]S39.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



261 



by the jn-essure of the opposite semi-arch is tlie lemt. This condition of 

 niiniuunii pressure at the key supplies niatheniatically all that is renuii'ed for 

 the complete deterniiiiation of that ]n-essure, and [icrfects tlie theory. 



The demonstration of it is easy. The pressure which an opposite scnu- 

 arch would produce ui>o-n the side AC of the key-stone, tig. 2, is ecpual to the 

 tendency of that semi-arch to revolve forwards upon the inferior edges of 

 one or more of its voussoirs. Now this tendency to motion is evidently equal 

 to the least force which would support this opposite semi-arch ; supposing 

 the semi-arehes, therefore, to be equal in every respect, and equally loaded, 

 it is equal to the least force which would support the serai-arch .\BDC. 



Fiy.\. 



Suppose the mass ABDC, tig. 1, to be acted upon liy ajiy number of force 

 among which is the force Q being the resultant of certain resistances, suj)- 

 plied by different points in a surface BU, connuou to the intersected mass 

 aiul to an immoveable obstacle BE. 



Now it is clear that under these circumstances we may vai'y the force P, 

 Ijoth as to its amount, direction, and point of apiiHeatiou, without disturbing 

 the equilibrium, provided only the form and direction of the line of resistance 

 continue to satisfy the conditions imjiosed by the equilibrium of the system. 



These have been sliown to be tlie following, — tliat it no where cut the 

 surface of the mass, except at P, anrl within the sjiace BD, and that it no 

 where cut any section MN of the mass, or the common surface BD of the 

 mass and obstacle, at an angle with the perpendicular to that surface, greater 

 than the limiting angle of resistance. 



Thus, varying the force P, we may destroy the equiUbriuni, either, first, by 

 causing the line of resistance to take a direction without the limits prescribed 

 by the resistance of any section MN through which it passes, that is, witliout 

 the cone of resistance at the jjoiut w here it intersects that sm-face ; or, 

 secondly, by causing the point Q to fall irithoiit the surface BD, in which 

 case no resintatice can be o\)posed to the resultant force acting in that point ; 

 or, thirdly, the point Q lying within the surface BD, we may destroy the 

 equilibrium by causing the line of resistance to cut the surface of the mass 

 somewhere between that point and P. 



Let us suppose the linnts of tlie variation of P within which the first two 

 conditions are satisfied, to be known ; and varying it, within tlujse limits, let 

 us consider what may be its least and greatest values so as to satisfy the third 

 condition. 



Let P act at a given point in AC and in a given direction. It is evident 

 that by diminishing it under these circumstances, the line of resistance will 

 be made continualh to assmne more nearly that direction which it would 

 have, if P were entirely removed. 



Provided then, that if P ivere thus removed, the line of resistance would 

 cut the surface, that is, provided the force P be necessary to the equilibrium ; 

 it follows that Ijy diminishing it, we may vary the direction and ciuvatnre of 

 the line of resistance until we at length make it touch some point or other in 

 the surface of the mass. 



And this is the limit ; for if the diminution be carried further, it will cut 

 the surface, and tlie equilibrium will be destroyed. It appears then that 

 under the circumstances supposed, when P, acting at a given point and in a 

 given direction, is the least possible, the line of resistance touches the interior 

 surface or intrados of the mass. 



In the same manner it may be showii, that when it is the greatest possible, 

 the line of pressure touches the exterior surface or extrados of the mass. 



I have here supposed the direction and point of application of P in AC to 

 be given; but by varying this direction and point of application, the contact 

 of the line of resistance with the intrados of the arch may be made to take 

 place in an infinite variety of different points, and each such variety supplies 

 a new value of P. Among these, therefo're, it remains to seek the at/solute 

 maximum and minimum values of that force. 



In respect to the direction of the force P, or its incUnation to AC, it is at 

 once apparent that the least value of that force is obtained, whatever be its 

 point of appUcation, when it is perpendicular to AC. 



There remain then two conditions to which P is to be subjected, and which 

 involve its condition of a mininunn. The first is, that its amount shall he 

 such as will give to the line of resistance a (loint of contact with the intrados. 

 The second, that its point of application in the key-stone AC shall be such as 

 to give it the least value which it can receive, subject to the first condition. 

 I have determined the value of P subject to these conditions in a paper 



read licfore the Candjridge Philosophical Society in May 1837, and published 

 iu the Olli volume of their Transactions. The equations involving that value 

 admit of a comiilcte solution, and determine it for every form and dimension 

 of the Inokeu or Gothic arch, and the complete segment, and for every cir- 

 cumstance of its loading. 



The condition however that the resultant jiressure upon the key-stone is 

 subject in respect to the position of its point of application on the key-stone 

 to the condition of a nnuimnm, is dejieudent upon hypothetical qualities of 

 the masonry. It supposes an unyielding material for the arcli-stones, and a 

 mathematical adjustment of their surfaces. These have no existence in prac- 

 tice. On the striking of the centres the arch invariably sinks at the crown, 

 its voussoirs there slightly opening at their lower edges, and pressing upon 

 one another cxelusixely by their upper edges. Practically the line of re- 

 sistance then, in an arch of unceynented stones, touches the extrados at the 

 crown ; so that only the first of the two conditions of the minimmn stated 

 above actually contains : that: namely, which gives to the line of resistance a 

 contact with the intrados of the arch. This condition being assumed, all 

 consideration of the yielding quality of the material of the arch and its abut- 

 ments is eliminated. It will thus be discussed in what remains of this paper. 



To simplify the analytical discussion of the question, I have hitherto as- 

 sumed the load upon the semi-arch to be placed over a single point of it X, 

 fig. 2. I now imagine it to be distributed in any way over the extrados, but 

 symmetrically iu respect to the two opposite semi-arehes. The centre of 

 gravity of this load on each senn-arch being determined, it is evident that 

 the horizontal thrust P on the key-stone of the arch will be the same if the 

 whole load upon it be imagined to be collecte<l in these two centres of 

 gravity. I determine then the horizontal thrust P on this hypothesis of a 

 concentrated loading : this determination being made, the data necessary to 

 the analvtical discussion of the question are complete, all the forces acting 

 upon a mass .\STD of the arch and its loading intercepted between the 

 crown and any inclined position CT of the radius are given, and the ciination 

 to the true line of resistance under any given circumstances of loading is de- 

 terminable in terms of the radius vector CK and the angle ACS. The equa- 

 tion deterndningthe value of P is unfortunately one of a lugh order, involving 

 circidar functions of complicated forms ; and the solution of it otherwise 

 than by approximation is perhaps to be despaired of. The small value of the 

 ratio of the depth AD of the voussoirs, in the majority of practical cases, to 

 the radius CA of the arch iu terms of w Inch ratio the value of P is expressed, 

 suggests a developement of the value of P in a scries of terms ascending by 

 l)owers of this ratio. To cfi'ect this developement 1 have called to my aid 

 the theorem of Lagrange, using two terms only of that theorem, and not 

 therefore extending the approximation beyond the first power of the ratio. 

 It nnght perhaps be expedient iu some cases to extend it to the second ; be- 

 vond this limit no inactical enquiry need however lie carrieil. 



The line of resistance being fully determined, the point Q, fig. 1 , where the 

 resultant jircssure of tire whole semi-arch intersects, the supporting surface 

 BD of the abutment becomes know n, and also the direction of this resultant 

 pressure. Now all the circumstances which determine the equilibrium of an 



