26 



RESISTANCE OF MATERIALS 



Since an area is not a solid and therefore does not possess inertia, 

 the shape factor /should not be called moment of inertia, but rather 

 the second moment of area, since the distance y occurs squared. 



To compute / for 

 any plane area, divide 

 the area up into small 

 elements kA (Fig. 21). 

 Then the first (or 

 static) moment of each 

 element with respect 

 to any axis 00 is y&A, 

 where y denotes the 

 distance of this ele- 

 ment from the given 

 axis. Now erect on AA 

 as base a prism of 



height y. If this is done for every element of the plane area, the 

 result will be a solid, or truncated cylinder, as shown in Fig. 21, 

 the planes of the upper and lower bases intersecting in the axis 

 00 at an angle of 45. 



Let V denote the volume of this moment solid, as it will be called, 

 and y Q the distance of its centroidal axis from 00. Then, by the 

 theorem of moments, 



,**- 



V 



FIG. 21 



Since AF=#A^4, the 

 right member becomes 



Hence 



(22) I=Vy Q . ^ \^ u -5-.. 



21. I for rectangle. FIG. 22 



Let it be required to 



find / for a rectangle of breadth b and height h with respect to an 

 axis through its centroid, or middle point, and parallel to the base 

 (Fig. 22). The moment solid in this case consists of a double wedge, 



