Geometric nonlinear time response analysis of shell structures by advanced finite elements

The accurate prediction of the in-service nonlinear response of highly flexible structures in the geometrically nonlinear regime are of paramount importance for the design and failure evaluation, for example, of elastic deployable space systems, antennas, high aspect ratio wings or smart devices. The use of advanced simulation tools is eventually needed whenever detailed analysis, e.g. large deflection response or internal stress state evaluation, of such structures are demanded. Nevertheless, the available methodologies may be inappropriate either because simplified kinematics or approximated theories of structures are employed. In this work, a full geometric nonlinear refined shell theory is developed and coupled with a direct integration scheme to analyze the time response analysis of highly flexible shell structures. The Carrera Unified Formulation (CUF) is utilized along with a recursive notation to implement two-dimensional shell structural theories with arbitrarily rich kinematics. Indeed, low- to high- order, and eventually layer-wise, shell models can be implemented with CUF by expanding the thickness variables using generalized functions. The full nonlinear geometrical relations are derived for a doubly curved shell in the orthogonal parallel curvilinear coordinate system, whereas linear elastic materials are addressed in the proposed research. Thanks to CUF, and by using the principle of virtual work, the nonlinear governing equations are set in a total Lagrangian framework and are expressed in terms of the fundamental nuclei of the finite element arrays. The analytical expressions of the secant and the tangent stiffness matrices of the unified shell element are given for completeness purposes via the CUF and the three-dimensional Green-Lagrange strain components. Hence, a path-following Newton–Raphson linearization scheme based on the arc-length constraint is used to solve the full geometrically nonlinear problem. On the other hand, time response or dynamic response problems are addressed by employing the well established Newmark direct integration scheme. The proposed numerical assessments include, but are not limited to, the dynamic analysis and the large deflection response of doubly curved shells, single curvature structures and axisymmetric cylinders. The formulation is demonstrated to be effective for the time-varying loading response analysis of both metallic and composite shells. In particular, by employing the hierarchical characteristics of the CUF, the effect of different kinematics approximations is investigated, from low- order equivalent single layer structural models to layer-wise ones. The solutions obtained with the proposed approach are compared against the results available from the literature and from commercial finite element software tools, which may be ineffective in the case of large rotations or whenever detailed internal stress states are of interest for the analyst