### Static and free vibration analyses of laminated composite shells by cell-based smoothed discrete shear gap method (CS-DSG3) using three-node triangular elements

#### Abstract

A cell-based smoothed discrete shear gap method (CS-DSG3) using three-node triangular elements was recently proposed to improve the performance of the discrete shear gap method (DSG3) for static and free vibration analyses of isotropic Reissner-Mindlin plates and shells. In this paper, the CS-DSG3 is further extended for static and free vibration analyses of laminated composite shells. In the present method, the first-order shear deformation theory (FSDT) is used in the formulation due to the simplicity and computational efficiency. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of others available numerical results.

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A. Bhimaraddi. Free vibration analysis of doubly curved shallow shells on rectangular planform using three-dimensional elasticity theory. International Journal of Solids and Structures, 27, (7), (1991), pp. 897–913. doi:10.1016/0020-7683(91)90023-9.

A. Bhimaraddi. Three-dimensional elasticity solution for static response of orthotropic doubly curved shallow shells on rectangular planform. Composite Structures, 24, (1), (1993), pp. 67–77. doi:10.1016/0263-8223(93)90056-v.

K. P. Rao. A rectangular laminated anisotropic shallow thin shell finite element. Computer Methods in Applied Mechanics and Engineering, 15, (1), (1978), pp. 13–33. doi:10.1016/0045-7825(78)90003-8.

J. N. Reddy. Exact solutions of moderately thick laminated shells. Journal of Engineering Mechanics, 110, (5), (1984), pp. 794–809. doi:10.1061/(asce)0733-9399(1984)110:5(794).

S. J. Hossain, P. K. Sinha, and A. H. Sheikh. A finite element formulation for the analysis of laminated composite shells. Computers & Structures, 82, (20-21), (2004), pp. 1623–1638. doi:10.1016/j.compstruc.2004.05.004.

D. Chakravorty, J. N. Bandyopadhyay, and P. K. Sinha. Free vibration analysis of point supported laminated composite doubly curved shells - A finite element approach. Computers & Structures, 54, (2), (1995), pp. 191–198. doi:10.1016/0045-7949(94)00329-2.

J. N. Reddy. Mechanics of laminated composite plates and shells: Theory and analysis. CRC Press, (2004).

L. Librescu, A. A. Khdeir, and D. Frederick. A shear deformable theory of laminated composite shallow shell-type panels and their response analysis I: Free vibration and buckling. Acta Mechanica, 76, (1-2), (1989), pp. 1–33. doi:10.1007/bf01175794.

A. A. Khdeir, L. Librescu, and D. Frederick. A shear deformable theory of laminated composite shallow shell-type panels and their response analysis II: Static response. Acta Mechanica, 77, (1-2), (1989), pp. 1–12. doi:10.1007/bf01379740.

J. N. Reddy and C. F. Liu. A higher-order shear deformation theory of laminated elastic shells. International Journal of Engineering Science, 23, (3), (1985), pp. 319–330. doi:10.1016/0020-7225(85)90051-5.

R. K. Khare, T. Kant, and A. K. Garg. Closed-form thermo-mechanical solutions of higher order theories of cross-ply laminated shallow shells. Composite Structures, 59, (3), (2003), pp. 313–340. doi:10.1016/s0263-8223(02)00245-3.

M. Y. Yasin and S. Kapuria. An efficient layerwise finite element for shallow composite and sandwich shells. Composite Structures, 98, (2013), pp. 202–214. doi:10.1016/j.compstruct.2012.10.048.

G. Giunta, F. Biscani, S. Belouettar, and E. Carrera. Hierarchical modelling of doubly curved laminated composite shells under distributed and localised loadings. Composites Part B: Engineering, 42, (4), (2011), pp. 682–691. doi:10.1016/j.compositesb.2011.02.002.

A. J. M. Ferreira, E. Carrera, M. Cinefra, and C. M. C. Roque. Analysis of laminated doubly curved shells by a layerwise theory and radial basis functions collocation, accounting for through-the-thickness deformations. Computational Mechanics, 48, (1), (2011), pp. 13–25. doi:10.1007/s00466-011-0579-4.

C. H. Thai, H. Nguyen-Xuan, N. Nguyen-Thanh, T.-H. Le, T. Nguyen-Thoi, and T. Rabczuk. Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS-based isogeometric approach. International Journal for Numerical Methods in Engineering, 91, (6), (2012), pp. 571–603. doi:10.1002/nme.4282.

C. H. Thai, A. J. M. Ferreira, S. P. A. Bordas, T. Rabczuk, and H. Nguyen-Xuan. Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. European Journal of Mechanics-A/Solids, 43, (2014), pp. 89–108. doi:10.1016/j.euromechsol.2013.09.001.

C. H. Thai, H. Nguyen-Xuan, S. P. A. Bordas, N. Nguyen-Thanh, and T. Rabczuk. Isogeometric analysis of laminated composite plates using the higher-order shear deformation theory. Mechanics of Advanced Materials and Structures, 22, (6), (2015), pp. 451–469. doi:10.1080/15376494.2013.779050.

T. Rabczuk, P. M. A. Areias, and T. Belytschko. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 72, (5), (2007), pp. 524–548. doi:10.1002/nme.2013.

T. Rabczuk, R. Gracie, J.-H. Song, and T. Belytschko. Immersed particle method for fluidstructure interaction. International Journal for Numerical Methods in Engineering, 81, (1), (2010), pp. 48–71. doi:10.1002/nme.2670.

N. Nguyen-Thanh, J. Kiendl, H. Nguyen-Xuan, R. Wuchner, K.-U. Bletzinger, Y. Bazilevs, and T. Rabczuk. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 200, (47-48), (2011), pp. 3410–3424. doi:10.1016/j.cma.2011.08.014.

N. Nguyen-Thanh, N. Valizadeh, M. N. Nguyen, H. Nguyen-Xuan, X. Zhuang, P. Areias, G. Zi, Y. Bazilevs, L. De Lorenzis, and T. Rabczuk. An extended isogeometric thin shell analysis based on Kirchhoff–Love theory. Computer Methods in Applied Mechanics and Engineering, 284, (2015), pp. 265–291. doi:10.1016/j.cma.2014.08.025.

G. R. Liu and T. Nguyen-Thoi. Smoothed finite element methods. CRC Press, Taylor and Francis Group, New York, (2010).

G. R. Liu, K. Y. Dai, and T. T. Nguyen. A smoothed finite element method for mechanics problems. Computational Mechanics, 39, (6), (2007), pp. 859–877. doi:10.1007/s00466-006-0075-4.

G. R. Liu, T. Nguyen-Thoi, H. Nguyen-Xuan, K. Y. Dai, and K. Y. Lam. On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering, 77, (13), (2009), pp. 1863–1869. doi:10.1002/nme.2587.

T. T. Nguyen, G. R. Liu, K. Y. Dai, and K. Y. Lam. Selective smoothed finite element method. Tsinghua Science & Technology, 12, (5), (2007), pp. 497–508. doi:10.1016/s1007-0214(07)70125-6.

G. R. Liu, H. Nguyen-Xuan, and T. Nguyen-Thoi. A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates. International Journal for Numerical Methods in Engineering, 84, (10), (2010), pp. 1222–1256. doi:10.1002/nme.2941.

G. R. Liu, T. T. Nguyen, K. Y. Dai, and K. Y. Lam. Theoretical aspects of the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering, 71, (8), (2007), pp. 902–930. doi:10.1002/nme.1968.

D. T. Hau, N. T. M. Hanh, and N. T. Trung. A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) for dynamic response of laminated composite plate subjected to blast loading. Vietnam Journal of Mechanics, 37, (2), (2015), pp. 81–90. doi:10.15625/0866-7136/37/2/5019.

G. R. Liu, T. Nguyen-Thoi, H. Nguyen-Xuan, and K. Y. Lam. A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Computers & Structures, 87, (1-2), (2009), pp. 14–26. doi:10.1016/j.compstruc.2008.09.003.

T. Nguyen-Thoi, G. R. Liu, H. Nguyen-Xuan, and C. Nguyen-Tran. Adaptive analysis using the node-based smoothed finite element method (NS-FEM). International Journal for Numerical Methods in Biomedical Engineering, 27, (2), (2011), pp. 198–218. doi:10.1002/cnm.1291.

T. Nguyen-Thoi, G. R. Liu, and H. Nguyen-Xuan. Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems. International Journal of Computational Methods, 6, (04), (2009), pp. 633–666. doi:10.1142/s0219876209001954.

G. R. Liu, T. Nguyen-Thoi, and K. Y. Lam. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. Journal of Sound and Vibration, 320, (4-5), (2009), pp. 1100–1130. doi:10.1016/j.jsv.2008.08.027.

T. Nguyen-Thoi, G. R. Liu, and H. Nguyen-Xuan. An n-sided polygonal edge-based smoothed finite element method (nES-FEM) for solid mechanics. International Journal for Numerical Methods in Biomedical Engineering, 27, (9), (2011), pp. 1446–1472. doi:10.1002/cnm.1375.

T. Nguyen-Thoi, G. R. Liu, K. Y. Lam, and G. Y. Zhang. A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements. International journal for numerical methods in Engineering, 78, (3), (2009), pp. 324–353. doi:10.1002/nme.2491.

T. Nguyen-Thoi, P. Phung-Van, H. Luong-Van, H. Nguyen-Van, and H. Nguyen-Xuan. A cell based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates. Computational Mechanics, 51, (1), (2013), pp. 65–81. doi:10.1007/s00466-012-0705-y.

X. Y. Cui, G. R. Liu, G. Y. Li, X. Zhao, T. Nguyen-Thoi, and G. Y. Sun. A smoothed finite element method (SFEM) for linear and geometrically nonlinear analysis of plates and shells. Comput Model Eng Sci, 28, (2), (2008), pp. 109–125.

C. H. Thai, L. V. Tran, D. T. Tran, T. Nguyen-Thoi, and H. Nguyen-Xuan. Analysis of laminated composite plates using higher-order shear deformation plate theory and nodebased smoothed discrete shear gap method. Applied Mathematical Modelling, 36, (11), (2012), pp. 5657–5677. doi:10.1016/j.apm.2012.01.003.

H. H. Phan-Dao, H. Nguyen-Xuan, C. Thai-Hoang, T. Nguyen-Thoi, and T. Rabczuk. An edge-based smoothed finite element method for analysis of laminated composite plates. International Journal of Computational Methods, 10, (01), (2013). doi:10.1142/s0219876213400057.

H. Nguyen-Xuan, G. R. Liu, C. Thai-Hoang, and T. Nguyen-Thoi. An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates. Computer Methods in Applied Mechanics and Engineering, 199, (9-12), (2010), pp. 471–489. doi:10.1016/j.cma.2009.09.001.

T. Nguyen-Thoi, T. Bui-Xuan, P. Phung-Van, S. Nguyen-Hoang, and H. Nguyen-Xuan. An edge-based smoothed three-node Mindlin plate element (ES-MIN3) for static and free vibration analyses of plates. KSCE Journal of Civil Engineering, 18, (4), (2014), pp. 1072–1082. doi:10.1007/s12205-014-0002-8.

H. Nguyen-Xuan, T. Rabczuk, N. Nguyen-Thanh, T. Nguyen-Thoi, and S. Bordas. A nodebased smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates. Computational Mechanics, 46, (5), (2010), pp. 679–701. doi:10.1007/s00466-010-0509-x.

T. Nguyen-Thoi, T. Bui-Xuan, P. Phung-Van, H. Nguyen-Xuan, and P. Ngo-Thanh. Static, free vibration and buckling analyses of stiffened plates by CS-FEM-DSG3 using triangular elements. Computers & Structures, 125, (2013), pp. 100–113. doi:10.1016/j.compstruc.2013.04.027.

T. Nguyen-Thoi, P. Phung-Van, C. Thai-Hoang, and H. Nguyen-Xuan. A cell-based smoothed discrete shear gap method (CS-DSG3) using triangular elements for static and free vibration analyses of shell structures. International Journal of Mechanical Sciences, 74, (2013), pp. 32–45. doi:10.1016/j.ijmecsci.2013.04.005.

K.-U. Bletzinger, M. Bischoff, and E. Ramm. A unified approach for shear-locking-free triangular and rectangular shell finite elements. Computers & Structures, 75, (3), (2000), pp. 321–334. doi:10.1016/s0045-7949(99)00140-6.

O. C. Zienkiewicz and R. L. Taylor. The finite element method for solid and structural mechanics. Butterworth-Heinemann, (2005).

T. Nguyen-Thoi, P. Phung-Van, H. Nguyen-Xuan, and C. Thai-Hoang. A cell-based smoothed discrete shear gap method using triangular elements for static and free vibration analyses of Reissner–Mindlin plates. International Journal for Numerical Methods in Engineering, 91, (7), (2012), pp. 705–741. doi:10.1002/nme.4289.

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