DOME is usually understood to mean a roof which is round or polygonal horizontally, and of which any vertical section is either a round or a pointed arch. There happen to be none of elliptical or any other section than these. But some, especially in the East, have what is called an ogival outline, convex below and concave towards the top, and these are generally called cupolas, though there is no real distinction. Most of the great European domes have an opening or eye at the top, on which stands a lan-tern, except in the Pantheon at Borne, where the eye is open. Until modern times all the domes worth notice were of masonry, i.e., stone, brick, tiles, or pots, which last were used for lightness. Probably the first large wooden dome was St Paul's, of which the construction is peculiar, the inner dome visible in the church being of brick only 18 inches thick, except near the bottom where it grows out of a cone of the same thickness going up outside it and carrying the stone lantern, which looks right down into the church through an eye in the internal dome. Outside the cone is built a wooden dome covered with lead. The domes of St Peter's at Bome and Florence Cathedral are of two stone shells near together, and connected by some vertical ribs, and also carrying lanterns. But Wren's construction is infinitely stronger, since a cone sufficiently tied at the bottom cannot give way until it is ab-solutely crushed, while the bursting pressure of a weight on the top of a dome increases the bursting force enormously. St Peter's dome is cracked in several places, and held to-gether by bands, and it is covered with lead, and therefore looks no better than St Paul's, and indeed on the whole not near so well, for various reasons which may be seen in archi-tectural books; and the lantern is smaller in proportion. The only full mathematical investigation of the theory of domes with practical results, that we know of, is in a paper by Sir Edmund Beckett (then Mr Denison) in the Memoirs of the Boyal Lnstitute of British Architects of February 1871, and two shorter ones by Mr E. W. Tarn, architect, in the Civil Engineer's Journal of March 1868, which substantially agree, so far as they deal with the same points. The investigation is long and complicated, and can only be done approximately, because the introduction of the thickness deranges all the ordinary trigonometrical relations, and so we only give the principal results of those calcu-lations. Some more of them are given in Sir E. Beckett's Book on Building. It is easy to prove by strict mathe-matics that the upper 52° (nearly) of a hemispherical dome would be absolutely stable, or have no tendency to fall in or burst out, without any sensible thickness, if only tied strongly enough round the base, where the tension would be -3 of the weight of the complete hemisphere, disregard-ing the bonding effect of mortar and friction. The weight of a thin hemispherical shell is the same as that of a cylin-der of the same height and thickness standing on the same base, and is twice that of the area which the dome covers, of the same thickness, provided that bears only a small pro-portion to the diameter. The weight of any zone of the dome is proportional to its height. A hemisphere of ordi-nary stone 100 feet wide at mid-thickness and 1 foot thick weighs about 1000 tons. It is also demonstrable that a dome spreading at the bottom a little more than a hemi-sphere, so as not to start vertically, and rather flattened at the top, would stand without any sensible thickness; and so would sundry other curves, and especially an inverted catenary, which will stand even as an independent arch without thickness, for a dome is far more stable than an arch or a barrel vault of the same thickness.
The essential difference between them is that the mathematical element of a dome is not an arch of any uni-form breadth, but one whose breadth, and therefore weight, decreases upwards to nothing, being in fact a lune enclosed between two meridians very close together. And it was shown in the R.I.B.A. paper, and also by models exhibited, that a dome is stable with a thickness of only o023 of its diameter, while an independent round arch or a barrel vault requires three times as much thickness, or o072 of its diameter. Therefore a barrel vault 100 feet wide must be 7 feet thick to be stable, while a dome of that diameter need only be 27 inches ; besides which, the strength of the dome can be increased to almost any extent by building in iron bands in the lower courses, while a barrel vault cannot be so helped. Bands would be of no use whatever in a dome above 52° from the top, as the pressure above that point is entirely inward, assuming it to be tied there, and from thence it gradually increases towards the bottom, where the tension is '215 of the weight of the hemisphere. It may seem paradoxical that it should be less there than at 52° ; but the explanation is that the tension bears a higher proportion to the weight in a thin dome than a thick one, and it was an infinitely thin one which had the tension of -3 of its weight at 52°; and such a dome cannot be carried lower without bands. In a dome of the required thickness ties would have very little to do above 68°.
As the tension at the bottom is rather more than a fifth of the weight, a dome of proper thickness would be stable standing on a conical drum, with a slope inwards of about 1 to 5, or 12°, of which the tangent is -215, if the drum itself has foundations which cannot spread. The thickness requisite, and also the tension at the bottom, may evidently be greatly diminished by gradually tapering the dome upwards. If it is half as thick at the top as at the bottom, with the thickness increasing downwards as the height, it need only weigh ^ of the lightest uniform dome of the same size, and only need be 20 inches thick at the bottom for 100 feet diameter.
Pointed domes are also much stronger than hemispheres, having lost the flat top which has the greatest bursting pressure. A dome generated by the revolution of an equilateral arch, or one of 60°, requires a thickness only =-0137 diameter, or 16J inches for 100 feet; and one of 70° requires 20 inches. The tension at the bottom of a 60° dome is only -15 of its weight, which weight, how-ever, is D372 of a hemisphere on the same base, their heights being as 1 '73 to 1.
For the same reason pointed domes are fittest for carry-ing a lantern, but they are not much benefited by tapering, having already lost the most oppressive part. The Florence dome, across the flat sides of the polygon, is about 70° of the circle of its curvature. It is shown in the R.I.B.A. paper that both in hemispherical and pointed domes the weight of the lantern they will carry varies practically as the cube of the thickness. Moreover a lanterned dome requires tieing much higher up than a plain one. In short, the cone is the only proper way of carrying a stone lantern. The cone at St Paul's has a great chain round the base, which is probably superfluous, as the drum below it seems thick enough to contain the requisite slope, and visibly leans in-wards besides.
Ribs inside a dome weaken more than strengthen it, as some persons imagine, unless they are themselves deep enough to be stable as independent arches, or unless they decrease in width and weight upwards like a lune, as those in the Pantheon do, which also is so euormously thick at the haunches that it has superabundant stability. Some of the Indian domes are thick enough for arches, and they have neither eyes nor lanterns. Polygonal domes may be considered as composed of a small number of widish lunes, and only differ from round ones in being rather weaker for any given thickness and size.
Domes require no wooden centring to build them on as arches do, until you get near the top, i.e., so long as each stone laid on the ring of stones below it will not slide inwards. And if they are notched to prevent sliding the whole dome may be built without centring. The dome of Mousta in Malta was so built in this century by a common mason, who must, however, have been a man of genius. There would be no difficulty in building a dome of almost any size of bricks or stones, with the help of hoop iron in all the lower courses up to about 22° from the bottom, and then less up to 52°, and higher if it has to carry a lantern. There is no masonry dome in the world wider than 142 feet. But there have been several larger iron ones, which are an easy piece of engineering, inasmuch as iron has enormous tensile strength, while stone has very little, and mortar practically none; and all the calculations above mentioned assume the domes to be composed of nar-row lunes having no lateral bond or tie; but on the other hand all the stones are assumed to go right through the thickness and not to be liable to crush at the edges. Building the lowest courses with horizontal beds, which some architects suggested, was shown to be exactly the opposite of what is mathematically required, as there would be nothing to prevent their sliding over each other, where-as the essence of dome-construction is that the lower courses should confine the upper. It is not however prac-tically expedient to make the beds lean inwards so much as to involve acute angles of the stones, as such angles in stone will bear very little pressure. Brick domes over wells or tanks, which should always be round for strength, are usually built on mere mounds of earth for centring, and they are always of flat section, or only about the upper half of a hemisphere, and are consequently stable with very little thickness, as the earth round them forms a strong abutment.
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