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Influence of the distortional-lateral buckling mode on the interactive buckling of short channels
Thin–Walled Structures 109 (2016) 296–303
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Influence of the distortional-lateral buckling mode on the interactive buckling of short channels
crossmark
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Zbigniew Kołakowski , Mariusz Urbaniak Lodz University of Technology Department of Strength of Materials (K12), PL-90-924 Lodz, ul. Stefanowskiego 1/15, Poland
A R T I C L E I N F O
A BS T RAC T
Keywords: Short channel Bending Distortional-lateral mode Interactive buckling Semi-analytical method Experimental tests
The present paper deals with an influence of the distortional-lateral buckling mode on the interactive buckling of thin-walled short channels with imperfections subjected to the bending moment when the shear lag phenomenon and distortional deformations are taken into account. A plate model (2D) is adopted for the channel section. The structure is assumed to be simply supported at the ends. A method of the modal solution to the coupled buckling problem within the Koiter's asymptotic theory, using the semi-analytical method (SAM) and the transition matrix method, was applied. The calculations and experimental preliminary tests were carried out for short channels.
1. Introduction Beams are the fundamental element in steel structures that carry loads mainly via bending. In the majority of cases, a possibility to manage comparatively high loads with thin-walled beams is limited not by their strength only but mainly by stability. The major development of research on stability of thin-walled isotropic structures took place in the 1970s and the 1980s. The exemplary papers dealing with local buckling of thin-walled structures are those written by Davids and Hancock [12]. Since the late 1980s, the Generalized Beam Theory (GBT) was developed extensively. At that time numerous studies employing such theories as: GBT, FSM (Finite Strip Method) and DSM (Direct Strength Method) [5,7,9,11,15,28,29,43] originated. In Refs. [2–4], a method for calculation of critical forces for pure bending of thin-walled beams, implemented on the basis of the Finite Strip Method (FSM), is presented, whereas decomposition of the buckling mode was conducted on the basis of assumptions taken from the GBT. This new method was referred to as the constrained Finite Strip Method (cFSM). The newest theoretical development trends in steel thin-walled structures are discussed in, e.g., [1,6,14,26,27,32–36]. Davies [13] described the development in research and analysis of thin-walled beams. The advancement in the buckling theory of thinwalled beams under buckling and compression was presented in Ref. [16]. In Ref. [10], models employed in the finite strip method are studied. The strength and local buckling of channel section and Zsection beams under bending were investigated numerically and experimentally in Ref. [46]. Stability of thin-walled beams and frames ⁎
was investigated numerically and described analytically in Ref. [44]. Finite-element (FE) software packages have long been used to analyse thin-walled structures. FE models can simulate the actual structural behaviour closely and, therefore, replace experiments. FE models are used to produce data for evaluation and revision of current design formulae [30,31]. The results of experimental investigations of steel thin-walled beams are shown [40–42]. In Refs. [38,39], special attention was paid to the distortional-global interaction buckling. Within the research project entitled”Experimental and numerical investigations of nonlinear stability of thin-walled composite structures” (DEC-2011/03/B/ST8/06447, founded by the National Centre for Sciences, Poland), experimental investigations were carried out for short composite channels subject to bending in the web plane (Fig. 1) as there is a lack of investigations devoted to the buckling analysis of such channel sections. The results pointed out especially to an effect of the global distortional-lateral buckling mode on postbuckling equilibrium paths. It was decided to explain that phenomenon in composite channels for a simpler case referring to steel channel sections under bending. In the present study, an influence of the distortional-lateral buckling mode on the interactive buckling of short steel channels is presented. The results of calculations are compared to the results of preliminary experimental tests.. The concept of interactive buckling (i.e., coupled buckling) that involves the general asymptotic theory of stability is fundamental for theoretical considerations. Among all versions of the general nonlinear theory, the Koiter's theory [17,18] of conservative systems is the most popular one, owing to its general character and development, even
Corresponding author. E-mail addresses: [email protected] (Z. Kołakowski), [email protected] (M. Urbaniak).
http://dx.doi.org/10.1016/j.tws.2016.10.002 Received 24 September 2015; Received in revised form 21 September 2016; Accepted 1 October 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.
Thin–Walled Structures 109 (2016) 296–303
Z. Kołakowski, M. Urbaniak
tage of this method is such that a complete range of behaviour of thinwalled structures from all global to the local stability [20,21,23,37,45] can be described. 2. Formulation of the problem A prismatic thin-walled isotropic channel built of plates connected along longitudinal edges and subjected to bending moment (Fig. 1) was considered. The girder was simply supported at its ends. In order to account for all modes of global, local and coupled buckling, a plate model of thin-walled structures was assumed. It was assumed that the isotropic material (i.e., steel) the structure was made of obeyed Hooke's law. For each plate component, precise geometrical relationships were assumed in order to enable the consideration of both out-of-plane and in-plane bending of the i-th plate [22–24]: 1
εxi = ui, x + 2 (wi2, x + vi2, x + ui2, x ) 1
εyi = vi, y + 2 (wi2, y + ui2, y + vi2, y ) 2εxyi = γxyi = ui, y + vi, x + wi, xwi, y + ui, xui, y + vi, xvi, y
(1)
and
κxi = − wi, xxκ yi = − wi, yyκxyi = − 2wi, xy
(2)
where: ui , vi , wi – components of the displacement vector of the i-th plate in the xi , yi , zi axis direction, respectively, and the plane xi − yi overlaps the central plane before its buckling. The nonlinear problem of stability was solved with the asymptotic perturbation method. Let λ be a load factor. The displacement fields U and the sectional force fields N (Koiter's type expansion for the buckling problem [8,17,18]) were expanded into power series with respect to the dimensionless amplitude of the r-th mode deflection ζr (normalized in the given case by the condition of equality of the maximum deflection to the thickness of the first component plate h1) (see [19–24,37,45]):
U ≡ (u , v , w ) = λU0 + ζrUr + ζr2Urr + ... N ≡ (Nx , Ny , Nxy ) = λN0 + ζrNr + ζr2Nrr + ...
(3)
where the prebuckling (i.e., unbending) fields are U0, N0 , the first nonlinear order fields are Ur , Nr (eigenvalues problems) and the second nonlinear order fields – Urr , Nrr , respectively. The range of indices is [1,J], where J is the number of interacting modes. The boundary conditions referring to the simply supported beamcolumns at their ends (i.e. x = 0; ℓ ) are assumed to be:
Fig. 1. Thin-walled channel subjected to bending in the web plane.
more so after Byskov and Hutchinson [8] formulated it in a convenient way. The nonlinear stability of thin-walled channels in the first order approximation of Koiter's theory is solved with the modified analyticalnumerical method (ANM) presented in Ref. [22]. The analyticalnumerical method (ANM) should consider also the second order approximation of the theory in the analysis of postbuckling of elastic structures. The second order postbuckling coefficients were estimated with the semi-analytical method (SAM) [19] modified by the solution method in Ref. [24]. The attained results were compared to the results of the”complete” second-order analysis for isotropic structures [19,21,24]. In the present study, a plate model (2D) of the short channel is adopted to describe all buckling modes. Instead of the finite strip method, the exact transition matrix method and the numerical method of the transition matrix using Godunov's orthogonalization is used. The differential equilibrium equations were obtained from the principle of virtual works taking into account: Lagrange's description, full Green's strain tensor for thin-walled plates and the second Piola-Kirchhoff's stress tensor. The interaction between all the walls of structures being taken into account, the shear lag phenomenon and also the effect of cross-sectional distortions were included. The most important advan-
b
b
∫0 Nx(x = 0, y)dy= ∫0 Nx(x = ℓ, y)dy= bNx(0) v(x = 0, y ) = v(x = ℓ, y ) = 0 w(x = 0, y ) = w(x = ℓ, y ) = 0 Mx (x = 0, y ) = Mx (x = ℓ, y ) = 0
(4)
The first condition in Eq. (4) means that the external loading is not subjected to any additional increment. If the structure contains the geometric imperfections U (only the linear initial imperfections determined by the shape of the r-th buckling modes), where U = ζr*Ur , then the total potential energy can be written in the form [8,19–24,37,45]:
Π=−
+
1 2 1 M a0 + 2 2 1 4
J
⎛
J
∑ arζr2⎜1 − ⎝
r =1 J
∑ brrrrζr3 − ∑ r
r
1 M⎞ ⎟+ 3 Mr ⎠
J
J
J
∑ ∑ ∑ apqrζpζqζr p
q
r
M ar ζr*ζr Mr
(5)
When the following notations are introduced:
apqr = apqr / ar brrrr = brrrr / ar 297
(6)
Thin–Walled Structures 109 (2016) 296–303
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then, the equilibrium equations corresponding to Eq. (5) are as follows:
⎛ M⎞ M * ξr r = 1, …, J ⎜1 − ⎟ξr + apqr ξpξq + brrrr ξr3 = Mr ⎠ Mr ⎝
Table 1 Buckling stresses σr for channels.
(7)
where: M is a magnitude of the applied bending moment, Mr , ζr , ζr* – the buckling moment instead of the load parameter λr of the r-th buckling mode, the dimensionless amplitude of the r-th buckling mode and the dimensionless amplitude of the initial deflection corresponding to the r-th buckling mode, respectively. The coefficients a 0 , ar , apqr and brrrr can be determined with the equations described in the literature [8,19,21,22,37,45]. In the semi-analytical method (SAM), one postulates to determine approximated values of the brrrr coefficients Eq. (7) on the basis of the linear buckling problem. This approach allows the values of the apqr coefficients Eq. (7) to be precisely determined, according to the applied nonlinear Byskov and Hutchinson theory [8,20,21]. On the basis of Castigliano's theorem, a relative angle of rotation of the girder in bending on the support as a function of the M / M1 load was determined through differentiation of the expression for the potential energy Eq. (5) with respect to M / M1 [20,21,37,45]:
M1 α M ⎡⎢ = 1+ α1 M1 ⎢⎣ Ma 0
J
∑ r =1
⎤ M1 ar ζr (0.5ζr + ζr*)⎥ ⎥⎦ Mr
t [mm]
index r
m
σr SAM [MPa]
σr CUFSM[47] [MPa]
0.5
1 2 3
3 1 1
25.7 64.0 1782
25.9 64.1 1766
1.2
1 2 3
3 1 1
148.1 360.9 4394
151.9 370.1 4618
E=200 GPa, ν =0.3. In the prebuckling state beams are subjected to linearly variable stresses caused by a bending moment in the web plane (i.e., the upper flange is compressed, whereas the lower one is subject to tension). In Table 1, values of critical loads with the corresponding number of halfwaves m of buckling along the longitudinal direction of the channel for the two cases of thickness (i.e., t=0.5; 1.2 mm) are presented. The following r index notations are introduced: 1 – the lowest value of the critical load corresponding to the local buckling mode for m ≠ 1, 2 – the value of the critical load corresponding to the local buckling mode form = 1, 3 – the value of the critical load corresponding to the global distortional-lateral buckling mode for m = 1. The results obtained with the free-to-use CUFSM software program [47] are presented as well. Very good conformity of the numerical results has been attained. As can be easily seen, the most significant differences between the two codes occur in the values of stresses σ3 for the case t=1.2 mm and they are equal to 5.1%. In Figs. 3–6, buckling modes and distributions of internal sectional
(8)
where α1 – minimal critical angle of rotation of the beam under pure bending, corresponding to the minimal value of the critical moment M1. 3. Analysis of the calculation results A detailed analysis of the calculations is conducted for thin-walled channels of two thicknesses with the following dimensions (Fig. 2):. b1=b3=40 mm, b2=80 mm, t=0.5 mm and 1.2 mm, ℓ =275 mm,
Fig. 2. Thin-walled channels.
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Fig. 5. Distributions of internal sectional forces Nx 2, Ny2, Nxy2 for t=0.5 mm.
Fig. 3. Buckling modes of channel for t=0.5 mm.
Fig. 6. Distributions of internal sectional forces Nx3, Ny3, Nxy3 for t=0.5 mm.
have a global distortional-lateral mode due to the fact that the plate model was adopted. The components of the forces Nxr are dominant (Figs. 5, 6), except for the lowest local mode (Fig. 4), when maximum values of components are comparable. For the case t=1.2 mm (Fig. 7), the buckling modes for r=1; 2; are distortional-local modes, similar to the mode in Fig. 3, whereas the mode for r=3 is a global distortional-lateral mode. This mode differs significantly from the global mode for t=0.5 mm (Fig. 2r=3). The component of the forces Nxr is then dominant also in this case (Figs. 8– 10). Detailed calculations for the nonlinear problem were carried out as well. The postbuckling equilibrium paths for the channels under
Fig. 4. Distributions of internal sectional forces Nx1, Ny1, Nxy1 for t=0.5 mm.
forces Nxr , Nyr , Nxyr are presented, respectively, for t=0.5 mm, whereas Figs. 7–10 show analogous plots for t=1.2 mm. Due to disproportions in components of the sectional forces Nxr , Nyr , Nxyr , a magnitude scale of the given sectional force is indicated for each buckling mode in each figure... For the beam of the wall thickness t=0.5 mm, the buckling modes (Fig. 3) for r=1; 2; are distortional-local modes, whereas for r=3 – we
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Fig. 9. Distributions of internal sectional forces Nx 2, Ny2, Nxy2 for t=1.2 mm.
Fig. 7. Buckling modes of channel for t=1.2 mm.
Fig. 10. Distributions of internal sectional forces Nx3, Ny3, Nxy3 for t=1.2 mm.
consideration were determined. The imperfections Eq. (7) assumed are: ζ1* = 0.2 , ζ2* = 1.0 , ζ3* = 1.0 . In each case, the sign of the imperfections was selected in the most unfavourable way [20–24,45]. In Figs. 11, 12 results of the postbuckling equilibrium path in the coordinate system representing the ratio of the bending moment to the minimal value of the critical moment M / M1 = σ / σ1 as a function of the angle of rotation of the channel on the support to the critical angle of rotation corresponding to the minimal value of the moment α / α1, determined from relationship Eq. (8), are depicted. Each figure shows the curves for the following approach:
Fig. 8. Distributions of internal sectional forces Nx1, Ny1, Nxy1 for t=1.2 mm.
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curve for the case (a) (curve 1,2) lies below, whereas for M / M1 > 5 – above the curve representing the case (b) (curve 1,3). Attention should be paid to the fact that for the case (a), we have σ2 / σ1 = 2.44 , whereas for the case (b) - σ3 / σ1 = 29.7, respectively. As is well-known from the literature, a turbulent interaction of buckling modes occurs for the cases when 0.8 ≤ σ2 /σ1 ≤ 1.2 or 0.8 ≤ σ3 /σ1 ≤ 1.2 . Thus, for the case (b), we have a very strong interaction of the distortional-local mode with the distortional-lateral mode. In the case of an interaction of three modes (J=3), unsteady postbuckling equilibrium paths were obtained. To presents them, a nonlinear system of equations describing the postbuckling equilibrium paths was solved with two methods. The first method consisted in solving the system via controlling an increment in external load (the curve denoted as 1,2,3-L in Fig. 12), in the second one, an increment in the local displacement amplitude ζ1 was controlled (the curve marked as 1,2,3-D in Fig. 12). One can easily notice that an interaction of two modes corresponding to the number of halfwaves m=1 (i.e., r=2 and r=3) and the local mode (r=1) makes unsteady equilibrium paths occur, which is an unexpected effect in the conducted analysis. It follows from much higher internal sectional forces, which affects directly values of the coefficientapqr in Eq. (7) than in the case t=0.5 mm. In the case (t=0.5 mm), an effect of nonsymmetrical equilibrium paths was been observed (Fig. 11). The method employed to solve the nonlinear stability problem is based on the modal approach within the elastic range, which makes an interpretation of the attained results much easier. These issues caused that it was decided to conduct preliminary experimental investigations for both cases of the channel thickness to confirm the assumptions taken in the theoretical considerations and the numerical models.
Fig. 11. Postbuckling paths in the coordinate system M /M1 as a function α /α1 for t=0.5 mm.
Fig. 12. Postbuckling paths in the coordinate system M /M1 as a function α /α1 for
4. Experimental investigations
t=1.2 mm.
• • • •
The channels under bending were subject to preliminary experimental investigations. Two models for each thickness of steel plates were made (i.e., t=0.5; 1.2 mm). For each sheet, the average values of material constants were determined, namely: E = 197 GPa, σY = 475 MPa. The sheet plate was cut with the water jet method, and the models were cold formed. The tests were conducted on an Instron universal testing machine modernized by Zwick-Roel, equipped with specially designed grips. The investigated composite beams were subjected to pure bending. This type of load was applied in a four-point bending test. A scheme of the performed bending test with dimensions describing the span of support and the span of load is presented in Fig. 13.. A special grip (Fig. 13) was designed and manufactured [25] to avoid stress concentration in the place where the beams were supported and loaded. The grips were used to ensure the load corresponding to pure bending. The tests were performed at a constant velocity of the cross-bar equal to 1.5 mm/min. The values of the loading force applied to the system and the
three-modal (J=3 in Eqs. (5) and (7)), i.e., when an interaction of all three possible modes is considered (i.e. the curve denoted as 1,2,3); two-modal (J=2) for two cases under consideration, accounting for an interaction of two modes: for r=1 and 2 (curve 1,2); (b) for r=1 and 3 (curve 1,3); one-modal (J=1) for r=1 (curve 1).
Each curve is described with the respective indices for the given buckling modes. In Fig. 11 (t=0.5 mm), the curves obtained for the one-modal approach (curve 1) and the two-modal approach in the case (b) (i.e., curve 1,3) are plotted. The curves overlap practically, whereas the curve referring to the three-modal approach (curve 1,2,3) has the lowest position, as should be expected. Fig. 12 shows more complex plots of the postbuckling equilibrium paths. The curve corresponding to the one-modal approach (curve 1) has the highest position, whereas for the two cases of the two-modal approach, the curves are very close to each other. For M / M1 ≤ 5, the
Fig. 13. Scheme of the load and the support for four-point bending test (all dimensions in mm).
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Fig. 14. The results of bending moment M and the angle of rotation α for t=0.5 mm, where: E-1, E-2 – experimental curve for test no. 1 and 2, respectively; T-0.5 – theoretical curve for three mode approach (for r=1, r=2 and r=3).
Fig. 16. The results of bending moment M and the angle of rotation α for t=1.2 mm, where: E-3, E-4 – experimental curve for test no. 3 and 4, respectively; 1–8 – theoretical curve for three mode approach for different combinations of the characters imperfections ζ1* = 0.2 , ζ2* = 1.0 , ζ3* = 1.0 .
displacement in the points where the load was applied were obtained directly from machine sensors. The models were subjected to testing until failure. The values of loads and vertical displacements of the machine traverse were recorded. These quantities were then recalculated into values of the bending moment M [Nm] and the angle of rotation α [deg] on the support of the channel investigated. Figs. 14 and 15 show collective results of the experimental investigations of the channel models under consideration. The results of the theoretical numerical calculations for the assumed values of imperfections are also presented in these figures to facilitate an interpretation of the results obtained with these two methods. The results of the theoretical calculations of the bending moment M and the angle of rotation α were transformed to the distance equal to 515 mm (Fig. 13) between the supports of the models. In Fig. 14 (for the case t=0.5 mm), the theoretical curve for the three-modal approach (curve T-0.5) agrees well with the two experimental curves (curve denoted as E-1, E-2). In Fig. 15 (for the case t=1.2 mm), a theoretical performance curve for three-mode approach (T-1.2) to control an increment in the external load is presented. The experimental investigations do not show the phenomenon of jump as suggested by the theoretical considerations for the three-mode approach. A shape of the theoretical curve can indicate a considerable effect of the distortional-lateral buckling mode on the interactive buckling, and, thus, on the postbuckling equilibrium paths. We can see a lack of conformity of the results obtained theoretically and experimentally. The experimental curves (E3, E-4) can correspond to other values of initial imperfections that the ones assumed in the theoretical considerations. In Fig. 16 a comparison of the experimental results (E-3, E-4) for the case t=1.2 mm with the results of numerical simulations for the
assigned absolute values of imperfections and all combinations of their signs is presented additionally. Those curves were obtained solving the system via controlling an increment in external load. In Fig. 16, the following notations of the curves for the combinations of imperfections signs were introduced, namely: 1 – ζ1* = 0.2 , ζ2* = − 1.0 , ζ3* = − 1.0 ; 2 – ζ1* = 0.2 , ζ2* = − 1.0 , ζ3* = 1.0 ; 3 – ζ1* = 0.2 , ζ2* = 1.0 , ζ3* = 1.0 ; 4 – ζ1* = 0.2 , ζ2* = 1.0 , ζ3* = − 1.0 ; 5 – ζ1* = − 0.2 , ζ2* = − 1.0 , ζ3* = − 1.0 ; 6 – ζ1* = − 0.2 , ζ2* = − 1.0 , ζ3* = 1.0 ; 7 – ζ1* = − 0.2 , ζ2* = 1.0 , ζ3* = 1.0 ; 8 – ζ1* = − 0.2 , ζ2* = 1.0 , ζ3* = − 1.0 . While solving nonlinear systems of Eq. (7) for cases 1–6 and 8, values of the Jacobian Eq. (7) equal to zero were obtained, and then a jump to a new equilibrium path took place. As one can easily observe, for cases 4 and 8 a double jump to new equilibrium paths occurs. In case 2, a jump into a new more rigid equilibrium path takes place. Only in case 7, along with an increase in the bending momentM , a monotonous increase in the angle of rotation α occurs. In this case, values of the deflection amplitude ζ3 are insignificant, and, moreover, the value of the Jacobian Eq. (7) is always positive and also grows monotonously with an increase in M . Sighs of the imperfection decrease considerably the sensitivity of interactions between modes 2 (the local buckling mode form = 1) and 3 (the distortional-lateral buckling mode for m = 1). The next step in the investigations should consist in the FEM analysis as a very universal and broad method, which will allow for verification of the presented theoretical considerations, whereas the attained results of the experimental investigations will enable the verification of both the numerical methods.
5. Conclusions An influence of the global distortional-lateral buckling mode on the interactive buckling of short channels subjected to bending in the web plane is presented. A nonlinear stability problem was solved with the SAM using Koiter's asymptotic theory in the modal description. Special attention was paid to theoretical unsteady postbuckling equilibrium paths. Preliminary experimental investigations were performed for steel channels. An effect of the buckling mode for m=1 on the results of investigations was observed. The theoretical and experimental investigations should be treated as an introduction to a more thorough analysis both in terms of theoretical considerations, employing the analytical-numerical methods and the FEM, as well as experimental investigations. Then, an influence of global modes on the interactive bucking of short channels under bending in the web plane can be comprehensively described.
Fig. 15. The results of bending moment M and the angle of rotation α for t=1.2 mm, where: E-3, E-4 – experimental curve for test no. 3 and 4, respectively; T-12 – theoretical curve for three mode approach (for r=1, r=2 and r=3).
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Acknowledgements The investigations were conducted under the DEC-2011/03/B/ ST8/06447 research project financed by the National Centre for Science, Poland. References [1] S. Adany, Flexural buckling of simply-supported thin-walled columns with consideration of membrane shear deformations: analytical solutions based on shell model, Thin-Walled Struct. 74 (2014) 36–48. [2] S.AdanyN.SilvestreB.W.SchaferD.Camotim, Buckling mode identification of thinwalled members: A comparison between cFSM and GBT approaches. Rasmussen K., Wilkinson T. (Eds. in: )Proceedings of the 5th InternationalConference on Coupled Instabilities in Metal Structures, Vol. 2, The University of Sydney, Australia, CIMS249-256, 2008. [3] S. Adany, B.W. Schafer, Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation, Thin-Walled Struct. 44 (2006) 563–584. [4] S. Adany, B.W. Schafer, Buckling mode decomposition of single-branched open cross-section members via finite strip method: application and examples, ThinWalled Struct. 44 (2006) 585–600. [5] C.BasagliaD.CamotimN.Silvestre, GBT-based analysis of the global buckling behaviour of thin-walled steel frames subjected to arbitrary loadings. in: Proceedings of the 5th InternationalConference on Advances in Steel Structures, Vol. 3, Singapore, 5–7 December309-315, 2007. [6] C. Basaglia, D. Camotim, N. Silvestre, Post-buckling analysis of thin-walled steel frames using generalised beam theory (GTB), Thin-Walled Struct. 62 (2013) 229–242. [7] P.BebinoD.DinisN.SilvestreD.Camotim, On the application of the direct strength method to cold-formed steel beams subjected to non-uniform bending. Proceedings of 5th International Conference on Advances in Steel Structures, Vol. 3, Singapore, 5-7 December 322–327, 2007. [8] E. Byskov, J.W. Hutchinson, Mode interaction in axially stiffened cylindrical shells, AIAA J. 15 (7) (1977) 941–948. [9] J. Becque, M. Lecce, K.J.R. Rasmussen, The direct strength method for stainless steel compression members, J. Constr. Steel Res. 64 (2008) 1231–1238. [10] D.CamotimN.Silvestre, Distortional buckling before and after Greg Hancock. Rasmussen K., Wilkinson T. (Eds. in: )Proceedings of the 5th InternationalConference on Coupled Instabilities in Metal Structures, Vol. 2, The University odf Sydney, Australia, CIMS 155–167, 2008. [11] Y.K. Cheung, L.G. Tham, Finite strip method, CRC Press, Boca Raton, Boston, London, New York, Washington, D.C., 1998. [12] A.J. Davids, G.J. Hancock, Compression tests of long welded I-section columns, J. Struct. Eng. 112 (10) (1986) 2281–2297. [13] J.M. Davies, Recent research advances in cold-formed steel structures, J. Constr. Steel Res. 55 (2000) 267–288. [14] P.B. Dinis, B. Young, D. Camotim, Local–distortional interaction in cold-formed steel rack-section columns, Thin-Walled Struct. 81 (2014) 185–194. [15] R. Greiner, M. Kettler, Interaction of bending and axial compression of stainless steel members, J. Constr. Steel Res. 64 (11) (2008) 1217–1224. [16] G.J. Hancock, Cold-formed steel structures, J. Constr. Res. 59 (2003) 473–487. [17] W.T.Koiter, General theory of mode interaction in stiffened plate and shell structures. Delft University, WTHDReport 590, 41, 1976. [18] W.T.KoiterM.Pignataro, A general theory for the interaction between local and overall buckling of stiffened panels. Delft University, WTHDReport 556, 49, 1976. [19] Z. Kolakowski, A semi-analytical method for the analysis of the interactive buckling of thin-walled elastic structures in the second order approximation, Int. J. Solids Struct. 33 (25) (1996) 3779–3790. [20] Selected problems of instabilities in composite structures, in: Z. Kolakowski, K. Kowal-Michalska (Eds.), A Series of Monographs, Technical University of Lodz, Poland, 1999. [21] Z. Kolakowski, M. Krolak, K. Kowal–Michalska, Modal interactive buckling of thin– walled composite beam–columns regarding distortional deformations, Int. J. Eng. Sci. 37 (1999) 1577–1596.
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Influence of the distortional-lateral buckling mode on the interactive buckling of short channels
crossmark
⁎
Zbigniew Kołakowski , Mariusz Urbaniak Lodz University of Technology Department of Strength of Materials (K12), PL-90-924 Lodz, ul. Stefanowskiego 1/15, Poland
A R T I C L E I N F O
A BS T RAC T
Keywords: Short channel Bending Distortional-lateral mode Interactive buckling Semi-analytical method Experimental tests
The present paper deals with an influence of the distortional-lateral buckling mode on the interactive buckling of thin-walled short channels with imperfections subjected to the bending moment when the shear lag phenomenon and distortional deformations are taken into account. A plate model (2D) is adopted for the channel section. The structure is assumed to be simply supported at the ends. A method of the modal solution to the coupled buckling problem within the Koiter's asymptotic theory, using the semi-analytical method (SAM) and the transition matrix method, was applied. The calculations and experimental preliminary tests were carried out for short channels.
1. Introduction Beams are the fundamental element in steel structures that carry loads mainly via bending. In the majority of cases, a possibility to manage comparatively high loads with thin-walled beams is limited not by their strength only but mainly by stability. The major development of research on stability of thin-walled isotropic structures took place in the 1970s and the 1980s. The exemplary papers dealing with local buckling of thin-walled structures are those written by Davids and Hancock [12]. Since the late 1980s, the Generalized Beam Theory (GBT) was developed extensively. At that time numerous studies employing such theories as: GBT, FSM (Finite Strip Method) and DSM (Direct Strength Method) [5,7,9,11,15,28,29,43] originated. In Refs. [2–4], a method for calculation of critical forces for pure bending of thin-walled beams, implemented on the basis of the Finite Strip Method (FSM), is presented, whereas decomposition of the buckling mode was conducted on the basis of assumptions taken from the GBT. This new method was referred to as the constrained Finite Strip Method (cFSM). The newest theoretical development trends in steel thin-walled structures are discussed in, e.g., [1,6,14,26,27,32–36]. Davies [13] described the development in research and analysis of thin-walled beams. The advancement in the buckling theory of thinwalled beams under buckling and compression was presented in Ref. [16]. In Ref. [10], models employed in the finite strip method are studied. The strength and local buckling of channel section and Zsection beams under bending were investigated numerically and experimentally in Ref. [46]. Stability of thin-walled beams and frames ⁎
was investigated numerically and described analytically in Ref. [44]. Finite-element (FE) software packages have long been used to analyse thin-walled structures. FE models can simulate the actual structural behaviour closely and, therefore, replace experiments. FE models are used to produce data for evaluation and revision of current design formulae [30,31]. The results of experimental investigations of steel thin-walled beams are shown [40–42]. In Refs. [38,39], special attention was paid to the distortional-global interaction buckling. Within the research project entitled”Experimental and numerical investigations of nonlinear stability of thin-walled composite structures” (DEC-2011/03/B/ST8/06447, founded by the National Centre for Sciences, Poland), experimental investigations were carried out for short composite channels subject to bending in the web plane (Fig. 1) as there is a lack of investigations devoted to the buckling analysis of such channel sections. The results pointed out especially to an effect of the global distortional-lateral buckling mode on postbuckling equilibrium paths. It was decided to explain that phenomenon in composite channels for a simpler case referring to steel channel sections under bending. In the present study, an influence of the distortional-lateral buckling mode on the interactive buckling of short steel channels is presented. The results of calculations are compared to the results of preliminary experimental tests.. The concept of interactive buckling (i.e., coupled buckling) that involves the general asymptotic theory of stability is fundamental for theoretical considerations. Among all versions of the general nonlinear theory, the Koiter's theory [17,18] of conservative systems is the most popular one, owing to its general character and development, even
Corresponding author. E-mail addresses: [email protected] (Z. Kołakowski), [email protected] (M. Urbaniak).
http://dx.doi.org/10.1016/j.tws.2016.10.002 Received 24 September 2015; Received in revised form 21 September 2016; Accepted 1 October 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.
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Z. Kołakowski, M. Urbaniak
tage of this method is such that a complete range of behaviour of thinwalled structures from all global to the local stability [20,21,23,37,45] can be described. 2. Formulation of the problem A prismatic thin-walled isotropic channel built of plates connected along longitudinal edges and subjected to bending moment (Fig. 1) was considered. The girder was simply supported at its ends. In order to account for all modes of global, local and coupled buckling, a plate model of thin-walled structures was assumed. It was assumed that the isotropic material (i.e., steel) the structure was made of obeyed Hooke's law. For each plate component, precise geometrical relationships were assumed in order to enable the consideration of both out-of-plane and in-plane bending of the i-th plate [22–24]: 1
εxi = ui, x + 2 (wi2, x + vi2, x + ui2, x ) 1
εyi = vi, y + 2 (wi2, y + ui2, y + vi2, y ) 2εxyi = γxyi = ui, y + vi, x + wi, xwi, y + ui, xui, y + vi, xvi, y
(1)
and
κxi = − wi, xxκ yi = − wi, yyκxyi = − 2wi, xy
(2)
where: ui , vi , wi – components of the displacement vector of the i-th plate in the xi , yi , zi axis direction, respectively, and the plane xi − yi overlaps the central plane before its buckling. The nonlinear problem of stability was solved with the asymptotic perturbation method. Let λ be a load factor. The displacement fields U and the sectional force fields N (Koiter's type expansion for the buckling problem [8,17,18]) were expanded into power series with respect to the dimensionless amplitude of the r-th mode deflection ζr (normalized in the given case by the condition of equality of the maximum deflection to the thickness of the first component plate h1) (see [19–24,37,45]):
U ≡ (u , v , w ) = λU0 + ζrUr + ζr2Urr + ... N ≡ (Nx , Ny , Nxy ) = λN0 + ζrNr + ζr2Nrr + ...
(3)
where the prebuckling (i.e., unbending) fields are U0, N0 , the first nonlinear order fields are Ur , Nr (eigenvalues problems) and the second nonlinear order fields – Urr , Nrr , respectively. The range of indices is [1,J], where J is the number of interacting modes. The boundary conditions referring to the simply supported beamcolumns at their ends (i.e. x = 0; ℓ ) are assumed to be:
Fig. 1. Thin-walled channel subjected to bending in the web plane.
more so after Byskov and Hutchinson [8] formulated it in a convenient way. The nonlinear stability of thin-walled channels in the first order approximation of Koiter's theory is solved with the modified analyticalnumerical method (ANM) presented in Ref. [22]. The analyticalnumerical method (ANM) should consider also the second order approximation of the theory in the analysis of postbuckling of elastic structures. The second order postbuckling coefficients were estimated with the semi-analytical method (SAM) [19] modified by the solution method in Ref. [24]. The attained results were compared to the results of the”complete” second-order analysis for isotropic structures [19,21,24]. In the present study, a plate model (2D) of the short channel is adopted to describe all buckling modes. Instead of the finite strip method, the exact transition matrix method and the numerical method of the transition matrix using Godunov's orthogonalization is used. The differential equilibrium equations were obtained from the principle of virtual works taking into account: Lagrange's description, full Green's strain tensor for thin-walled plates and the second Piola-Kirchhoff's stress tensor. The interaction between all the walls of structures being taken into account, the shear lag phenomenon and also the effect of cross-sectional distortions were included. The most important advan-
b
b
∫0 Nx(x = 0, y)dy= ∫0 Nx(x = ℓ, y)dy= bNx(0) v(x = 0, y ) = v(x = ℓ, y ) = 0 w(x = 0, y ) = w(x = ℓ, y ) = 0 Mx (x = 0, y ) = Mx (x = ℓ, y ) = 0
(4)
The first condition in Eq. (4) means that the external loading is not subjected to any additional increment. If the structure contains the geometric imperfections U (only the linear initial imperfections determined by the shape of the r-th buckling modes), where U = ζr*Ur , then the total potential energy can be written in the form [8,19–24,37,45]:
Π=−
+
1 2 1 M a0 + 2 2 1 4
J
⎛
J
∑ arζr2⎜1 − ⎝
r =1 J
∑ brrrrζr3 − ∑ r
r
1 M⎞ ⎟+ 3 Mr ⎠
J
J
J
∑ ∑ ∑ apqrζpζqζr p
q
r
M ar ζr*ζr Mr
(5)
When the following notations are introduced:
apqr = apqr / ar brrrr = brrrr / ar 297
(6)
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then, the equilibrium equations corresponding to Eq. (5) are as follows:
⎛ M⎞ M * ξr r = 1, …, J ⎜1 − ⎟ξr + apqr ξpξq + brrrr ξr3 = Mr ⎠ Mr ⎝
Table 1 Buckling stresses σr for channels.
(7)
where: M is a magnitude of the applied bending moment, Mr , ζr , ζr* – the buckling moment instead of the load parameter λr of the r-th buckling mode, the dimensionless amplitude of the r-th buckling mode and the dimensionless amplitude of the initial deflection corresponding to the r-th buckling mode, respectively. The coefficients a 0 , ar , apqr and brrrr can be determined with the equations described in the literature [8,19,21,22,37,45]. In the semi-analytical method (SAM), one postulates to determine approximated values of the brrrr coefficients Eq. (7) on the basis of the linear buckling problem. This approach allows the values of the apqr coefficients Eq. (7) to be precisely determined, according to the applied nonlinear Byskov and Hutchinson theory [8,20,21]. On the basis of Castigliano's theorem, a relative angle of rotation of the girder in bending on the support as a function of the M / M1 load was determined through differentiation of the expression for the potential energy Eq. (5) with respect to M / M1 [20,21,37,45]:
M1 α M ⎡⎢ = 1+ α1 M1 ⎢⎣ Ma 0
J
∑ r =1
⎤ M1 ar ζr (0.5ζr + ζr*)⎥ ⎥⎦ Mr
t [mm]
index r
m
σr SAM [MPa]
σr CUFSM[47] [MPa]
0.5
1 2 3
3 1 1
25.7 64.0 1782
25.9 64.1 1766
1.2
1 2 3
3 1 1
148.1 360.9 4394
151.9 370.1 4618
E=200 GPa, ν =0.3. In the prebuckling state beams are subjected to linearly variable stresses caused by a bending moment in the web plane (i.e., the upper flange is compressed, whereas the lower one is subject to tension). In Table 1, values of critical loads with the corresponding number of halfwaves m of buckling along the longitudinal direction of the channel for the two cases of thickness (i.e., t=0.5; 1.2 mm) are presented. The following r index notations are introduced: 1 – the lowest value of the critical load corresponding to the local buckling mode for m ≠ 1, 2 – the value of the critical load corresponding to the local buckling mode form = 1, 3 – the value of the critical load corresponding to the global distortional-lateral buckling mode for m = 1. The results obtained with the free-to-use CUFSM software program [47] are presented as well. Very good conformity of the numerical results has been attained. As can be easily seen, the most significant differences between the two codes occur in the values of stresses σ3 for the case t=1.2 mm and they are equal to 5.1%. In Figs. 3–6, buckling modes and distributions of internal sectional
(8)
where α1 – minimal critical angle of rotation of the beam under pure bending, corresponding to the minimal value of the critical moment M1. 3. Analysis of the calculation results A detailed analysis of the calculations is conducted for thin-walled channels of two thicknesses with the following dimensions (Fig. 2):. b1=b3=40 mm, b2=80 mm, t=0.5 mm and 1.2 mm, ℓ =275 mm,
Fig. 2. Thin-walled channels.
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Fig. 5. Distributions of internal sectional forces Nx 2, Ny2, Nxy2 for t=0.5 mm.
Fig. 3. Buckling modes of channel for t=0.5 mm.
Fig. 6. Distributions of internal sectional forces Nx3, Ny3, Nxy3 for t=0.5 mm.
have a global distortional-lateral mode due to the fact that the plate model was adopted. The components of the forces Nxr are dominant (Figs. 5, 6), except for the lowest local mode (Fig. 4), when maximum values of components are comparable. For the case t=1.2 mm (Fig. 7), the buckling modes for r=1; 2; are distortional-local modes, similar to the mode in Fig. 3, whereas the mode for r=3 is a global distortional-lateral mode. This mode differs significantly from the global mode for t=0.5 mm (Fig. 2r=3). The component of the forces Nxr is then dominant also in this case (Figs. 8– 10). Detailed calculations for the nonlinear problem were carried out as well. The postbuckling equilibrium paths for the channels under
Fig. 4. Distributions of internal sectional forces Nx1, Ny1, Nxy1 for t=0.5 mm.
forces Nxr , Nyr , Nxyr are presented, respectively, for t=0.5 mm, whereas Figs. 7–10 show analogous plots for t=1.2 mm. Due to disproportions in components of the sectional forces Nxr , Nyr , Nxyr , a magnitude scale of the given sectional force is indicated for each buckling mode in each figure... For the beam of the wall thickness t=0.5 mm, the buckling modes (Fig. 3) for r=1; 2; are distortional-local modes, whereas for r=3 – we
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Fig. 9. Distributions of internal sectional forces Nx 2, Ny2, Nxy2 for t=1.2 mm.
Fig. 7. Buckling modes of channel for t=1.2 mm.
Fig. 10. Distributions of internal sectional forces Nx3, Ny3, Nxy3 for t=1.2 mm.
consideration were determined. The imperfections Eq. (7) assumed are: ζ1* = 0.2 , ζ2* = 1.0 , ζ3* = 1.0 . In each case, the sign of the imperfections was selected in the most unfavourable way [20–24,45]. In Figs. 11, 12 results of the postbuckling equilibrium path in the coordinate system representing the ratio of the bending moment to the minimal value of the critical moment M / M1 = σ / σ1 as a function of the angle of rotation of the channel on the support to the critical angle of rotation corresponding to the minimal value of the moment α / α1, determined from relationship Eq. (8), are depicted. Each figure shows the curves for the following approach:
Fig. 8. Distributions of internal sectional forces Nx1, Ny1, Nxy1 for t=1.2 mm.
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curve for the case (a) (curve 1,2) lies below, whereas for M / M1 > 5 – above the curve representing the case (b) (curve 1,3). Attention should be paid to the fact that for the case (a), we have σ2 / σ1 = 2.44 , whereas for the case (b) - σ3 / σ1 = 29.7, respectively. As is well-known from the literature, a turbulent interaction of buckling modes occurs for the cases when 0.8 ≤ σ2 /σ1 ≤ 1.2 or 0.8 ≤ σ3 /σ1 ≤ 1.2 . Thus, for the case (b), we have a very strong interaction of the distortional-local mode with the distortional-lateral mode. In the case of an interaction of three modes (J=3), unsteady postbuckling equilibrium paths were obtained. To presents them, a nonlinear system of equations describing the postbuckling equilibrium paths was solved with two methods. The first method consisted in solving the system via controlling an increment in external load (the curve denoted as 1,2,3-L in Fig. 12), in the second one, an increment in the local displacement amplitude ζ1 was controlled (the curve marked as 1,2,3-D in Fig. 12). One can easily notice that an interaction of two modes corresponding to the number of halfwaves m=1 (i.e., r=2 and r=3) and the local mode (r=1) makes unsteady equilibrium paths occur, which is an unexpected effect in the conducted analysis. It follows from much higher internal sectional forces, which affects directly values of the coefficientapqr in Eq. (7) than in the case t=0.5 mm. In the case (t=0.5 mm), an effect of nonsymmetrical equilibrium paths was been observed (Fig. 11). The method employed to solve the nonlinear stability problem is based on the modal approach within the elastic range, which makes an interpretation of the attained results much easier. These issues caused that it was decided to conduct preliminary experimental investigations for both cases of the channel thickness to confirm the assumptions taken in the theoretical considerations and the numerical models.
Fig. 11. Postbuckling paths in the coordinate system M /M1 as a function α /α1 for t=0.5 mm.
Fig. 12. Postbuckling paths in the coordinate system M /M1 as a function α /α1 for
4. Experimental investigations
t=1.2 mm.
• • • •
The channels under bending were subject to preliminary experimental investigations. Two models for each thickness of steel plates were made (i.e., t=0.5; 1.2 mm). For each sheet, the average values of material constants were determined, namely: E = 197 GPa, σY = 475 MPa. The sheet plate was cut with the water jet method, and the models were cold formed. The tests were conducted on an Instron universal testing machine modernized by Zwick-Roel, equipped with specially designed grips. The investigated composite beams were subjected to pure bending. This type of load was applied in a four-point bending test. A scheme of the performed bending test with dimensions describing the span of support and the span of load is presented in Fig. 13.. A special grip (Fig. 13) was designed and manufactured [25] to avoid stress concentration in the place where the beams were supported and loaded. The grips were used to ensure the load corresponding to pure bending. The tests were performed at a constant velocity of the cross-bar equal to 1.5 mm/min. The values of the loading force applied to the system and the
three-modal (J=3 in Eqs. (5) and (7)), i.e., when an interaction of all three possible modes is considered (i.e. the curve denoted as 1,2,3); two-modal (J=2) for two cases under consideration, accounting for an interaction of two modes: for r=1 and 2 (curve 1,2); (b) for r=1 and 3 (curve 1,3); one-modal (J=1) for r=1 (curve 1).
Each curve is described with the respective indices for the given buckling modes. In Fig. 11 (t=0.5 mm), the curves obtained for the one-modal approach (curve 1) and the two-modal approach in the case (b) (i.e., curve 1,3) are plotted. The curves overlap practically, whereas the curve referring to the three-modal approach (curve 1,2,3) has the lowest position, as should be expected. Fig. 12 shows more complex plots of the postbuckling equilibrium paths. The curve corresponding to the one-modal approach (curve 1) has the highest position, whereas for the two cases of the two-modal approach, the curves are very close to each other. For M / M1 ≤ 5, the
Fig. 13. Scheme of the load and the support for four-point bending test (all dimensions in mm).
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Fig. 14. The results of bending moment M and the angle of rotation α for t=0.5 mm, where: E-1, E-2 – experimental curve for test no. 1 and 2, respectively; T-0.5 – theoretical curve for three mode approach (for r=1, r=2 and r=3).
Fig. 16. The results of bending moment M and the angle of rotation α for t=1.2 mm, where: E-3, E-4 – experimental curve for test no. 3 and 4, respectively; 1–8 – theoretical curve for three mode approach for different combinations of the characters imperfections ζ1* = 0.2 , ζ2* = 1.0 , ζ3* = 1.0 .
displacement in the points where the load was applied were obtained directly from machine sensors. The models were subjected to testing until failure. The values of loads and vertical displacements of the machine traverse were recorded. These quantities were then recalculated into values of the bending moment M [Nm] and the angle of rotation α [deg] on the support of the channel investigated. Figs. 14 and 15 show collective results of the experimental investigations of the channel models under consideration. The results of the theoretical numerical calculations for the assumed values of imperfections are also presented in these figures to facilitate an interpretation of the results obtained with these two methods. The results of the theoretical calculations of the bending moment M and the angle of rotation α were transformed to the distance equal to 515 mm (Fig. 13) between the supports of the models. In Fig. 14 (for the case t=0.5 mm), the theoretical curve for the three-modal approach (curve T-0.5) agrees well with the two experimental curves (curve denoted as E-1, E-2). In Fig. 15 (for the case t=1.2 mm), a theoretical performance curve for three-mode approach (T-1.2) to control an increment in the external load is presented. The experimental investigations do not show the phenomenon of jump as suggested by the theoretical considerations for the three-mode approach. A shape of the theoretical curve can indicate a considerable effect of the distortional-lateral buckling mode on the interactive buckling, and, thus, on the postbuckling equilibrium paths. We can see a lack of conformity of the results obtained theoretically and experimentally. The experimental curves (E3, E-4) can correspond to other values of initial imperfections that the ones assumed in the theoretical considerations. In Fig. 16 a comparison of the experimental results (E-3, E-4) for the case t=1.2 mm with the results of numerical simulations for the
assigned absolute values of imperfections and all combinations of their signs is presented additionally. Those curves were obtained solving the system via controlling an increment in external load. In Fig. 16, the following notations of the curves for the combinations of imperfections signs were introduced, namely: 1 – ζ1* = 0.2 , ζ2* = − 1.0 , ζ3* = − 1.0 ; 2 – ζ1* = 0.2 , ζ2* = − 1.0 , ζ3* = 1.0 ; 3 – ζ1* = 0.2 , ζ2* = 1.0 , ζ3* = 1.0 ; 4 – ζ1* = 0.2 , ζ2* = 1.0 , ζ3* = − 1.0 ; 5 – ζ1* = − 0.2 , ζ2* = − 1.0 , ζ3* = − 1.0 ; 6 – ζ1* = − 0.2 , ζ2* = − 1.0 , ζ3* = 1.0 ; 7 – ζ1* = − 0.2 , ζ2* = 1.0 , ζ3* = 1.0 ; 8 – ζ1* = − 0.2 , ζ2* = 1.0 , ζ3* = − 1.0 . While solving nonlinear systems of Eq. (7) for cases 1–6 and 8, values of the Jacobian Eq. (7) equal to zero were obtained, and then a jump to a new equilibrium path took place. As one can easily observe, for cases 4 and 8 a double jump to new equilibrium paths occurs. In case 2, a jump into a new more rigid equilibrium path takes place. Only in case 7, along with an increase in the bending momentM , a monotonous increase in the angle of rotation α occurs. In this case, values of the deflection amplitude ζ3 are insignificant, and, moreover, the value of the Jacobian Eq. (7) is always positive and also grows monotonously with an increase in M . Sighs of the imperfection decrease considerably the sensitivity of interactions between modes 2 (the local buckling mode form = 1) and 3 (the distortional-lateral buckling mode for m = 1). The next step in the investigations should consist in the FEM analysis as a very universal and broad method, which will allow for verification of the presented theoretical considerations, whereas the attained results of the experimental investigations will enable the verification of both the numerical methods.
5. Conclusions An influence of the global distortional-lateral buckling mode on the interactive buckling of short channels subjected to bending in the web plane is presented. A nonlinear stability problem was solved with the SAM using Koiter's asymptotic theory in the modal description. Special attention was paid to theoretical unsteady postbuckling equilibrium paths. Preliminary experimental investigations were performed for steel channels. An effect of the buckling mode for m=1 on the results of investigations was observed. The theoretical and experimental investigations should be treated as an introduction to a more thorough analysis both in terms of theoretical considerations, employing the analytical-numerical methods and the FEM, as well as experimental investigations. Then, an influence of global modes on the interactive bucking of short channels under bending in the web plane can be comprehensively described.
Fig. 15. The results of bending moment M and the angle of rotation α for t=1.2 mm, where: E-3, E-4 – experimental curve for test no. 3 and 4, respectively; T-12 – theoretical curve for three mode approach (for r=1, r=2 and r=3).
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