This section is from the book "A Treatise On Architecture And Building Construction Vol2: Masonry. Carpentry. Joinery", by The Colliery Engineer Co. Also available from Amazon: A Treatise On Architecture And Building Construction.
Pendentives. Another method of finishing the roof or ceiling over the intersection of two passageways, is to construct over the crossing a hemispherical dome, when the passages are of the same width, and an elliptical dome when they are of different widths. The effect of this treatment is shown in Fig. 87, where the passage A intersects and crosses the passage B, and the sides of each one, prolonged by dotted lines, forms the rectangle abcd, which is the plan of their intersection. The roof of each passage is vaulted over with a semicircular arch, as gke and e If, while from the points e, f, and g, where these arches spring, there also starts a hemispherical dome g in o nf. The corners e, f, and g gradually round over to the eye of the dome at o, and at the same time broaden out in a fan shape over each of the adjacent arches, as seen at ekl.
This spherical triangle e k l, which brings the curvature of the dome down to the corner of the passage at e, is called a pendentlve, and there is, of course, one of them over each of the corners a, b, c, and d.
213. To develop the curvature of the ribs and construct the frame work for the dome and pendentives, the plan of the intersecting passages is first drawn, as shown in Fig. 88, at a b c d; next, the axial lines of the passages A A and B B and the diagonal lines of their intersection, over which the ribs v' a', gg', a c, and b d are located, are drawn. Then with center o, and with a radius oa, draw a circle through the four corners of the plan abcd; this circle will represent the plan of the dome. If this dome is now cut on the line dc at right angles to its plan, the section produced will be the semicircle ced, which is the arch over passage A.
There will be four of these arch ribs, one over each of the lines a b, ad, cb, and cd, and that part of the dome which would extend into each of the passages, as at a v d, is cut off by the arches over the entrances to the passages A and B. Now, as the dome is a hemisphere, any section of it through the center of its plan will be a semicircle, whose radius is the same as that of the circular plan, and we can consider the semicircle bcd as a revolved section through the dome on the diagonal rib bd; bcuy will then represent that rib as it would appear when cut out of its plank with its lower end b resting on the corner, and its upper end c standing over the center o, a distance of o c higher than the end b.
Four of these ribs will constitute the diagonals ac and b d, and the jack-ribs as og, o h, o i, etc. will have the same curve exactly as far as they reach, but their upper ends will be beveled and pointed to fit in around o, while the lower ends will require a plumb-cut and a cheek-cut to fit them against the arch pieces at ab, ad, cb, and cd.
214. To find the lengths and cuts for these jack-ribs, from the points s and s, where the sides of the ribs oa' and o b' intersect the side of the diagonal o b, draw z u and s t parallel to oc. The line zn, where it crosses the rib bcny, will mark the shoulder from which to bevel the end of the rib o a', and the line s t will mark the shoulder from which to bevel the rib o b'; the end piece u c will be cut off, as the rib o b' does not reach to the center o.
Now with o as a center, and with the radii og', o h', and ox, describe the arcs g'1, h'2, and x3, which will measure the lengths of each side of these jack-ribs on the diagonal rib o b; and the lines 1 /, 2 m, and 3 n perpendicular to ob will cut the rib b c ny to the required lengths for the jack-rafters; culk willbe the pattern for four of the jack-ribs og', oa', og, and ov', the cheek-cuts being determined as explained above, and c u np will be the pattern for the other jack-ribs, whose plumb-cut, where they join the arch ribs c e d, cfb, etc. will be the line p n on one side of the rib, and the line r m on the other side, while a line drawn on the top or bottom of the rib joining the ends of the plumb-cut lines will mark the cheek-cut or bevel.
215. In lathing the ribs of the passages, ordinary wood lath may be used, but the pendenfives and dome must have metallic lath so that they may be smoothly finished in plaster.