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THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



LMav, 



subject is best discussed by first supposing; that the materials are 

 incompressible, and tracinf; the conditions of stability on this hypo- 

 thesis, and then by examining? in what respect those conditions are 

 modified by the limited strentfth of the materials. 



Art.6. — Inapplyiufj the following^methods to analyse the strenf^h 

 of any given structure, tlie first (piestion to lie solved is : Is the 

 structure, when acted on by tlie Riven pressure, on the balance be- 

 tween standinsr and falling ? The problems determine if this is 

 the case, and if not, if the tendency is towards stal)ility or insta- 

 bility. If the structure be on the balance between standing and 

 falling, then the slightest alteration in tlie jiressures may cause it 

 to fail, and it would therefore be condemned as unsafe. If the 

 tendency be towards instability, unquestionably the arch will not 

 stand. If, on the other hand, the tendency be towards stability, 

 then another question arises : How great a degree of strength does 

 the structure possess ? When it is decided in what terms this 

 strength is to be measured, the problems in the following pages 

 can be applied to answer the tpiestion. Thus, the strength may be 

 measured by the weight in dift'erent positions and directions, that 

 will be recpiired to produce the state of unstable equilibrum, or the 

 balance between standing and falling. Or, again, the strength of 

 the material may be hypothetically diminished, until this unstable 

 equilibrum is produced, and thus a measure of strength is obtained ; 

 as for instance, if the hypothetically diminished strength of the 

 material is one-tenth the actual strength of the material used, 

 then the structure is ten times stronger than is theoretically ne- 

 cessary. 



Section II. 



Art. 7. — On the conditions of stability of an arch whose voussoirs 

 are incompressible ; the form of which, and the pressures sustained 

 by it, as regards position, direction, and amount, being similar on 

 either side of the crown of the arch. 



In such an arch, the conditions of failure are, as before stated, 

 the first and second ; that is, the voussoirs may slip on one another, 

 or turn over on their edges : the latter condition wiU first be dis- 

 cussed. 



It need not he proved, that if in one part of an arch the vous- 

 soirs turn over on their edges at the extrados, causing the joint to 

 open at the intrados, then at some other positions, other voussoirs 

 must turn over on their edges at the intrados and the joints open 

 at the extrados. Also it need hardly be proved, that if the arch is 

 similar in form and similarly loaded on eitlier side of tlie crown, 

 that if failure takes place, in the manner above described, one of 

 the points of rupture will be at the crown of the arch : this is 

 nearly self-evident, and may be proved by experiments on any 

 model of an arch ; it is, however, proved geometrically by applica- 

 tion of the problem in Section 4. If the arch fails at the crown, 

 by the voussoirs turning on their edges A,, at the extrados, as in 

 diagram 7, then at some point in the haunches, the voussoirs will 

 at the same time be turning on their edges A,,, at the intrados, in 

 which case the crown will sink and the haunches will spread. 



If the arch fails at the crown, by the voussoirs turning on their 

 •dges at the intrados, as in diagram 8, then at simie point in the 

 naunches, the voussoirs will, at the same time, be turning on their 

 edges Aj, at the extrados, in which case the haunches will sink 

 and the crown of the arch will rise. 



Art. 8. — When the arch is failing, as shown in diagrams 7 and 8, 

 then the points of application of the resultant pressures at the 

 places of failure are beyond the edge of the voussoir, as shown in 



sultant pressure must be at the extreme edge of the voussoir, and 

 its direction must also be that of the tangent to the intrados, or 

 extrados, at A,, Aj, &c., because if not, the line of resistance 

 passes without the boundary of the voussoirs, either on one or 

 otlier side of the point A, and tlie structure has already failed, by 

 the turning over of some other voussoir. Therefore, wlien the 

 arch is in the condition of unstable equilibrium, then, at all the 

 points of rupture, the directions of the resultant pressures are tan- 

 gents to the intrados, or extrados. 



Art. 9. Pruhlem 2. — To find the second point of rupture, in an 

 arch whose voussoirs are incompressible, the form of whicli and the 

 pressure sustained by it, as regards position, direction, and amount, 

 being similar on either side of the crown of the arch. 



Also to find the amount of pressure at the crown and at the 

 second point of rupture. 



Take for example an arch with a backing, or superstructure, 

 diagram 9. It is required to find the second point of rupture, that 



Diagram 7. 



Diagram 6. 



diagram 3. But when the arch is in the condition of unstable 

 equilibrium, that is, when it is on the balance between standing 

 and falling, and when the voussoirs are on the point of turning on 

 their edges at A„ Aa, &c., then the point of application of the re- 



Diagram 9. 



is, that point in the haunches, at which the voussoirs will he about 

 to turn on their edges, when the arch is in the condition of un- 

 stable equilibrium. 



As the form of the structure leads to the supposition, that, if 

 failure take place, it will be by the sinking of the crown and the 

 spreading of the haunches, let it be first assumed that the arch is 

 about to fail in that manner. Then the point C, in the extrados, 

 at the crown, will be the first point at wliich the voussoirs are 

 about to turn ; and the horizontal line C E, will represent the 

 direction and position of the pressure upon the side of the arch 

 drawn in the figure, caused by the weight of the opposite and 

 similar side : see Art. 8. 



Choose some point R,, in the intrados, and, for trial, suppose 

 that to be the second point of rupture. Then the voussoirs will 

 be on the point of turning on their edges at R,, and the resultant 

 pressure will act through R,, in the direction R, T,, of the tan- 

 gent to the intrados : see Art. 8. Draw the joint or normal to the 

 intrados R, N,, and the vertical line N, U,. Find the centre of 

 gravity of the mass A DR., N, B, ; and draw the vertical line 

 G, W,, and produce it till it intersects C E, at the point I,. 

 Then the only pi-essures acting on the point R,, are the pressure 

 of the opposite arch, acting in the direction C E, and the weight 

 of the mass A D 11, N, B,, acting in the direction I, W, ; and 

 since the direction of these two pressures intersect in the point 

 1 1, therefore, by the well-known law of Statics, the direction of 

 their resultant also passes through the point I, ; but when the 

 arch is about to fail at the point R,, R, T, is the direction of the 

 resultant, and tliis does not, if continued, pass througli the point 

 I,. Therefore, R, is not the second point of rupture, and some 

 other point must be tried. If the line R, I, he drawn, it will be 

 seen that its direction is less inclined to the vertical than R, T, ; 

 and this leads to the supposition that the point of rupture is lower 

 down, at some point where the tangent to the curve is less inclined 

 to the vertical. Therefore, choose some other point R^, and pur- 



