A COMMON EVALUATION OF THE MULTILAYERED COMPOSITE PLATE AND SHELLS ANALYSIS

Year 2014, Volume: 32 Issue: 2, 176 - 188, 01.06.2014

K. Ahmet Haşim

Abstract

This review article is devoted to the developments and ideas for the analysis of multilayered composite plate and shells. In the first part; the paper presents a through review of the literature involving the use in the modeling of multilayered plates and shells. A second part reviews relevant key points that should be considered for an accurate stress and strain fields, herein referred to as C^0 requirements (zig-zag form of the displacement field in the thickness direction and continuity of transverse normal and shear stresses at each layer interface). In the third part, the paper explains one of the mixed variation principles Hellinger-Reissner (HR) in view of C^0 requirements. It is then shown that Reissner’s mixed variational theorem (RMVT) can be simply constructed by adding the constraint equations (Lagrange multiplier) for the transverse stresses to Hellinger Reissner Principle (HR). The mixed form of Hooke’s law has also been derived in this section. The final part of the paper is devoted to giving an overview with selected results of numerical performance that can be acquired by RMVT applications; comparison to elasticity solutions and other significant analyses, based on classical and refined approaches are given. It is concluded that RMVT leads to a better description than classical analysis formulated with only displacement variables.

Keywords

FSDT, HSDT, layerwise, laminated composite, RMVT, C0 requirements.

References

• [1] Kirchhoff G., “Uber das Gleichgewicht und die Bewegung einer elastishen Scheibe”, J.Angew Math., 40, 51-88, 1850.
• [2] Love AEH., “The Mathematical Theory of Elasticity”,4th Edition, Cambridge Univ Press, Cambridge, 1927.
• [3] Cauchy AL., “Sur l’equilibre et le mouvement d’une plaque solide”, Exercises de Mathematique., 3, 328-355, 1828.
• [4] Poisson SD., “Memoire sur l’equilibre et le mouvement des corps elastique”, Mem. Acad. Sci., 8, 357, 1829.
• [5] Reissner E., “The effect of transverse shear deformation on the bending of elastic plates”, ASME J. Appl. Mech., 12, 69-76, 1945.
• [6] Mindlin RD., “Influence of rotary inertia and shear in flexural motions of isotropic elastic plates”, ASME J. Appl. Mech., 18, 1031-1036, 1951.
• [7] Koiter WT., “A consistent first approximations in the general theory of thin elastic shells”, Proc. of Symp. on the Theory of Thin Elastic Shells, North-Holland, Amsterdam, Aug, 1959, 12-23.
• [8] Jones RM., “Mechanics of Composite Materials”, Mc Graw Hill, New York, 1975.
• [9] Whitney J., “The effects of transverse shear deformation on the bending of laminated plates”, J. Compos. Mater., 3, 534-547, 1969.
• [10] Lo KH, Christensen RM, Wu EM., “A higher order theory of plate deformation, Part 2: Laminated plates”, ASME J. Appl. Mech., 44, 669-676, 1977.
• [11] Reddy JN., “Mechanics of Laminated Composite Plates. Theory and Analysis”, CRC Press, Boca Raton FL, 1997.
• [12] Pagano N.J., “Exact Solutions for Composite Laminates in Cylindrical Bending”, Journal of Composite Materials, 3, 398-411, 1969.
• [13] Pagano N.J., “Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates”, Journal of Composite Materials, 4, 20-34, 1970.
• [14] Pagano, N.J., “Elastic Behavior of Multilayered Bidirectional Composites, AIAA Journal, 10, 931-933, 1972.
• [15] Srinivas S., Joga Rao C.V., Rao A.K., “An Exact Analysis for Vibration of Simply Supported Homogenous and Laminated Thick Rectangular Plates”, Journal of Sound and Vibration, 12, 187-199, 1970.
• [16] Srinivas S., Rao A.K., “Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates”, International Journal of Solids and Structures, 6, 1463-1481, 1970.
• [17] Noor A.K., “Mixed Finite Difference Scheme for Analysis of Simply Supported Thick Plates”, Computers and Structures, 3, 967-982, 1973.
• [18] Noor AK., “Free Vibrations of Multilayered Composite Plates”, AIAA Journal, 11, 1038-1039, 1973.
• [19] Srinivas S., “A refined analysis of composite laminates”, Journal of Sound and Vibration, 30, 495-507, 1973.
• [20] Cho KN., Bert CW., Striz AG., “Free vibrations of laminated rectangular plates analyzed by higher order individual layer theory”, Journal of Sound and Vibration, 98, 157-170, 1985.
• [21] Lo KH., Christensen RM., Wu EM., “A higher order theory of plate deformation, Part 2: laminated plates”, Journal of Applied Mechanics, 44, 669-676, 1977.
• [22] Murakami H., “A laminated beam theory with interlayer slip”, Journal of Applied Mechanics, 51, 551-559, 1984.
• [23] Murakami H., “Laminated composite plate theory with improved in-plane responses”, Journal of Applied Mechanics, 53, 661-666, 1986.
• [24] Carrera E., “Mixed layerwise models for multilayered plates analysis”, Composite Structures, 43, 57-70, 1998.
• [25] Carrera E., “Evaluation of layerwise mixed theories for laminated plates analysis”, American Institute of Aeronautics and Astronautics Journal, 26, 830-839, 1998.
• [26] Carrera E., Kröplin B., “Zig-zag and interlaminar equilibria effects in large deflection and postbuckling analysis of multilayered plates”, Mechanics of Composite Materials and Structures, 4, 69-94, 1997.
• [27] Carrera E., Krause H., “An investigation on nonlinear Dynamics of multilayered plates accounting for C^0 requirements”, Computers and Structures, 69, 463-486, 1998.
• [28] Moleiro F., Mota Soares C.M., Mota Soares C.A., et.al., “A layerwise Mixed Least-Squares Finite Element Model for Static Analysis of Multilayered Composite Plates”, Computers and Structures, 89, 1730-1742, 2011.
• [29] Lekhnitskii S.G., “Strength calculation of composite beams”, Vestn. inzhen. i tekhnikov, 9, 540-544, 1935.
• [30] Ren JG., “A new theory for laminated plates”, Composite Sci and Technol, 26, 225-239, 1986.
• [31] Di Scuvia M., “A general quadrilateral multilayered plate element with continuous interlaminar stresses”, Computers & Structures, 47, 91-105, 1993.
• [32] Idlbi A., Karama M., Touratier M., “Comparison of various laminated plate theories”, Composite Struct, 37, 173-184, 1997.
• [33] Reissner E., “Reflections on the theory of elastic plates”, Appl. Mech. Rev., 38, 1453-1464, 1985.
• [34] Carrera E., “A priori vs. a posteriori evaluation of transverse stresses in multilayered orthohropic plates”, Composite Structures, 48, 245-260, 2000.
• [35] Reissner E., “On a certain mixed variational theorem and a proposed application”, International Journal for Numerical Methods in Engineering, 20, 7, 1366-1368, 1984.
• [36] Whitney J.M., Pagano N.J., “Shear Deformation in Heterogeneous Anisotropic Plates”, J. of Appl. Mech., 37, 1031-1036, 1970.
• [37] Carrera, E., “ C^0 Reissner-Mindlin multilayered plate elements including zig-zag and interlaminar stresses continuity”, Int. J. Numer. Methods Eng., 39, 1797-1820, 1996.
• [38] Reissner E., “On a certain mixed variational theorem and on laminated elastic shell theory”, Proc of Euromech-Colloquium, 219, 17-27, 1986.
• [39] Toledano A., Murakami H., “A higher order laminated plate theory with improved in-plane responses”, Int. J. Solids Struct., 23, 111-131, 1987.
• [40] Toledano A., Murakami H., “A composite plate theory for arbitrary laminate configurations”, ASME J. Appl. Mech., 54, 181-189, 1987.
• [41] Bhaskar K., Varadan T.K., “A higher order theory for bending analysis of laminated shells of revolution”, Comput. Struct., 40, 815-819, 1991.
• [42] Jing H., Tzeng K.G., “Refined shear deformation theory of laminated shells”, AIAA J., 31, 765-773, 1993.
• [43] Carrera E., “A Reissner’s mixed variational theorem applied to vibration analysis of multilayered shells”, ASME J. Appl. Mech., 66, 69-78, 1999.
• [44] Carrera E., “A study of transverse normal stress effects on vibration of multilayered plates and shells”, J. Sound Vib., 225, 803-829, 1999.