from the well-known expression for the area of the zone of a sphere it appears that ad, will represent the weight of that part of the lune above ad. Similarly is the weight of the lune ag,; ad, the weight of aa,, etc.

au

This method of obtaining the weight applies of course in case the dome is any segment of a sphere less than a hemisphere and of uniform thickness. If the thickness increases from the crown, the weights of the zones cut by equi-distant horizontal planes increase directly as the thickness. In case the dome is not spherical the weights must be determined by some process suited to the form of the dome and its variation in thickness.

of the equation gives the height of it above bas (5-1) r, corresponding to about 51°49'. Now consider any zone, as, for example, that whose meridian section is g,a,: the upper edge is subjected to a thrust whose radial horizontal component is proportional to ut, while the horizontal thrust against its lower edge is proportional to ds, and the difference S2x2 between these radial forces produces a hoop compression around the zone proportional to s,,. It will be seen that these differences which are of the type sx or ty, change sign at t. Hence all parts of the dome above 51° 49′ from the crown, are subjected to a hoop compression which vanishes at that distance from Now the weight of the lune aa, is sus- a, while all parts of the dome below tained by a horizontal thrust which is this are subjected to hoop tension. This the resultant of the horizontal pressures may be stated by saying that a thin in the meridian planes by which it is dome of masonry would be stable under bounded, and by a thrust, as before re- hoop compression as far as 51° 49′ from marked, in the direction of the tangent the crown, but unstable below that, being at a. Draw a horizontal line through d,, liable to crack open along its meridian and through a a parallel to the tangent sections. A thick dome of masonry, at a: these intersect at s,, then is ads, however, does not have the resultant the triangle of forces which hold in thrust at every point of its meridian equilibrium the lune aa,. Similarly, section in a direction which is tangential aut, is the triangle of forces holding the to its surface, this will be discussed lune ag, in equilibrium, etc. Draw a curve st through the points thus determined. This curve is a well-known cubic which when referred to ba as the axis of x and bg, as that of y has for its equation

8

On being traced at the right of a it has in the other quadrant of the dome a part like that here drawn forming a loop; it passes through b at an inclination of 45° and the two branches below 6 finally become tangent to a horizontal line drawn tangent to the circle aa of the dome. The curve has this remarkable property:-If any line be drawn from a, cutting the curve here drawn and, also, the part below bg,, the product of these two radii vectores of the curve from the pole a is constant, and the locus of the intersection of the normals at these two points is a parabola.

Draw a vertical tangent to this curve: the point of contact is very near t,, and g,, the corresponding point of the dome is almost 52° from the crown a. A determination of this maximum point by means

later.

It is necessary to determine the actual hoop tension or compression in any ring in order to determine the thickness of the dome such that the metal may not be subjected to too severe a stress.

The rule for obtaining hoop tension (we shall use the word tension to include both tension and compression) is: Multiply the intensity of the radial pressure by the radius of the hoop, the product is the tension at any meridian section of the hoop. The correctness of this rule appears at once from consideration of fluid pressure in a tube, in which it is seen that the tensions at the two extremities of a diameter prevent the total pressure on that diameter from tearing the tube asunder.

Now in the case before us ty, is the radial force distributed along a certain lune. The number of degrees of which the lune consists is at present undetermined let it be determined on the supposition that it shall be such a number of degrees as to cause that the total radial force against it shall be equal to the hoop tension. Call the total radial force P and the hoop tension T, then the lune

is to be such that PT. Also let be the number of degrees in the lune, then 90° is the number of lunes in a quarter of the dome, and 90 P÷0 is the radial force against a quarter of the dome, which last must be divided by 27 to obtain the hoop tension; because if p is the intensity of radial pressure, arp is the total pressure against a quadrant and rp, as previously stated, is the hoop tension. The ratio of these is 7, and by this we must divide the total radial pressure in every case to obtain hoop tension

θπ

π

for P=T.. 0=57°.3—

CHAPTER XVI.

SPHERICAL DOME OF MASONRY.

Let the dome treated be that in Fig. 18 in which the uniform thickness of the masonry is one-sixteenth of the internal diameter or one-eighth of the radius of the intrados. Divide ab the radius of the center line into any convenient number of equal parts, say eight, at u,, u, etc.: a much larger number would be preferable in actual construction. At the points a,, a,, etc., on the same levels with u,, u,, etc. pass conical joints normal to the dome, so that b is the vertex of each of the cones.

If we consider a lune between meridian planes making a small angle with each other, the center of gravity of the parts This is the number of degrees of which of the lune between the conical joints lie the lune must consist in order that when at g1, g, etc. on the horizontal midway ab represents its weight, ty, shall rep-between the previous horizontals. These resent the hoop tension in the meridian points are not exactly upon the central section a,g,. The expression we have line aa, but if the number of horizontals found is independent of the radius of the is large, the difference is inappreciable. ring, and hence holds for any other ring We assume them upon aa. That they as ga, in which s,, is the hoop tension, fall upon the horizontals through d,, d, etc. To find what fraction this lune is etc., midway between those through u, of the whole dome, divide ✪ by 360° u,, etc., is a consequence of the equality in area between spherical zones of the same height.

Ꮎ 180 1 4

=

=

=

360 360π 2π 25

nearly,

from which the scale of weight is easily found, thus; let W be the total weight of the dome and r its radius, then

2πr: W::1: n, the weight per unit, or the hoop tension per unit of the distances ty or sx.

In finding the volume of a sphere it may be considered that we take the sum of a series of elementary cones whose bases form the surface of the sphere, and whose height is the radius. Hence, if any equal portions of the surface of a sphere be taken and sectorial solids be formed on them as bases and having their vertices at the center, then the sectorial solids have equal volumes. The lunes of which we treat are equal fractions of such equal solids.

Distances at or as, on the same scale, represent the thrust tangential to the dome in the direction of the meridian Draw the verticals of the type bg sections, and uniformly distributed over through the centers of gravity 9, 9, etc. an arc of 570.3-: e.g. if we divide at The weights applied at these points are measured as a force by Xug, measured equal and may be represented by au as a distance we shall obtain the intensi-u,u,=ww2, etc. Use a as the pole and ty of the meridian_compression at the ww, as the weight line; and, beginning joint cut from the dome by the horizon- at the point f, draw the equilibrium tal plane through a,. polygon c due to the weights.

Ө

Analogous constructions hold for domes not spherical and not of uniform thickness. Approximate results may be obtained by assuming a spherical dome, or a series of spherical zones approximating in shape to the form which it is desired to treat.

We have used for pole distance the greatest horizontal thrust which it is possible for any segment of the dome to exert upon the part below it, when the hoop compression extends to 51° 49′ from the crown.

Below the point where the compression

vanishes we shall not assume that the bond of the masonry is such that it can resist the hoop tension which is developed. The upper part of the dome will be then carried by the parts of the lunes below this point by their united action as a series of masonry arches standing side by side.

92, 9,, etc. by horizontals through c1, c,, etc. Through these points draw the curve qq, whose ordinates are of the type gh. Some one of these ordinates is to be elongated to its corresponding ph, and in such a manner that no qh shall then become longer than its corresponding ph. To effect this, draw oq, tangent Now it is seen that the curve of equi- to the curve 99; then will og, enable us librium c, drawn with this assumed hori- to effect the required elongation: e.g. let zontal thrust falls within the curve of the the horizontal through c, cut oq, at j lune, which signifies that the dome will and then the vertical through j, cuts fo not exert so great a thrust as that as-at i,, then is e, (which is on the same sumed. By the principle of least resist-level with i) the new position of c. ance, no greater horizontal thrust will Similarly, we may find the remaining be called into action than is necessary to points of the curve e; but it is better to cause the dome to stand, if stability is determine the new pole distance, and use possible. If a less thrust than that just this method as a test only. employed be all that is developed in the dome, then the point where the hoop compression vanishes is not so far as 51° 49' from the crown, and a longer portion of the lune acts as an arch, than has been supposed by previous writers on this subject,* none of whom, so far as known, have given a correct process for the solution of the problem, although the results arrived at have been somewhat approximately correct.

The curve 99 made use of in this construction for finding the ratio lines for so elongating the ordinates of the curve c, that the new ordinates shall be those of a curve e tangent to the exterior line of the inner third, may be applied with equal facility to the construction for the arch of masonry. This furnishes us with a direct method in place of the tentative one employed in connection with Fig.

14.

To ensure stability, the equilibrium To find the new pole distance, draw curve must be inscribed within the inner fj || oq, cutting ww at j, then will the third of that part of the meridian section intersection of the horizontal through j, of the lune which is to act as an arch; as he the new position of the weight line vv, appears from the same reasons which having its pole distance from a diminishwere stated in connection with arches of ed in the required ratio. masonry.

And, further, the hoop compression will vanish at that level of the dome where the equilibrium curve, in departing from the crown, first becomes more nearly vertical than the tangent of the meridian section; for above that point the greatest thrust that the dome can exert, cannot be so great as at this point where the thrust of the arch-lune is equal to that of the dome.

Now to determine in what ratio the ordinates of the curve c must be elongated to give those of the curve e which fulfills the required conditions, we draw the line fo, and cut it at P,, P2, etc. by the horizontals m,p,, m,p,, etc., the quantities mb being the ordinates of exterior of the inner third. Again draw verticals through P,, P,, etc., and cut them at q,, * See a paper read before the Royal Inst. of British Architects, on the Mathematical Theory of Domes," Feb. 6th, 1871. By Edmund Beckett Denison, L.L.D.,

Q.C., F.R.A.S.

The equilibrium curve e will be parallel to the curve of the dome at the points where the new weight line vv cuts the curve st. It should be noticed that the pole distance which we have now determined is still a little too large because the polygon e is circumscribed about the true equilibrium curve; and as the polygon has an angle in the limiting curve mm the equilibrium curve is not yet high enough to be tangent to the limiting curve. If the number of divisions had originally been larger (which the size of our Figure did not permit) this matter would be rectified.

The polygon e is seen at e, to fall just without the required limits, this would be partly rectified by slightly decreasing the pole distance as just suggested; the point, however, would still remain just without the limit after the pole distance is decreased, and by so much is the dome unstable. Á dome of which the thick

ness is one fifteenth of the internal dia- in the metallic dome. It will be noticed meter, is almost exactly stable.

that the addition of very small weight at the crown will cause the point m, of no hoop tension in the dome of masonry to approach almost to the crown, so that then the lunes will act entirely as stone arches with the exception of a very small segment at the crown.

On the contrary, the removal of a segment at the crown, or the decrease of the thickness, or any device for making the upper part of the dome lighter will remove the point of no hoop tension further from the crown, both for the dome of metal and of masonry. In any dome of masonry the thickness above the point of no hoop tension, as determined by the curve st, need be only such as to withstand the two compressions to which it is subjected, viz; hoop compression and meridian compression: while below that the lunes acting as arches must be thick enough to cause a horizontal thrust equal to the maximum radial thrust of the dome above the point of no hoop tension.

It is a remarkable fact that a semicylindrical arch of uniform thickness and without surcharge must be almost exactly three times as thick, viz., the thickness must be about one fifth the span in order that it may be possible to inscribe the equilibrium curve within the inner third. The only large hemispherical dome, of which I have the dimensions, which is thick enough to be perfectly stable without extraneous aid such as hoops or ties, is the Gol Goomuz at Beejapore, India. It has an internal diameter of 1373 feet, and a thickness of 10 feet, it being slightly thicker than necessary, but it probably carries a load upon the crown which requires the additional thickness. The hemispherical dome of uniform thickness is a very faulty arrangement of material. It is only necessary to make the dome so light and thin for 51° 49′ from the crown that it cannot exert so great a horizontal thrust as do the thicker lunes below, to take complete advantage of the real strength of this form Several large domes are constructed of of structure. A dome whose thickness more than one shell, to give increased gradually decreases toward the crown security to the tall lanterns surmounting takes a partial advantage of this, but them: St. Peter's, at Rome, is double, nothing short of a quite sudden change and the Pantheon, at Paris, is triple. near this point appears to be completely The different shells should all spring

effective.

The necessary thickness to withstand the hoop compression and the meridian thrust can be found as previously shown in the dome of metal.

Domes are usually crowned with a lantern or pinnacle, whose weight must be first laid off below the pole a after having been reduced to the same unit as that of the zones of the dome.

Likewise when there is an eye, at the crown or below, the weight of the material necessary to fill the eye must be subtracted, so that a is then to be placed below its present position. The construction is then to be completed in the same manner as in Fig, 18.

It is at once seen that the effect of an additional weight, as of a lantern, at the crown, since it moves the point a upward a certain distance, will be to cause the curve st to have all its points except b to the left of their present position, and especially the points in the upper part of the curve, thus making the point of no hoop tension much nearer the crown than

from the same thick zone below the point of no hoop tension; and the lunes of this thick zone should be able to afford a horizontal thrust equal to the sum of the radial thrusts of all the shells standing upon it.

Attention to this will secure the stability in itself of any dome of masonry spherical or otherwise; and, though I here offer no proof of the assertion, I am led to believe that this is the solution of the problem of constructing the dome of a minimum weight of material, on the supposition that the meridian joints can afford no resistance to hoop tension.

Now, in fact, it is a common device to ensure the stability of large domes by encircling them with iron hoops or chains, or by embedding ties in the masonry; and this case appears to be of sufficient importance to demand our attention.

If the hoop encircles the dome at 51° 49' or any other less distance from the crown the dome will be a true dome at all points above the hoop. Suppose the