Document Type : Original Article

Authors

Shahid Beheshti University

Abstract

In this paper, a method based on Chebyshev polynomials is developed for examination of geometrically nonlinear behaviour of thin rectangular composite laminated plates under end-shortening strain. Different boundary conditions and lay-up configurations are investigated and classical laminated plate theory is used for developing the equilibrium equations. The equilibrium equations are solved directly by substituting the displacement fields with equivalent finite double Chebyshev polynomials. Using this method allows one to analyze the composite laminated plates with combination of different boundary conditions on all edges. The final nonlinear system of equations is obtained by discretizing both equilibrium equations and boundary conditions with finite Chebyshev polynomials. Nonlinear terms caused by the product of variables are linearized by using quadratic extrapolation technique to solve the system of equations. Since number of equations is always more than the number of unknown parameters, the least squares technique is used to solve the system of equations. Some results for angle-ply and cross-ply composite plates with different boundary conditions are computed and compared with those available in the literature, wherever possible.

Keywords

Main Subjects

Article Title [Persian]

Post-buckling response of thin composite plates under end-shortening strain using Chebyshev techniques

Authors [Persian]

• M. Barekati

Shahid Beheshti University

Abstract [Persian]

In this paper, a method based on Chebyshev polynomials is developed for examination of geometrically nonlinear behavior of thin rectangular composite laminated plates under end-shortening strain. Different boundary conditions and lay-up configurations are investigated and classical laminated plate theory is used for developing the equilibrium equations. The equilibrium equations are solved directly by substituting the displacement fields with equivalent finite double Chebyshev polynomials. Using this method allows one to analyze the composite laminated plates with a combination of different boundary conditions on all edges. The final nonlinear system of equations is obtained by discretizing both equilibrium equations and boundary conditions with finite Chebyshev polynomials. Nonlinear terms created as the product of variables are linearized by using quadratic extrapolation technique to solve the system of equations. Since the number of equations is always more than the number of unknown parameters, the least squares technique is used to solve the system of equations. Some results for angle-ply and cross-ply composite plates with different boundary conditions are computed and compared with those available in the literature, wherever possible.

Keywords [Persian]

• Post-buckling
• Chebyshev polynomials
• End-shortening
• Composite plates
• Least squares technique
###### ##### References
[1]     Turvey, G.J., Marshall, IH.,  Buckling and post buckling of composite plates. Springer Science & Business Media 1995.
[2]     Argyris, J., Tenek, L.,  "Recent advances in computational thermo structural analysis of composite plates and shells with strong nonlinearities." Appl. Mech. Rev., 1997,50(5): 285-306.
[3]     Ovesy, HR., Hajikazemi,  M., Assaeeb,  H.,  "A novel semi energy finite strip method for post-buckling analysis of relatively thick anti-symmetric laminated plates." Adc Eng Softw, 2012,48, 32-39.
[4]     Komur, M.A., Sen,  F., Ataş,  A.,   "Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM." AdcEngSoftw, 2010, 41(2): 161–164.
[5]     Dawe, D.J., Lam, S.S.E., Azizian, Z.G.,  "Non-linear finite strip analysis of rectangular laminates under end shortening using classical plate theory." Int J Numer Meth Eng, 1992, 35(5): 1087-1110.
[6]     Wang, S., Dawe, D.J., "Spline FSM postbuckling analysis of shear deformable rectangular laminates." Thin Wall Struct, 1999, 34(2): 163–178.
[7]     Ovesy, H.R., GhannadPour, S.A.M., Morada,  G., "Post-buckling behavior of composite laminated plates under end shortening and pressure loading  using two versions of finite strip method." Compos Struct, 2006,75(1): 106-113.
[8]     Ghannadpour, S.A.M., Ovesy, H.R., "The application of an exact finite strip to the buckling of symmetrically laminated composite rectangular plates and prismatic plate structures." Compos Struct, 2009,89 (1): 151-158.
[9]     Ghannadpour, S.A.M., Ovesy, H.R., "An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts." Int J MechSci, 2008, 50(9): 1354-1364.
[10]  Ghannadpour, S.A.M., Ovesy, H.R., "Exact post-buckling stiffness calculation of box section struts." Eng Computation, 2009, 26(7): 868-893.
[11]  Ghannadpour, S.A.M., Ovesy, H.R., Nassirnia , M., "High accuracy postbuckling analysis of box section struts." Journal of Applied Mathematics and Mechanics Zeitschrift Angewandte Mathematik und Mechanik, 2012, 92(8): 668-680.
[12]  Ghannadpour, S.A.M., Ovesy, H.R., "High accuracy postbuckling analysis of channel section struts." Nt J Nonlinear Mech, 2012, 47(9): 968-974.
[13]  Ovesy, H.R, Ghannadpour, S.A.M., Zia-Dehkordi,  E., "Buckling Analysis of Moderately Thick Composite Plates and Plate Structures Using an Exact Finite Strip." Compos Struct, 2013, 95, 697-704.
[14]  Ghannadpour, S.A.M., Ovesy, H.R., Zia-Dehkordi,  E., "An exact finite strip for the calculation of initial post-buckling stiffness of shear-deformable composite laminated plates." Compos Struct, 2014,108, 504-513.
[15]  Shen, H., Postbuckling, "analysis of orthotropic rectangular plates on nonlinear elastic foundations." EngStruct, 1995,17(6): 407-412.
[16]   Wang, Z.X., Shen, H.S.,  "Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations." Compos Struct, 2011, 93(10): 2521-2532.
[17]   Shen, H.S., Zhang, C.L., "Non-linear analysis of functionally graded fiber reinforced composite laminated plates," Part II: Numerical results.Int J Nonlinear Mech, 2011,47(9): 1055-1064.
[18]   Shen, H.S., Zhu, Z.H., "Postbuckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations."  European Journal of Mechanics-A/Solids 2012,35.
[19]   Ghannadpour, S.A.M., Shakeri, M., "A New Method to Investigate the Progressive Damage of Imperfect Composite Plates Under In-Plane Compressive Load." AUT J. Mech. Eng. 2017, 1(2): 159-168.
[20]   Ghannadpour, S.A.M., Shakeri, M., "Energy based collocation method to predict progressive damage behavior of imperfect composite plates under compression." Lat. Am. J. Solids Struct., 2018, 15(4): 1-25.
[21]   Ghannadpour, S.A.M., Barvaj, A.K., Tornabene, F., "A semi-analytical investigation on geometric nonlinear and progressive damage behavior of relatively thick laminated plates under lateral pressure and end-shortening." Comp. Struct. 2018, 194: 598-610.
[22]   Ghannadpour, S.A.M., Mehrparvar, M., "Energy effect removal technique to model circular/elliptical holes in relatively thick composite plates under in-plane compressive load." Comp. Struct. 2018, doi.org/10.1016/j. compstruct. 2018.05.026.
[23]   Mehrparvar,  M., Ghannadpour, S.A.M., "Plate assembly technique for nonlinear analysis of relatively thick functionally graded plates containing rectangular holes subjected to in-plane compressive load." Comp. Struct. 2018, doi.org/10.1016/j.compstruct. 2018. 04. 053.
[24]   Ghannadpour, S.A.M., Kiani, P. and Reddy, J.N., "Pseudo spectral method in nonlinear analysis of relatively thick imperfect laminated plates under end-shortening strain." Comp. Struct. 2017, 182: 694-710.
[25]   Ghannadpour, S.A.M., Kiani, P., "Nonlinear spectral collocation analysis of imperfect functionally graded plates under end-shortening." Struct.Eng. Mech. 2018, 66(5): 557-568.
[26]   Alwar, R.S., Nath, Y., "Application of Chebyshev polynomials to the nonlinear analysis of circular plates." Int J MechSci, 1976,18(11): 589-595.
[27]   Nath, Y., Kumar, S., "Chebyshev series solution to non-linear boundary value problems in rectangular domain." Comput Method Appl M , 1995,125(1): 41-52.
[28]   Shukla, K.K., Nath,  Y., "Nonlinear analysis of moderately thick laminated rectangular plates." J Eng. Mech, 2000,126(8): 831-838.
[29]   Ghannadpour, S.A.M., Barekati. B., "Initial imperfection effects on postbucklingresponse of laminated plates under end-shortening strain using Chebyshev techniques." Thin-WalledStruct. 2016,106: 484-494.
[30]   Reddy, J.N., Mechanics of laminated composite plates and shells: theory and analysis.  CRC press 2004.
[31]   Fox, L., Parker, I.B., Chebyshev polynomials in numerical analysis. Oxford Univ. Press 1968.
[32]   Shukla, K.K., Nath, Y., "Analytical solution for buckling and post-buckling of angle-ply laminated plates under thermomechanical loading," Int J NonlinarMech, 2001, 36(7): 1097-1108.
[33]   Mason, J.C. and Handscomb, D.C., Chebyshev polynomials. CRC Press 2002.
[34] Ovesy, H.R., Ghannadpour, S.A.M., "Non-linear analysis of composite laminated plates under end shortening using finite strip method, " Proceedings of the fourth Australasian congress on applied mechanics, Melbourne, Australia 2005