270 STRENGTH OF MATERIALS 



Problem 181. Find the single central load that can be supported by the beam 

 in Problems J78 and 179, assuming the values given as safe values. 



PI 



HINT. Use the formula M = - 

 4 



Problem 182. In Problem 178 suppose that the metal reenforcement has an 

 elastic limit of 45,000 lb./in. 2 but that its area is not known, the rest of the data 

 remaining the same. Neglecting the tensile strength of the concrete, find the amount 

 of steel reenforcement necessary to make the beam equally strong in tension and 

 compression. 



219. Linear variation of stress. Assuming the straight-line law of 

 the distribution of stress (Articles 41 and 42), the equations corre- 

 sponding to (111) and (114) are 



(115) i^c=?j(l-^ t + ^,, 

 and 



(116) M = J bh*(l - v)*p t + J l>)rrp c + brh*(u - v)p t , 



while the expressions for p c and p t given in equation (112) are 

 unchanged. Substituting the values of p c and p, from (112) in (115), 

 the expression for the location of the neutral axis becomes 



(U7) P =- rtf +(1 + 2 



(it 1) 



The above formulas are somewhat simpler than those obtained 

 under the parabolic assumption as to the distribution of stress, and 

 for this reason are preferred by many engineers. In cither case the 

 additional assumption that the beam is also reeiilWvd on the com- 

 pression side adds considerably to the complexity of the analysis. 



The following table gives the calculated stress in the steel ivi : n- 

 forcement for the tests reported.* In these calculations the para- 

 bolic variation of stress was assumed for the compression side. The 

 assumption of the straight-line variation was found to give sti > 

 differing by only 2 per cent. 



Numerous empirical formulas for the calculation of reenforced con- 

 crete beams have been proposed by Christophe, Thacher, A. L. Johns, m, 

 and others. For a complete discussion of the subject the student is 

 referred to Chapter II of Buel and Hill's book on reenforced concrete.! 



* Proc. Amer. Soc. for Testing Materials, 1905. 



t Buel and Hill, Reenforced Concrete. Engineering News Publishing Company, 1904. 



