This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
We shall apply the above principles to the design of a segmental arch having a span of 60 feet and a rise of 15 feet. To find the radius for the intrados which will fulfil these conditions, we may note from Fig. 232 that the angle O'B'C is measured by one-half of the arc O'C, and therefore O'B'C is one-half of the angle a; but the angle O'B'C is an angle whose natural tangent equals 15 ÷ 30, or precisely 0.5. The angle whose tangent has this value is 26° 34', and therefore a equals 53° 8'. To find the radius, we must divide the half-span (30) by the sine of 53° 8', and we find that the radius equals 37.50 feet.
Fig. 232. Reinforced-Concrete Arch Rib, Fixed Ends.
For the depth of the keystone we can employ only empirical rules. The depth as computed from Equation 47 would call for a keystone depth of about 27 inches, which would be proper for an ordinary masonry arch; but considering recent successful practice in rein-forced-concrete arches, and the far greater reliability and higher permissible unit-stresses which may be adopted, we may select about 2/3 of this - or, say, 18 inches - as the depth of the arch ring at the crown.
We shall compute the depth the arch ring should have at various points, according to the tabular form in Article 427, so that the moment of inertia will vary in the ratio ds/dx, which will make Equations
56 to 58 applicable. An arc of 1 degree equals .0175 of the radius, and therefore an arc of 1 degree on a circle with a radius of 37.5 feet will have a length of .6545 foot. At a distance of 15 degrees from the center, or at a distance of 9.82 feet, the depth is one per cent greater than the depth at the center, or it is 18.18 inches deep.
At 30 degrees from the center, or at a distance of 19.63 feet measured on the arc, the depth is 5 per cent greater, or it is 18.90 inches deep. At a distance of 40 degrees from the center, or 26.18 feet measured on the arc, the value of h is 9 per cent greater, or it is 19.62 inches. At 45 degrees from the center, or 29.45 feet measured on the arc, h is 12 per cent greater than the center depth, or the depth is 20.16 inches. At 50 degrees from the center, or 32.72 feet measured on the arc, h is 16 per cent greater, or its thickness is 20.88 inches. At the abutment, which is 53° 8' from the center, the thickness should be (by interpolation) about 18 per cent greater than at the center, or it should be 21.24 inches, or say 21 1/4 inches.
Laying off these various distances from the center on the in-trados, and measuring radial distances at each point to represent the proper thickness at the several points, we may join the various points and obtain the curve of the extrados. Bisecting each one of these several arch thicknesses will give us a series of points which are points on the center line of the arch rib. We thus find that the actual span may be considered as 61.40 feet, and that the rise is scaled at 15.25 feet.