Linear response of thin axysimmetric cross-ply structure under a static load: Numerical and analythical comparisons
Salvatore Saputo, Erasmo Carrera, Volodymyr V. Zozulya
download PDFAbstract. Thin-walled mechanical components, such as beams, plates and shells, are widely used as structural components in several engineering fields, in particular mechanical, aeronautical and aerospace sectors. The purpose of this work is to analyse the cross-ply bending behaviour of cylindrical and spherical shell structures using the finite element method. Hence, numerical models, realized using commercial software, were realized using the shell and solid approaches and were compared with numerical and analytical methods to appreciate their advantages. In this research, a Navier solution in close form for high-order theories, developed using the Carrera Unified Formulation (CUF) approach, has been reported, where the high-order elastic shell model has been developed using the variational principle of virtual work for three-dimensional linear theory equations and the analytical results were obtained using the Mathematica software. The results furnished by the numerical method such as the elasticity solutions given in the literature using Navier’s method are used as a benchmark for comparing the finite element method results in terms of maximum displacement and stress distribution along the principal structure direction. However, the numerical shell model cannot provide sufficient data to describe the tensional and deformational state at all points and especially along the laminate thickness. Wishing to obtain a complete description of the plate’s mechanical behaviour, it is necessary to use a three-dimensional approach with the associated increase in calculation time. In contrast, the numerical solution based on the CUF approach shows a very efficient description of the composite structure behaviour and its use should be preferred to the classical lamination approach if an accurate description of the structure is necessary.
Keywords
Carrera Unified Formulation, Higher-Order Theories, Analytical Solution, Bending, Laminated Composite Plate
Published online 8/10/2023, 18 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA
Citation: Salvatore Saputo, Erasmo Carrera, Volodymyr V. Zozulya, Linear response of thin axysimmetric cross-ply structure under a static load: Numerical and analythical comparisons, Materials Research Proceedings, Vol. 31, pp 596-613, 2023
DOI: https://doi.org/10.21741/9781644902592-62
The article was published as article 62 of the book Advanced Topics in Mechanics of Materials, Structures and Construction
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
References
[1] Ambartsumyan VA. In: Ashton JE, editor. Theory of anisotropic plates. Tech Pub Co; 1969. Translated from Russian by T Cheron.
[2] Reddy JN. Mechanics of laminated composite plates, theory and analysis. CRC Press; 1997.
[3] Noor AK, Rarig PL. Three-dimensional solutions of laminated cylinders. Computer Methods in Applied Mechanics and Engineering 1974; 3:319-334. https://doi.org/10.1016/0045-7825(74)90017-6
[4] Malik M. Differential quadrature method in computational mechanics: new development and applications. Ph.D Dissertation, University of Oklahoma, Oklahoma, 1994.
[5] Liew KM, Han B, Xiao M. Differential quadrature method for thick symmetric cross-ply laminates with first-order shear flexibility. International Journal of Solids and Structures 1996; 33:2647-2658. https://doi.org/10.1016/0020-7683(95)00174-3
[6] Ferreira AJM, Roque CC, Carrera E, Cinefra M. Analysis of thick isotropic and cross-ply laminated plates by radial basis functions and unified formulation. Journal of Sound and Vibration 2011; 330:771-787. https://doi.org/10.1016/j.jsv.2010.08.037
[7] Ferreira AJM, Roque CC, Carrera E, Cinefra M, Polit O. Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations. Composites Part B 2011; 42:1276-1284. https://doi.org/10.1016/j.compositesb.2011.01.031
[8] Reddy JN, Robbins DH. Theories and computational models for composite laminates. Applied Mechanics Review 1994; 47:147-165. https://doi.org/10.1115/1.3111076
[9] Varadan TK, Bhaskar K. Review of different theories for the analysis of composites. Journal of Aerospace Society of India 1997; 49:202-208.
[10] Carrera E. Developments, ideas and evaluations based upon Reissner’s mixed variational theorem in the modeling of multilayered plates and shells. Applied Mechanics Review 2001; 54:301-329. https://doi.org/10.1115/1.1385512
[11] Argyris JH. Matrix displacement analysis of plates and shells, Prolegomena to a general theory, part I. Ingeniur-Archiv 1966; 35:102-142. https://doi.org/10.1007/BF00536183
[12] Sabir AB, Lock AC. The application of finite elements to the large deflection geometrically non-linear behaviour of cylindrical shells. In Variational Methods in Engineering 2, Southampton University Press: 1972; 7/66-7/75.
[13] Wempner GA, Oden JT, Kross DA. Finite element analysis of thin shells. Journal of Engineering Mechanics ASCE 94 1968; 94:1273-1294. https://doi.org/10.1061/JMCEA3.0001039
[14] Abel JF, Popov EP. Static and dynamic finite element analysis of sandwich structures. Proceedings of the Second Conference of Matrix Methods in Structural Mechanics, Vol. AFFSL-TR-68-150, Wright-Patterson Air Force Base, Ohio, USA, 1968; 213-245.
[15] Monforton GR, Schmidt LA. Finite element analyses of sandwich plates and cylindrical shells with laminated faces. Proceedings of the Second Conference of Matrix Methods in Structural Mechanics, Vol. AFFSL-TR-68-150, Wright-Patterson Air Force Base, Ohio, USA, 1968; 573-308.
[16] Noor AK. Finite element analysis of anisotropic plates. American Institute of Aeronautics and Astronautics Journal 1972; 11:289-307. https://doi.org/10.1002/nme.1620110206
[17] Hughes TJR, Tezduyar T. Finite elements based upon Mindlin plate theory with particular reference to the four-node isoparametric element. Journal of Applied Mechanics 1981; 48:587-596. https://doi.org/10.1115/1.3157679
[18] Parisch H. A critical survey of the 9-node degenerated shell element with special emphasis on thin shell application and reduced integration. Computer Methods in Applied Mechanics and Engineering 1979; 20:323-350. https://doi.org/10.1016/0045-7825(79)90007-0
[19] Zienkiewicz OC, Taylor RL, Too JM. Reduced intergration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering 1973; 3:275-290. https://doi.org/10.1002/nme.1620030211
[20] Naghdi PM. The theory of shells and plates. In Handbuch der Physic, Vol. 4. Springer: Berlin, 1972; 425-640. https://doi.org/10.1007/978-3-642-69567-4_5
[21] Bathe KJ, Dvorkin EN. A four node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation. International Journal for Numerical Methods in Engineering 1985; 21:367-383. https://doi.org/10.1002/nme.1620210213
[22] Arnold D.N., Brezzi F. Locking-free finite element methods for shells. Mathematics of Computation 1997; 66(217):1-14 https://doi.org/10.1090/S0025-5718-97-00785-0
[23] Bischoff M, Ramm E. On the physical significance of higher-order kinematic and static variables in a threedimensional shell formulation. International Journal of Solids and Structures 2000; 37:6933-6960. https://doi.org/10.1016/S0020-7683(99)00321-2
[24] Simo JC, Rafai S. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering 1990; 29:1595-1638. https://doi.org/10.1002/nme.1620290802
[25] Kant T, Owen DRJ, Zienkiewicz OC. Refined higher order C0 plate bending element. Computer & Structures 1982; 15:177-183 https://doi.org/10.1016/0045-7949(82)90065-7
[26] Polit O, Touratier M. A new laminated triangular finite element assuring interface continuity for displacements and stresses. Composite Structures 1997; 38(1-4):37-44. https://doi.org/10.1016/S0263-8223(97)00039-1
[27] Tessler A. A higher-order plate theory with ideal finite element suitability. Computer Methods in Applied Mechanics and Engineerings 1991; 85:183-205. https://doi.org/10.1016/0045-7825(91)90132-P
[28] Reddy JN. Mechanics of Laminated Composite Plates and Shells, Theory and Analysis. CRC Press: New York, USA, 1997
[29] Rao KM, Meyer-Piening HR. Analysis of thick laminated anisotropic composites plates by the finite element method. Composite Structures 1990; 15:185-213. https://doi.org/10.1016/0263-8223(90)90031-9
[30] Noor AK, Burton WS. Assessment of computational models for multi-layered composite shells. Applied Mechanics Review 1990; 43:67-97. https://doi.org/10.1115/1.3119162
[31] Reddy JN. An evaluation of equivalent single layer and layer-wise theories of composite laminates. Composite Structures 1993; 25:21-35. https://doi.org/10.1016/0263-8223(93)90147-I
[32] Rammerstorfer FG, Dorninger K, Starlinger A. Composite and sandwich shells. In Nonlinear Analysis of Shells by Finite Elements, Rammerstorfer FG (ed.). Springer-Verlag: Vienna, 1992; 131-194. https://doi.org/10.1007/978-3-7091-2604-2_6
[33] Cinefra M., Carrera E. Shell finite elements with different through-the-thickness kinematics for the linear analysis of cylindrical multilayered structures. International journal for numerical methods in engineering. 93:(2013), pp. 160-182. https://doi.org/10.1002/nme.4377
[34] Carrera E., Ciuffreda A., A unified formulation to assess theories of multilayered plates for various bending problems. Composite Structures 69 (2005) 271-293 https://doi.org/10.1016/j.compstruct.2004.07.003
[35] Carrera E. Theories and finite elements for multilayered anisotropic composite plates and shells. Archives of Computational Methods in Engineering 2002; 9:87-140. https://doi.org/10.1007/BF02736649
[36] 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 2003; 10(3):215-296. https://doi.org/10.1007/BF02736224
[37] Carrera E. A class of two-dimensional theories for multi-layered plates analysis. Atti Accad Sci Tor, Mem Sci Fis 1995;19-20:49-87.
[38] Murakami H. Laminated composite plate theory with improved in-plane response. J Appl Mech 1986; 53:661-6. https://doi.org/10.1115/1.3171828
[39] Meyer-Piening HR. Experiences with ‘Exact’ linear sandwich beam and plate analyses regarding bending, instability and frequency investigations. In: Proceedings of the Fifth International Conference On Sandwich Constructions, Zurich, Switzerland, September 5-7, vol. I.; 2000. p. 37-48.
[40] Pagano NJ. Exact solutions for composite laminates in cylindrical bending. J Compos Mater 1969;3:398-411. https://doi.org/10.1177/002199836900300304
