[ 446 ] 

 LVII. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from p. 379.] 



June 16, 1870. — General Sir Edward Sabine, K.C.B., President, in 



the Chair. 

 r PHE following communications were read : — 



" On the Theory of Continuous Beams." By John Mortimer 

 Heppel, M. Inst. C.E. 



In venturing to present to the Royal Society a paper on a subject 

 which has engaged the attention, more especially in France, of some 

 of the most eminent engineers and writers on Mechanical Philosophy, 

 the author feels it to be incumbent on him to state the nature of the 

 claim to their attention which he hopes it may be found to possess 

 in point of originality or improvement on the method of treatment. 



To do this clearly, however, it will be necessary to advert to the 

 principal steps by which progress in the knowledge of this subject has 

 been made, both in France and in this country. 



The theory of continuous beams appears to have first attracted at- 

 tention in France about 1825, when a method of determining all 

 the conditions of equilibrium of a straight beam of uniform section 

 throughout, resting on any number of level supports at any dis- 

 tances apart, each span being loaded uniformly, but the uniform 

 loads varying in any manner from one span to another, was inves- 

 tigated and published by M. Navier. This method, although per- 

 fectly exact for the assumed conditions, was objectionable from the 

 great labour and intricacy of the calculations it entailed. Messrs. 

 Molinos and Pronnier, in their work entitled ' Traite Theorique et 

 Pratique de la construction des Ponts Metalliques,' describe this 

 process fully, and show that for a bridge of n openings, the solution 

 must be effected of 3^+1 equations, involving as many unknown 

 quantities, these equations being themselves of a complex character ; 

 and they observe, " Thus, to find the curve of the moments of rup- 

 ture for a bridge of 6 spans, 1 9 equations must be operated on ; such 

 calculations would be repulsive ; and when the number of spans is at 

 all considerable this method must be abandoned." 



The method of M. Navier, however, remained the only one avail- 

 able till about 1849, when M. Clapeyron, Ingenieur des Mines, and 

 Member of the Academy of Sciences, being charged with the con- 

 struction of the Pont d'Asnieres, a bridge of five continuous spans 

 over the Seine, near Paris, applied himself to seek some more ma- 

 nageable process. He appears to have perceived (and, so far as the 

 writer is informed, to have been the first to perceive) that if the 

 bending-moments over the supports at the ends of any span were 

 known as well as the amount and distribution of the load, the entire 

 mechanical condition of this portion of the beam would become known 

 just as if it were an independent beam. Upon this M. Clapeyron pro- 

 ceeded to form a set of equations involving as unknown quantities 

 the bending-moments over the supports, with a view to their deter- 

 mination. He found himself, however, obliged to introduce into 



