38 GRAPHIC STATICS 



Moment of Inertia of Areas. The moment of inertia of an 

 area about an axis in the same plane is equal to the summation of the 

 products of the differential areas which compose the area and the 

 squares of the distances of the differential areas from the axis. 



The moment of inertia of an area about a neutral axis (axis 

 through center of gravity of the area) is less than that about any parallel 

 axis, and is the moment of inertia used in the fundamental formula for 

 flexure in beams 



SI 

 M = 



where 



M = bending moment at point in inch-pounds ; 



= extreme fibre stress in pounds ; 



/ = moment of inertia of section in inches to the fourth power; 



c = distance from neutral axis to extreme fibre in inches. 



An approximate value of the moment of inertia of an area may 

 be obtained by either of the preceding methods by dividing the area 

 into laminae and assuming each area to be a force acting through the 

 center of gravity of the lamina, the smaller the laminae the greater the 

 accuracy. The true value may be obtained by either of the above 

 methods if each one of the forces is assumed to act at a distance from 

 the given axis equal to the radius of gyration of the area with reference 

 to the axis, d = i/a z -{- r 2 , where a is the distance from the given axis to 

 the center of gravity of the lamina and r is the radius of gyration of the 

 lamina about an axis through its center of gravity. If A is the area 

 of each lamina the moment of inertia of the lamina will be 



/ = A d 2 = A a 2 + A r 2 = A a* + I cg> 



which is the fundamental equation for transferring moments of inertia 

 to parallel axes. 



