Chap. XXIII.
1359
of the binding-piece, for this weight separates at this extremity into two forces, the oneacting in the direction of this piece, and the other in the direction of the beam, and doesnot tend to make the other struts, which are held hv the binding-pieces, yield; all thesestruts then support only longitudinal pressures, whilst in the system fig. 2176., besides thelongitudinal pressures, which are the same as in fig. 2174., the inclined timbers have also tosupport transversal pressures, since the weight carried by the binding-piece then decomposesinto two forces, the one acting in the direction of the great rafter, and the other perpen-dicular to it. This observation leads to a new modification : as the timbers yield lesseasily as they are shorter, there must be an advantage of substituting for two inclined timbersa system of three pieces, and although these pieces, making more obtuse angles, cause theweight to be decomposed in stronger longitudinal pressures, more must be gained by thediminution of their length than is lost by the effect of the augmentation of their pressures;thus we obtain the system fig. 2177., which is that of the bridge built by Palladio overthe Cismone: in placing, as in fig. 2178., four inclined timbers instead of three, theirstrength would be augmented, and by thus multiplying their number and diminishingtheir length, we should have pieces too short to bend, as seen in fig. 2179., andwhich could only yield by crushing; this is the arrangement for the arches projected byPerronet for the bridge of the Salpetriere , and for those executed at Lyons in that of theMulaticre. In order clearly to show where these last systems are wrong, we shall makesome observations on the nature of constructions in carpentry, which naturally find placehere.
'Whatever may be the combination, in constructions of carpentry the object should alwaysbe the transmitting of certain efforts to points of support, by means of a system of levers,subjected one to the other: now we can distinguish in such a system two different equi-libriums, which we shall designate as Equilibrium of Position , and the Equilibrium ofliesistance. The first takes place when the efforts exercised on each lever are so combinedwith their respective situations that the system does not tend to produce any move-ment : it is that which forms the objects of statics, of which science teaches us to regulate theconditions. The equilibrium of resistance consists in making each lever, which in staticsis considered as an inflexible line, have the necessary force for the resistance of thepressures exercised in it; the conditions of this latter equilibrium are the object of thetheory of the resistance of solids, and, according to what we have already described, we canalways express these conditions with sufficient exactness in constructions of oak.
A system of levers may be maintained in repose by two different means; first, it may bedisposed in such a manner that the conditions of the equilibrium of position should beexactly satisfied ; secondly, the levers should be so subjected to each other by framing, orby auxiliary pieces, as binding pieces or braces, that it should be impossible for them tochange situations with regard to each other as long as the resistance of this framework issuperior to the efforts which tend to make the form of the system vary : in this latter casewe are not required to satisfy the conditions of the equilibrium of position; the system mustbe regarded as formed by one single piece, and the equilibrium of resistance should alonebe taken into consideration. If these ideas be applied to wooden bridges, and the systemrepresented in fig. 2179. be taken, having only regard to the weight of the carpentry andthe pavement, the levers may be placed in the situation which suits the equilibrium ofposition, so that they do not tend to any change of form under the load : but we mustremark, that the equilibrium will often be found deranged by the passage of carriages;and secondly, that this not being stable, the slightest derangement would involve thefalling of the bridge, whence we may conclude that in the system of levers which composethe principals of a bridge, the equilibrium of position can never be exactly attained : it isthen necessary that these levers should be subjected and framed to each other, and that ateach articulation the framing should be opposed to their various angles with a force equalto that which produces the variation, according to the effect of the weight; thus the systemof the carpentry of a bridge must necessarily be in the second of the two cases alreadystated. Tile consideration of the equilibrium of position can never be employed, but toassure ourselves that the framing of the articulations has the force necessary to opposeevery change of form ; this indispensible condition fulfilled, the system changes its nature;it should be regarded as forming only a single piece, and its resistance valued in conse-quence.
In examining the principles of figs. 2174, 2175, 2176., we see that in whatever mannerthe load is distributed, it does not tend to make any change of form, and among thedifferent framings of carpentry, that which has the figure of a triangle is the only onewhich can have this property ; we ought, then, to carry every other system to this, by properlyfortifying the articulations, with the exception of that of the summit, so that, whatever maybe the arrangement of a principal, its two halves may be considered as two rafters made of asingle piece, which at the summit abut against each other; it is an indispensable conditionin order that the system should have stability, and no bridge can subsist if this conditionbe not fulfilled.