TY - JOUR
T1 - Displacement-based formulation of Koiter's method
T2 - Application to multi-modal post-buckling finite element analysis of plates
AU - Castro, S. G.P.
AU - Jansen, E. L.
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
KW - Buckling
KW - Classical formulation
KW - Displacement-based
KW - Finite elements
KW - Koiter method
KW - Plate
KW - Post-buckling
UR - http://www.scopus.com/inward/record.url?scp=85096444618&partnerID=8YFLogxK
U2 - 10.1016/j.tws.2020.107217
DO - 10.1016/j.tws.2020.107217
M3 - Article
AN - SCOPUS:85096444618
SN - 0263-8231
VL - 159
JO - Thin-Walled Structures
JF - Thin-Walled Structures
M1 - 107217
ER -