LI I. Discussion of a Differential Equation relating to the breaking of Railway Bridges. 

 B)j G. G. Stokes, M. A., Fellow of Pembroke College, Cambridge. 



[Read May 21, 1849.] 



To explain the object of the following paper, it will be best to relate the circumstance which 

 gave rise to it. Some time ago Professor Willis requested my consideration of a certain differential 

 equation in which he was interested, at the same time explaining its object, and the mode of ob- 

 taining it. The equation will be found in the first article of this paper, which contains the sub- 

 stance of what he communicated to me. It relates to some experiments which have been 

 performed by a Royal Commission, of which Professor Willis is a member, appointed on the 

 27th of August, 184.7, " for the purpose of inquiring into the conditions to be observed by eno-ineers 

 in the application of iron in structures exposed to violent concussions and vibration." The object 

 of the experiments was to examine the effect of the velocity of a train in increasing or decreasing 

 the tendency of a girder bridge over which the train is passing to break under its weight. In order 

 to increase the observed effect, the bridge was purposely made as slight as possible : it consisted in 

 fact merely of a pair of cast or wrought iron bars, nine feet long, over which a carriage, variously 

 loaded in different sets of experiments, was made to pass with different velocities. The remarkable 

 result was obtained that the deflection of the bridge increased with the velocity of the carriage, 

 at least up to a certain point, and that it amounted in some cases to two or three times the central 

 statical deflection, or that which would be produced by the carriage placed at rest on the middle 

 of the bridge. It seemed highly desirable to investigate the motion mathematically, more especially 

 as the maximum deflection of the bridge, considered as depending on the velocit)' of the carriage, 

 had not been reached in the experiments*, in some cases because it corresponded to a velocity 

 greater than any at command, in others because the bridge gave way by the fracture of the bars 

 on increasing the velocity of the carriage. The exact calculation of the motion, or rather a cal- 

 culation in which none but really insignificant quantities should be omitted, would however be 

 extremely difficult, and would require the solution of a partial differential equation with an ordinary 

 differential equation for one of the equations of condition by which the arbitrary functions would 

 have to be determined. In fact, the forces acting on the body and on any element of the bridge 

 depend upon the positions and motions, or rather changes of motion, both of the body itself and 

 of every other element of the bridge, so that the exact solution of the problem, even when the de- 

 flection is supposed to be small, as it is in fact, appears almost hopeless. 



In order to render the problem more manageable, Professor Willis neglected tiic inertia of tl)o 

 bridge, and at the same time regarded the moving body as a heavy particle. Of course the masses 

 of bridges such as are actually used must be considerable; but the mass of the bars in the ex- 

 periments was small comjiared with that of the carriage, and it was reasonable to expect a near 

 accordance between the tlieory so sini])lified and experiment. Tliis simplification of tlie pnil)leni 

 reduces the calculation to an ordinary (Hdrrential ecjuation, whicii is that which has Ijcen already 

 mentioned ; and it is to tiie discussion of this equation that the present pajier is mainly devoted. 



This equation cannot apparently be integrated in finite terms, except for an infinite number 

 of particular values of a certain constant involved in it ; but I have investigated rapidly con- 

 vergent .series whereby numerical results may lie obtained. Hy merely altering the scale of the 



The detail* of the cxpcrimentN will be found in the Keport nt' the Comiulniion, to wliicli ihc renter in rcrorrcd. 



