CHAP. VI.] MOMENT OF INERTIA. 61 



CHAPTER VI. 



MOMENT OF INERTIA. 



52. THUS far we have seen that by the graphic method we 

 can in any practical case determine the moment of the exterior 

 forces acting upon a piece at any cross-section of that piece. 

 But the exterior forces give rise to and are resisted by molecu- 

 lar or interior forces. Now the moment of the exterior forces 

 being found, the cross-section of the piece at any point being 

 known, and one of the dimensions of this cross-section being 

 assumed, it is required to find the other dimension, so that the 

 strain per unit of area of cross-section shall be less than the 

 recognized safe strain of the material as found by experiment. 



The moment of the exterior forces at any cross-section we 

 call the moment of rupture ; and designate it by M. Let d = 

 the depth of cross-section.* 



y = the variable distance of any fibre above or below the 

 neutral axis. * 



ft = the breadth of the section at the distance y from the 

 neutral axis, and consequently a variable, except in the case of 

 rectangular sections. 



s = the horizontal unit strain exerted by fibres in the cross- 

 section at a given distance c from the neutral axis. 



Then since the fibres exert forces which are proportional tc 

 their distance from the neutral axis or to their change of length, 

 the unit strain in any fibre at a distance y from the neutral 



S* ?/ 



axis will be . Let the depth of this fibre be d y, then, since 



G 



the breadth of section is ft, the total horizontal force exerted 







by the fibres in the breadth ft, will be fty dy. The moment 







Q 



of this force about the neutral axis will be ft if d y, and the 



G 

 * Theory of Strains, Stoney, p. 43, Art. 67. 



