REENFORCED CONCRETE 



169 



w&x. 



~. Therefore, equating the tension in the hoop to the sum of 

 the components of the radial stress perpendicular to the plane of 

 the section, we have 



whence 



P 



Now let A' denote the cross-sectional area of an equivalent 

 amount of radial reenforcement. Then, since the length of the 

 arc considered is TTT and the radial stress is of amount w per unit 

 of length, the total amount 

 of radial reenforcement re- 

 quired would be given by 

 the equation 



pA 1 = irrw ; 



whence A' = 



7TTW 



P 



Comparing these expres- 

 sions for A and A 1 , it is found 



that ' 



FIG. 114 



Consequently, the theoretical efficiency of tensile reenforcement 

 in the form of hoops is 3.14 times as great as the same cross- 

 sectional area of direct, or radial, reenforcement. The amount of 

 metal in a hoop of radius r, however, is 2 irrA, whereas that in the 

 radial reenforcement is A f 2 r, and since A' = TrA, these volumes are 

 equal. Consequently, there is no saving in material effected by 

 making the reenforcement in the form of hoops. But when there 

 is such a complex system of reenforcement as that shown in 

 Fig. Ill, some of the metal may be used to better advantage hi 

 the form of hoops, as this lessens somewhat the congestion of metal 

 at the columns. 



101. Maximum moment. For a continuous beam of span Z, 

 carrying a total uniform load of amount W, the moment at the 



supports is - ; whereas the moment at midspan is one half this 





