RECTANGULAR BEAMS 



59 



reinforced concrete beam. The form of parabola used has its 

 axis vertical in the stress-deformation diagram, Fig. 10, and its 

 vertex at the point S of the curve representing the ultimate 

 strength. 



The assumption is made in deriving formulas for ultimate loads, 

 that the amount of reinforcement is sufficient to develop the full 

 compressive strength of the concrete without stressing the steel 

 beyond its yield point. Failure under such conditions will occur 

 by crushing the concrete, with the yield point of the steel not 

 exceeded. Then, the parabola representing the variation of com- 

 pression is a full parabola, the upper end being the vertex and 

 the axis horizontal. (Fig. 29.) 



Average compressive 

 stress - | f c 



Total compressive 

 stress- | f c bKOL 



FIG. 29. 



When ultimate loads are considered, the secant modulus for 

 the concrete should not be used. The initial modulus should be 

 employed and will be denoted in this article by E c . Considering 

 Fig. 29 as drawn to scale, 



AB 



It is a well-known property of the parabola that 



Since AO represents the deformation A A.' (Fig. 28). 



9f 9.f 



E c = 



In the present connection, the two following properties of a 

 parabola (Fig. 29) are useful: (1) the average abscissa of the 

 parabolic arc equals two-thirds the greatest (f c ) ; (2) the distance 

 from the center of gravity of the parabolic area to its top 

 equals three-eighths the total height kd. 



Following the same procedure as employed in deriving formulas 

 for a rectangular variation of stress in the concrete and using 



