Analytical Model for Vertical Dynamic Interaction between Circular Raft and Saturated Soils
Publication: International Journal of Geomechanics
Volume 21, Issue 4
Abstract
The raft–soil dynamic interaction problem is of great significance in the fields of marine geotechnical engineering and traffic engineering. This paper develops an analytical model to study the dynamic response of an elastic raft embedded in saturated soil bonded to a rigid base. The model combines the classical theory of elastic thin plate and the analytical layer element method (ALEM). The raft was divided into finite circular elements that are governed by the elastic thin plate theory. Fundamental solutions for the layered saturated soils with compressible constituents were obtained by employing ALEM. By considering the compatible conditions at the interface, the dynamic behavior of the elastic raft was investigated. The present method was verified by comparing with existing solutions. The influences of the vibration frequency, embedded depth of the raft, raft–soil stiffness ratio and the layer property of soil were studied in the numerical analysis. The present model is capable of simulating the dynamic response of the elastic raft resting on the surface or embedded in layered saturated soils, and can also be employed to analyze general raft–soil interaction problems encountered in practical engineering.
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Acknowledgments
The authors acknowledge the support of the Scientific Research Foundation (Zhejiang Sci-Tech University, No. 19052148-Y) and the National Natural Science Foundation of China (Nos. 52078458 and 51908225). The support from Prof. Z.Y. Ai at Tongji University on the development of ALEM is highly appreciated.
Notation
The following symbols are used in this paper:
Definitions for Variables in Numerical Calculation
- a
- radius of raft;
- b
- parameter accounting for the internal friction;
- flexural rigidity of raft;
- E
- elastic modulus of soil;
- Ep
- elastic modulus of raft;
- g
- gravity acceleration;
- H
- thickness of soil;
- h
- embedment of raft;
- ht
- thickness of raft;
- h*
- reinforced depth of soil;
- k
- permeability of soil;
- m
- density-like parameter;
- n
- porosity of soil;
- P
- vertical load applied on the rigid disk;
- vertical displacement for an elastic raft on elastic half-space;
- α, M
- compressible parameters of soil constituents;
- stiffness ratio of raft–soil;
- Δ
- vertical displacement of the rigid disk;
- η0
- viscosity of the pore fluid;
- λ
- Lame's constant of soil;
- μ
- shear modulus of soil;
- μp
- Poisson's ratio of raft;
- ν
- Poisson's ratio of soil;
- ρ
- density of soil;
- ρf
- density of pore fluid;
- ρp
- density of raft;
- ρ0 = ρpht/ρa
- mass ratio of raft–soil; and
- ω
- circular frequency.
Dimensionless Variables in Numerical Calculation
- b
- η0/k;
- ;
- h/a;
- ;
- M/μ;
- m
- ρf/n;
- m/ρ;
- p0/μ;
- w/a;
- ;
- δ
- ;
- η0
- nρfg;
- λ /μ; and
- ρf/ρ.
References
Ai, Z. Y., and C. L. Liu. 2014. “Axisymmetric vibration of an elastic circular plate bonded on a transversely isotropic multilayered half-space.” Soil Dyn. Earthquake Eng. 67: 257–263. https://doi.org/10.1016/j.soildyn.2014.09.006.
Ai, Z. Y., and L. H. Wang. 2017a. “Influences of Biot’s compressible parameters on dynamic response of vertically loaded multilayered poroelastic soils.” Soil Dyn. Earthquake Eng. 94: 7–12. https://doi.org/10.1016/j.soildyn.2016.12.010.
Ai, Z. Y., and L. H. Wang. 2017b. “Effect of compressible parameters on vertical vibration of an elastic pile in multilayered poroelastic media.” Comput. Geotech. 89: 195–202. https://doi.org/10.1016/j.compgeo.2017.05.002.
Ai, Z. Y., and L. J. Wang. 2015. “Axisymmetric thermal consolidation of multilayered porous thermoelastic media due to a heat source.” Int. J. Numer. Anal. Methods Geomech. 39 (17): 1912–1931. https://doi.org/10.1002/nag.2381.
Ai, Z. Y., and L. J. Wang. 2016. “Three-dimensional thermo-hydro-mechanical responses of stratified saturated porothermoelastic material.” Appl. Math. Modell. 40 (21–22): 8912–8933. https://doi.org/10.1016/j.apm.2016.05.034.
Amiri-Hezaveh, A., M. Eskandari-Ghadi, M. Rahimian, and A. K. Ghorbani-Tanha. 2013. “Impedance functions for surface rigid rectangular foundations on transversely isotropic multilayer half-spaces.” J. Appl. Mech. 80 (5): 051017. https://doi.org/10.1115/1.4023626.
Bhaduri, A., and D. Choudhury. 2020. “Serviceability-based finite-element approach on analyzing combined pile–raft foundation.” Int. J. Geomech. 20 (2): 04019178. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001580.
Biot, M. A. 1956. “Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range.” J. Acoust. Soc. Am. 28 (2): 168–178. https://doi.org/10.1121/1.1908239.
Biot, M. A. 1962. “Mechanics of deformation and acoustic propagation in porous media.” J. Appl. Phys. 33 (4): 1482–1498. https://doi.org/10.1063/1.1728759.
Cattoni, E., and C. Tamagnini. 2020. “On the seismic response of a propped r.c. diaphragm wall in a saturated clay.” Acta Geotech. 15 (4): 847–865. https://doi.org/10.1007/s11440-019-00771-4.
Celep, Z., and K. Güer. 2007. “Axisymmetric forced vibrations of an elastic free circular plate on a tensionless two parameter foundation.” J. Sound Vib. 301 (3–5): 495–509. https://doi.org/10.1016/j.jsv.2006.09.029.
Chen, L. 2016. “Dynamic interaction between rigid surface foundations on multi-layered half space.” Int. J. Struct. Stab. Dyn. 16 (5): 1550004. https://doi.org/10.1142/S0219455415500042.
Chen, S. L. 2009. “Vertical vibration of a flexible foundation resting on saturated layered soil half-space.” Int. J. Geomech. 9 (3): 113–121. https://doi.org/10.1061/(ASCE)1532-3641(2009)9:3(113).
Chen, S. L., L. Z. Chen, and J. M. Zhang. 2006. “Dynamic response of a flexible plate on saturated soil layer.” Soil Dyn. Earthquake Eng. 26 (6–7): 637–647. https://doi.org/10.1016/j.soildyn.2006.01.014.
Chen, S. S., and J. G. Hou. 2009. “Modal analysis of circular flexible foundations under vertical vibration.” Soil Dyn. Earthquake Eng. 29 (5): 898–908. https://doi.org/10.1016/j.soildyn.2008.10.004.
Eskandari-Ghadi, M., and A. Ardeshir-Behrestaghi. 2010. “Forced vertical vibration of rigid circular disc buried in an arbitrary depth of a transversely isotropic half space.” Soil Dyn. Earthquake Eng. 30 (7): 547–560. https://doi.org/10.1016/j.soildyn.2010.01.011.
Eskandari-Ghadi, M., M. Fallahi, and A. Ardeshir-Behrestaghi. 2010. “Forced vertical vibration of rigid circular disc on a transversely isotropic half-space.” J. Eng. Mech. 136 (7): 913–922. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000114.
Eskandari-Ghadi, M., S. Masoud Nabizadeh, and A. Ardeshir-Behrestaghi. 2014. “Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space.” Soil Dyn. Earthquake Eng. 61–62 (2): 135–139. https://doi.org/10.1016/j.soildyn.2014.01.022.
Gucunski, N., and R. Peek. 1993. “Vertical vibrations of circular flexible foundations on layered media.” Soil Dyn. Earthquake Eng. 12 (3): 183–192. https://doi.org/10.1016/0267-7261(93)90045-S.
Iguchi, M., and J. E. Luco. 1981. “Dynamic response of flexible rectangular foundations on an elastic half-space.” Earthquake Eng. Struct. Dyn. 9 (3): 239–249. https://doi.org/10.1002/eqe.4290090305.
Jin, B. 1999. “The vertical vibration of an elastic circular plate on a fluid-saturated porous half space.” Int. J. Eng. Sci. 37 (3): 379–393. https://doi.org/10.1016/S0020-7225(98)00073-1.
Lamb, H. 1904. “On the propagation of tremors over the surface of an elastic solid.” Philos. Trans. R. Soc. London, Ser. A 203: 359–371. https://doi.org/10.1098/rsta.1904.0013.
Lin, G., Z. J. Han, and J. B. Li. 2013. “An efficient approach for dynamic impedance of surface footing on layered half-space.” Soil Dyn. Earthquake Eng. 49 (8): 39–51. https://doi.org/10.1016/j.soildyn.2013.01.008.
Lin, Y. J. 1978. “Dynamic response of circular plates resting on viscoelastic half space.” J. Appl. Mech. 45 (2): 379–384. https://doi.org/10.1115/1.3424306.
Lu, J. F., and A. Hanyga. 2005. “Fundamental solution for a layered porous half space subject to a vertical point force or a point fluid source.” Comput. Mech. 35 (5): 376–391. https://doi.org/10.1007/s00466-004-0626-5.
Luco, J. E. 1974. “Impedance functions for a rigid foundation on a layered medium.” Nucl. Eng. Des. 31 (2): 204–217. https://doi.org/10.1016/0029-5493(75)90142-9.
Masın, D. 2005. “A hypoplastic costitutive model for clays.” Int. J. Numer. Anal. Methods Geomech. 29 (4): 311–336. https://doi.org/10.1002/nag.416.
Masın, D., C. Tamagnini, G. Viggiani, and D. Costanzo. 2006. “Directional response of a reconstituted fine-grained soil. Part II: Performance of different constitutive models.” Int. J. Numer. Anal. Methods Geomech. 30 (13): 1303–1336. https://doi.org/10.1002/nag.527.
Miriano, C., E. Cattoni, and C. Tamagnini. 2016. “Advanced numerical modeling of seismic response of a propped r.c. diaphragm wall.” Acta Geotech. 11 (1): 161–175. https://doi.org/10.1007/s11440-015-0378-8.
Pak, R. Y. S., and A. T. Gobert. 1991. “Forced vertical vibration of rigid discs with arbitrary embedment.” J. Eng. Mech. 117 (11): 2527–2548. https://doi.org/10.1061/(ASCE)0733-9399(1991)117:11(2527).
Pan, E. 1999. “Green’s functions in layered poroelastic half-spaces.” Int. J. Numer. Anal. Methods Geomech. 23 (13): 1631–1653. https://doi.org/10.1002/(SICI)1096-9853(199911)23:13%3C1631::AID-NAG60%3E3.0.CO;2-Q.
Pietruszczak, S., and G. Pande. 1996. “Constitutive relations for partially saturated soils containing gas inclusions.” J. Geotech. Eng. 122 (1): 50–59. https://doi.org/10.1061/(ASCE)0733-9410(1996)122:1(50).
Quinlan, P. M. 1954. “The elastic theory of soil dynamics.” Symp. Dyn. Test. Soils 156: 3–34.
Rahimian, M., A. K. Ghorbani-Tanha, and M. Eskandari-Ghadi. 2006. “The reissner-sagoci problem for a transversely isotropic half-space.” Int. J. Numer. Anal. Methods Geomech. 30 (11): 1063–1074. https://doi.org/10.1002/nag.512.
Rajapakse, R. K. N. D. 1989. “Dynamic response of elastic plates on viscoelastic half space.” J. Eng. Mech. 115 (9): 1867–1881. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:9(1867).
Reissner, E. 1936. “Stationäre, axialsymmetrische, durch eine schüttelnde masse erregte schwingungen eines homogenen elastischen halbraumes.” Ingenieur-Archiv 7 (6): 381–396. https://doi.org/10.1007/BF02090427.
Robertson, I. A. 1966. “Forced vertical vibration of a rigid circular disc on a semi-infinite elastic solid.” Math. Proc. Cambridge Philos. Soc. 62 (3): 547–553. https://doi.org/10.1017/S0305004100040184.
Roy, J., A. Kumar, and D. Choudhury. 2018. “Natural frequencies of piled raft foundation including superstructure effect.” Soil Dyn. Earthquake Eng. 112: 69–75. https://doi.org/10.1016/j.soildyn.2018.04.048.
Schmidt, H., and S. Krenk. 1982. “Asymmetric vibrations of a circular elastic plate on an elastic half space.” Int. J. Solids Struct. 18 (2): 91–105. https://doi.org/10.1016/0020-7683(82)90019-1.
Senjuntichai, T., and Y. Sapsathiarn. 2003. “Forced vertical vibration of circular plate in multilayered poroelastic medium.” J. Eng. Mech. 129 (11): 1330–1341. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:11(1330).
Sneddon, I. N. 1972. The use of integral transform. New York: McGraw-Hill.
Sung, T. Y. 1954. “Vibrations in semi-infinite solids due to periodic surface loading.” Symp. Dyn. Test. Soils 156: 35–68.
Thomson, W. T., and T. Kobori. 1963. “Dynamical compliance of rectangular foundations on an elastic half-space.” J. Appl. Mech. 30 (4): 579–584. https://doi.org/10.1115/1.3636622.
Wang, L. H., L. J. Wang, and C. L. Liu. 2020. “Vertical dynamic analysis of a rigid disk embedded in layered saturated soils with compressible fluid.” Comput. Geotech. 119: 103347. https://doi.org/10.1016/j.compgeo.2019.103347.
Wang, L. J., and L. H. Wang. 2020. “Semianalytical analysis of creep and thermal consolidation behaviors in layered saturated clays.” Int. J. Geomech. 20 (4): 06020001. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001615.
Wong, H. L., and J. E. Luco. 1976. “Dynamic response of rigid foundations of arbitrary shape.” Earthquake Eng. Struct. Dyn. 4 (6): 579–587. https://doi.org/10.1002/eqe.4290040606.
Yu, L., H. S. Shen, and X. P. Huo. 2007. “Dynamic responses of Reissner–Mindlin plates with free edges resting on tensionless elastic foundations.” J. Sound Vib. 299 (1–2): 212–228. https://doi.org/10.1016/j.jsv.2006.07.015.
Zeng, X., and R. K. N. D. Rajapakse. 1999. “Vertical vibrations of a rigid disk embedded in a poroelastic medium.” Int. J. Numer. Anal. Methods Geomech. 23 (15): 2075–2095. https://doi.org/10.1002/(SICI)1096-9853(19991225)23:15%3C2075::AID-NAG50%3E3.0.CO;2-P.
Zheng, P., B. Ding, S. X. Zhao, and D. Ding. 2013a. “Dynamic response of a multilayered poroelastic half-space to harmonic surface tractions.” Transp. Porous Media 99 (2): 229–249. https://doi.org/10.1007/s11242-013-0182-6.
Zheng, P., B. Ding, S. X. Zhao, and D. Ding. 2013b. “3D dynamic Green′s functions in a multilayered poroelastic half-space.” Appl. Math. Modell. 37 (24): 10203–10219. https://doi.org/10.1016/j.apm.2013.05.041.
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Published In
International Journal of Geomechanics
Volume 21 • Issue 4 • April 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Mar 6, 2020
Accepted: Oct 27, 2020
Published online: Jan 29, 2021
Published in print: Apr 1, 2021
Discussion open until: Jun 29, 2021
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