+ ^ [V + Wg] + ^ [ V ,-W] [ ..[13] 



Us „ w z w s w,v u*w 



_ UtVs WW, Vwo, W,W aS UgW, 



R'RRR + R* 

 We shall simplify these complicated expressions to the following 





[14] 



For the changes in curvature, we shall use the quantities pre- 

 viously used by Donnell (8) for perfect cylinders, viz. 



X, - W,„ xe 



R 



[15] 



The work done W D by the uniform external pressure acting on 

 all sides of the cylinder is given by the product of the pressure 

 and the change in volume of the cylinder. Thus there follows 

 (16) 



'-rrt'-? 



RU. 

 2 



+ -r VJLW + w.) — \ U(W, + w„ z ) 



+ t u '( w > + «*>«) + 4 Vi'w + «*>) ■ 



Wvx , 

 R 



— + » (W r + IOo) . 



V 1 



- j tff,. - j c/,'y, - i c/'7,J ijdide 



,, Jo 



E/d0 



". IV 



/'2i 

 o U - d 



[16] 



We shall simplify this complicated expression for W D to the follow- 

 ing 



^ " '/."/.* [^-T-T 9 ]*"*-' • [171 



It is alBo convenient to consider the £/, V, and If displacements 

 to be made up of two parts 



U - U + u 



V - V + v 

 W - W + w 



.[18] 



where !7, P, and IP are the displacements which would occur in 

 the equilibrium problem of a perfectly circular cylinder under 

 uniform external pressure. 



The total potential of the Bystem U T is then obtained by add- 

 ing the extensional and bending-etrain energies of the shell and 

 subtracting the work done by the external pressure. In calculat- 

 ing the extensional energy, we retained the terms in u, v, to, too 

 through the second order and the terms in 0, V, W (directly pro- 

 portional to the applied pressure) through the first order only; 

 also, we neglected the effect of the deflection of the shell between 

 supports on the displacements and stresses of the perfect cylinder. 



Variation of U T with respect to u, v, and u> then gives the dif- 

 ferential equations of the problem. Further manipulation of these 

 equations results in the following 



V*u 



R> ' 



V'v 



(2_+j0 

 R' 



+ h 



(o) 



(6) 



2r, 



+ -JT (W + U»o),8 + — 



R 



i,(w + I0o)„ 



(to + 100)99 (c) 



[19] 



where i t , 9 e , and f,, are the membrane stresses which would 

 occur in a perfectly circular cylinder and which were assumed to 

 be constant in deriving the foregoing equations. 



For the case of uniform external hydrostatic pressure applied 

 on all sides of a perfectly circular cylinder, and neglecting the de- 

 flection of the shell between supports, there resultB 



pR 



■T' »■ 



pR 



2h ' 



T.9 - 



[20] 



Substituting the Relations ]20] into Equation [19c] yields 



Eh 

 DV*w + — - V _4 u>„ 



■a 



+ pR \~- (to + «,„), 



■p -0 



.[21] 



Now define a stress function of the total membrane stresses 

 F, such that 



** m r %* 



[22] 



h " R' ' h ' '" h 



Using Equation [14] and the stress-strain relations it is then 

 easy to show that 



,W SS 



I R " R' fl» 



2W l) w tl , s W„wm W m wo zz \ 



+ ^ «i rt) !231 



As we are interested here in the linear problem, and as we have 

 previously neglected the deflection of the shell between supports, 

 Equation [23] reduceB to 



V'F - — 



E 



.[24] 



Equations [19a], [196], [21], and [24] together with the appro- 

 priate boundary conditions define the problem. 



i 



