Dr. Rankine on Mr. Heppel's Theory of Continuous Beams, 459 



required may be computed for any particular span as follows : — 

 The inclination T at a point of support by equation (5) ; the shear- 

 ing-force P at the same point by equation (4) ; the deflection y 

 and moment <J> at any point in that span by equations (3) and (2). 

 Points of maximum and minimum bending-moment are of course 



found by making — =0; and points of inflection by making <fc=0. 



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 formulae (1) take the 

 following values : — 



m=B £f!. n=—- <, = — • F=-^=^ . . . (7) 

 m 2 ' U 2EI' q 6EI'* 24EI 2* KO 



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

 making oc=l. 



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

 of m and F, the load fi per unit of span has a greater and a less 

 value alternately. Let w be the weight per unit of span of the 

 girder with its fixed load, w 1 that of the travelling load (increased, 

 if necessary, to allow for the additional straining effect of motion) ; 

 then the alternate values of fi are 



{i=w ; fx'=w + w 1 (8) 



The moments at the points of support are all equal ; that is, 



Equation (6) now becomes the following (the common factor P 

 having been cancelled) : — 



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



giving for the bending-moment at each point of support 



F ' +F -'-^ ** +, \*> (9) 



2^ 24 I 



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 re- 

 ferred 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 : — 



Shearing- force, light load, P=-^-; 



Shearing-force, heavy load, P l =-^— — \l. 



2H2 



(10) 



