180 Mr. M. Merriman on the Flexure 



line. From the first two hypotheses stated above its differential 

 equation is easily derived and given in all books on the theory 

 of beams. This equation is 



d 2 y _ m 

 dx*~%V 



m being the moment of the molecular forces in the section whose 

 abscissa is x, E the modulus of elasticity, and I the moment of 

 inertia of the beam. To obtain from this the equation for any 

 particular case, it is only necessary to substitute for m and I 

 their values in terms of x and integrate the equation twice. In 

 what follows, I will be regarded as constant. Let o q represent 



A*— -a — *-\ \ A-*— ■«'-*" 1 A 



o P ? P' r 



a span of a continuous girder with level supports whose length 

 is /. Let P be a single concentrated load at a distance a from 

 the support o ; also let the span q r be equal to /', and the load 

 P ; be placed at a distance a! from the support q. Let M, M', 

 and M" denote the moments at the supports o, q, and r respec- 

 tively. Then ail the exterior forces which act upon the beam to 

 the left of the point may be replaced by the horizontal couple 

 M and a vertical shearing-force S. Since equilibrium prevails, 

 we have for a section between P and q the equation of moments 



M.-Sx+V{x-a)-m=0 (I.) 



Making in this x = l, m becomes M', and we have 



s= M-M' + P( ! - £ ) = M-M !+p(1 _ /t)) ; (r) 



a being replaced by kl, where k denotes any fraction. Insert 

 now the value of m in the differential equation of the curve, and 

 integrate it twice. The constant for the first integration is t, 

 the tangent of the angle which the curve at the origin makes 

 with the axis of abscissae ; the constant for the second integra- 

 tion is zero ; then the required equation is 



y = tx+ g^j[3M^-S.2? 3 + P(^-.fl) 3 ]. . . (II.) 



Insert in this the value S in terms of M, M', and P ; also make 

 x=l and put a — kl\ then y — } and we get the expression 



6EI/= -2M/-M'/+P/ 2 (2£-3A; 2 + * 3 ). 



u/u dv 



Now in the value of -j- make x = l; -j- then becomes /',the tan- 



