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Static and dynamic analysis of two-layer Timoshenko composite beams by weak-form quadrature element method. (English) Zbl 1480.74181

Summary: To improve the efficiency in predicting the dynamic mode and static response of the two-layer partial interaction composite beams, this paper utilizes the differential quadrature technique to approximate derivatives of the primary unknowns with adaptive order of precision, rather than the low and constant order of interpolation used in the conventional finite element method (FEM). A degree-of-freedom-adaptive weak-form quadrature element (WQE) for dynamic analysis is formulated and implemented based on the principle of virtual work. For the purpose of comparison, a parabolic displacement-based finite element is also provided, thus (1) the predicted deflections and natural frequencies of the composite beams are verified; (2) the smoothness of the internal forces and stresses generated by WQE method and FEM are compared, and (3) the convergent rates of higher order free vibration modes are also examined. Numerical results show that the efficiency of the proposed WQE method has, on the one hand, significantly triumphed over that of FEM on analyses including static response, natural frequencies and higher order free vibration modes, on the other hand, the smoothness of results, including internal forces and stresses, is greatly refined.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] Newmark, N. M.; Siess, C. P.; Viest, I. M., Tests and analysis of composite beams with incomplete interaction, Proc. Soc. Exp. Stress Anal., 9, 1, 75-92 (1951)
[2] Xu, R.; Wang, G., Variational principle of partial-interaction composite beams using Timoshenko’s beam theory, Int. J. Mech. Sci., 60, 1, 72-83 (2012)
[3] Xu, R.; Wang, G., Bending solutions of the Timoshenko partial-interaction composite beams using Euler-Bernoulli solutions, J. Eng. Mech., ASCE., 139, 12, 1881-1885 (2013)
[4] Ecsedi, I.; Baksa, A., Analytical solution for layered composite beams with partial shear interaction based on Timoshenko beam theory, Eng. Struct., 115, 107-117 (2016)
[5] Schnabl, S.; Saje, M.; Turk, G.; Planinc, I., Analytical solution of two-layer beam taking into account interlayer slip and shear deformation, J. Struct. Eng., ASCE., 133, 6, 886-894 (2007)
[6] Girhammar, U. A.; Pan, D. H.; Gustafsson, A., Exact dynamic analysis of composite beams with partial interaction, Int. J. Mech. Sci., 51, 8, 565-582 (2009)
[7] Ranzi, G.; Bradford, M. A., Direct stiffness analysis of a composite beam-column element with partial interaction, Comput. Struct., 85, 15-16, 1206-1214 (2007)
[8] Nguyen, Q. H.; Martinelli, E.; Hjiaj, M., Derivation of the exact stiffness matrix for a two-layer Timoshenko beam element with partial interaction, Eng. Struct., 33, 2, 298-307 (2011)
[9] Nguyen, Q. H.; Hjiaj, M.; Guezouli, S., Exact finite element model for shear-deformable two-layer beams with discrete shear connection, Finite Elem. Anal. Des., 47, 7, 718-727 (2011)
[10] Carrera, E.; Giunta, G.; Petrolo, M., Beam Structures: Classical And Advanced Theories (2011), John Wiley & Sons: John Wiley & Sons New Delhi · Zbl 1238.74001
[11] Carrera, E.; Cinefra, M.; Petrolo, M.; Zappino, E., Finite Element Analysis of Structures Through Unified Formulation (2014), John Wiley & Sons: John Wiley & Sons New Delhi · Zbl 1306.74001
[12] 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)
[13] Pagani, A.; Carrera, E.; Boscolo, M.; Banerjee, J. R., Refined dynamic stiffness elements applied to free vibration analysis of generally laminated composite beams with arbitrary boundary conditions, Compos. Struct., 110, 305-316 (2014)
[14] Giunta, G.; Biscani, F.; Belouettar, S.; Ferreira, A. J.M.; Carrera, E., Free vibration analysis of composite beams via refined theories, Compos. Part B-Eng., 44, 1, 540-552 (2013)
[15] Pagani, A.; Carrera, E., Large-deflection and post-buckling analyses of laminated composite beams by Carrera Unified Formulation, Compos. Struct., 170, 15, 40-52 (2017)
[16] Stein, E., Adaptive Finite Elements in Linear and Nonlinear Solid and Structural Mechanics (2005), Springer: Springer New York · Zbl 1110.74010
[17] Szabo, B. A.; Mehta, A. K., \(p\)-Convergent finite element approximations in fracture mechanics, Int. J. Numer. Meth. Eng., 12, 3, 551-560 (1978) · Zbl 0369.73097
[18] Bellman, R. E.; Casti, J., Differential quadrature and long-term integration, J. Math. Anal. Appl., 34, 2, 235-238 (1971) · Zbl 0236.65020
[19] Bellman, R.; Kashef, B. G.; Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys., 10, 1, 40-52 (1972) · Zbl 0247.65061
[20] Quan, J. R.; Chang, C. T., New insights in solving distributed system equations by the quadrature method—I. Analysis, Comput. Chem. Eng., 13, 7, 779-788 (1989)
[21] Shu, C.; Richards, B. E., Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 15, 7, 791-798 (1992) · Zbl 0762.76085
[22] Ghasemi, M., High order approximations using spline-based differential quadrature method: implementation to the multi-dimensional PDEs, Appl. Math. Model., 46, 63-80 (2017) · Zbl 1443.65100
[23] Wang, X.; Yuan, Z., Accurate stress analysis of sandwich panels by the differential quadrature method, Appl. Math. Model., 43, 548-565 (2017) · Zbl 1446.74063
[24] Kudela, P.; Krawczuk, M.; Ostachowicz, W., Wave propagation modelling in 1D structures using spectral finite elements, J. Sound Vibr., 300, 1-2, 88-100 (2007)
[25] Jin, C.; Wang, X., Accurate free vibration analysis of Euler functionally graded beams by the weak form quadrature element method, Compos. Struct., 125, 41-50 (2015)
[26] Wang, X., Differential Quadrature and Differential Quadrature Based Element Methods Theory and Applications (2015), Butterworth-Heinemann: Butterworth-Heinemann Oxford · Zbl 1360.74003
[27] Wang, Y.; Wang, X., Free vibration analysis of soft-core sandwich beams by the novel weak form quadrature element method, J. Sandw. Struct. Mater., 18, 3, 294-320 (2016)
[28] Wang, Y.; Wang, X., Static analysis of higher order sandwich beams by weak form quadrature element method, Compos. Struct., 116, 841-848 (2014)
[29] Wang, X.; Wang, Y., Static analysis of sandwich panels with non-homogeneous soft-cores by novel weak form quadrature element method, Compos. Struct., 146, 207-220 (2016)
[30] Zhong, H.; Yue, Z., Analysis of thin plates by the weak form quadrature element method, Sci. China Ser. G., 55, 5, 861-871 (2012)
[31] Shen, Z.; Zhong, H., Geometrically nonlinear quadrature element analysis of composite beams with partial interaction, Eng. Mech., 30, 3, 270-275 (2013), (in Chinese)
[32] Du, H.; Lim, M. K.; Lin, R. M., Application of generalized differential quadrature method to structural problems, Int. J. Numer. Meth. Eng., 37, 1881-1896 (1994) · Zbl 0804.73076
[33] Xu, R.; Wu, Y., Two-dimensional analytical solutions of simply supported composite beams with interlayer slips, Int. J. Solids Struct., 44, 1, 165-175 (2007) · Zbl 1118.74018
[34] Girhammar, U. A.; Gopu, V. K.A., Composite beam-columns with interlayer slip — exact analysis, J. Struct. Eng., ASCE., 119, 4, 1265-1282 (1993)
[35] Xu, R.; Wu, Y., Static, dynamic, and buckling analysis of partial interaction composite members using Timoshenko’s beam theory, Int. J. Mech. Sci., 49, 10, 1139-1155 (2007)
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