Abstract
This article presents the derivation of the elastic buckling equations and boundary conditions of shear-deformable plates in the frame of the extended Kantorovich method (EKM). Surveying the literature shows that those stability equations are often obtained using a wrong derivation by confusing them with the linear equilibrium condition. This work aims at providing the correct derivation that is built on the stability of the equilibrium condition. Buckling equations are derived for three different plate theories, namely, the first-order shear deformation plate theory (FSDT), the refined-FSDT, and the refined plate theory (RPT). This article is the first to implement the EKM based on a refined theory. Also, it is the first time to implement the refined-FSDT in buckling analysis. For the generic FGM plates, buckling equations derived based on the FSDT and refined-FSDT are both found to be simple and contain only the lateral displacements/rotations variations. On the other hand, those of the RPT, have coupled lateral and in-plane displacement variations, even if the physical neutral plate is taken as the reference plane. The considered plate is rectangular and under general in-plane loads. The properties of the plate are continuously varying through its thickness which is assumed to change smoothly with a separable function in the two in-plane directions. The von Kármán nonlinearity is considered. The stability equations are derived according to the Trefftz criterion, using the variational calculus. The solution methods of the obtained equations are out of the scope of this article, however, a brief on the solution strategy is presented.
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References
Chen PC (1972) Buckling analysis of a rectangular plate by the kantorovich method. Int J Mech Sci 14(1):15–24. https://doi.org/10.1016/0020-7403(72)90003-3
Do VT, Pham VV, Nguyen HN (2020) On the development of refined plate theory for static bending behavior of functionally graded plates. Math Probl Eng. https://doi.org/10.1155/2020/2836763
Efraim E, Eisenberger M (2007) Exact vibration analysis of variable thickness thick annular isotropic and fgm plates. J Sound Vib 299(4):720–738. https://doi.org/10.1016/j.jsv.2006.06.068
Eisenberger M, Alexandrov A (2003) Buckling loads of variable thickness thin isotropic plates. Thin-Walled Struct 41(9):871–889. https://doi.org/10.1016/S0263-8231(03)00027-2
Eisenberger M, Shufrin I (2009) Buckling of plates by the multi-term extended Kantorovich method. In: 7th EUROMECH solid mechanics conference
Eslami MR (2018) Buckling and postbuckling of beams, plates, and shells. Springer, Berlin
Farahmand H (2020) Analytical solutions of bending and free vibration of moderately thick micro-plate via two-variable strain gradient theory. J Braz Soc Mech Sci Eng 42:1–11. https://doi.org/10.1007/s40430-020-02341-2
Grimm TR, Gerdeen JC (1975) Instability analysis of thin rectangular plates using the kantorovich method. J Appl Mech 42(1):110–114. https://doi.org/10.1115/1.3423499
Hassan AHA, Kurgan N (2020) Buckling of thin skew isotropic plate resting on pasternak elastic foundation using extended kantorovich method. Heliyon 6(6):e04236. https://doi.org/10.1016/j.heliyon.2020.e04236
Hassan AHA, Kurgan N (2020) The relations between the various critical temperatures of thin fgm plates. J Appl Comput Mech 6:1404–1419. https://doi.org/10.22055/jacm.2020.34697.2459
Jones RM (2006) Buckling of bars, plates, and shells. Bull Ridge Corporation
Kantorovich L, Krylov V (1958) Approximate Methods of Higher Analysis. Interscience, New York
Kerr AD (1968) An extension of the kantorovich method. Q Appl Math 26(2):219–229
Kerr AD (1969) An extended kantorovich method for the solution of eigenvalue problems. Int J Solids Struct 5(6):559–572. https://doi.org/10.1016/0020-7683(69)90028-6
Kerr AD, Alexander H (1968) An application of the extended kantorovich method to the stress analysis of a clamped rectangular plate. Acta Mech 6(2–3):180–196. https://doi.org/10.1007/BF01170382
Lopatin A, Morozov E (2011) Buckling of the sscf rectangular orthotropic plate subjected to linearly varying in-plane loading. Compos Struct 93(7):1900–1909. https://doi.org/10.1016/j.compstruct.2011.01.024
Lopatin A, Morozov E (2013) Buckling of a uniformly compressed rectangular sscf composite sandwich plate. Compos Struct 105:108–115. https://doi.org/10.1016/j.compstruct.2013.04.036
Lopatin A, Morozov E (2014) Approximate buckling analysis of the ccff orthotropic plates subjected to in-plane bending. Int J Mech Sci 85:38–44. https://doi.org/10.1016/j.ijmecsci.2014.05.003
Nguyen HN, Hong TT, Vinh PV, Quang ND, Thom DV et al (2019) A refined simple first-order shear deformation theory for static bending and free vibration analysis of advanced composite plates. Materials 12(15):2385. https://doi.org/10.3390/ma12152385
Nguyen TK, Sab K, Bonnet G (2007) Shear correction factors for functionally graded plates. Mech Adv Mater Struct 14(8):567–575. https://doi.org/10.1080/15376490701672575
Nguyen TK, Sab K, Bonnet G (2008) First-order shear deformation plate models for functionally graded materials. Compos Struct 83(1):25–36. https://doi.org/10.1016/j.compstruct.2007.03.004
Qi Y, Knight NF Jr (1996) A refined first-order shear-deformation theory and its justification by plane strain bending problem of laminated plates. Int J Solids Struct 33(1):49–64. https://doi.org/10.1016/0020-7683(95)00010-8
Reddy JN (2006) Theory and analysis of elastic plates and shells. CRC Press, Boca Raton
Shariyat M, Alipour M (2014) A novel shear correction factor for stress and modal analyses of annular FGM plates with non-uniform inclined tractions and non-uniform elastic foundations. Int J Mech Sci 87:60–71. https://doi.org/10.1016/j.ijmecsci.2014.05.032
Shimpi RP (2002) Refined plate theory and its variants. AIAA J 40(1):137–146. https://doi.org/10.2514/2.1622
Shufrin I, Eisenberger M (2005) Stability and vibration of shear deformable plates - first order and higher order analyses. Int J Solids Struct 42(3):1225–1251. https://doi.org/10.1016/j.ijsolstr.2004.06.067
Shufrin I, Eisenberger M (2005) Stability of variable thickness shear deformable plates - first order and high order analyses. Thin-Walled Struct 43(2):189–207. https://doi.org/10.1016/j.tws.2004.07.013
Shufrin I, Eisenberger M (2006) Buckling of plates with variable in-plane forces. In: Analysis and Design of Plated Structures, 2: 26–55. Woodhead Publishing https://doi.org/10.1533/9781845690960.26
Shufrin I, Eisenberger M (2007) Shear buckling of thin plates with constant in-plane stresses. Int J Struct Stab Dyn 7(02):179–192. https://doi.org/10.1142/S021945540700223X
Shufrin I, Rabinovitch O, Eisenberger M (2008) Buckling of laminated plates with general boundary conditions under combined compression, tension, and shear-a semi-analytical solution. Thin-Walled Struct 46(7):925–938
Shufrin I, Rabinovitch O, Eisenberger M (2008) Buckling of symmetrically laminated rectangular plates with general boundary conditions - a semi analytical approach. Compos Struct 82(4):521–531. https://doi.org/10.1016/j.compstruct.2007.02.003
Shufrin I, Rabinovitch O, Eisenberger M (2008) Buckling of symmetrically laminated rectangular plates with general boundary conditions-a semi analytical approach. Compos Struct 82(4):521–531. https://doi.org/10.1016/j.compstruct.2007.02.003
Singhatanadgid P, Singhanart T (2019) The kantorovich method applied to bending, buckling, vibration, and 3d stress analyses of plates: A literature review. Mech Adv Mater Struct 26(2):170–188. https://doi.org/10.1080/15376494.2017.1365984
Tran HQ, Tran MT, Nguyen-Tri P et al (2020) A new four-variable refined plate theory for static analysis of smart laminated functionally graded carbon nanotube reinforced composite plates. Mech Mater 142:103294. https://doi.org/10.1016/j.mechmat.2019.103294
Xie WC, Elishakoff I (2000) Buckling mode localization in rib-stiffened plates with misplaced stiffeners - kantorovich approach. Chaos Solitons Fractals 11(10):1559–1574. https://doi.org/10.1016/S0960-0779(99)00078-8
Yu T, Zhang L (1986) The elastic wrinkling of an annular plate under uniform tension on its inner edge. Int J Mech Sci 28(11):729–737. https://doi.org/10.1016/0020-7403(86)90020-2
Yuan S, Jin Y (1998) Computation of elastic buckling loads of rectangular thin plates using the extended kantorovich method. Comput Struct 66(6):861–867. https://doi.org/10.1016/S0045-7949(97)00111-9
Yuan S, Zhang Y (1992) Further extension of the extended kantorovich method. Comput Methods Eng Adv Appl 2:1240–1245
Zhang DG (2013) Modeling and analysis of fgm rectangular plates based on physical neutral surface and high order shear deformation theory. Int J Mech Sci 68:92–104. https://doi.org/10.1016/j.ijmecsci.2013.01.002
Zhang DG, Zhou YH (2008) A theoretical analysis of fgm thin plates based on physical neutral surface. Comput Mater Sci 44(2):716–720. https://doi.org/10.1016/j.commatsci.2008.05.016
Zhang L, Yu T (1988) The plastic wrinkling of an annular plate under uniform tension on its inner edge. Int J Solids Struct 24(5):497–503. https://doi.org/10.1016/0020-7683(88)90004-2
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Hassan, A.H.A., Kurgan, N. & Can, N. The correct derivation of the buckling equations of the shear-deformable FGM plates for the extended Kantorovich method. Meccanica 57, 441–456 (2022). https://doi.org/10.1007/s11012-021-01441-0
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DOI: https://doi.org/10.1007/s11012-021-01441-0