426 



critical load is exceeded. 



Although the formulation of the dynamic theory of buckling is 

 somewhat more involved than that of the usual static theory and yet leads 

 to the same results for static loading, it has several advantages. First, 

 it may be generalized in a natural way to yield conditions of stability 

 under dynamic loading. Second, it shows that the critical pressure of 

 the static theory really defines the limit of stability for the structxxre 

 under small perturbations. This is important in connection with the 

 nonlinear buckling theory of von Karman, ■2/ since it shows that, even if 



V von Karman, Enz. d. Math. IViss., Bd. IV, U?.^ (1910) 



Friedricks and Stoker, Am. Jour. J.'ath., vol. LXIII, 839 (19A1) 



static modes of buckling are possible at lower loads than the critical 



load of the linear theory, they could only be excited by large perturbations, 



IVe consider a cylindrical shell of radius R, thickness a, and 

 length L, closed at both ends and in equilibrium with a uniform external 

 hydrostatic pressure P. 'Te denote by u, v, w the ajxial, circumferential, 

 and inwardly directed radial components of the supplementary displacements 

 produced by additional loading due to an excess pressure p(8,2, t) where 

 and z are cylindrical coordinates specifying position on the surface of 

 the undeformed shell, and t is the time. Buckling can of course be excited 

 by out-of -roundness as well as by a supplementary load. However, the 

 excitation function for out-of-roundness can be regarded as an equivalent 

 excess pressure. Supplementing the Epstein^ shell equations with the 



ijj Epstein, Tfork cited in footnote 2, 



