178 



On the Theory of Continuous Beams. 



[Jan. 27, 



only known to Mr. Pole through the examples of its application given in 

 Moseley's work, and the results obtained were identical with those which 

 would have followed from the application of the method of Clapeyron in 

 its most improved and generalized form. 



In 1858, the present writer, being then in India, had occasion to con- 

 sider the condition of a continuous girder of five spans, and finding the me- 

 thod of Navier unmanageable, was forced to seek for some other. He 

 first came upon the equation which he afterwards found had been for some 

 years known in France as the " Theorem of the three Moments," and 

 afterwards extended it, so as to take in all the conditions of the Britannia 

 bridge and to verify all Mr. Pole's results. In this form it was absolutely 

 identical with the equation given by M. Belanger, and nearly so with that 

 of M. Bresse. 



The great defect in all this theory up to the present time has been that, 

 in order to avoid an inextricable complexity, it has been necessary to con- 

 sider the load in each span as uniformly distributed over it, and the mo- 

 ment of inertia of the section as uniform throughout each span. 



In many cases these hypotheses are false, notably so in the case of the 

 Britannia ; and the conclusions are affected by their falsity, to what extent 

 being a matter of uncertainty, though good grounds have been shown for 

 believing that the errors cannot attain to importance. 



The method now given treats these conditions, it is hoped, rigo- 

 rously ; and although the equations obtained are such as necessarily re- 

 quire some laborious computation to obtain numerical results, they are 

 certainly by no means inextricable. 



It is satisfactory to find that in the case of the Britannia, where these 

 new conditions enter with much greater force than in most cases, their 

 effect on the resulting stresses is very unimportant ; so that the inference 

 may legitimately be drawn that in all ordinary cases the method of 

 Bresse may be confidently applied. 



It is scarcely possible in a short abstract to give an idea of an ana- 

 lytical investigation. The equations obtained are of the same form as those 

 of the previous methods, each containing, as unknown quantities, the 

 bending moments over three consecutive supports ; but the coefficients are 

 somewhat involved functions of the varying loads and sections. An ab- 

 breviated functional notation has, wherever possible, been used, by means 

 of which a certain degree of clearness and symmetry is preserved in ex- 

 pressions which would otherwise become inextricably complex. 



IV. "Remarks on Mr. HeppeFs Theory of Continuous Beams." 

 By W. J. Macquorn Rankine, C.E., LL.D., F.R.S. Received 

 December 22, 1869. 



(Abstract.) 



The author states that the advantages possessed by Mr. Heppel's method 

 will probably cause it to be used both in practice and in scientific study. 



