Buckling and stability problems for thin shell structures using high performance finite elements

A hierarchical co-rotational theory for analysis of geometrically non-linear shell structures has been developed that addresses attributes identified as desirable: self-equilibrium, consistency, invariance, symmetrizability and element-independence. The unified formulation recovers different existing co-rotational formulations by making certain simplifying kinematic and static assumptions. This unification offers additional flexibility to finite element developers in that tradeoffs between simplicity, robustness and generality can be more clearly understood.;The nonlinear response of the co-rotational finite element models is obtained by incremental/iterative continuation methods. The equilibrium-path-following algorithm combines a standard arc length predictor phase with two alternative versions of a true-Newton corrector phase: the normal-plane corrector of Riks-Wempner and the orthogonal trajectory accession corrector proposed by Fried. The algorithm is treated with scaling techniques that aim to make the solution algorithm performance insensitive to discretization changes. Numerical experiments indicate that the orthogonal trajectory corrector in general outperforms the normal plane corrector in terms of robustness and allowance of larger stepsizes when tracing smooth response paths.;A modification of the predictor-corrector continuation algorithm to detect and handle traversal of bifurcation points has been developed. The modified algorithm relies on linearized buckling analysis carried out at two "bracketing" configurations in the neighborhood of the bifurcation point. The estimated buckling mode is used to initiate branch switching into the outgoing (secondary) path. The normal plane corrector constraint is modified to avoid the "switch-back" to the incoming (primary) path. This modification has proven to be robust in handling symmetric bifurcation points in the test problems reported here.;A new four-noded quadrilateral shell element has been developed based on the Assumed Natural-coordinate Deviatoric Strains formulation. This element is derived with reference to a flat geometry defined by the medians of the generally-warped quadrilateral. Projector matrices are used to fulfill self-equilibrium conditions in the warped geometry. The numerical results indicate that the new quadrilateral element delivers modeling accuracy similar to that of existing Free Formulation elements, but without the burden of numerical inversion to form the higher order stiffness.