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A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. (English) Zbl 1147.74330

Summary: A mixed method, which combines the state space method and the differential quadrature method, is proposed for bending and free vibration of arbitrarily thick beams resting on a Pasternak elastic foundation. Based on the two-dimensional state equation of elasticity, the domain along the axial direction is discretized according to the principle of differential quadrature (DQ). As a result, the state equations about the variables at discrete points are established. With consideration of the end conditions and the upper and lower boundary conditions in the derived state equations, governing equations for bending and free vibration problems are formulated. Numerical results prove that the present approach is very efficient and reliable. The effects of Poisson’s ratio and foundation parameters on the natural frequencies are discussed.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74J99 Waves in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
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[1] Hetenyi, M., A general solution for the bending of beams on an elastic foundation of arbitrary continuity, J. appl. phys., 21, 55-58, (1950) · Zbl 0035.40302
[2] Eisenberger, M.; Clastornik, J., Vibrations and buckling of a beam on a variable Winkler elastic foundation, J. sound vib., 115, 233-241, (1987) · Zbl 1235.74065
[3] Ding, Z., A general solution to vibrations of beams on variable Winkler elastic foundation, Comput. struct., 47, 83-90, (1993) · Zbl 0799.73045
[4] Eisenberger, M.; Yankelevsky, D.Z.; Clastornik, J., Stability of beams on elastic foundations, Comput. struct., 24, 135-140, (1986) · Zbl 0589.73046
[5] Au, F.T.K.; Zheng, D.Y.; Cheung, Y.K., Vibration and stability of non-uniform beams with abrupt changes of cross-section by using C1 modified beam vibration functions, Appl. math. model., 29, 19-34, (1999) · Zbl 0959.74027
[6] Clastornik, J.; Eisenberger, M.; Yankelevsky, D.Z., Beams on variable Winkler elastic foundation, J. eng. mech. ASCE, 53, 925-928, (1986) · Zbl 0607.73116
[7] Farghaly, S.H.; Zeid, K.M., An exact frequency equation for an axially loaded beam-mass-spring system resting on a Winkler elastic foundation, J. sound vib., 185, 357-363, (1995) · Zbl 1048.74523
[8] P.L. Pasternak, On a new method of analysis of an elastic foundation by means of two foundation constants, Gos. Izd. Lip. po Strait i Arkh. Moscow, 1954 (in Russian)
[9] De Rosa, M.A., Free vibrations of Timoshenko beams on two-parameter elastic foundation, Comput. struct., 57, 151-156, (1995) · Zbl 0900.73355
[10] Naidu, N.R.; Rao, G.V., Vibrations of initially stressed uniform beams on two-parameter elastic foundation, Comput. struct., 57, 941-943, (1995)
[11] Ayvaz, Y.; Özgan, K., Application of modified Vlasov model to free vibration analysis of beams resting on elastic foundations, J. sound vib., 255, 111-127, (2002)
[12] Lee, S.Y.; Kes, H.Y., Free vibrations of non-uniform beams resting on non-uniform elastic foundation with general elastic end restraints, Comput. struct., 34, 421-429, (1990) · Zbl 0716.73054
[13] Franciosi, C.; Masi, A., Free vibrations of foundation beams on two-parameter elastic soil, J. sound vib., 47, 419-426, (1993) · Zbl 0774.73068
[14] Wang, C.M.; Lam, K.Y.; He, X.Q., Exact solutions for tiomoshenko beams on elastic foundations using Green’s functions, Mech. struct. Mach., 26, 101-113, (1998)
[15] De Rosa, M.A.; Maurizi, M.J., The influence of concentrated masses and Pasternak soil on the free vibrations of Euler beams–exact solution, J. sound vib., 212, 573-581, (1998)
[16] Ho, S.H.; Chen, C.K., Analysis of general elastically end restrained non-uniform beams using differential transform, Appl. math. model., 22, 219-234, (1998)
[17] Chen, C.N., Vibration of prismatic beam on an elastic foundation by the differential quadrature element method, Comput. struct., 77, 1-9, (2000)
[18] Davies, J.M., An exact finite element for beam on elastic foundation problems, J. struct. mech., 14, 489-499, (1986)
[19] Kar, R.C.; Sujata, T., Parametric instability of Timoshenko beam with thermal gradient resting on a variable Pasternak foundation, Comput. struct., 36, 659-665, (1990)
[20] Chen, C.N., Solution of beam on elastic foundation by DQEM, J. eng. mech. ASCE, 124, 1381-1384, (1998)
[21] Wang, X.; Bert, C.W.; Striz, A.G., Differential and quadrature analysis of deflection, buckling and free vibration of beams and rectangular plates, Comput. struct., 48, 473-479, (1993)
[22] Laura, P.A.A.; Gutierrez, R.H., Analysis of vibrating Timoshenko beams using the method of differential quadrature, Shock and vibration, 1, 89-93, (1993)
[23] Siddiqi, Z.A.; Kukreti, A.R., Analysis of eccentrically stiffened plates with mixed boundary conditions using differential quadrature method, Appl. math. model., 22, 251-275, (1998)
[24] Chen, C.N., A derivation and solution of dynamic equilibrium equations of shear undeformable composite anisotropic beams using the DQEM, Appl. math. model., 26, 833-861, (2002) · Zbl 1042.74030
[25] Chen, C.N., Buckling equilibrium equations of arbitrarily loaded nonprismatic composite beams and the DQEM buckling analysis using EDQ, Appl. math. model., 27, 27-46, (2003) · Zbl 1042.74016
[26] Lur’e, A.I., Three-dimensional problems of theory of elasticity, (1964), Interscience Publishers New York · Zbl 0122.19003
[27] Shu, C.; Richards, B.E., Application of generalized differential quadrature to solve two-dimensional incompressible navier – stokes equations, Int. J. num. methods fluids, 15, 791-798, (1992) · Zbl 0762.76085
[28] Bert, C.W.; Malik, M., Differential quadrature method in computational mechanics: a review, Appl. mech. rev., 49, 1-28, (1996)
[29] Sherbourne, A.N.; Pandey, M.D., Differential quadrature method in the buckling analysis of beams and composite plates, Comput. struct., 40, 903-913, (1991) · Zbl 0850.73355
[30] Ding, H.J.; Chen, W.Q.; Xu, R.Q., On the bending, vibration and stability of laminated rectangular plates with transversely isotropic layers, Appl. math. mech., 22, 16-22, (2001)
[31] Ding, H.J.; Chen, W.Q., Nonaxisymmetric free vibrations of a spherically isotropic spherical shell embedded in an elastic medium, Int. J. solids struct., 33, 2575-2590, (1996) · Zbl 0900.73390
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