1848.1 



THE CIVIL ENGINEER ANP ARCHITECT'S JOURNAL. 



2i<) 



question, which has at lenfrth attracted the attention due to its 

 influence on tlie secui-ity of railu-ay traffic. The necessity of 

 further inquiry seems to be generally acknowledged among engi- 

 neers ; and by the recommendation of the Commissioners of llail- 

 «-ays, in a published minute of the 29th of June IS-IT, a government 

 commission has Ijeen appointed for the very purpose. Tlie minute 

 exi)resses a doubt " whethei' the experimental data and the theo- 

 retical principles at present known are adequate " for the " design- 

 ing iron bridges, when these are to be traversed by loads of extra- 

 ordinary weight at great velocities." 



There seems to exist gi'Bcit discrepancy of opinion as to the 

 effect of the velocity of transit. Some have imagined tiiat it may 

 become a source of safety, by causing the railway train to pass 

 over before the girder has had time to yield. Others, again, have 

 estimated tiie effect of the moving load as highly as six or seven 

 times that of the same load at rest. In the following investiga- 

 tion, both these opinions will be shown to be incorrect : they are 

 here cited merely as indications of the extreme uncertainty pre- 

 valent on the subject. 



The method of inquiry about to be explained consists, not in 

 determining the dynamical strain absolutely, but by comparison 

 witli the corresponding statical strain. The results will conse- 

 quently be much simpler than they would be if the dynamical strain 

 were estimated independentl}'. The deflection which a given load 

 ai reM upon a girder produces, will be always taken as one of the 

 known data of the problem. The determination of this .statical de- 

 flection, as it forms the basis of all the remaining calculations, is 

 the first point of inquiry. 



When a beam is not affected by a permanent set or defect of 

 elasticity, it appears, both from theory and actual experiment, 

 that the deflection by a weight resting at its centre is very nearly 

 proportional to that weight — that is, if a given number of tons 

 deflect it one inch, double the number of tons will deflect it two 

 inches. This result is arrived at by Professor Moseley in his " i\Ie- 

 chanical Principles of Engineering," and M. Navier in his 

 " Resume de Lemons de Construction," by independent methods. 

 Its near accordance with practical truth has been abundantly con- 

 firmed by experiment, as may be verified by reference to numerous 

 published accounts of actual observations on the subject, and es- 

 pecially to Mr. Hodgkinson's invaluable " Experimental Re- 

 searches on the Strength of Cast-iron." This work gives the 

 results of an exceedingly large number of experiments, made 

 by the author and others, on the transverse strength of beams 

 loaded at their centres ; and although these beams were of very 

 different forms and dimensions, the law indicated is nearly observed 

 in all of them. Whether the section of the beam be rectangular, 

 triangular, or T-shaped, with the vertical rib either upwards or 

 downwards, the constant ratio, in each beam, of each deflection to 

 the corresponding load is nearly maintained : and the same remark 

 applies to beams of the form most useful for railway purposes — 

 that of an upper and lower flange connected by a vertical rib. 



It will be found, however, by reference to the tables in Mr. 

 Hodgkinson's %vork, that the actual deflections are somewhat more 

 than the theoretical law would make them. This discrepancy may 

 be accounted for by attributing it to the defect of elasticity, which 

 the ordinary theory of beams does not consider. As this defect 

 is not generally very great, it will here in the first instance be 

 neglected : the deflections will primarily be estimated as if the 

 elasticity were perfect ; and subsequently the modifications due 

 to defect of elasticity will be taken into consideration. 

 Work Done on the Deflection of a Beam. 



The " work done" by a moving force may be defined to be the 

 product of that force into the distance through which it acts, A 

 familiar instance of the use of this measure is the Steam-Engine ; 

 where the work done receives the particular name of Horse-Power. 

 If the pressure on the piston were uniform, that pressure(in pounds) 

 multiplied by the distance through which it is exerted (in feet) 

 would, if divided by 33,000, give the horse-povver. But in the 

 steam-engine, and all other practical instances, where the pressure 

 is not uniform, but varying, it is impossible to calculate the work 

 done by this direct multiplication. Where the value of the 

 moving force is constantly altering, we may resort to either of the 

 following methods of ascertaining the work done by it, — we may 

 multiply its average value by the distance through which it acts ; 

 or, when that average cannot be ascertained, we may consider the 

 whole distance divided into elementary portions, so small that it 

 may be supposed without sensible error — that the pressure is at 

 least uniform while it acts through each portion in succession. 

 The aggregate work done, is the sum of the work done on each of 

 these portions — that is, it is the sum of the products of each por- 

 tioa of the distance and the corresponding pressure. 



This process of summation, when carried out with the greatest 

 possible accuracy, is e(iuivak'nt to that of matliematical integra- 

 tion ; in which case, the work done by a varying pressure may be 

 defined, in mathematical language, to be the integral of the jiroduct 

 ot the pressure, and its " virtual velocity." Tiie w(u-k done in 

 deflecting a beam by pressure at its centre' is easily ascertained, if 

 tliat pressure be assumed proportional to the deflection. Calling 

 the deflection .r, and therefore the pressure a x (where a is a con- 

 stant depending on the dimensions &c. of tlie beam) we have- 



work done 



- y O' 



X dx = 



a ,v^ 

 ~2^ 



2 ■ 



, T ax . 



jNow — IS the pressure or weight which would statically maintain 



half the deflection x. Hence, the work done in producing a given 

 deflection is equal to the weig/it which would .stuticolhi maintain half 

 the deflection, multiplied by tlie ichole distance of deflection. 

 The value of this rule will appear hereafter. 



Distinction of Gradual and Instantaneous Loading. 



AV'Iien experiments are made on the strength and deflection of 

 beams, they are generally loaded very gradually at their centres. 

 Each addition to the load is allowed to produce its full efl'ect 

 before more be imposed. Consequently, at every stage of the ex- 

 periment, tlie beam is in a state of statical equilibrium : the pres- 

 sure of the load on the beam is always just equal to its weight, 

 and is never increased by any momentum arising from downward 

 velocity. 



But if the whole load be suddenly and at once placed on the 

 beam, while it is as yet undeflected, the cfi'ects are entirely altered. 

 Tlie deflection is greater than tlie same load would produce if 

 gradually applied : for when the beam lias reached the point of 

 statical deflection, the momentum acquired bythe downward motion 

 urges it further; and the descent of load continues till it be brought 

 w/^ (so to speak) by the increased resistance of the beam. After- 

 wards, the be.am and load rise again, as the deflection has been 

 carried beyond the degree at which it can be statically maintained. 



In the case here supposed of instantaneous loading, nothing 

 like impact or sudden collision occurs. The pressure at tlie centre 

 of the beam is finite and continuous. The load does not fall upon 

 the beam — it is merely supposed to be placed originally in close 

 contact with the beam, and then suffered to instantaneously rest 

 upon it. 



F(U' the sake of elucidation, one or two instances of analogous 

 action may be cited. If a common balance have its fulcrum above 

 the points of suspension of the scales, and a weight suddenly rest 

 in one of the scales, tlie lever will turn through a much greater 

 angle than if the same weight were applied in small successive 

 portions. 



If an elastic string suspend vertically a weight from one end of 

 it, the string will be more stretched if the whole weight be suffered 

 to act at once, than if applied in small portions. It will be found, 

 that if the extension of the string be proportional to the stretch- 

 ing force, the extension produced by the descending freight will 

 be twice that due to tlie gradual effect of the same weight. 



A light cylinder of wood, hiaded at its lower end, and floating 

 vertically in water, furnishes another iUustration. If the cylinder 

 be raised a little above its position of equilibrium and then let go, 

 it will sink twice the distance it has been raised, if tlie motion be 

 so small that the resistance is equal to the hydrostatical jiressure. 



In the same way, in a perfectly elastic horizontal beam, loaded 

 at its centre, the effect of instantaneous loading is double that of 

 gradual loading. For, by a known principle of mechanics, when a 

 material system moves from one position of rest to another position 

 of rest, the work done by the retarding forces is equal to the work 

 done by the accelerating forces. For any small deflection of a beam 

 by instantaneous loading, its position of ultimate deflection is one of 

 instantaneous rest, for immediately before it arrives at that posi- 

 tion, all the parts of the beam descend, and immediately after, 

 ascend. Also, the work done by the accelerating force is the 

 weight actually resting on the beam, multiplied by the space of 

 deflection : and the work done by the retarding forces is, by what 

 has been said above, " equal to the weight which would statically 

 maintain half the deflection, multiplied by the whole distance of 

 deflection." Therefore, putting the two amounts of work done equal 

 to one another, we see that the weight actually upon tliie beam is 

 that which would statically maintain half the deflection. In other 

 words, the deflectUm is doubled by instantaneous loading. 



Transit of a Single Weight. 



We now proceed to examine the effect of the transit of a single 



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