**4.2. Segments of the girders**

238 Nonlinearity, Bifurcation and Chaos – Theory and Applications

= 0.01, where \*

with different boundary conditions on non-loaded edges

according to Volmir criterion was assumed as cr= 1.

different boundary conditions on non-loaded edges [35]

(Figure 6a) and for plate with geometrical imperfection with amplitude \*

with amplitude \*

The geometrical imperfection was assumed in the shape corresponding to the buckling mode

thickness. The Figure 6 presents postbuckling equilibrium paths for ideal flat composite plate

**Figure 6.** Postbuckling equilibrium paths for square ideal plates (a) and plates with imperfection (b)

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

**Figure 7.** Nondimensional deflection (a) and maximum radius *r*max (b) vs. DLF for square plates with

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)

is an amplitude of deflection divided by the plate

=0.01 (Figure 6b).

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.

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

The geometrical imperfection was assumed in the shape corresponding to the buckling mode with amplitude \* = 0.01. The static buckling load and fundamental flexural natural frequency obtained with analytical-numerical method for girders made of composite with different fibre fraction are presented in Tables 4. The static critical buckling loads are presented in Table 4 and postbuckling equilibrium paths are presented in Figure 10.

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 241

**Figure 11.** Shapes of pulse loading: three triangular impulses T1 (a), T2 (b), T3 (c), rectangular R (d) and

01 2345

The results presented in Figure 12 were obtained with the proposed analytical-numerical method (ANM) and compared with FEM computations. They are similar for both assumed shapes of pulse loading (Figure 11): rectangular (R) and sinusoidal (S). Nevertheless, for rectangular shape pulse loading some small differences in deflection are visible for *DLF* greater than 2. It should be noted that obtained curves (Figure 12) from both methods allow to find the same critical dynamic load factor *DLFcr* using Budiansky-Hutchinson or Volmir criterion – for rectangular pulse loading *DLFcr* ≈1.4 (both criteria) and for sinusoidal pulse

In Figure 13 the dynamic response comparison of girders made of composite material (*f* = 0.5) with different cross-section subjected to triangular T3 pulse was presented. The curves for omega cross-section and channel cross-section with inner stiffeners cover each other's.

Dynamic responses for girder with channel cross-section with inner stiffeners for different pulse loading are presented in Figure 14. The curves denoted by *S=R* were obtained for sinusoidal pulse loading with the same area as rectangular pulse loading (for the same pulse duration the amplitude was higher for sinusoidal pulse). The highest deflection was

**Figure 12.** Dimensionless deflection vs. *DLF* for channel girder – results comparison obtained with

DLF

sinusoidal S (e)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 analytical-numerical method ANM and finite element method FEM [56]

loading *DLFcr* ≈2.1 (Budiansky-Hutchinson) or 2.3 (Volmir).

R ANM R, FEM S, ANM S, FEM


**Table 4.** Critical load and natural frequencies for analysed girder's segments

**Figure 10.** Postbuckling equilibrium paths for segment of girders

Buckling load and natural frequency for girder segment with omega and stiffened channel cross-section are similar – it is true only for local buckling case. For girder with channel cross-section the buckling was caused by flanges – this is a reason why for this cross-section the buckling load and natural frequency are smaller than for two others analysed crosssections. Looking at obtained results (Table 4) it can be seen that increasing the volume fibre fraction *f* leads to an increasing the buckling loads as well as the natural frequencies. The postbuckling equilibrium paths for stiffened cross-section (channel with inner stiffeners and omega) overlap. The postbuckling path for channel cross-section lies below the equilibrium paths of girders with stiffened cross-section – it is obvious because the girders with stiffened cross-section have similar stiffness and the girder with channel cross-section is more flexible.

In dynamic buckling case the time of pulse duration *Tp* was assumed as the period of natural fundamental flexural vibration corresponding to the local buckling mode. Considered shapes of pulse loading are presented in Figure 11.

240 Nonlinearity, Bifurcation and Chaos – Theory and Applications

mode with amplitude \*

cross-section

is more flexible.

The geometrical imperfection was assumed in the shape corresponding to the buckling

frequency obtained with analytical-numerical method for girders made of composite with different fibre fraction are presented in Tables 4. The static critical buckling loads are

channel (Figure 9a) 1526 2281 1076 1076 channel with inner stiffeners (Figure 9c) 2821 4217 1308 1308 omega (Figure 9b) 2819 4214 1308 1308

Buckling load and natural frequency for girder segment with omega and stiffened channel cross-section are similar – it is true only for local buckling case. For girder with channel cross-section the buckling was caused by flanges – this is a reason why for this cross-section the buckling load and natural frequency are smaller than for two others analysed crosssections. Looking at obtained results (Table 4) it can be seen that increasing the volume fibre fraction *f* leads to an increasing the buckling loads as well as the natural frequencies. The postbuckling equilibrium paths for stiffened cross-section (channel with inner stiffeners and omega) overlap. The postbuckling path for channel cross-section lies below the equilibrium paths of girders with stiffened cross-section – it is obvious because the girders with stiffened cross-section have similar stiffness and the girder with channel cross-section

In dynamic buckling case the time of pulse duration *Tp* was assumed as the period of natural fundamental flexural vibration corresponding to the local buckling mode. Considered

presented in Table 4 and postbuckling equilibrium paths are presented in Figure 10.

volume fibre fraction *f*:

**Table 4.** Critical load and natural frequencies for analysed girder's segments

**Figure 10.** Postbuckling equilibrium paths for segment of girders

shapes of pulse loading are presented in Figure 11.

= 0.01. The static buckling load and fundamental flexural natural

critical load *Pcr* [N]

natural frequency [rad/s]

0.4 0.6 0.4 0.6

**Figure 11.** Shapes of pulse loading: three triangular impulses T1 (a), T2 (b), T3 (c), rectangular R (d) and sinusoidal S (e)

**Figure 12.** Dimensionless deflection vs. *DLF* for channel girder – results comparison obtained with analytical-numerical method ANM and finite element method FEM [56]

The results presented in Figure 12 were obtained with the proposed analytical-numerical method (ANM) and compared with FEM computations. They are similar for both assumed shapes of pulse loading (Figure 11): rectangular (R) and sinusoidal (S). Nevertheless, for rectangular shape pulse loading some small differences in deflection are visible for *DLF* greater than 2. It should be noted that obtained curves (Figure 12) from both methods allow to find the same critical dynamic load factor *DLFcr* using Budiansky-Hutchinson or Volmir criterion – for rectangular pulse loading *DLFcr* ≈1.4 (both criteria) and for sinusoidal pulse loading *DLFcr* ≈2.1 (Budiansky-Hutchinson) or 2.3 (Volmir).

In Figure 13 the dynamic response comparison of girders made of composite material (*f* = 0.5) with different cross-section subjected to triangular T3 pulse was presented. The curves for omega cross-section and channel cross-section with inner stiffeners cover each other's.

Dynamic responses for girder with channel cross-section with inner stiffeners for different pulse loading are presented in Figure 14. The curves denoted by *S=R* were obtained for sinusoidal pulse loading with the same area as rectangular pulse loading (for the same pulse duration the amplitude was higher for sinusoidal pulse). The highest deflection was

channel omega channel with inner

2.1 1.6 1.4 3.1 2.5 2.5 stiffeners

2.1 1.6 1.4 3.1 2.5 2.5

width ratio *l*/ *b*1 = 4; 6 and 8 and for the following dimensions of the cross-section: width of

the web to its thickness *b*1/*h* = 100, width of the flange to its thickness *b2*/*h* = 50.

2.0 1.6 1.4 3.1 2.3 2.4

**Figure 15.** Nondimensional deflection (a) and maximum radius *r*max (b) as a function of DLF for

channel beam-columns with different length ratio *l*/ *b*1 and pulse duration *Tp* [35]

**Table 5.** The *DLFcr* value for different shape of applied pulses

analysed cross-section:

**S R S=R T1 T2 T3**

type of pulse

**Figure 13.** Nondimensional deflection as a DLF function for girder with different channel crosssection subjected to T3 pulse loading [56]

obtained for pulse denoted by *S=R* because this pulse has the highest amplitude. The rest of compared pulses have the same amplitude and the same duration. Analysing the curves (*DLF*) for rectangular, sinusoidal and three triangular impulses it can be seen that the highest increment of deflection for the smallest *DLF* takes place for rectangular pulse loading.

**Figure 14.** Dimensionless deflection vs. dynamic load factor for different shapes of pulse loading channel with inner stiffeners, composite material *f* = 0.7 [56]

The comparison of *DLFcr* obtained using Budiansky-Hutchinson criterion for girders with different cross-section made of composite materials (*f* = 0.5) are presented in Table 5. Only the average values from the obtained critical ranges are presented. The dynamic load factors for different cross-sections are in the same relation as buckling loads (Table 4) – the same or similar *DLFcr* for cross-sections with stiffeners (omega and channel with inner stiffeners).

### **4.3. Columns**

Next, the exemplary results for the dynamic interactive buckling of channel cross-section the columns are presented. The columns subjected to rectangular compressive pulse loading were analysed. The calculation was carried out for columns with various length to web



**Table 5.** The *DLFcr* value for different shape of applied pulses

C C-stiff omega

section subjected to T3 pulse loading [56]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

**Figure 13.** Nondimensional deflection as a DLF function for girder with different channel cross-

**Figure 14.** Dimensionless deflection vs. dynamic load factor for different shapes of pulse loading -

The comparison of *DLFcr* obtained using Budiansky-Hutchinson criterion for girders with different cross-section made of composite materials (*f* = 0.5) are presented in Table 5. Only the average values from the obtained critical ranges are presented. The dynamic load factors for different cross-sections are in the same relation as buckling loads (Table 4) – the same or similar *DLFcr* for cross-sections with stiffeners (omega and channel with inner stiffeners).

0123456

Next, the exemplary results for the dynamic interactive buckling of channel cross-section the columns are presented. The columns subjected to rectangular compressive pulse loading were analysed. The calculation was carried out for columns with various length to web

channel with inner stiffeners, composite material *f* = 0.7 [56]

S R S=R T1 T2 T3

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

**4.3. Columns** 

obtained for pulse denoted by *S=R* because this pulse has the highest amplitude. The rest of compared pulses have the same amplitude and the same duration. Analysing the curves (*DLF*) for rectangular, sinusoidal and three triangular impulses it can be seen that the highest increment of deflection for the smallest *DLF* takes place for rectangular pulse loading.

012345

DLF

DLF

**Figure 15.** Nondimensional deflection (a) and maximum radius *r*max (b) as a function of DLF for channel beam-columns with different length ratio *l*/ *b*1 and pulse duration *Tp* [35]

The interaction between global buckling mode *m* = 1 and the first local buckling mode *m* > 1 was considered. The geometrical imperfection were assumed in the shape corresponding to the buckling mode with amplitude \* ≡ 2\* equals 1/100 wall thickness for local mode and 1\* equal to length to one thousand wall thickness (*l*/1000*h*) for global mode. Time of pulse duration *Tp* was assumed as *T1* equal to the period of natural fundamental flexural vibration or *Tm* (where *m* is a number of half waves of local buckling mode) equal to the period of natural vibration with mode corresponding to the local buckling mode.

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 245

in the middle cross-section of the columns and in the middle of the web (point 1 – Figure 16)

**Figure 16.** Deflection in time for channel columns subjected to rectangular pulse loading with the DLF

Analysing the results presented in Figure 16, it can be said that the global buckling appears for channel cross-section columns. The column edge deflections are greater than deflections of the middle part of the web. The FEM results of calculations presented in Figure 16 have initiated the need for a multimodal buckling analysis also for short columns subjected to

Results for linear buckling and modal analyses obtained with proposed analytical-numerical method are presented in Table 7. As it will be presented below (see Figure 17) the four modes should take into consideration in ANM to obtain similar results to this obtained with FEM. The finite element method gives results (global mode) even in the case when only one buckling mode as the initial imperfection (for example, the local buckling mode *m*=3) has

A comparison between the results obtained with the analytical-numerical method and the finite element method on plots presenting a dimensionless deflection vs. a dynamic load factor for columns with channel cross-sections are shown in Figure 17. Some differences appear because the analytical-numerical model has only a few degrees of freedom in contrary to FE model, which has thousands DoF. However the curves presented in Figure 17 are different the critical DLF values estimated using the Budiasky-Hutchinson criterion is

Mode *cr* [MPa] *n* [Hz] local mode *m* = 3 53 614 primary local mode *m* = 1 123 312 secondary local mode *m* = 1' 972 880 global mode *m* = 1'' 5122 2001

**Table 7.** Buckling stress and natural vibration for the channel column

similar. From the ANM, the *DLFcr* = 2.7 and from the FEM, the *DLFcr* = 2.6.

and on the edge between the web and the flange (point 2 – Figure 16).

= 1.6

pulse loading.

been taken into account [56].

Figure 15 presents dimensionless deflection as a function of dynamic load factor and maximal radius *r*max calculated for maximal characteristic root of Jacoby matrix as a function of DLF.

From curves presented in Figure 15a the critical value of dynamic load factor based on Volmir (V) or Budiansky-Hutchinson (B-H) criterion can be found. The curves presented in Figure 15b help to find critical DLF value based on proposed criterion (PC) [35]. The obtained critical DLF's according to mentioned above criteria are presented in Table 6.


**Table 6.** Comparison of DLFcr obtained from different criteria for interactive buckling

The comparison of the obtained results shows that they are in good agreement. In all cases with pulse duration equal to the period of natural vibration of the form corresponding to local buckling mode the results obtained from the proposed criterion (PC) are between the results obtained from Volmir (V) and Budiansky-Hutchinson (B-H) criteria. For loading with time of pulse duration equal to the period of natural fundamental vibration T1 the critical DLF values obtained using the proposed criterion (PC) are equal or a bit greater (about 6%) than the critical dynamic load factors from Budiansky-Hutchinson (B-H) or Volmir (V) criteria.

Should be pointed out that in the dynamic buckling problem also for the short columns the multimodal buckling analysis should be carried out. It has been proven on exemplary channel columns with following dimensions: *b1*/*h* = 100, *b2*/*h* = 50, *b3*/*h* = 25 and *l*/*b1* = 4.

The problem has been calculated with the analytical-numerical method and the finite element method [56].

The dimensionless deflection as a function of dimensionless time (time divided by pulse duration) for channel column is presented in Figure 16. The characteristic points are located in the middle cross-section of the columns and in the middle of the web (point 1 – Figure 16) and on the edge between the web and the flange (point 2 – Figure 16).

244 Nonlinearity, Bifurcation and Chaos – Theory and Applications

pulse duration *Tp* [ms]

the buckling mode with amplitude \* ≡ 2\*

of DLF.

criteria.

element method [56].

columns length ratio

The interaction between global buckling mode *m* = 1 and the first local buckling mode *m* > 1 was considered. The geometrical imperfection were assumed in the shape corresponding to

equal to length to one thousand wall thickness (*l*/1000*h*) for global mode. Time of pulse duration *Tp* was assumed as *T1* equal to the period of natural fundamental flexural vibration or *Tm* (where *m* is a number of half waves of local buckling mode) equal to the period of

Figure 15 presents dimensionless deflection as a function of dynamic load factor and maximal radius *r*max calculated for maximal characteristic root of Jacoby matrix as a function

From curves presented in Figure 15a the critical value of dynamic load factor based on Volmir (V) or Budiansky-Hutchinson (B-H) criterion can be found. The curves presented in Figure 15b help to find critical DLF value based on proposed criterion (PC) [35]. The obtained critical DLF's according to mentioned above criteria are presented in Table 6.

*l*/*b1* = 4 T1= 1.6 m=1; 3 local 1.01.15 1.07 1.16 *l*/ *b1* = 6 T1= 1.9 m=1; 5 local 0.951.0 1.01 1.08 *l*/ *b1* = 8 T1= 2.7 m=1; 6 local 1.11.15 1.09 1.09 *l*/ *b1* = 4 T3= 0.8 m=1; 3 local 1.61.75 1.40 1.58 *l*/ *b1* = 6 T5= 0.7 m=1; 5 local 1.61.75 1.39 1.59 *l*/ *b1* = 8 T6= 0.8 m=1; 6 local 1.62.05 1.43 1.51

The comparison of the obtained results shows that they are in good agreement. In all cases with pulse duration equal to the period of natural vibration of the form corresponding to local buckling mode the results obtained from the proposed criterion (PC) are between the results obtained from Volmir (V) and Budiansky-Hutchinson (B-H) criteria. For loading with time of pulse duration equal to the period of natural fundamental vibration T1 the critical DLF values obtained using the proposed criterion (PC) are equal or a bit greater (about 6%) than the critical dynamic load factors from Budiansky-Hutchinson (B-H) or Volmir (V)

Should be pointed out that in the dynamic buckling problem also for the short columns the multimodal buckling analysis should be carried out. It has been proven on exemplary

The problem has been calculated with the analytical-numerical method and the finite

The dimensionless deflection as a function of dimensionless time (time divided by pulse duration) for channel column is presented in Figure 16. The characteristic points are located

channel columns with following dimensions: *b1*/*h* = 100, *b2*/*h* = 50, *b3*/*h* = 25 and *l*/*b1* = 4.

buckling

**Table 6.** Comparison of DLFcr obtained from different criteria for interactive buckling

natural vibration with mode corresponding to the local buckling mode.

equals 1/100 wall thickness for local mode and 1\*

modes B-H V PC

**Figure 16.** Deflection in time for channel columns subjected to rectangular pulse loading with the DLF = 1.6

Analysing the results presented in Figure 16, it can be said that the global buckling appears for channel cross-section columns. The column edge deflections are greater than deflections of the middle part of the web. The FEM results of calculations presented in Figure 16 have initiated the need for a multimodal buckling analysis also for short columns subjected to pulse loading.

Results for linear buckling and modal analyses obtained with proposed analytical-numerical method are presented in Table 7. As it will be presented below (see Figure 17) the four modes should take into consideration in ANM to obtain similar results to this obtained with FEM. The finite element method gives results (global mode) even in the case when only one buckling mode as the initial imperfection (for example, the local buckling mode *m*=3) has been taken into account [56].


**Table 7.** Buckling stress and natural vibration for the channel column

A comparison between the results obtained with the analytical-numerical method and the finite element method on plots presenting a dimensionless deflection vs. a dynamic load factor for columns with channel cross-sections are shown in Figure 17. Some differences appear because the analytical-numerical model has only a few degrees of freedom in contrary to FE model, which has thousands DoF. However the curves presented in Figure 17 are different the critical DLF values estimated using the Budiasky-Hutchinson criterion is similar. From the ANM, the *DLFcr* = 2.7 and from the FEM, the *DLFcr* = 2.6.

Good agreement between the results obtained with ANM and FEM is possible because the interactive dynamic buckling problem has been solved in the analytical-numerical method. Four modes have been taken into account. The buckling stress and the natural frequency obtained with the analytical-numerical method for all the modes assumed in the multimodal analysis are listed in Table 7. The buckling modes taken into consideration in the multimodal analysis are presented in Figure 18, correspondingly.

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 247

composite materials. As it was shown not only static load can be considered but also dynamic load with intermediate velocity – the dynamic buckling can be analysed using

The proposed analytical–numerical method gives almost the same results for eigenvalue problem (buckling loads, natural frequencies with corresponding modes) and similar results for dynamic buckling as the finite element method. However the dimensionless deflection versus dynamic load factor relation obtained with both (proposed and FEM) methods are not identical (especially for higher DLF value). These relations allow to find similar critical value of DLF taking into consideration one of the well-known criterion. The differences in the dimensionless deflection ξ appear because the numerical model in the FEM has more degrees of freedom than the model in the analytical–numerical method, but the results from the ANM are obtained in a significantly faster way than those from the finite element

This publication is a result of the research work carried out within the project subsidized over the years 2009-2012 from the state funds designated for scientific research (MNiSW - N

[1] Bernoulli J., Euler L. (1910) Abhandlungen uber das Gleichegewicht und die Schwingungen der Ebenen Elastischen Kurven, Wilhelm Engelmann, Nr 175, Leipzig. [2] Timoshenko S.P., Gere J.M. (1961) Theory of Elastic Stability, McGraw-Hill Book

[3] Volmir A.S. (1967), Stability of Deformation Systems, Moscow, Nauka, Fizmatlit /in

[4] Ali M.A., Sridharan S. (1988) A Versatile Model for Interactive Buckling of Columns and Beam-Columns, International Journal of Solids and Structures, 24(5): 481–486. [5] Benito R., Sridharan S. (1985) Mode Interaction in Thin-Walled Structural Members,

[6] Byskov E. (1988) Elastic Buckling Problem with Infinitely Many Local Modes,

[7] Koiter WT, Pignataro M. (1974) An Alternative Approach to the Interaction Between Local and Overall Buckling in Stiffened Panels, Buckling of structures Proceedings of

assumed plate theory and the proposed method of solution.

*Department of Strength of Materials, Lodz University of Technology, Poland* 

Company, Inc. NewYork, Toronto, London.

Journal of Structural Mechanics, 12(4): 517–542.

IUTAM Symposium, Cambridge: 133–148.

Mechanics and Structures Machines, 15(4): 413–435.

method.

**Author details** 

**Acknowledgement** 

Tomasz Kubiak

N501 113636).

**6. References** 

Russian/.

**Figure 17.** Dimensionless edge deflection vs. the *DLF* for channel columns subjected to rectangular pulse loading *Tp* = *T3* = 1.6 ms [56]

**Figure 18.** Buckling modes for the channel cross-section column
