ANALYSIS OF STRESS IN BEAMS 43 



If the moment of inertia I g with respect to a gravity axis is required, 

 then, since by Article 46 (A), 1= I g + c 2 F, we have I g = / c*F; and 

 hence, by substituting the values of / and c from the above, 



i 

 I 

 i 

 i 

 10 



The above method is due to Nehrs, and furnishes an easy method 

 of calculating the moment of inertia of any cross section by simply 

 measuring the area F of the original section and the area F' } F" of 

 the transformed sections by means of a planimeter, and then substi- 

 tuting these values in the above formulas. 



48. Moment of inertia of non-homogeneous sections. The stand- 



M c 

 ard formula for calculating the stress in beams, p = , assumes 



that the material of which the beam is composed is homogeneous 

 throughout. If, then, a beam is com- \ Y 



posed of two different materials, such, 

 for instance, as concrete and steel, it is 

 necessary to modify this formula some- 

 what before applying it. 



To exemplify this, consider a rectan- 

 gular concrete beam, reen forced by steel 

 rods near the bottom, as shown in cross 

 section in Fig. 27. Let p e and p t denote 

 the stresses on a fiber of concrete and 

 of steel respectively, at the same distance y from the neutral axis, and 

 let E t and E t denote the moduli of elasticity for concrete and steel 

 Then, by Hooke's law, 



fr = y = 7r; 



' ' 



whence 



E. 



fore, if dF is an infinitesimal area of steel at the distance y 

 from the neutral axis, the moment of the stress acting on this area is 



I 

 Fio. 27 



