Chap. XXIII.

CENTERING.

HOD

moves itself from the centre, the first joints of the extradostowards the point L are seen to open, and this effect con-tinually augments with the settlement of the centre, until theplacing of the keystone ; then the effects change their nature,because the voussoirs of the upper part being able to sustainthemselves, without the assistance of the centre, the equi-

F

x

librium tends to establish itself in the entire arch, and no / / j

longer in the portion A S. The point of rupture is then no / /f j

longer situated in KL; it rises higher, and therefore at this pNv *

period the first joints, which had opened at the extrados, are / / •

closed, and others open at joints farther removed from the | // j

springing. It is now evident that the lower part A It of the A | f „ |

centre will find itself entirely disengaged, but that the part ..

RC, besides the weight of the voussoirs It, X, will still sup- ' 8 ‘ 0

port a certain pressure, resulting from the thrust of the rampant arch AS: it will beeasy, knowing the weight of the portion K S, to have the value of that pressure whichwill be directed, following the line S F perpendicular to the joint S It, which will meetthe vertical in F, passing by the common centre of gravity It X ; then, supposing thepressure which exerts itself according to S F , and the weight of the voussoir It X, ap-plied at the point of meeting, F, of their directions we should take the result of thesetwo forces, which will be directed, following a line such as FII, and will represent thetotal burden supported by the centre : an equal force will be furnished by the other halfof the arch.

The effects alluded to are equally evident in a semicircle, or the segment of any kind,unless the latter is so flat that the first voussoir from the springing leans entirely on thecentre; in this case the weight of the centre is due solely to the weight of the voussoirs, andthe calculation of this weight becomes simple, and may be readily continued for everyperiod of the construction; thus the first part of the study is resolved by considerations,exactly agreeing with the observations and experiments that have been made, and form anecessary addition to our knowledge on the theory of arches.

The forces which act on the centre at the different epochs of the construction beingknown, it will be very easy to determine the loads supported by the different pieces ofcarpentry, whether parallel or perpendicular to their length, and it would only remain toproportion their dimensions to these loads. Gauthey has given an example of this kind ofcalculation, which he has applied to the bridge at Nemours. The ell'ort of the weightof the arch is transmitted first to the first row of inclined timbers, but the pieces of whichit is composed, slanting very little towards each other, can scarcely otter any resistance ; theslightest settlement carries back the weight on the second and third rows, so that the firstmust be considered as destined only to receive the voussoirs, and to transmit their weightto the pendant binding-pieces : by observing that the principals of the centres support2*05 metres of length of arch, that the specific weight of masonry is 2*6, and having regardto the different heights of the section of the voussoirs carried by each binding-piece, it isfound that the weight of the binding-piece ab is 18,789 kilogrammes, and that of thebinding-piece g h is 14,696 kilogrammes.

We must remark that each of these binding-pieces bears on the meeting of two piecesof one of the lower rows of inclined timbers, and on the middle one in the other row:now if the pressure acting on the binding-piece ab, (fig. 2341.) for example, cannot fail tomake the piece cd bend, and greater strength would be obtained than by augmenting theangle of the two pieces be and bf, though it is not possible to know precisely what part ofthe weight carried by the binding-pieces is supported on one side by the piece cd and on theother by the pieces be and bf, because this in part depends on the settlement of the centreand the goodness of the framing : it is evident that these two last pieces will carry almostthe whole, and in the impossibility of regulating exactly the manner in which the weight isdistributed, the most natural and the nearest to the truth consists in supposing it issupported entirely at the point b, that the piece cd does not carry any weight, and that ithas no other effort to sustain but the pressure which it exercises in the direction of itslength : it is always easy to render the work conformable to this supposition, by leaving alittle play in the notch of the binding-piece at the meeting of this piece. Naming P theweight supported by ab, p its composante following be, p' its composante following bf athe angle ebf C and Y the angles formed by each of the pieces be and bf with the vertical,we shall have

sin. Y

sin. C

making in these equations P= 18,789 kilogrammes, and putting for the angles aCY the

4 X