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It seems to me (David Bushnell) that the “m” numbers (axial half-waves?) do not correspond to the buckling mode shapes shown here. The top looks like m=13 axial half-waves; the middle looks like m - 1 axial half-wave; the bottom looks like m = 3 axial half-waves. Probably “m” does not refer to the number of axial half-waves.
FROM:
Michael Joachim Andreassen, “Distortional mechanics of thin-walled structural elements”, Ph.D. dissertation, Civil Engineering Dept., DTU Byg, Technical University of Denmark, September 2012
ABSTRACT: In several industries such as civil, mechanical, and aerospace, thin-walled struc- tures are often used due to the high strength and effective use of the materials. Because of the increased consumption there has been increasing focus on opti- mizing and more detailed calculations. However, finely detailed calculations will be very time consuming, if not impossible, due to the large amount of degrees of freedom needed. The present thesis deals with a novel mode-based approach concerning more detailed calculation in the context of distortion of the cross section which model distortion by a limited number of degrees of freedom. This means that the classical Vlasov thin-walled beam theory for open and closed cross sections is generalized as part of a semi-discretization process by including distortional displacement fields. A novel finite-element-based displacement ap- proach is used in combination with a weak formulation of the shear constraints and constrained wall widths. The weak formulation of the shear constraints enables analysis of both open and closed cell cross-sections by allowing constant shear flow. Variational analysis is used to establish and identify the uncoupled set of homogeneous and non-homogeneous differential equations and the related solutions. The developed semi-discretization approach to Generalized Beam Theory (GBT) is furthermore extended to include the geometrical stiffness terms for column buckling analysis based on an initial stress approach. Through varia- tions in the potential energy a modified set of coupled homogeneous differential equations of GBT including initial stress is establish and solved. In this context instability solutions are found for simply supported columns and by solving the reduced order differential equations the cross-section displacement mode shapes and buckling load factor are given. In order to handle arbitrary boundary conditions as well as the possibility to add concentrated loads as nodal loads the formulation of a generalized one- dimensional semi-discretized thin-walled beam element including distortional contributions is developed. From the full assembled homogenous solution as well as the full assembled non-homogeneous solution the generalized displacements of the exact full solution along the beam are found. This new approach is a considerable theoretical development since the ob- tained GBT equations including distributed loading found by discretization of the cross section are now solved analytically and the formulation is valid with- out special attention and approximation also for closed single or multi-cell cross sections. Furthermore, the found eigenvalues have clear mechanical meaning, since they represent the attenuation of the distortional eigenmodes and may be used in the automatic meshing of approximate distortional beam elements. The magnitude of the eigenvalues thus also gives the natural ordering of the modes. The results are compared to results found using other computational meth- ods taking distortion of the cross section into account. Thus, the results are compared to results found using the commercial FE program Abaqus as well as the free available software GBTUL and CUFSM concerning conventional GBT and the finite strip method, respectively. Reasonable matches are obtained in all cases which confirm that this new approach to GBT provides reasonable re- sults with a very small computational cost making it a good alternative to the classical FE calculations and other available methods.
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