MI;, i; v. SOITTHWKI.I. ON THK KKNKKAL TIIKOUY OF ELASTIC STABILITY. ijy 



:ind we may assume that the system i.f strain which is introduced at collapse will lie 

 two-dimensional, so that 



^ = ^=0, - = const. (19) 



z 



The third equation <>f utmtnil stability (for the direction Oz) is then satisfied 



G* 

 identically, and the other two equations become (if we neglect terms of order -^u' ...J 



rn-2 



and 



m-2 



m-2 



m-2 



4C 



G_ 

 4C 



k . . (20) 



Let us assume a solution of the form 



/ = 2[V. cos a 



"1 

 J ' 



(21) 



where U. and V. are functions of y only. It is easy to show that this assumption as 

 to the phase-relation of u' and v' is justified. We have then 



and 



-. . . (22) 



The solution of these equations is of the form 



U. = (P// + Q) sinh ay + (R// + S) cosh ay, 



G 



3i-4 G\ 

 m-2 4C \ R 



__G la 



- > cosh a?/, 



. . (23) 



-2 4C / 



where P, Q, R, S are constants. 



The boundary conditions now demand attention. It is clear that the stresses 

 introduced by ', t 1 ', w' must vanish at the surfaces of the plate. Hence these 

 surfaces will still be planes of principal stress, and, moreover, the normal stress upon 



