24 



RESISTANCE OF MATERIALS 



Fie. 17 



18. Axis of symmetry. If a figure has an axis of symmetry, then 

 to any element of the figure on one side of the axis there must 



correspond an equidistant 

 element on the opposite side, 

 and since the moments of 

 these equal elements about 

 the axis of symmetry are 

 equal in amount and oppo- 

 site in sign, their sum is 

 zero (Fig. 17). Since the 

 moment of each pair of ele- 

 ments with respect to the axis 

 of symmetry is identically 

 zero, the total moment is also zero, and hence the centroid of the 

 figure must lie on the axis of symmetry. 



When a figure has two or more axes of symmetry, their inter- 

 section completely determines the centroid. 



19. Centroid of composite figures. To determine the centroid 

 of a figure made up of several parts, the centroid of each part 

 may first be determined separately. Then, assuming that the 

 area of each part is concentrated at its centroid, the centroid of 

 the entire figure may be deter- 

 mined by equating its moment to 



the sum of the moments of the 

 several parts. 



To illustrate this method, let it 

 be required to find the centroid of 

 the I -shape shown in Fig. 18. Since 

 the figure has an axis of symmetry 

 MN, the centroid must lie some- 

 where on this line. To find its 

 position, divide the / into three 

 rectangles, as indicated by the 

 dotted lines in the figure. The 

 centroids of these rectangles are at their centers , >, c. Therefore, 

 denoting these three areas of the rectangles by A, B, (7, respectively, 

 and taking moments with respect to the base line, the distance of the 



To 



I 



i 

 i 



