1847.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



97 



STRAIN ON THE PLATFORM OF A SUSPENSION BRIDGE. 

 In the following paper it is proposed to examine the nature and 

 amount of the strain to which the platform of a suspension bridge is 

 subjected, by its connection with the chain? and piers, and a load 

 equally or unequally distributed throughout its length. We shall 

 assume that the platform is rigid, the curve of the chain a catenary, 

 the links indefinitely short compared with the length of the platform, 

 and the rods indefinitely close to each other and inextensible. 



Let O, the centre of the platform, be taken for origin; the axis of 

 the platform, which we suppose horizontal, for axis of x ; and a ver- 

 tical through O, for the axis of y. Let dt he taken to represent 

 the tension of the rod applied at point (x y) of the chain; T the ten- 

 sion of the chain at that point; / the weight of a unit's length of 

 chain, — then we shall have 



ds ds dx c 



But in the common catenary, 



dy 



j- = c' s, c' being an arbitrary oonstant ; 



.'.< must =: ^s, where (1 is some constant. Consequently, the re- 

 sultant of all the tensions of the rods, attached to any portion of the 

 chain, passes through the centre of gravity of that portion. If, now, 

 the platform be supposed uniformly loaded throughout, and perfectly 

 rigid, it would be impossible to determine whether its weight were 

 wholly supported by the chains, or wholly by the platform, — or how 

 it might be divided between them; but as the nature of the mate- 

 rials we are considering is only so far rigid, that neither the flexi- 

 bility of the platform, nor the extensibility of the rods and chains, 

 are supposed to be sufficiently great to affect the curve which the 

 chains assume, a very little consideration will be sufficient to show 

 that the weight of the platform will be so distributed, that the tend- 

 ency to bend it will be a minimum. When the platform is unequally 

 loaded, if we suppose the load not sufficiently great sensibly to de- 

 flect it, it will be hereafter shown that a pressure will be generated 

 on that pier nearest to the centre of gravity of the p latform. In 

 practice, however, if the load were much increased and unequally 

 distributed, the platform would bend, and the curve assumed by the 

 chains would be modified; the point where the resultant of all the 

 vertical tensions of the rods meets the platform, approaching nearer 

 to tlie centre of gravity of the platform and load, and, in case disrup- 

 tion ensued, actually and suddenly coinciding with it. 



To find the strain on any point of a platform equally loaded 

 throughout: — 



Let P be a point in the platform, and PQ vertical thereto ; AP ;= 

 x; AO = OB=a; CQ=rS'QD = S" 



W = weight of platform and load; T' = tension of rods from P to 

 A; K distance of the centre of gravity of CQ from P. 



Let T' = V S, where V is determined from the equation 

 VS'+VS" = W. 



Then the moment tending to turn AP about P, which measures the 

 straia at P, is given by the equation — 



No. llj.— Vol. X.— April, 1347. 



Moment of strain =VS'K-W, 



Cx 



ia ' 



If the load be unequally distributed, — 

 Let G be the centre of gravity of load and platform ; S the whole 

 length of the chain. 



Then a pressure will be exerted where the platform rests on the 

 pier nearest to G. Let X = this pressure. 



Taking moments about O, if OG = A; WA = aX'; W=X-i-VS; 



:W 



fe')- 



And for the strain at P, if W' = weight of platform, AP, and its 

 load, — 



p the distance of its centre of gravity from P ; 

 Moment teniling to turn A P round P ^ 



W-a: + VKS'-W'u=W -a:+W.^— ^^^' -W'p. 



Deductions from the above formulae : — 



1st. When the platform is equally loaded throughout, the strain 



will be least when the chain has but a slight depression; for then. 



Car 

 V S' K will most nearly, cosleris paribus, equal W — . 



4a 



2nd. The strain of a load, unequally and uusymmetrically distri- 

 buted, will always be greater than the strain produced by the same 

 load equally distributed. 



J. H. R. 



[In the remarks appended to Sir Howard Douglas's paper in our last numljer, it was 

 observed that several topics iver« pnssed over for tlie sake of brevity. Lest it should be 

 inferred that there were still wide ditferences of opinion ;\vhich is nut the case), it may be 

 rem arkf^d that the topics referred to were ai>t of a contruversial nature, and that this 

 question of the struiu on the platform of a suapsosion bridge vras one of the most im- 

 portant of them.] 



ON THE MOTION OF FLUIDS. 



The discrepancy between theory and experiment in all problems 

 concerning the flow of water has been universally acknowledged. This 

 extraordinary fact has hitherto been accounted for on the supposition 

 of the imperlect character of the fluidity of that liquid ; whereas, as 

 we shall pre^ieiitly show, it is not the water but the analysis — not na- 

 ture but the philosophers who are at fault. In the present paper We 

 shall point out some of the fundamental errors of analytical hydro- 

 dynamics, and endeavour to show how theory and practice can be recon- 

 ciled. Some time since, one of the most eminent of living mathema- 

 ticians pointed out to us the incorrectness of certain analysis connected 

 with the motion of a wave along a canal, in which, as he clearly proved, 

 the hypotheses adopted were inconsistent (with themselves; that is^ 

 parallel motion and perfect continuity were assumed to co-exist. Our 

 attention has since been more recently directed to the subject, and 

 having taken Professor Miller's work as a text- book, we were asto- 

 nished to find the same two assumptions vitiating the whole of the 

 chapter on fluid motion. 



In section V. of Miller, the first sentence runs thus— 



"When an incompressible fluid flows through a tube, the velocities 

 of the fluid at any two points, are inversely proportional to the areas 

 of the perpendicular sections of the tube at those points; supposing 

 the tube to continue always full, and the velocities at all points in the 

 same section to be equal to one another, and perpendicular to the 

 section." 



The two hypotheses with which this paragraph concludes are in- 

 consistent. Let the tube be of variable bore and its axis straight, let 

 this axis be the axis of x, and let u, v, m, be the velocities of any fluid 

 particle parallel to the axes of x, y, jjr, respectively ; then, by the equa- 

 tion of continuity for incompressible homogeneous fluids, we have 



dx dy d z ~ 

 Now and m both = 0, by the hypotliesis ; 

 — which is absurd. 



du 



-0. 



. uz=e, 



14 



