Displacement-based formulation of Koiter's method: Application to multi-modal post-buckling finite element analysis of plates. (February 2021) - Castro, S.G.P.; Jansen, E.L.
Abstract: Koiter's asymptotic method enables the calculation and deep understanding of the initial post-buckling behaviour of thin-walled structures. For the single-mode asymptotic analysis, Budiansky (1974) presented a clear and general formulation for Koiter's method, based on the expansion of the total potential energy function. The formulation from Budiansky is herein revisited and expanded for the multi-modal asymptotic analysis, of primordial importance in structures with clustered bifurcation modes. Given the admittedly difficult implementation of Koiter's method, especially for multi-modal analysis and during the evaluation of the third– and fourth–order tensors involved in Koiter's analysis; the presented study proposes a formulation and notation with close correspondence with the implemented algorithms. The implementation is based on state-of-the-art collaborative tools: Python, NumPy and Cython. The kinematic relations are specialized using von Kármán shell kinematics, and the displacement field variables are approximated using an enhanced Bogner-Fox-Schmit (BFS) finite element, modified to reach third-order interpolation also for the in-plane displacements, using only 4 nodes per element and 10 degrees-of-freedom per node, aiming an accurate representation of the second-order fields. The formulation and implementation are verified by comparing results for isotropic and composite plates against established literature. Finally, results for multi-modal displacement fields with up to 5 modes and corresponding post-buckling factors are reported for future reference. Graphical abstract: Image 1 Highlights: Derivation of a displacement-based formulation of Koiter's method for multi-mode analysis of isotropic and composite plates. Achievement of a transparent correspondence between formulation and numerical implementation of Koiter's method. Development of an enhanced Bogner-Fox-Schmit (BFS) finite element with enriched in-plane displacements. Verification by comparison with literature in single-mode and multi-mode initial post-buckling analysis. Calculation and report of reference values for multi-modal expansion coefficients and second-order fields.