In the preceding paragraphs we have considered the subject of domical roofs covering buildings of circular plan, which is the simplest possible case, but unfortunately not the most usual one. It very often happens that a domical roof must be erected over a building which is square or rectangular in plan, in which case a new and difficult problem must be considered, namely, that of the pendentives. A horizontal section taken through a dome must in every case show a circle or possibly an ellipse. If, then, we consider the horizontal section cut from a domical roof by the plane of the top of the wall, it must usually be a circle and can not exactly coincide with the section cut from the wall of the building by the same plane, unless the building is circular in plan. This is shown in Fig. 256 in which A B C D represents the section cut from the wall of the building by a horizontal plane, and the circle E F G H represents the section which would be cut from a domical roof covering the building if the framing for the dome were carried down to meet this plane all the way around.

Fig. 255. Section through Domical Roof Showing One of the Ribs.

In order to cover every part of the building, the dome must be large enough to include the corners, and if made sufficiently large for this it must overhang the side walls of the building, by an amount A E B on each side, if the framing is carried down to the same horizontal plane all the way around. Horizontal sections taken through the dome at intervals throughout its height, however, show smaller and smaller circles as they are taken nearer and nearer to the top of the dome. Some one of these sections will cut out from the dome a circle which will appear in plan as though it were inscribed in the square formed by the walls of the building.

Fig. 256. Diagram Showing Relation of Dome Roof of Square Building.

Such a circle is shown at I J K L in Fig. 256. A dome built up with this circle as a base would not cover the corners of the building, so that the triangular spaces like A I L would be kept open. These triangular spaces, or rather the coverings over them, are called the pendentives. Fig. 257 shows in perspective the outline of four pendentives E D H, H C G, etc.

We have seen that a dome built up on the circumscribed circle as a base is too large, while a dome built up on the inscribed circle is too small and will not completely cover the building. To overcome this difficulty it is customary to erect a dome on the smaller or inscribed circle, as a base, and to extend the ribs so as to fill up the corners and form a framework for the pendentives. This is shown in Fig. 258 which is a plan of the framework for a domical roof. The ribs will be of different lengths and will intersect the inside face of the wall at different heights, because as they are extended outward they must also be extended downward. Each one will be curved if the dome is spherical, and straight if the dome is conical. The upper ends of the ribs bear against the curb A, leaving a circular opening for a lantern or cupola. The lower ends may be supported on a masonry wall, or may rest on curved wood plates, as shown in Fig. 259. This is an elevation of a conical dome, and shows the straight ribs A.

Fig. 257. Perspective Outline of Pendentives.

Fig. 258. Dome Framing with Extended Rafters.

Fig. 260 shows an elevation of a spherical dome which has curved ribs A, as shown. Each of these ribs must be bent or shaped to the segment of a circle, in order that the edges may lie in a spherical surface.

If the design calls for a domical ceiling and the exterior may be of some other form, then only the inside edges of the ribs need be dressed to correspond with a spherical or conical surface, in order that they may receive the lathing or furring, and the outside may be left rough so that a false roof of any desired shape may be built. If the exterior must be of domical form, while on the interior there is a suspended or false ceiling of some kind, then only the outside edges of the ribs must lie in the conical or spherical surface, so as to receive the roof boarding, while the inside edges may be left rough or shaped to any other form. If both the exterior and interior must be domical, then both the inside and outside edges of the ribs must be dressed so as to lie in the domical surface.

Conical domes are very uncommon, but they are sometimes used. A conical dome is much easier to frame than a spherical dome because the ribs are straight. The shape of the curved plate which supports the lower ends of the ribs may be easily determined, since it must conform to the line of intersection between the conical or spherical surface of the dome, and the plane of the face of the wall.

Fig. 259. Diagram Showing Construction for Apex of Conical Dome.

Fig. 260. Diagram Showing Construction for Spherical Dome.