2 GO 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



LSeptembeb, 



weight along the pirder, and first of all to show tliiit its effect 

 cnnnot exceed that which it has just heen estimated to produce, if 

 stationed at the centre of the girder and an()we<l to descend freely 

 from the undeflected ])Osition — in otiier words, it will l)e proved tliat 

 at whatever rate tlie weight may travel over the girder, its ulti- 

 mate strain and deflection cannot be more than double the corre- 

 sponding statical effects produced when it restn at the centre of 

 the girder. 



There is a general rule of constant nse in engineering which, 

 expressed in practical language, states that power is never gained, 

 but only modiiied, by the intervention of machinery. This rule 

 may be more scientifically expressed and extended by tracing it to 

 its origin — it is a particular case of the ])rinciple known in theo- 

 retical mechanics, as the Conservation of Vii Vim. This ]>rinciple 

 may be very conveniently enunciated by employing tlie term 

 '• work done," as defined aliove : and it then assumes this form of 

 enunciation— that the vU viva gained or lost by a system in moving 

 from one position to another, is equal to twice the difference 

 between the work done by the accelerating, and that done by the 

 retarding, forces in the same interval. 



From this it follows, that where there is no gain or loss of vis 

 viva, there is no difference between the work done by the accele- 

 rating and retarding forces respectively. Hence, if the i)arts of 

 the system be moving at the same velocity in the second position 

 as in the first — or if both ])Ositions be positions of rest — the ag- 

 gregate work done in the interval by the retarding forces is equal 

 to that done by the accelerating forces. 



A very simple case will illustrate this theorem. If a locomo- 

 tive-engine travel a mile along a railway, and its velocity at tlie 

 end of the mile be the same as at tlie beginning of the mile, the 

 work done by all the forces which have resisted its motion is in 

 the aggregate just equal to the horse-power developed in the 

 Kteam-cylinders. And this equality holds good, however the 

 engine have moved in the interval — whether on a straight level 

 road, or on severe curves and gradients — whether the speed were 

 uniform or very irregular — whether the steam were on the whole 

 time, or the engine during large parts of the journey moved by its 

 momentum only. The intermission of the moving force and all 

 other irregularities disappear in the result. To establish equality 

 l>etween the work done in moving, and that done in retarding, the 

 engine, all that is necessary is that tlie engine be moving neither 

 faster nor slower at the end of the mile, than at the beginning 

 of it. 



Another illustration will serve to show the extreme generality of 

 the princi])le in question. If a certain quantity of water have to 

 be raised a certain lieight, the amount of work actually requisite 

 for effecting the object is in all cases equal to the weight of water 

 multiplied by the vertical height. This amount of necessary 

 power or work is incapable of being diminished by any mechanical 

 ui- hydraulic contrivance. The water may be contained in a vessel 

 vtIucIi is drawn up perpendicularly, as from a well, or which is 

 drawn up an inclined plane or by a sjiiral path ; or the water may 

 be raised by an Archimedian screw, or by buckets attached to the 

 periphery of a revolving wheel, or by a hydraulic-ram, or by a 

 force-pump ; or lastly, it may be thrown up in a jet, as from a 

 fountain or fire-engine. But it is pliysically impossible, by these 

 or any other methods, to diminish the requisite amount of labour. 

 Jt is, of course, easy to increase the amount by a waste or unpro- 

 fitable expenditure of labour, such as is caused by friction of the 

 machinery, or the mutual action of the particles of water among 

 themselves. But supposing no waste of force to occur — supposing 

 all the power usefully employed in simply raising the water witli- 

 out doing anything else; then the amount of that power is in all 

 cases just what has been stated — the weight of water multiplied 

 by the vertical distance through which it is raised. 



The rule is of universal application, and tliere is no otlier prin- 

 ciple of dynamics of such great and constant utility in practical 

 science ; for it embraces all those cases of motion witli whicli the 

 engineer happens to be concerned — cases where the motion either 

 ceases, or has the same values, at regularly-recurring intervals. 



The case before us, of the transit of a weight along a girder, 

 is a striking exemplification of this Principle of the Conservation 

 of Work. For tliis principle enables us immediately to compare 

 the effect of a weight moving along the girder, and that of the 

 game weight stationed at its centre, and descending. If the de- 

 flection lie the same in both cases, the work done by the descent of 

 the load in both cases is the same — namely, the weight multiplied 

 by tlie vertical descent : and this is true, whatever be tlie patli of 

 descent. Now, it has already been shown, that in the case of in- 

 stantaneous loading, the work done by the descent of the weight is 

 equal to that necessary to produce in the beam the deflection whicli 



twice the weight would statically maintain. Hence, the travelling 

 weiglit can do no more. 



The value of this conclusion appears the greater, when it is con- 

 sidered that it avoids all hazardous hypotheses as to the forms 

 assumed by the beam during the transit. However the beam 

 may lie bent — whatever may be the nature of its vibrations and 

 internal action, this is certain, — that when its elasticity is un- 

 impaired, a weiglit travelling along it cannot, under any circum- 

 stances wliatever, more than double its corresponding statical de- 

 flection. To suppose it capable of doing more, is to suppose the 

 physical inipossilnlity of a gain of power. 



But though the travelling weiglit cannot, under any circum- 

 stances, produce more than double the statical deflection, it is 

 quite possible that it may do less: A large portion of the work 

 done by the weight may be absorbed in producing lateral vibra- 

 tions and other irregularities of motion in the beam. All these 

 concomitant operations act by way of diminution, and tend to make 

 the dynamical deflection less than double the statical central de- 

 flection. 



In determining the actual amount of this diminution, the velo- 

 city of transit must be taken into account. For that there is 

 some particular velocity for which the deflection is a maximum, 

 is obvious from this simple consideration — that when the weight 

 travels exceedingly slowly along the beam, it always exerts n 

 statical pressure, and does not tend to increase by momentum the 

 deflection beyond its statical amount ; — and, on the other hand, 

 when the weight travels with excessive rapidity, it may not have 

 time during the transit, to sink even the distance of statical 

 deflection. To take the limiting case, when the velocity is inde- 

 finitely great, the descent of the weight must be indefinitely 

 small ; for even if it fell freely, and there were no beam to support 

 it, the distance of descent in an indefinitely short time is inap- 

 preciable. 



Effect of the Inertia of the Beam. 



There is, then, between the exceedingly high and the exceed- 

 ingly low velocity, some particular intermediate speed which pro- 

 duces the greatest possible deflection. Before, however, consider- 

 ing what that velocity is, or endeavouring to establish a direct 

 relation between the velocity and the deflection, it is necessary to 

 examine more particularly the case j ust referred to — where the 

 velocity of transit is so great, that the weight has not time to 

 sink beyond a certain degree. 



Now, there are two ways in which this consideration of time 

 might be supposed to affect the amount of deflection. The first is 

 that already stated, where the period of transit is so short, that 

 even if the weight descended freely, w ithout support from the beam, 

 its descent would be inconsiderable. This case may, however, be at 

 once excluded, when it is considered that at all practicable railway 

 velocities, the time of transit over a long girder (50 to 80 feet) 

 could not be much less than one second, that a body would fall 

 freely upwards of IG feet in tliat time, and that its actual descent 

 (equal to the deflection of the girder) is only a few inches. 



But there is another way in which the consideration of time 

 might be supposed to affect the deflection : there might not be 

 time enough to overcome the inertia of the beam. This case 

 requires more particular examination. 



A person skating over a weak piece of ice may sometimes, by 

 moving rapidly, glide over it safely before it have time to break — 

 that is, before the pressure of his body have impressed on the ice 

 the downward motion sufficient for it to attain the point of 

 fracture while he is passing over it. Now, by the general princi- 

 ples of mechanics, the same pressure which, acting for a given 

 time, would produce a great velocity in a small mass, w ill produce 

 proportionably little velocity in a large mass. In order then 

 that the inertia of the ice may, in the case supposed, be a cause o£ 

 safety, it must be large in comparison w ith the pressure acting on 

 it ; that is, the mass of ice acted upon must greatly exceed the 

 mass of the man's body. 



In the same way, in order that the inertia of a girder might be 

 a cause of security, the mass of the girder must be very much 

 greater than that of the train passing over. But it will be shown 

 tliat the mass of the former does not, for heavy loads, exceed 

 tliat of the latter so greatly as to perceptibly diminish the deflec- 

 tion. It has sometimes been found useful to add to the inertia of 

 the girder by laying on it heavy ballast, and by this means the 

 structure is rendered steadier, — that is, the slight lateral oscilla- 

 tions and other irregularities of motion are reduced. But it 

 is only these smaller or subsidiary movements that can be 

 diminished by adding to the weight of the girder. Its mass, and 

 that of the permanent load upon it, is not in generalso large as to 

 materially influence the main, or vertical, deflection, when pro- 

 duced by nearljr as heavy a load as it will safely bear. 



