l&i6.J 



THE CIVIL ENGINEER AND ARCHITECTS JOUR^JAL. 



263 



Defect of Elasticity. 



It now remains that something be said of the defect of elasticity, 

 and the modilications of the above results wlien applied to jointed 

 or compound structures. The ordinary mathematical theory of 

 the girder is based on the law of perfect elasticity, known as 

 Ilooke's law — namely, that the elastic force is proportional to the 

 extension or compression. 



It appears from experimental inquiries, subsequent to, and more 

 axtensive than. Dr. Hooke's, that this law is not quite true. The 

 elastic force is in reality less than tlie law would assijrn it to be. 

 Mr. Hodg-kinson, in his recently published " Researches on Cast- 

 iron," Art. lOG, seems inclined to tliink that the elastic force may be 

 expressed by a j — 6 .r-, where x is tlie measure of compression or 

 extension, and a and b constant empirical co-efficients. That this 

 hypothesis is near the truth may be inferred from the considera- 

 tion, that if the elastic force be expressed by a series in ascending 

 j>owers of .r, all terms involving high powers must be very small, 

 as the elastic force is always nearly equal to the first term, and j: 

 is very small. It may, however, he worth while to remark, that if 

 ax — bx' he taken to express correctly the elastic force, the 

 same value of or, which reckoned positively gives the tension, will 

 not, when reckoned negatively, give the same value for the pres- 

 sure. In order that this may be the case, only uneven powers of 

 X must be involved. 



But whatever law of elasticity be assumed, this is easily ascer- 

 tained — that where the elasticity is imperfect (that is, where it is 

 less than in proportion to the extension or compression), the de- 

 flecting pressure of a girder will be less than in proportion to the 

 deflection. In cast-iron girders, cast in one piece and in metal of 

 good quality, the defect of elasticity is small ; and consequently 

 the deflection is pretty nearly proportional to the pressure. But 

 in jointed structures, compounded of several parts connected by 

 rivets or bolts, this is by no means the case. In them, the defect 

 of elasticity must he great ; and the deflection will therefore in- 

 crease at a considerably higher rate than in proportion to the 

 external pressure. If a load of 200 tons produce in a compound 

 girder 2 inches deflection, 300 tons will pi-oduce considerably more 

 than 3 inches deflection. Or, if 300 tons produce 3 inches de- 

 flection, 400 tons will produce considerably more than 4- inches 

 deflection. How much more can be ascertained only from actual 

 exjieriment. 



It is very important that this distinction between simple and 

 compound girders should be always taken into consideration, for 

 the neglect of it would lead to very erroneous conclusions respect- 

 ing the strength of structures of the latter kind. As cases in 

 point, may be instanced calculations respecting the strength of 

 girders formed in three pieces and supported by tension-rods. 

 Formulce which determine the strength of simple, unjointed 

 girders, are inapplicable to these structures, and are not likely to 

 give even an approximation to the amount of their real strength. 



Where, however, the compound-girder is so well constructed, 

 that its curvature, ivhen deflected, is regular and free from sudden 

 inflections, the formula given above for the dynamical pressure of 

 long trains on perfectly elastic beams, will apply with considerable 

 accuracy. For the deflection being previously assigned, is a safe- 

 guard against any very great error. That deflection being small, 

 the curvature will also be small ; only, on account of the defect of 

 elasticity, it increases cccteris paribus more rapidly towards the 

 centre of the beam, than it would if the beam were perfectly 

 elastic. Consequently, the pressure towards the centre is compa- 

 ratively greater in the compound, than in the simple girder : and 

 pressure towards the centre is more efi^ectual in producing deflec- 

 tion than pressure near the ends of the girder. 



Consequently, there are two reasons why ^'elocity increases the 

 deflection of a compound, more than that of a simple, girder. In the 

 first place, on account of the defect of elasticity in tliejointed struc- 

 ture, its deflection increases in a higher degree than in proportion to 

 the external pressure. Secondly, that external pressure is of neces- 

 sity greater for the jointed than for the simple girder, because in 

 the former the curve is sharper towards its centre. Velocity of 

 transit has therefore much greater influence on the security of 

 girders of the former, than of the latter kind. 



It would have been satisfactory to have been able to confirm the 

 results of these investigations, by reference to actual experiments. 

 Unfortunately, however, there are at present but very scanty data 

 for the purpose. An account of two experiments made on the 

 Dee-bridge, of the Holyhead Railway, is all that can be cited. 

 These experiments are described in a Report to the Commissioners 

 of Ridlways, 15th June, 1847. An engine and tender, about 30 

 tons weight, passing over the bridge at 15 miles an hour, produced 

 " a deflection nearly the same as with the engine at xest — vizq 



and train of 48 tons, at rest, gave a deflection of 2-4 inches ; while 

 the deflection caused by the same train at a speed of 15 to 20 

 miles an liour, was only Ig of an inch." 



These accounts do not however furnish much information 

 suited to our present purpose. In the first place, the experiments 

 were made on a jointed structure of a complex nature, and of 

 which the deflection appears, even from this brief account, to have 

 followed no sim)(le law. Moreover, in the first experiment, the 

 deflection is not actually determined : it is merely said to have been 

 from I of an inch to tliree-sixteenths more; and in both, the 

 mass of the girder greatly exceeded the mass sustained. All the 

 inference that can be dra\s n is, that velocity did not very materially 

 influence the deflection, but tliat the deflection was diminished at 

 the highest velocity, the load sustained being comparatively light. 



Means of Diminishing the Dynamical Pressure. 



AVhen a ball moves along a perfectly horizontal surface, the 

 pressure on its under side is just equal to its weight, for this 

 simple reason — that if the pressure were greater, the ball would 

 rise ; if less, sink. 



In the same way, if a train moved along the surface of a girder 

 which remained perfectly horizontal during the transit, its pres- 

 sure would be just equal to its weight. But the train generally 

 sinks a little, and acquires a momentum downwards, which has to 

 be destroyed by increased pressure. The simplest precaution 

 against this eff'ect is — not to remedy it — but to prevent its exist- 

 ence. Suppose it be found that, when a certain weight travels 

 along a certain girder which is originally perfectly horizontal, it 

 produces a deflection of three inches at its centre : then, if the 

 rails had a rise given them of three inches towards the centre, it 

 is clear, tliat when the same weight travelled over them, it Mo\ild 

 be no lower when at the centre, than when at either end, of the 

 beam. 



Suppose now the reverse case — that there is a hollow or de- 

 pression originally in the beam. Then, when the weight jiasses 

 over the beam, it sinks the distance of this original depression, in 

 addition to the deflection produced by pressure. Hence, the 

 downwiird momentum is materially greater than if the beam had 

 been perfectly horizontal originally. Or, to take another ^'iew of 

 the question, the original hollow or depression, added to the de- 

 flection, increases the curvature of the beam, and therefore the 

 centrifugal pressure of the loacL Either way, then, of viewing 

 the eff'ect of the hollow, eitlier in inci-easing the momentum down- 

 wards, or in increasing the centrifugal pressure, leads to the same 

 result — that tlie pressure is increased. Mathematically, these two 

 views of the case coincide. 



It is seen, then, how extremely important it is that there should 

 be no original hollow in the beam. On the contrary, it is advan- 

 tageous that its surface should be convex, instead of concave — or 

 should have a camber. In this case, the centrifugal pressure would 

 act upwards instead of downwards ; so that the pressure, instead of 

 being greater than the weight, would be less at high speeds. 



There is a very simple way of calculating this diminution, or of 

 estimating the centrifugal force. And it may be remarked, paren- 

 thetically, that the method about to be given is useful for many 

 purposes besides that to whicli we are to apply it. For example, 

 it furnishes most simple and ready means of ascertaining the 

 horizontal pressure on the flanges of the wheels of carriages going 

 round railway curves. 



If a be the length of the chord of a circular arc, which is of 

 large radius, and x its lineal versed-sine, or the length of the per- 

 pendicular drawn from tlie centre of the arc to the chord, it will 



be found that the radius nearly equals — -^ 



Now, if V be the velocity of a mass m, moving round this curve, 



its centrifugal force becomes m V* — j ; and if T be the number of 



seconds in which any part of the mass describes the distance n, 



V' I 



— =; - , . Substituting this value, and putting g = 32, the cen- 

 trifugal force is equal to -~^=:, X the weight. This formula appliee 

 41* 



to horizontal lailway curves, as well as to the vertical cur\'es of a 

 beam. Confining attention liowever to this application of it, we 

 see that a very considerable reduction of the pressui-e of the 

 weight may be effected by curving the upper surface of the beam. 

 Suppose, for instance, that the time of transit were one second 

 (T = 1), and that it werejjracticable to give the -rails such a 



