Abstract
The investigation concerns local buckling of compressed flanges of axially compressed composite channel columns. Cooperation of the member flange and web is taken into account here. The buckling mode of the member flange is defined by rotation angle a flange about the line of its connection with the web. The channel column under investigation is made of unidirectional fibre-reinforced laminate. Two approaches to member orthotropic material modelling are performed: the homogenization with the aid of theory of mixture and periodicity cell or homogenization upon the Voigt–Reuss hypothesis. The fundamental differential equation of local buckling is derived with the aid of the stationary total potential energy principle. The critical buckling stress corresponding to a number of buckling half-waves is assumed to be a minimum eigenvalue of the equation. Some numerical examples dealing with columns are given here. The analytical results are compared with the finite element stability analysis carried out by means of ABAQUS software. The paper is focused on a close analytical solution of the critical buckling stress and the associated buckling mode while the web–flange cooperation is assumed.
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Abbreviations
- A :
-
Column cross-section area
- b :
-
Flange width
- \(D_\mathrm{l}\) :
-
Modulus of elasticity in longitudinal direction
- \(D_\mathrm{t}\) :
-
Modulus of elasticity in transverse direction
- \(E_\mathrm{f}\) :
-
Modulus of elasticity of fibre
- \(E_\mathrm{m}\) :
-
Modulus of elasticity of matrix
- \(E_\mathrm{l}\) :
-
Young’s modulus of elasticity in longitudinal direction
- \(E_\mathrm{t}\) :
-
Young’s modulus of elasticity in transverse direction
- G :
-
Shear modulus
- f :
-
Fibre volume fraction
- h :
-
Web height
- \(J_d\) :
-
Free torsion moment of inertia of the flange cross section
- \(J_\mathrm{w}\) :
-
Moment of inertia of the web cross section
- \(J_{x}, J_{y}\) :
-
Moment of inertia of the flange cross section
- L :
-
Length of the column
- \(L_0\) :
-
Characteristic length of the column
- n :
-
Number of half-waves of a buckling mode
- \(\delta \) :
-
Wall thickness
- V :
-
Total potential energy
- \(V_\mathrm{e}\) :
-
Elastic strain energy
- \(V_\mathrm{l}\) :
-
Potential energy of applied loads
- \(V_\mathrm{s}\) :
-
Potential energy of uniformly distributed springs
- \(V_\mathrm{w}\) :
-
Potential energy of web
- x, y, z :
-
Cartesian coordinate system
- \(\eta \) :
-
Ratio of the member length to characteristic one
- \(v_{\mathrm{lt}}, v_{\mathrm{tl}}\) :
-
Poisson’s ratios
- v(z):
-
Deflection of the flange due to the rotation angle
- u(z):
-
Deflection of the web due to the rotation angle
- \(\mu \) :
-
Coefficient of the flange–web cooperation in longitudinal direction
- \(\chi \) :
-
Coefficient of the flange–web cooperation in transverse direction
- \(\sigma _{\mathrm{cr}}\) :
-
Critical local buckling stress
- \(\sigma _{\mathrm{cr,min}}\) :
-
Minimum critical local buckling stress
- \(\sigma _\mathrm{b}\) :
-
Local buckling stress
- \(\theta (z)\) :
-
Angle of rotation of the flange
References
Bambach, M.: Compression strength of natural fibre composite plates and sections of flax, jute and hemp. Thin-Walled Struct. 119, 103–113 (2017)
Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0626-x
Berthelot, J.: Composite Materials—Mechanical Behaviour and Structural Analysis. Springer, New York (1999)
Daniel, L., Ishai, O.: Engineering Mechanics of Composite Materials. Oxford University Press, Oxford (2006)
Debski, H., Kubiak, T., Teter, A.: Buckling and post-buckling behaviour of thin-walled composite channel section column. Comput. Struct. 100, 195–204 (2013)
Debski, H., Kubiak, T., Teter, A.: Experimental investigation of channel-section composite profiles behavior with various sequences of plies subjected to static compression. Thin-Walled Struct. 71, 147–154 (2013)
Debski, H., Kubiak, T., Teter, A.: Numerical and experimental studies of compressed composite columns with complex open cross-sections. Comput. Struct. 118, 28–36 (2014)
dell’Isola, F., Steigmann, D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118, 113–125 (2015)
Gelfand, I., Fomin, S.: In: Silverman, R.A. (ed.) Calculus of Variations. Selected Russian Publications in the Mathematical Science. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1963)
Giorgio, I., Corte, A.D., del’Isola, F., Steigmann, D.: buckling modes in pantographic lattices. Comptes Rendus Mec. 344(7), 487–501 (2016)
Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A Math. Phys. Eng. Sci. 473, 2207 (2017)
Habbit, D., Karlsson, B., Sorensen, P.: ABAQUS analysis user’s manual. Hibbit, Karlsson, Sorensen Inc
Jones, R.: Mechanics of Composites Materials. Taylor & Francis, Milton Park (1999)
Kaw, A.: Mechanics of Composite Materials. Taylor & Francis, Milton Park (2006)
Kelly, A. (ed.): Concise Encyclopedia of Composite Materials. Pergamon Press, Oxford (1989)
Kim, N., Jeon, C.: Coupled static and dynamic analyses of shear deformable composite beams with channel-sections. Mech. Based Des. Struct. Mach. 41, 489–511 (2013)
Kołakowski, Z., Kowal-Michalska, K. (eds.): Selected Problems of Instabilities in Composite Structures. A Series Monographs. Technical University of Lodz (1999)
Kollar, L., Springer, G.: Mechanics of Composite Structures. Cambridge University Press, Cambridge (2003)
Królak, M., Mania, R. (eds.): Stability of Thin-Walled Plate Structures. A Series of Monographs. Lodz University of Technology (2016)
Mania, R. (ed.): Buckling of Plate Structures in Analytical, Numerical and Experimental Investigations. A Series of Monographs. Lodz University of Technology (2016)
Pignataro, M., Luongo, A.: Asymmetric interactive buckling of thin-walled columns with initial imperfections. Thin-Walled Struct. 5(5), 365–386 (1987)
Pignataro, M., Luongo, A., Rizzi, N.: On the effect of the local-overall interaction on the postbuckling of uniformly compressed channels. Thin-Walled Struct. 3(4), 293–321 (1985)
Pignataro, M., Rizzi, N., Luongo, A.: Bifurcation, Stability and Postcritical Behaviour of Elastic Structures. Elsevier, Amsterdam (1991)
Ranzi, G., Luongo, A.: A new approach for thin-walled member analysis in the framework of GBT. Thin Walled Struct. 49(11), 1404–1414 (2011)
Rasheed, H., Al-Musri, R., Alali, B.: Closed form stability solution of simply supported anisotropic laminated composite plates under axial compression compared with experiments. Eng. Struct. 151, 327–336 (2017)
Scerrato, D., Giorgio, I., Rizzi, N.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für angewandte Mathematik und Physik 67, 3 (2016)
Silvestre, N., Comotim, D.: GBT buckling analysis of pultruded FRP lipped channel members. Comput. Struct. 81, 1889–1904 (2003)
Singer, J., Arbocz, J., Weller, T.: Buckling experiments. Experimental methods in buckling of thin-walled structure, vol. 1 and 2. John Wiley & Sons Inc. (1998, 2002)
Szymczak, C., Kujawa, M.: Buckling of thin-walled columns accounting for initial geometrical imperfections. Int. J. Non-Linear Mech. 95, 1–9 (2017)
Taig, G., Ranzi, G., D’Annibale, F.: An unconstrained dynamic approach for the generalised beam theory. Contin. Mech. Thermodyn. 27(4–5), 879–904 (2015)
Taig, G., Ranzi, G., Luongo, A.: GBT pre-buckling and buckling analyses of thin-walled members under axial and transverse loads. Contin. Mech. Thermodyn. 28(1–2), 41–66 (2016)
Trahair, N.: Flexural-Torsional Buckling of Structure. E&FN Spon, London (1993)
Vasiliev, V., Morozov, E.: Mechanics and Analysis of Composites Materials. Elsevier, Amsterdam (2001)
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Communicated by Francesco dell’Isola.
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Szymczak, C., Kujawa, M. Local buckling of composite channel columns. Continuum Mech. Thermodyn. 32, 555–567 (2020). https://doi.org/10.1007/s00161-018-0674-2
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DOI: https://doi.org/10.1007/s00161-018-0674-2