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Torsional buckling of generally laminated conical shell

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Abstract

Buckling of generally laminated conical shells under uniform torsion with simply-supported boundary conditions is investigated. The Donnel type strain displacement relations are used to obtain potential strain energy of the shell and membrane stability equation is applied to acquire the external work done by torsion. The Ritz method is used to solve the governing equations and critical buckling loads are obtained. The accuracy of the results is validated in comparison of with other investigations and finite element method. The effects of lamination sequence, semi-vertex angle and length to radius ratio of the cone are evaluated and mode shapes are presented for two types of lamination sequences. To find a design criterion, effects of lamination angles and semi-vertex angle for two types of lamination sequence on torsional buckling of conical shells are investigated.

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References

  1. Seide P (1962) On the buckling of truncated conical shells in torsion. J Appl Mech 29:321–328

    Article  MATH  Google Scholar 

  2. Yamaki N, Tani J (1969) Buckling of truncated conical shells under torsion. J Appl Math and Mech 49:471-480

    Google Scholar 

  3. Weingarten VI (1964) Stability of internally pressurized conical shells under torsion. AIAA J 2:1782–1788

    Article  ADS  Google Scholar 

  4. Gnoffo PA, Maccalden P, Matthiesen R (1967) Combination torsion and axial compression tests of conical shells. AIAA J 5:305–309

    Article  ADS  Google Scholar 

  5. Chehil DS, Cheng S (1968) Elastic buckling of composite cylindrical shells under torsion. J Spacecr Rockets 5:973–978

    Article  ADS  Google Scholar 

  6. Yamaki N, Matsuda K (1976) Postbuckling behavior of circular cylindrical shells under torsion. Ing Arch 45:79–89

    Article  MATH  Google Scholar 

  7. Simitses GJ, Shaw D, Sheinman I (1985) Inperfection sensitivity of laminated cylindrical shells in torsion and axial compression. Compos Struct 4:335–360

    Article  Google Scholar 

  8. Hui D, Du IHY (1987) Initial postbuckling behavior of imperfect, antisymmetric cross-ply cylindrical shells under torsion. J Appl Mech 54:174–180

    Article  MATH  Google Scholar 

  9. Stavsky Y, Greenberg JB, Sabag M (1989) Buckling of edge-damaged filament-wound composite cylindrical shells under combined torsional/axial loads. Compos Struct 13:21–34

    Article  Google Scholar 

  10. Teng JG, Rotter JM (1989) Non-symmetric bifurcation of geometrically nonlinear elastic-plastic axisymmetric shells under combined loads including torsion. Comput Struct 32:453–475

    Article  MATH  Google Scholar 

  11. Park HC, Cho C, Choi Y (2001) Torsional buckling analysis of composite cylinders. AIAA J 39:951–955

    Article  ADS  Google Scholar 

  12. Sofiyev AH (2003) Torsional buckling of cross-ply laminated orthotropic composite cylindrical shells subject to dynamic loading. Eur J Mech A Solids 22:943–951

    Article  MATH  Google Scholar 

  13. Sofiyev AH, Schnack E (2004) The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading. Eng Struct 26:1321–1331

    Article  Google Scholar 

  14. Sofiyev AH (2005) The torsional buckling analysis for cylindrical shell with material non-homogeneity in thickness direction under impulsive loading. Struct Eng Mech 19:231–236

    Article  Google Scholar 

  15. Shen H-S (2008) Boundary layer theory for the buckling and postbuckling of an anisotropic laminated cylindrical shell, Part III: prediction under torsion. Compos Struct 82:371–381

    Article  Google Scholar 

  16. Shen H-S, Xiang Y (2008) Buckling and postbuckling of anisotropic laminated cylindrical shells under combined axial compression and torsion. Compos Struct 84:375–386

    Article  Google Scholar 

  17. Shen H-S (2009) Torsional buckling and postbuckling of FGM cylindrical shells in thermal environments. Int J Non Linear Mech 44:644–657

    Article  MATH  Google Scholar 

  18. Huang H, Han Q (2010) Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment. Eur J Mech A Solids 29:42–48

    Article  MathSciNet  Google Scholar 

  19. Najafov AM, Sofiyev AH, Kuruoglu N (2013) Torsional vibration and stability of functionally graded orthotropic cylindrical shells on elastic foundations. Meccanica 48:829–840

    Article  MathSciNet  MATH  Google Scholar 

  20. Sofiyev AH, Kuruoglu N (2013) Torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium. Compos B Eng 45:1133–1142

    Article  Google Scholar 

  21. Sofiyev A, Deniz A, Avcar M, Özyigit P, Omurtag M (2013) Effects of the non-homogeneity and elastic medium on the critical torsional load of the orthotropic cylindrical shell footnotemark. Acta Phys Pol A 123:728–730

    Article  Google Scholar 

  22. Dung DV, Hoa LK (2015) Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment. Compos B Eng 69:378–388

    Article  Google Scholar 

  23. Tani J (1984) Buckling of truncated conical shells under combined pressure and heating. J Therm Stress 7:307–316

    Article  Google Scholar 

  24. Zhou H, Liu D (1991) Mechanical analysis of the elastic buckling of an orthogonal anisotropic circular conical shell under uniform external pressure. Int J Press Vessels Pip 48:111–122

    Article  Google Scholar 

  25. Tong L, Wang TK (1992) Simple solutions for buckling of laminated conical shells. Int J Mech Sci 34:93–111

    Article  MATH  Google Scholar 

  26. Pariatmono N, Chryssanthopoulos M (1995) Asymmetric elastic buckling of axially compressed conical shells with various end conditions. AIAA J 33:2218–2227

    Article  ADS  MATH  Google Scholar 

  27. Liu R-H (1996) Non-linear buckling of symmetrically laminated, cylindrically orthotropic, shallow, conical shells considering shear. Int J Non Linear Mech 31:89–99

    Article  MATH  Google Scholar 

  28. Ross CTF, Sawkins D, Johns T (1999) Inelastic buckling of thick-walled circular conical shells under external hydrostatic pressure. Ocean Eng 26:1297–1310

    Article  Google Scholar 

  29. Spagnoli A, Chryssanthopoulos MK (1999) Elastic buckling and postbuckling behaviour of widely-stiffened conical shells under axial compression. Eng Struct 21:845–855

    Article  Google Scholar 

  30. Spagnoli A, Chryssanthopoulos MK (1999) Buckling design of stringer-stiffened conical shells in compression. J Struct Eng 125:40

    Article  Google Scholar 

  31. Chung Y (2001) Buckling of composite conical shells under combined axial compression, external pressure, and bending. New Jersey Institute of Technology, Department of Mechanical Engineering

  32. Wu CP, Chen CW (2001) Elastic buckling of multilayered anisotropic conical shells. J Aerosp Eng 14(1):29–36

    Article  Google Scholar 

  33. Duc ND, Cong PH, Anh VM, Quang VD, Tran P, Tuan ND et al (2015) Mechanical and thermal stability of eccentrically stiffened functionally graded conical shell panels resting on elastic foundations and in thermal environment. Compos Struct 132:597–609

    Article  Google Scholar 

  34. Duc ND, Cong PH (2015) Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations. Eur J Mech A Solids 50:120–131

    Article  MathSciNet  Google Scholar 

  35. Sofiyev AH (2015) On the vibration and stability of shear deformable FGM truncated conical shells subjected to an axial load. Compos B Eng 80:53–62

    Article  Google Scholar 

  36. Ventsel E, Krauthammer T (2001) Thin plates and shells: theory, analysis, and applications. Marcel Dekker, New York City

    Book  Google Scholar 

  37. Amabili M (2008) Nonlinear vibrations and stability of shells and plates. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  38. Hibbitt, Karlsson, Sorensen (2001) ABAQUS/Standard user’s manual, vol 1. Hibbitt, Karlsson & Sorensen, Pennsylvania State University

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, Semnan University, P.O. Box 35131-19111, Semnan, Iran

    M. Shakouri

  2. Department of Aerospace Engineering, Sharif University of Technology, P.O. Box 11155-8639, Tehran, Iran

    H. Sharghi

  3. Department of Aerospace Engineering and Center of Excellence in Aerospace Systems, Sharif University of Technology, P.O. Box 11155-8639, Tehran, Iran

    M. A. Kouchakzadeh

Authors
  1. M. Shakouri
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  2. H. Sharghi
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  3. M. A. Kouchakzadeh
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Corresponding author

Correspondence to M. A. Kouchakzadeh.

Appendix

Appendix

Stretching coupling

$$\begin{array}{*{20}l} {{\text{U}}_{\text{s}} = \frac{1}{{R(x)^{2} }}\left( {2A_{12} R\left( x \right)\frac{{\partial U\left( {x,\theta } \right)}}{\partial x}\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \right.} \hfill \\ {\quad \quad - \;2A_{26} \left( { - R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} - \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } + \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right)\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \hfill \\ {\quad \quad + \;2A_{16} R\left( x \right)\frac{{\partial U\left( {x,\theta } \right)}}{\partial x}\left( {R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } - \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right) + A_{66} \left( {R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } - \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right)^{2} } \hfill \\ {\quad \quad + \;A_{11} R\left( x \right)^{2} \left( {\frac{{\partial U\left( {x,\theta } \right)}}{\partial x}} \right)^{2} + \left. {A_{22} \left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)^{2} } \right)} \hfill \\ \end{array}$$
(17)

Bending coupling

$$\begin{array}{*{20}l} {{\text{U}}_{\text{b}} = \frac{1}{{R(x)^{4} }}\left( {4D_{16} R\left( x \right)^{2} \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}\left( {R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta } - \sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta }} \right)} \right.} \hfill \\ {\quad \quad + \;4D_{26} \left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)\left( {R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta } - \sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta }} \right)} \hfill \\ {\quad \quad + \;4D_{66} \left( {\sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta } - R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta }} \right)^{2} + 2D_{12} R\left( x \right)^{2} \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}\left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)} \hfill \\ {\quad \quad + \;D_{11} R\left( x \right)^{4} \left( {\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}} \right)^{2} + \left. {D_{22} \left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)^{2} } \right)} \hfill \\ \end{array}$$
(18)

Stretching-bending coupling

$$\begin{array}{*{20}l} {U_{bs} = \frac{1}{{R(x)^{4} }}\left( {B_{16} \left( {4R\left( x \right)^{2} \frac{{\partial U\left( {x,\theta } \right)}}{\partial x}} \right.\left( {\sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta } - R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta }} \right)} \right.} \hfill \\ {\quad \quad - \;2R\left( x \right)^{3} \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}\left. {\left( {R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } - \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right)} \right)} \hfill \\ {\quad \quad + \;2B_{26} R\left( x \right)\left( { - 2R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta }} \right.\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \hfill \\ {\quad \quad - \;\sin \left( \alpha \right)R\left( x \right)\frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta }\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \sin^{2} \left( \alpha \right)R\left( x \right)V\left( {x,\theta } \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x}} \hfill \\ {\quad \quad + \;\sin \left( \alpha \right)R\left( x \right)^{2} \left( { - \frac{{\partial V\left( {x,\theta } \right)}}{\partial x}} \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} - R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x}\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \hfill \\ {\quad \quad + \;2\sin^{2} \left( \alpha \right)U\left( {x,\theta } \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta } - \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta }\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }} + \sin \left( \alpha \right)V\left( {x,\theta } \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \hfill \\ {\quad \quad + \;2\sin \left( \alpha \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta }\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta } - \left. {2\sin \left( \alpha \right)\cos \left( \alpha \right)W\left( {x,\theta } \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta }} \right)} \hfill \\ {\quad \quad - \;4B_{66} R\left( x \right)\left( {R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta } - \sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta }} \right)\left( {R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } - \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right)} \hfill \\ {\quad \quad + \;B_{12} \left( { - 2R\left( x \right)^{3} \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}} \right.\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \hfill \\ {\quad \quad - \;2R\left( x \right)^{2} \frac{{\partial U\left( {x,\theta } \right)}}{\partial x}\left. {\left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)} \right)} \hfill \\ {\quad \quad - \;2B_{22} R\left( x \right)\left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \hfill \\ {\left. {\quad \quad - \;2B_{11} R\left( x \right)^{4} \frac{{\partial U\left( {x,\theta } \right)}}{\partial x}\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}} \right)} \hfill \\ \end{array}$$
(19)

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Shakouri, M., Sharghi, H. & Kouchakzadeh, M.A. Torsional buckling of generally laminated conical shell. Meccanica 52, 1051–1061 (2017). https://doi.org/10.1007/s11012-016-0429-8

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