70 Remarks on Mr. HeppeVs Theory of Continuous Beams. 



2. Case of a uniform girder with an indefinite number of equal spans, 

 uniformly loaded; loads alternately light and heavy. — The supposition 

 just described forms the basis of the formulae given in a treatise called 'A 

 Manual of Civil Engineering,' page 288 ; and it therefore seems to me 

 desirable to test those formulae by means of Mr. Heppel's method. 



The cross-section of the whole girder and the load on a given span 

 being uniform, the definite integrals of the formulee (1) take the following 

 values : — 



fi\ _ x 2 _ x s 

 2 _; n ~2EL ; ? ~6Ei' " 24E1 2x 



iix J x" x" „ ux~ mq 



m= ^ ; n= ; q= ; F=£--=-±. . . . (7) 



The values of those integrals for the complete span are expressed by 

 making x=l. 



The values of n and q are the same for every span. In the values of m 

 and F, the load jjl per unit of span has a greater and a less value alter- 

 nately. Let iv be the weight per unit of span of the girder, with its fixed 

 load, w x that of the travelling load (increased, if necessary, to allow for 

 the additional straining effect of motion) ; then the alternate values of fi 

 are 



P=w ; } i , =w +w x (8) 



The moments at the points of support are all equal ; that is, $> =® 1 



=*->• 



Equation (6) now becomes the following (the common factor I 3 having 

 been cancelled) : — 



0=-2$ w 1 + F 1 + F^ 1 -^j 



giving for the bending moment at each point of support 

 R + F —tl 2w +w, t 



If t be made =0, so that the continuity is perfect, this equation exactly 

 agrees with the formula at page 289 of the treatise just referred to ; and 

 the same is the case with the following formulae for the shearing-forces 

 and slopes close to a point of support, and for the moments and deflections 

 at other points : — 



wJ 



Shearing-force, light load, P= 



Shearing-force, heavy load, F 1 == W °^~ W \ l. 



4 j 



Slope, light load, T= ». ' '-J-^; j 



Slope, heavy load, -T'=|+^. j 



(10) 



