Mathematical and Physical Papers. ■ 217 



NOTE ON CERTAIN FORMULAS FOR THE DESIGN OF 

 REENFORCED CONCRETE BEAMS. 



By A. K. Hubbard, Lawrence. 



IN chapter II of the book on Reenforced Concrete, by A. W. 

 Buel and C. S. Hill, some formulas and deductions are given 

 which are incorrect. It is the object of this paper to point out the 

 mistakes in these formulas and deductions, and to illustrate the 

 errors involved by numerical examples. 



Before proceeding to a consideration of the formulas, it will be 

 well to notice the assumptions on which the formulas are based. It 

 is not proposed to enter into any discussion concerning the correct- 

 ness of these assumptions : 



1. The strain of any fiber is directly proportional to the distance 



of that fiber from the neutral axis of the beam. 



« 



2. The stress in any fiber is proportional to its distance from the 

 neutral axis. 



3. Any variation of stress or strain near the reenforcement is 

 neglected. 



Referring to figure 1, the author uses the following notation : 



M = bending moment in inch-pounds. 



1 = length of beam in inches, center to center of supports. 



h = depth of beam in inches, out to out concrete. 



b = breadth of beam in inches. 



A = total area of cross-section of beam = hb. 

 As =: area of steel reenforcement in tension. 

 As^ = area of steel reenforcement in compression. 

 Ac = A— (As rf- As') = area of concrete. 



X = distance from neutral axis to outer compression fiber of concrete. 



y = distance from neutral axis to outer tension fiber of concrete. 



u = distance from neutral axis to outer compression fiber of steel. 



z = distance from neutral axis to outer tension fiber of steel. 



t = distance from neutral axis to center of steel sections in compression. 



v = distance from neutral axis to center of steel sections in tension. 



y==z-|-d; x = u-fdi; h = x-l-y; hi = u + z; hii = t-(-v; y = v + di; 

 x=:t + dii. 



