**1.1. Static buckling**

The buckling and postbuckling of thin-walled structures subjected to static load have been investigated by many authors for more than one hundred years. To the group of precursors of the investigation on the stability of thin-walled structures problem should be included following scientists: Euler [1], Timoshenko [2] and Volmir [3].

© 2012 Kubiak, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This chapter considered the thin plate or thin-walled structures composed of flat plates. Such structures have a various of buckling modes which can differ from one another both in quantitative (e.g., by the number of half-waves) and in qualitative (e.g., by global and local buckling) way.

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 221

examined but the concept of dynamic stability is broader and applies also to the stability of motion, which for thin-walled structures can be found for example in [22, 23]. The dynamic buckling occurs when the loading process is of intermediate amplitude and the pulse duration is close to the period of fundamental natural flexural vibrations (in range of milliseconds). In such case the effects of dumping are neglected [24]. Damping neglecting is

It should be noted that dynamic stability loss may occur only for structures with initial geometric imperfections; therefore the dynamic bifurcation load does not exist. For the ideal structures (without geometrical imperfection) the critical buckling amplitude of pulse loading tends to infinity [26]. The dynamic buckling load should be defined on the basis of

The precise mathematical criteria were formulated for structures having unstable postcritical equilibrium path or having limit point [26, 27]. But for the structures having stable postbuckling equilibrium path (thin plate, thin-walled beam-columns with minimal critical load corresponding to local buckling) the precise mathematical criterion have not

Therefore Simitses [27] suggested not to define the dynamic buckling for the structures with stable postbuckling behaviour, but rather it should be defined as a dynamic response to

It is a reason why in world literature a lot of criteria can be found. In the sixties of the twenty century Volmir [28] proposed a criterion for plates subjected to in-plane pulse loading. The Volmir criterion - considered the easiest to use - states that *the dynamic critical load corresponds to the amplitude of pulse force (of constant duration) at which the maximum plate* 

In many publications the dynamic buckling load is determined on the basis of stability criterion of Budiansky and Hutchinson [26, 29, 30]. However, this criterion was formulated for shell structures but also it can be used for the plate structures [31-34]. Budiansky and Hutchinson noticed that in some range of the amplitude value, the deflection of structures grows more rapidly than in other. Budiansky and Hutchinson formulated the following criterion: *Dynamic stability loss occurs when the maximum deflection grows rapidly with the small* 

In the end of 90's Ari-Gur and Simonetta [20] analysed laminated plates behaviour under impulse loading and formulated four own criteria of dynamic buckling, two of them of collapse-type conditions. One of them states: *Dynamic buckling occurs when a small increase in* 

The failure criterion was proposed by Petry and Fahlbush [34], who suggest that for structures with stable postbuckling equilibrium path the Budiansky-Hutchinson criterion is conservative because it does not take into account load carrying- capacity of the structure.

Based on examples [35] it was noticed that for the thin-walled structures subjected to pulse loading, which lose their stability according to Budiansky-Hutchinson criterion or Volmir

*deflection is equal to some constant value k (k - one half or one plate thickness)* [28].

*the pulse intensity causes a decrease in the peak lateral deflection* [20].

only possible for problem solved in elastic range [25].

the assumed buckling criterion.

*variation of the load amplitude* [26].

been defined till now.

pulse loads.

The stability loss or buckling is a system transition from one equilibrium to another (the bifurcation point), or jump from the stable to the unstable equilibrium path (the limit point). Load resulting in the loss of stability is called the critical load. The behaviour of the structure subjected to load higher than the critical one can be described by a stable (the grow of displacement is caused by increased load) or unstable (displacements grow with decreasing load) postbuckling equilibrium path.

The postbuckling behaviour of the structures depends on their type. For example, the cylindrical shells subjected to axial compression change their equilibrium stage (buckling) by unstable bifurcation point or limit point. Long rods or columns subjected to axial compression have usually a sudden global buckling (bifurcation point of passage to the unstable postbuckling equilibrium path). Thin plates supported on all edges lose their stability having the local buckling mode and the stable postbuckling equilibrium path. Mentioned above type of buckling and postbuckling behaviour for given thin-walled structures are the same for ideal structures as for structures with geometrical imperfection. Columns made of thin prismatic plates could have the local buckling mode, global (flexural, torsional or distorsional) one or coupled. The structures after local buckling are able to sustain further load, because increasing the displacement is only possible by increasing the load value (stable postbuckling equilibrium path), further increasing the load leads to plasticity or reaching the new, this time unstable bifurcation point (global buckling). The dangerous form of stability loss is the interactive buckling (coupled buckling), which usually causes the structure transition to the unstable equilibrium path what leads to the destruction of the structure with load lower than the critical load corresponding to each mode separately. The interaction of different buckling modes occurs when the critical loads corresponding to the different buckling modes are close to each other.

A more comprehensive review of the literature concerning the interactive buckling analysis of an isotropic structure can be found for example in Ali and Sridharan [3], Benito and Sridharan [5], Byskov [6], Koiter and Pignataro [7], Kolakowski [8–10], Manevich [11], Moellmann and Goltermann [12], Pignataro et al. [13], Pignataro and Luongo [14, 15], Sridharan and Ali [16, 17]. The interactive buckling of orthotropic structures can be found for example in [18, 19].
