Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 1, pp. 177-192, Warsaw 2009 STATIC AND DYNAMIC INTERACTIVE BUCKLING OF COMPOSITE COLUMNS Zbigniew Kołakowski Technical University of Łódź, Department of Strength of Materials and Structures, Łódź, Poland e-mail: zbigniew.kolakowski@p.lodz.pl The static and dynamic problem of interaction of global bucklingmodes in compressed columnswith complex open and closed cross-sectionswas considered in the paper. The columnsmade of laminate compositeswere assumed to be simply supported at both loaded ends. A plate model was adopted in the analysis. The equations of motion of individual pla- tes (Schokker et al., 1996; Sridharan and Benito, 1984) were obtained from Hamilton’s Principle, taking into account all components of iner- tia forces (Teter and Kołakowski, 2005). Within the frame of the first order nonlinear approximation, the dynamic problem of modal interac- tive buckling was solved by the transition matrix using a perturbation method. Distortions of cross-sections and the shear-lag phenomenonwe- re taken into consideration in the problem solution. A modification of the quasi-bifurcationdynamicKleiber-Kotula-Sarancriterion (Kleiber et al., 1987) was proposed. A comparison of the proposed modification to the Budiansky-Hutchinson criterion (Budiansky and Hutchinson, 1966; Hutchinson and Budiansky, 1966) was presented for a rectangular pulse loading. Key words: lamina, composite, buckling, pulse loading, interaction 1. Introduction Thin-walled structures, composed of plate elements, havemany different buc- klingmodes that vary inquantitative (e.g. thenumber of halfwaves) andquali- tative (e.g. global and local buckling) aspects. In these cases, the postcritical behaviour cannot be described anymore by a single generalized displacement. When the postcritical behaviour of each individual mode is stable, their inte- raction can lead to unstable behaviour, and thus to an increase in the imper- fection sensitivity (Byskov, 1987-8; Camotion and Prola, 1996; Kołakowski, 178 Z. Kołakowski 1993; Kołakowski and Kowal-Michalska, 1999; Kołakowski and Teter, 2000; Kołakowski, 1993; Kołakowski and Kowal-Michalska, 1999; Kołakowski and Teter, 2000; Teter andKołakowski, 2005). A nonlinear stability theory should describe allmodes of global, local, distortional and interactive buckling, taking into consideration the structure imperfection. The theory of interactive buckling of thin-walled structures subjected to static and dynamic loading has been already widely developed for over forty years. Although the problem of static coupled buckling can be treated as well recognized, the analysis of dynamic interactive buckling is limited in practice to columns (adopting their beammodel), single plates and shells. In theworld literature, a substantial lack of the nonlinear analysis of dynamic stability of thin-walled structures with complex cross-sections can be felt. In this study, ananalysis of static anddynamic stability of composite struc- tures with complex cross-sections is presented. Special attention is focused on coupled buckling of various global buckling modes. 1.1. Static interactive buckling A plate model of the column has been adopted in the study to describe global buckling, which leads to lowering the theoretical value of the limit lo- ad. In this case, buckling characteristics for the independent global mode are nonsymmetrical, and thus the equilibrium is unsteady (Kołakowski, 1993; Ko- łakowski and Kowal-Michalska, 1999; Kołakowski and Teter, 2000; Teter and Kołakowski, 2005). When components of the displacement state for the first order approximation are taken in account, it can be followed by a decrease in values of global loads. In the case, the critical values corresponding to global buckling modes are significantly lower than local modes, then their interac- tion can be considered within the first nonlinear approximation (Kołakowski, 1993; Kołakowski and Kowal-Michalska, 1999; Kołakowski and Teter, 2000; Teter andKołakowski, 2005). It is possible as the post-buckling coefficient for uncoupled buckling is equal to zero for the second order global mode in the Euler columnmodel, and in the case of an exact solution it is very often of lit- tle significance. The theoretical static load-carrying capacity, obtained within the frame of the asymptotic theory of the nonlinear first order approximation, is always lower than theminimumvalue of critical load for the linear problem. The solution method assumed in this study allows for analysing interactions of all bucklingmodes. Dubina (1996) paid special attention to the interaction of global modes of buckling (flexural, flexural-torsional, lateral) with a distortional and/or loca- lized (Byskov, 1987-8) mode. Static and dynamic interactive buckling... 179 1.2. Dynamic interactive buckling Thedynamic pulse load of thin-walled structures can be divided into three categories, namely: impact with accompanying perturbation propagation (a phenomenon that occurs with the soundwave propagation speed in the struc- ture), dynamic load of a mean amplitude and a pulse duration comparable to the fundamental flexural vibration period, and quasi-static load of a low am- plitude and a load pulse duration approximately twice as long as the period of fundamental natural vibrations. As for dynamic load, effects of damping can be neglected in practice. This study is devoted to the stability problem of a rectangular dynamic pulse load. Dynamic buckling of a columncanbe treated as reinforcement of imperfec- tions, initial displacements or stresses in the column through dynamic loading in such a manner that the level of the dynamic response becomes very high. When the load is low, the column vibrates around the static equilibrium po- sition. On the other hand, when the load is sufficiently high, then the column can vibrate very strongly or canmove divergently, which is caused by dynamic buckling. In the literature on this problem, various criteria concerning dynamic sta- bility have been adopted. One of the simplest is the criterion suggested by Volmir (1972). The most widely used is the Budiansky-Hutchinson criterion (Budiansky andHutchinson, 1966;Hutchinson andBudiansky, 1966), inwhich it is assumed that the loss of dynamic stability occurs when the velocity with which displacements grow is the highest for a certain force amplitude. Other criteria were discussed in papers Ari-Gur and Simonetta (1997), Cui et al., 2000, 2001, 2002; Gantes et al., 2001; Hao et al., 2000; Huyan and Si- mitses, 1997; Kleiber et al., 1987; Kowal-Michalska et al., 2004; Papazoglou and Tsouvalis (1995), Petry and Fahlbusch, 2000; Schokker et al., 1996; Sri- dharan and Benito, 1984; Weller et al., 1989; Zhang and Taheri, 2002), for instance. A diversity of dynamic stability loss criteria follows from a lack of a gene- rally assumed, accurate, explicit mathematical definition. One of a few excep- tions, known to the author [e.g. Budiansky and Hutchinson, 1966; Gantes et al., 2001; Hutchinson and Budiansky, 1966), is the quasi-bifurcation criterion of dynamic buckling for a step-like load (Heaviside’s function) and that one concerning the critical pulseduration (Kleiber-Kotula-Saran criterion (Kleiber et al., 1987)). This criterion is based on the condition that the tangentmatrix of the system stiffness is zero, that is to say, all the Jacobian matrices are equal to zero. Eigenvalues of this matrix have to be computed. 180 Z. Kołakowski In this study, the followingmodification of theKleiber-Kotula-Saran crite- rion (Kleiber et al., 1987) as the criterion of dynamic stability loss for a pulse loading of finite duration has been proposed: Dynamic stability loss occurs when during the time (0, t0) of the pulse load and in its vicinity 0¬ t¬ 1.3t0, the minimum eigenva- lue of the tangent stiffnessmatrix (Jacobianmatrix) is lower than, for example, ”−1” (i.e. ρmin <−1). A dynamic response to the rectangular pulse load with the duration time corresponding to the fundamental period of flexural and flexural-distortional free vibrations of the unloaded column (i.e. t0 = T1) and for t0 = T1/2 has been analysed. 2. Formulation of the problem Prismatic thin-walled columns with open and closed cross-sections, subjected to axial compression, have been considered. Cross-sections of the elements underanalysis are built of rectangular plates interconnected along longitudinal edges and simply supported at both ends. All component plates are made of the same laminate composite subject to Hooke’s law. The attention has beendrawn to the necessity of considering the full strain tensor and all the components of inertial forces in order to carry out a proper dynamic analysis in the whole range of length of the structures. For thin-walled structures with initial deflections, Lagrange’s equations of motion for the case of interaction of N eigenmodes canbewrittenas (Schokker et al., 1996; Sridharan and Benito, 1984; Teter and Kołakowski, 2005) 1 ω2r ζr,tt+ ( 1− σ σr ) ζr+aijrζiζj − ζ ∗ r σ σr + . . .=0 (2.1) for r=1, . . . ,N, where ζr is the dimensionless amplitude of the r-th buckling mode (maximumdeflection referred to the thickness of the first plate), σr,ωr, ζ∗r – critical stress, circular frequency of free vibrations and dimensionless amplitude of the initial deflection corresponding to the r-th buckling mode, respectively. The expressions for aijr are to be found in Byskov (1987-8), Byskov and Hutchinson (1977), Kołakowski andKowal-Michalska (1999), Teter andKoła- kowski (2005). In equations of motion (2.1), inertia forces of the pre-buckling Static and dynamic interactive buckling... 181 state and second order approximations have been neglected (Schokker et al., 1996; Sridharan and Benito, 1984; Teter and Kołakowski, 2005). The initial conditions have been assumed in the following form ζr(t=0)=0 ζr,t(t=0)=0 (2.2) The static problem of interactive buckling of thin-walled multilayer co- lumns (i.e. for ζr,tt = 0 in (2.1)) has been solved with the method presented inKołakowski andKowal-Michalska (1999), Teter andKołakowski (2005), the frequencies of free vibrations have been determined analogously as in Teter andKołakowski (2005), whereas the problem of interactive dynamic buckling (2.1) have been solved by means of the Runge-Kutta numerical method mo- dified byHairer andWanner (with differentiation formulas of a variable order and automatic time stepping). 3. Analysis of the calculation results 3.1. Open section columns 3.1.1. Static buckling A detailed analysis of the calculations was conducted for compressed co- lumns with the following dimensions of open cross-sections (Fig.1) b1 =100mm b2 =50mm b3 =15mm bS =15mm h1 =h2 =h3 =12hlay =1.5mm Fig. 1. Open cross-sections of columns with a central intermediate stiffener: (a) outer omega, (b) inner omega Each plate is made of a twelve-layer composite with the symmetric ply alignment [45/−45/0/0/0/0]S. Each layer of the thickness hlay =0.125mm 182 Z. Kołakowski is characterized by the followingmechanical properties (Teter andKołakowski, 2005) E1 =140GPa E2 =10.3GPa G=5.15GPa ν12 =0.29 ρ=1600kg/m 3 The geometrical dimensions of intermediate stiffeners and the ply align- ment were selected in such a way that the critical values of local loads were considerably higher than the critical values of global loads within the variabi- lity range of the column length ℓ under consideration. For global bucklingmodes, the imposing of the symmetry conditions of the bucklingmode corresponds to flexural or flexural-distortional buckling,where- as the antisymmetry conditions entail flexural-torsional or flexural-torsional- distortional buckling (Camotion and Prola, 1996; Dubina, 1996; Kołakowski and Kowal-Michalska, 1999). Local buckling modes correspond to short co- lumns. The interactive buckling of the columns with open cross-sections shown in Fig.1 was analysed within the first order approximation for global buc- kling modes and five various lengths ℓ = 2500, 2000, 1500 and 1000mm. The interaction of buckling modes can occur among several buckling modes symmetric with respect to the symmetry axis of the cross-section and also be- tween a symmetric mode and pairs of antisymmetric modes (Byskov, 1987-8; Kołakowski, 1993; Kołakowski and Kowal-Michalska, 1999; Kołakowski and Teter, 2000; Teter and Kołakowski, 2005). In order to consider a possible ef- fect of the localized buckling (Byskov, 1987-8; Dubina, 1996; Kołakowski and Kowal-Michalska, 1999) for the assumed length ℓ, a global flexural-distortional mode (m = 1), a flexural-torsional-distortional mode (m = 1) and higher global modes: flexural-distortional and flexural-torsional-distortional modes, respectively, with the halfwave number m= 3, were analysed. The following index symbols were introduced: 1 – flexural-distortional mode for m = 1; 2 – flexural-torsional-distortional mode for m = 1; 3 – flexural-distortional mode for m=3; 4 – flexural-torsional-distortional mode for m=3. Buckling modes of the open section columns (Fig.1) for two lengths ℓ = 2500 and 1000mm are shown in Figs.2 and 3, respectively. The adop- tion of the plate model allowed us to account for all buckling modes for co- lumns of different shapes and flexural rigidities. This can help in their rational designing. Detailed results of the interactive buckling analysis are presented in Table 1 for the columns with the cross-section shown in Fig.1a and Fig.1b. The following imperfections were assumed: ζ∗1 = ζ ∗ 2 = |ℓ/(1000h)|, Static and dynamic interactive buckling... 183 Fig. 2. Buckling modes of open section columns (Fig.1a) for two lengths: (a) ℓ=2500mm, (b) ℓ=1000mm Fig. 3. Buckling modes of open section columns (Fig.1b) for two lengths: (a) ℓ=2500mm, (b) ℓ=1000mm ζ∗3 = ζ ∗ 4 = |ℓ/(2000h)|. In each case, the signs of the imperfection ζ ∗ i (where i=1, . . . ,4) were selected in the most unfavourable way, that is to say, as to obtain the lowest theoretical limit load-carrying capability σS for the given level of imperfection when the interaction of bucklingmodes is accounted for. The modes under consideration can be identified according to the adopted index symbols. It was demonstrated in each case under analysis that themost dangerous interaction of buckling modes occurred between global modes for m = 1 and m = 3. A decrease in the limit load capacity σS/σmin (where σmin =min(σi); i=1, . . . ,4) did not exceed 30% practically (see Table 1). Owing to the above-mentioned reasons, the further analysis was limited to the interaction of four global buckling modes. The signs of imperfections, defined in the static analysis, were next employed in the dynamic stability analysis. 184 Z. Kołakowski Table 1. Critical loads and theoretical limit load-carrying capacity for the columns shown in Fig.1 ℓ σ1 σ2 σ3 σ1 σS/σmin (m=1) (m=1) (m=3) (m=3) [mm] [MPa] [MPa] [MPa] [MPa] [–] Fig.1a (outer omega) 2500 52.2 29.3 131.9 119.0 0.699 2000 79.5 42.1 127.0 112.8 0.708 1500 129.0 66.9 138.7 136.7 0.719 1000 143.9 109.2 208.1 210.2 0.785 Fig.1b (inner omega) 2500 50.9 42.2 106.3 189.1 0.768 2000 74.5 62.5 112.2 175.7 0.701 1500 104.9 103.5 135.7 174.2 0.685 1000 107.7 180.4 219.3 242.1 0.765 3.1.2. Linear dynamic analysis Values of the natural frequencies of vibrations corresponding to the global bucklingmodes under analysis for different column lengths ℓ are presented in Table 2. The same index symbols were adopted as for the interactive static buckling. Vibration frequencies were determined, taking into account all com- ponents of inertia forces (Teter and Kołakowski, 2005) (in-plane ρu,tt, ρv,tt and out-of-plane ρw,tt). Table 2.Natural frequency of the columns shown in Fig.1 ℓ Fig.1a (outer omega) Fig.1b (inner omega) ω1 ω2 ω3 ω4 ω1 ω2 ω3 ω4 [mm] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s] 2500 227 170 1082 1027 224 204 971 1295 2000 350 254 1327 1305 339 310 1247 1561 1500 594 428 1849 1835 536 532 1828 2072 1000 942 820 3396 3412 814 1054 3485 3662 3.1.3. Dynamic stability Further on, an analysis of dynamic interactive buckling of the columns under consideration was conducted. Identically as in the static analysis, the Static and dynamic interactive buckling... 185 interaction of the same global bucklingmodeswas considered.A detailed ana- lysis was conducted for a rectangular load pulse σ(t)= { σD for 0¬ t¬ t0 0 for t0