266. A dome or cupola is a roof, of which the base is a circle, an ellipse, or a polygon, and its vertical section a curve line, concave towards the interior. Hence domes are called circular, elliptical, or polygonal, according to the figure of the base.
The most usual form for a dome is the spherical, in which case its plan is a circle, and section a segment of a circle.
The top of a large dome is often finished with a lantern, which is supported by the framing of the dome.
267. The interior and exterior forms of a dome are not often alike, and in the space between, a staircase to the lantern is generally made. According to the space left between the external and internal domes, the framing must be designed. Sometimes the framing may be trussed with ties across the opening; but often the interior dome rises so high that ties cannot be introduced: in the latter case, the observations made on the equilibrium of domes in Sect. 1, Arts. 69, 73, should be attended to.
Accordingly, the construction of domes may be divided into two cases, viz. domes which admit of horizontal ties, and domes without such ties.
Domes which admit of Horizontal Ties.
268. A truss for a dome where a horizontal tie can be introduced is shown by Fig. 1, Plate XXIX. In this figure A A is the tie; B B posts, which may be continued to form the lantern; C,C, are continued curbs in two thicknesses, with the joints crossed and bolted together; D D, a curved rib to support the rafters. This design is calculated for a span of about 60 feet, and may be extended to 120 feet.
Two principal trusses may be placed across the opening parallel to each other, and at a distance apart equal to the diameter of the lantern, as A B, C D (Fig. 2), with a sufficient number of half-trusses to reduce the bearing of the rafters to a convenient length.
Or, the two principal trusses may cross each other at right angles in the centre of the dome, the one being placed so much higher than the other as to prevent the ties interfering. This disposition is represented in Fig. 3,'and is the same as that adopted for the Dome des Invalides, at Paris, of which the external diameter is nearly 90 English feet.
As the dimensions of the parts must depend chieny on the weight of the lantern, it is scarcely possible to give any general rules for them which would be satisfactory. The dimensions of the timbers may however be easily ascertained for any particular design, from the rules and principles laid down in Sects. I. and II.
Domes without Horizontal Ties.
269. The construction of domes without horizontal cross-ties is not difficult when there is a sufficient tie round the base. The most simple method, and one which is particularly useful in small domes, is to place a series of curved ribs so that the lower ends of those ribs stand upon the curb at the base, and the upper ends meet at the top, with a sufficient number of intermediate braces to prevent the ribs from yielding laterally.
When the pieces are long, and so much curved that they cannot be cut out of timber otherwise than across the grain, which reduces their strength, they should be put together in thicknesses, with the joints crossed, and well nailed together; or, in very large domes, they should be bolted or keyed together. The manner of forming these ribs has been already described, as applied to roofs (see Sect. IV., Art. 211). The method of making curved ribs in thicknesses has been adopted in the construction of centres for arches from the earliest period of arch building; and it was first applied to the construction of domes by Philibert de Lorme,* who gives the following scantlings for different sized domes: -
For domes of 24 feet diameter 8 inches by 1 inch.
These ribs are formed of two thicknesses of the scantlings given above, and are placed about 2 feet apart at the base. The rafters are notched upon them for receiving the boarding, and also horizontal ribs are notched on the inside, which gives a great degree of stiffness to the whole.† Fig. 1, Plate XXX., is a section of a dome constructed in this manner, and Fig. 2 a projection of a part of the dome, with the rafters and inside ribs.
When a dome is of considerable magnitude, the curve of equilibrium should pass through the middle of the depth of the ribs, particularly if a heavy lantern rests upon them.-This curve will be found by means of Art. 71 or 72, Sect. 1. Otherwise the ribs should be within the curve of equilibrium, and they ought to be strutted to prevent their bending in. Or, if it be necessary for the external appearance of the dome
* See his 'Nouvelles Inventions pour bien Batir a Petits Frais.' 1561.
† Mr. Price proposes a similar mode of forming bridges and domes in his ' British Carpenter.' that the curvature of the ribs should be without the curve of equilibrium, then an iron hoop may be put round at about one-third of the height to prevent the dome bursting outwards. This latter method was adopted in the external dome of the Church de la Salute at Venice; the outside dimensions of which are 80 feet diameter, 40 5 high, and the lantern 39.5 feet high; but the lantern is supported by a brick dome, which is considerably below the wooden one. The ribs of this dome are 96 in number, and each rib is in four thicknesses; the four together make 5.5 inches, so that each rib is 8 • 5 inches by 5 • 5 inches. The iron hoop is 4.5 inches wide, and half an inch in thickness, and is placed at one-third of the height of the dome.
270. One of the finest applications of the system of De Lorme was the cupola over the Original Halle au Ble at Paris, completed in 1783 (Plate XXXI.). Although 129 feet in diameter* its thickness did not much exceed 1 foot, notwithstanding which it stood in perfect safety until destroyed by fire in 1802. The ribs of which it was composed were formed of planks in 9 feet lengths, 13 inches wide, and 3 inches thick, bolted and tied together after the manner shown by Fig. 80, Art. 241.
At about one-third of the height of the dome from the springing every third rib was discontinued to admit of an opening, which was glazed. The ribs were about 2 1/2 feet apart at the base, and those next the openings were formed with four thicknesses of planks, all the others having only three. At the top of the dome the ribs were framed into a circular ring of timber, leaving an open space which was protected by a glazed canopy, with perforations for the ventilation of the building.
No modern example, executed in wood, has surpassed this
* Emy, Traile de Charpenterie.' dome, either for simplicity or strength; and the facilities afforded at the present day by the use of wrought iron has probably rendered the execution of domes of this magnitude in wood a thing of the past, except for temporary purposes.
271. When a dome is intended to support a heavy lantern, it may require the principal ribs to be stronger than can be obtained out of a single piece of timber; but the framing may always be made sufficiently strong by using two ribs, with braces between, and tied together by radial pieces across from rib to rib. A truss of this form is shown by Fig. 3, Plate XXX., which would sustain a very heavy lantern if the curve of equilibrium were to pass in the middle between the ribs, as the dotted line does in the figure. The proper form for the curve will be found by the equations in Art. 72, Sect. 1.
Trusses somewhat similar to that shown by Fig. 3, Plate XXX., were used for the roof and semi-domes of the Dublin Exhibition building in 1853. Each truss was formed of two concentric vertically laminated ribs about 5 feet apart, with intermediate diagonal framing, in which both struts and ties were formed of timber. The upper or outer rib consisted of ten lamina 1 1/2 to 2 inches in thickness, and 4 to 18 inches in depth. The breadth of the rib at top was 18 inches, and at bottom only 3 inches, each ply being stepped back from the lower edge of the preceding. The inner rib was formed of six 1 1/2 and 2 inch lamina, and was 12 inches deep and 10 inches wide. The span of the semi-domes of the great hall was 100 feet, and the principal trusses were 25 feet apart.*
272. Where a light dome is required, without occupying much space, the ribs may be placed so near to each other that the boards can be fixed to them without rafters, or short struts may be placed between the ribs, as shown by Fig. 4.
* Mallet's ' Record of the International Exhibition of 1862.'