62 MOMENT OF INEETIA. [CHAP. VI. 



integral of this quantity will be the sum of the moments of all 

 the horizontal elastic forces in the cross-section round the neu- 

 tral axis, that is, equal to the moment of rupture of the section 

 in question. We have therefore 



For a rectangular cross-section, for instance, ft is constant 

 and equal to the breadth b. Representing the depth by d we 



b d* s 



have M = - , or if we make c the distance of the extreme 

 12 c 



fibres = -5 



sbd* 

 M = 



from which M being known, as also s, if we assume b we can 

 find d or the reverse. 



The integral //? y^ d y is the moment of inertia of the cross- 



section, and may be defined as the sum of the products obtained 

 by multiplying the mass of each elementary particle by the 

 square of its distance from the axis. [See Supplement to Chap- 

 ter VII., Art. 10.] 



From the above, we see its importance in determining the 

 strain at any distance from the neutral axis, or in proportioning 

 the cross-section, so that the resulting strain shall be less than 

 a given quantity at any point. We see also that for a rectan- 



b d 3 

 gular cross -section the moment of inertia is -yr-, where b is the 



breadth and d the depth. 



53. Graphical Determination. We have already seen 

 that the moment of a force, as P t (PI. 6, Fig. 20) with reference 

 to any point, as 0, is given by the ordinate n m multiplied by 

 the constant H (Art. 38). The ordinate n m then represents 

 the product of P! multiplied by the horizontal distance of b 



from n. But the area of the triangle bnmiBmny.^ b n = 



PI x -7 b n*, that is, the area of the triangle b n m represents 



"A 



one-half the moment of inertia of PI with respect to o. Just 

 as the exterior ordinates of the equilibrium polygon have been 

 shown to have a certain significance, and to represent the mo- 



