**4. Exemplary results of calculations**

234 Nonlinearity, Bifurcation and Chaos – Theory and Applications

where cr is the critical load value.

defined by the equation:

**3.5. Natural frequencies** 

**3.6. Lagrange equations** 

the Lagrange equations can be written as [56]:

equations of motion may be written in the form:

 

2

 2 3\*

 

2

 (36)

(37)

.. *h u* 

0 and

.. *h v* 0).

(38)

*cr cr a b*

In a special case, i.e. for the so-called ideal structure without initial imperfections (\*=0) and when the equilibrium path (*a*111) is symmetrical, the postbuckling equilibrium path is

<sup>1111</sup> 1

Determination of the natural frequencies is similar to the determination of critical buckling

Natural frequencies of thin-walled structures were determined by solving a dynamic problem, which uses the approach proposed by Koiter in his asymptotic stability theory of

To determine the natural frequencies [55] of the structure the adopted equilibrium equations (26) contain cross-sectional inertia forces acting in the direction normal to the middle surface

In the dynamic analysis (while finding the frequency of natural vibrations [55]), the independent non-dimensional displacement and the load factor become a function dependent on time, and dynamic terms were added to equations describing postbuckling equilibrium path. Neglecting the forces associated with the inertia terms of prebuckling state and the second-order approximations, and taking into account the orthogonality conditions for the displacement field in the first ( )*<sup>i</sup> U* and second-order approximation ( ) *ij U* ,

..

 

where s is a natural frequency with mode corresponding to buckling mode; a*ijs* and *bijks* are the coefficients (34) describing the postbuckling behaviour of the structure (independent of time); however the parameters of load and the displacement are the functions of time *t*.

For the uncoupled buckling, i.e. the single-mode buckling (where index *s* = *N* = 1), the

*ab s N*

   

 

<sup>1</sup> <sup>1</sup> ; 1,2, , *<sup>s</sup> s ijs i j ijks i j k s*

*s s s*

  *b*

111 1111 1

*cr*

load and the natural frequencies are found by solving the eigenvalue problem.

conservative systems in the first-order approximation [52].

of the plate (column wall) and in the middle plane of plate (i.e.

 

> The exemplary results of numerical calculation are presented in this sub-chapter. All results are obtained using explained above proposed analytical-numerical method (ANM) based on the nonlinear orthotropic plate theory.

> The material properties (E – Young modulus, ν – Poisson ratio, G=E/[2(1+ν)] – Kirchhoff modulus; – density) for materials taken into account are presented in Table 1.


**Table 1.** Assumed material properties

The fibre composite material was modelled as orthotropic but for components (resin and fibre) the isotropic material properties (Table 1) was assumed. Necessary equations for material properties homogenization based on theory of mixture [57, 58] are as follows:

$$\begin{aligned} E\_x &= E\_m \left( 1 - f \right) + E\_f f, \\ E\_y &= E\_m \frac{E\_m \left( 1 - \sqrt{f} \right) + E\_f \sqrt{f}}{E\_m \left[ 1 - \sqrt{f} \left( 1 - \sqrt{f} \right) \right] + E\_f \sqrt{f} \left( 1 - \sqrt{f} \right)}, \\ \nu\_{yx} &= \nu\_m \left( 1 - \sqrt{f} \right) + \nu\_f \sqrt{f}, \\ G &= G\_m \frac{G\_m \sqrt{f} \left( 1 - \sqrt{f} \right) + G\_f \left[ 1 - \sqrt{f} \left( 1 - \sqrt{f} \right) \right]}{G\_m \sqrt{f} + G\_f \left( 1 - \sqrt{f} \right)}. \end{aligned} \tag{42}$$

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 237

*Pcr* [kN] [rad/s]

4423 (*m* = 1) 8363 (*m* = 2) 4423 8344

ANM FEM ANM FEM

ss 7.23 7.24 3016 3010

se 2.53 2.54 1784 1784 ce 2.99 2.99 1935 1935 sc 10.38 10.41 3613 3607

ss 0.54 0.54 1703 1709 cc 0.93 0.95 2231 2237 se 0.34 0.35 1351 1351 ce 0.36 0.37 1389 1389 sc 0.69 0.70 1916 1923

Exemplary results were calculated for steel and epoxy glass composite (fibre volume factor *f* = 0.5) square plates subjected to rectangular compressive pulse loading. The buckling load for plate under analysis is presented in Table 2. The pulse duration Tp was equal to the

The dimensions of analysed plates were assumed as follows: the length (width) a= b= 100

15.7 14.0

The rectangular thin plates simply supported on loaded edges with different boundary conditions along the unloaded ones were considered (Figure 5). On the longitudinal edges five different boundary condition cases were taken into account. Following notations is used

in Figure 5: s – simply supported edge, c – clamped edge, e – free edge.

**Figure 5.** Analysed plates with different boundary conditions

cc 15.6 (*m* = 1)

**Table 2.** Critical load *Pcr* and natural frequencies for analysed plates

period of natural vibration with mode corresponding to the buckling mode.

13.9 (*m* = 2)

boundary condition

**4.1. Plates** 

material:

steel

composite *f* = 0.5

mm and thickness *h* = 1 mm.

where *E*m and *E*f are the Young's modulus of elasticity for matrix and fibre, respectively, *G*<sup>m</sup> and *G*f are the shear modulus for matrix (subscript m) and fibre (subscript *f*), νm and νf are the Poisson's ratios for matrix and fibre and *f* = *V*f /(*V*m + *V*f) is the fibre volume fraction.

For static buckling the critical buckling load and corresponding modes are presented as well as the postbuckling equilibrium paths.

For dynamic buckling the proposed by Budiansky and Hutchinson parameter called Dynamic Load Factor DLF is introduced. The DLF is defined as a ratio of pulse loading amplitude to static buckling load. The results are presented of nondimensional deflection ξ versus DLF. The critical dynamic load factor *DLF*cr corresponding to dynamic buckling has been estimated using different criteria – the obtained results were compared.

For the proposed method the validation of the results was made by comparison with the other Authors [34] calculations (Figure 4) or with the results obtained with FEM [38]. The results presented in Figure 4 were obtained for thin (ratio length to thickness equals 200) aluminium square plate simply supported at all edges and subjected to sinusoidal pulse load. The time of pulse duration was equal to the period of natural vibration of the plate. The considered plate has a geometrical imperfection corresponding to buckling mode with amplitude equal to 0.05 of the plate thickness.

**Figure 4.** The results of different calculation comparison
