**4.1. Plates**

236 Nonlinearity, Bifurcation and Chaos – Theory and Applications

*xm f*

*E E f Ef*

*y m*

*E E*

*m*

*G G*

as the postbuckling equilibrium paths.

amplitude equal to 0.05 of the plate thickness.

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

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 ,

*yx m f*

1 ,

*f f*

 

*m f*

11 1

*E f f Ef f*

*m f*

*E f Ef*

1 11

*Gf fG f f*

*G fG f*

1

*m f*

*m f*

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

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

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

DLF

proposed ANM

results obtained by Petry-

FEM

Fahlbush

been estimated using different criteria – the obtained results were compared.

,

(42)

.

1

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


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

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

The dimensions of analysed plates were assumed as follows: the length (width) a= b= 100 mm and thickness *h* = 1 mm.

The geometrical imperfection was assumed in the shape corresponding to the buckling mode with amplitude \* = 0.01, where \* is an amplitude of deflection divided by the plate thickness. The Figure 6 presents postbuckling equilibrium paths for ideal flat composite plate (Figure 6a) and for plate with geometrical imperfection with amplitude \* =0.01 (Figure 6b).

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 239

are in the range obtained from Budiansky-Hutchinson criterion (B-H). For the case denoted as **se** the greatest differences between critical dynamic load factors from the proposed and

> se m=1 1.51.6 1.55 1.35 ce m=1 1.31.4 1.58 1.35 ss m=1 1.51.6 1.43 1.51 sc m=1 1.41.8 1.46 1.58 cc m=2 1.41.5 1.51 1.39

> > 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

The comparison for postbuckling behaviour of the rectangular plate simply supported at all edges subjected to static and dynamic load are obtained using proposed analytical-

As a next example the static and dynamic buckling of composite (epoxy glass composite with different volume fibre fraction *f*) girders with open cross-section (Figure 9) is presented. The assumed boundary conditions on loaded edges correspond to simply support. The calculation was carried out for short segment of girder with length to web width ratio *l*/*b1* = 1 and for the following dimensions of the cross-section: *b1*/*h* = 50, *b2*/*h* = 25 and *b3*/*h* = 12.5.

DLF P/Pcr

Budiansky-Hutchinson criterion were obtained but these differences are less than 10%.

for unloaded edges mode B-H V PC

**Table 3.** Comparison of DLFcr obtained from different criteria for compressed plate

**Figure 8.** Nondimensional deflection vs. DLF for simply supported square plate

dynamic static

0.0 0.5 1.0 1.5 2.0 2.5

numerical method and presented in Figure 8.

**Figure 9.** Cross-sections of analysed segment of the girders

**4.2. Segments of the girders** 

boundary conditions

**Figure 6.** Postbuckling equilibrium paths for square ideal plates (a) and plates with imperfection (b) with different boundary conditions on non-loaded edges

In the dynamic buckling case the results are shown as graphs presenting nondimensional deflection or radius *r* calculated from real and imaginary part of maximal characteristic root of Jacoby matrix as a function of dynamic load factor DLF. The graphs mentioned above allow to find critical amplitude of pulse loading using the proposed criterion (PC) [35] and to compare the obtained results with Budiansky-Hutchinson (B-H) or Volmir (V) criteria. In brackets the notation used in Figures and Tables is given. The critical deflection according to Volmir criterion was assumed as cr= 1.

**Figure 7.** Nondimensional deflection (a) and maximum radius *r*max (b) vs. DLF for square plates with different boundary conditions on non-loaded edges [35]

Basing on curves presented in Figure 7 the critical value of dynamic load factor can be found. The comparison of obtained critical DLF values using different criteria is presented in Table 3. All critical DLF values except the case denoted as *se* obtained from the proposed criterion (PC) are in the range obtained from Budiansky-Hutchinson criterion (B-H). For the case denoted as **se** the greatest differences between critical dynamic load factors from the proposed and Budiansky-Hutchinson criterion were obtained but these differences are less than 10%.


**Table 3.** Comparison of DLFcr obtained from different criteria for compressed plate

**Figure 8.** Nondimensional deflection vs. DLF for simply supported square plate

The comparison for postbuckling behaviour of the rectangular plate simply supported at all edges subjected to static and dynamic load are obtained using proposed analyticalnumerical method and presented in Figure 8.
