158 CONTINUOUS BEAMS. Art. 96. 



conditions determine the four constants of integration, R lt 

 and M . 



Since it is necessary to use the equation of the elastic line 

 to determine the reactions and end moments, they are statically 

 indeterminate, that is, they depend upon the deformation of 

 the beam. In the above case, knowing any two of the quantities 

 R lt J? 2 , M!, and M 2 , the moment and shear at any section are 

 statically determinate, that is, they are gotten by means of the 

 equations of equilibrium. When ^ = // 2 , R 1 = R 2 and M = M 

 by symmetry, that is, the equations for the two segments become 

 identical. 



Knowing M l and R lt the points of contra-flexure are located 

 by putting the above values for the moments equal to zero and 

 solving for x. Knowing the location of the points of contra- 

 flexure, it may be convenient to treat the beam as made up of 

 two cantilevers supporting a beam similar to that shown in 

 Fig. 96 (89). 



96. Bending Moments and Shears for Continuous Beams. 

 Fig. 108 shows a two-span continuous beam. If the middle sup- 

 port be lowered, the beam will deflect and the load on the middle 

 support will decrease, finally becoming zero when the deflection 



is equal to ^4EI ' the ^ eam becomes discontinuous and now 

 has a single span L = 2l (Fig. 101). 



If the middle support be raised, the load on the end sup- 

 ports will decrease, finally becoming zero when the end deflec- 



tions are -^U and the beam becomes a double cantilever 



(Fig. 100). 



These considerations show that the reactions, and therefore 

 the shears and moments depend upon the deformation of the 

 beam, that is, upon the elastic line, just as for beams having 

 fixed ends, and are therefore statically indeterminate; beams 

 with fixed ends may, in fact, be considered as special cases of 

 continuous beams, as will now be shown from the simple case 

 of Fig. 108. 



In this case, the elastic line must be horizontal at the middle 

 support on account of symmetry. Imposing this condition by 

 means of the equation of the elastic line, together with the con- 

 dition of zero deflection at the supports, the reactions are deter- 

 mined and the moments may be gotten as in ordinary beams. 



