100 Dr. J. Prescott on the Equations of Equilibrium 



Now let 



E = Young's modulus for the material of the plate, 

 cr = Poisson's ratio, 

 n = the modulus of rigidity, 

 2A = the thickness of the plate, 

 P l5 P 2 = the tensional stresses in the middle surface in the 

 directions of dx and dy, 

 S = the shear stress in the middle surface on the faces 

 on which T > 1 and P 2 act. 



The stresses P 2 , P 2 , S, are shown in fig. 1. It is under- 

 stood that, as dx and dy approach zero, the stresses Si', S 2 '> 

 S/', S 2 ", all approach the limit S. 



Fig. 1. 





j 



% 2 













D dx C 

 A B 



i.s 1 , 1 



pj 



P, 













s' 2 



fPi 



Now, assuming that the tensional stress in the e-direction 

 is negligible in comparison with P x and P 2 , the usual 

 equations of elasticity give 





— f 3« 9« 'dw 5«-"> 

 \ By B^ ~dx ' ~di/ / 



(5) 

 (6) 

 (7) 



Since the stresses P l5 P 2 , S, can be proved to be the mean 

 values of the stresses at x, y, from ~= — h to .s = .-f h 9 arid 

 since we are assuming that there are no external forces 



