VAU

517

VAU

cal section of a magazino arch balanced in all its parts, in"diich the span or width a M is 20 feet, the pitch or heightD Q is 10 feet, thickness at the crown d k = 7 feet, and theangle of the ridge lkn 112°37', or the half of it l k d =°6° 18-Jj-' the complement of which, or the elevation KIR, is^3° 41^-', the tangent of which is = §, which will thereforebe the value of t in the foregoing investigation. The values°f the other letters will be as follows, viz. dk = « = 7 ;^ Q = 5 = 10; DQ = /i = 10; a L = c = lOj = A —

kg. of 7 = .8450980; C = 4 X log. of ^ -— =

b a

ijs log. of + / - 5 - 20 - = * log. of 2.56207 = .0408591;Cy + a = .0408591 y + .8450980 = log. of n. From the

Q? |— Q?

general equation, then, viz. ci = w = — -= -1 - \n,

2 n 2 n *

by assuming y successively equal to 1,2, 3, 4, &c., thencefinding the corresponding values of cy + A or .0408591 y+ .8450980, and to these, as common logs, taking out thecorresponding natural numbers, which will be the values ofn ; then the above theorem will give the several values of to° r C i, as they are here arranged in the annexed table, from' v hieh the figure of the curve is to be constructed, by thusfinding so many points in it.

Value ofy or c p.

Value of w or c i.

i

7.0309

2

7.1243

3

7.2806

4

7 5015

5

7.7888

6

8.1452

7

8.5737

8

9.0781

9

9.6028

10

10.3333

“ Otherwise : Instead of making n the number of the log.c y + a, if we put m = the natural number of the log. cy

w + s/ (w 7 — O ,

v -- and am — w zr

- 2 a mw + w 2 = w 1

°«ly ; then m =(w — a 2 ), or by

a 7 , and hence w =

a

squaring, &c. a 2 nr

m 2 + 1

2 m

X a ; to which the numbers

being applied, the very same conclusions result as in the fore-going calculation and table.”

The following description of a method of stone-roofing inl be Southern Concan, in the East Indies, was communicated*° the Institution of Civil Engineers by Lieutenant Outram,°1' the Bombay Engineers. Though hardly applicable to theSituate of this country, it seems of considerable value asRelating to an important part of the British empire. The’ e ' v houses which had been constructed on this plan weretQ Und to answer so well, at the time this paper was written,kat government had given orders to construct, on this prin-Cl ple, all the public buildings, wherever suitable materialsc °uld be found.

The roofing with stone (iron • clay, or laterite) in theSouthern Concan, is of a compound nature, consisting of twobinds of arches ; the first being parallel to each other, from^o3 feet apart, and very light, their average section being

fr

°m 12 by 10 inches to 15 by 12 ; i. e.voi.. n. 6 Q

for roofs of from 25

to 35 feet span ; so that when any two of these arches or ribsare complete, they are strong enough to bear slabs of stone 5or 6 inches thick, extending a few inches over each, begin-ning from the wall and meeting at the top, thus forming asecond complete arch, and making, with the ribs, a compoundmuch stronger than vaulting of equal solidity over the sameextent, made in the usual way.

The lateral thrust of the arches of one room are counter-acted by those of the rooms on its sides, and so on for anyextent; those of the end-rooms being counteracted on theirouter sides by buttresses, or by the walls of baths, &c., sothat the walls are required to be only sufficiently strong tosupport the mere weight of the masonry of the roofs, whichhas an average thickness of about 9 inches, excepting theplaster or tiles, and, therefore, in rooms of 400 square feet,would be about one-fifth the weight of the upper walls of atwo-storied house. As the roof itself is of considerable alti-tude, the walls supporting it need not be of more than two-thirds the usual height.

One advantage of the lightness of these roofs is, that ofwhatever form the arches may be, very little loading willsuffice ; of course some arches would require no loading, butsuch are not the most convenient for roofs in general. Thebest appears to be a compound of two segments of a circle of50° or 55°, their chords intersecting at an angle of about 100°,such compound arch requiring a little loading at the top andthe haunches, which, when duly added, gives an outer surfaceof two inclined planes to each roof, which may be then eitherplastered or tiled. But instead of loading the haunchesthroughout with solid rubble, it is better to do so partly withhollow masonry, to the upper surface of which may be givenany slopes which, by the connection of the opposite slopes ofany two adjacent roofs, form a gutter of the securest kind.The average height of this gutter should be about one-thirdthat of the roof, if to be plastered, but not so much if theroof is to be tiled.

The expense of these roofs, including the outer plaster,has been found by myself and successor, in the Concan, tobe much less than that of tiled roofs over the same extent.The walls should cost no more than those of a substantialbungalow ; for although the transverse walls have a greaterweight to support, yet, as they need be only two-thirds theheight, their total expense should not be greater than that ofthe walls of a substantial house. The only part of which thecomparative expense remains to be considered, is the ceiling.The inner surface of the stone roofs, when finely plastered,forms an excellent ceiling ; being light and cleanly, and mostdurable. The expense of this plastering, if not much orna-mented, is below one-third that of the lath-and-plaster gene-rally used. Hence it is plain, and has been practically found,that the total expense of stone-roofed houses in the Concan, ifproperly constructed, is less than that of tiled houses of thesame size ; but the sums saved in annual and special repairsare of far greater consideration. In the Deccan, wheretimber is so expensive, the comparative cost of these build-ings would be still less, in all those parts of it where properstone is met with.

The principal cause of the cheapness of these stone roofsis the very little centering, &c. that is requisite. For asthe ribs, or primary arches, are very light, centering of thesimplest kind does for any one of them, and thus for all suc-cessively in either room. But as the centering cannot beremoved from any rib till its counteracting ribs are complete,there is of course required one centering for each room,which, when one series of the primary arches is complete,may be removed with ease for the next, till a convenientnumber are ready for the superior arching, which of course