361 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Dec' 



Dishes its deflection : — both these alterations tend to increase the horizontal 

 tension of the middle chain. At the same time it tends to diminish ihe 

 span of the side chain and increase its deflection: — both these alterations 

 tend lo diminish the horizontal tension of the side chain. So that on the 

 whole, if the horizontal tension of the side chaiu exceed that of the middle 

 chain and Ihe shifting saddle recede, there are four causes in operation to 

 establish equality of tension. Conversely, if the horizontal tension of the 

 middle chain exceed that of the side chain, and the saddle advance, there 

 are four causes in operation lo establish equality of tension. 



But there are two ways by which the establishment of this equality may 

 be prevented. Either the friction of the saddles on the piers may restrain 

 them from moving, or a sufficiently large range of motion may not have been 

 provided for in the construction of Ihe bridge. 



Let us consider first the eflect of friction. By reference to the last 

 volume of this Journal, p. 1G5, it will be seen that between the saddle and 

 Ihe top of the pier a number of rollers are interposed in such a manner that 

 the friction is wholly of Ihe nature of rolling friction : there is no rubbing 

 friction, such as that at the axle of a wheel, or the pin of a pulley. It is 

 known by experiment that rolling friction is exceedingly small, almost in- 

 considerable, compared with rubbing friction. 



From experiments by Babbage, it would appear that weights placed on 

 wooden rollers, 3 inches in diameter, can be moved with jJjth of the force 

 required to move them when placed on wooden sledges. The friction of 

 metallic rollers is a subject which, notwithstanding its importance, seems 

 to have been greatly neglected. Of all the laborious and careful experi- 

 ments upon friction which have been instituted by numerous investigators, 

 comparatively few relate to rolling friction. 



The rolling friction of any material depends not so much on its smooth- 

 ness as upon its hardness: for it principally arises from the slight flatten- 

 ing of the roller where it is in contact with the plane, and the slight de- 

 pression of the plane itself. Now in the shifting saddles of Hungerford 

 Bridge these causes must operate very slightly, for each saddle rests upon 

 50 rollers placed almost close together and arranged in an ingenious man- 

 ner so as to work with the utmost regularity. As we said, however, little 

 more than a rougli estimate of the friction can be arrived at ; but we 

 think that an inspection of the saddle itself will instantly satisfy any 

 one that the coellicient of friction cannot be nearly so great as in a railway 

 cirriage. Now the friction of a railway carriage is about ^^Ih of its weight. 

 This friction consists of two parts : the rolling friction at the circumferences 

 of the wheels, and the friction of the axles. These axles are about half the 

 diameter of the rollers at Hungerford, and the rubbing friction of llie former 

 must greatly exceed the rolling friction of the latter. Moreover,in Ihe railway 

 carriages the axle friction is only a part of the resistance, and tlierefore 

 would be represented by a smaller coeflicient than ji^ ; the roller friction 

 of Hungerford Bridge would be represented by a still smaller coeflicient. 

 If then we make the friction on each saddle half the friction of railway 

 wheels, or ^ of the weight, we probably shall have considerably over- 

 estimated it. Now the total weight on each pier will be scarcely SOU* Ions 

 supposing the passengers packed together as closely as they can stand' 

 (a most improbable supposition.) In this case the utmost resistance to the' 

 motion of the saddle will be less than a ton. Consequently, if the " power 

 represented by a — a'" exceed 1 ton, the saddle will begin lo adjust it- 

 self. 



It is impossible to suppose that masses of brickwork such as those which 

 form Ihe piers of Hungerford Bridge can be overthrown by a force of one 

 ton. We believe, however, that we have greatly over-estimated the amount of 

 this force. In addition we must remember that the vertical pressure on the 

 pier has an cll'ect as well as the horizontal, and that the former tends to 

 the stability of the pier. In the extracts given, ante p. 3JS, there is a dia- 

 gram in which the line A I) represents the magnitude and direction of the 

 resultant of the forces acting on the head of the pier. Let us suppose the 

 diagram slightly altered, by constructing a parallelogram in which the 

 rcrtical and horizontal pressures, instead of the tensions of the chains, are 

 represented. Then A D represents the resultant of those pressures, and it 

 is Ihe diagonal of a parallelogram, of which the sides are in the proportion 

 1 (ton) ; 8U0 (tons). Me see then how extremely oblique is Ihe direc- 

 tion of A D. If when produced it fell with the base of the pier, the pier 

 would be stable, even if it had no weight, and the morlar no cohesion. 



To illustrate this important point a little more fully, let us actually 

 construct the diagram. Let A B C L) be the parallelogram of Ihe forces 



* The total weight of thf centre ctiain when loaded is 1000 tons. One-half of this 

 weight presses each abutment, and 3U0 tons more may be added for the side chaiu. 



applied by the saddle to the top of the pier, an I 

 let AB be Ihe resultant of the vertical pressures 

 exerted by the rollers : this rejultant will always 

 be somewhere near the centre of the tower. Let 

 A D represent the horizontal friction of the roll- 

 ers : then the diagonal joining A and C will re- 

 present the direction and magnitude of the result- 

 ant. Now, B C is only j^th part of A B, ainl 

 therefore, since the pier is 8U feet in height, tha 

 diagonal A C will, at the base of the pier, be 

 only the ^th of a foot out of the vertical. The 

 base of the pier is 40 feet wide. If, therefore, 

 the position of A B be supposed to be exactly 

 central, the friction or horixjntal force must be 

 two hundred times what we have supposed it to 

 be, before the re:;ultaut cau fall bt-yonU the base. 

 If we take into account the ell'ect of the weight 

 and cohesion of masonry itself, the horizontal force must be sliU further 

 multiplied, before the stability of the pier cau be endangered. 



Limits of the Motion of the Shifting Saddle. 



The next point lo be considered is, whether the motion of the shifting 

 saddle has a sufficient range. Me think that it may be satisfactorily 

 shown that the saddle has ample room to adjust itself to any variation of 

 the horizontal tensions of the chains which is likely to occur. It is diffi- 

 cult, in a subject so complicated as the theory of catenaries, to work out 

 general solution of the problem ; a little attention, however, to the case 

 before us, and an allowable simplilication of it, will furnish all the in- 

 formation required. 



Let us first see the effect of a slight diminution of the span of the cen- 

 tral chain, everytliiug else remaining unaltered. In the tract on '• Metro- 

 politan Bridges," p. 10, the following useful approximate relations are 

 given : — 



2x^ = 3y{,-y) (1) a = — ^— - (2) 



where s is half the length of the chain, y the half span, jr the deflection, 

 and a the length which measures the horizontal tension. In the case of 

 the Hungerford Bridge (supposing the curve to be a common cateuar) j, 

 Ihe values of these quantities are — 



Half the central span, J .... 338-25 feet. 



Deflection, x 50- 



Half length of the chain, s . . . 343- 

 Length mt-asuring horizontal tension, a . 1152'5 

 where the first two quantities are found by actual measurement, and Ihe 

 two latter are those calculated by Sir Howard Douglas from his for- 

 mulie. 



Now let us suppose the half span to become 337 feet — that is, to be 

 diminished by 1^ foot ; and the length of the chaiu being unaltered, let us 

 see what will be the new value of a. By equation (Ij, 



2 x' = 3 X 337 X 6 ; or, x = 3 X ""^ Ssf. 



Therefore, by equation (2), 



3 X (337)^ + 9 X 337 



" = « X 3 X ■^m = * " ^^^ ^ ^"^ + ^) = ^°**' '■^^t- 



It follows, therefore, that by the small diminution of the span here sup- 

 posed, a very great alteration of the horizontal teusiou is produced — • 

 namely, from 11.52 to 1054, or in a ratio of about 11 ; 10. The lengtli a 

 indicates not the actual value of the tension, but iis proportion to the weight 

 of the chain. Hence, whatever proportion Ihe horizontal tension bora 

 to the weight of the chaiu in the first case, it would beari^ths of ihat 

 proportion in the second case. Now, in the construction of the Bridge, 

 provision has been m.ide for giving the saddles on each pier a play or range 

 of three feet. And we have made the case far too unfavourable for 

 our own view of it; for we have neglected the consideration that, when 

 the saddle moves, the horizontal tension of the side ciiaius is increased at 

 the same time that the horizontal tension of the central chain is diminislted. 

 So that the adjustment would take place much sooner thau has been here 

 supposed. 



The adjustment neglected in the above calculation is by far the most im- 

 portant in point of magnitude. For a variation of span would not nearly 

 so much alier the tension of the main chain as that of the sides chains, of 

 which the curvature is very small. The latter are stnlched almost in a 

 .straight liue between their points of attachment, at the banks of the river 

 and the top of the piers. So that it tjay be doubted whetlier an increase 



