1848.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



The form of the analysis is such that it applies to this case as 

 well as the last, and leads to a similar conclusion — that the indirect 

 or bent connection, DEB, reciuires more material than the direct 

 or rectilinear connection, D B. Now it is evident, that what is 

 true of the several parts of the system individually, is true of the 

 whole collectively — that if less material be required for each of the 

 direct connections than for each corresponding- indirect connection, 

 the total material reijuired for all the former will be less than the 

 total material required for the whole of the latter. In other words, 

 the St/stem of sm-jjension. in fiff. 1 m" the moxt economical. 



The same result might have been obtained by supposing B E a 

 simple, undivided bar, and tlie amount of material given by that 

 hj'pothesis would be the same as that on the liypothesis here 

 adopted. But the method of investigation given above leads more 

 easily to the general results for which we are seeking. It has the 

 advantage of admitting immediately, and without any more mathe- 

 matical analysis, tlie following important 



CoROLLABV. — The method of suspension (fig. 1) is more economical 

 than the method (fig. 2), for any number of points of support in tlie 

 platform. 



For the reasoning given above is not affected by supposing the 

 rod B E to divaricate at E into three or more radial bars proceed- 

 ing to as many points of support in the roadway. W'hatever might 

 be the number of indirect connections by this method, eacli of 

 tliem would require more material than the corresponding direct 

 connection of fig. 1 : and therefore the total quantity of metal 

 required by the former method exceeds the total quantity required 

 by the latter method. 



We have hitherto considered, in the second or indirect methml, 

 only one point of divarication, E : the inquiry will be completed 

 by considering several other such points to exist — ^as at B, B', B ", 

 B ", &c., fig. 3. 



Fig. a. 



The connection of A and D with B, by bars meeting at E, has 

 been already considered. Less material would have been required 

 to support the weight at A and D, if, instead of the method shown 

 in fig. 3, there had been separate straight rods from A and D to B. 

 In this latter case, B' (the next point of suspension) ivould be 

 connected with the three points A, D, ft, by three straiglit rods, 

 divaricating from the end B of a common rod B B'. 



The " Corollary" given aljove shows that this triple divarication 

 involves a waste of material. Had there been in the place of it, 

 three straight rods from A, D, and ft, to B', less material would 

 have been required to support the corresponding weight. But this 

 triple <livarication itself requires less material than the metliod 

 shown in fig. 3. Hence, s, fortiori, the direct connection with B' 

 would require still less material than the method shown in the figure. 

 And so, by continuing the same mode of reasoning for the otlier 

 points, B", B'", &c., we come at last to the general conclusion 

 that, if all the points of support had been directly and indepen- 

 dently connec4:ed with K (the ultimate point of suspension), less 

 material would have been required to sustain given loads than by 

 the method shown in fig. 3. 



This conclusicm is independent of the inclination of the rods 

 E D, Bft, B ft', B ft", &c., and remains true when they are vertical. 

 Hence, in the common suspension bridges, such as those at Chiuing- 

 cross, HammersmitlL, &:c., with a main chain or catenary hanging 

 between tlie abutments, and connected with the platform by xerti- 

 C;il rods, there is a waste of material. The same conclusion applies 

 to all suspension bridges having radial bars radiating from any 

 point except the points of ultimate suspension at the abutments — 

 and, therefore, hold with respect to the bridges on Dredge's jirin- 

 ciple, some of which are erected in the Regent's-park, and of wliich 

 one recently gave way and was destroyed near Calcutta. 



The amount of saving effected by connecting all the points in the 

 platform with the abutments by independent straight rods, may be 

 best shown by an example. The Hungerford bridge, at Charing- 

 cross, may be taken as a familiar example — and we will, therefore, 



proceed to compare the material required for that bridge by the 

 method actually adopted, and the quantity which would be required 

 by the method here advocated. 



Tiie quantity of material required for suspending the bridge by 

 a catenary and vertical rods will first be considered. The position 

 of the centre of gravity of the half-span depends on the form and 

 weight of the chain, and the manner in which the load is distri- 

 buted along the platform. When the load is small compared with 

 the weight of the chain, the centre of gravity of half the bridge 

 and load will be nearer the abutment than the centre of the bridge • 

 for the curvature of the chain, its increase of thickness near fhe 

 point of suspension, and the increased length of the vertical rods 

 all tend to make the weight preponderate towards the abutment' 

 But wlien the bridge is supposed to be loaded with a breaking 

 weight greatly exceeding the weight of tlie chain, and uniformlV 

 distributed along the platform, it may be assumed, without sensible 

 error that the iiorizontal distribution of the wei^j-ht of the whole 

 system is uniform. In this case, tlie centre of gravity of the half- 

 span will be midway between the abutment and centre of the 

 bridge. 



At this latter point the tension of the catenary is horizontal. 

 Let moments be taken about tlie point of suspension for the equi- 

 librium of the half-span : then, since the horizontal tension in 

 question acts below the point of suspension, at a vertical distance 

 equal to the defiection of the chain, and since the weiu-ht acts at a 

 horizontal distance from the same point equal to the quarter-span 

 the products of each of these forces into the corresponding distance 

 will, by the Principles of Moments, be equal. Hence, calling AV 

 the total weight of the half-span (including tlie half-cliaiii), 'l^the 

 horizontal tension, d the defiection, u the quiu-ter-span, — it follows 

 that 



W a = Td: or T =; ^V 



.(1). 



That is, the horiz:intal tension — the weiyht of the half-span multiplied 

 by the ratio of the <iuartcr-spun to the defection,- a simple rule, from 

 which tlie horizontal tension of the chain of any suspension bridu^ 

 loaded h itii its breaking weight may generally be calculated with 

 sulficient accuracy. 



It has been assumed that the load is uniformly distributed, or 

 that any portion of the weight is proportional to the length of the 

 corresponding jiortioii of toe platform. It follows, that if any 

 distance, j, be measured along the platform from the lowest point 



of the chain, the weight corresponding to that distance is W*- • 



2 a 

 Also, if y be the vertical ordinate of the chain at the same dis- 

 tance, a known principle which applies to catenaries of every form 

 gives 



<!/ ^w^- ^T = ^^ (2) 



dx 2a T2« ^ ' 



By another known principle which also applies to all kinds of 

 catenaries, the tension at the point (.c, y) is equal to 



■■■■(■+ a*- 



And since the sectional area of the chain at any point is supposed 

 proportional to the strain at the same point, we have, if K and A' 

 be sectional area at point (,r, (/), and the lowest point of the chain 

 respectively 



The mass of each small portion of the length of the chain is the 

 product of that element of length, and the corresponiling sectiomil 

 area : hence it will be easily seen that the 



mass of the half-chain -z^ k I 1 1 4- - - \ d x. 



Jo ^ rf.i'V 



And this quantity by substitution from (2) will be found equal to 



2a/i; ll -|- ;r7rr). Fimilly, if the tension per square inch be /, 



and consequently T = k t, and if a be put ^ 1 70 feet, and d ^ SO, 

 it will be readily ascertained that the 



mass of the half-chain =. X 1IS9-3. 



(which are almost exactly the values of those quantities in the 

 Hungerford Bridge.) 



To obtain the whole quantity of material required for the pm-- 

 poses of suspension, we must add to the quantity last obtained, the 



2* 



