FLAT PLATES 137 



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 (83) is therefore well established 

 both theoretically and practically. 



Problem 140. The cylinder of a locomotive is 20 in. internal diameter. What 

 must be the thickness of the steel end plate if it is required to withstand a pres- 

 sure of 160 lb./in. 2 with a factor of safety of 6 ? 



Problem 141. A circular cast-iron valve gate J in. thick closes an opening 6 in. 

 in diameter. If the pressure against the gate is due to a water head of 150 ft., 

 what is the maximum stress in the gate ? 



117. Maximum stress in homogeneous circular plate under con- 

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

 aid 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- 



P 2r 



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



P w 

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



4r 

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



3 7T 



at the section is p r 2 Pr Pr / 2 



TT 3 TT TT y 3 r 



Assuming that the stress is uniformly distributed throughout the plate, 

 the stress due to the external moment M is given by the formula 



= Me 

 I 

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



rh* h 



/= _ and , = _. 



^/l_^Lo\^ 

 Me TT \ 3r/2 t 



/ yh? 



s* 

 whence 



