130 REINFORCED CONCRETE CONSTRUCTION 



in Fig. 63. The fixing at the ends is effected in this case by build- 

 ing the beam for some distance into the wall. The same result, 

 as far as the effect on the beam is concerned, might be effected as 

 follows: having merely supported the beam and placed upon it 

 the load's it has to bear, load the ends outside of the supports 

 just enough to make the tangents at the supports horizontal. 

 In reinforced concrete construction the designer must use his judg- 

 ment of how near any particular case approaches the case above 

 mentioned, the bending moment formulas for fixed beams being 

 based upon the direction of the neutral surface as just described. 



With restrained beams, a negative bending moment occurs 

 over a fixed end; that is, at such a point the upper surface of 

 the beam is in tension, and the bottom is in compression. The 

 negative moment at each support, for a beam uniformly loaded 



w 



FIG. 63. 



and fixed at both ends, is 1/12 wl 2 . For one end free, the 

 negative moment at the fixed end is l/8wl 2 . If a beam in 

 all-concrete construction runs into a column or heavy wall girder, 

 the end in question is practically fixed. Top reinforcement will 

 be required for the negative moment, and rods running into the 

 support must be bent or otherwise anchored. 



54. Continuous Beams. The bending moments and shears 

 on beams which are continuous over one or more intermediate 

 supports cannot be derived from the principles of statics, since 

 the unknown conditions in the case of the reactions are greater 

 in number than the statical conditions of equilibrium. To find 

 the reactions for continuous beams, an additional condition is to 

 be introduced from the theory of elasticity, by means of the 

 equation of the elastic curve. After the reactions are computed, 

 the shears and moments due to given loads are readily deter- 

 mined, either analytically or graphically. Only the results 

 of such study will be stated here, since the study itself is not 

 important in this connection. 



In general practice, such as building construction, the live 



