Mr. J. M. Heppel on the Theory of Continuous Beams. 451 



General Investigation of the B ending-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 /. 

 x, y the coordinates of the deflection-curve, the origin being 



at the point 1. 

 a and b particular values of x. 

 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 



<$ 



Pv f*2> H-3 lo aas P er unit of length in the same spaces. 



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



joining 1 and 2, its positive value being taken upwards. 

 <p v <j> 2 bending-moments at I and 2. 

 P shearing force at 1. 

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

 between 1 and a be called F x "(d?), 

 between a and b be called F 2 "(o?), 

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

 and let the part of this bending-moment which results alone from 

 the load on the beam between 1 and x be called, 



between 1 and a,f"(x), 

 between a and 6,/ 2 "(<z), 

 between b and 2, f 3 '(x) ; 



and let the first and second integrals of these functions, as of F/'(#)j 

 //'(», be denoted by F/O), //O), and F^x^f^x), and the value 

 of any one, as F^o?), for a particular value of a?, as a, by F x (a) ; then 



/,»-ftJ. ( [ ) 



fl ;X.)-H>(— £)+!>.£=£*) (2) 



Also, from equality of moments about the point (x . y\ 



F/'C^ft-Ptf+ZiVX (4) 



F 2 » = 1 -Pa;+/ 2 "(^ (5) 



F,»=k- p *+/.»5 (6) 



and, from equality of moments about the point 2, 



P=4(fc-*.+/."(0) (7) 



* + *H*&=». • (3) 



