The Problem of Center Construction

Notice that by equation 26, N = 0, if tan a = p; that is to say, N = 0, if a = 30°. This shows that as the arch stones are placed upon one another they would not begin to press upon the center rib until the plane of the lower face of the top one reaches an angle of 30° with the horizon.

Table 92 gives the value of the radial pressure of the several portions of the arch upon the center; and also shows the difference between applying equation 26 and equation 27. As a rough approx imation, for a full-centered arch, equation 27 may be applied for the first 30° from the crown, although it gives results slightly greater than the real pressures; and for the second 30°, equation 26 may be employed, although it gives results less than the actual pressure; and for the third 30°, the arch stones may be considered self-sup porting.

Example. To illustrate the method of using Table 92, assume that it is required to find the pressure on a back-piece of a 20-foot semicircular arch which extends from 30° to 50° from the horizontal, the ribs being 5 feet apart, and the arch stones being 2 feet deep and weighing 150 pounds per cubic foot. Take the sum of the decimals in the middle column of Table 92, from 30° to 50° inclusive, which is 2.20. This must be multiplied by the weight of the arch re Sting on 2° of the centre. An arc of 1° is equal to 0.0175 times the radius. The radius to the middle of the voussoir is 11 feet, and the length of 2° of arc is 0.38 feet. The volume of 2' is 0.38 X 5 X 2 = 3.8 cubic feet; and the weight of 2° is 3.8 X 150 = 570 pounds. Therefore the pressure on the back-piece is 570 X 2.20 = 1,254 pounds.

Solid Wooden Rib.

For flat arches of 10-foot span or under, the rib may consist of a plank, aa, Fig. 205, page 654, 10 or 12 inches wide and 1+ or 2 inches thick, set edgewise, and another, b, of the same thickness, trimmed to the curve of the intrados and placed above the first. The two should be fastened together by nailing on two cleats of narrow boards as shown. These centers may be placed 2 or 3 feet apart.

Built Wooden Rib.

For flat arches of 10 to 30 feet span, the rib may consist of two or three thicknesses of short boards, fitted and nailed (or bolted) together as shown in Fig. 206. An iron plate is often bolted on over the joints (see Fig. 226, page 669), which adds materially to the rigidity of the rib. Centers of this form have an astonishing strength. Trautwine gives the two following examples which strikingly illustrate this.

"In the first of these examples, this form of center was employed for a semicircular arch of 35 feet span, having arch stones 2 feet deep. Each rib consisted of two thicknesses of 2-inch plank, in

lengths of about 6.5 feet, treenailed together so as to break joint. Each piece of plank was 12 inches deep at the middle, and 8 inches at each end, the top edge being cut to suit the curve of the arch. The treenails were 1.25 inches in diameter, and 12 of them were used to each length. These ribs were placed 17 inches apart from center to center, and were steadied together by a bridging piece of 1-inch board, 13 inches long, at each joint of the planks, or about 3.25 feet apart. Headway for traffic being necessary under the arch, there were no chords to unite the opposite feet of the ribs. The ribs were covered with close board-lagging, which also assisted in steadying them together transversely. As the arch approached about two thirds of its height on each side, the ribs began to sink at the haunches and rise at the crown. This was rectified by loading the crown with stone to be used in completing the arch, which was then finished without further trouble." The other example was an elliptic arch of 60 feet span and 15 feet rise, the arch stones being 3 feet deep at the crown and 4 feet at the springing. " Each frame 'of the center was a simple rib 6 inches thick, composed of three thicknesses of 2-inch oak plank, in lengths (about 7 to 15 feet) to suit the curve and at the same time to preserve a width of about 16 inches at the middle of each length and 12 inches at each of its ends. The segments broke joints, and were well treenailed together with from ten to sixteen treenails to each length. There were no chords. These ribs were placed 18 inches from center to center, and were steadied against one another by a board bridging-piece, 1 foot long, at every 5 feet. When the arch stones had approached to within about 12 feet of each other, near the middle of the span, the sinking at the crown and the rising at the haunches had become so alarming that pieces of 12- by 12 inch oak were hastily inserted at intervals and well wedged against the arch stones at their ends. The arch was then finished in sections between these timbers, which were removed one by one as the arch was completed." Although the above examples can not be commended as good construction—the flexibility of the ribs being so great as to endanger the stability of the arch during erection and to break the adhesion of the mortar, thus decreasing the strength of the finished arch,— they are very instructive as showing the strength attainable by this method of construction.