FLAT PLATES 



139 



If the load is uniformly distributed over the entire plate, then 

 = r and P = 7rr 2 w, where w is the load per unit of area. In this 

 case formula (195) becomes 



HH' 



which agrees with the result of the preceding article. 



86. Dangerous section of elliptical plate. Consider a homogeneous 

 elliptical plate of semi-axes a and b and thickness h, and suppose 

 that an axial cross is cut out of the plate, composed of two strips 

 AB and CD, each of unit width, in- 

 tersecting in the center of the plate, 

 as shown in Fig. 102. 



Now suppose that a single concen- 

 trated load acts at the intersection 

 of the cross and is distributed to the 

 support in such a way that the two 

 beams AB and CD each deflect the 

 same amount at the center. Since 

 AB is of length 2 a, from article 40, equation (54), the deflection 



P(2a 



From symmetry, the reac- 



at the center of AB is Z> = 



4SUI 



tions at A and B are equal. Therefore, if each of these reactions is 

 denoted by R^ 2R 1 = P and, consequently, 





BJSI 



Similarly, if R z denotes the equal reactions at C and Z>, the deflec- 



tion D Z of CD at its center is 



R 



If the plate remains intact, the two strips AB and CD must deflect 

 the same amount at the center. Therefore D 1 = D 2 , and hence 



(197) 



