Abstract
Distortional buckling of axially compressed columns of box-like composite cross sections with and without internal diaphragms is investigated in the framework of one-dimensional theory. The channel members are composed of unidirectional fibre-reinforced laminate. Two approaches to the member orthotropic material are applied: homogenization based on the theory of mixture and periodicity cells, and homogenization based on the Voigt–Reuss hypothesis. The principle of stationary total potential energy is applied to derive the governing differential equation. The obtained buckling stress is valid in the linear elastic range of column material behaviour. Numerical examples address simply supported columns, and analytical critical stress formulas are derived. The analytical and FEM solutions are compared, and sufficient accuracy of the results is observed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- a :
-
Height of cross section
- f :
-
Fibre volume fraction
- n :
-
Number of half-waves of a buckling mode
- \(r_0\) :
-
Polar radius of gyration
- u :
-
Displacement of cross section corner
- \(v_{\mathrm{lt}}\) :
-
Homogenized Poisson’s ratio
- \(v_\mathrm{l}\) :
-
Poisson’s ratio in the longitudinal direction
- \(v_\mathrm{t}\) :
-
Poisson’s ratio in the transverse direction
- \(v_\mathrm{m}\) :
-
Poisson’s ratio of the matrix
- \(v_\mathrm{f}\) :
-
Poisson’s ratio of fibres
- x; y; z :
-
Cartesian coordinate system
- A :
-
Area of cross section
- \(D_\mathrm{l}\) :
-
Elastic modulus in the longitudinal direction
- \(D_\mathrm{t}\) :
-
Elastic modulus in the transverse direction
- \(E_\mathrm{l}\) :
-
Homogenized Young’s modulus in the longitudinal direction
- \(E_\mathrm{t}\) :
-
Homogenized Young’s modulus in the transverse direction
- \(E_\mathrm{m}\) :
-
Young’s modulus of the matrix
- \(E_\mathrm{f}\) :
-
Young’s modulus of fibres
- G :
-
Homogenized shear modulus
- \(G_\mathrm{m}\) :
-
Shear modulus of the matrix
- \(G_\mathrm{f}\) :
-
Shear modulus of fibres
- \(J_0\) :
-
Polar moment of inertia
- \(J_\mathrm{g}\) :
-
Moment of inertia of wall cross section in the longitudinal direction
- \(J_\mathrm{p}\) :
-
Moment of inertia of wall cross section in the transverse direction
- \(J_\mathrm{s}\) :
-
Free torsion moment of inertia of wall cross section
- \(K_\mathrm{s}\) :
-
Torsional stiffness of cross section
- \(K_\mathrm{g}\) :
-
Longitudinal stiffness of cross section
- \(K_{\mathrm{\gamma }}\) :
-
Distortional stiffness of cross section
- \(\overline{K}_{\mathrm{\gamma }}\) :
-
Diaphragm stiffness
- L :
-
Length of column
- \(L_0\) :
-
Characteristic length of column
- \(M_\mathrm{p}\) :
-
Bending moment of walls in the transverse direction
- \(M_\mathrm{g}\) :
-
Bending moment of walls in the longitudinal direction
- P :
-
Compressive axial load
- \(P_{\mathrm{cr}}\) :
-
Critical distortional buckling load
- \(U^I\) :
-
Potential energy of compressive load due to bending
- \(U^{II}\) :
-
Potential energy of compressive load due to torsion
- V :
-
Elastic strain energy
- \(V_\mathrm{g}\) :
-
Potential energy of elastic bending
- \(V_\mathrm{p}\) :
-
Potential energy of cross-sectional distortion
- \(V_\mathrm{s}\) :
-
Potential energy of torsion
- \(\gamma \) :
-
Distortion angle
- \(\delta \) :
-
Wall thickness
- \(\eta \) :
-
Coefficient of characteristic length of column
- \(\sigma _\mathrm{b}\) :
-
Buckling stress
- \(\sigma _{\mathrm{cr}}\) :
-
Critical buckling stress
- \(\sigma _{\mathrm{cr},\mathrm{min}}\) :
-
Minimum critical buckling stress
- \(\varPi \) :
-
Total potential energy
References
Abambres, M., Camotim, D., Silvestre, N.: \(\rm GBT\)-based elastic-plastic post-buckling analysis of stainless steel thin-walled members. Thin Walled Struct. 83, 85–102 (2014)
Abambres, M., Camotim, D., Silvestre, N., Rasmussen, K.: \(\rm GBT\)-based structural analysis of elastic-plastic thin-walled members. Comput. Struct. 136, 1–23 (2014)
Ádány, S., Silvestre, N., Schafer, B., Camotim, D.: GBT and cFSM: two modal approaches to the buckling analysis of unbranched thin-walled members. Adv. Steel Constr. 5(2), 195–223 (2009)
Berthelot, J.: Composite Materials—Mechanical Behaviour and Structural Analysis. Springer, Berlin (1999)
Camotim, D., Basaglia, C., Silvestre, N.: \(\rm GBT\) buckling analysis of thin-walled steel frames: a state-of-the-art report. Thin Walled Struct. 48, 726–743 (2010)
Cheung, Y.: Finite Strip Method in Structural Analysis. Elsevier, Amsterdam (1976)
Cheung, Y., Tham, L.: Finite Strip Method. CRC Press, Boca Raton (1998)
Chudzikiewicz, A.: Stability loss due to the deformation of the cross-section. Eng. Trans. 7(1), 45–61 (1960)
Daniel, I., Ishai, O.: Engineering Mechanics of Composite Materials. Oxford University Press, Oxford (2006)
Davies, J.: Recent research advances in cold-formed steel structures. J. Constr. Steel Res. 55, 267–288 (2000)
Dinis, P., Camotim, D., Silvestre, N.: \(\rm GBT\) formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Thin Walled Struct. 44, 20–38 (2006)
Dow, N., Rosen, B.: Evaluation of Filament-reinforced Composites for Aerospace Structural Applications. NASA Contractor Report. General Electric Company, Philadelphia (1965)
Eliseev, V., Vetyukov, Y.: Finite deformation of thin-walled shells in the context of analytical mechanics of material surfaces. Acta Mech. 209(1–2), 43–57 (2010)
Gonçalves, R., Camotim, D.: \(\rm GBT\) deformation modes for curved thin-walled cross-sections based on a mid-line polygonal approximation. Thin Walled Struct. 103, 231–243 (2016)
Habbit, D., Karlsson, B., Sorensen, P.: ABAQUS Analysis User’s Manual. Hibbit, Karlsson, Sorensen Inc, Providence (2007)
Hancock, G., Pham, C.: Buckling analysis of thin-walled sections under localised loading using the semi-analytical finite strip method. Thin Walled Struct. 86, 35–46 (2015)
Jones, R.: Mechanics of Composites Materials. Taylor & Francis, Abingdon (1999)
Kaw, A.: Mechanics of Composite Materials. Taylor & Francis, Abingdon (2006)
Kelly, A. (ed.): Concise Encyclopaedia of Composite Materials. Pergamon Press, Oxford (1989)
Kollar, L., Springer, G.: Mechanics of Composite Structures. Cambridge University Press, Cambridge (2003)
Królak, M., Mania, R. (eds.): Stability of Thin-Walled Plate Structures. Technical University of Łódź, Łódź (2011)
Kujawa, M., Szymczak, C.: Elastic distortional buckling of thin-walled bars of closed quadratic cross-section. Mech. Mech. Eng. 17(2), 119–126 (2013)
Li, Z., Abreu, J., Leng, J., Schafer, B.: Review: constrained finite strip method developments and applications in cold-formed steel design. Thin Walled Struct. 81, 2–18 (2014)
de Miranda, S., Gutiérrez, A., Miletta, R., Ubertini, F.: A generalized beam theory with shear deformation. Thin-Walled Struct. 67, 88–100 (2013)
Philips, L.: Design with Advanced Composite Materials. Springer, Berlin (1989)
Pietraszkiewicz, W., Górski, J. (eds.): Shell Structures: Theory and Applications, vol. 3. CRC Press, Boca Raton (2014)
Pietraszkiewicz, W., Kreja, I. (eds.): Shell Structures: Theory and Applications, vol. 2. CRC Press, Boca Raton (2010)
Pietraszkiewicz, W., Szymczak, C. (eds.): Shell Structures: Theory and Applications, vol. 1. Taylor & Francis, Abingdon (2005)
Pietraszkiewicz, W., Witkowski, W. (eds.): Shell Structures: Theory and Applications, vol. 4. CRC Press, Boca Raton (2018)
Schardt, R.: Generalized beam theory—an adequate method for coupled stability problems. Thin Walled Struct. 19, 161–180 (1994)
Silvestre, N., Camotim, D.: Second-order generalised beam theory for arbitrary orthotropic materials. Thin Walled Struct. 40, 791–820 (2002)
Szymczak, C., Kujawa, M.: Distortional buckling of thin-walled columns of closed quadratic cross-section. Thin Walled Struct. 113, 111–121 (2017)
Szymczak, C., Kujawa, M.: Local buckling of composite channel columns. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00,161-018-0674-2
Thompson, J., Hunt, G.: A General Theory of Elastic Stability. Wiley, Hoboken (1973)
Timoshenko, S., Gere, J.: Theory of Elastic Stability. McGraw-Hill, International Book Company, New York (1961)
Vasiliev, V., Morozov, E.: Mechanics and Analysis of Composites Materials. Elsevier, Amsterdam (2001)
Vetyukov, Y.: Direct approach to elastic deformations and stability of thin-walled rods of open profile. Acta Mech. 200, 167–176 (2008)
Vetyukov, Y.: Nonlinear Mechanics of Thin-Walled Structures. Springer, Berlin (2014)
Waszczyszyn, Z. (ed.): Modern Methods of Stability Analysis of Structures. Ossolineum, Wrocław (1981)
Waszczyszyn, Z. (ed.): Selected Problems of Stability of Structures. Ossolineum, Wrocław (1987)
Zienkiewicz, O., Taylor, R.: The Finite Element Method, 7th edn. Elsevier, Amsterdam (2013)
Acknowledgements
The calculations presented in this paper were carried out at the TASK Academic Computer Centre in Gdańsk, Poland.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that he has no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Szymczak, C., Kujawa, M. Distortional buckling of composite thin-walled columns of a box-type cross section with diaphragms. Acta Mech 230, 3945–3961 (2019). https://doi.org/10.1007/s00707-019-02406-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-019-02406-x