Coupled Static and Dynamic Buckling Modelling of Thin-Walled Structures in Elastic Range Review of Selected Problems

Abstract

A review of papers that investigate the static and dynamic coupled buckling and post-buckling behaviour of thin-walled structures is carried out. The problem of static coupled buckling is sufficiently well-recognized. The analysis of dynamic interactive buckling is limited in practice to columns, single plates and shells. The applications of finite element method (FEM) or/and analytical-numerical method (ANM) to solve interaction buckling problems are on-going. In Poland, the team of scientists from the Department of Strength of Materials, Lodz University of Technology and co-workers developed the analytical-numerical method. This method allows to determine static buckling stresses, natural frequencies, coefficients of the equation describing the post-buckling equilibrium path and dynamic response of the plate structure subjected to compression load and/or bending moment. Using the dynamic buckling criteria, it is possible to determine the dynamic critical load. They presented a lot of interesting results for problems of the static and dynamic coupled buckling of thin-walled plate structures with complex shapes of cross-sections, including an interaction of component plates. The most important advantage of presented analytical-numerical method is that it enables to describe all buckling modes and the post-buckling behaviours of thin-walled columns made of different materials. Thin isotropic, orthotropic or laminate structures were considered.Key words: Interaction, Buckling, Thin-Walled Structures, FEM, Analytical-Numerical Method, Review COUPLED BUCKLING OF THIN-WALLED STRUCTURESThe theory of coupled or interactive buckling of thin walled structures has been already developed widely for over sixty years. Thin-walled structures, especially plates, columns and beams, have many different buckling modes that vary in quantitative and qualitative aspects. In these cases, nonlinear buckling theory should describe all buckling modes from global (i.e., flexural, flexural-torsional, lateral, distortional and their combinations) to local and the coupled buckling as well as the determination of their load carrying capacity taking into consideration the structure imperfection. Coupling between modes occur for columns of such length where two or more eigenvalues loads of a structure are nearly identical (Fig. 1). The local buckling takes place for the short columns. On the other hand, the long columns are subject to global buckling.The concept of coupled or interactive buckling involves the general asymptotic nonlinear theory of stability. Among all versions of the general nonlinear theory, the Koiter theory of conservative systems (Koiter, 1976; van der Heijden, 2009) is the most popular one, owing to its general character and development, even more so after Byskov and Hutchinson (1997) formulated it in a convenient way. The details descriptions of this method can be found in the monographs: van der Heijden (2009), Thompson and Hunt (1973) or Kubiak (2013). Applicability of an asymptotic expansion for elastic buck...

Numerical modelling of thin-walled Z-columns made of general laminates subjected to uniform shortening

“…The analytical-numerical method is based on Koiter's asymptotic method [13][14][15]. An eigenvalue problem of perfect structures and a post-buckling analysis were performed.…”

“…If the number of interacting buckling modes is one, the solution is a one-mode approach (i.e., uncoupled buckling). In this case, the dimensionless post-buckling equilibrium path can be written as [13]:…”

“…Finally, one can distinguish global buckling which is typical of long structures. The numerical modelling of the buckling behaviour of plate structures can be performed with the analytical-numerical method (ANM) [13][14][15] based on Koiter's asymptotic method (Fig. 1a) or the finite element method [16][17] (Fig.…”

Catastrophic Influence of Global Distortional Modes on the Post-Buckling Behavior of Opened Columns

“…Expressions for coefficients: a o , a r , a pqr , b rrrr are calculated by known formulae [ 1 , 2 , 3 , 32 , 37 ]. The post-buckling coefficients a pqr depend only on the buckling modes, whereas the coefficients b rrrr also depend on the second-order field.…”

“…The key problem consists of a selection of eigenmodes, which are accounted for in the multimodal analysis. In the literature, it is easy to find many monographs, e.g., [ 1 , 2 ], and papers, e.g., [ 3 , 4 , 5 ], presenting the state of the art in the field of the nonlinear stability of thin-walled structures. Despite this fact, these issues have not been well recognized so far and are the subject of interest for numerous researchers.…”