Skip to main content
SpringerLink
Log in

Static analysis of functionally graded composite shells on elastic foundations with nonlocal elasticity theory

  • Original Article
  • Published:
Archives of Civil and Mechanical Engineering Aims and scope Submit manuscript

Abstract

The aim of this present work is to study the higher-order modelling of a cylindrical nano-shell resting on Pasternak’s foundation based on nonlocal elasticity theory. Third-order shear deformation theory is developed for modelling the kinematic relations, and nonlocal elasticity theory is developed for size-dependent analysis. The principle of virtual work is applied to derive static governing equations. The solution is presented for simply supported boundary conditions in terms of various important parameters. The numerical results including lower- and higher-order longitudinal and radial displacements are presented in terms of nonlocal parameter, two parameters of Pasternak’s foundation and some dimensionless geometric parameters such as length-to-radius ratio and length-to-thickness ratio.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Yang J, Shen H-S. Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels. J Sound Vib. 2003;261:871–93.

    Article  Google Scholar 

  2. Chen WQ, Bian ZG, Lv CF, Ding HJ. 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. Int J Solids Struct. 2004;41:947–64.

    Article  Google Scholar 

  3. Lam KY, Qian WU. Vibrations of thick rotating laminated composite cylindrical shells. J Sound Vib. 1999;225(3):483–501.

    Article  Google Scholar 

  4. Asgari M, Akhlaghi M. Natural frequency analysis of 2D-FGM thick hollow cylinder based on three-dimensional elasticity equations. Eur J Mech A Solid. 2011;30:72–81.

    Article  Google Scholar 

  5. Shakeri M, Akhlaghi M, Hoseini SM. Vibration and radial wave propagation velocity in functionally graded thick hollow cylinder. Compos Struct. 2006;76:174–81.

    Article  Google Scholar 

  6. Mohammadi K, Mahinzare M, Ghorbani Kh, Ghadiri M. Cylindrical functionally graded shell model based on the first order shear deformation nonlocal strain gradient elasticity theory. Microsyst Technol. 2018;24:1133–46.

    Article  Google Scholar 

  7. Dastjerdi S, Abbasi M, Yazdanparast L. A new modified higher-order shear deformation theory for nonlinear analysis of macro- and nano-annular sector plates using the extended Kantorovich method in conjunction with SAPM. Acta Mech. 2017;228(10):3381–401.

    Article  MathSciNet  Google Scholar 

  8. Razavi S. Magneto-electro-thermo-mechanical response of a multiferroic doubly-curved nano-shell. J Solid Mech. 2018;10(1):130–41.

    Google Scholar 

  9. Shooshtari A, Razavi S. Vibration analysis of a magnetoelectroelastic rectangular plate based on a higher-order shear deformation theory. Lat Am J Solids Struct. 2016;13(3):554–72.

    Article  Google Scholar 

  10. Raissi H, Shishesaz M, Moradi S. Applications of higher order shear deformation theories on stress distribution in a five layer sandwich plate. J Appl Comput Mech. 2017;48(2):233–52.

    Google Scholar 

  11. Ansari R, Hasrati E, Torabi J. Vibration analysis of pressurized sandwich FG-CNTRC cylindrical shells based on the higher-order shear deformation theory. Res Express. 2019;6:045049.

    Article  Google Scholar 

  12. Mehraliana F, Beni YT. A nonlocal strain gradient shell model for free vibration analysis of functionally graded shear deformable nanotubes. Int J Eng Appl Sci. 2017;9(2):88–102.

    Google Scholar 

  13. Beni YT, Mehralian F, Zeverdejani MK. Free vibration of anisotropic single-walled carbon nanotube based on couple stress theory for different chirality. J Low Freq Noise Vib Active Control. 2017;36(3):277–93.

    Article  Google Scholar 

  14. Zhang Y, Zhang F. Vibration and buckling of shear deformable functionally graded nanoporous metal foam nanoshells. Nanomaterials. 2019;9:271.

    Article  Google Scholar 

  15. Rouhi H, Ansari R, Darvizeh M. Exact solution for the vibrations of cylindrical nanoshells considering surface energy effect. J Ultrafine Grained Nanostruct Mater. 2015;48(2):113–24.

    Google Scholar 

  16. Soleimani YT, Beni MB. Dehkordi, Finite element vibration analysis of nanoshell based on new cylindrical shell element. Struct Eng Mech. 2018;65(1):33–41.

    Google Scholar 

  17. Ghadiri M, Safarpour H. Free vibration analysis of a functionally graded cylindrical nanoshell surrounded by elastic foundation based on the modified couple stress theory. Amirkabir J Mech Eng. 2018;49(4):721–30.

    Google Scholar 

  18. Hajilak ZE, Pourghader J, Hashemabadi D, Bagh FS, Habibi M. Multilayer GPLRC composite cylindrical nanoshell using modified strain gradient theory. Mech Based Des Struct. 2019. https://doi.org/10.1080/15397734.2019.1566743.

    Article  Google Scholar 

  19. Razavi H, Babadi AF, Beni YT. Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory. Compos Struct. 2017;160:1299–309.

    Article  Google Scholar 

  20. Chakraborty S, Dey T, Kumar R. Stability and vibration analysis of CNT-Reinforced functionally graded laminated composite cylindrical shell panels using semi-analytical approach. Compos B Eng. 2019;168:1–14.

    Article  Google Scholar 

  21. Mehralian F, Beni YT. Size-dependent torsional buckling analysis of functionally graded cylindrical shell. Compos B Eng. 2016;94:11–25.

    Article  Google Scholar 

  22. Habibi M, Taghdir A, Safarpour H. Stability analysis of an electrically cylindrical nanoshell reinforced with graphene nanoplatelets. Compos B Eng. 2019;175:107125.

    Article  Google Scholar 

  23. Jabbari M, Sohrabpour S, Eslami MR. Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int J Press Vessels Pip. 2002;79(7):493–7.

    Article  Google Scholar 

  24. Jabbari M, Bahtui A, Eslami MR. Axisymmetric mechanical and thermal stresses in thick short length FGM cylinders. Int J Press Vessels Pip. 2009;86(5):296–306.

    Article  Google Scholar 

  25. Jabbari M, Sohrabpour S, Eslami MR. General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to nonaxisymmetric steady-state loads. J Appl Mech. 2003;70(1):111–8.

    Article  Google Scholar 

  26. Arefi M, Zenkour AM. Size-dependent thermoelastic analysis of a functionally graded nanoshell. Mod Phys Lett B. 2018;32(03):1850033.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was financially supported by the University of Kashan (Grant Number: 574613/026). M. Arefi would like to thank the Iranian Nanotechnology Development Committee for their financial support. Second author also would like to thank the Research Center for Interneural Computing of China Medical University of Taiwan.

Author information

Authors and Affiliations

  1. Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-51167, Iran

    Mohammad Arefi

  2. Research Center for Interneural Computing, China Medical University, Taichung, Taiwan

    O. Civalek

Authors
  1. Mohammad Arefi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Civalek
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

MA and OC contributed to study conception and design. MA helped in the acquisition of data, analysis and interpretation of data. MA and OC contributed to drafting of manuscript and critical revision.

Corresponding author

Correspondence to O. Civalek.

Ethics declarations

Conflict of interest

All authors declare that they have no conflict of interest.

Ethical statement

Authors state that the research was conducted according to ethical standards.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

$$\left\{ {A_{1} ,A_{2} ,A_{3} ,A_{4} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \left( {1 - \nu } \right)\left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{5} ,A_{6} ,A_{7} ,A_{8} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{9} ,A_{10} ,A_{11} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \nu \left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{12} ,A_{13} ,A_{14} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \left( {1 - \nu } \right)\left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{15} ,A_{16} ,A_{17} ,A_{18} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu \left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{19} ,A_{20} ,A_{21} ,A_{22} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{23} ,A_{24} ,A_{25} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \frac{{\left( {1 - 2\nu } \right)}}{2}\left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{26} ,A_{27} ,A_{28} ,A_{29} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \frac{{\left( {1 - 2\nu } \right)}}{2}\left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{30} ,A_{31} ,A_{32} ,A_{33} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \left( {1 - \nu } \right)z\left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{34} ,A_{35} ,A_{36} ,A_{37} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{38} ,A_{39} ,A_{40} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z\left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{41} ,A_{42} ,A_{43} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \left( {1 - \nu } \right)z\left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{44} ,A_{45} ,A_{46} ,A_{47} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z\left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{48} ,A_{49} ,A_{50} ,A_{51} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{52} ,A_{53} ,A_{54} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \frac{{\left( {1 - 2\nu } \right)}}{2}z\left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{55} ,A_{56} ,A_{57} ,A_{58} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \frac{{\left( {1 - 2\nu } \right)}}{2}z\left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{59} ,A_{60} ,A_{61} ,A_{62} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \left( {1 - \nu } \right)z^{2} \left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{63} ,A_{64} ,A_{65} ,A_{66} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z^{2} \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{67} ,A_{68} ,A_{69} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z^{2} \left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{70} ,A_{71} ,A_{72} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \left( {1 - \nu } \right)z^{2} \left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{73} ,A_{74} ,A_{75} ,A_{76} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z^{2} \left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{77} ,A_{78} ,A_{79} ,A_{80} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z^{2} \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{81} ,A_{82} ,A_{83} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \frac{{\left( {1 - 2\nu } \right)}}{2}z^{2} \left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{84} ,A_{85} ,A_{86} ,A_{87} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \frac{{\left( {1 - 2\nu } \right)}}{2}z^{2} \left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z$$
$$\left\{ {A_{88} ,A_{89} ,A_{90} ,A_{91} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \left( {1 - \nu } \right)z^{3} \left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{92} ,A_{93} ,A_{94} ,A_{95} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z^{3} \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{96} ,A_{97} ,A_{98} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z^{3} \left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{99} ,A_{100} ,A_{101} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \frac{{\left( {1 - 2\nu } \right)}}{2}z^{3} \left( {R + z} \right)\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{102} ,A_{103} ,A_{104} ,A_{105} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \frac{{\left( {1 - 2\nu } \right)}}{2}z^{3} \left( {R + z} \right)\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{106} ,A_{107} ,A_{108} ,A_{109} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \frac{{\lambda \left( {1 - \nu } \right)}}{{\left( {R + z} \right)}}\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{110} ,A_{111} ,A_{112} ,A_{113} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{114} ,A_{115} ,A_{116} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \nu \left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{117} ,A_{118} ,A_{119} ,A_{120} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \frac{{\lambda \left( {1 - \nu } \right)}}{{\left( {R + z} \right)}}z\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{121} ,A_{122} ,A_{123} ,A_{124} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z\left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{125} ,A_{126} ,A_{127} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \nu z\left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{128} ,A_{129} ,A_{130} ,A_{131} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \frac{{\lambda \left( {1 - \nu } \right)}}{{\left( {R + z} \right)}}z^{2} \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{132} ,A_{133} ,A_{134} ,A_{135} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z^{2} \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{136} ,A_{137} ,A_{138} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \nu z^{2} \left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$
$$\left\{ {A_{139} ,A_{140} ,A_{141} ,A_{142} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \frac{{\lambda \left( {1 - \nu } \right)}}{{\left( {R + z} \right)}}z^{2} \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{143} ,A_{144} ,A_{145} ,A_{146} } \right\} = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \lambda \nu z^{2} \left\{ {1,z,z^{2} ,z^{3} } \right\}{\text{d}}z,$$
$$\left\{ {A_{147} ,A_{148} ,A_{149} } \right\} = \mathop \int \limits_{ - h/2}^{h/2} \lambda \nu z^{2} \left\{ {1,2z,3z^{2} } \right\}{\text{d}}z,$$

Appendix B

$$\begin{aligned} K_{11} & = + A_{1} \lambda_{n}^{2} ,K_{12} = + A_{2} \lambda_{n}^{2} ,K_{13} = + A_{3} \lambda_{n}^{2} ,K_{14} = + A_{4} \lambda_{n}^{2} ,K_{15} = - A_{5} \lambda_{n} ,K_{16} = - \left( {A_{6} + A_{9} } \right)\lambda_{n} , \\ K_{17} & = - \left( {A_{7} + A_{10} } \right)\lambda_{n} ,K_{18} = - \left( {A_{8} + A_{11} } \right)\lambda_{n} \\ K_{21} & = + A_{30} \lambda_{n}^{2} ,K_{22} = + A_{31} \lambda_{n}^{2} + A_{23} ,K_{23} = + A_{32} \lambda_{n}^{2} + A_{24} ,K_{24} = + A_{33} \lambda_{n}^{2} + A_{25} ,K_{25} = + \left( {A_{26} - A_{34} } \right)\lambda_{n} \\ K_{26} & = + \left( {A_{27} - A_{35} - A_{38} } \right)\lambda_{n} ,K_{27} = + \left( {A_{28} - A_{36} - A_{39} } \right)\lambda_{n} ,K_{28} = + \left( {A_{29} - A_{37} - A_{40} } \right)\lambda_{n} , \\ K_{31} & = + A_{59} \lambda_{n}^{2} ,K_{32} = + A_{60} \lambda_{n}^{2} + 2A_{52} ,K_{33} = A_{61} \lambda_{n}^{2} + 2A_{53} ,K_{34} = + A_{62} \lambda_{n}^{2} + 2A_{54} , \\ K_{35} & = + \left( {2A_{55} - A_{63} } \right)\lambda_{n} ,K_{36} = \left( {2A_{56} - A_{67} - A_{64} } \right)\lambda_{n} ,K_{37} = \left( {2A_{57} - A_{65} - A_{68} } \right)\lambda_{n} ,K_{38} = \left( {2A_{58} - A_{66} - A_{69} } \right)\lambda_{n} , \\ K_{41} & = + A_{88} \lambda_{n}^{2} ,K_{42} = + A_{89} \lambda_{n}^{2} + 3A_{81} ,K_{43} = + A_{90} \lambda_{n}^{2} + 3A_{82} ,K_{44} = + A_{91} \lambda_{n}^{2} + 3A_{83} ,K_{45} = \left( {3A_{84} - A_{92} } \right)\lambda_{n} \\ K_{46} & = + \left( {3A_{85} - A_{96} - A_{93} } \right)\lambda_{n} ,K_{47} = \left( {3A_{86} - A_{94} - A_{97} } \right)\lambda_{n} ,K_{48} = \left( {3A_{87} - A_{95} - A_{98} } \right)\lambda_{n} , \\ K_{51} & = - A_{110} \lambda_{n} ,K_{52} = - \left( {A_{111} - A_{23} } \right)\lambda_{n} ,K_{53} = - \left( {A_{112} - A_{24} } \right)\lambda_{n} ,K_{54} = - \left( {A_{113} - A_{25} } \right)\lambda_{n} ,K_{55} = A_{26} \lambda_{n}^{2} + A_{106} \\ K_{56} & = A_{27} \lambda_{n}^{2} + \left( {A_{107} + A_{114} } \right),K_{57} = A_{28} \lambda_{n}^{2} + \left( {A_{108} + A_{115} } \right),K_{58} = A_{29} \lambda_{n}^{2} + \left( {A_{109} + A_{116} } \right) \\ K_{61} & = - \left( {A_{15} + A_{121} } \right)\lambda_{n} ,K_{62} = - \left( {A_{16} + A_{122} - A_{52} } \right)\lambda_{n} ,K_{63} = - \left( {A_{123} + A_{17} - A_{53} } \right)\lambda_{n} , \\ K_{64} & = - \left( {A_{124} - A_{54} + A_{18} } \right)\lambda_{n} ,K_{65} = + \left( {A_{117} + A_{19} } \right) + A_{55} \lambda_{n}^{2} ,K_{66} = \left( {A_{12} + A_{20} + A_{118} + A_{125} } \right) + A_{56} \lambda_{n}^{2} \\ K_{67} & = + \left( {A_{13} + A_{21} + A_{119} + A_{126} } \right) + A_{57} \lambda_{n}^{2} ,K_{68} = \left( {A_{14} + A_{120} + A_{22} + A_{127} } \right) + A_{58} \lambda_{n}^{2} \\ K_{71} & = - \left( {2A_{44} + A_{132} } \right)\lambda_{n} ,K_{72} = - \left( {A_{133} + 2A_{45} - A_{81} } \right)\lambda_{n} ,K_{73} = - \left( {A_{134} + 2A_{46} - A_{82} } \right)\lambda_{n} \\ K_{74} & = - \left( {A_{135} + 2A_{47} - A_{83} } \right)\lambda_{n} ,K_{75} = + A_{84} \lambda_{n}^{2} + \left( {2A_{48} + A_{128} } \right),K_{76} = + A_{85} \lambda_{n}^{2} + \left( {2A_{41} + 2A_{49} + A_{129} + A_{136} } \right) \\ K_{77} & = + A_{86} \lambda_{n}^{2} + \left( {A_{130} + 2A_{42} + 2A_{50} + A_{137} } \right),K_{78} = + A_{87} \lambda_{n}^{2} + \left( {A_{131} + A_{138} + 2A_{43} + 2A_{51} } \right) \\ K_{81} & = - \left( {3A_{73} + A_{143} } \right)\lambda_{n} ,K_{82} = - \left( {3A_{74} + A_{144} - A_{99} } \right)\lambda_{n} ,K_{83} = - \left( {3A_{75} + A_{145} - A_{100} } \right)\lambda_{n} \\ K_{84} & = - \left( {A_{146} - A_{101} + 3A_{76} } \right)\lambda_{n} ,K_{85} = \left( {3A_{77} + A_{139} } \right) + A_{102} \lambda_{n}^{2} ,K_{86} = + A_{103} \lambda_{n}^{2} + \left( {A_{140} + 3A_{78} + 3A_{70} + A_{147} } \right) \\ K_{87} & = \begin{array}{*{20}c} { + A_{104} \lambda_{n}^{2} + \left( {A_{141} + A_{148} + 3A_{71} + 3A_{79} } \right),K_{88} = A_{105} \lambda_{n}^{2} + \left( {A_{149} + 3A_{80} + A_{142} + 3A_{72} } \right)} \\ \end{array} \\ \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arefi, M., Civalek, O. Static analysis of functionally graded composite shells on elastic foundations with nonlocal elasticity theory. Archiv.Civ.Mech.Eng 20, 22 (2020). https://doi.org/10.1007/s43452-020-00032-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43452-020-00032-2

Keywords

Search

Navigation