60 Mr. J. M. Heppel on the Theory of Continuous Beams. 



way, he had occasion to investigate the conditions of a bridge of five 

 continuous spans over the River Palar. Having in India no books to refer 

 to but those of Moseley and Edwin Clark, he found himself unable to extend 

 the treatment of the cases there given to that of a beam with an increased 

 number of openings and varying loads. After many attempts and failures, 

 the same idea occurred to him which appears to have struck M. Clapeyron 

 nine or ten years before, that if the bending moments over the supports 

 were known, the whole conditions would become known. 



Following this clue, he was fortunate enough to succeed in at once 

 eliminating the other unknown quantities, which M. Clapeyron had been 

 obliged to retain in his equations for many years after his original discovery 

 of the method, and thus to arrive at an equation precisely identical with 

 that which had been first published in France by M. Bertot in 1856, and 

 was known as the " Theorem of the three Moments." 



This was sufficient for the immediate purpose, as the beams in question 

 were straight and of uniform section throughout, conditions to which this 

 theorem is strictly applicable without any modification whatever. 



As, however, the writer was at this time under the impression that he 

 was using an entirely new mode of analysis, he was naturally anxious to 

 check its results by comparison with tbose obtained in some well-known 

 case by other means. Fortunately he had at hand that of the Britannia 

 Bridge, perhaps the best that could have been selected ; but for this purpose 

 it became necessary to import into the fundamental equation the conditions 

 of varying sections in the different spans and imperfect continuity. This, 

 however, presented no great difficulty, and by means of an equation thus 

 modified, he had the satisfaction of reproducing all Mr. Pole's results, and 

 thus convincing himself of the trustworthiness of the method in question. 



The equation thus generalized is absolutely identical with that arrived at 

 by M. Belanger in the work above referred to*. 



It would appear, then, that the theory of this subject was independently 

 advanced to about the same state of perfection in France and in England, 

 though as regards the development of its application to practice no doubt 

 very much the more has been done in the former country. 



The writer will now advert to some inherent defects of this theory, the 

 cure of which is the principal object of the investigation which follows. 



The chief one, which is admitted by all writers on the subject, is the 

 necessity for supposing the moment of inertia of the section constant 

 throughout each span ; any more general hypothesis, it is said, would render 

 the calculation inextricable. Still it is certain that the conclusions arrived 

 at on the hypothesis of a constant section cease to be true if a variation of 

 section is introduced, and the amount of error thereby induced, though 

 considered to be probably small, is still a matter of uncertainty. 



The next defect is the assumption of uniformity of load throughout 



* A paper on this subject by the writer was published in the Minutes of Proceedings 

 Inst. C.E. vol. xix. 1859-60. 



