×

Analysis of plates on Pasternak foundations by radial basis functions. (English) Zbl 1288.74030

Summary: This paper addresses the static and free vibration analysis of rectangular plates resting on Pasternak foundations. The Pasternak foundation is described by a two-parameter model. The numerical approach is based on collocation with radial basis functions. The model allows the analysis of arbitrary boundary conditions and irregular geometries. It is shown that the present method, based on a first-order shear deformation theory produces highly accurate displacements and stresses, as well as natural frequencies and modes.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74K20 Plates
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pasternak PL (1954) On a new method of analysis of an elastic foundation by means of two foundation constants. Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow, USSR, pp 1–56 (in Russian)
[2] Winkler E (1867) Die Lehre von der Elasticitaet und Festigkeit. Prag, Dominicus
[3] Leissa AW (1973) The free vibration of plates. J Sound Vib 31: 257–293 · Zbl 0268.73033 · doi:10.1016/S0022-460X(73)80371-2
[4] Lam KY, Wang CM, He XQ (2000) Canonical exact solutions for levy-plates on two-parameter foundation using green’s functions. Eng Struct 22: 364–378 · doi:10.1016/S0141-0296(98)00116-3
[5] Xiang Y, Wang CM, Kitipornchai S (1994) Exact vibration solution for initially stressed Mindlin plates on pasternak foundation. Int J Mech Sci 36: 311–316 · Zbl 0799.73048 · doi:10.1016/0020-7403(94)90037-X
[6] Omurtag MH, Ozutok A, Akoz AY (1997) Free vibration analysis of kirchhoff plates resting on elastic foundation by mixed finite element formulation based on gateaux differential. Int J Numer Methods Eng 40: 295–317 · doi:10.1002/(SICI)1097-0207(19970130)40:2<295::AID-NME66>3.0.CO;2-2
[7] Matsunaga H (2000) Vibration and stability of thick plates on elastic foundations. J Eng Mech ASCE 126: 27–34 · doi:10.1061/(ASCE)0733-9399(2000)126:1(27)
[8] Shen HS, Yang J, Zhang L (2001) Free and forced vibration of Reissner–Mindlin plates with free edges resting on elastic foundations. J Sound Vib 244: 299–320 · doi:10.1006/jsvi.2000.3501
[9] Ayvaz Y, Daloglu A, Dogangun A (1998) Application of a modified Vlasov model to earthquake analysis of plates resting on elastic foundations. J Sound Vib 212: 499–509 · doi:10.1006/jsvi.1997.1394
[10] Liew KM, Teo TM (2002) Differential cubature method for analysis of shear deformable rectangular plates on pasternak foundations. Int J Mech Sci 44: 1179–1194 · Zbl 1021.74016 · doi:10.1016/S0020-7403(02)00167-4
[11] Liew KM, Han JB, Xiao ZM, Du H (1996) Differential quadrature method for Mindlin plates on winkler foundations. Int J Mech Sci 38: 405–421 · Zbl 0841.73075 · doi:10.1016/0020-7403(95)00062-3
[12] Han JB, Liew KM (1997) Numerical differential quadrature method for Reissner–Mindlin plates on two-parameter foundations. Int J Mech Sci 39: 977–989 · Zbl 0910.73075 · doi:10.1016/S0020-7403(97)00001-5
[13] Zhou D, Cheung YK, Lo SH, Au FTK (2004) Three-dimensional vibration analysis of rectangular thick plates on pasternak foundation. Int J Numer Methods Eng 59: 1313–1334 · Zbl 1041.74517 · doi:10.1002/nme.915
[14] Civalek O, Acar MH (2007) Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations. Int J Pres Vessel Pip 84: 527–535 · doi:10.1016/j.ijpvp.2007.07.001
[15] Chucheepsakul S, Chinnaboom B (2002) An alternative domainboundary element technique for analyzing plates on two-parameter elastic foundations. Eng Anal Bound Elem 26: 547–555 · Zbl 1087.74646 · doi:10.1016/S0955-7997(02)00007-3
[16] Kansa EJ (1990) Multiquadrics–a scattered data approximation scheme with applications to computational fluid dynamics. i: Surface approximations and partial derivative estimates. Comput Math Appl 19(8/9): 127–145 · Zbl 0692.76003 · doi:10.1016/0898-1221(90)90270-T
[17] Hon YC, Lu MW, Xue WM, Zhu YM (1997) Multiquadric method for the numerical solution of byphasic mixture model. Appl Math Comput 88: 153–175 · Zbl 0910.76059 · doi:10.1016/S0096-3003(96)00309-8
[18] Hon YC, Cheung KF, Mao XZ, Kansa EJ (1999) A multiquadric solution for the shallow water equation. J Hydr Eng ASCE 125(5): 524–533 · doi:10.1061/(ASCE)0733-9429(1999)125:5(524)
[19] Wang JG, Liu GR, Lin P (2002) Numerical analysis of biot’s consolidation process by radial point interpolation method. Int J Solids Struct 39(6): 1557–1573 · Zbl 1061.74014 · doi:10.1016/S0020-7683(02)00005-7
[20] Liu GR, Gu YT (2001) A local radial point interpolation method (LRPIM) for free vibration analyses of 2-d solids. J Sound Vib 246(1): 29–46 · doi:10.1006/jsvi.2000.3626
[21] Liu GR, Wang JG (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Methods Eng 54: 1623–1648 · Zbl 1098.74741 · doi:10.1002/nme.489
[22] Wang JG, Liu GR (2002) On the optimal shape parameters of radial basis functions used for 2-d meshless methods. Comput Meth Appl Mech Eng 191: 2611–2630 · Zbl 1065.74074 · doi:10.1016/S0045-7825(01)00419-4
[23] Chen XL, Liu GR, Lim SP (2003) An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape. Compos Struct 59: 279–289 · doi:10.1016/S0263-8223(02)00034-X
[24] Dai KY, Liu GR, Lim SP, Chen XL (2004) An element free Galerkin method for static and free vibration analysis of shear-deformable laminated composite plates. J Sound Vib 269: 633–652 · doi:10.1016/S0022-460X(03)00089-0
[25] Liu GR, Chen XL (2002) Buckling of symmetrically laminated composite plates using the element-free Galerkin method. Int J Struct Stabil Dyn 2: 281–294 · Zbl 1205.74162 · doi:10.1142/S0219455402000634
[26] Liew KM, Chen XL, Reddy JN (2004) Mesh-free radial basis function method for buckling analysis of non-uniformity loaded arbitrarily shaped shear deformable plates. Comput Meth Appl Mech Eng 193: 205–225 · Zbl 1075.74700 · doi:10.1016/j.cma.2003.10.002
[27] Huang YQ, Li QS (2004) Bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method. Comput Meth Appl Mech Eng 193: 3471–3492 · Zbl 1068.74090 · doi:10.1016/j.cma.2003.12.039
[28] Liu L, Liu GR, Tan VCB (2002) Element free method for static and free vibration analysis of spatial thin shell structures. Comput Meth Appl Mech Eng 191: 5923–5942 · Zbl 1083.74610 · doi:10.1016/S0045-7825(02)00504-2
[29] Ferreira AJM (2003) A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates. Compos Struct 59: 385–392 · doi:10.1016/S0263-8223(02)00239-8
[30] Ferreira AJM (2003) Thick composite beam analysis using a global meshless approximation based on radial basis functions. Mech Adv Mater Struct 10: 271–284 · doi:10.1080/15376490306743
[31] Ferreira AJM, Roque CMC, Martins PALS (2003) Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Compos Part B Eng 34: 627–636 · doi:10.1016/S1359-8368(03)00083-0
[32] Madich WR, Nelson SA (1990) Multivariate interpolation and conditionally positive definite functions. ii. Math Comp 54(189): 211–230 · Zbl 0859.41004 · doi:10.1090/S0025-5718-1990-0993931-7
[33] Yoon J (2001) Spectral approximation orders of radial basis function interpolation on the sobolev space. SIAM J Math Anal 33(4): 946–958 · Zbl 0996.41002 · doi:10.1137/S0036141000373811
[34] Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 176: 1905–1915 · doi:10.1029/JB076i008p01905
[35] Buhmann MD (2000) Radial basis functions. Acta Numer 9: 1–38 · Zbl 1004.65015 · doi:10.1017/S0962492900000015
[36] Platte RB, Driscoll TA (2004) Computing eigenmodes of elliptic operators using radial basis functions. Comput Math Appl 48: 561–576 · Zbl 1063.65117 · doi:10.1016/j.camwa.2003.08.007
[37] Reddy JN (1997) Mechanics of laminated composite plates: theory and analysis. CRC Press, Boca Raton · Zbl 0899.73002
[38] Liew KM, Huang YQ, Reddy JN (2003) Vibration analysis of symmetrically laminated plates based on fsdt using the moving least squares differential quadrature method. Comput Meth Appl Mech Eng 192: 2203–2222 · Zbl 1119.74628 · doi:10.1016/S0045-7825(03)00238-X
[39] Kobayashi H, Sonoda K (1989) Rectangular Mindlin plates on elastic foundation. Int J Mech Sci 31(9): 679–692 · Zbl 0704.73054 · doi:10.1016/S0020-7403(89)80003-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.