Evaluation of the Best New Cross-ply Laminated Plate Theories through the Axiomatic/Asymptotic Approach

Document Type : Research Paper


1 Faculty of Mechanical Engineering, Universidad de Ingeniería y Tecnología (UTEC), Medrano Silva 165, Barranco, Lima, Perú

2 Department of Mechanical Engineering, University of New Mexico, Albuquerque 87131, USA


This paper presents Best Theory Diagrams (BTDs) constructed from various non-polynomial theories for the static analysis of thick and thin symmetric and asymmetric cross-ply laminated plates. The BTD is a curve that provides the minimum number of unknown variables necessary for a fixed error or vice versa. The plate theories that belong to the BTD have been obtained by means of the Axiomatic/Asymptotic Method (AAM). The different plate theories reported are implemented by using the Carrera Unified Formulation (CUF). Navier-type solutions have been obtained for the case of simply- supported plates loaded by a bisinuisoidal transverse pressure with different length-to-thickness ratios. The BTDs built from non-polynomials functions are compared with BTDs using Maclaurin expansion. The results suggest that the plate models obtained from the BTD using non-polynomial terms can improve the accuracy obtained from Maclaurin expansions for a given number of unknown variables of the displacement field.


Main Subjects

[1] Kirchoff, G., Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe, Journal fur die Reine und Angewandte Mathematik, 40, 1850, 51-88.
[2] Love, A. E. H., A treatise on the mathematical theory of elasticity, Cambridge university press, 2013
[3] Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics, 12, 1945,  69-77.
[4] Mindlin, R. D., Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics, 18, 1031-1036.
[5] Vlasov, B. F., On the equations of bending of plates, Dokla Ak Nauk Azerbeijanskoi-SSR, 3, 1957, 955-979.
[6] Jemielita, G., Technical theory of plates with moderate thickness, Rozprawy Ink, 23(3), 1975, 483-499.
[7] Schmidt, R., A refined nonlinear theory of plates with transverse shear deformation, Journal of Industrial Mathematics Society, 27(1), 1977, 23-38.
[8] Reddy, J. N., A general non-linear third-order theory of plates with moderate thickness, International Journal of Non-Linear Mechanics, 25(6), 1990, 677-686.
[9] Levy M., Mémoire sur la théorie des plaques élastiques planes, Journal de Mathématiques Pures et Appliquées, 30, 1877,  219-306
[10] Stein M., Nonlinear theory for plates and shells including the effects of transverse shearing, AIAA Journal, 24(9), 1986, 1537-1544.
[11] Shimpi, R. P., Ghugal, Y. M., A new layerwise trigonometric shear deformation theory for two-layered cross-ply beams, Composites Science and Technology, 61(9), 2001, 1271-1283.
[12] Touratier M., An efficient standard plate theory, International Journal of Engineering Science, 29(8), 1991, 901-916.
[13] Soldatos K., A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica, 94(3), 1992, 195-220.
[14] Ferreira, A. J. M., Roque, C. M. C., Jorge, R. M. N., Analysis of composite plates by trigonometric shear deformation theory and multiquadrics, Computers & Structures, 83(27), 2005, 2225-2237.
[15] Karama, M., Afaq, K. S., Mistou, S., A new theory for laminated composite plates, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 223(2), 2009, 53-62.
[16] Akavci S., Two new hyperbolic shear displacement models for orthotropic laminated composite plates, Mechanics of Composite Materials, 46(2), 2010, 215-226.
[17] Mantari, J. L., Oktem, A. S., Soares, C. G., A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates International Journal of Solids and Structures, 49(1), 2012, 43-53.
[18] Mantari, J. L., Soares, C. G., Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory, Composite Structures, 98(8), 2012, 2640-2656.
[19] Thai, C. H., Ferreira, A. J. M., Bordas, S. P. A., Rabczuk, T., Nguyen-Xuan, H., Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory, European Journal of Mechanics-A/Solids, 43, 2014, 89-108.
[20] Zenkour, A. M., Thermal bending of layered composite plates resting on elastic foundations using four-unknown shear and normal deformations theory, Composite Structures, 122, 2015, 260-270.
[21] Grover, N., Maiti, D. K., Singh, B. N., A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates, Composite Structures, 95, 2013, 667-675.
[22] Sarangan, S., Singh, B. N., Higher-order closed-form solution for the analysis of laminated composite and sandwich plates based on new shear deformation theories, Composite Structures, 138, 2016, 391-403.
[23] Mahi, A., Tounsi, A., A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates, Applied Mathematical Modelling, 39(9), 2015, 2489-2508.
[24] Carrera, E., A class of two-dimensional theories for anisotropic multilayered plates analysis, Atti della accademia delle scienze di Torino. Classe di scienze fisiche matematiche e naturali, 19, 1995, 1-39.
[25] Carrera, E., Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking, Archives of Computational Methods in Engineering, 10(3), 2003, 215-296.
[26] Carrera, E., Giunta, G., Petrolo, M., Beam structures: classical and advanced theories, John Wiley & Sons, 2011.
[27] Carrera, E., Cinefra, M., Petrolo, M., Zappino, E., Finite element analysis of structures through unified formulation, John Wiley & Sons, 2014.
[28] Carrera, E., Petrolo, M., Guidelines and recommendations to construct theories for metallic and composite plates, AIAA Journal, 48(12), 2010, 2852-2866.
[29] Carrera, E., Petrolo, M., On the effectiveness of higher-order terms in refined beam theories, Journal of Applied Mechanics, 78(2), 2011, 1-17.
[30] Carrera, E., Miglioretti, F., Petrolo, M., Guidelines and recommendations on the use of higher order finite elements for bending analysis of plates, International Journal for Computational Methods in Engineering Science and Mechanics, 12(6), 2011, 303-324.
[31] Carrera, E., Miglioretti, F., Petrolo, M., Accuracy of refined finite elements for laminated plate analysis, Composite Structures, 93(5), 2011, 1311-1327.
[32] Carrera, E., Miglioretti, F., Petrolo, M., Computations and evaluations of higher-order theories for free vibration analysis of beams, Journal of Sound and Vibration, 331(19), 2012, 4269-4284.
[33] Carrera, E., Miglioretti, F., Selection of appropriate multilayered plate theories by using a genetic like algorithm, Composite Structures, 94(3), 2012, 1175-1186.
[34] Mashat, D. S., Carrera, E., Zenkour, A. M., Al Khateeb, S. A., Use of axiomatic/asymptotic approach to evaluate various refined theories for sandwich shells, Composite Structures, 109, 2014, 139-149.
[35] Mashat, D. S., Carrera, E., Zenkour, A. M., Al Khateeb, S. A., Axiomatic/asymptotic evaluation of multilayered plate theories by using single and multi-points error criteria, Composite Structures, 106, 2013, 393-406.
[36] Mashat, D. S., Carrera, E., Zenkour, A. M., Al Khateeb, S. A., Lamberti, A., Evaluation of refined theories for multilayered shells via Axiomatic/Asymptotic method, Journal of Mechanical Science and Technology, 28(11), 2014, 4663-4672.
[37] Cinefra, M., Lamberti, A., Zenkour, A. M., Carrera, E., Axiomatic/asymptotic technique applied to refined theories for piezoelectric plates, Mechanics of Advanced Materials and Structures, 22(1-2), 2015, 107-124.
[38] Carrera, E., Cinefra, M., Lamberti, A., Zenkour, A. M., Axiomatic/asymptotic evaluation of refined plate models for thermomechanical analysis, Journal of Thermal Stresses, 38(2), 2015, 165-187.
[39] Carrera, E., Cinefra, M., Lamberti, A., Petrolo, M., Results on best theories for metallic and laminated shells including Layer-Wise models, Composite Structures, 126, 2015, 285-298.
[40] Petrolo, M., Cinefra, M., Lamberti, A., Carrera, E., Evaluation of mixed theories for laminated plates through the axiomatic/asymptotic method, Composites part B: Engineering, 76, 2015, 260-272.
[41] Petrolo, M., & Lamberti, A., Axiomatic/asymptotic analysis of refined layer-wise theories for composite and sandwich plates, Mechanics of Advanced Materials and Structures, 23(1), 2016, 28-42.
[42] Petrolo, M., Lamberti, A., Miglioretti, F., Best theory diagram for metallic and laminated composite plates, Mechanics of Advanced Materials and Structures, 23(9), 2016, 1114-1130.
[43] Cinefra, M., Carrera, E., Lamberti, A., Petrolo, M., Best theory diagrams for multilayered plates considering multifield analysis, Journal of Intelligent Material Systems and Structures, 28(16), 2017, 2184-2205.
[44] Neves, A. M. A., Ferreira, A. J. M., Carrera, E., Cinefra, M., Jorge, R. M. N., Soares, C. M. M., Buckling analysis of sandwich plates with functionally graded skins using a new quasi‐3D hyperbolic sine shear deformation theory and collocation with radial basis functions, ZAMM-Zeitschrift fur Angewandte Mathematikund Mechanik, 92(9), 2012, 749-766.
[45] Carrera, E., Filippi, M., Zappino, E., Laminated beam analysis by polynomial, trigonometric, exponential and zig-zag theories, European Journal of Mechanics-A/Solids, 41, 2013, 58-69.
[46] Carrera, E., Filippi, M., & Zappino, E., Free vibration analysis of laminated beam by polynomial, trigonometric, exponential and zig-zag theories, Journal of Composite Materials, 48(19), 2014, 2299-2316.
[47] Filippi, M., Carrera, E., Zenkour, A. M., Static analyses of FGM beams by various theories and finite elements, Composites Part B: Engineering, 72, 2015, 1-9.
[48] Filippi, M., Petrolo, M., Valvano, S., Carrera, E., Analysis of laminated composites and sandwich structures by trigonometric, exponential and miscellaneous polynomials and a MITC9 plate element, Composite Structures, 150, 2016, 103-114.
[49] Mantari, J. L., Ramos, I. A., Carrera, E., Petrolo, M., Static analysis of functionally graded plates using new non-polynomial displacement fields via Carrera Unified Formulation, Composites Part B: Engineering, 89, 2016, 127-142.
[50] Ramos, I. A., Mantari, J. L., Zenkour, A. M., Laminated composite plates subject to thermal load using trigonometrical theory based on Carrera Unified Formulation, Composite Structures, 143, 2016, 324-335.
[51] Pagano, N. J., Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of Composite Materials, 4(1), 1970, 20-34.
[52] Candiotti, S., Mantari, J. L., Yarasca, J., Petrolo, M., Carrera, E., An axiomatic/asymptotic evaluation of best theories for isotropic metallic and functionally graded plates employing non-polynomic functions, Aerospace Science and Technology, 68, 2017, 179-192.