represented in the same manner, by an additional area on top of the reduced line for the earth pressure, the depth of that area being made in proportion to the intensity of the live load compared with the unit-weight of stone. For example, if the earth filling weighs 100 pounds per cubic foot, and the stone of the arch weighs 160 pounds per cubic foot, then each ordinate for the earth load would be la of the actual depth of the earth. Likewise, if the live load per square foot on the arch equals 120 pounds, then the area representing the live load would be -11 ,(),- of a foot, according to the scale adopted for the arch. The weight of the paving, if there is any, should he similarly allowed for. If we draw from the upper end of each joint a vertical line extending to the top of the reduced load line, then the area between the weight at the scale adopted for the drawing, and at the unit-value for the weight per cubic foot (MO pounds per cubic foot, as suggested above) actually pressing on that particular voussoir. A line through the center of gravity of the stone itself gives the line of action of the force of gravity on the voussoir. An approximation to the position of this center of gravity, which is usually amply accurate, is the point which is midway between the two joints, and which is also on the arch curve which lies in the middle of the depth of each voussoir. The center of gravity of the load on the voussoir is approximately in the center of its width. The re sultant of two parallel forces, such as I' and L, Fig. 221, equals in amount their sum R, and its line of action is between them and at distances from them such that: ac : be :: force L: force V.
Usually the horizontal space between the forces and I, is so very small that the position of their resultant 11 can be drawn by estimation as closely as the possible accuracy of drawing will permit, without recourse to the theoreticallyaccurate method just given. - iven The amount of the result-
ant is determined by measuring the areas, and multiplying the sum of the two areas by the weight per cubic foot of the stone. This gives the weight of a section of the arch ring one foot thick (parallel with the axis of the arch). The area of the voussoir practically equals the length (between the joints of that section) of the middle curve, times the thickness of the arch ring. The area of the load trapezoid equals the horizontal width between the vertical sides, times its middle height. The student should notice that several of the above statements regarding areas, etc., are not sions of the voussoirs to the span of the arch, the errors involved by the approximations are harmless, while the additional labor necessary for a more accurate solution would not be justified by the inappreciable difference in the final results.
407. Depth of Keystone. The proper depth of keystone for an arch should theoretically depend on the.total pressure on the key stone 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 com puted, 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 eases 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 he seldom, if ever, that the depth of the keystone, as determined by this rule, would need to be altered. The rule is as follows: Had.+ half-span Depth of Keystone, in feet = + 0.2 foot • . (47) 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 per missible 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), then any extra depth which may subsequently be given to the key stone should be considered as mere ornamentation and as not af fecting the mechanics of the problem.