of an Elastic Plate under Normal Pressure. 101 



parallel to the middle surface of the plate, the equilibrium of 

 the element dx x dy (fig. 1) requires that 



2A(P 1 "_P/) t / i/ + 2A(S 2 "-S 2 , >^ = 



and 2h(F/'-¥ 2 ')dx+2h(S 1 ' , -S 1 ')di/=0, 



But P/'-P/=^<fo, 



ox 



b 2 — S 2 = -^ dy* 



Therefore the above equations give 



SPi + |S =0 (8) 



8=-E^-, . . : . . (10) 



and ^ + |§=0 (9) 



dy o« 



If we now choose a function <j>, such that 



oxoy 



then equations (8) and (9) show that 



F I= Ep ; P 2 =E0 (11) 



Writing ^-. r for n in equation (7), and then elimin- 

 ating u and v from (5), (6), (7), we get 



That is, 



B^> 9 _B*i_ , ^ = /JB^\ 2 _B^ ^ 

 Ba' 4 d;r% 2 3/ Vd^'W Ba; 2 " By 



B 2 B 2 



Writing V 2 for 4^ + =^-9 5 tnis last equation may be 



u;„„ & B# 2 Br 



written 



v *-(b^)-b^'V ' ' ' (1) 



This is one of the equations of equilibrium of the plate. 

 The other equation is a modified form of Poisson's equation, 

 which form we shall now find. 



