Art. 72. 



MOMENTS OF INERTIA. 



103 



The same result may be gotten as above by integrating 



d+vi 

 IQ = ( v' 2 dA, and these operations are applicable to any kind 



J di 

 of area, Figs. 71 to 74, for example. 



Moments of inertia of irregular areas may be gotten ap- 

 proximately by dividing them into a number of rectangles, or 

 mechanically by means of a moment planimeter. The moments 

 of inertia of angles, channels, Z bars, etc. should be taken from 

 the handbooks. 



To find the moment of inertia of the cross section shown in 

 Fig. 72, about its neutral axis NN, equations (15) and (16) are 

 used, the area being treated as two rectangles as was done in 

 finding d. 



Ad 2 = 3X3.Q9 2 +8X 1.16 8 = 39.41 



/for axis NN=82.U in.* 

 For the section of Fig. 73 refer to Cambria. 



I=s X I + 12 3 (for plate) + 2 X 2.58 (for angles) 



=-72.0+5.16 = 77.16 



Ad~ = 6X(^Vs 0.75 2.39) 2 +7.5X2.39 2 == 96.30 

 / for axis NN =173.46 in. 4 



It is often more convenient to find / by subtraction than by 

 addition ; thus for a ring whose external radius is r and internal 

 radius r 2 , 1= y 4 TT (r 4 r 2 4 ). 



73. Oblique Loading. In what precedes it was assumed 

 that the neutral axis is perpendicular to the plane of the bending 

 moment. In Fig. 67 the plane of the bending moment cuts the 

 cross section in W, an axis of symmetry. On account of this 

 symmetry, the plane of the resisting moment is evidently coinci- 

 dent with that of the bending moment, and it follows that the 

 neutral axis is perpendicular to this plane, since the moment of 

 resistance is a moment about it. This may also be true if the 

 section has no axis of symmetry as is shown by the following 

 consideration. In Fig. 67, the origin of a rectangular sytem of 

 coordinates is at the center of gravity of the section and the axis 

 of Z is coincident with the neutral axis. Since the external forces 

 lie in the plane whose trace is the axis of V, the sum of their 



