FLEXURE OF BEAMS 83 



From symmetry, R a = #c, and consequently 



R b = i wl. 



Problem 89. A continuous beam of four equal spans is uniformly loaded. Find 

 the bending moments and reactions at the supports. 



Solution. The system of simultaneous equations to be solved in this case is 



-Vi = 0, 



?//- 

 Hi + 4Jf,+ *,:=-, 



-V, = 0, 

 the solution of which gives 



Problem 90. A continuous beam of five equal spans is uniformly loaded. Find 

 the moments and reactions at the supports. 



72. Work of deformation. In changing the shape of a body the 

 points of application <>f the external forces necessarily move, and 

 therefore do a certain amount of work called 

 the work of deformation. 



To tind the amount of this work of defor- 

 mation for a prismatic beam, consider t\v. adja- 

 cent cross sections of the beam at a distance 

 dx apart (Fig. 66). Suppose one of these cross 

 sections rem;. -nary and the otlin turns 



through an angle d& with reference to the first. 

 Tht-n the change in length of a fiber at a dis- 

 tance y from the neutral B -^. ;md therefore, by Hooke's law, 



K 



wh.-re f> \< tin- intensity of the stress on the fiber. By the straight- 

 line law, p = - 



- and hence 



Since one of the cross sections is assumed to be stationary, the stress 

 _C on it does no work. On the other cross section the normal 



