lRi.S.1 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



3«1 



When the train passes over the g-irder, the centre of gravity of 

 the whole system sinks, the impressed moving force downwards 

 being the weight of the train, and the motion of the centre of 

 gravity being retarded by the elastic force of the girder. 



To take a case every way unfavourable to the conclusion which we 

 wish to establish, let tlie' greatest deflection he 3 inches, and the 

 velocity of the transit so great that the weight passes over the girder 

 in a second of time. This would be the time of transit over a girder 

 88 feet long, at the rate of a mile a-minute. Now, tlie extreme de- 

 flection may be supposed to be accomplished in half the time of 

 transit, or the centre of the girder sinks 3 inches in half a second. 

 The centre of gravity of the whole system at no time sinks so 

 much as the centre of tlie beam sinks, for its two ends do not sink 

 at all. On the whole, it seems an ample allowance to suppose the 

 maximum vertical descent of the centre of gravity of the beam Ig 

 inch. Now, to find the work which would alone produce this velo- 

 city, we must have an equation of vis viva, excluding the retarding 

 force. 



By the ordinary rules for calculating the rectilineal motion of 

 bodies, if a given mass j\l originally at rest be acted upon at its 

 centre of gravity by one iiniform force /moving through a space 

 J in the time (, 



2 M a; =/(''. 



Suppose the force to be that of a small weight. The mass of this 

 weight will be found (on substituting numerical values and putting 

 gravity r= 32) to be only the 32d part of M, if the latter move 

 through 1^ inch in half a second. 



The beam is usually constructed to bear a pressure considerably 

 exceeding its own weight. In that case less then one S2d of the 

 work actually exerted by the travelling weight would suffice for 

 the mere acceleration of the beam : and we come to tlie conclusion, 

 that even at the highest practicable velocity, the power required 

 to set the beam in motion subtracts very little of the power 

 producing deflection. In other words, when the mass of the load 

 is not small compared with that of the beam, the deflection is 

 never materially influenced by the inertia of the beam. 



Influence of the Velocity of Transit on the Deflection 

 in the case of a Single Weight. 



Having arrived at the important conclusion that when the 

 travelling weight is large, the inertia of the beam is an immaterial 

 consideration, or that the efl^ective moving forces are inconsiderable 

 compared with the impressed forces, we might suppose the mutual 

 pressure between the beam and the weight statically equal to the 

 force which the former by its elasticity exerts in an upward 

 direction to resist deflection. 



But, in fact, the mutual pressure between the beam and the 

 weight is an unknown force, not generally susceptible of exact de- 

 termination. During the first part of the motion, the weight does 

 not, so to speak, exert its full pressure on the beam, for the 

 surface yields and recedes before it. During the latter part of the 

 descent, on the contrary, the pressure in question exerts a supe- 

 rior power, to destroy the momentum previously acquired by the 

 descending weight. The weight then moves downwards, first 

 with an accelerated, and subsequently with a retarded, velocity : 

 or the pressure on its under .side is in the former stage of motion, 

 less, and in the latter stage greater, than the effect of gravity. 



The path of the weight is likewise unknown, for the motion is 

 made up of two parts — the motion along the beam, and the motion 

 of the beam itself. If, indeed, it be assumed that the motion is 

 always along the beam, or that at every instant the curvature of 

 the beam has, at the point of mutual contact, the same tangent as 

 the path of the weight, the problem would be capable of soluti(m. 

 The investigations of Professor Moseley and M. Navier have de- 

 termined the curvature of the beam sufficiently to aiford means of 

 tracing the curve described by the moving weight ; and therefore 

 its pressure, which is equal to its centrifugal force -|- the effect of 

 gravity, might be ascertained. 



The hypothesis which would lead to these results is, however, 

 arbitrary and unsafe : and besides, the curvature of the beam as 

 mathematically determined, is not exactly that which occurs in 

 actual practice, where the elasticity is always more or less imperfect. 

 The difficulty is however of no great importance, because, as will 

 be presently shown, it does not occur where the moving body is not 

 a single weight, but a long train. And the subject is here referred 

 to, merely to show the almost insuperable difficulties of determin- 

 ing the motion of a single weight along an elastic beam. 

 Uniformly Distributed Load. 



We have hitherto considered the effect of a single weight press- 

 ing only at one point of the girder. The more important practi- 

 cal case, where the pressure is applied to a considerable surface, 

 remains to be examined. 



In entering upon this inquiry, the consideration of horizontal 

 motion will be in the first instance excluded. The distinction 

 between the effects of gradual and instantaneous loading lias been 

 already pointed out, in reference to a single weight ; and the 

 comparison may now be extended to the case of an uniformly 

 distributed load. If this load be gradually laid on, it produces less 

 deflection than when laid on all at once. A series of weights applied 

 simultaneously all along the undeflected girder, will move vertically 

 downwards, and acquire momentum which has to be destroyed by 

 an increased exertion of the elastic forces of the girder. In this 

 case, as in that of a single weight, the ultimate deflection and 

 pressure will be doubled, as will be demonstrated by analogous 

 principles. 



The beam being, as before, supposed to be perfectly elastic, the 

 central deflection is proportional to the weight of the uniformly 

 distributed load. If a be the length of the beam, x the central 

 deflection, u the weight of a unit of length of the load, we may 

 put u a ^ a .r, where a is a determinate constant. 



Let x\ y' be co-ordinates of any point in the surface of the 

 beam, which is supposed to have the same curvature as its neutral 

 axis. Then as the curvature is always exceedingly small, u dy' is 

 the weight of an element of the load, anAudy' dx' represents 

 the product of this weight and its virtual velocity, when the 

 centre of the beam is displaced through a small vertical distance 

 d.r. The product of all the pressures and the corresponding 

 virtual velocities is equivalent to 



/ 



ud x' d y". 



taking y' between the limits and a. It may mathematically ba 

 shown that this integral is equal to u a d.r, where .r is the vertical 

 ordinate of the centre of gravity of the load. Also, from the 

 equations given by Professor Moseley, in his " Principles of En- 

 gineering," Art. 374, it may be shown by a simple process, which is 

 here omitted for the sake of brevity, that x is proportional to x, 

 and may therefore be put = /3 x, where 3 is a determinate constant. 

 Hence, ua d x =: a fix d x. 

 The integral of this, between limits and x, is the total work 

 done in producing the central deflection. This integral is equal to 



x' 

 oiS „ = — - 



a X ~ 

 -.X. 



Or the " work done" is equivalent to that produced by a weight 

 which would statically maintain half the deflection, moving 

 through the whole space which the centre of gravity actually de- 

 scribed. Hence, by the same reasoning as applied to the case of a 

 single weight, the statical deflection and pressure are doubled by in- 

 ttantuneous loading. 



Transit of a Continuous Load. 



By combining the conclusion just arrived at with the principle 

 which has here been termed the Principle of the Conservation of 

 Work, it is readily seen that the statical strain and deflection 

 cannot be more than doubled by the transit, at any horizontal 

 velocity, of a uniform load of the same lengrth as the girder. 

 Indeed', the dynamical will in general be considerably less than 

 double the corresponding statical effects. 



It has been shown that where the weight of the load sustained 

 is nearly as great or greater than that of the beam, the force 

 required to produce motion in the beam is inconsiderable compared 

 with the actual deflecting forces. The beam itself, therefore, is then 

 always nearly in a state of equilibrium, and its form nearly the 

 same as that which would be statically produced by the external 

 pressures. If this be assumed to be strictly true, it follows that 

 the curve of deflection is concave in every part, and therefore that 

 no part of the beam sinks while another part is rising— that all 

 the parts sink together, and all rise together. The' vertical 

 motion is so extremely small and gradual, that there can be no 

 danger in assuming that all the parts arrive at their lowest 

 positions at the same instant. It follows then, as previously to 

 that instant the motion was downwards, and subsequently upw ards, 

 the beam in its lowest position is at rest, either instantaneously or 

 for a definite period. 



In this position of rest, the pressures on the surface of the 

 beam are in statical equilibrium with its internal forces. At the 

 same time, the pressure produced by the travelling load is the 

 same as if the curve of deflection were a flxed curve. 

 Effect of Centrifugal Force. 



AVhen a body, moving along a fixed curve, is acted upon by no 

 forces but the' pressure of the curve and its own weight, the 

 pressure on the curve (by the known principles of mechanics) is 



