294 CONCRETE DESIGN 



These solutions give the most economical design; that is, 

 when the allowable unit stress in the steel and the allowable 

 unit stress in the concrete are realized under the proposed 

 load. This condition is determined by the value used for p. 

 However, other values of p are sometimes used, and in such 

 cases the procedure is as follows: 



Assume a value of n. Find the value of k by the for- 

 mula k= -^2 pn + (pn) 2 pn. Then find the value of ; by 

 the formula j 1 $ k. Next assume values for F s and F c . 

 Find M from the conditions of the problem. Assume 

 values for either b or d. and find the other by means of the 

 formula F s = M-*-pjbd 2 . Also find the same dimension by 

 the formula F c = 2 M + jkbd 2 , and use the largest value found 

 by either of these equations. 



Sometimes the problem is thus: b, d, and M are given. 

 Assume F s , then find A by the formula M = F s Ajd, using 

 for j. Solve p = A-*-bd for p, which should, by changing 6, 

 if required, be kept less than the value of p that gives the 

 most economical design mentioned above. Then to check, 

 find k and j as on page 292 from the value of p obtained and 

 find F s and F c accurately by the formulas at the foot of 

 page 292. These values must not be excessive. 



To investigate a beam already built, proceed as follows: 

 Measure the value of b, d, and A. Calculate the value of p 

 as well as the moment of the loads on the beam, or that are 

 to be put on the beam, including the weight of the beam 

 itself. Assume a value for , and find the value of k by the 

 formula k= "^2 pn + (pn) 2 -pn. Find the value of / by the 

 formula ;'=! J k. Find the stress in the steel by the for- 

 mula F s = M-r-Ajd, and' then the stress in the concrete by 

 the formula F c 2 M-s-jkbd 2 . Neither of these values should 

 exceed the safe allowable limit. 



Continuous Beams. If W is the total uniform load on a 

 beam and / is its length, then, for a simple beam, the moment 

 is Wl-t-8. In building construction, many beams and floor 

 slabs are continuous, and in this case the external moment 

 at the center of the span is decreased and there is produced 

 a negative moment over each support. Sufficient steel 



