Bart derived the following relevant partial differential equations in Refer- 

 ence 44, using the principle of minimum potential energy: 



4 Eh -4 



DV w + — - V w, + pR 



Z xxxx 



R 



h \ 1 / 



Hw + w ;, + — —(w+w ) , ^- 

 c o XX Z o 0G 



- R _ 



p = (119) 



^4 1 



Vu = - 



l/w 



» ~ w > 



XXX Z X09 



R 



IZO) 





[Z+l/)w, „ + — rw.„„„ 

 'xx0 Z '000 



R 



(IZl) 



^4^ E 



VF = - -w. 



R XX 



IZZ) 



where 



A / 2 , Z \Z 



ax^ R^ 90^' 



4 -4 



V V f = f 



F is the stress function of the total membrane 



stresses, and 

 Wq the initial out-of-roundness, is + inward toward 



the BJcis of the cylinder. 

 The comma after the dependent variables denotes partial differentations 

 with respect to the independent variable which follows, so that 



76 



