1403

THEORY AND PRACTICE OF ENGINEERING.

Book II.

These researches were made in the year 1726, and inserted by Pitot in the memoir ofthe Academy ; they are founded on the hypothesis that the voussoirs are very numerous,and capable of sliding without any friction ; consequently Pitot has valued the weight,therefore, which rests on the centre as too high.

Couplet has also examined the weight supported by centres, in his memoir of 1729 :referring to arches composed of polished voussoirs, he remarks, and with reason, that oneach side, starting with the springing, there is a certain number of voussoirs, which donot bear any weight upon the centre; and seeking the length of the arc embraced by thesevoussoirs in the semicircular vault, he finds it equal to the third of a quarter of a circle,a result which, being independent of the thickness of the arch, could only be exact in aparticular case. With regard to the relation between the load which the upper part ofthe arch makes the centre support and its absolute weight, Couplet arrives at a resultdifferent from that of Pitot, and points out the error into which this latter had fallen :we will not enter further into these details, the inaccuracy of which is generally ac-knowledged.

It is known that in an arch extradossed the first courses support each other, withoutthe assistance of a centre, so that it does not receive any load from it, until theplane of the joint makes with the horizon anangle, the trigonometrical tangent of which isequal to the relation of the friction to thepressure. It appears that when the voussoirsare placed on wedges, and laid dry over eachother, this relation should be equally valuednearly as 0 is to 8, whence it results that thevoussoir which begins to make the centre ex-perience a certain pressure should have theplane of its joint inclined about 42° to thehorizon : as to the value of this pressure, whichacts perpendicularly to the surface of the centre,calling a the angle of inclination of the plane ofthe inferior joint, m the weight of the voussoir, f the relation of the friction to the pressure,it will be expressed thus,

m ( sin. a ~ f cos. a)

The weight of the centre may be valued by calculating the value of this expression, for thedifferent courses which shall be successively placed, until we arrive at a voussoir, M, soinclined that the vertical passing by its centre of gravity falls on the edge, R, of the voussoirimmediately beneath it. There then happens one of two things, if the angle T R D, formedby the surface of the centre with a horizontal line D R, be less than 42 degrees, thevoussoir M will sustain itself on the centre, without leaning on the inferior voussoirs ; but ifthat angle be greater, the inferior voussoirs will carry a part of the weight of the voussoirJVI. It is easy to convince ourselves that the latter of these two cases hardly ever occurs,and that the first is the only one necessary to take into consideration. Starting from thevoussoir M, the load of the centre may be estimated by the entire weight of the voussoirs:the position of the voussoir M depends moreover on the curve of the vault, and on therelation of the dimensions of the voussoirs.

Let us suppose that the head of the voussoir M be a rectangle, U R will be the verticalpassing by the centre of gravity, and the triangles U R T and T R D being similar, we11T

shall have tang. T RD = r jr--jj: but if the angle TRD be the angle of friction, the tangent

RT

of this angle will be =0'8 ; then we shall have also, -^^ = 0 - 8, that is to say, that the width

of the voussoir will be $ of its length. Now the width of a voussoir, in large arches, isat the most only ^ or J of its length ; consequently the angle TRD will be less than theangle of the friction ; but if in the first moment the voussoir M, and those which followit, do not exert any action on the inferior voussoirs, it is not the case when the settlingof the centre takes place. As this settling cannot continue without obliging the arc A S(fig. 2351.) to become shorter, this arc resists, as a portion of the arch, the pressure whichexerts itself at its superior extremity; as long as that pressure is not strong enough, it pre-serves its form and position ; it is the voussoirs situated above INI which yield by spreading,but as soon as there is placed above M a sufficient number of courses for their stabilityto be superior to the resistance of the portion of the arch A S, this latter breaks in twoparts, which tears the centre asunder; two points of rupture are the result, at the extre-mities AB and R S, and a third at a certain point, such as KL, the position of whichlatter joint is determined by the condition that the momentum of stability of the twoparts A L and L R, taken with relation to the point B, be equal.

'When an arch is constructed at the moment the rupture takes place, and the arc A S re-

c

Fig. 2350.