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



each span ; for although as far as rolling load is concerned no more correct 

 hypothesis could be made, the weight of the bridge itself, if a large one, 

 usually varies considerably in the different parts of the same span. 



The equation given by M. Bresse, as has been stated, provides for certain 

 kinds of variable loads by the use of integrals, but the writer is not aware 

 that they have been applied, even by that author himself, to the purposes 

 of calculation, and it seems to him that in most cases the attempt to make 

 such an application would be beset with difficulties. 



It will, however, it is hoped, be seen from what follows, that the dealing 

 with variations of the above elements does not in fact present any very 

 formidable difficulty, though no doubt the labour of calculation is greater ; 

 but what the writer regards as most satisfactory is the very small difference 

 in the principal results in the case of the Britannia (where these variations 

 greatly exceed in amount those usually occurring), whether obtained by 

 the approximate method hitherto followed, or by the more rigorous one to 

 be explained, affording a strong presumption that in all ordinary cases the 

 former method may be confidently employed without risk of any important 

 error. 



Should the following treatment of the case be deemed successful, the 

 author would remark that its success is mainly due to the use of an 

 abbreviated functional notation, by which a great degree of clearness and 

 symmetry is preserved in expressions which would otherwise have become 

 inextricably complex. 



General Investigation of the Bending Moments and Deflections of 

 Continuous Beams. 



A a b A 



1 2 



Let 1 . 2 represent any span of a continuous beam, the length «of the 



span being I. 



cc . y the coordinates of the deflection curve, the origin being at the 



point 1. 

 a and b particular values of cc* 



e v e 2 , e 3 reciprocals of the products of the moments of inertia of the 

 sections in the spaces (1 .a), (a .b), (b .2), about their neutral 

 axes, by the modulus of elasticity of the material 



fi v p 2 y /; 3 loads per unit of length in the same spaces. 

 T tangent of inclination of deflection curve at 1, to straight line join- 

 ing 1 . 2, its positive value being taken upwards. 

 <f> l3 (f>. 2 bending moments at 1 . 2. 

 P shearing force at 1 . 

 Now let the bending moment at any point (x . y) 



between 1 and a be called ¥"(x) s 

 between a and b be called F,"(a?), 

 between b and 2 be called F 3 ' f (>) ; 



