Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 1, pp. 147-168, Warsaw 2012 50th anniversary of JTAM SOME ASPECTS OF THE AXIAL EXTENSION MODE IN AN ELASTIC THIN-WALLED BEAM-COLUMN Zbigniew Kołakowski Technical University of Lodz, Department of Strength of Materials and Structures, Łódź, Poland e-mail: zbigniew.kolakowski@p.lodz.pl The present paper deals with the influence of the axial extension mode on static and dynamic interactive buckling of a thin-walled beam-column with imperfections subjected to uniform compression when the shear lag pheno- menon and distortional deformations are taken into account. A platemodel (2D) is adopted for the beam-column. One- and two-dimensional models of the elements are compared, too. The structure is assumed to be simply supported at the ends. Equations of motion of the component plates are obtained from Hamilton’s Principle, taking into account all components of inertia forces. Within the frame of the first order nonlinear approximation, thedynamicproblemofmodal interactivebuckling is solvedby the transition matrix using the perturbationmethod andGodunov’s orthogonalization. Key words: axial extension mode, eigenvalue problem, dynamic interactive buckling, thin-walled structure 1. Introduction Thin-walled structures composed of plate elements have many different buc- klingmodes that vary in quantitative and qualitative aspects. The analysis of buckling of conservative systems belongs to the main problems in mechani- cal sciences. In the case of finite displacements, different buckling modes are interrelated even with the loads close to their critical values (eigenvalues of the respective boundaryproblem).The investigation of stability of equilibrium states requires application of anonlinear theory that enables us to estimate the influence of different factors on the structure behavior.When the static post- critical behavior of each individualmode is stable, their interaction can lead to unstable behavior, and thus to an increase in the imperfection sensitivity (Ali and Sridharan, 1988; Benito and Sridharan, 1984-85; Byskov, 1987-88; Byskov 148 Z. Kołakowski and Hutchinson, 1977; Kolakowski, 1993; Kolakowski and Kowal-Michalska, 1999; Kolakowski and Krolak, 2006; Kolakowski et al., 1999; Kolakowski and Teter, 2000; Manevich, 1985, 1988; Moellmann and Goltermann, 1989; Pi- gnataro and Luongo, 1987; Pignataro et al., 1985; Sridharan and Ali, 1986; Sridharan andBenito, 1984; Sridharan andPeng, 1989; Teter andKolakowski, 2004). The concept of static interactive buckling (the so-called coupled buckling) involves the general asymptotic theory of stability. Among all versions of the general nonlinear theory, Koiter’s theory (Koiter, 1976; Koiter andPignataro, 1976; Koiter and van der Neut, 1980) of conservative systems is the most popular one, owing to its general character and development, even more so after Byskov and Hutchinson (1977) formulated it in a convenient way. The theory is based on asymptotic expansions of the postbuckling path and is capable of considering nearly simultaneous bucklingmodes. The theory of interactive buckling of thin-walled structures subjected to static and dynamic loading has been already developed widely for over forty years.Although the problemof static coupled buckling can be treated as fairly well recognized, the analysis of dynamic interactive buckling is in practice limited to columns (adopting their beammodel), single plates and shells (Ari- GurandSimonetta, 1997;HuyanandSimitses, 1997;Kołakowski, 2010;Kowal- Michalska, 2007; Papazoglou andTsouvalis, 1995; Petry andFahlbusch, 2000; Schokker et al., 1996; Volmir, 1972). In the world literature, a substantial lack of the nonlinear analysis of dy- namic stability of thin-walled structures with complex cross-sections can be felt. In this study, special attention is focused on the influence of the axial extension mode on static and dynamic coupled buckling with local buckling modes of the thin-walled beam-column with closed cross-section. The axial extension mode should be included in dynamic analysis (Kołakowski, 2010). 1.1. Static interactive buckling When components of the displacement state for the first order approxi- mation are taken into account, this can be followed by a decrease in values of global loads. The theoretical static load carrying capacity, obtainedwithin the frame of the asymptotic theory of the nonlinear first order approximation, is always lower than the minimum value of critical load for the linear problem, and the imperfection sensitivity can only be obtained. According to the assumptions made in Byskov and Hutchinson’s theory (Byskov, 1987-88; Byskov andHutchinson, 1977), local bucklingmodes do not Some aspects of the axial extension mode... 149 interact explicitly. However, an interaction occurs through an interaction of each of them with the global modes. It can be noticed that the global fle- xural (Euler) buckling can interact with an even number of modes that are symmetric or antisymmetric but the global flexural-torsional mode interacts only with pairs of symmetric and antisymmetric modes (Kolakowski, 1993; Kolakowski and Kowal-Michalska, 1999; Kolakowski et al., 1999; Kolakowski andTeter, 2000; Sridharan andAli, 1986; Sridharan andBenito, 1984; Sridha- ran and Peng, 1989; Teter and Kolakowski, 2004). However, the global axial extension mode can interact even with one symmetric mode or with a pair of antisymmetric modes with respect to the axis of overall bending. Since the late 1980’s, the Generalized Beam Theory (GBT) (Basaglia et al., 2008; Camotim et al., 2008; Camotin and Dinis, 2009; Davies, 2000; Si- lva et al., 2008) has been developed extensively. Recently, a new approach has been proposed, i.e., the constrained Finite StripMethod (cFSM) (Adany and Schafer, 2006a,b; Dinis et al., 2007; Schafer, 2006). These two alternative modal approaches to analyze the elastic buckling behavior were compared in Adany et al. (2006, 2007, 2008). In the current decade, inmore andmore numerous publications byAdany and Schafer (2006a,b, 2008), Basaglia et al. (2008), Camotim et al. (2008), Dinis et al. (2007), Schafer (2006), Silva et al. (2008), attention has been paid to the global axial mode, which is considered only in the theoretical aspect in linear issues, that is to say, under critical loads. Adany and Schafer (2006a), have said that “it should be noted that this axial mode is a theoretically possible bucklingmode, even though it has little practical importance”. In the axial extensionmode, longitudinal displacements of the cross-section dominate and this mode can be referred to as the shortening one (Fig.1). The axial mode is symmetric with respect to the cross-section axis of symmetry and it is symmetric with respect to the axis of overall bending. The axial extension mode should be included in dynamic analysis (Kołakowski, 2010). Fig. 1. Longitudinal displacement distributions of the axial mode of the beam-column 150 Z. Kołakowski A trial to dispute with the statement included in the paper byAdany and Schafer (2006a), has been undertaken.Here special attention has been focused on the eigenvalue problem of the axial mode and on the interaction of the global axial extensionmodewith local bucklingmodes for a square box of the column in the first order nonlinear approximation of the perturbationmethod (Kołakowski and Kowal-Michalska, 2011). In thepresent study, aplatemodel (2D) of the columnhasbeenadopted to describe all buckling modes. Instead of the finite strip method, the numerical method of the transition matrix using Godunov’s orthogonalization is used in this case. In the solution obtained, the interaction between all the walls of structures being taken into account, the shear lag phenomenonandalso the ef- fect of cross-sectional distortions are included. Themost important advantage of thismethod is such that it enables us to describe a complete range of beha- vior of thin-walled structures from all global to the local stability (Kolakowski and Kowal-Michalska, 1999; Kolakowski and Krolak, 2006; Kolakowski et al., 1999; Kolakowski and Teter, 2000). The solution method has been partially presented in paper by Kolakowski and Krolak 2006. In the present study, in order to simplify the analysis of the eigenvalue problem, the results obtained for the two-dimensional element have been com- pared to the results for the one-dimensional element. 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. Thedynamicbucklingof abeam-columncanbe treated as reinforcement of imperfections, initial displacements or stresses in the column throughdynamic loading in such amanner that the level of the dynamic response becomes very high.When the load is low, the column vibrates around the static equilibrium position.Ontheotherhand,whenthe load is sufficiently high, then the column can vibrate very strongly or canmove divergently, which is caused by dynamic buckling (i.e., dynamic response). The effects of damping can be neglected in practice. Some aspects of the axial extension mode... 151 In the literature on this problem, various criteria concerning dynamic sta- bility have been adopted. Themost widely used is the Budiansky-Hutchinson criterion (BudianskyandHutchinson, 1966;Hutchinson andBudiansky, 1966), inwhich it is assumed that a loss of dynamic stability occurswhen 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), Gan- tes et al. (2001), Huyan and Simitses (1997), Kleiber et al. (1987), Kowal- Michalska (2007), Papazoglou and Tsouvalis (1995), Petry and Fahlbusch (2000), Volmir (1972), for instance. A dynamic response to the rectangular pulse load of the duration corre- sponding to the fundamental period of flexural free vibrations has been ana- lyzed. 2. Formulation of the problem 2.1. Two-dimensional model – plate model A long prismatic thin-walled beam-column built of panels connected along longitudinal edges has been considered.Thebeam-column is simply supported at its ends. In order to account for all modes and coupled buckling, a plate model of the thin-walled beam-column has been assumed. The material the column is made of is subject to Hooke’s law. For each plate component, precise geometrical relationships are assumed to enable the consideration of both out-of-plane and in-plane bending of the i-th plate (Kolakowski andKrolak, 2006; Kowal-Michalska, 2007) εxi =ui,x+ 1 2 (w2i,x+v 2 i,x+u 2 i,x) εyi = vi,y+ 1 2 (w2i,y+u 2 i,y +v 2 i,y) 2εxyi = γxyi =ui,y +vi,x+wi,xwi,y+ui,xui,y+vi,xvi,y (2.1) and κxi =−wi,xx κyi =−wi,yy κxyi =−2wi,xy (2.2) where: ui,vi,wi – components of the displacement vector of the i-th plate along the xi,yi,zi axis direction, respectively, and the plane xiyi overlaps the central plane before its buckling. In the majority of publications devoted to stability of plate structures, the terms (v2i,x +u 2 i,x), (u 2 i,y + v 2 i,y) and (ui,xui,y + vi,xvi,y) are neglected for εxi,εyi,γxyi =2εxyi, correspondingly, in strain tensor components (2.1). 152 Z. Kołakowski The main limitation of the assumed theory lies in the assumption of li- near relationships between curvatures (2.2) and second derivatives of the di- splacement w (discussed in detail in, e.g., Opoka and Pietraszkiewicz, 2004; Pietraszkiewicz, 1989). This is themost often applied limitation in the theory of thin-walled structures. From Hamilton’s principle, having taken into account relationships (2.1) and (2.2), the following equations of motion are obtained Nx,x+Nxy,y+{(Nxu,x),x+(Nyu,y),y +(Nxyu,x),y+(Nxyu,y),x} +[−hρu,tt] = 0 Nxy,x+Ny,y +{(Nxv,x),x+(Nyv,y),y+(Nxyv,x),y +(Nxyv,y),x} +[−hρv,tt] = 0 Mx,xx+My,yy +2Mxy,xy+(Nxw,x),x+(Nyw,y),y+(Nxyw,x),y +(Nxyw,y),x+ [ −hρw,tt+ 1 12 h3ρ(w,xxtt+w,yytt) ] =0 (2.3) 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. In the case static issues are analyzed, all dynamic components should be neglected, that is to say, all terms in square brackets should be omitted in (2.3). After expanding the fields of displacements U and the fields of sectional forces N into a power series with respect to the mode amplitudes ζj (the di- mensionless amplitude of the j-thmode), Koiter’s asymptotic theory (Koiter, 1976; Koiter andPignataro, 1976; Koiter and van derNeut, 1980; Kolakowski and Kowal-Michalska, 1999; Kolakowski et al., 1999; Kowal-Michalska, 2007) has been employed U =λU (0) i + ζjU (j) i + . . . N =λN (0) i + ζjN (j) i + . . . (2.4) where λ is the scalar load parameter, U (0) i , N (0) i are the zero (e.g., pre- buckling) state fields, and U (j) i , N (j) i – j-th mode fields for the i-th plate. The range of indices is [1,J], where J is the number of interacting modes. For the case, the uncoupled mode for the axial extension mode are j= J =1. In all earlier publications by the author and his co-workers, it has been assumed that all the modes are normalized so that the maximum normal displacement of component plates of the j-th mode is equal to the first plate thickness h1. Due to the fact that the axial mode is taken into consideration, Some aspects of the axial extension mode... 153 it is assumed in the present paper that the absolute maximum value of one of the components of the displacement field of the j-thmode is equal to the first plate thickness h1. The zero approximation describes the pre-buckling state, whereas the first order approximation allows for determination of eigenvalue problems (critical loads/natural frequencies and buckling modes/eigenvectors), being the linear problem of stability and initial post-buckling equilibrium paths. Theobtained static systemof homogeneous ordinarydifferential equations, with the corresponding conditions of the interaction of walls, has been solved by the transition matrix method, having integrated numerically the equili- brium equations along the circumferential direction in order to obtain rela- tionships between the state vectors on two longitudinal edges. During the in- tegration of the equations, Godunov’s orthogonalization method is employed. For a more detailed analysis of the solution in the first nonlinear approxi- mation, see papers Kolakowski and Kowal-Michalska (1999), Kolakowski and Krolak (2006), Kolakowski et al. (1999), Kolakowski and Teter (2000), Te- ter andKolakowski (2004). The solutions for components of the displacement fields for the pre-buckling (zero) and the first order approximation are presen- ted in Kolakowski and Krolak (2006). The natural frequencies of free vibrations have been determined analogo- usly as in Teter andKolakowski (2003). If the components of membrane forces and displacements within the first order approximation are taken into account, a shear lag phenomenon and distortions of cross-sections can be considered then. For thin-walled structures with initial deflections, Lagrange’s equations of motion for the case of the interaction of J eigenmodes can be written as papers by Kowal-Michalska (2007), Schokker et al. (1996), Sridharan and Benito (1984), Teter and Kolakowski (2003) 1 ω2r ζr,tt+ ( 1− σ σr ) ζr+aijrζiζj − ζ∗r σ σr + . . .=0 for r=1, . . . ,J (2.5) where: ζr – dimensionless amplitude of the r-th buckling mode, σr = λr∆E (at r = 1,2, . . .) – critical stress instead of the load parameter λr of the r-th buckling mode, ωr, ζ ∗ r – frequency of vibrations and the dimensionless amplitudeof the initial deflection correspondingto the r-thmode, respectively. Theexpressions for aijr are tobe found inpapersbyByskov (1987-88), By- skov andHutchinson (1977), Kolakowski andKowal-Michalska (1999), Kowal- Michalska (2007). In equations of motion (2.5), the inertia forces of the pre- buckling state have been neglected (Kowal-Michalska, 2007; Schokker et al., 154 Z. Kołakowski 1996; Sridharan and Benito, 1984). The initial conditions have been assumed in the form ζr(t=0)=0 ζr,t(t=0)=0 (2.6) The static problem of interactive buckling of the thin-walled channel (i.e., for ζr,tt = 0 in (2.5)) has been solved with the method presented in Kola- kowski andKrolak (2006), the frequencies of vibrations have been determined analogously as in Teter and Kolakowski (2003), whereas the problem of inte- ractive dynamic buckling (2.5) has been solved bymeans of the Runge-Kutta numerical methodmodified by Hairer andWanner. At the point where the load parameter for static problems σ reaches its maximum value σs (the so-called theoretical load carrying capacity) for the imperfect structure with regard to the imperfection of the mode with the amplitude ζ∗r , the Jacobian of nonlinear system of equations (2.5) is equal to zero. 2.2. One-dimensional model – beam-column To simplify the analysis of the axial extension mode, a one-dimensional model of the beam-column has been assumed as well. In the cases of one-dimensional structural elements, the relationships are reduced significantly. When this element is loaded by the axial compression load only, one speaks of a column butwhen it is under flexural loads, one calls this element a beam. For one-dimensional structural elements, precise geometrical relationships (2.1) and equation (2.2) are reduced to the following form εx =u,x+ 1 2 [w2,x+u 2 ,x] κx =−w,xx (2.7) whereas equilibrium equations (2.3) take the form F̂,x+(F̂u,x),x−ρAu,tt =0 M̂,xx+(F̂w,x),x−ρAw,tt =0 (2.8) where: F̂ – longitudinal force, M̂ – bendingmoment. To consider the axial extensionmode for the one-dimensional column only, the bending effect has been neglected, i.e., it has been assumed that equations (2.7) and (2.8) are reduced to the form εx =u,x+ 1 2 [u2,x] F̂,x+(F̂u,x),x−ρAu,tt ≡ F̂,x+ F̂,xu,x+ F̂u,xx−ρAu,tt =0 (2.9) Some aspects of the axial extension mode... 155 To compare the results for the one- and two-dimensional models of the column, identical boundary conditions have been assumed (i.e., the same as for the plate model) u(x= ℓ/2)=0 u(x=0)=−u(x= ℓ) (2.10) which corresponds to both endsmoving and the bar being fixed in themiddle of its length ℓ. Thus, the zero and the first order solutions have been assumed u(0) =(ℓ/2−x)∆ u(1) =U(1) cos mπx ℓ (2.11) where ∆ is measure of applied pressure. Having considered (2.4) and (2.11), the strain and the longitudinal force can be expressed as εx =u,x+ 1 2 u2,x ≈−λ∆+ ζ1u(1),x (1−λ∆)+0(λ2,ζ21,λ2ζ1)=λε(0)x + ζ1ε(1)x F̂ =EAεx ≈EA(λε(0)x + ζ1ε(1)x )=λF̂(0)+ ζ1F̂(1) (2.12) and equation of motion (2.9)2 for the first-order approximation (i.e., with respect to ζ1 – the dimensionless amplitude of the axial extension mode) takes the form u(1),xx(1−3λ∆)− 1 c2 u (1) ,tt +0(λ 2,ζ21,λ 2ζ1)= 0 (2.13) where: c2 =E/ρ. In the above-mentioned equation, the inertia forces of the zero state have been neglected, which is a commonly applied reduction. For the static problem, the dynamic term, i.e., u (1) ,tt =0, should be neglec- ted in relationship (2.13). Considering (2.11) in the static version of equation (2.13), the critical eige- invalue is the nontrivial solution λcr = 1 3∆ (2.14) and the respective critical compression force and the critical stress are then equal to F̂cr = |(F̂)cr|=EAλcr∆= EA 3 σcr = F̂cr A = E 3 (2.15) 156 Z. Kołakowski The above relationship defines the critical stress corresponding to the axial extension mode. As can be easily seen, the static critical value corresponding to the axial mode neither depends on the bar length nor on the number of halfwaves that occur along the bar length ℓ. In the case of the linear dynamic problem, the frequency of free vibrations can be determined from relation (2.13). To achieve this, the forecast solution u(1) =U (1) eiωmtcos mπx ℓ (2.16) has been substituted into (2.13) and the relationship that determines the free vibration frequency corresponding to the axial extension mode for the bar under compression is obtained ωm = mπ ℓ √ E ρ (1−3λ∆) (2.17) For the bar that is not subject to the longitudinal force (i.e., F̂ =0or λ=0), we have ωm = mπ ℓ √ E ρ (2.18) As can be seen, the frequency of longitudinal vibrations of the bar corre- sponding to the axial extension mode is directly proportional to the number of halfwaves m and inversely proportional to the length ℓ. In equation (2.13), only the terms standing at ζ1 and λζ1 have been taken into account, whereas the terms of the type λ2ζ1 have been omitted. If we additionally consider the terms λ2ζ1 in (2.13), then the lowest critical value is expressed by the formula (λ1)cr = 3− √ 3 3∆ = 0.423 ∆ σcr =E(λ1)cr∆=0.423E (2.19) In the developed computer code, according to the characteristics of the solutionmethod presented inKolakowski andKrolak (2006), the critical value is equal to (λ1)cr = 3− √ 5 2∆ = 0.382 ∆ σcr =E(λ1)cr∆=0.382E (2.20) Some aspects of the axial extension mode... 157 Flexural vibrations of the beam To determine the frequency of free flexural vibrations of the beam, it has beenassumed that F̂ =0and the longitudinal inertia force has beenneglected (i.e., ρAu,tt = 0) in equations (2.8). In such a case, the system of equations (2.8), having taken into account that M̂x = −EIzw,xx, is reduced to the equation EIzw,xxxx+ρAw,tt =0 (2.21) When the following boundary conditions are assumed for the beam w(x=0)=w(x= ℓ)= 0 (2.22) then, the frequency of flexural free vibrations of the beam equals ̟m = (mπ ℓ )2 √ EIz ρA (2.23) The frequency of beam flexural vibrations is inversely proportional to the squared length of the beam and directly proportional to the squared number of halfwaves that occur along its length. 3. Analysis of the calculation results Toverify the convergence of the calculation results obtainedwith two different methods, the results obtained by Camotim and Denis (2009) have been com- pared to the author’s results. The results in Camotim and Denis (2009) were obtained for the elastic post-buckling behavior of the cold-formed steel lipped channel (Table 1 – case 3 inCamotim andDenis (2009)) and simply supported columns affected by coupled instabilities with local/distortional/global mode interaction effects throughABAQUS shell finite element analyses.Avery good agreement of the results has been received when the axial extension mode is neglected in the considerations. 3.1. Eigenvalue problem The most difficult for analysis within the first order approximation ca- se of the column cross-section having at least two axes of symmetry, i.e., a thin-walled square cross-section, has been assumed. A detailed analysis of the 158 Z. Kołakowski Table 1. Natural frequencies of free vibrations, critical stresses of the axial extension mode and ratios of maximum values of displacement components for the plate model corresponding to different lengths ℓ ωpl ωcol σpl σcol ∣ ∣ ∣ vmax umax ∣ ∣ ∣ ∣ ∣ ∣ wmax umax ∣ ∣ ∣ [mm] [rad/s] [rad/s] [MPa] [MPa] 100000 158.5 158.5 76393 66666 (Eq. (2.15)) 0.00026 0.00026 10000 1585 1585 76392 0.0026 0.0029 7500 2114 2114 76392 0.0035 0.0043 5000 3170 3171 76391 0.0052 0.0085 2500 6342 6342 76398 0.01 0.047 2000 7928 7928 76393 0.013 0.021 1000 15839 15857 76379 76400 (Eq. (2.20)) 0.026 0.026 750 21139 21143 76362 0.035 0.041 500 31642 31714 76447 0.051 0.238 250 62851 63429 76141 0.11 0.15 200 77423 79286 76352 0.13 0.31 100 175675 158573 75217 0.29 0.32 calculations is conducted for the compressed beam-column of a square cross- section with the following dimensions (Fig.2) (Kolakowski, 1993; Kolakowski et al., 1999; Teter and Kolakowski, 2004) b1 = b2 = b3 =100mm h1 =h2 =h3 =1mm Each plate is made of steel characterized by the following mechanical proper- ties E =210GPa ν =0.3 ρ=7850kg/m 3 Fig. 2. Geometry of the thin-walled beam-column In Table 1, values of the natural frequencies of vibrations and the criti- cal stresses for the axial extension mode corresponding to different lengths Some aspects of the axial extension mode... 159 100¬ ℓ¬ 100000mm of the column under investigation are shown. Here ωpl, ωcol denote thenatural frequencies for theplate and the columnmodel, respec- tively; σpl, σcol are the critical stresses for the plate and the column model, respectively. The table also includes the ratios of maximum values of displa- cement components for eachmode, i.e., |vmax/umax| and |wmax/umax| for the platemodel of the column. Themaximum values of displacement components umax, vmax, wmax for the particular column length may appear in different points of the beam-column cross-section. For the plate model of the column, vibration frequencies have been deter- mined, taking into account all components of inertia forces (Kowal-Michalska, 2007; Teter and Kolakowski, 2003). Values of free vibrations determined for both the models under investigation are identical in practice, except for the length ℓ=100mm,where the difference equals approx. 10%.Values of critical stresses, which are identical for the lengths considered for the assumedmodel, are collected. Values of critical stresses for both themodels in thewhole range of ℓ are virtually the same if σcol is determined with relationship (2.18). Values of critical stresses for the axial extension mode determined for the elastic range have very high stresses and one should totally agree with Adany and Schafer’s statement quoted in Introduction. For the plate model of the column whose length is 750¬ ℓ¬ 100000mm, maximum displacements in the cross-section plane (i.e., v, w) are equal to approx. 4% of the longitudinal displacements (i.e., u) at most. The displace- ments u are practically constant for the cross-section, for example x=0. For the axial mode, the displacements are equal to the thickness h1 (that is to say, umax ∼=h1). For the column length ℓ¬ 500mm (i.e., for the case when the dimensions of the cross-section are comparable to the length ℓ), the maximum values of displacement components |vmax|, |wmax| aremore than 10%|umax|. Here, the assumptionof a one-dimensionalmodel to determine the eigenvector can prove insufficient. A case of static and dynamic interactive buckling for which validity of this statement can be questioned is presented. Next, detailed computations for the analyzed beam-column of the given length, i.e., ℓ=2750mm, were conducted. In Table 2, values of the critical stresses σr, values of the natural frequ- encies of free vibrations ωr and their corresponding numbers of halfwaves m for the square beam-column are shown. 160 Z. Kołakowski Now, the following index symbols have been introduced: 1 – flexural global mode for m = 1; 2 – primary local mode for m = 28; 3 – secondary local mode m=28; 4 – axial extension mode for m=1. Table 2.Critical stresses σr and natural frequency ωr of the beam-column Index m σr [MPa] ωr [rad/s] 1 1 427.3 266.3 2 28 72.3 3069 3 28 102.7 3659 4 1 76415 5766 As obviously follows from Table 2, the vibration frequencies ω2 and ω4 satisfy the relation ω4 =1.88ω2 ≈ 2ω2, which in the case of system vibrations is veryunfavorable taking into account the energy exchangebetween theglobal axial extension mode and the primary local buckling mode. InTable 3, subsequentvalues of thenatural frequencies of freevibrations ω for m = 1 and the ratios of maximum values of displacement components for each mode, i.e., |vmax/umax| and |wmax/umax|, are displayed. The table includes also the condition on the symmetric axis of global bending corre- sponding to this mode. The following index symbols are introduced: A and S – antisymmetric and symmetric condition on the symmetric axis of global bending, respectively. The natural frequency of free vibrations corresponding to the axial extension mode is the fifth value of the free vibration frequency (ω=5766) for m=1. Table 3.Natural frequencies of vibrations ω for m=1 ω (m=1) ∣ ∣ ∣ wmax umax ∣ ∣ ∣ ∣ ∣ ∣ vmax umax ∣ ∣ ∣ Condition on the symmetric [rad/s] axis of global bending 266.3 17.9 17.7 A 1509 145967 18.9 S 2970 41.6 17.6 A 3416 205 0.129 S 5766 0.0186 0.0171 S (axial extension mode) 7646 325 16.2 A 3.2. Static interactive buckling Detailed results of the static interactive buckling analysis of the first order nonlinear approximation are presented in Table 4 for the beam-column. The imperfections assumed are: ζ∗1 = |1.0|, ζ∗2 = |0.2|, ζ∗3 = ζ∗4 =0. Some aspects of the axial extension mode... 161 In each case, the sign of the imperfections has been selected in the most unfavorable way, that is to say, as to obtain the lowest theoretical load carry- ing capability σS for the given level of imperfection when the interaction of buckling modes is accounted for. Table4 shows ratios of the theoretical load carryingcapacity to theprimary local critical stress σs/σ2 for the first order nonlinear approximation. It can be easily seen in Table 4 that when the axial mode is accounted for in the interaction, then the theoretical load carrying capacity σs is decre- ased considerably. A decrease in the load carrying capacity σS/σ2 does not exceed 33%. Table 4.Theoretical static load carrying capacity σs/σ2 and critical dynamic load factors DLFcr =σ BH D /σ2 J-mode Index of the σs/σ2 DLFcr approach coupled mode 4 1; 2; 3; 4 0.6730 0.593 3 1; 2; 3 0.820 0.840 3 1; 2; 4 0.6733 0.602 2 2; 4 0.6733 0.610 In the case of the three-mode approach only (i.e., global flexural mode, primary and secondary local modes) for the square cross-section, the coef- ficients responsible for the interaction of buckling modes a122 and a133 are identical with zero due to vanishing integrals denoted for the opposedwalls of the cross-section (accounting for the symmetry of the local buckling mode). This requires consideration of the interaction of the global flexural modewith the two local ones having the same number of buckling halfwaves m along the longitudinal direction. Apart from the fundamental local mode, the nontrivial local mode considered is referred to as the secondary one (Koiter and Pigna- taro, 1976; Koiter and van der Neut, 1980; Kolakowski andKowal-Michalska, 1999; Kolakowski and Krolak, 2006; Kolakowski and Teter, 2000; Moellmann andGoltermann, 1989; Sridharan andAli, 1986; Teter andKolakowski, 2004). The term a123, i.e., the term a123ζ1ζ2ζ3 in the potential energy expression, is responsible for the total effect of the interaction between the global flexural mode and local modes. When additionally the axial extension mode is accounted for in the inte- raction, the two nonlinear coefficients of system of equations (2.5), namely: a123 and a422, exert the main influence on the decrease in the load σs. 162 Z. Kołakowski The key role in the interaction of the buckling mode is played by the co- efficient a422, i.e., the term a422ζ4ζ 2 2 of the third order in the expression for potential energy. It is related to the term σ(4)L2(U (2)), following the nota- tion of Byskov and Hutchinson (Byskov, 1987-88; Byskov and Hutchinson, 1977; Kowal-Michalska, 2007). This term arises from the product stress σ(4) associatedwith the axialmode and the term representing themidsurface stra- in L2(U (2)) associated with the primary local mode and integrating the same over the structure. The integral tends to vanish unless the longitudinal wave lengths (e.g., the number of halfwaves m) are the same. It makes the coef- ficient a422 play such a key role for the first order approximation (compare cases 1, 3, 4 in Table 4). In this case, the global axial extension mode can interact even with one symmetric local buckling mode. This interaction of the axial extension mode with the primary local buc- klingmode can be better visualized by the fact that the additional axial com- pression N (4) ix (i.e., σ (4)) dramatically increases the deflections corresponding to the primary local bucklingmode w (2) i (i.e., L2(U (2))). Thus, it can be stated that the most dangerous mode in the nonlinear problemmay not comply with the linear analysis and themaximum values of the coefficients aijr should be defined on the basis of the nonlinear first order approximation. It follows from this comparison that the consideration of the axialmode in the interaction is necessary as it results in a visible decrease in the theoretical load carrying capacity in the first order approximation. 3.3. Dynamic response Further on, an analysis of dynamic interactive buckling (i.e., dynamic respon- se) of the square box beam-column under consideration is conducted. Identi- cally as in the static analysis, the interaction of the samemodes is considered. A detailed analysis is carried out for a rectangular pulse load σ(t) = σD for 0¬ t¬T1 and σ(t) = 0 for T1 < t. This case corresponds to the pulse dura- tion equal to the period of fundamental flexural free vibrations T1 =2π/ω1. InTable 4, values of the critical dynamic load factors DLFcr =σ BH D /σ2 for the squarebox for the same imperfectionsunderanalysis are given,where σBHD denotes the critical value of dynamic stress determined from the Budiansky- Hutchinson criterion (Budiansky and Hutchinson, 1966; Hutchinson and Bu- diansky, 1966). For the case of the beam-column dimensions under analysis, the obtained values of DLFcr are lower than the respective values of the theoretical static load carrying capacity σs/σ2, except for case 2 in Table 4. Some aspects of the axial extension mode... 163 As canbe easily seen inTable 4, the axial extensionmode shall be included in dynamic stability analysis. 4. Conclusion In the present paper, a way ofmodeling and eigenvalues of the axial extension mode for the one- and the two-dimensional models of the structural elements have been presented. A very good agreement between the results of critical loads and free vibrations has been obtained. Whendimensionsof the cross-section are comparable to the column length, the determined eigenvectors can differ considerably for both the models. In the study, special attention has been focused on static and dynamic coupled buckling of the local buckling mode with the global axial extension mode in the first nonlinear approximation of the perturbation method. In the world literature, it is probably the first study, to the author’s knowledge, devoted to dynamic interactive buckling of the global axial extension mode and the primary local mode. 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VolmirS.A., 1972,NonlinearDynamics ofPlates andShells, Science,Moscow, p.432 [in Russian] Pewne aspekty osiowej postaci wzdłużnej w sprężystej cienkościennej belce-słupie Streszczenie Wprezentowanej pracy omówionowpływ osiowej wzdłużnej postaci na statyczne i dynamiczne interakcyjne wyboczenie cienkościennej belki-słupa z niedokładnościa- mi poddanej równomiernemu ściskaniu przy uwzględnieniu zjawiska shear-lag oraz dystorsyjnej deformacji. Przyjęto płytowymodel (2D) belki-słupa. Porównano jedno- i dwuwymiarowe modele elementów. Konstrukcja jest przegubowo podparta na obu końcach. Równania ruchu płyt składowych otrzymano z zasady Hamiltona, biorąc 168 Z. Kołakowski pod uwagę wszystkie składowe sił bezwładności. Dynamiczne zagadnienie modalne- go interakcyjnego wyboczenia w ramach pierwszego rzędu nieliniowej aproksymacji rozwiązanometodąmacierzyprzeniesienia,wykorzystującmetodę perturbacyjną i or- togonalizację Godunova. Manuscript received April 18, 2011; accepted for print May 12, 2011