**2.1. Basic assumptions**

The basic assumption for thin plate are given by Kirchhoff for linear and by von Kármán and Marquerre for nonlinear thin plate theory. They made their assumption for isotropic material; lots of authors extended these assumptions for orthotropic or even for orthotropic multilayer thin plate [18, 48]. The assumptions are as follows:


*even if one characteristic root* 

of Jacoby matrix is equal or greater than unity in complex plane.

*in the complex plane outside the circle with radius equal to unity*.

*minimum value of the pulse load such that phase portrait is an open curve*.

dynamic buckling of plated structures can be found in [46, 47].

multilayer thin plate [18, 48]. The assumptions are as follows:

**2. Thin orthotropic plate theory** 

**2.1. Basic assumptions** 

criterion, the maximal radius *r*max calculated from characteristic root =a+jb (where j= 1 )

Therefore the criterion for thin-walled structures proposed by author [35] can be formulated as follows: *Thin-walled structures subjected to pulse loading of finite duration lose their stability* 

Teter [36, 37] in his works analysed the long columns with longitudinal stiffeners and basing on the phase portrait for dynamic response of these structures defined the following criterion: *The dynamic buckling load for the tracing time of solutions has been defined as the* 

The dynamic buckling problem has been well known in the literature for over 50 years and was the subject of numerous works [20, 24, 26-34]. The extensive list of work dealing with dynamic buckling can be found for example in the book edited by Kowal-Michalska [38] or written by Simitses [39] or Grybos [40]. It seems that the analysis of dynamic buckling of thin-walled structures, especially structures with flat walls is not sufficiently investigated. There is a lack of both single-and multimodal analysis of dynamic buckling of columns with complex cross-sections made of thin flat walls. The author of this paper decided to fill this gap presenting a method for the analysis of the local (single mode) and interactive (coupled mode – local and global) buckling of thin-walled structures subjected to pulse loading.

It should be mentioned that the presented method can be used only if the structures are in the elastic range. The case of dynamic buckling in elasto-plastic range including the viscoplastic effect has been investigated by Mania and Kowal-Michalska [25, 41-43]. In world literature it is also possible to find the paper dealing with the dynamic buckling of thin-walled structures subjected to combined load [44]. Czechowski [45] modelled the girder subjected to twist and bending considering only one plate subjected to shear and compression. The general summary showing which parameters have an influence on

The thin isotropic or orthotropic plates with constant or widthwise variable material properties are considered. The thin-walled beam-columns or girders composed of mentioned above plates are also analysed. In order to taken into account all buckling modes (global, local and their interaction) the plate model was adopted to the analysed structures.

The basic assumption for thin plate are given by Kirchhoff for linear and by von Kármán and Marquerre for nonlinear thin plate theory. They made their assumption for isotropic material; lots of authors extended these assumptions for orthotropic or even for orthotropic

*=a+jb of Jacoby matrix find for every time moment from 0 to 1.5Tp lies* 


Additionally, it is assumed that principal axes of orthotropy are parallel to the edges of analysed structures (plate, beam, column, beam-column or girder).
