REINFO}{CEI) COXCRETE STRENGTH OF BEAMS. 647 



On this are base<l the formula of Talbot and Hatt for beams 

 ^vliich necessarily are ultimate strength fornmlte ; that is to say, the 

 load under which a beam will fail is calculated by these fornuila; and 

 dividetl by some factor of safety to o1)tain a working load. 



A more rational method is to adopt a safe working stress, and to 

 consider the nujdulus as constant, as it practically is up to that 

 intensity of stress in the extreme fibres, and in fact to adopt Hooke's 

 Law. To recognise the very small change in the niodulus in beams 

 which are not called upon to sustain a compressive stress of more 

 than 600 or 700 lb. per square inch would be an unnecessary refine- 

 ment. Tlie important thing is that concrete can be regarded as an 

 elastic material up to certain limits, inasmuch as repetitions of stress 

 •do not cause increasing deformations after the first few applications. 



Reinforced concrete beams, liaving the reinforcement placed so 

 as to take the tensile stresses, are reailily calcidated from diagrams 

 and tables published in many text books. The best of these neglect 

 the tensile strength of the concrete, adopt Hooke's Law, and use 

 various ratios for the moduli of elasticity of steel and of concrete. 

 Upon these ratios depend the amount of stress carried by the 

 reinforcements, as it is clear that if concrete and steel are so com- 

 bined as to inidergo the same strain, the stresses in each by Hooke's 

 Law will be proportional to its modulus of elasticity. Fig-ure 1 will 

 show the effect of this. To calculate the moment of resistance of any 

 section, we proceed as follows : — 



Total stress C in concrete = ^/ x kd x b 

 „ „ T in steel := a x f 

 whence/' x a = \f x kd x b since total ten -ion must be equal to 

 total compression. 



D i- _ « = ^>^^ (^ — ^*') = "^ — ^ by Hooke's Law 



/; , i-d ~ ^' ' 



n X X a ^^\kd X b. 



k 



If a, (1, n, and h are known, we can now calculate k. 



The aiiiouut of resistance may now l)e calculated in one of three 

 T\ays-^ 



1. By conqjuting the moment of the compressive stress about 



the centroid of the tensile stress. 



2. Bj' computing in the samenuuiuer the moment of the tensile 

 ■ sti'essea. 



."!. By a.<lding the moments ol' the comjiressive and tensile 

 stresses about the neutral axis. 



The third method is of but little use, since we must know both 



tensile and conq)ressive stresses. The bettei- plan is to fii-st determine 



whether /', or/'^ fii'st i-eaches its niisximum value, and calculate 

 resistance foi- that value. 



