**1.2. Dynamic buckling**

In literature a quantity of ''pulse intensity'' [20] or ''pulse velocity'' [21] is introduced. The analysis of dynamic stability of plates under in-plane pulse loading can be divided into three categories depending on pulse duration and magnitude of its amplitude. For pulses of high intensity the impact phenomenon is observed whereas for pulses of low intensity the problem becomes quasi-static. The phenomenon of dynamic stability and dynamic buckling are often confused with each other. In this chapter the dynamic buckling phenomenon is 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 only possible for problem solved in elastic range [25].

220 Nonlinearity, Bifurcation and Chaos – Theory and Applications

load) postbuckling equilibrium path.

for example in [18, 19].

**1.2. Dynamic buckling** 

buckling) way.

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

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

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

In literature a quantity of ''pulse intensity'' [20] or ''pulse velocity'' [21] is introduced. The analysis of dynamic stability of plates under in-plane pulse loading can be divided into three categories depending on pulse duration and magnitude of its amplitude. For pulses of high intensity the impact phenomenon is observed whereas for pulses of low intensity the problem becomes quasi-static. The phenomenon of dynamic stability and dynamic buckling are often confused with each other. In this chapter the dynamic buckling phenomenon 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 assumed buckling criterion.

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 been defined till now.

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 pulse loads.

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

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 variation of the load amplitude* [26].

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

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 criterion, the maximal radius *r*max calculated from characteristic root =a+jb (where j= 1 ) of Jacoby matrix is equal or greater than unity in complex plane.

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 223

the plate is homogeneous (for example orthotropic homogenisation is made for fibre

the plate is thin – other dimensions (length and width) are at least 10 times higher than

 the plane stress state is considered for the plate – stress acting in the plate plane dominates the plate behaviour, stress acting in normal to plate plane direction are

all strains (normal and shear) in plate plane are small compared to unity and they are

 the strains of the plate to its normal direction are neglected (thickness of the plate do not change after deformation) – this assumption are made according to the Kirchhoff-

straight lines normal to the mid-surface of the plate remain straight and normal to the

Additionally, it is assumed that principal axes of orthotropy are parallel to the edges of

A plate model has been assumed for a thin plates and thin-walled beam-columns or girders. For easier explanation the plate (Figure 1a) or each *i*-th strip (Figure 1b) of the plate (or wall

To describe the middle surface strains for each plate the following strain tensor have been

, , , , ,, ,,

,

*ix iy ixy*

 

 (1)

correspondingly,

1 222 , ,,, 2 1 222 , ,,, 2

*ix i x i x i x i x*

 

*u wuv v wuv*

*iy i y i y i y i y*

, , ( ) *iy iy u v* and , , ,, ( ) *ix iy ix iy uu vv* are in general neglected for , , *mm m*

( ), ( ),

*ixy i y i x i x i y i x i y i x i y*

where: *u*i, *v*i, *w*i - displacements parallel to the respective axes *x*i, *y*i, *z*i of the local Cartesian system of co-ordinates, whose plane *x*i*y*i coincides with the middle surface of the *i*-th plate

In the majority of publications devoted to structure stability, the terms 2 2 , , ( ) *ix ix u v* , 2 2

The change of the bending and twisting curvatures of the middle surface are assumed

*u v ww uu vv*

 there is no interaction in normal direction between layers parallel to middle surface; deflections of the plate can be considerable in terms of nonlinear geometrical relations;

analysed structures (plate, beam, column, beam-column or girder).

of the girder) or each *i*-th wall of the girder (Figure 1c) are called plate.

composite – resin matrix and fibre-reinforcement)

the material of the plate subjects to Hooke's law;

plate thickness;

assumed to be zero;

Love hypothesis;

mid-surface after deformation

**2.2. Geometrical equations for thin plate** 

*m*

before its buckling (Figure 1).

in (1) in the strain tensor components.

according to [48, 49] as follows:

*m*

*m*

linear;

assumed:

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 even if one characteristic root =a+jb of Jacoby matrix find for every time moment from 0 to 1.5Tp lies in the complex plane outside the circle with radius equal to unity*.

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

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