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Analytical and FEM solutions for free vibration of joined cross-ply laminated thick conical shells using shear deformation theory

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Abstract

In this study, the free vibration analysis of two joined laminated conical shells is investigated. Five equilibrium equations for each conical shell have been derived in a particular coordinate system; using Hamilton’s principle and first-order shear deformation theorem. The analytical solutions are obtained in the form of power series based on separation of variables method. The boundary conditions at both ends of the joined shells and the continuity conditions, at the conical shells contact, are extracted from energy formulations. As a result, the non-dimensional natural frequencies are studied for cross-ply laminated shells. The effects of semi-vertex angle, circumferential modes, number of layers, thickness, length of shells and different boundary conditions on non-dimensional frequencies are considered. As a comparing result, the non-dimensional frequencies and mode shapes are extracted using finite element method (FEM). The results are compared and verified with the previous available results in other researches. The results reveal a good agreement among analytical solutions, FEM and other results. The output of this paper can be used for analyzing cylindrical–conical shells in addition to joined conical shells.

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References

  1. Ugural, A.C.: Stresses in Plates and Shells. McGraw-Hill, New York (1981)

    Google Scholar 

  2. Cooper, R.M., Naghdi, P.M.: Propagation of elastic waves in cylindrical shells, including the effects of transverse shear and rotary inertia. J. Acoust. Soc. 28, 56–63 (1956)

    Article  Google Scholar 

  3. Tong, L.: Free vibration of composite laminated conical shells. Int. J. Mech. Sci. 35, 47–61 (1993)

    Article  Google Scholar 

  4. Hosseini Hashemi, Sh., Fadaee, M.: On the free vibration of moderately thick spherical shell panel—a new closed-form procedure. J. Sound Vib. 330, 4352–4367 (2011)

  5. Hosseini Hashemi, Sh., Ilkhani, M.R.: Exact solution for free vibrations of spinning nanotube based on nonlocal first order shear deformation shell theory. Compos. Struct. 157, 1–11 (2016)

  6. Talebitooti, M., Ghayour, M., Ziaei-Rad, S., Talebitooti, R.: Free vibrations of rotating composite conical shells with stringer and ring stiffeners. Arch. Appl. Mech. 80(3), 201–215 (2010)

    Article  Google Scholar 

  7. Talebitooti, M.: Three-dimensional free vibration analysis of rotating laminated conical shells: layerwise differential quadrature (LW-DQ) method. Arch. Appl. Mech. 83(5), 765–781 (2013)

    Article  Google Scholar 

  8. Sofiyev, A.H.: The buckling of FGM truncated conical shells subjected to combined axial tension and hydrostatic pressure. Compos. Struct. 92(2), 488–498 (2010)

    Article  Google Scholar 

  9. Talebitooti, M.: Analytical and finite-element solutions for the buckling of composite sandwich conical shell with clamped ends under external pressure. Arch. Appl. Mech. 87(1), 59–73 (2017)

    Article  Google Scholar 

  10. Shadmehri, F., Hoa, S.V., Hojjati, M.: Buckling of conical composite shells. Compos. Struct. 94, 787–792 (2012)

    Article  Google Scholar 

  11. Lashkari, M., Weingarten, V.: Vibration of segmented shells. Exp. Mech. 13, 120–125 (1973)

    Article  Google Scholar 

  12. Hu, W.C., Raney, J.: Experimental and analytical study of vibrations of joined shells. AIAA J. 5, 976–980 (1976)

    Article  Google Scholar 

  13. Alwar, R.S., Ramamurti, V.: Asymmetric bending of cylindrical conical shell junction. Nucl. Eng. Des. 12, 97–121 (1970)

    Article  Google Scholar 

  14. Rose, J.L., Mortimer, R.W., Blum, A.: Elastic-wave propagation in a joined cylindrical–conical–cylindrical shell. Exp. Mech. 13, 150–6 (1973)

    Article  Google Scholar 

  15. Irie, T., Yamada, G., Muramoto, Y.: Free vibration of joined conical–cylindrical shells. J. Sound Vib. 95, 31–9 (1984)

    Article  Google Scholar 

  16. Tavakoli, M.S., Singh, R.: Eigensolutions of joined/hermetic shell structures using the state space method. J. Sound Vib. 130, 97–123 (1989)

    Article  Google Scholar 

  17. Sivadas, K.R., Ganesan, N.: Free vibration analysis of combined and stiffened shells. Comput. Struct. 46, 537–546 (1993)

    Article  Google Scholar 

  18. Zhao, Y., Teng, J.G.: A stability design proposal for cone–cylinder intersections under internal pressure. Int. J. Press. Vessels Pip. 80, 297–309 (2003)

    Article  Google Scholar 

  19. Teng, J.G.: Elastic buckling of cone–cylinder intersection under localized circumferential compression. Eng. Struct. 18, 41–48 (1996)

    Article  Google Scholar 

  20. Teng, J.G., barbagallo, M.: Shell restraint to ring buckling at cone–cylinder intersections. Eng. Struct. 19, 425–431 (1997)

    Article  Google Scholar 

  21. Benjeddou, A.: Vibration of complex shells of revolution using B-spline finite elements. Comput. Struct. 74, 429–440 (2000)

    Article  Google Scholar 

  22. Kamat, S., Ganapathi, M., Patel, B.P.: Analysis of parametrically excited laminated composite jointed conical–cylindrical shells. Comput. Struct. 79, 65–76 (2001)

    Article  Google Scholar 

  23. El Damatty, A.A., Saafan, M.S., Sweedan, A.M.I.: Dynamic characteristics of combined conical–cylindrical shells. Thin Walled Struct. 43, 1380–1397 (2005)

    Article  Google Scholar 

  24. Efraim, E., Eisenberger, M.: Exact vibration frequencies of segmented axisymmetric shells. Thin Walled Struct. 44, 281–289 (2006)

    Article  Google Scholar 

  25. Patel, B.P., Shukla, K.K., Nath, Y.: Thermal post buckling analysis of laminated cross-ply truncated circular conical shells. Compos. Struct. 71, 101–114 (2005)

    Article  Google Scholar 

  26. Patel, B.P., Ganapathi, M., Kamat, S.: Free vibration characteristics of laminated composite joined conical–cylindrical shells. J. Sound Vib. 237, 920–930 (2000)

    Article  Google Scholar 

  27. Caresta, M., Kessissoglou, N.J.: Free vibrational characteristics of isotropic coupled cylindrical–conical shells. J. Sound Vib. 329, 733–751 (2010)

    Article  Google Scholar 

  28. Ma, X., Jin, G., Xiong, Y., Liu, Z.: Free and forced vibration analysis of coupled conical–cylindrical shells with arbitrary boundary conditions. Int. J. Mech. Sci. 88, 122–137 (2014)

    Article  Google Scholar 

  29. Shakouri, M., Kouchakzadeh, M.A.: Stability analysis of joined isotropic conical shells under axial compression. Thin Walled Struct. 74, 20–27 (2013)

    Article  Google Scholar 

  30. Kouchakzadeh, M.A., Shakouri, M.: Analytical solution for axisymmetric buckling of joined conical shells under axial compression. Struct. Eng. Mech. 54, 649–664 (2015)

    Article  Google Scholar 

  31. Shakouri, M., Kouchakzadeh, M.A.: Free vibration analysis of joined conical shells: analytical and experimental study. Thin Walled Struct. 85, 350–358 (2014)

    Article  Google Scholar 

  32. Sarkheil, S., Foumani, M.S., Navazi, H.M.: Theoretical and experimental analysis of the free vibrations of a shell made of n cone segments joined together. Thin Walled Struct. 108, 416–427 (2016)

    Article  Google Scholar 

  33. Sarkheil, S., Foumani, M.S., Navazi, H.M.: Free vibrations of a rotating shell made of p joined cones. Int. J. Mech. Sci. 124–125, 83–94 (2017)

    Article  Google Scholar 

  34. Tong, L.: Free vibration of laminated conical shells including transverse shear deformation. Int. J. Solids Struct. 31, 443–456 (1994)

    Article  Google Scholar 

  35. Wu, C.P., Lee, C.V.: Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int. J. Mech. Sci. 43, 1853–1869 (2001)

    Article  Google Scholar 

  36. Shu, C.: Free vibration analysis of composite laminated conical shells by generalized differential quadrature. J. Sound Vib. 194, 587–604 (1996)

    Article  Google Scholar 

  37. Kouchakzadeh, M.A., Shakouri, M.: Free vibration analysis of joined cross-ply laminated conical shells. Int. J. Mech. Sci. 78, 118–125 (2014)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran

    Mohammad Hadi Izadi, Shahrokh Hosseini-Hashemi & Moharam Habibnejad Korayem

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  1. Mohammad Hadi Izadi
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  2. Shahrokh Hosseini-Hashemi
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  3. Moharam Habibnejad Korayem
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Correspondence to Shahrokh Hosseini-Hashemi.

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Appendix

Appendix

The partial differential operators :

$$\begin{aligned} L_{11}= & {} A_{11} \frac{\partial ^{2}}{\partial s^{2}}+\frac{A_{11} \sin \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{A_{22} \sin ^{2}\alpha }{R^{2}(s)}+\frac{A_{33} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}} \\ L_{12}= & {} \frac{\left( {A_{12} +A_{33} } \right) }{R(s)}\frac{\partial ^{2}}{\partial s\partial \theta }-\frac{\left( {A_{22} +A_{33} } \right) \sin \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{13}= & {} \frac{A_{12} \cos \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{A_{22} \sin \alpha \cos \alpha }{R^{2}(s)} \\ L_{14}= & {} B_{11} \frac{\partial ^{2}}{s^{2}}+\frac{B_{11} \sin \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{B_{22} \sin ^{2}\alpha }{R^{2}(s)}+\frac{B_{33} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}} \\ L_{15}= & {} \frac{\left( {B_{12} +B_{33} } \right) }{R(s)}\frac{\partial ^{2}}{\partial s\partial \theta }-\frac{\left( {B_{22} +B_{33} } \right) \sin \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{21}= & {} \frac{\left( {A_{12} +A_{33} } \right) }{R(s)}\frac{\partial ^{2}}{\partial s\partial \theta }+\frac{\left( {A_{22} +A_{33} } \right) \sin \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{22}= & {} A_{33} \left[ {\frac{\partial ^{2}}{\partial s^{2}}+\frac{\sin \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{\sin ^{2}\alpha }{R^{2}(s)}} \right] +\frac{A_{22} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}}-\frac{A_{44} \cos ^{2}\alpha }{R^{2}(s)} \\ L_{23}= & {} \frac{\left( {A_{22} +A_{44} } \right) \cos \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{24}= & {} \frac{\left( {B_{12} +B_{33} } \right) }{R(s)}\frac{\partial ^{2}}{\partial s\partial \theta }+\frac{\left( {B_{22} +B_{33} } \right) \sin \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{25}= & {} B_{33} \left[ {\frac{\partial ^{2}}{\partial s^{2}}+\frac{\sin \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{\sin ^{2}\alpha }{R^{2}(s)}} \right] +\frac{B_{22} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}}+\frac{A_{44} \cos \alpha }{R(s)} \\ L_{31}= & {} -\frac{A_{12} \cos \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{A_{22} \sin \alpha \cos \alpha }{R^{2}(s)} \\ L_{32}= & {} -\frac{\left( {A_{22} +A_{44} } \right) \cos \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{33}= & {} A_{55} \left[ {\frac{\partial ^{2}}{\partial s^{2}}+\frac{\sin \alpha }{R(s)}\frac{\partial }{\partial s}} \right] +\frac{A_{44} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}}-\frac{A_{22} \cos ^{2}\alpha }{R^{2}(s)} \\ L_{34}= & {} \left( {A_{55} -\frac{B_{12} \cos \alpha }{R(s)}} \right) \frac{\partial }{\partial s}+\frac{A_{55} \sin \alpha }{R(s)}-\frac{B_{22} \sin \alpha \cos \alpha }{R^{2}(s)} \\ L_{35}= & {} \left( {\frac{A_{44} }{R(s)}-\frac{B_{22} \cos \alpha }{R^{2}(s)}} \right) \frac{\partial }{\partial \theta } \\ L_{41}= & {} B_{11} \frac{\partial ^{2}}{\partial s^{2}}+\frac{B_{11} \sin \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{B_{22} \sin ^{2}\alpha }{R^{2}(s)}+\frac{B_{33} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}} \\ L_{42}= & {} \frac{\left( {B_{12} +B_{33} } \right) }{R(s)}\frac{\partial ^{2}}{\partial s\partial \theta }-\frac{\left( {B_{22} +B_{33} } \right) \sin \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{43}= & {} -\left( {A_{55} -\frac{B_{12} \cos \alpha }{R(s)}} \right) \frac{\partial }{\partial s}-\frac{B_{22} \sin \alpha \cos \alpha }{R^{2}(s)} \\ L_{44}= & {} D_{11} \frac{\partial ^{2}}{\partial s^{2}}+\frac{D_{11} \sin \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{D_{22} \sin ^{2}\alpha }{R^{2}(s)}+\frac{D_{33} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}}-A_{55} \\ L_{45}= & {} \frac{\left( {D_{12} +D_{33} } \right) }{R(s)}\frac{\partial ^{2}}{\partial s\partial \theta }-\frac{\left( {D_{22} +D_{33} } \right) \sin \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{51}= & {} \frac{\left( {B_{12} +B_{33} } \right) }{R(s)}\frac{\partial ^{2}}{\partial s\partial \theta }+\frac{\left( {B_{22} +B_{33} } \right) \sin \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{52}= & {} B_{33} \left[ {\frac{\partial ^{2}}{\partial s^{2}}+\frac{\sin \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{\sin ^{2}\alpha }{R^{2}(s)}} \right] +\frac{B_{22} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}}+\frac{A_{44} \cos \alpha }{R(s)} \\ L_{53}= & {} -\left( {\frac{A_{44} }{R(s)}-\frac{B_{22} \cos \alpha }{R^{2}(s)}} \right) \frac{\partial }{\partial \theta } \\ L_{54}= & {} \frac{\left( {D_{12} +D_{33} } \right) }{R(s)}\frac{\partial ^{2}}{\partial s\partial \theta }+\frac{\left( {D_{22} +D_{33} } \right) \sin \alpha }{R^{2}(s)}\frac{\partial }{\partial \theta } \\ L_{55}= & {} D_{33} \left[ {\frac{\partial ^{2}}{\partial s^{2}}+\frac{\sin \alpha }{R(s)}\frac{\partial }{\partial s}-\frac{\sin ^{2}\alpha }{R^{2}(s)}} \right] +\frac{D_{22} }{R^{2}(s)}\frac{\partial ^{2}}{\partial \theta ^{2}}-A_{44} \end{aligned}$$

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Izadi, M.H., Hosseini-Hashemi, S. & Korayem, M.H. Analytical and FEM solutions for free vibration of joined cross-ply laminated thick conical shells using shear deformation theory. Arch Appl Mech 88, 2231–2246 (2018). https://doi.org/10.1007/s00419-018-1446-y

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