Mixed Finite Elements for Higher-order Laminated Cylindrical and Spherical Shells

Abstract

This paper presents a mixed finite element (MFE) formulation for studying the linear static behavior of both thin and relatively thick laminated composite cylindrical and spherical shells. The method employs the Higher Order Shear Deformation Theory to account for cross-section warping due to transverse shear stress. It ensures the stationarity of the system's functional using the Hellinger-Reissner principle. Finite element discretization is accomplished with four-noded quadrilateral two-dimensional elements. The MFE formulation offers the advantage of directly obtaining displacements and stress resultants at the nodes. Comparison and convergence analyses are performed considering various shear functions, boundary conditions, and geometrical configurations.

Keywords

• Higher order shear deformation theory
• laminated composite shell
• hellinger-reissner principle
• mixed finite element method
• static analysis

References

1. M. Dorduncu, Peridynamic modeling of delaminations in laminated composite beams using refined zigzag theory, Theoretical and Applied Fracture Mechanics 112 (2021) 102832. https://doi.org/10.1016/j.tafmec.2020.102832.
2. M. Dorduncu, Stress analysis of laminated composite beams using refined zigzag theory and peridynamic differential operator, Composite Structures 218 (2019) 193–203. https://doi.org/10.1016/j.compstruct.2019.03.035.
3. J.N. Reddy, K. Chandrashekhara, Geometrically non-linear transient analysis of laminated, doubly curved shells, International Journal of Non-Linear Mechanics 20 (1985) 79–90. https://doi.org/10.1016/0020-7462(85)90002-2.
4. S.J. Hossain, P.K. Sinha, A.H. Sheikh, A finite element formulation for the analysis of laminated composite shells, Computers & Structures 82 (2004) 1623–1638. https://doi.org/10.1016/j.compstruc.2004.05.004.
5. E. Asadi, M.S. Qatu, Static analysis of thick laminated shells with different boundary conditions using GDQ, Thin-Walled Structures 51 (2012) 76–81. https://doi.org/10.1016/j.tws.2011.11.004.
6. A.A. Khdeir, Comparative dynamic and static studies for cross-ply shells based on a deep thick shell theory, IJVNV 7 (2011) 306. https://doi.org/10.1504/IJVNV.2011.043192.
7. S.M. Mousavi, M. Aghdam, Static bending analysis of laminated cylindrical panels with various boundary conditions using the differential cubature method, J. Mech. Mater. Struct. 4 (2009) 509–521. https://doi.org/10.2140/jomms.2009.4.509.
8. B. Sobhaniaragh, R.C. Batra, W.J. Mansur, F.C. Peters, Thermal response of ceramic matrix nanocomposite cylindrical shells using Eshelby-Mori-Tanaka homogenization scheme, Composites Part B: Engineering 118 (2017) 41–53. https://doi.org/10.1016/j.compositesb.2017.02.032.

9. J.N. Reddy, C.F. Liu, A higher-order shear deformation theory of laminated elastic shells, International Journal of Engineering Science 23 (1985) 319–330. https://doi.org/10.1016/0020-7225(85)90051-5.
10. A.S. Sayyad, Y.M. Ghugal, Static and free vibration analysis of laminated composite and sandwich spherical shells using a generalized higher-order shell theory, Composite Structures 219 (2019) 129–146. https://doi.org/10.1016/j.compstruct.2019.03.054.
11. J.L. Mantari, A.S. Oktem, C. Guedes Soares, Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory, Composite Structures 94 (2011) 37–49. https://doi.org/10.1016/j.compstruct.2011.07.020.
12. A. Kumar, A. Chakrabarti, M. Ketkar, Analysis of laminated composite skew shells using higher order shear deformation theory, Lat. Am. j. Solids Struct. 10 (2013) 391–919. https://doi.org/10.1590/S1679-78252013000500003.
13. G. Giunta, F. Biscani, S. Belouettar, E. Carrera, Hierarchical modelling of doubly curved laminated composite shells under distributed and localised loadings, Composites Part B 42 (2011) 682–91. https://doi.org/10.1016/j.compositesb.2011.02.002.
14. E. Asadi, W. Wang, M.S. Qatu, Static and vibration analyses of thick deep laminated cylindrical shells using 3D and various shear deformation theories, Composite Structures 94 (2012) 494–500. https://doi.org/10.1016/j.compstruct.2011.08.011.
15. H.L. Ton-That, H. Nguyen-Van, T. Chau-Dinh, An Improved Four-Node Element for Analysis of Composite Plate/Shell Structures Based on Twice Interpolation Strategy, Int. J. Comput. Methods 17 (2020) 1950020. https://doi.org/10.1142/S0219876219500208.
16. H. Zuo, Y. Chen, F. Jia, Z. Yang, Unified wavelet finite element formulation for static and vibration analysis of laminated composite shells, Composite Structures 272 (2021) 114207. https://doi.org/10.1016/j.compstruct.2021.114207.
17. M. Yaghoubshahi, E. Asadi, S.J. Fariborz, A higher-order shell model applied to shells with mixed boundary conditions, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 225 (2011) 292–303. https://doi.org/10.1243/09544062JMES2050.
18. R.M.J. Groh, P.M. Weaver, Static inconsistencies in certain axiomatic higher-order shear deformation theories for beams, plates and shells, Composite Structures 120 (2015) 231–245. https://doi.org/10.1016/j.compstruct.2014.10.006.
19. K. Bhaskar, T.K. Varadan, A higher-order theory for bending analysis of laminated shells of revolution, Computers & Structures 40 (1991) 815–819. https://doi.org/10.1016/0045-7949(91)90310-I.
20. T.N. Doan, D. Van Thom, N.T. Thanh, P. Van Chuong, N.C. Tho, N.T. Ta, H.N. Nguyen, Analysis of stress concentration phenomenon of cylinder laminated shells using higher-order shear deformation Quasi-3D theory, Composite Structures 232 (2020) 111526. https://doi.org/10.1016/j.compstruct.2019.111526.
21. E. Viola, F. Tornabene, N. Fantuzzi, Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories, Composite Structures 101 (2013) 59–93. https://doi.org/10.1016/j.compstruct.2013.01.002.
22. I.F. Pinto Correia, C.M. Mota Soares, C.A. Mota Soares, J. Herskovits, Analysis of laminated conical shell structures using higher order models, 62 (2003) 383–390. https://doi.org/10.1016/j.compstruct.2003.09.009.
23. F. Tornabene, N. Fantuzzi, E. Viola, R.C. Batra, Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory, Composite Structures 119 (2015) 67–89. https://doi.org/10.1016/j.compstruct.2014.08.005.
24. M. Yaqoob Yasin, S. Kapuria, An efficient layerwise finite element for shallow composite and sandwich shells, Composite Structures 98 (2013) 202–214. https://doi.org/10.1016/j.compstruct.2012.10.048.
25. A. Gupta, S. Pradyumna, Geometrically nonlinear bending analysis of variable stiffness composite laminated shell panels with a higher-order theory, Composite Structures 276 (2021) 114527. https://doi.org/10.1016/j.compstruct.2021.114527.
26. G.M. Kulikov, S.V. Plotnikova, Advanced formulation for laminated composite shells: 3D stress analysis and rigid-body motions, Composite Structures 95 (2013) 236–246. https://doi.org/10.1016/j.compstruct.2012.07.020.
27. M.S. Qatu, A. Algothani, Bending analysis of laminated plates and shells by different methods, Computers & Structures 52 (1994) 529–539. https://doi.org/10.1016/0045-7949(94)90238-0.
28. E.E. Karataş, R.F. Yükseler, Snap-through Buckling of Shallow Spherical Shells under Ring Loads, Teknik Dergi 32 (2021) 10695–10716. https://doi.org/10.18400/tekderg.565095.
29. A. Sofiyev, A. Deniz, M. Avcar, P. Özyigit, M. Omurtag, Effects of the non-homogeneity and elastic medium on the critical torsional load of the orthotropic cylindrical shell footnotemark, Acta Physica Polonica A 123 (2013) 728–730.
30. A. Yadav, M. Amabili, S. Kumar Panda, T. Dey, Instability analysis of fluid-filled angle-ply laminated circular cylindrical shells subjected to harmonic axial loading | Elsevier Enhanced Reader, European Journal of Mechanics - A/Solids 97 (2023). https://doi.org/10.1016/j.euromechsol.2022.104810.
31. M.C. Ray, Exact solutions of elasticity theories for static analysis of doubly curved antisymmetric angle-ply composite shells, Mechanics of Advanced Materials and Structures (2023) 1–15. https://doi.org/10.1080/15376494.2023.2246223.
32. Md.I. Alam, M.K. Pandit, A.K. Pradhan, A modified higher-order zigzag theory for predicting flexural behavior of laminated composite and sandwich shell, Mechanics of Advanced Materials and Structures (2023) 1–16. https://doi.org/10.1080/15376494.2023.2231445.
33. A.B. Arumugam, M. Subramani, M. Dalakoti, P. Jindal, R. Selvaraj, E. Khalife, Dynamic characteristics of laminated composite CNT reinforced MRE cylindrical sandwich shells using HSDT, Mechanics Based Design of Structures and Machines 51 (2023) 4120–4136. https://doi.org/10.1080/15397734.2021.1950550.
34. A. Ozutok, E. Madenci, Static analysis of laminated composite beams based on higher-order shear deformation theory by using mixed-type finite element method, Int. J. Mech. Sci. 130 (2017) 234–243. https://doi.org/10.1016/j.ijmecsci.2017.06.013.
35. A. Kutlu, M. Dorduncu, T. Rabczuk, A novel mixed finite element formulation based on the refined zigzag theory for the stress analysis of laminated composite plates, Composite Structures 267 (2021) 113886. https://doi.org/10.1016/j.compstruct.2021.113886.
36. A. Kutlu, Mixed finite element formulation for bending of laminated beams using the refined zigzag theory, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications 235 (2021) 1712–1722. https://doi.org/10.1177/14644207211018839.
37. A. Kutlu, G. Meschke, M.H. Omurtag, A new mixed finite-element approach for the elastoplastic analysis of Mindlin plates, J Eng Math 99 (2016) 137–155. https://doi.org/10.1007/s10665-015-9825-7.
38. U.N. Aribas, M. Ermis, N. Eratli, M.H. Omurtag, The static and dynamic analyses of warping included composite exact conical helix by mixed FEM | Elsevier Enhanced Reader, Composites Part B: Engineering 160 (2019) 285–297. https://doi.org/10.1016/j.compositesb.2018.10.018.
39. U.N. Aribas, M. Ermis, A. Kutlu, N. Eratli, M.H. Omurtag, Forced vibration analysis of composite-geometrically exact elliptical cone helices via mixed FEM, Mechanics of Advanced Materials and Structures (2020) 1–19. https://doi.org/10.1080/15376494.2020.1824048.
40. U.N. Aribas, M. Ermis, M.H. Omurtag, The static and stress analyses of axially functionally graded exact super-elliptical beams via mixed FEM, Arch Appl Mech 91 (2021) 4783–4796. https://doi.org/10.1007/s00419-021-02033-w.
41. A. Kutlu, M. Hakkı Omurtag, Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method, International Journal of Mechanical Sciences 65 (2012) 64–74. https://doi.org/10.1016/j.ijmecsci.2012.09.004.
42. M.H. Omurtag, A.Y. Aköz, Isoparametric mixed finite element formulation of orthotropic cylindrical shells, Computers & Structures 55 (1995) 915–924. https://doi.org/10.1016/0045-7949(94)00450-H.
43. Y. Bab, A. Kutlu, Mixed finite element formulation based on higher order theory for stress calculation of laminated composite beams, in: Proceedings 22. National Mechanics Congress, Adana, Turkey, 2021.
44. Y. Bab, A. Kutlu, Stress analysis of laminated HSDT beams considering bending extension coupling, Turkish Journal of Civil Engineering 34 (2023) 1–23. https://doi.org/10.18400/tjce.1206777.
45. M. Touratier, An efficient standard plate theory, International Journal of Engineering Science 29 (1991) 901–916. https://doi.org/10.1016/0020-7225(91)90165-Y.
46. J.N. Reddy, A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics 51 (1984) 745–752. https://doi.org/10.1115/1.3167719.
47. E. Reissner, On transverse bending of plates, including the effect of transverse shear deformation, International Journal of Solids and Structures 11 (1974) 569–573.
48. H. Nguyen-Xuan, C.H. Thai, T. Nguyen-Thoi, Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory, Composites Part B: Engineering 55 (2013) 558–574. https://doi.org/10.1016/j.compositesb.2013.06.044.
49. J.N. Reddy, Mechanics of Laminated Composite Plates and Shells, CRC Press, Boca Raton, 2003.
50. Y. Bab, Mixed finite element formulations for laminated beams and plates based on higher order shear deformation theories, Master’s Thesis, Istanbul Technical University, 2021.
51. A.A. Khdeir, L. Librescu, D. Frederick, A shear deformable theory of laminated composite shallow shell-type panels and their response analysis II: Static response, Acta Mechanica 77 (1989) 1–12. https://doi.org/10.1007/BF01379740.