138 RESISTANCE OF MATEKIALS 



Foppl has shown that the arbitrary assumption made in deriving 

 this formula can be avoided, and the same result obtained, by a more 

 rigorous analysis than the preceding, and Bach has verified the 

 formula experimentally. Formula (194) is therefore well established 

 both theoretically and practically. 



85. Maximum stress in homogeneous circular plate under con- 

 centrated load. Consider a flat, circular plate of homogeneous mate- 

 rial, and suppose that it bears a single concentrated load P which is 

 distributed over a small circle of radius r Q concentric with the plate. 

 Taking a section through the center of the plate and regarding either 

 half as a cantilever, as in the preceding article, the total rim pres- 



sure is , and it is applied at a distance of -- from the center. The 



P " 

 total load on the semicircle of radius r Q is , and it is applied at a dis- 



4 T 



tance of - from the section. Therefore the total external moment M 

 3 TT 



at the section is Pr _ 2Pr _ Pr L _ 2r \ 



~~~^~' STT == ~^\ "37/ 





Assuming that the stress is uniformly distributed throughout the 

 plate, the stress due to the external moment M is given by the 

 formula ^ 



P= -. 



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



rh 8 h 



I= _ and e = _. 



Therefore , 2 



_Me_ TT \ 3r/2 



7" fJ) 



whence 



(195) p = 



If r = 0, that is to say, if the load is assumed to be concentrated 

 at a single point at the center of the plate, formula (195) becomes 



(196) 



