494 



X. STATICS OF DEFORMABLE BODIES. 



so that 133) gives, M = 0. Thus the moment of the applied force X 

 and of the couple about the origin is zero, while the moment of 

 the couple alone due to the forces Z z symmetrically applied is 

 M 1 = El}^Iy2. It is to be noted that while the rod is bent down 



by the couple M' (Fig. 157), the 

 tendency of the force X is to 

 pull the end up. We see also 



that -o for the line of centers 

 does not vanish in general at the 

 origin, being equal to &i( a 1 ) ' 



The solution of the practical 

 problem of the deflection of a 

 beam by a transverse force applied 

 at one end is obtained by a com- 

 bination of the results for uniform 



and non- uniform flexure, as just investigated. Since the former 

 requires a couple which is the same for all sections, the latter one 

 which is proportional to #, by a suitable combination of the two 

 with opposite signs, we may make the couple upon the end section 

 equal to zero, so that we have to apply only a force. Since the 

 couple due to the uniform flexure is Ea^ly, and that due to the 

 non-uniform flexure Eb^I y Zj if we put a t -f- b l = 0, where I is the 

 length of the beam, there will be no couple to be applied to the 

 end of the beam, but only the force X. Determining b from 137) 

 and adding equations 124) and 129), we obtain the shifts, 



Fig. 157. 



138) vscfn 1 



W = -^r 



The equation of the central line is 



and the deflection of the end of the beam 



- 



El 



