1502
THEORY AND PRACTICE OF ENGINEERING.
Book II.
0*22 metres (0 '72 feet) in length. With these materials arches -were constructed semicircularlywith three centres, and others flattened a third and a fourth in the arc of a circle, the versedsine of which was the fourth, the eighth, and the seventeenth of the opening: they wereconstructed on a wooden centre, and their rupture took place on lowering it vertically,either when loaded at the top or when the thickness of the abutments was diminished.Each of the arches was submitted to three principal trials; in the first they were extra-dossed 4 inches of thickness, and as that thickness was not sufficient for them to supportthemselves when the centres were lowered, a certain number of voussoirs of the upper partdescended vertically with it, and were carried on its summit; the two inferior parts of thearch then had the effect of two ramping arches, and divided into two parts; the last jointof each part opened at the intrados, near the upper extremity and near the springing, andthe rupture appeared towards the middle, where the voussoirs did not touch the centre, andwhere the joints opened at the extrados.
In the second trial, where the arches were again extradossed, on each side a certain num-ber of voussoirs of the lower part were embraced by a cord which leant on their extrados,and was extended by a weight: the pressure which this cord produced on the last voussoiropposed their separation, which the upper parts of the arch tended to produce; and it wasconstantly observed that if the weight occasioned by the tension of the two cords was notsufficient for the equilibrium, the arch broke by opening at the intrados, near the key andnear the springing, where the last voussoir tended to swing round its‘external edo-e, andat the extrados on the haunches. Secondly, that where the weight was sufficient to main-tain the equilibrium, the joints opened in the same manner, on account of the inevitablesinking, but that the action of the weights tending to tighten them, they opened and shutalternately by an oscillating movement, in which the parts of the arch turned alternatelyround the points of support which the edges of the consecutive voussoirs presented tothem. Thirdly, that when the tension of the cord was sufficiently considerable for thepressure which it exerted over the inferior parts of the arch to be capable of raising up theupper parts, the same effects were manifested in a contrary sense, that is to say, that thearch broke at the key, opening at the extrados, and at the haunches it opened at theintrados, and at the springing the last voussoir turned round the interior edge. Whenthe arches were raised on piers, the effects were exactly similar, except that the piersbecame a body with the inferior parts of the arch, which in falling over tended to turnround the external edge of the base of the piers, instead of that of the voussoir placed atthe springing.
In the third trial abutments were established, and the haunches of the arch filled inwith masonry, levelled at the summit, where this vault was 4 inches in thickness. Whenthe stability of the abutments was sufficient to resist the thrust, the arch preserved its primi-tive figure after striking the centre; when the weight of the upper part was increased, thearch broke as usual, and the position of the joints of rupture in the haunches was de-termined by the value of the weight, by the manner in which it was distributed on thesummit of the arch, and by the height of piers on which this arch was sometimes raised;it has generally been observed that when the arches were not raised on piers, the rupturetended towards the angle of 30° of the half circle described by the semicircle, or towardsthe angle of 50° of the little arc described in the arch struck with three arcs of a circle:the point of rupture tended to rise when the height of the piers was augmented, and whenthe summit of the arch was more loaded.
All these experiments confirm the principle already established on the nature of themovement of arches, from whence it results that in the research for the conditions of theirequilibrium, we may without sensible error consider them as a system of four levers, KD,DE, Ed, dK, each loaded with the respective weights of the parts of arches which cor-respond with them (see fig. 2737.), and being able to turn round the points of supportK, D, E, d, k, where they form turning joints; the position of the points d, D dependfor each particular case on the figure of the arch, and the distribution of the weights withwhich it is loaded.
The same conclusion is applicable to the arcs of a circle, and to the flat arch, where thearch and its piers form a system, the nature of which is similar : we must here remark withregard to the figure of this arch, that the position of the joints of rupture is found naturallyfixed at the springings, unless it be very slightly extradossed, or the versed sine of the arcof the circle according to which it is described be almost equal to the radius. The ex-periments of M. Boistard proved that an arc, the versed sine of which is the quarter ofthe chord, has its joints of rupture at the springing, when the haunches are filled withmasonry.
The application of the calculation to the preceding theory presents no difficulty: thepoints N,M, m,n, being those where the levers are met by the verticals which pass by thecentres of gravity of the parts corresponding to the arch, we may suppose that these leversare loaded in these points with four weights equal to those of the masonry, and we must