Barlow's edition of Tredgold's work, from 34° to 36°, is repeated. G. Rennie, in some Careful experiments at London Bridge, found that dressed voussoirs commenced sliding, without mortar, at an angle of 33° 30; and, with mor ar fresh laid, at 25° 30.

1496. Well may it be said that the thrust of an arch is the constant dread of an architect; but it depends entirely on the method employed in the construction. It is only dangerous where the precautions indicated in the foregoing examples have had no attention paid to them. It has been seen that the least fracture in too thin an arch of equally deep voussoirs may cause its ruin; and we shall here add, that this defect is more dangerous in arches wherein the number of joints is many, such as those constructed in brick; for when they are laid in mortar they are too often rather heaped together than well fitted to each other.

1497. Whatever materials are used in the construction of vaults, the great object is to prevent separation, which, if it occur, must be immediately met by measures for making the resistance of the lower parts capable of counterbalancing the effort of the upper parts. Those fractures which occur in cylindrical arches are the most dangerous, because they take place in straight lines which run along parallel to the walls bearing them. To avoid the consequences of such failures, it is well to fill up the haunches to the height where the fracture is usually to be found, as in K, K', K", K"" (fig. 590.) and diminish the thickness towards the key. 1498. Rondelet found, and so indeed did Couplet before him, that the least thickness which an arch of equal voussoirs ought to have, to be capable of standing, was one fiftieth part of the radius. But as the bricks and stone employed in the construction of arches are never so perfectly formed as the theory supposes the least thickness which can be used for cylindrical arches from 9 to 15 feet radius is 4 inches at the vertex if the lower course be laid with a course of brick on edge or two courses flatwise, and 5 inches when the material used is not a very hard stone, intreasing the thickness from the keystone to the point where the extrados leaves the walls or piers. But if the haunches are filled up to the point N (fig. 590.), it will be found that for the pointed arch in the figure the thickness need not be more than the of the radius, and for the semicircular arch, . For arches whose height is less than their opening or that are segmental the thickness should be part of the versed sine; a practice also applicable to Gothic vaults and semicircular cylindrical arches, to which for vaults cemented with plaster one line should be added for each foot in length, or part of the chord subtended by the extrados. With vaults executed in mortar may be added, the thickness of the arch increasing till it reaches the point N, where the arch becomes detached from the haunches, and where it should be once and a half the thickness of the key. It was in this way the arches throughout the Pantheon at Paris were regulated, and a very similar sort of expedient is practised in the dome of the Pantheon at Rome. A like diminution at the keystone may be used in groined, coved, and spherical vaults.

Fig. 590.

0"

1499. For vaultings of large openings, Rondelet (and we fully concur with him) thinks wrought stone preferable to brick or rubble stone, because it has the advantage of being liable to less settlement and stands more independent of any cementitious medium employed. It is indeed true that this cannot connect wrought stone so powerfully as it does rubble; but in the former we can employ cramps and dowells at the joints, which are useful in doubtful cases to prevent derangement of the parts. In many Roman ruins the surfaces of the voussoirs were embossed and hollowed at the joints, for the purpose of preventing their sliding upon each other; and expedients of the same nature are frequently found in Gothie ruins.

1499a. The figure 590. is one that has been found to perplex students, as it is herein given without much explanation of it. In Rondelet's work it is engraved for the purpose of elucidating certain tables of thicknesses of the keystones, the parts KN, and the piers, for ready reference in designing arched constructions. As a proper understanding of the above system is of immense importance to the effective carrying out of buildings, we append an explanation from Rondelet, but in a much abridged form.

E

14996. Having laid down the half of the curve required, draw B 4, forming an angle of 45° with the vertical B 6; place on this line from B to 4 the thickness shown in the table (in his work) for a cylindrical vault Y, of the diameter and thickness required, and describe the quarter circie 1, 4, 6; draw the chord C"B, prolonged to meet the circle at 4; then through the point in which it (the chord) will le cut, as 4; draw a line parallel to B6; then 4c will represent the thickness required for the wall of the vault. For instance, il in the segmental vault X, its chord C′ B, be prolonged, it will cut the circle at 3; through 3 draw the vertical 3b, and it will be the thickness to be given to the wall for such a vault. When the thickness at the key, and towards the middle of the extrados, is required either stronger or weaker than those indicated in the tables, then, if the portion of the extradossed line be of an equal thickness, take the square root of the double thickness of this portion, multiplied by mL; place it from B to 4, and describe the quarter of the circle 1, 4, 6, which will determine, by the length of the chord prolonged beyond the point B, the thickness of the wall pier.

1499c. Take a vault of 30 ft. span; the extrados being half on a level and half of an equal thickness, which it is intended to make only 6 in, thick at the key, instead of 10 in., as indicated in the tables. The radius being 15 ft., we have KL 15x70-10-6, and i K= 15 – 106 = 44, which gives mL=6′2, which multiplied by 1 foot, or double the thickness of the keystone, will give 6-2, the square foot of which is 2:49, or a trifle under 2 ft. 6 in., instead of 2 ft. 8 in. and 9 lines, marked in the tables. This measure of 24

G

G

99

49 ft.,

or 2 ft. 6 in., is to be placed from B to 4, and the quarter circle and chord line drawn according to the rise of the arch.

1499d. The geometrical method of drawing it will be to place the double of the thickness of the vault from B to n, and mL from B to h, and describing on nh, as a diameter, a semicircle which shall cut the horizontal BO, giving the "L" thickness to be placed from B to 4 on the chord line; the remainder of the operation will be as above described.

1499e. If the thickness CD and KN of an extrados portion of a vault, be not the same as indicated in the tables, the sum of the thicknesses intended to be given is to be placed from B to n, and mL from B to h, and then the process goes on as above described. The letters also refer to the preceding diagrams.

1499f. These observations, however, do not apply to fig. 590., for it will be observed that the arches therein shown are not of equal thickness. On drawing out these arches, according to the directions given (1436, and fig. 582.), for an extrados which increases towards the springing (fig. 590a), we find that the chord lines are not properly drawn; that the thicknesses of the walls vary; and that the two arches W and X, which are less in height than the semicircular arch Y, are treated in the same manner as Y, instead of the line BF being drawn as a tangent to the curve as directed (1898, and fig. 573.); this would have caused the walls to be of a less thickness the more the arch was depressed, and therefore would evidently have been wrong in principle.

CONSTRUCTION OF DOMES.

1499g. From the Remarks on Theatres, 1809, by Samuel Ware, we extract the portion relating to the now little studied subject of construction of domes. "It may with propriety be asked," he writes, " and it is a question of much importance, what are the properties in the construction of a dome, by which its vaulting may have that extreme tenuity, by which its lateral thrust becomes so extremely small in comparison with cylindrical vaulting, while the stone furthest from the supports may be of extraordinary gravity, compared with any other part of the vaulting, or it and any part below it contiguous may be wholly omitted, and yet the equilibrium of the dome be not affected."

1499h, "In analysing a dome, it will be found that it is nothing more than rib-vaulting taried to its maximum, that it consists of as many ribs as there are vertical sections to be made in the dome, or is composed wholly of ribs abutting against each other, in direct opposition, by which the force of each is destroyed. In the ceilings of King's College Chapel, Cambridge, and Henry VII.'s Chapel, London, this most admirable invention is exemplified. The author ventures an hypothesis, that, in an equilibrated dome, the thickness of the vaulting will decrease from the vertex to the springing, and assigns the following reason theoretically, and the Gothic vaulting practically, in confirmation.

* 14992. "The parts of a circular wall compose a horizontal arch; but the whole gravity of each part is resisted by the bed on which it rests, therefore the parts cannot be in mutual opposition; and, although the parts are posited like those of an arch, a circular wall has not the properties of one. In a semi-spherical dome the first course answers this description, no part gravitating in the direction of its radius. When the beds are oblique on which the parts of the wall rest, each course may then be called an oblique arch, as it then assumes the property of an arch, by having a double action, the one at right angles to, or on the bed, and the other in the direction of the radius; and if this arch be of equal thickness throughout, and has an equal inclination to the horizon, it will be an arch of equilibration. All the courses in a dome are oblique arches of equilibration, of various inclinations between the horizontal line at the springing, and the perpendicular at its vertex.

1499j. "A dome is comprised of as many vertical arches as there are diameters, and as many oblique arches as there are chords. The actions of the parts of a vertical arch are eccentric, an oblique arch concentric; consequently they will be in opposition, and the greater force will lose power equal to that of the less. An oblique arch bears the same relation to a dome as a voussoir does to an arch; when the vertical arches are not in equilibration, the action is upon the whole oblique arch, not upon the voussoirs separately; although a whole course or oblique arch (which must be the case, or no part of it, admitting that each course in itself is similar and equal throughout) be thrust outwards by the equilibration of the vertical arches; the incumbent oblique arches will descend perpendicularly, keeping the same congruity of their own parts.

1499k. "As the voussoirs of each oblique arch are in equilibration, no one can approach bearer to the centre of the dome than another, unless the other voussoirs squeeze or crush, which, in investigations of subjects of this nature, are always assumed perfectly rigid ; therefore, in their position in the dome, they have obtained their concentration.

Hence

we obtain the essential distinction between an arch and a dome, that no part of the latter an fall inwardly. Since no part of a dome can fall inwards, it resembles an arch resting on the centre on which it has been constructed, and the resistance which the vertical arch meets with from that centre is similar to the opposition of the oblique arches to the vertical arches. If this deduction be just, the mechanician will be able to describe the extrados of equilibration to a dome and its abutment wall, with the same facility as he may to an arch and its abutment piers."

14991. Pasley has likewise stated that "as soon as any course is completed all round, the stones or bricks composing it form a circular arch like that of a cone, which cannot by any means fall inwards. Hence there is an important difference between the dome and the common arch, which latter cannot stand at all without its centering, unless the whole curve be completed, and when finished, the crown or upper segment tends to overset the haunches or lower segments. The dome, on the contrary, is perfectly strong, and is a complete arch without its upper segment; and thus, as the pressure acts differently, there is less the haunches and abutments of a dome, than on those of a common arch of the Hence a sufficient dome may be constructed with much thinner materials than would be proper for a common arch of the same section. The dome of St. Paul's Cathedral offers a fine specimen of this kind of work." It has been described in par. 472.

strain upon same curve.

The

1499m. The Pantheon, at Paris, has a dome formed of three portions. The first, or interior one. is a regular hemisphere of about 66 ft. 9 in. span, with a circular opening at lop of about 31 ft. 44 in. in diameter. It is built of cut stones, varying from 18 in. thick at bottom, to 10 in. at top. Thus the thickness is only about rd part of the span. intermediate dome is a catenarian curve having a span of about 70 ft. with a rise of 50 ft.; and it has to support considerable weight at top. It has four large openings in its sides to give light, about 37 ft. high by 31 ft. wide, arched at top in a somewhat parabolic form. The outer dome has an external diameter of 78 ft. Its height is not stated, but it appears to be a moderately pointed Gothic arch had it been continued, without forming an opening at top for the sides of a lantern, which it was intended to support. The thickness of the Stone at bottom is about 28 in. and 14 in. at top. A great part of the surface is only half the above thickness, as the dome is laid out internally in piers, supporting three tiers of parched recesses, or niches, of less substance, and showing like the panels in joiners' work. See figs. 177 and 178.)

1499. Partington, in the British Cyclopædia, 1835, expresses the opinion that "the eight of the dome may force out its lower parts, if it rises in a direction too nearly ver

CC

tical; and supposing its form to be spherical, and its thickness equal, it will require to b confined by a hoop or chain as soon as the span becomes eleven fourteenths of the whol diameter. But if the thickness of the dome be diminished as it rises, it will not requir to be bound so high. Thus, if the increase of thickness in descending begins at abou 30° from the summit, and be continued until at about 60°, the dome becomes little mor than twice as thick as at first, the equilibrium will be so far secure. At this distance i would be proper to employ either a chain or some external pressure to prove the stability since the weight itself would require to be increased without limit, if it were the onl source of pressure on the lower parts. The dome of the Pantheon, at Rome, is nearly circular, and its lower parts are so much thicker than its upper parts, as to afford a suffi cient resistance to their pressure; they are supported by walls of great thickness, and furnished with many projections, which answer the purpose of abutments and buttresses." 14990. Keeping to the theory of the dome, we must avoid noticing its history, beyon pointing out the papers which have of late years treated on the subject. These are pub lished in the Transactions of the Royal Institute of British Architects. The first was by J. Fergusson, On the Architectural Splendour of the City of Beejapore, November 1854 the discussion in December following, when J. W. Papworth detailed his interesting and novel theory, to be presently noticed; and two papers by T. H. Lewis, Some Remark on Domes, June 1857; and On the Construction of Domes, May 1859, in which, however great care must be taken by the reader to separate the arch from the dome constructions as in our opinion they are treated therein as of one principle. The question of a Gothi dome was much discussed without a solution in the journals of the period named. Dome and pendentives are illustrated in Fergusson's Handbook of Architecture. The very in teresting paper On the Mathematical Theory of Domes, by E. B. Denison, Q C., read a the Institute on 6th February, 1871, should be consulted by all students on this difficul subject; as well as the papers by E. W. Tarn, M. A., printed in the Civil Engineer an Architect's Journal of March 1868 and February 1870.

1499p. On the occasion referred to, Mr. Papworth asserted that a dome was not an arel and that domes were not governed by the same laws as vaults. He then entered into cal culations on the causes of the stability of domes, showing that in domes of great thicknes the upper half of each gore was only about one-third in weight of the lower half, and ad duced the possibility of loading the crown to a certain extent. He produced a series drawings of domes, constructed upon principles which ought theoretically, if they wer arches, to lead to their failure, but which had nevertheless proved perfectly sound; hi views being fortified by Mr. Fergusson's concurrence as to the absence of examples c failure where the bases were stable. He then alluded to the following arguments of other and explained his reasons for not agreeing with them. Such as, that the dome of the Par theon, at Rome, had been built on the principle of a bridge, i.e. of an arch; that it w impossible to plan a large dome without great thickness of walls, i.e. greater than suff cient to bear the weight and its consequences; that it was necessary for the exterior of dome to stand flush with the wall of the building to which it belonged; that it was de sirable to append heavy corbelling to the inside of the wall to counteract the thrust of th

M

N

E

Figs. 5906 and 500c.

C

A

A

dome, with special reference to some circular tambour of which he exhibited sketches; to the supposed un necessarily great weight on the top of some examples and to the supposed beauty of principle exhibited in th dome of Sta. Maria, at Florence, which he characte ised as a piece of octagonal vaulting and not a dom He also explained that domes which had failed had n Leen supported on a stable foundation; that he sa great beauty in the idea of forming an eye in so large dome as that of the Gol Goomuz, at Beejapore, whe the centre of the curve on each side of the section w in the edge of the eye; that the outer face of the spring ing of the dome might be within the inside of the squa enclosing wall of the building; that if the principles vaulting were applied, the wagon-headed section of th Gol Goomuz dome would not be expected, theoreticall to stand; and concluded by some observations in expl nation of his illustrations, as to the requisite thickne of domes. All writers, so far as he had seen, consider the dome as a case of vaulting on principles deduc from their experiments on arches, which was a mo repudiated by him.

14999. The causes of the stability of domes, as th put forward for the first time, by Mr. Papworth, are t following::-Let the plan (fig. 590b.), of a semicircular dome be divided, say, into twel or more equal parts, and the section (fig. 590c.), say, into nine or more. Give a thickn

by an inner line for stone or brick work. Then it will be at once perceived that the lower black K has to support a mass L of less dimensions as to horizontal length; that the Shock L supports a still less mass M; that M supports a much less mass N; and that N @pports a mass of but a small length in comparison with K, whilst in breadth it dimiishes 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 4134 ft. cube; L 368 ft. cube; M 274 ft. cube; N 146 ft. cube; and the half block O 223 ft. cube. The fact has to be remembered, that all domes re 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. 14997. Rankine, Applied 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

following manner. Let fig. 590d. represent a vertical 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 of 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

D

Fig. 590d.

C

B

CD, whose vertical height is dr, and the difference of its lower and upper radii dy; so

dy

that

ds =cotan i;

dr

=cosec i. Let P 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

dz

ds

dr

=

Pr 'cosec i; and the total

horizontal component of that radiating thrust is P.dy=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 π y ( = 6·2832 y), we have py Pr cotan i

=

P. cotan i
2 п

1499s. 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 boop, being denoted by Py, is given 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 Py-dPy. 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 DD, and whose intensity per unit of length of the arc CD is pz (Pr cotan i.) 1499 Every ring of brickwork for which pz 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 pz becomes negative, that is, when Py 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 P=0, and below which that quantity becomes negative, is the angle of rupture of the dome; 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 have a circular opening 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

Rankine concludes with examples of "spherical," and "truncated conical,” domes.

1499. 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 conruered without centering or support, provided the joints be made to radiate towards the The courses should be laid perpendicular to the sides of the proposed cone.

Centre.

A