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by an inner line for stone or brickwork. Then it will be at once perceived that the lower block K has to support a mass I, of less dimensions as to horizontal length; that the block L supports a still less mass M ; that M supports a much less mass N.; and that N supports a mass of but a small length in comparison with K, whilst in breadth it dini. mishes from a few feet to nothing at the apex. If the dimensions of a dome were worked out, say of 50 ft. internal diameter, and of 4 ft. in thickness, it would be found that the block K would be about 413 ft. cube; L 3683 ft. cube; M 2743 ft. cube; N 146; it. cube; and the half block O 223 ft. cube. The fact has to be remembered, that all domes are built in courses of stones which are bonded one into the other, forming circular rings; and that even if a dome be cut down into four quarters, each quarter will stand of itself. 1499r. Rankine, Appled Mechanics, 1858, points out that the tendency of a dome to spread at its base is resisted by the stability of a cylindrical wall, or of a series of buttresses surrounding the base of the domes, or by the tenacity of a metal hoop encircling the base of the dome. The conditions of stability of a dome are ascertained by him in the followingmanner. Let fig. 59ca. represent a ver- o tical section of a dome springing from a cylindrical wall BB. The shell of the dome is supposed to be thin as compared with its external and internal dimensions. Let the centre of the crown cf the dome, O, be taken as origin of coordinates; let r be the depth of any circular joint in the shell, such as CC; and y the radius of that joint Let i be the angle of inclination of the shell at C to the horizon, and ds the length of an elementary are of the vertical section of the dome, such as Fig. 500d. CD, whose vertical height is dr, and the difference of its lower and upper radii dy; so that '=cotan i; '-cosec i. Let Pr be the weight of the part of the dome above the circular joint CC. Then the total thrust in the direction of a set of tangents to the dome,
radiating obliquely downwards all round the joint CC, is P. #- P. cosec i, and the total
horizontal component of that radiating thrust is P'= P, cotan i. Let py denote the intensity of that horizontal radiating thrust, per unit of periphery of the joint CC; then because the periphery of that joint is 2 w y (= 6.2832 y), we have p, = 1: - - - - - - - ." w
14.99s. If there be an inward radiating pressure upon a ring of a given intensity per unit of arc, there is a thrust exerted all round that ring, whose amount is the product of that intensity into the radius of the ring. The same proposition is true, substituting an outward for an inward radiating pressure, and a tension all round the ring for a thrust. If, therefore, the horizontal radiating pressure of the dome at the joint CC be resisted by the tenacity of a hoop, the tension at each point of that hoop, being denoted by Py, is given
- P. cotani
by the equation Py=ypy =#. Now conceive the hoop to be removed to the circular joint DD, distant by the arc ds from CC, and let its tension in this new position be l', -d Py. The difference, dPy, when the tension of the hoop at CC is the greater, represents a thrust which must be exerted all round the ring of brickwork CC D1), and
whose intensity per unit of length of the arc CD is p. = # =: • # P. cotan i.) 1499t. Every ring of brickwork for which p2 is either nothing or positive, is stable, independently of the tenacity of cement; for in each such ring there is no tension in any direction. ...When p2 becomes negative, that is, when P, has passed its maximum and begins to diminish, there is tension horizontally round each ring of brickwork, which, in order to secure the stability of the dome, must be resisted by the tenacity of cement, or of external hoops, or by the assistance of abutments. Such is the condition of the stability of a dome. The inclination to the horizon of the surface of the dome at the joint where l's =0, and below which that quantity becomes negative, is the angle of rupture of the done; and the horizontal component of its thrust at that joint, is its total horizontal thrust against the abutment, hoop or hoops, by which it is prevented from spreading. A dome may have a circular orening in its crown. Oval-arched openings may also be made at lower points, provided at such points there is no tension; and the ratio of the horizontal to the inclined axis of any such opening should be fixed by the equation
inclined aris " c -v.: Itankine concludes with examples of “spherical,” and “truncated conical,” domes.
1499n. Cones.—These are used in tile-kilns, glass-houses, and such like. A building in the shape of a hollow cone forms everywhere a species of circular arch, which may be constructed without centering or support, provided the joints be made to radiate towards the centre. The courses should be laid perpendicular to the sides of the proposed cone. A
rod of variable length, turning on a pivot, must be stretched all round from time to time, upon a moveable centre, rising as the work procee's, in order to regulate the internal outline. Such is the strength of this form that the highest kilns are seldom built more than one brick thick, although this dimension would be altogether insufficient for a common wall of the same height. It is, probably, this principle which has conduced to the existence of the Round Towers of Ireland. That of Kilkenny, for example, 100 ft. in height, was built on, or very near the surface, for at 2 ft. below it, wood coffins with skeletons were found partly under the walls, thus affording an unstable foundation.
PoinTED ARch WAULTING.
1499r. We now proceed to enter into a view of the general forms of groining in pointed architecture, observing, by the way, that the groins at the arrises, up to the twelfth century, were seldom moulded with more than a simple torus or some fillets. In the twelfth - century, however, the torus is doubled, and the doubling parted by a fillet. Towards the end of the twelfth century, three tori often occur; and at the beginning of the thirteenth, the moulded arrises become similar to the moulded archivolts of the arches, both in their form and arrangement. In France, until the middle of the fifteenth century, the arrises of the groins only were moulded; but in this country the practice took place much earlier, for, instead of simple groining, the introduction of a number of subdivisions in the soffits of arches had become common. In fig. 590e is given a plan of the soflit of a vault of this kind, in which A is an arc doubleau (by which is understood an arc supposited below another at cer. – tain intervals, and concentric with the latter); B is an upper arch, called by the French antiqua. ries formeret; C, the wall arch, or formeret du mur; D is a diagonal rib, or croisée d'ogire; E, intermediate rib or tierceron; FF, summit ribs or liernes; G, the key or boss, clef de voute. Mr. Willis has used the French terms here given, and as we have no simple terms to express them in English, it may be convenient to adopt the practice. 1499ne. The ribs formed by the intersections of the groins perform the office of supporting the vaulting which lies upon them, they in their turn being borne by the pillars. Thus, in the simple groin (fig. 590f), the arches AA, and diagonal rib C, carry the vaulting BB, a rebate being formed at the lower part of the ribs on which the vaulting lies. This figure exhibits the simplest form of groining in any species of vaulting, the intersecting arches being of equal height. The contrivance in its earliest state was ingenious, and the study attractive, and we cannot be surprised at Dr. Robison observing, in respect of the artists of the thirteenth and two following centuries, that “an art so multifarious, and so much out of the road of ordinary thought, could not but become an object of fond study to the architects most eminent for ingenuity and invention: becoming thus the dupes of their own ingenuity, they were - fond of displaying it where not necessary.” This --------4-- observation would be fully verified had we room for Fig. 59Qf. showing the reader the infinite number of devices that ingenuity has created: he will, however, from the few elementary ones that we do give, be enabled to see the germs of countless others. 1499r. Ware, in his Tracts on l'aults and Bridges, 1822—a work which, notwithstanding the quaint method in which the subject is treated, contains extremely valuable matter, —has made some remarks which we must introduce at length, or justice would not be done to them. “In the vaulting,” he says, “of the aisles of Durham and Canterbury cathedrals are to be observed the arcs doubleaux and groined ribs in round-headed vaults. In the naves of the same buildings is the same character of vaulting, except that the arch of the vault is pointed. Some vaults of this kind are to be distinguished from others by the
positing of the stones of the vault between the ribs, which, instead of being parallel to each side of the plan, as in Roman groined vaults, take a mean direction between the groined rib and the ribs of the arches over the sides; whence they meet at the vertex at an acute angle, and are received by spones running along the vertex, cut in the form of a ratchet. The advantage of this method consists in requiring less centering, and originates in the position of the ribs at the springing.” “From these beginnings vaulting began to assume those practical advantages which the joint adaptation of the pointed arch and ribs was calcul ited to produce.” “The second step differed from the first, inasmuch as at the vertex of the vault a continued keystone or ridge projects below the surface of the vault, and forms a feature similar to the ribs. But here it was necessary that the ridge should be a stone ot great length, or having artificially that property, because its suspension by a thinner vault than itself would be unsafe, unless assisted by the rib arches over the diagonals and side, a distance equal to half the width of the vault. To obviate this objection, other ribs were introduced at intervals, which may be conceived to be groined ribs over various oblongs, one side continually decreasing. This practice had a further advantage, as the panels or vaults between the ribs might become proportionally thinner as the principal supports increased. It is now that the apparent magic hardiness of pointed vaulting and the high embowered roof began to display itself; from slender columns to stretch shades as broad as those of the oak's thick branches, and, in the levity of the panel to the rib, to imitate that of the leaf to the branch.” “On comparing rib-pointed vaulting with Roman vaulting, it will be invariably found that the rib itself is thinner than the uniform thickness of the Roman vault under similar circumstances; and that the panel, which is the principal part of the vault in superficial quantity, sometimes does not exceed one ninth part of the rib in thickness. The Gothic architects, it has been expressively said, have given to stone an apparent flexibility equal to the most ductile metals, and have made it forget its nature, weaning it from its fondness to descend to the centre.”
Fig. 590g. Fig. 590h. 14.99g. In the second example (fig. 590g.), another rib, a b, is introduced, which on plan produces the form of a star of four points. The forms of these thus inserted ribs result from curves of the lines on the plan in the space to be vaulted. As many radii are drawn
from the angles of the plan as there are ribs intended, until they mutually intersect each other. The curvatures of the ribs will be elongated as they recede from the primitive arch, till they reach the centre on the place where the groins cross, and where of course the elongated curve is a maximum The ribs thus form, when they are of the same curvature, portions of an inverted conoid. 14992. In the next example (fig. 590h.), the primitive arches are unequal in height, the arch A being higher than B The plan remains the same as in that immediately preceding: but from the inequality of height, a d, c b, must be joined by curved lines, determined on one side by the point a where e a intersects the longer arch. A curved summit rib, as well longitudinally as transversely, may occur with equal or unequal heights of primitive arches (as in fig. 593i.); but the stellar form on the plan still remains, though differently modified, with the same, or a less or greater, number of ribs on the plan (fig. 590k.). By truncating, as it were, the summit ribs, level or otherwise, with the tops of the primitive arches, and introducing on the plan a polygon or a circle touching quadrants inscribed in the square, we obtain, by means of the rising conoidal quadrants, figures which perform the office of a keystone. In this, as we have above observed, the construction of the work is totally different from rib vaulting, inasmuch as each course, in rising, supports the next, after the manner of a dome, and is not dependent on ribs for carrying the filling-in pieces. Hence the distinction between fanwork and radiating rib work so judiciously made by Mr. Willis. 1499aa. The sixth example (fig. 590l.) has primitive arches of different heights, forming an irregular star on plan, that is to say, the points are of different angles. The figure will scarcely need explanation after what has been already said in relation to the subject 1499bb. A polygonal space may be vaulted in three different ways. First, by a central column serving for the reception of the ribs of the vault, the :olumn or pillar performing in such case the office of a wall, as in the chapter-houses of Worcester, Salisbury, Wells, and Lincoln. This mode evidently admits of the largest space being covered, on account of the subdivision of the whole area by means of the central pillar. The second mode is by a pendent for the reception of the arches, as in the Lady Chapel at Caudebec, (given in the section MAsoNRY). This mode is necessarily restricted in practice to small spans, on account of the limits attached to the power of materials; albeit in theory its range is as extensive as the former. The last method is by at
once vaulting the space from wall to wall, as in fig. 590m., like the vaulting to the kitchen of the monastery of Durham Cathedral, or fig. 590m., similar to the chapter-house at York, of which, the upper part being of wood, Ware quaintly observes, “The people of Yorkshire fondly admire and justly boast of their cathedral and chapter-house. The principle of vaulting at the chapter-house may be admired and imagined in stone; not so the vault of the nave; it is manifestly one of those sham productions which cheat where there is no merit in deceiving.” The principle, as Ware justly observes, is perfectly misonic, and might be easily carried out with stone ribs and panel stones, it being nothing more than an extension of that exhibited in the third example of simple groining (fig. 590f.) above given; and the same remark applies to the Durham kitchen. 1499cc We propose to offer explanations of the nature of the vaulting at King's College Chapel at Cambridge, and the silly story related by Walpole of Sir Christopher Wren, saying, “that if any man would show him where to place the first stone he would engage to build another" (vault like it). The vault of the chapel in question is divided into oblong severies, whose shorter sides are placed longitudinally (fig. 5900.) It must be evident that the curves of the inverted quadrants must intersect each other previous to the whole quadrant of the circle being completed. Hence these intersections form a curved summit line lowest against the windows or smaller sides of the oblong. This summit line of the vaulting of the building in the direction of its length forms a series of curves, though from the angle under which it is seen it is scarcely perceptible. Mr. Ware says, “It is observable, in the construction of this vault, that the principle of using freestone for the ribs, and tufa for the panels, has not been followed; but the whole vault has been got out of the same description of stone, and with an uniform face, and the panels worked afterwards, and reduced to a tenuity hardly credible except from measurement. The artists of this building might be trusted in the decoration of a vault with what is now called tracery; they knew how to render it the chief support, and what was the superfluous stone to be taken away : every part has a place, not only proper, but necessary; and in the ribs which adorn the vault we may in vain look for false positions. This is the ocular music which affords universal pleasure.” 1499dd. We now return to the consideration of two more modes of simple vaulting. In England, the summit ribs of the vault are almost always found running longitudinally and transversely in the various examples. In Germany the summit ribs are more frequently omitted than introduced. Thus in the example fig. 590l, the scheme is merely a square diagonally placed within the severy, subdivided into four parts and connected with the basepoints of the groins by ribs not parallel to the alternate sides of the inserted square. This, however, sometimes occurs in English buildings, as in the monument of Archbishop Stratford, at Canterbury Cathedral; though in that the central portion is not domical. It is to be remarked that the intersecting arches are not of equal height, otherwise the arrangement could not occur. 1499ee. In the example fig. 590p, the arrangement completely assumes what Mr. Willis calls the stellal form. Here in the soffit a star of six points is the figure on which the projection depends, the points radiating from the angles of an hexagon, and thus forming a cluster of lozeng's whose middle longitudinal sides produce another longitudinal lozenge to connect the centres of the pattern. The longitudinal arches are, as in the preceding figure. lower than the transverse arches. Mr. Willis says, “the principal distinction between these and our own fanvaulting is the substitution of lozenge-headed compartments in the fans, for the English horizontal transom rib. We have also lozenge-headed compartments in our early vaulting, but they are never so symmetrically arranged in stars throughout.” 1499ff: From the simple lines or principles above given, it is easy to perceive through what numberless ramifications of form they may be carried. Another form is that called hexpartite vaulting, where the ribs spring from the angles, and two others from a shaft placed in the middle of each long side, thus making six divisions. This is a step beyond the quadripartite groining shown in fig. 590f Examples of hexpartite vaulting are scarce in England, but it may be seen in the chapel of St. Blaise in Westminster Abbey, the choir of Canterbury Cathedral, and in many parts of Lincoln Minster. 1499.gg. It would be difficult to find a system of vaulting more unlike any English example than that in Anjou generally, of which the Hospital at Angers is a fair specimen. It is always excessively domical in its sections, both longitudinal and transverse; and has eight ribs, the cells being filled in with stones exactly parallel with the centre or ridge of each cell: the ribs are edge-roll mouldings. 1499.hh. Besides the books named above, Prof. Willis On Vaulting, and by T. Eagles, 1874, both read at the Royal Institute of British Architects, the Dictionnaire by Violletle-Duc, the Lectures by Sir G. G. Scott, R.A., and the paper by W. H. Wood, in Builder for 1883, xliv, 55, should be referred to. A very complete outline of the subject has been printed by Prof. Babcock, of the Cornell University, Ithaca, New York, for his courses of lectures.