Plates of Variable Thickness. 957 



Substituting (4) and (5) in (6) and (7), the terms in £~ 2 and 

 t' 1 vanish. From the constant term in (6) we obtain 



B 2 U 2 3X + 4/* 





d* 2 * 



whence, using (B), 



2(\ + 2,u) 





Also, the constant term in (7) vanishes. 



Furthermore, the surface tractions must vanish on the 

 free surfaces z = x, z= — a ; hence (Love, p. 76) the following 

 equations must hold on these surfaces : — 



X v = X* cos (#, v) + Xy cos (y, v) + X z cos (~, v) — 0, \ 

 Y v = Y z cos (a?, i/)+Y y cos(y, v) + Y*cos (*, v) = 0, I (8) 

 Z„ = Z z cos (a?, v) + Z y cos (?/, v) + Z s cos (z, v) — 0. ) 



If we consider a tangent plane at an arbitrary point of the 

 surface z = a, the direction of the normal is denoted by v, 

 and (8) gives the tractions across the tangent plane in terms 

 of the stress-components across planes parallel to the rectan- 

 gular coordinate planes. For cylindrical coordinates, we 

 have (Love, pp. 100, 141) : — 



X x =\A + 2^ xx = \A + 2^^, 

 Yy=XA + 2 t ie yy =\^ + 2fJL — , 



Z S = \A + 2 A 6^ = XA + 2^ 1 |^, 



t QZ 



Y s = Zy = jLl<?yz = 0. 



iBu , a. 



Xy = Yx = iie xy — 0, 





r her< 



r to- 



To determine the direction cosines, we return to our 

 original variables and observe that tan 6 — ta,'. 



