## Closed form solutions for an anisotropic composite beam on a two-parameter elastic foundation.(English)Zbl 1475.74077

Summary: Beams resting on elastic foundations are widely used in engineering design such as railroad tracks, pipelines, bridge decks, and automobile frames. Laminated composite beams can be tailored for specific design requirements and offer a desirable design framework for beams resting on elastic foundations. Therefore, the analysis of flexural behaviour of laminated composite beams on elastic foundations is of important consequence. Exact solutions for flexural deflection of composite beams with coupling terms between stretching, shearing, bending and twisting, resting on two-parameter elastic foundations for various types of loading and boundary conditions, are presented for the first time. The proposed new formulation is based on Euler-Bernoulli beam theory having four degrees of freedom, namely bending in two principal directions, axial elongation and twist. Governing equations and boundary conditions are derived from the principle of virtual work and expressed in a compact matrix-vector form. By decoupling bending in both principal directions from twist and axial elongation, the fourth-order differential equation for bending is derived and transformed into a system of first-order differential equations. An exact solution of this system of equations is obtained using a fundamental matrix approach. Fundamental matrices for different configurations of elastic foundation are provided. The ability of the presented mathematical model in predicting flexural behaviour of beams on elastic foundations is verified numerically by comparison with results available in the literature. In addition, the deflection of anisotropic beams is analysed for different types of stacking sequences, boundary and loading conditions. The effect of elastic foundation coefficients on the flexural behaviour is also investigated and discussed.

### MSC:

 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74G05 Explicit solutions of equilibrium problems in solid mechanics
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### References:

 [1] Afshin, M.; Taheri-Behrooz, F., Interlaminar stresses of laminated composite beams resting on elastic foundation subjected to transverse loading, Comput. Mater. Sci., 96, 439-447 (2015) [2] Akgöz, B.; Civalek, Ö., Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity, Compos. Struct., 134, 294-301 (2015) [3] Al-Shujairi, M.; Mollamahmutoğlu, Ç., Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect, Composites B, 154, 292-312 (2018) [4] Atmane, H. A.; Tounsi, A.; Bernard, F., Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations, Int. J. Mech. Mater. Des., 13, 1, 71-84 (2017) [5] Babaee, A.; Sadighi, M.; Nikbakht, A.; Alimirzaei, S., Generalized differential quadrature nonlinear buckling analysis of smart SMA/FG laminated beam resting on nonlinear elastic medium under thermal loading, J. Therm. Stresses, 41, 5, 583-607 (2018) [6] Babaei, H.; Kiani, Y.; Eslami, M., Geometrically nonlinear analysis of shear deformable FGM shallow pinned arches on nonlinear elastic foundation under mechanical and thermal loads, Acta Mech., 229, 7, 3123-3141 (2018) · Zbl 1459.74102 [7] Babaei, H.; Kiani, Y.; Eslami, M. R., Large amplitude free vibration analysis of shear deformable FGM shallow arches on nonlinear elastic foundation, Thin-Walled Struct., 144, Article 106237 pp. (2019) [8] Babaei, H.; Kiani, Y.; Eslami, M. R., Thermal buckling and post-buckling analysis of geometrically imperfect FGM clamped tubes on nonlinear elastic foundation, Appl. Math. Model., 71, 12-30 (2019) · Zbl 1481.74206 [9] Bourada, F.; Bousahla, A. A.; Tounsi, A.; Bedia, E.; Mahmoud, S.; Benrahou, K. H.; Tounsi, A., Stability and dynamic analyses of SW-CNT reinforced concrete beam resting on elastic foundation, Comput. Concr., 25, 6, 485-495 (2020) [10] Bousahla, A. A.; Bourada, F.; Mahmoud, S.; Tounsi, A.; Algarni, A.; Bedia, E.; Tounsi, A., Buckling and dynamic behavior of the simply supported CNT-RC beams using an integral-first shear deformation theory, Comput. Concr., 25, 2, 155-166 (2020) [11] Chaabane, L. A.; Bourada, F.; Sekkal, M.; Zerouati, S.; Zaoui, F. Z.; Tounsi, A.; Derras, A.; Bousahla, A. A.; Tounsi, A., Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation, Struct. Eng. Mech., 71, 2, 185-196 (2019) [12] Chen, W.; Lü, C.; Bian, Z., A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation, Appl. Math. Model., 28, 10, 877-890 (2004) · Zbl 1147.74330 [13] Doeva, O.; Masjedi, P. K.; Weaver, P. M., Exact solution for the deflection of composite beams under non-uniformly distributed loads, (AIAA Scitech 2020 Forum (2020)), 0245 [14] Doeva, O.; Masjedi, P. K.; Weaver, P. M., Static deflection of fully coupled composite Timoshenko beams: An exact analytical solution, Eur. J. Mech. A Solids, 81, Article 103975 pp. (2020) · Zbl 1475.74076 [15] Hadji, L.; Bernard, F., Bending and free vibration analysis of functionally graded beams on elastic foundations with analytical validation, Adv. Mater. Res., 9, 1, 63-98 (2020) [16] Li, Z.; Xu, Y.; Huang, D., Accurate solution for functionally graded beams with arbitrarily varying thicknesses resting on a two-parameter elastic foundation, J. Strain Anal. Eng. Des., 55, 7-8, 22-236 (2020) [17] Li, Z.-M.; Zhao, Y.-X., Nonlinear bending of shear deformable anisotropic laminated beams resting on two-parameter elastic foundations based on an exact bending curvature model, J. Eng. Mech., 141, 3, Article 04014125 pp. (2015) [18] Luo, Y., An efficient 3D Timoshenko beam element with consistent shape functions, Adv. Theor. Appl. Mech., 1, 3, 95-106 (2008) · Zbl 1156.74342 [19] Mahmoudpour, E.; Hosseini-Hashemi, S.; Faghidian, S., Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model, Appl. Math. Model., 57, 302-315 (2018) · Zbl 1480.74118 [20] Masjedi, P. K.; Doeva, O.; Weaver, P. M., Closed-form solutions for the coupled deflection of anisotropic Euler-Bernoulli composite beams with arbitrary boundary conditions, Thin-Walled Struct., 161, Article 107479 pp. (2021) [21] Masjedi, P. K.; Maheri, A., Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation, Eur. J. Mech. A Solids, 66, 329-340 (2017) · Zbl 1406.74304 [22] Masjedi, P. K.; Maheri, A.; Weaver, P. M., Large deflection of functionally graded porous beams based on a geometrically exact theory with a fully intrinsic formulation, Appl. Math. Model., 76, 938-957 (2019) · Zbl 1481.74444 [23] Masjedi, P. K.; Ovesy, H. R., Chebyshev collocation method for static intrinsic equations of geometrically exact beams, Int. J. Solids Struct., 54, 183-191 (2015) [24] Masjedi, P. K.; Ovesy, H. R., Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations, Acta Mech., 226, 6, 1689-1706 (2015) · Zbl 1325.74086 [25] Masjedi, P. K.; Weaver, P. M., Analytical solution for the fully coupled static response of variable stiffness composite beams, Appl. Math. Model., 81, 16-36 (2020) [26] Masjedi, P. K.; Weaver, P. M., Variable stiffness composite beams subject to non-uniformly distributed loads: An analytical solution, Compos. Struct., 256, Article 112975 pp. (2020) [27] Murin, J.; Aminbaghai, M.; Kutis, V.; Hrabovsky, J., Modal analysis of the FGM beams with effect of axial force under longitudinal variable elastic Winkler foundation, Eng. Struct., 49, 234-247 (2013) [28] Naik, N., Composite beams on elastic foundations, J. Thermoplastic Compos. Mater., 13, 1, 2-11 (2000) [29] Pai, P. F., Highly Flexible Structures: Modeling, Computation, and Experimentation (2007), American Institute of Aeronautics and Astronautics [30] Pakar, I.; Bayat, M.; Cveticanin, L., Nonlinear vibration of unsymmetrical laminated composite beam on elastic foundation, Steel Compos. Struct., 26, 4, 453-461 (2018) [31] Phuong, N. T.B.; Tu, T. M.; Phuong, H. T.; Van Long, N., Bending analysis of functionally graded beam with porosities resting on elastic foundation based on neutral surface position, J. Sci. Technol. Civ. Eng. (STCE)-NUCE, 13, 1, 33-45 (2019) [32] Robinson, M. T.A.; Adali, S., Buckling of nonuniform and axially functionally graded nonlocal Timoshenko nanobeams on Winkler-Pasternak foundation, Compos. Struct., 206, 95-103 (2018) [33] Sayyad, A. S.; Ghugal, Y. M., An inverse hyperbolic theory for FG beams resting on Winkler-Pasternak elastic foundation, Adv. Aircr. Spacecr. Sci., 5, 6, 671-689 (2018) [34] Shen, H.-S., Nonlinear analysis of functionally graded fiber reinforced composite laminated beams in hygrothermal environments, part I: Theory and solutions, Compos. Struct., 125, 698-705 (2015) [35] Shen, H.-S.; Lin, F.; Xiang, Y., Nonlinear bending and thermal postbuckling of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations, Eng. Struct., 140, 89-97 (2017) [36] Shen, H.-S.; Lin, F.; Xiang, Y., Nonlinear vibration of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations in thermal environments, Nonlinear Dynam., 90, 2, 899-914 (2017) [37] Shenoi, R.; Wang, W., Through-thickness stresses in curved composite laminates and sandwich beams, Compos. Sci. Technol., 61, 11, 1501-1512 (2001) [38] Sobhy, M.; Zenkour, A. M., The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations, Mech. Adv. Mater. Struct., 27, 7, 525-538 (2020) [39] Trabelssi, M.; El-Borgi, S.; Fernandes, R.; Ke, L.-L., Nonlocal free and forced vibration of a graded Timoshenko nanobeam resting on a nonlinear elastic foundation, Composites B, 157, 331-349 (2019) [40] Wang, C. M.; Lam, K. Y.; He, X. Q., Exact solutions for Timoshenko beams on elastic foundations using Green’s functions, J. Struct. Mech., 26, 1, 101-113 (1998) [41] Wattanasakulpong, N.; Ungbhakorn, V., Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation, Comput. Mater. Sci., 71, 201-208 (2013) [42] Yang, J.; Huang, X.-H.; Shen, H.-S., Nonlinear flexural behavior of temperature-dependent FG-CNTRC laminated beams with negative Poisson’s ratio resting on the Pasternak foundation, Eng. Struct., 207, Article 110250 pp. (2020) [43] Ying, J.; Lü, C. F.; Chen, W. Q., Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations, Compos. Struct., 84, 3, 209-219 (2008) [44] Zenkour, A. M.; Allam, M. N.M.; Sobhy, M., Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak’s elastic foundations, Acta Mech., 212, 3-4, 233-252 (2010) · Zbl 1397.74124
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