J 847.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



69 



Let ABFH represent the 

 vertical section of the half arch; 

 the part of the load below B 

 (in the spindril) bearing a con- 

 siderable proportion to the 

 part above B. And to take all 

 the variations of circumstances 

 at once, let us suppose that it 

 has been found necessary (from 

 considerations which will be 

 referred to hereafter) to load 

 the arch with heavier materials 

 in one part than in another ; 

 for instance, with light sandy 

 ballast near the vertex, and 

 heavy granite rubble towards 

 the springing. 



Instead of taking the load, 

 as before, as a single mass, 



consider it made up of several portions, as in the figure. Let the 

 part in the spandril be taken as two (nearly) triangular masses or 



-J B 



prisms, A E C, C D B. The forms of the rest of the loading will de- 

 pend upon the nature of the superstructure. But where the upper 

 line, F GH, is a horizontal straight line, we may consider the rest of 

 the loading as rectangular masses, E C G F, D G H B. 



One of the advantages of this hypothetical division of the loading 

 is that it enables us to estimate the effect of variations in the density 

 of the ballast. For example, AEC may be one kind of ballast, 

 CDB a second, ECGF a third, and DGHB a fourth. The first 

 operation in the calculation will be to find out the weight of each 

 portion as nearly as possible. This will easily be done (the form of 

 the arch and superstructure being already determined upon) by esti- 

 mating the cubic content of each portion, and multiplying this quan- 

 tity by the weight of each cubic foot of the material employed. It 

 is scarcely necessary to observe, that the weight of the voussoirs 

 themselves must be included in the lower or triangular portions. 



The weights being found, there will be very little difficulty in as- 

 certaining their effect or moment in producing the thrust. Let g be 

 the position of the centre of gravity of the lower triangle. By a 

 known property of the centre of gravity of triangles, a g, ^ iEC. 

 Now, ag^ is equal to the horizontal distance of the centre of gravity 

 from A, and therefore determines the moment. Hence the weight cf 

 the triangle AEC, multiplied by i E C, is the moment of that portion 

 about A. 



Similarly, in the triangle CDB, if gj be the «entre of gravity, 

 «^a=iDB. Therefore, the liorizontal distance of gj from A is 



E C + cg^ : the moment of the triangle CDB is, consequently, its 

 weight multiplied by (E C + i D B). 



The two parts EF GC and DGHB, being rectangles, their centres 

 of gravity may be considered as situated in the centres of those 

 figures. Consequently, 6^3= iFG; and dg, = ^GH. On the 

 whole, then, if we call the first mentioned weight W^ ; the second 

 third, and fourth, W^, W^, W4, respectively, we have for the sum of, 

 the moments, the expression 



W,xiEC + W,x(EC-|-iDB)-J-\V3XjEC-^W^X(EG-(-^DB) 

 Adding these moments together, and dividing by the height of the 

 arch, the result or quotient is the amount of the thrust. 



This method, like the one first explained, merely suffices to indicate 

 the limits within which the value of the thrust lies. There will be, 

 as before, a maximum and mininum value, but the difference between 

 them will be very small, except where the voussoirs are of great 

 depth compared with the other dimensions of the arch. The maxi- 

 mum will be determined by estimating the horizontal distances of the 

 weight from the springing of the extrados, and by giving the rise its 

 least value, namely, the vertical height of the vertex of the intrados 

 above the springing of the extrados: and conversely, for the minimum 

 value of the thrust, the horizontal distances of the weight must be 

 measured from the springing of the intrados, and the rise must have 

 its greatest value, namely, the vertical height of the vertex of the 

 extrados above the springing of the intrados. With these limitations, 

 the following is the general rule for calculating the thrust :— " Con- 

 sider the loading as composed of triangular and rectangular portions. 

 Multiply the height of each portion by the horizontal distance of its 

 centre of gravity from the springing. The sum of the products di- 

 vided by the rise of the arch gives the value of the horizontal 

 thrust." 



To take an instance in illustration of the rule, suppose that in the 

 last figure, the rise of the arch is 19 feet, and E C = 6 feet, and D B 

 also = G feet, in this case the rise of the arch will be about 9 feet less 

 than the span. Also let W, = 4 tons, W^ == 3 tons, W, = 6 tons, 

 and W^ = 5 tons. The moment of W, will be 4 X i E C = 8. The 

 moment of W^ will be 3 X (E C + i D B) = 24. The moment of 

 W3 will be 6 X i E C= 18. The moment of W^ will be 5 x 

 (EC+iDB)=45. The sum of these moments is 95, and this 

 quantity divided by 19, the rise, gives 5 tons for the value of the 

 horiiontal thrust. If the above admeasurements be supposed to cor- 

 respond to the maximum value of the thrust, a second calculation 

 must be made, as above explained, with the admeasurements cor- 

 responding to a minimum value. The true value will lie between 

 these two results. 



Of course, where further accuracy is required, the load in spandrils 

 may be considered as divided into three or more triangular portions, 

 with as many rectangular portions above them. As the divisions are 

 perfectly arbitrary and hypothetical, they may not only be of any num- 

 ber most convenient, but also the intersecting lines may be taken 

 wherever they afford the greatest facility of calculation. In a four- 

 centred arch, for instance, a vertical division may be made where the 

 segment of short radius ends and the segment of large radius begins : 

 or if there be an abrupt change from a heavier to a lighter loading, 

 the vertical line at the place of change may be adopted in the calcu- 

 lation. 



It must be carefully borne in mind, that the whole of these investi- 

 gations presuppose that the only external forces are vertical weights. 

 Where the loading rests firmly on the arch, and has no tendency to 

 slide down the side of it, the hypothesis is strictly true ; but in the case 

 of a series of arches, as in bridges, the spandrils which adjoin at each 

 pier are filled up simultaneously by throwing in the ballast, till it 

 reach the intended height of the roadway. In this case it is clear 

 that unless the ballasting were rammed hard, or concreted, the re- 

 moval of the portion in one of these spandrils would cause the portion 

 in the other spandril to slide down. Here, it is obvious that the two 

 portions of the loading exert a mutual horizontal pressure, by which 

 each prevents the other from slipping. This horizontal pressor* 

 must, in considering the equilibrium of each half arch separately (as 



