52 RESISTANCE OF MATERIALS 



section. Now, if A^4 denotes an element of area of the cross section, 

 and p' the intensity of the normal stress acting on it, the total stress 

 on this area is p'&A. If, then, y is the distance of this stress, or 

 internal force, from the neutral axis of the section, and M denotes 

 the resultant moment of the external forces about this axis, for 

 equilibrium 



Now let p denote the stress on the extreme fiber and e the distance 

 of this fiber from the neutral axis. Then, by the straight-line law, 



=, 



y e 



and, inserting this value of p' in the above equation, it becomes 



The quantity / 2 A^4, however, is the moment of inertia, /, of the 

 cross section (article 20). Therefore 



(33) M = . 



^/ 



The right member of this equation, __ , is the resultant internal 



c> 



stress couple, and is called the moment of resistance of the beam. 

 Since e denotes the distance of the extreme fiber of the beam 



from the neutral axis, the ratio - is also a function of the shape 



e 



and size of the cross section, and is therefore called the section 

 modulus. Let this section modulus be denoted by Z. Then Z= y 



P 



and the fundamental formula becomes 



(34) M = pZ. 



Since this is an equality between the resultant external moment M 

 and the product of the working stress p by the section modulus Z, 

 it expresses the fact that the strength of a beam depends jointly on 

 the shape and size of the cross section and the allowable stress for 

 the material. 



35. Calculation and design of beams. For a beam of given size 

 and loading the maximum external moment M, acting at any point 

 along the beam, is first determined by the methods explained in 



