Journal of Applied Mathematics

Soil-Pile Interaction in the Pile Vertical Vibration Based on Fictitious Soil-Pile Model

Guodong Deng, Jiasheng Zhang, Wenbing Wu, Xiong Shi, and Fei Meng

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By introducing the fictitious soil-pile model, the soil-pile interaction in the pile vertical vibration is investigated. Firstly, assuming the surrounding soil of pile to be viscoelastic material and considering its vertical wave effect, the governing equations of soil-pile system subjected to arbitrary harmonic dynamic force are founded based on the Euler-Bernoulli rod theory. Secondly, the analytical solution of velocity response in frequency domain and its corresponding semianalytical solution of velocity response in time domain are derived by means of Laplace transform technique and separation of variables technique. Based on the obtained solutions, the influence of parameters of pile end soil on the dynamic response is studied in detail for different designing parameters of pile. Lastly, the fictitious soil-pile model and other pile end soil supporting models are compared. It is shown that the dynamic response obtained by the fictitious soil-pile model is among the dynamic responses obtained by other existing models if there are appropriate material parameters and thickness of pile end soil for the fictitious soil-pile model.

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J. Appl. Math., Volume 2014 (2014), Article ID 905194, 11 pages.

First available in Project Euclid: 2 March 2015

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Deng, Guodong; Zhang, Jiasheng; Wu, Wenbing; Shi, Xiong; Meng, Fei. Soil-Pile Interaction in the Pile Vertical Vibration Based on Fictitious Soil-Pile Model. J. Appl. Math. 2014 (2014), Article ID 905194, 11 pages. doi:10.1155/2014/905194.

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  • M. Novak and Y. O. Beredugo, “Vertical vibration of embedded footings,” Journal of the Soil Mechanics and Foundations Division ASCE, vol. 98, no. SM12, pp. 1291–1131, 1972.
  • T. Nogami and K. Konagai, “Time domain axial response of dynamically loaded single piles,” Journal of Engineering Mechanics ASCE, vol. 112, no. 11, pp. 1241–1252, 1986.
  • K. H. Wang, K. H. Xie, and G. X. Zeng, “Analytical solution to vibration of finite length pile under exciting force and its application,” Chinese Journal of Geotechnical Engineering, vol. 19, no. 6, pp. 27–35, 1997 (Chinese).
  • D. S. Kong, M. T. Luan, and Q. Yang, “Review of dynamic Winkler model applied in pile-soil interaction analyses,” World Information on Earthquake Engineering, vol. 21, no. 1, pp. 12–17, 2005 (Chinese).
  • D. J. Liu, “Longitudinal waves in piles with exponentially varying cross sections,” Chinese Journal of Geotechnical Engineering, vol. 30, no. 7, pp. 1066–1071, 2008 (Chinese).
  • K. H. Wang, W. B. Wu, Z. Q. Zhang, and C. J. Leo, “Vertical dynamic response of an inhomogeneous viscoelastic pile,” Computers and Geotechnics, vol. 37, no. 4, pp. 536–544, 2010.
  • T. Nogami and M. Novak, “Soil-pile interaction in vertical vibration,” Earthquake Engineering and Structural Dynamics, vol. 4, no. 3, pp. 277–293, 1976.
  • A. S. Veletsos and K. W. Dotson, “Vertical and torsional vibration of foundations in inhomogeneous media,” Journal of Geotechnical Engineering Division ASCE, vol. 114, no. 9, pp. 1002–1021, 1988.
  • G. Militano and R. K. N. D. Rajapakse, “Dynamic response of a pile in a multi-layered soil to transient torsional and axial loading,” Geotechnique, vol. 49, no. 1, pp. 91–109, 1999.
  • H. D. Wang and S. P. Shang, “Research on vertical dynamic response of single-pile in radially inhomogeneous soil during the passage of Rayleigh waves,” Journal of Vibration Engineering, vol. 19, no. 2, pp. 258–264, 2006 (Chinese).
  • K. H. Wang, D. Y. Yang, and Z. Q. Zhang, “Study on dynamic response of pile based on complex stiffness transfer model of radial multizone plane strain,” Chinese Journal of Rock Mechanics and Engineering, vol. 27, no. 4, pp. 825–831, 2008 (Chinese).
  • R. K. N. D. Rajapakse, Y. Chen, and T. Senjuntichai, “Electroelastic field of a piezoelectric annular finite cylinder,” International Journal of Solids and Structures, vol. 42, no. 11-12, pp. 3487–3508, 2005.
  • T. Senjuntichai, S. Mani, and R. K. N. D. Rajapakse, “Vertical vibration of an embedded rigid foundation in a poroelastic soil,” Soil Dynamics and Earthquake Engineering, vol. 26, no. 6-7, pp. 626–636, 2006.
  • K. H. Wang, Z. Q. Zhang, C. J. Leo, and K. H. Xie, “Dynamic torsional response of an end bearing pile in transversely isotropic saturated soil,” Journal of Sound and Vibration, vol. 327, no. 3–5, pp. 440–453, 2009.
  • C. B. Hu and X. M. Huang, “A quasi-analytical solution to soil-pile interaction in longitudinal vibration in layered soils considering vertical wave effect on soils,” Journal of Earthquake Engineering and Engineering Vibration, vol. 26, no. 4, pp. 205–211, 2006 (Chinese).
  • L. C. Liu, Q. F. Yan, and X. Yang, “Vertical vibration of single pile in soil described by fractional derivative viscoelastic model,” Engineering Mechanics, vol. 28, no. 8, pp. 177–182, 2011 (Chinese).
  • J. Lysmer and F. E. Richart, “Dynamic response of footing to vertical load,” Journal of the Soil Mechanics and Foundations Division ASCE, vol. 2, no. 1, pp. 65–91, 1966.
  • R. Y. Liang and A. I. Husein, “Simplified dynamic method for pile-driving control,” Journal of Geotechnical and Geoenvironmental Engineering ASCE, vol. 119, no. 4, pp. 694–713, 1993.
  • G. G. Meyerhof, “Bearing capacity and settlement of pile foundations,” Journal of Geotechnical Engineering Division ASCE, vol. 102, no. GT3, pp. 195–228, 1976.
  • D. Y. Yang and K. H. Wang, “Vertical vibration of pile based on fictitious soil-pile model in inhomogeneous soil,” Journal of Zhejiang University (Engineering Science), vol. 44, no. 10, pp. 2021–2028, 2010 (Chinese).
  • W. B. Wu, K. H. Wang, D. Y. Yang, S. J. Ma, and B. N. Ma, “Longitudinal dynamic response to the pile embedded in layered soil based on fictitious soil pile model,” China Journal of Highway and Transport, vol. 25, no. 2, pp. 72–80, 2012 (Chinese).
  • W. B. Wu, K. H. Wang, Z. Q. Zhang, and J. Z. Chen, “Accuracy and application of virtual soil pile model in half space foundation,” Journal of Basic Science and Engineering, vol. 20, no. 1, pp. 121–129, 2012 (Chinese). \endinput