Kinematically admissible folding mechanisms for the progressive collapse of foam filled conical frusta

  • Fan Yang
  • S. A. Meguid
  • A. M. S. Hamouda


In this paper, the progressive collapse of foam filled conical frusta is investigated analytically using four different kinematically admissible folding mechanisms with varied straight folds. Comparisons are made between these four kinematically admissible mechanisms; specifically, pure inward folding, pure outward folding, first inward followed by outward folding, and first outward followed by inward folding. The instantaneous force as well as the mean crushing force was derived based on the principle of energy conversation, and the crushing energy was absorbed by the plastic deformation of the shell, the crushing of foam filler and the foam/shell interaction. The resulted upper bound solution of the four different mechanisms is compared with the finite element predictions of the same system. Our parametric study reveals that first outward then inward folding mechanism is generally energy favorable except for cases involving greater foam resistance, thin shell thickness, and/or large taper angle in which the pure outward folding mechanism may be preferable.


Kinematically admissible mechanism Progressive collapse Foam filled Frusta 



This research effort was made possible by NPRP Grant # (7-236-3-053) from the Qatar National Research Fund (a member of Qatar Foundation), National Natural Science Foundation of China (11402173), and Innovation Program of Shanghai Municipal Education Commission (15zz018). Additional support from the Natural Sciences and Engineering Research Council of Canada and Shanghai Supercomputer Center is also gratefully acknowledged.


  1. Abbas, H., Tyagi, B.L., Arif, M., Gupta, N.K.: Curved fold model analysis for axi-symmetric axial crushing of tubes. Thin-Walled Struct. 41, 639–661 (2003)CrossRefGoogle Scholar
  2. Abramowicz, W., Jones, N.: Dynamic axial crushing of circular tubes. Int. J. Impact Eng. 2, 263–281 (1984a)CrossRefGoogle Scholar
  3. Abramowicz, W., Jones, N.: Dynamic axial crushing of square tubes. Int. J. Impact Eng 2, 179–208 (1984b)CrossRefGoogle Scholar
  4. Abramowicz, W., Wierzbicki, T.: Axial crushing of foam-filled columns. Int. J. Impact Eng. 30, 263–271 (1988)Google Scholar
  5. Ahmad, Z., Thambiratnam, D.P.: Application of foam-filled conical tubes in enhancing the crashworthiness performance of vehicle protective structures. Int. J. Crashworthiness 14(4), 349–363 (2009a)CrossRefGoogle Scholar
  6. Ahmad, Z., Thambiratnam, D.P.: Dynamic computer simulation and energy absorption of foam-filled conical tubes under axial impact loading. Comput. Struct. 87, 186–197 (2009b)CrossRefGoogle Scholar
  7. Alexander, J.: An approximate analysis of the collapse of thin cylindrical shells under axial loading. Q. J. Mech. Appl. Math. 13, 10–15 (1960)MathSciNetCrossRefMATHGoogle Scholar
  8. Aktay, L., Johnson, A.F., Toksoy, A.K., Kröplin, B.H., Güden, M.: Modeling the progressive axial crushing of foam-filled aluminum tubes using smooth particle hydrodynamics and coupled finite element model/smooth particle hydrodynamics. Mater. Des. 29(3), 569–575 (2008)CrossRefGoogle Scholar
  9. Azimi, M.B., Asgari, M.: Energy absorption characteristics and a meta-model of miniature frusta under axial impact. Int. J. Crashworthiness 21(3), 222–230 (2016)CrossRefGoogle Scholar
  10. Chen, W., Wierzbicki, T.: Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption. Thin Walled Struct. 39, 287–306 (2001)CrossRefGoogle Scholar
  11. Desjardins, S.P.: The evolution of energy absorption systems for crashworthy helicopter seats. J. Am. Helicopter Soc. 51(2), 150–163 (2006)CrossRefGoogle Scholar
  12. El-Sobky, H., Singace, A.A., Petsios, M.: Mode of collapse and energy absorption characteristics of constrained frusta under axial impact loading. Int. J. Mech. Sci. 43, 743–757 (2001)CrossRefMATHGoogle Scholar
  13. Fyllingen, Ø., Hopperstad, O., Hanssen, A., Langseth, M.: Modelling of tubes subjected to axial crushing. Thin Walled Struct. 48, 134–142 (2010)CrossRefGoogle Scholar
  14. Ghamarian, A., Zarei, H.R., Farsi, M.A., Ariaeifar, N.: Experimental and numerical crashworthiness investigation of the empty and foam-filled conical tube with shallow spherical caps. Strain 49, 199–211 (2013)CrossRefGoogle Scholar
  15. Goel, M.D.: Deformation, energy absorption and crushing behavior of single-, double- and multi-wall foam filled square and circular tubes. Thin Walled Struct. 90, 1–11 (2015)CrossRefGoogle Scholar
  16. Guillow, S.R., Lu, G., Grzebieta, R.H.: Quasi-static axial compression of thin-walled circular aluminium tubes. Int. J. Mech. Sci. 43, 2103–2123 (2001)CrossRefMATHGoogle Scholar
  17. Gupta, N.K., Abbas, H.: Mathematical modeling of axial crushing of cylindrical tubes. Thin Walled Struct. 38, 355–375 (2000)CrossRefGoogle Scholar
  18. Gupta, N.K., Abbas, H.: Some considerations in axi-symmetric folding of metallic round tubes. Int. J. Impact Eng. 25, 331–344 (2001)CrossRefGoogle Scholar
  19. Hanssen, A.G., Langseth, M., Hopperstad, O.S.: Static crushing of square aluminium extrusions with aluminium foam filler. Int. J. Mech. Sci. 41, 967–993 (1999)CrossRefMATHGoogle Scholar
  20. Hanssen, A.G., Langseth, M., Hopperstad, O.S.: Static and dynamic crushing of square aluminium extrusions with aluminium foam filler. Int. J. Impact Eng. 24, 347–383 (2000a)CrossRefMATHGoogle Scholar
  21. Hanssen, A.G., Langseth, M., Hopperstad, O.S.: Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler. Int. J. Impact Eng 24, 475–507 (2000b)CrossRefMATHGoogle Scholar
  22. Hong, W., Lai, C., Fan, H.: Frusta structure designing to improve quasi-static axial crushing performances of triangular tubes. Int. J. Steel Struct. 16(1), 257–266 (2016)CrossRefGoogle Scholar
  23. Hosseini, M., Abbas, H., Gupta, N.K.: Straight fold analysis for axisymmetric crushing of thin walled frusta and tubes. Lat. Am. J. Solids Struct. 3, 345–360 (2006)Google Scholar
  24. Hosseini, M., Abbas, H., Gupta, N.K.: Change in thickness in straight fold models for axisymmetric crushing of thin-walled frusta and tubes. Thin Walled Struct. 47, 98–108 (2009)CrossRefGoogle Scholar
  25. Li, Z., Yu, J., Guo, L.: Deformation and energy absorption of aluminum foam-filled tubes subjected to oblique loading. Int. J. Mech. Sci. 54, 48–56 (2012)CrossRefGoogle Scholar
  26. Li, Q.F., Liu, Y.J.,Wang, H.D., Yan, S.Y.: Finite element analysis and shape optimization of automotive crash-box subjected to low velocity impact. In: International Conference on Measuring Technology andMechatronics Automation, ICMTMA’09, pp. 791–794. Zhangjiajie China, April 11–12 (2009)Google Scholar
  27. Mahmoodia, A., Shojaeefard, M.H., Googarchin, H.S.: Theoretical development and numerical investigation on energy absorption behavior of tapered multi-cell tubes. Thin Walled Struct. 102, 98–110 (2016)CrossRefGoogle Scholar
  28. Mamalis, A.G., Manolakos, D.E., Ioannidis, M.B., Kostazos, P.K., Dimitriou, C.: Finite element simulation of the axial collapse of metallic thin-walled tubes with octagonal cross-section. Thin Walled Struct. 41(10), 891–900 (2003)CrossRefGoogle Scholar
  29. Mamalis, A.G., Manolakos, D.E., Saigal, S., Viegelahn, G., Johnson, W.: Extensible plastic collapse of thin-wall frusta as energy absorbers. Int. J. Mech. Sci. 28, 219–229 (1986)CrossRefGoogle Scholar
  30. Meguid, S.A., Attia, M.S., Monfort, A.: On the crush behaviour of ultralight foam-filled structures. Mater. Des. 25, 183–189 (2004a)CrossRefGoogle Scholar
  31. Meguid, S.A., Attia, M.S., Stranart, J.C.: Solution stability in the dynamic collapse of square aluminium columns. Int. J. Impact Eng. 34(2), 348–359 (2007)CrossRefGoogle Scholar
  32. Meguid, S.A., Stranart, J., Heyerman, J.: On the layered micromechanical three-dimensional finite element modelling of foam-filled columns. Finite Elem. Anal. Des. 40, 1035–1057 (2004b)CrossRefGoogle Scholar
  33. Meguid, S.A., Yang, F., Verberne, P.: Progressive collapse of foam-filled conical frustum using kinematically admissible mechanism. Int. J. Impact Eng. 82, 25–35 (2015)CrossRefGoogle Scholar
  34. Meguid, S.A., Yang, F., Hou, P.: Crush behaviour of foam-filled thin-walled conical frusta: analytical, numerical and experimental studies. Acta Mech. 227, 3391–3406 (2016)MathSciNetCrossRefGoogle Scholar
  35. Nia, A.A., Parsapour, M.: Comparative analysis of energy absorption capacity of simple and multi-cell thin-walled tubes with triangular, square, hexagonal and octagonal sections. Thin Walled Struct. 74, 155–165 (2014)CrossRefGoogle Scholar
  36. Niknejad, A., Abedi, M.M., Liaghat, G.H., Nejad, M.Z.: Prediction of the mean folding force during the axial compression in foam-filled grooved tubes by theoretical analysis. Mater. Des. 37, 144–151 (2012)CrossRefGoogle Scholar
  37. Pugsley, A.G.: On the crumpling of thin tubular struts. Q. J. Mech. Appl. Math. 32, 1–7 (1979)CrossRefGoogle Scholar
  38. Reid, S.R., Reddy, T.Y.: Static and dynamic crushing of tapered sheet metal tubes of rectangular cross-section. Int. J. Mech. Sci. 28, 623–637 (1986)CrossRefGoogle Scholar
  39. Reid, S., Reddy, T., Gray, M.: Static and dynamic axial crushing of foam-filled sheet metal tubes. Int. J. Mech. Sci. 28, 295–322 (1986)CrossRefGoogle Scholar
  40. Rezvani, M.J., Nouri, M.D.: Axial crumpling of aluminum frusta tubes with induced axisymmetric folding patterns. Arab. J. Sci. Eng. 39, 2179–2190 (2014)CrossRefGoogle Scholar
  41. Singace, A.A., El-Sobky, H., Petsios, M.: Influence of end constraints on the collapse of axially impacted frusta. Thin Walled Struct. 39, 415–428 (2001)CrossRefMATHGoogle Scholar
  42. Song, J.: Numerical simulation on windowed tubes subjected to oblique impact loading and a new method for the design of obliquely loaded tubes. Int. J. Impact Eng. 54, 192–205 (2013)CrossRefGoogle Scholar
  43. Song, X., Sun, G., Li, G., Gao, W., Li, Q.: Crashworthiness optimization of foam-filled tapered thin-walled structure using multiple surrogate models. Struct. Multidisc. Optim. 47, 221–231 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. Wierzbicki, T., Abramowicz, W.: On the crushing mechanics of thin-walled structures. J. Appl. Mech. 50(4a), 727–734 (1983)CrossRefMATHGoogle Scholar
  45. Yang, P., Meguid, S.A., Zhang, X.: Accurate modelling of the crush behaviour of thin tubular columns using material point method. Sci. China Phys. Mech. Astron. 56(6), 1209–1219 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina
  2. 2.Mechanics and Aerospace Design LaboratoryUniversity of TorontoTorontoCanada
  3. 3.Department of Mechanical and Industrial EngineeringQatar UniversityDohaQatar

Personalised recommendations