24 ANNUAL OF SCIENTIFIC DISCOVERY. 



exceedingly small compared with that of the load. Having obtained 

 these under different relations of the length of the bridge, its statical 

 deflection and the velocity of the passing load, \ve proceeded to inves- 

 tigate, in addition, the effect which a greater proportional mass of the 

 bar or bridge has upon the deflections. Unfortunately, the extreme 

 difficulty of the problem has rendered its solution unattainable, except- 

 ing in the case in which the mass of the bridge is supposed to be ex- 

 ceedingly small compared with that of the load, and in the opposite 

 case, in which the mass of the load is supposed to be small compared 

 with that of the bridge. The examples that occur in practice lie be- 

 tween these two extremes ; for in the experiments of the commission, 

 performed at Portsmouth, with the inclined plane already described, 

 the weight of the load was from three to ten times that of the bar ; 

 but this is a much greater proportion than that, which occurs in bridges, 

 partly on account of the necessity for employing in experiments very 

 flexible bars, to render the changes of deflection sufficiently apparent, 

 and partly on account of the great difference in length ; for if bars bear- 

 ing the same ratio of weight to that of the load were employed in exper- 

 iment, the deflection would become so small as to be scarcely apprecia- 

 ble. This will readily be perceived when it is stated that, in a bridge 

 thirty-three feet long, a deflection not greater than one fourth of an 

 inch is usually allowed, which deflection is only one 1584th part of its 

 length ; whereas, in experiment, it is necessary to employ deflections 

 of two or more inches. In actual bridges of about forty feet span, the 

 weight of the engine and tender is very nearly the same as the weight 

 of that half of the bridge over which it passes ; and in large bridges 

 the weight of the load is much less than that of the bridge. :; 



The commissioners then state a mathematical investigation made at 

 their request, and continue: "It is thus shown that the enormous 

 increase of deflection produced by velocity in the Portsmouth experi- 

 ments cannot occur with real bridges, since it appears that the phe- 

 nomena in question are developed to a great extent when the magni- 

 tude of the structure is diminished. The total increase of the statical 

 deflection, when the inertia of the bridge is taken into account, will 

 be found much greater for short bridges than for long bridges. Sup- 

 posing, for example, the mass of the travelling load and of the bridge 

 to be nearly equal, the increase of the statical deflection at the highest 

 velocities, for bridges of twenty feet in length, and of the ordinary 

 degree of stiffness, may be more than one half; whereas, for bridges 

 of fifty feet in length, the increase will not be greater than one sev- 

 enth, and will rapidly diminish as greater lengths are taken. But as 

 it has been shown that the increase, catcris paribus, is diminished by 

 increasing the stiffness of the bridge, we always have it in our power 

 to reduce its amount within safe limits. Hence, in estimating the 

 strength of a railway bridge, this increase of the statical deflection 

 must be taken into account, by calculating it from the greatest load 

 which is likely to pass over the bridge, and from the highest possible 

 velocity. It must be remembered, also, that this deflection is liable 

 to be increased by jerks produced by the passage of the train over the 

 joints of tho rails. 



