THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[.Tamahy, 



NOTES ON ENGINEERING.— No. VIII. 



Ry HoniKKSiiA.M Co.\, R.A. 



Oil the most Ecnnomicul Furms nf Sunpiiishm liriilges. 

 Of all kinds of liritlj^es suspension liri(lp;es iiro cjipalile of being 

 ronstruotod h itli the preatest span. Notwitlistandiiijj; tliis advan- 

 fnjfe and tlie facility of constnution, tlie use of tliese structures 

 lias been restricted t)y their flexibility and tendency to undulate. 

 'I'hey have fallen into disrepute in the modern practice of enj^ineer- 

 inp, on account of the difficulty (generally deemed insuperable) of 

 making them sufficiently rigid for the )>urposes of heavy traffic, 

 such as that upon railways. Anotlier, though less (divious, olrjec- 

 tion is that tlie ordinary methods of calculating the jiroper form 

 and dimensions of suspension bridges, and the different strains to 

 which they are subjected, are exceedingly complicated. The in- 

 tricacy of the investigations leaves a degree of uncertainty and 

 distrust as to the actual strength which the several parts of a sus- 

 pension bridge may be assumed to possess. 



The object of tlie present paper is to examine how far these 

 difficulties may be removed, and to show what method of arranging 

 the dift'erent jiarts of the structure secures the greatest amount of 

 strength for tlie wliole. 



Suspension bridges may he distinguished generally into two 

 classes: 1st., those of the ordinary form, that of a main chain or 

 catenary, with the roadway suspended from it by vertical rods ; 

 'ind., those in which the roadway is suspended directly from the 

 abutments liy straight rods, the catenary or curve chain being 

 altogether dispensed with. It will be shown, on strict statical 

 ])riiiciples, that the first method involves a great waste of material, 

 and that, by a pro])er arrangement oi -strdight rods, a given amount 

 of strength may be secured with a smaller quantity of iron, or a 

 greater amount of strength with a given ipiantity of iron, than by 

 the use of a main catenary. Of course, methods of using straight 

 rods may be employed which involve greater waste of material 

 than even the employment of the curved chain. The most econo- 

 mical arrangement of straight rods is not a merely arbitrary mat- 

 ter, but depends, like every other branch of engineering, on sound 

 deductions from the laws of mechanics. 



Refore proceeding with the investigation, it may be as well to 

 remind tlie reader that the object of these Notes on Engineering 

 is to simplify the practical ajiplications of theory, and to exjdain 

 them, as far as possible, in familiar, untechnical language. This 

 important rule should be constantly remembered by all who teach 

 and all who study the mathematics of engineering — tliat long fnr- 

 mii/re are never j/iit into practiee. In practice, simple general prin- 

 ciples are far more useful, because capable of being applied with 

 far more certainty and facility, than the most elaborate results of 

 scientific research. 



We now proceed to establish the following important general 



Proposition. — In a susijciifiion briilye the niateriut required to sus- 

 tain a given had will be the least when each point of support in the 

 roadway is directhj conneeted with a point of suspension in the nearest 

 ahntnient by one independent straight rod. 



To begin w ith the simplest case, it will be first of all supposed 

 that only two points of support in the roadway are connected with 

 the point of suspension. Suppose that R (fig. 1) is the point of 



Fig. 1. 



Fig. 2. 



suspension ; A and D the two points of support in the horizontal 

 platform A C. Tlien it will be shown that to sustain a given load, 

 the most economical arrangement of the suspensicm bars consists in 

 connecting R with A 1) independently and directly by two rectili- 

 near rods, AR and DR. If, however, as in fig, 2, the connection 

 he indirectly made by suspension rods meeting at an intermediate 

 point E, more material will be required for a given amount of 

 strength. 



In order to prove this proposition, which has so important a rela- 

 tion to the most usual methods of constructing suspension bridges, 

 it is necessarv to ascertain tlie (|iiaiitities <if material in the rod 

 A R and R D'(fig. 1), and the rods A E, D E, and R E (fig. 2), and 

 to compare the aggregate annuint of material used in both cases. 

 It is, of course, ])resupposed that the strength of the rods is pro- 

 portioned to tlie strain upon them. In ascertaining the thickness 

 to be given to the rods of a suspension bridge, the first point to be 

 settled is the amount of strain which the material will bear on each 

 square inch of the sectional area. For tlie purpose of mere cmn- 

 parixon, it is indifierent what amount be assumed : it may therefore 

 be supposed that the rods are to be calculated to bear a strain or 

 tension of t lb. per square indi of their sectional area. Conse- 

 quently, multiplying the sectional area of any rod by /, we have 

 the whole strain to which it is subjected. Further, for )iurposes of 

 comparison it is indifierent what be the load on the bridge, so that 

 in both cases the weights at corresponding points of tlie platform 

 be supposed the same : let it therefore he assumed that both in 

 fig. 1 and fig. 2 the jioint A has to sustain a vertical weight ;/-, and 

 also (for the sake of simplicity) that the point D in both figures 

 has to sustain the same weight u\ 



It will (at first) be taken for granted that the jilatform contri- 

 butes nothing by its rigidity to sustain the load ; that the whole 

 weight is borne by the suspension rods, which are ke]it in their 

 oblique positiiui by the connection of the platform. The amount 

 of material requisite to support w at the point A will first be con- 

 sidered. 



Commencing with the case of fig. 1, we have, since the rod A R 

 sustains the weight w at A, the vertical component of tlie tension 

 of A R equal to «;. Supposing the sectional area of this rod to he 

 k square inches, its tension, by what has been already said, will 

 be ki. 



, ■ T, ^ r. , B C W A R 



.-.«, = A/ smR AC =ht.^^; k=j.j^f,. 



Consequently, the mass of the rod = its sectional area multiplied 



by Its length = - g^ (1). 



Proceeding now to the case of fig. 2, and still confining attention 

 to the suspension of the point A, by reasoning exactly the same as 



w A E' 



that for fig. 1, the mass of the rod A E := = — , (Ee being 



t IL e 

 drawn vertical.) 



It is clear that the connection between the point R and the point 

 E may be supposed to be established, not by a simple bar, but hy a 

 compound bar of two or more parallel lengths. In fact, this 

 method is that usually adopted in actual practice, the several links 

 of the chain commonly consisting of se\eral bars or iron plates 

 laid side by side, and connected at their extremities. Their relative 

 thickness is a matter of indift'erence, provided that the total thick- 

 ness be sufficient to sustain the strain. In fig. 2 the rod R E, pro- 

 vided it have the thi<'kness necessary to sustain the united effects 

 of the two weights at A and D, may be supposed to be made up of 

 any number of parallel bars of any relative tliickness whatever. 

 Now, supposing R E to be a compound bar, let A' he the sectional 

 area or thickness of metal due to the eft'ect of the weight at A, 

 kf' the thickness due to the weight at D : k' -\- hf' will be the total 

 thickness of R E. 



Taking the thickness k' to be that requisite to sustain w at A, 

 and k't the consequent amount of tension of that part of the com- 

 pound bar, we have the vertical component of k't (= vertical com- 

 ponent of tension along AE) = lo. Hence, if E/ be drawn hori- 



Bf . . , _ 'f ER 

 ER' '' t Bf 



Multiplying this quantity by the length E R, and adding the mass 

 of the rod ascertained above, we have the total mass of metal re- 



ae: ERn 



Ee + Bfi ^ > 



Hence subtracting the expression (1) from the expression (2), it 

 will he easily found by some simple analysis, which is here omitted 

 for the sake of brevity, that the mass required for the indirect con- 

 nection A E R, fig. 2, exceeds tlie mass required for the direct or 

 rectilinear connection, A R fig. 1, by a quantity 

 (RC.E/ - AC.Ee) ^ w 

 RC.R/./C ■ t ' 



which is positive in all cases. Hence, more material is always re- 

 quired for the indirect than for the direct connection of A and R, 

 The same mode of reasoning apjilies to the weight suspended at D. 



zontal. 



quired to connect A and H 



= ?{■ 



