Nonlinear Asymmetric Dynamic Buckling of Isotropic/Laminated Orthotropic Spherical Caps

Abstract

Here, the nonlinear dynamic behavior of clamped isotropic/laminated composite spherical caps under suddenly appliedloadsisstudiedusingathree-nodedaxisymmetriccurvedshellelementbasedone eldconsistencyapproach. The formulation is based on e rst-order shear deformation theory, and it includes the in-plane and rotary inertia effects.Geometricnonlinearity is introduced in theformulationusing von Karman’ sstrain-displacementrelations. The governing equations obtained are solved employing the Newmark’ s integration technique coupled with a modie edNewton‐ Raphsoniteration scheme.Theload beyond which themaximum averagedisplacementresponse shows signie cant growth in the time history of the shell structure is taken as dynamic buckling pressure. The present model is validated against theavailableanalytical solutions and also with the results evaluated using threedimensional e nite element method. A detailed parametric study is carried out to bring out the effects of shell geometries, orthotropicity, and the number of layers in the cross-ply laminates on the axisymmetric/asymmetric dynamic buckling load of shallow spherical shells. I. Introduction T HIN spherical shells form an important class of structural components, with many signie cant applications in engineering e elds. These shells subjected to dynamic load could encounter dee ections of the order of the thickness of the shell. The dynamic response of such shells can lead to the phenomenon of dynamic snapping or dynamic buckling. Because these kinds of responses cannot be determined accurately using small displacement theory, nonlinear dynamic analysis is required, and such study has received considerable attention in the literature. However, most of the available works are related to axisymmetric behavior of homogeneous, isotropic, or single-layered orthotropic spherical shells subjected to the step pressure load of ine nite duration. The present tendency to use e ber-reinforced composite materials for the structural components necessitates the analysis of shells made up of layers of such materials, leading to anisotropic behavior. Moreover, quite often, the asymmetric modes of these shells might be excited as a result of the introduction ofslight deviation in perfectaxisymmetric loading, geometric imperfection, and/or initial displacement/velocity to the shells. The anisotropic material properties coupled with the asymmetric structural behavior render the failure analysis of these shells quite complex. Hence, there is a growing appreciation of the importance of studying the dynamic response, in particular, dynamic bucklingoflaminatedcompositesphericalshellsandhasconstituted a major e eld of research in structural mechanics. First, a brief review of important contributions to the axisymmetric dynamic snap-through buckling of spherical case is presented here. The analysis of isotropic shallow spherical shells has been carried out by Budiansky and Roth, 1 Simitses, 2 Huang, 3 Stephens andFulton, 4 BallandBurt, 5 andStricklinandMartinez. 6 Budiansky and Roth 1 have employed the Galerkin method, whereas Simitses 2 adoptedtheRitz ‐Galerkinprocedure.Ae nitedifferenceschemehas been introduced in the method of solution by Huang, 3 Stephens and Fulton, 4 andBallandBurt, 5 whereasStricklinandMartinez 6 utilized