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THEORY AND PRACTICE OF ENGINEERING. Book II.

points D and F, andthe second conditionfurnishes two new li-mits between whichthe value of thehorizontal thrustshould be com-prised : after that, ifwe suppose that thefriction of the jointsbe sufficiently con-siderable for theupper part not to

slide on D F, the Ksecond condition of

Fig. 2737.

the equilibrium will

be the only one necessary to take into consideration, and the value of the horizontal forcewill be found comprehended between the maximum of the expression:

and the minimum of the expression:

in which B represents that force, <p the weight of the portion of the vault DE, and Dpand lq the horizontal distances of its centre of gravity, to the points D and F, round whichit tends to turn; these maximum and minimum being taken by varying the angle formedby the joints F, D with the vertical. This analysis leaves nothing to desire, and to make itcoincide with that to which we are conducted by recent experiments on the stability ofarches, it is sufficient to express that the horizontal thrust of the arch, such as has been justdetermined, cannot displace the piers D, K, whether it tend to turn round the externaledge K, or to slide horizontally on its base.

Bossut published in 1774, in the “ Memoires de l’Academie des Sciences,” some researcheson the equilibrium of arches, in which he took up the hypothesis of the infinitely polishedjoints, and developed it to determine the conditions of the equilibrium of semicirculararches and of domes. He endeavoured to ascertain, after admitting the hypothesis ofLahire, the manner in which the rupture of the arch is made, and by what means itsthrust is exercised against the piers; also to establish the form to be given to the ex-ternal face of the pier, that it may offer equal resistance in all parts of its height.

M. de Prony in the first volume of his Hydraulic Architecture, published in 1790, afterhaving given the different equations for establishing the thickness of the abutments, thelength of the joints of the voussoirs, the form of an extradossed arch, which in the receivedhypothesis should be that of a catenarian curve, that the voussoirs should be in equili-brio, he introduced into the question the condition of the friction of the joints, and, showed that, on account of the modification which results from this in the equation ofequilibrium, it is not necessary to attribute to the horizontal joint an infinite length, asrequired by the first hypothesis. M. de Prony also insists on the necessity of giving tothe key of an arch a sufficient thickness for it to resist the pressure it supports, and in-dicates the manner of determining this thickness: with a view of enabling builders to judgeof the boldness of their works, he establishes a relation between the opening of an arch, andthe smallest length which can be given to the key.

A memoir entitled “ Dissertations sur les Degradations du Pantheon Franjais," publishedby Gauthey, contained researches on spherical vaults, in which are indicated the truemethods by which the thrust of cylindrical vaults may be estimated, deduced from knownobservations.

M. Boistard, chief engineer of the Ponts et Chauss^es, has since repeated at large, andwith all the care and exactness possible, the experiments on the stability of arches, thedescription of which is given in a memoir placed in the school of the Ponts et Chaussees,to which is subjoined a theoretical essay on the most important conclusions to which theymay lead. Such are the principal researches which have been made on the theory 6f semi-circular arches. Several experiments upon this subject were made by M. Lecreulx, on amodel of the Pont Fouchard, which was on a scale of about a sixtieth of the original, andthe material employed was freestone.

Fig. 2738. shows the abutments formed of a single stone; one was 24 feet thick, theother 18 feet 6 inches : after striking the centre the equilibrium was maintained, but whena weight equal to one-twentieth part of that of the arch was added, it fell: when both