Journal of Applied and Computational Mechanics
Generalized 2-Unknown’s HSDT to Study Isotropic and Orthotropic Composite Plates
Document Type : Research Paper
Authors
1 Department of Civil Engineering, Universidad Peruana de Ciencias Aplicadas (UPC), Surco, Lima, Peru
2 Faculty of Mechanical Engineering, Universidad de Ingeniería y Tecnología (UTEC), Barranco, Lima, Peru
3 Department of Mechanical Engineering, University of New Mexico, Albuquerque 87131, USA
Abstract
The present study introduces a generalized 2-unknown’s higher order shear deformation theory (HSDT) for isotropic and orthotropic plates. The well-known Shimpi’s two-unknown's HSDT is reproduced as a special case. Reddy’s shear strain shape function (SSSF) is also adapted to the present generalized theory. The results show that both Shimpi and the adapted Reddy’ HSDT are essentially the same, i.e., both present the same static results. This is due to the fact that both theories use polynomial SSSFs. This study presents a new optimized cotangential SSSF. The generalized governing equation obtained from the principle of virtual displacement is solved via the Navier closed-form solution. Results show that transverse shear stresses can be improved substantially when non-polynomial SSSFs are utilized. Finally, this theory is attractive and has the potential to study other mechanical problems such as bending in nanoplates due to its reduced number of unknown’s variables.
Keywords
Main Subjects
[1] R.P. Shimpi. Refined plate theory and its variants. AIAA J, 40(1) (2002) 137-46.
[2] R.P. Shimpi, H.G. Patel. A two variable refined plate theory for orthotropic plate analysis. Int J Solids Struct, 43(22) (2006) 6783-99.
[3] R.P. Shimpi, H.G. Patel. Free vibrations of plate using two variable refined plate theory. J Sound Vib, 296(4-5) (2006) 979-99.
[4] I. Mechab, H. Ait Atmane, A. Tounsi, H.A. Belhadj, E.A. Adda Bedia, A two variable refined plate theory for bending of functionally graded plates, Acta Mech Sin, 26(6) (2010) 941.
[5] H.H. Abdelaziz, H.A. Atmane, I. Mechab, L. Boumia, A. Tounsi, A.B.E. Abbas, Static Analysis of Functionally Graded Sandwich Plates Using an Efficient and Simple Refined Theory, Chinese Journal of Aeronautics, 24 (2011) 434-448.
[6] M.S.A. Houari, S. Benyoucef, I. Mechab, A. Tounsi, E.A. Adda bedia, Two variable refined plate theory for thermoelastic bending analysis of functionally graded sandwich plates, J Therm Stresses, 34 (2011) 315-34.
[7] A. Hamidi, M. Zidi, M.S.A. Houari, A. Tounsi, A new four variable refined plate theory for bending response of functionally graded sandwich plates under thermomechanical loading, Composites: Part B, (2012), DOI:10.1016/j.compositesb.2012.03.021.
[8] B. Mechab, I. Mechab, S. Benaissa, Static and dynamic analysis of functionally graded plates using Four-variable refined plate theory by the new function, Composites: Part B, 45 (2013) 748-757.
[9] H.T. Thai, S.E. Kim, A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates. Composite Structures, 99 (2013) 172-180.
[10] J.L.Mantari, C Guedes Soares. A trigonometric plate theory with 5-unknowns and stretching effect for advanced composite plates. Composite Structures, 107 (2014) 396-405.
[11] J.L.Mantari, C Guedes Soares. Optimized sinusoidal higher order shear deformation theory for the analysis of functionally graded plates and shells. Composites Part B: Engineering, 56 (2014) 126-136.
[12] J.L.Mantari, C Guedes Soares. Five-unknowns generalized hybrid-type quasi-3D HSDT for advanced composite plates. Applied Mathematical Modelling, 39 (2015) 5598–5615.
[13] A.M. Zenkour, Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate, Applied Mathematical Modelling, 77 (2007) 197-214.
[14] A.M. Zenkour. A simple four-unknown refined theory for bending analysis of functionally graded plates. Applied Mathematical Modelling, 37(20-21) (2013) 9041-9051.
[15] J.L.Mantari, C Guedes Soares. Four-unknown quasi-3D shear deformation theory for advanced composite plates. Composite Structures, 109 (2014) 231-239.
[16] J.L.Mantari, C Guedes Soares. A quasi-3D tangential shear deformation theory with four unknowns for functionally graded plates. Acta Mech, 226 (2015) 625-642.
[17] J.L. Mantari. A simple polynomial quasi-3D HSDT with four unknowns to study FGPs. Reddy's HSDT assessment. Composite Structures. 137 (2016) 114–120.
[18] J.N. Reddy, C.F. Liu. A higher-order shear deformation theory of laminated elastic shells. Int J Eng Sci, 23 (1985) 319–30.
[19] R.A. Chaudhuri. On boundary-discontinuous double Fourier series solution to a system of completely coupled P.D.E.’S. Int J Eng Sci, 27(9) (1989) 1005-22.
[20] R.A. Chaudhuri. On the roles of complementary and admissible boundary constraints in Fourier solutions to boundary-value problems of completely coupled rth order P.D.E.'s. J Sound Vib, 251 (2002) 261–313.
[21] R.A. Chaudhuri, H.R.H. Kabir. Fourier Solution to higher order theory based laminated shell boundary-value problem. AIAA J, 33 (1995) 1681-88.
[22] A.S. Oktem, R.A. Chaudhuri. Levy type analysis of cross-ply plates based on higher-order theory. Compos Struct, 78 (2007) 243-53.
[23] A.S. Oktem, R.A. Chaudhuri. Fourier solution to a thick Levy type clamped plate problem. Compos Struct, 79 (2007) 481-92.
[24] A.S. Oktem, R.A. Chaudhuri. Fourier analysis of thick cross-ply Levy type clamped doubly-curved panels. Compos Struct, 80 (2007) 489-503.
[25] A.S. Oktem, R.A. Chaudhuri. Boundary discontinuous Fourier analysis of thick cross-ply clamped plates. Compos Struct, 82 (2008) 539-48.
[26] A.S. Oktem, R.A. Chaudhuri. Effect of in-plane boundary constraints on the response of thick general (unsymmetric) cross-ply plates. Compos Struct, 83 (2008) 1-12.
[27] S. Srinivas. Three dimensional analysis of some plates and laminates and a study of thickness effects. Ph.D. Thesis, Dept. of Aeronautical Engineering, Indian Institute of Science, Bangalore, India, 1970.
[28] S. Srinivas, A.K. Rao. Bending, vibration and buckling of simply-supported thick orthotropic rectangular plates and laminates. International Journal of Solids and Structures, 6(11) (1970) 1463–1481.
[28] J.N. Reddy. A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures, 20(9-10) (1984) 881–896.
[30] E. Carrera, M. Cinefra, P. Nali, MITC technique extended to variable kinematic multilayered plate elements, Composite Structures, 92 (2010) 1888–1895.
[31] A. Ferreira, E. Carrera, M. Cinefra, C. Roque, Radial basis functions collocation for the bending and free vibration analysis of laminated plates using the Reissner-Mixed variational theorem, European Journal of Mechanics - A/Solids, 39 (2012) 104–112.
[32] J. Reddy, A simple higher order theory for laminated composite plates, J Appl Mech, 51 (1984) 745–752.
[33] J. Reddy, W. Chao, A comparison of closed-form and finite-element solutions of thick laminated anisotropic rectangular plates, Nuclear Engineering and Design, 64 (1981) 153–167.
[34] M. Sobhy. Hygrothermal vibration of orthotropic double-layered grapheme sheets embedded in an elastic medium using the two-variable plate theory. Applied Mathematical Modelling, 40 (2016) 85–99.
[35] I. Senjanović, N Vladimir, M Tomić. On new first-order shear deformation plate theories. Mechanics Research Communications, 73 (2016) 31–38.
[36] M. Bouazza, A. Lairedj, N. Benseddiq, S. Khalki. A refined hyperbolic shear deformation theory for thermal buckling analysis of cross-ply laminated plates. Mechanics Research Communications, 73 (2016)117–126.
[37] Y.M. Ghugal, A.S. Sayyad. Flexure of thick orthotropic plates by exponential shear deformation theory. Latin American Journal of Solids and Structures, 10 (2013) 473-490.
[38] Y. M. Ghugal, A. S. Sayyad. Free vibration analysis of thick orthotropic plates using trigonometric shear deformation theory. Latin American Journal of Solids and Structures, 8 (2011) 229-243.
[39] D. Lanc, T.P. Vo, G. Turkalj, J. Lee. Buckling analysis of thin-walled functionally graded sandwich box beams. Thin-Walled Structures, 86 (2015) 148–156.
[40] I. Senjanovic, S. Tomasevic, N. Vladimir. An advanced theory of thin-walled girders with application to ship vibrations. Mar Struct, 22(3) (2009) 387–437.
[41] I. Senjanovic, I. Catipovic, S. Tomasevic. Coupled flexural and torsional vibrations of ship-like girders. Thin-Wall Struct, 45(12) (2007) 1002–21.
[42] A.H. Sofiyev, N. Kuruoglu. Buckling and vibration of shear deformable functionally graded orthotropic cylindrical shells under external pressures. Thin-Walled Structures, 78 (2014) 121-130.
[43] A.H. Sofiyev, The effect of elastic foundations on the nonlinear buckling behavior of axially compressed heterogeneous orthotropic truncated conical shells. Thin-Walled Structures, 80 (2014) 178-191.
[44] H.T. Thai, D.H. Choi. A refined plate theory for functionally graded plates resting on elastic foundation. Composites Science and Technology, 71 (2016) 1850-1858.
[45] H. Asadi, A.H. Akbarzadeh, Q. Wang. Nonlinear thermo-inertial instability of functionally graded shape memory alloy sandwich plates. Composite Structures, 120(1) (2015) 496-508.
[46] H. Asadi, M. Eynbeygi, Q. Wang. Nonlinear thermal stability of geometrically imperfect shape memory alloy hybrid laminated composite plates. Smart Materials and Structures, 23(7) (2014) 075012.
[47] S.F. Nikrad, H. Asadi. Thermal postbuckling analysis of temperature dependent delaminated composite plates. Thin-Walled Structures, 97 (2015) 296-307.
[48] S.F. Nikrad, H. Asadi, A.H. Akbarzadeh, ZT Chen. On thermal instability of delaminated composite plates. Composite Structures, 132(1) (2015) 1149-1159.
[49] T.R. Mahapatra, V.R. Kar, S.K. Panda, K. Mehar. Nonlinear thermoelastic deflection of temperature-dependent FGM curved shallow shell under nonlinear thermal loading. Journal of Thermal Stresses, 40(9) (2017) 1184-1199.
[50] T.R. Mahapatra, V.R. Kar, S.K. Panda. Large Amplitude Free Vibration Analysis of Laminated Composite Spherical Panel under Hygrothermal Environment. International Journal of Structural Stability and Dynamics, 16(3) (2016) 1450105.
[51] T.R. Mahapatra, S.K. Panda, V.R. Kar. Geometrically nonlinear flexural analysis of hygro-thermo-elastic laminated composite doubly curved shell panel, International Journal of Mechanics and Materials in Design, 12(2) (2015) 942-959.
[52] J.L. Mantari, E.V. Granados, M.A. Hinostroza, C.G. Soares. Modelling advanced composite plates resting on elastic foundation by using a quasi-3D hybrid type HSDT. Composite Structures, 118 (2014) 455-471.
[53] J.L. Mantari. Computational Development of a 4-Unknowns Trigonometric Quasi-3D Shear Deformation Theory to Study Advanced Sandwich Plates and Shells. International Journal of Applied Mechanics, 8(4) (2016) 1650049.
[54] J.L. Mantari. General recommendations to develop 4-unknowns quasi-3D HSDTs to study FGMs. Aerospace Science and Technology, 58 (2016) 559-570.
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