The unit-pressure on any joint is assumed to varv in accordance with the location of the center of pressure, as is illustrated in Fig. 219. In the first case, where the center of pressure is over the center of the face of the joint and is perpendicular to it, the pressure will be uniformly distributed, and may be represented, as in Fig.219a, by a series of arrows which are all made equal, thus representing equal unit-pressures. As the center of pressure varies from the center of the joint, the unit-pressure on one side increases and the unit-pressure on the other side decreases, as shown in Fig. 219 b. The trapezoid in this diagram has the same area as the rectangle of the first diagram (a), and the center of pressure passes through the center of gravity of the trapezoid. As the center of pressure continues to move away from the center of the joint, the unit-pressure on one side becomes greater, and on the other side less, until the center of pressure is at a point 1/6 of the width of the joint away from the center. In this case (c),the center of pressure is at the extreme edge of the middle third of the joint. The group of pressures illustrated in diagram c becomes a triangle, which means that the pressure at one side of the joint has become just equal to zero, and that the maximum pressure at the other side of the joint is twice the average pressure. If the line of pressure varies still further from the center of the joint, the diagram of pressures will always be a triangle whose base is always three times the distance of the center of pressure from the nearest edge of the joint. If the total pressure on that joint remains constant, then the intensity of pressure on one side of the joint becomes extreme, and may be sufficient to crush the stone. Also, since the elasticity of the stone (or of the mortar between the stones) will cause the stone (or mortar) to yield, the yielding being proportional to the pressure, the joint will open at the other side, where there is no pressure. In accordance with this principle of the distribution of pressure, it is always specified that a design for an arch cannot be considered safe unless it is possible to draw a line of pressure (an equilibrium polygon) which shall at every joint pass through the middle third of that joint. If the line of pressure at any joint does not pass through the middle third, it means that such a joint will inevitably open, and make a bad appearance, even though the unit-pressure on the other end of that joint is not so great that the masonry is actually crushed.

Fig. 219. Distribution of Pressure

Fig. 219. Distribution of Pressure.

Since the actual crushing strength of stone is a rather uncertain and variable quantity, a larger factor of safety is usually employed with stone than with other materials of construction. This factor is usually made ten; and therefore, whenever the line of pressures passes through the edge of the middle third, the average unit-pressure on the joint should not be greater than 1/20 of the crushing strength of the stone.

A table of these ultimate values has been given in Table I, Part I (page 10). They vary from about 3,000 pounds per square inch, for a sandstone found in Colorado, up to 28,000 pounds per square inch for a granite found in Minnesota. The weaker stone would hardly be selected for any important work. Usually a stone whose ultimate strength is 10,000 pounds per square inch or more, would be selected for a stone arch. Such a stone could be used with a working pressure of 500 pounds per square inch at any joint, assuming that the line of pressure does not pass outside of the middle third at any joint.

40G. 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 essential 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 uncertain, 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 (see 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 100/160 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 120/160 of a foot, according to the scale adopted for the arch. The weight of the paving, if there is any, should be 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 these two verticals and between the arch and the load line represents the weight at the scale adopted for the drawing, and at the unit-value for the weight per cubic foot (160 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 resultant of two parallel forces,such as V 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: bc:: force L: force V.

Fig. 220. Determination of Reduced Load Line.

Fig. 220. Determination of Reduced Load Line.

Usually the horizontal space between the forces V and L is so very small that the position of their resultant R can be drawn by estimation as closely as the possible accuracy of drawing will permit, without recourse to the theoretically accurate method just given. The amount of the resultant 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 theoretically accurate; but, with the usual proportions of the dimensions 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.

Fig. 221. Graphical Determination of Cir cular Arch, Span and Rise Being Known.

Fig. 221. Graphical Determination of Cir cular Arch, Span and Rise Being Known.