FLAT PLATES 



137 



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

 either half of the plate as a cantilever (Fig. 101). Then if r is the 



2 



radius of the plate, the total load on this semicircle is -^-w, and 

 its resultant is applied at the center of gravity of the semicircle, 



which is at a distance of - from AB. The moment of this result- 



^ ?rr 2 4 r 2 r*w 



ant about the support AB is therefore - w , or - Simi- 



2i O 7T O 



larly, the resultant of the supporting forces at the edge of the 



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



2r 

 the semi-circumference, which is at a distance of - - from AB. The 



moment of this resultant about AB is therefore 



9 9 f 



. , or r 8 w. Hence the total external 



2 TT 

 moment M at the support is 



2 r s w r s w 



FIG. 101 



Now assume that the stress at any point of 

 the plate is independent 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, 



_Me 

 I 



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

 of the section is b = 2 r, 



Consequently, 



whence 

 (194) 



