DESIGN OF REINFORCED CONCRETE ARCH Selection of Dimensions.—In designing an arch, it is neces sary to first assume dimensions for the arch ring, and then investigate for the strength of the arch and the suitability of the assumed dimen sions to the conditions of service. The methods of investigation usually employed are indicated in Art. 46. The investigation will show whether changes in form or thickness should be made in the arch ring. The shape of the arch should be such as to fit as closely as possible the lines of pressure, and the thickness should be such as to give allowable stresses under all conditions of loading.
Example.—As an illustration of the method of investigation, we will assume an arch of 60 feet elear span and 12 feet rise, to carry a live load of 100 pounds per square foot of road and a solid spandrel filling, 2 feet deep over the crown, weighing 100 pounds per cubic foot. For ordinary arches with solid spandrel filling, a three-centered intrados, with radii at. the sides from three-fifths to three-fourths that at the crown, is apt to give better results than a segmental intrados. We will use an intrados composed of three arcs tangent. to each other at the quarter points with radii of 52.5 feet and 31.25 feet respect ively (see Fig. 91).
Weld's formula 1 for the crown thickness is in which crown thickness in inches; L = clear span in feet; TV = live load in pounds per square foot; 1r = weight of fill at crown per square foot.
Applying this formula, we find a crown thickness of 16 inches. The thickness of the arch ring should increase from the crown to the spring ing line; the thickness at the quarter point may be made a little greater than that at the crown (about 11 to 13 times). We will assume a thickness at the quarter point of 21 inches, and at the spring ing line of 45 inches. The extrados will now consist of three arcs tangent to each other at the quarter points and giving the desired thicknesses. Fig. 117 shows the arch ring as assumed.
The reinforcement may be assumed at from .4 to .7 per cent of the area of the section at the crown, to be placed at both extrados and intrados. We will use i-inch round bars spaced 7 inches apart.
167. Division of Arch Ring.—Having chosen the form and dimen sions of the arch ring, it is necessary to divide the ring so that the lengths of the divisions shall be directly proportional to the moments of inertia of their mid-sections, s/I = constant. This may be done by trial, assuming a division next the crown, determining the value of s/I for the assumed division, and finding the corresponding lengths of other sections toward the abutment. Then changing the first assumption as may seem necessary to make the division come out properly at the abutment.
More easily, the division may be made graphically as shown in Fig. 90. The line is laid off equal in length to half the arch axis (34.52 feet). The moments of inertia are then computed at several points along the arch axis, and their amounts laid off normally to the line o-k, and the curves of moment of inertia drawn through the points so located.
A trial diagonal is then drawn from A to intersection with the curve in the point B. A vertical from B is drawn to intersection with the upper curve, and a second diagonal parallel to A-B, cutting the lower curve in C. Continue successive diagonals and verticals until the encl k is reached. If these clo not cone out accurately at the end k the inclination of the diagonals may be varied until the division of a-k is made into the correct number of parts. This divides a-k into lengths which are proportional to the average of the moments of inertia at the ends of the divisions.
The lengths of the divisions, a-b, b-c, etc., are now transferred to the arch axis. The axis of the arch in Fig. 91 is thus laid off into ten divisions on each side of the crown section. The con stant ratio s/I is found to be 5.1, all measurements being taken in feet.