246 Professors Ayrton and Perry on 



with arch-ribs fixed at the ends ; and, indeed, it was when 

 putting Prof. Fuller's method before our students that we 

 discovered the following simple application of the principle to 

 beams. In spite of the apparent rashness of such a statement 

 about such a well-ventilated subject as the elasticity of beams, 

 we believe our method to be quite new. 



II. In the case of a horizontal beam fixed at the ends, the 

 angle between the end sections remains equal to nothing ; and 

 hence, in this case, 



2^=0 (1) 



Now, if the beam were merely supported at the ends, from a 

 given system of loading it is easy to find what the bending- 

 moment everywhere would be, either by numerical calculation 

 or a link-polygon method. Let us suppose the diagram of 

 this bending- moment m to be known. Then the bending- 

 moment M of a beam fixed at the ends with the same loading, 

 is m — c , where c is a constant ; so that it is only necessary 

 from the condition (1) to determine this constant c. 

 (l)is 



^m — c A m c 



2 -j— = 0, or Zj—Zjj 



m 

 or c = -4 ( 2 ) 



The rule then is, knowing m and I at every point : — Divide 



Tift 



the beam into any number, n, of equal parts ; find y at the 



middle of each part, and add all the values together : this is the 



numerator in (2). Fin dy at the middle of each part, and 



add the values together : this is the denominator in (2); so 

 that c is known. Subtract c from every value of m, or 

 diminish all the ordinates of the m diagram by the amount c, 

 and we have the M diagram ; that is, the real diagram of 

 bending-moment of the beam fixed at the ends, with any 

 distribution of loading and any variation in cross section. 



III. If the cross section is the same everywhere, it is 

 obvious that (2) becomes 



c = - 2 m, (3) 



n ' 



