This section is from the book "Safe Building", by Louis De Coppet Berg. Also available from Amazon: Code Check: An Illustrated Guide to Building a Safe House.
118) a o = g5 h5 = 4500 pounds and draw ob, oc, od, etc. We next construct the curve of pressure a K and find that it coincides as closely as possible with the centre line of the arch. This means that the pressure on each joint will be uniformly distributed. That on the lower joint is, of course, the largest and is = o f, which scales 4700 pounds. The area of the joint is, of course, a = 3. 12 = 36 square inches, therefore the greatest stress per square inch will be 4700/66 = 130 pounds compression.
As the tests gave us 300 pounds compression, per square inch, as safe stress of a sample only twelve days old, the arch is, of course, perfectly safe.
So far we have simply considered the danger of compression or tension at the joints of an arch; there is, however, another element of danger, though one that does not arise frequently, viz.: the danger of one voussoir sliding past the other. Where strong and quick-setting cements are used this danger is, of course, not very great. But in other cases, and particularly in large arches, it must be guarded against. The angle of friction of brick against brick (or stone against stone) being generally assumed at 30°, care must be taken that the angle formed by the curve of pressure at the joint, with a normal to the joint (at the point of intersection K) does not form an angle greater than 30°. If the angle be greater than 30° there is danger of sliding; if smaller, there is, of course, no danger. Thus, if in Fig. 114 we erect through K4 a normal K4 X to the joint, the angle X K4 i3 should not exceed 30°.
In arches with heavy-concentrated loads at single points, there might, in rare cases, be danger of the load shearing right through the arch. The resistance to shearing would, of course, be directly as the vertical area of cross-section of the arch, and in such cases this area must be made large enough to resist any tendency to shear.
Arches are frequently built shallower at the crown and increasing gradually in depth towards the spring, the amount being regulated, of course, by the curve of pressure and Formulas (44) and (45).
To establish the first experimental thickness at the crown of an arch, many engineers use the empyrical formula:
Depth at crown,
Where x = the depth of arch, at crown, in inches. Where r = the radius at crown, in inches. Where y = a constant, as follows:
For cut stone, in blocks:
When Portland cement is used, a somewhat lower value may be assumed for y. The depth thus established for crown is only exper-mental, of course, and must be varied by calculation of curve of pressure, etc.
In cases where the architect does not feel the necessity for such a close calculation of the arch, it will be sufficient to find the curve of pressure. If this curve of pressure comes within the inner third of arch-ring, at every point, the arch is safe, provided the thrust on each joint, divided by the area of joint, does not exceed one-half of the safe compressive stress of the material, or:
P/a = 1/2 (c/f) (68)
Where p = the thrust on joint, in pounds.
Where a = the area of joint, in square inches.
Where (c/f)= the safe resistance to compression, per square inch, of the material.
Vaulted and groined arches are calculated on the same principles as ordinary arches and domes; though, in groined arches, as a rule, the ribs do all the work, the spandrels between the ribs being filled with stone slabs or other light material; in some European examples even flower-pots, plastered on the under side, have been used for this purpose.
Vaulted and Croined Arches.