158 W. H. WARREN. 



Substitute for c and f from (1) and (2) we have : — 



5 Ecx't _ 8 f1 _ v p E a (u-x) 



~8E t (l-x)~10 {1 X)t + Ft E b (l-x) 



From which quadratic equation x may be found. 



Take moments about the point of application of the 

 resultant of the compressive stresses we have the moment 

 of resistance of a unit section : — 



M 8f /t v J 6 (A v , 2 I ,, /3u-a; 



,,= 4^(9-8,-.= ) + ^^) 



To find the moments of resistance for any intensity of 

 stress in the concrete we must substitute in equation (4) 

 tlie values of M & and E t for the particular stresses c and t 

 and find x which should be substituted in equation (5) to 

 find M. When a crack has developed on the tension face 

 of the concrete t = and equation (4) becomes : — 



J-* s =pj|(-*)...(6) 



Equation (5) becomes : — 



M , /3 u - x 



l„r =fP 3 . 

 Equation (6) may be solved for x thus : — 



v - 2 /lOp^" , 4j9 2 E. 2 _ 4 E* , g v 

 " 5 7 E c E c 2 5 J E c *" V ; 



If we apply these results to the concrete beams Nos. 2, 

 3, and 4, recorded in Table V. and Fig. V. we have the 

 following data : — 



^ = 12, M = 0*9, P = 0*018 



Tests made on the steel rods used in the reinforcement 

 gave the following results : — 



E s = 31,000,000 lbs. per square inch. 



