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Acta Mechanica

, Volume 230, Issue 11, pp 3945–3961 | Cite as

Distortional buckling of composite thin-walled columns of a box-type cross section with diaphragms

  • Czesław Szymczak
  • Marcin KujawaEmail author
Open Access
Original Paper

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.

List of symbols

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

xyz

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

Notes

Acknowledgements

The calculations presented in this paper were carried out at the TASK Academic Computer Centre in Gdańsk, Poland.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

References

  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    Berthelot, J.: Composite Materials—Mechanical Behaviour and Structural Analysis. Springer, Berlin (1999)zbMATHGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Cheung, Y.: Finite Strip Method in Structural Analysis. Elsevier, Amsterdam (1976)zbMATHGoogle Scholar
  7. 7.
    Cheung, Y., Tham, L.: Finite Strip Method. CRC Press, Boca Raton (1998)Google Scholar
  8. 8.
    Chudzikiewicz, A.: Stability loss due to the deformation of the cross-section. Eng. Trans. 7(1), 45–61 (1960)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Daniel, I., Ishai, O.: Engineering Mechanics of Composite Materials. Oxford University Press, Oxford (2006)Google Scholar
  10. 10.
    Davies, J.: Recent research advances in cold-formed steel structures. J. Constr. Steel Res. 55, 267–288 (2000)CrossRefGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Dow, N., Rosen, B.: Evaluation of Filament-reinforced Composites for Aerospace Structural Applications. NASA Contractor Report. General Electric Company, Philadelphia (1965)Google Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    Habbit, D., Karlsson, B., Sorensen, P.: ABAQUS Analysis User’s Manual. Hibbit, Karlsson, Sorensen Inc, Providence (2007)Google Scholar
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    Jones, R.: Mechanics of Composites Materials. Taylor & Francis, Abingdon (1999)Google Scholar
  18. 18.
    Kaw, A.: Mechanics of Composite Materials. Taylor & Francis, Abingdon (2006)zbMATHGoogle Scholar
  19. 19.
    Kelly, A. (ed.): Concise Encyclopaedia of Composite Materials. Pergamon Press, Oxford (1989)Google Scholar
  20. 20.
    Kollar, L., Springer, G.: Mechanics of Composite Structures. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  21. 21.
    Królak, M., Mania, R. (eds.): Stability of Thin-Walled Plate Structures. Technical University of Łódź, Łódź (2011)Google Scholar
  22. 22.
    Kujawa, M., Szymczak, C.: Elastic distortional buckling of thin-walled bars of closed quadratic cross-section. Mech. Mech. Eng. 17(2), 119–126 (2013)Google Scholar
  23. 23.
    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)CrossRefGoogle Scholar
  24. 24.
    de Miranda, S., Gutiérrez, A., Miletta, R., Ubertini, F.: A generalized beam theory with shear deformation. Thin-Walled Struct. 67, 88–100 (2013)CrossRefGoogle Scholar
  25. 25.
    Philips, L.: Design with Advanced Composite Materials. Springer, Berlin (1989)Google Scholar
  26. 26.
    Pietraszkiewicz, W., Górski, J. (eds.): Shell Structures: Theory and Applications, vol. 3. CRC Press, Boca Raton (2014)Google Scholar
  27. 27.
    Pietraszkiewicz, W., Kreja, I. (eds.): Shell Structures: Theory and Applications, vol. 2. CRC Press, Boca Raton (2010)zbMATHGoogle Scholar
  28. 28.
    Pietraszkiewicz, W., Szymczak, C. (eds.): Shell Structures: Theory and Applications, vol. 1. Taylor & Francis, Abingdon (2005)Google Scholar
  29. 29.
    Pietraszkiewicz, W., Witkowski, W. (eds.): Shell Structures: Theory and Applications, vol. 4. CRC Press, Boca Raton (2018)Google Scholar
  30. 30.
    Schardt, R.: Generalized beam theory—an adequate method for coupled stability problems. Thin Walled Struct. 19, 161–180 (1994)CrossRefGoogle Scholar
  31. 31.
    Silvestre, N., Camotim, D.: Second-order generalised beam theory for arbitrary orthotropic materials. Thin Walled Struct. 40, 791–820 (2002)CrossRefGoogle Scholar
  32. 32.
    Szymczak, C., Kujawa, M.: Distortional buckling of thin-walled columns of closed quadratic cross-section. Thin Walled Struct. 113, 111–121 (2017)CrossRefGoogle Scholar
  33. 33.
    Szymczak, C., Kujawa, M.: Local buckling of composite channel columns. Contin. Mech. Thermodyn. (2018).  https://doi.org/10.1007/s00,161-018-0674-2 CrossRefzbMATHGoogle Scholar
  34. 34.
    Thompson, J., Hunt, G.: A General Theory of Elastic Stability. Wiley, Hoboken (1973)zbMATHGoogle Scholar
  35. 35.
    Timoshenko, S., Gere, J.: Theory of Elastic Stability. McGraw-Hill, International Book Company, New York (1961)Google Scholar
  36. 36.
    Vasiliev, V., Morozov, E.: Mechanics and Analysis of Composites Materials. Elsevier, Amsterdam (2001)Google Scholar
  37. 37.
    Vetyukov, Y.: Direct approach to elastic deformations and stability of thin-walled rods of open profile. Acta Mech. 200, 167–176 (2008)CrossRefGoogle Scholar
  38. 38.
    Vetyukov, Y.: Nonlinear Mechanics of Thin-Walled Structures. Springer, Berlin (2014)CrossRefGoogle Scholar
  39. 39.
    Waszczyszyn, Z. (ed.): Modern Methods of Stability Analysis of Structures. Ossolineum, Wrocław (1981)Google Scholar
  40. 40.
    Waszczyszyn, Z. (ed.): Selected Problems of Stability of Structures. Ossolineum, Wrocław (1987)Google Scholar
  41. 41.
    Zienkiewicz, O., Taylor, R.: The Finite Element Method, 7th edn. Elsevier, Amsterdam (2013)zbMATHGoogle Scholar

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© The Author(s) 2019

Open AccessThis 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.

Authors and Affiliations

  1. 1.Department of Structural Engineering, Faculty of Ocean Engineering and Ship TechnologyGdańsk University of TechnologyGdańskPoland
  2. 2.Department of Structural Mechanics, Faculty of Civil and Environmental EngineeringGdańsk University of TechnologyGdańskPoland

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