Non-conventional 1D and 2D finite elements based on CUF for the analysis of non-orthogonal geometries. (English) Zbl 1481.74693

Summary: When dealing with innovative materials – such as composites and metamaterials with complex microstructure – or structural components with non-orthogonal beam/plate geometry, the Finite Element Method can become very costly in calculations and time because of the use of very fine 3D meshes. By exploiting the Node-Dependent Kinematic approach of the Carrera Unified Formulation and using Lagrange expanding functions, this work presents the implementation of non-conventional 1D and 2D elements mainly based on the 3D integration of the approximating functions and computation of 3D Jacobian matrix inside the element for the derivation of stiffness and mass matrices; substantially, the resulting elements are 3D elements in which the order of expansion can be different in the three spatial directions. The free vibration analysis of some typical components is performed and the results are provided in terms of natural frequencies. The present elements allow us to accurately study beam-like and plate-like structures with non-orthogonal geometries by employing much less degrees of freedom with respect to the use of classical 3D finite elements.


74S05 Finite element methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K20 Plates
Full Text: DOI


[1] Carrera, E., Theories and finite elements for multilayered, anisotropic, composite plates and shells, Arch. Comput. Methods Eng., 9, 2, 87-140 (2002) · Zbl 1062.74048
[2] Carrera, E., Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking, Arch. Comput. Methods Eng., 10, 3, 216-296 (2003) · Zbl 1140.74549
[3] Carrera, E.; Cinefra, M.; Petrolo, M.; Zappino, E., Finite Element Analysis of Structures through Unified Formulation (2014), John Wiley & Sons · Zbl 1306.74001
[4] Carrera, E.; Giunta, G., Refined beam theories based on Carrera’s Unified Formulation, Int. J. Appl. Mech., 2, 1, 117-143 (2010)
[5] Carrera, E.; Giunta, G.; Petrolo, M., Beam Structures: Classical and Advanced Theories, 45-63 (2011), John Wiley and Sons
[6] Carrera, E.; Maiarú, M.; Petrolo, M., Component-wise analysis of laminated anisotropic composites, Int. J. Solids Struct., 49, 1839-1851 (2012)
[7] Carrera, E.; Pagani, A., Analysis of reinforced and thin-walled structures by multi-line refined 1D/beam models, Int. J. Mech. Sci., 75, 278-287 (2013)
[8] Carrera, E.; Pagani, A., Free vibration analysis of civil engineering structures by component-wise models, J. Sound Vib., 333, 19, 4597-4620 (2014)
[9] Carrera, E.; Pagani, A., Evaluation of the accuracy of classical beam FE models via locking-free hierarchically refined elements, Int. J. Mech. Sci., 100, 169-179 (2015)
[10] Carrera, E.; Pagani, A.; Petrolo, M., Classical, refined and component-wise theories for static analysis of reinforced-shell wing structures, AIAA J., 51, 5, 1255-1268 (2013)
[11] Carrera, E.; Pagani, A.; Petrolo, M., Component-wise method applied to vibration of wing structures, J. Appl. Mech., 80, 4 (2013), art. no. 041012 1-15
[12] Carrera, E.; Pagani, A.; Petrolo, M., Use of Lagrange multipliers to combine 1D variable kinematic finite elements, Comput. Struct., 129, 194-206 (2013)
[13] Carrera, E.; Pagani, A.; Petrolo, M., Refined 1D finite elements for the analysis of secondary, primary, and complete civil engineering structures, J. Struct. Eng., 141, 4 (2014), art. no. 04014123
[14] Carrera, E.; Pagani, A.; Petrolo, M.; Zappino, E., Recent developments on refined theories for beams with applications, Mech. Eng. Rev., 2, 2, 14-00298 (2015)
[15] Carrera, E.; Pagani, A.; Valavano, S., Multilayered plate elements accounting for refined theories and node-dependent kinematics, Composites B, 114, 189-210 (2017)
[16] Carrera, E.; Petrolo, M., Refined beam elements with only displacement variables and plate/shell capabilities, Meccanica, 47, 3, 537-556 (2012) · Zbl 1293.74408
[17] Carrera, E.; Petrolo, M.; Zappino, E., Performance of CUF approach to analyze the structural behavior of slender bodies, J. Struct. Eng., 138, 2, 285-297 (2012)
[18] Carrera, E.; Zappino, E., One-dimensional finite element formulation with node-dependent kinematics, Comput. Struct., 192, 114-125 (2017)
[19] Carrera, E.; Zappino, E.; Li, G., Analysis of beams with piezo-patches by node-dependent kinematic finite element method models, J. Intell. Mater. Syst. Struct., 29, 7, 1379-1393 (2018)
[20] Carrera, E.; Zappino, E.; Li, G., Finite element models with node-dependent kinematics for the analysis of composite beam structures, Composites B, 132, Supplement C, 35-48 (2018)
[21] Cinefra, M., Formulation of 3D finite elements using curvilinear coordinates, Mech. Adv. Mater. Struct. (2020)
[22] Cinefra, M.; D’Amico, G.; De Miguel Garcia, A.; Filippi, M.; Pagani, A.; Carrera, E., Efficient numerical evaluation of transmission loss in homogenized acoustic metamaterials for aeronautical application, Appl. Acoust., 164, Article 107253 pp. (2020)
[23] Cinefra, M.; de Miguel, A. G.; Filippi, M.; Houriet, C.; Pagani, A.; Carrera, E., Homogenization and free-vibration analysis of elastic metamaterial plates by CUF finite elements, Mech. Adv. Mater. Struct., 1-10 (2019)
[24] Cinefra, M.; Valvano, S.; Carrera, E., A layer-wise MITC9 finite element for the free-vibration analysis of plates with piezo-patches, Int. J. Smart Nano. Mater., 6, 2, 85-104 (2015)
[25] Cinefra, M.; Valvano, S.; Carrera, E., A layer-wise MITC9 finite element for the free-vibration analysis of plates with piezo-patches, Int. J. Smart Nano. Mater., 6, 2, 85-104 (2015)
[26] D’Ottavio, M.; Vidal, P.; Valot, E.; Polit, O., Assessment of plate theories for free-edge effects, Composites B, 48, 111-121 (2013)
[27] Dozio, L., Natural frequencies of sandwich plates with FGM core via variable-kinematic 2-D Ritz models, Compos. Struct., 96, 561-568 (2013)
[28] Euler, L., De Curvis Elasticis (1744), Lausanne and Geneva: Bousquet
[29] Kapania, K.; Raciti, S., Recent advances in analysis of laminated beams and plates, part I: Shear effects and buckling, AIAA J., 27, 7, 923-935 (1989) · Zbl 0674.73047
[30] Kapania, K.; Raciti, S., Recent advances in analysis of laminated beams and plates, part II: Vibrations and wave propagation, AIAA J., 27, 7, 935-946 (1989) · Zbl 0674.73048
[31] Kirchhoff, G., Uber das gleichgewicht und die bewegung einer elastishen scheibe, Crelles J., 40, 51-88 (1850)
[32] Kulikov, G. M.; Mamontov, A. A.; Plotnikova, S. V.; Mamontov, S. A., Exact geometry solid-shell element based on a sampling surfaces technique for 3D stress analysis of doubly-curved composite shells, Curved Layer. Struct., 3, 1 (2016) · Zbl 1388.74070
[33] Kulikov, G. M.; Plotnikova, S. V., A method of solving three-dimensional problems of elasticity for laminated composite plates, Mech. Compos. Mater., 48, 1, 15-26 (2012)
[34] Li, G.; Carrera, E.; Cinefra, M.; de Miguel, A. G.; Pagani, A.; Zappino, E., An adaptable refinement approach for shell finite element models based on node-dependent kinematics, Composites B, 210, 1-19 (2019)
[35] Mindlin, R. D., Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates, J. Appl. Mech., 18, 28-31 (1951) · Zbl 0044.40101
[36] Moleiro, F.; Carrera, E.; Zappino, E.; Li, G.; Cinefra, M., Layerwise mixed elements with node-dependent kinematics for global-local stress analysis of multilayered plates using high-order Legendre expansions, Comput. Methods Appl. Mech. Engrg., 359, Article 112764 pp. (2020) · Zbl 1441.74265
[37] Pagani, A.; Valvano, S.; Carrera, E., Analysis of laminated composites and sandwich structures by variable-kinematic MITC9 plate elements, J. Sandw. Struct. Mater., 28, 20, 2959-2987 (2017)
[38] Reddy, J. N., A simple higher-order theory for laminated composite plates, J. Appl. Mech., 51, 4, 745-752 (1984) · Zbl 0549.73062
[39] Reddy, J. N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis (2004), CRC Press · Zbl 1075.74001
[40] Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech., 12, 69-76 (1945)
[41] Timoshenko, S. P., On the transverse vibrations of bars of uniform cross section, Phil. Mag., 43, 125-131 (1922)
[42] Washizu, K., Variational Methods in Elasticity and Plasticity (1974), Elsevier Science & Technology · Zbl 0164.26001
[43] Zappino, E.; Li, G.; Pagani, A.; Carrera, E., Global-local analysis of laminated plates by node-dependent kinematic finite elements with variable ESL/LW capabilities, Compos. Struct., 172, 1-14 (2017)
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