III. On Impact on Elastic Beams. By Homersham Cox, Esq., B.A. 



of Jesus College. 



[Read Dec. 10, 1849.] 



The modern use of iron girders for Railway Bridges has given great interest and im- 

 portance to the investigation of the strength of Elastic Beams when subjected to various 

 statical and dynamical tests. Unfortunately the theoretical examination of the dynamical 

 question is very difficult, and to supply additional knowledge respecting them an extensive 

 series of experiments has been instituted under the authority of the Royal Commission ap- 

 pointed to inquire into the Application of Iron to Railway Structures. 



Part of these experiments, related to the effects of impact in producing the deflection and 

 fracture of beams. In most cases the impact was given in a horizontal direction by means 

 of heavy balls moving in circular arcs as pendulums. The ball was suspended by a fine 

 inextensible cord from a point of support vertically over the centre of the beam which was 

 fixed in a horizontal position by bearings near its two ends. The ball descended by its own 

 weight, and when it reached the lowest point of its arc, struck the centre of the beam hori- 

 zontally in a direction perpendicular to its length. Care was taken to prevent the ends of the 

 beam from yielding to the blow, and to carefully mark the degree of deflection produced at 

 the centre. The velocity acquired by the ball before impact, was computed by a well known 

 formula from the chord of the arc of descent. The course of experiments comprised great 

 variations of the strength and size of the beam, the velocities of impact, and the relative masses 

 of the beam and ball. 



There were two ways in which these experiments might be made practically useful : 

 either (firstly) as confirmations of an independent theoretical investigation of the laws of im- 

 pact on beams — or (secondly) supposing such an independent investigation impossible, as 

 empirical data susceptible of theoretical generalisation. 



It is to the first of these purposes that the experiments will here be applied, and it will 

 be attempted to be shewn that the observed results might have been predicted with a very 

 great degree of confidence and accuracy from known dynamical principles. 



A most elaborate paper by Poisson, entitled Sur VEquilibre et le Mouvement des Corps 

 FAastiques, is given in the Memoirs of the French Academy of Sciences, Tome Till., and 

 the eighth chapter of the Traite de Mecanique of the same author, relates to vibrations of 

 elastic rods. It is assumed however in these investigations, that the initial velocity and 

 position of every particle are known. But the essential difficulty of the problem here to 

 be discussed, is to determine that which, for the purposes of the illustrious writer referred to, 

 might be considered as known — namely, the initial effects upon the several points of the 

 system produced by assigned external causes of motion. 



Vol. IX. Part I. 10 



