Buckling analysis of rotationally periodic structures using shell element with relative degrees of freedom

By considering the characteristics of deformation of rotationally periodic structures subjected to rotationally periodic loads, the periodic structure is divided into several identical substructures in this paper. If the structure is really periodic but not axisymmetric, the number of the substructures can be de=ned accordingly. If thestructureis axisymmetric (special in thecaseof thepe riodic), thestructurecan be divided into any number of substructures. It means, in this case, the number of substructures is independent of the number of buckling waves. The degrees of freedom (DOFs) of joint nodes between the neighbouring substructures are classi=ed as master and slave ones. The stress and strain conditions of thewholestructureareobtaine d by solving thee lastic static equations for only onesubstructureby introducing the displacement constraints between master and slave DOFs. The complex constraint method is used to get thebifurcation buckling load and modefor thewholerotationally periodic structureby solving thee igenvalueproble m for only onesubstructurewithout introducing any additional approximation. Finite element (FE) formulation of shell element of relative degrees of freedom (SERDF) in the buckling analysis is then derived. Di?erent measures of tackling internal degrees of freedom for di?erent kinds of buckling problems and di?erent stages of numerical analysis are presented. Some numerical examples are given to illustrate the high e@ciency and validity of this method.