136 



STRENGTH OF MATERIALS 



Now suppose a diametral section of the plate taken, and regard 

 either half of the plate as a cantilever (Fig. 97). Then if / N the 

 radius of the plate, the total load on this semi- 

 circle is - w t and its resultant is applied at 

 the center of gravity of the semicircle, which is 



at a distance of from AB. The moment of 

 3 TT 



this resultant about the support AB is therefore 



Similarly, the resultant 



FIG. 97 



rr 2 4 r 



w --- 



2 STT 



or 



3 



Trr 2 



of the supporting forces at the edge of the 

 plate is of amount w, and is applied at the center of gravity of 



2 2r 



the semi-circumference, which is at a distance of - - from . I /'. 



7T 



moment of this resultant about AB is therefore - 



7T 



Hence the total external moment M at the support is 



2 r*w r*w 

 3 3 



Now assume that the stress at any point >t' the plate is independ- 

 ent of the distance of this point from the center. Under this arbi- 

 trary assumption the stress in the' plate is given by the fundamental 

 formula in the theory of beams, namely, 



M. 



, t - r iv r w 

 M = T*W 



If the thickness of the plate is denoted by //, then, since the breadth 



of the section is b = 2 r, 



h 



Consequently, 



whence 

 (83) 



bit* rh* 



TTT = -r- and 



Me _ 3 2 

 / rh* 

 6 



-- 



