This section is from the book "The Building Trades Pocketbook", by International Correspondence Schools. Also available from Amazon: Building Trades Pocketbook: a Handy Manual of reference on Building Construction.
Begin by drawing, to scale, a diagram of one-half the arch. As in the previous example, the arch and its load is considered to be 1 ft. thick; and the brickwork weighs 140 lb. per cu. ft. A load of 9,200 lb. upon one-half the arch has been assumed. Lay off, to scale, a height of brickwork whose weight will represent this load. Commencing at the crown, divide the load into, say, sections of 2 ft., as far as possible. The weight of each slice will be its contents multiplied by 140 lb., and is marked on the diagram. Next, fix a point at the crown, and one at the spring of the arch, through which the pressure curve is assumed to pass. The points may lie anywhere within the middle third of the width; but the point a at the crown has been taken at the outer edge, and the point u at the spring at the inner edge, of the middle third. Lay off from a, on the vertical a d', the distances ab,bc,cd, etc., which represent the weight of the slices from the crown to the spring. Thus, if the scale were 1 in. per 1,000 lb., a b would be 1.1 in. long. Next, draw 45° lines from a and h, intersecting at i; and from i draw i b, i c, id, etc. Through the center of gravity of each slice, draw a vertical, as o v, pw,qx, etc. Starting from a, draw a v parallel to ai; from v, draw vw parallel to bi, etc. These lines form a broken line, which changes its direction on the vertical line through the center of gravity of each slice. From the last point k, draw kj parallel to ih, and intersecting ai, extended, at j; from j draw a vertical line jl, which will pass through the center of gravity of the half arch and load. From I, where the horizontal line al intersects lay off a distance Im equal to ah, which represents the weight of all the slices. From I draw a line through the point u; and from m, a horizontal line intersecting lu, extended, at n. Then m n will be the horizontal thrust at the crown, required to maintain the half arch in equilibrium when the other half is removed; and In will be the direction and amount of the oblique thrust at the skewback. On la extended, lay off, from a, a distance ab' equal to m n. From b', draw lines to 6, c, d, etc., which represent the thrusts at the center of gravity of each slice. From a, draw a o, parallel to b' a; from o, draw op, parallel to b' b, etc.; then a, o, p, etc. will be points on the line of pressure. If this line lies within the middle third, the arch will be stable, provided the pressure is within safe limits. The pressure at u is found by measuring b' h with the same scale as for a b, b c, etc., and is about 16,000 lb. Hard-burned brick, laid in cement mortar, will safely sustain a compressive stress of from 150 to 200 lb. per sq. in. The area at the skewback, 144 sq. in., multiplied by 200, gives 28,800 lb., which is well within the safe limit.
The stability of the abutments may be determined thus: Having calculated the weight of the pier or wall, lay off this weight on the vertical line from h to d', and draw d' b'. Draw a vertical line through the center of gravity of the pier, cutting In at c' ; also, a line from c', parallel to b' d'. The latter line will be the resultant thrust of the arch, after being influenced by the weight of the pier. If this line falls beyond the foot of the pier, at the ground line, the pier will be incapable of resisting the thrust of the arch. In order that a pier may be secure, this final or resultant line of thrust should fall on the ground line, well within the middle third of the base.