Acta Geotechnica

, Volume 14, Issue 6, pp 1857–1870 | Cite as

Study on longitudinal vibration of a pile with variable sectional acoustic impedance by integral transformation

  • Jun-tao Wu
  • Kui-hua WangEmail author
  • Liu Gao
  • Si Xiao
Research Paper


A study on longitudinal vibration of a pile with variable sectional acoustic impedance subjected to arbitrary form of external excitation is conducted. The analytical solution of transfer function is primarily solved by Laplace transform, and then its corresponding impulse response function is solved by residue method of inverse Laplace transformation. With impulse response function, the analytical solution of response at pile top can be obtained by convolution calculation, which overcomes the limit of previous analytical solutions subjected to prescribed time-harmonic load. The accuracy of presented solution is verified by compared with existing solutions subjected to semi-sine excitation and low-strain test of model piles. To take advantage of presented solution, this paper also proposes a case study of imposed excitation considering superimposed high-frequency interference component, and some conclusions derived from this case can provide practical guidance for engineering implementation.


Integral transformation Laplace transform Model pile Pile defect Time domain Variable sectional acoustic impedance 



  1. 1.
    Ai ZY, Li ZX, Wang LH (2016) Dynamic response of a laterally loaded fixed-head pile group in a transversely isotropic multilayered half-space. J Sound Vib 385:171–183CrossRefGoogle Scholar
  2. 2.
    Baranov VA (1967) On the calculation of excited vibrations of an embedded foundation. Voprosy Dynamiki Prochnocti 14:195–209Google Scholar
  3. 3.
    Boulanger RW, Curras CJ, Kutter BL, Wilson DW (1999) Seismic soil–pile–structure interaction experiments and analyses. J Geotech Geoenviron Eng 125(9):750–759CrossRefGoogle Scholar
  4. 4.
    El Naggar MH (2000) Vertical and torsional soil reactions for radially inhomogeneous soil layer. Struct Eng Mech 10(4):299–312CrossRefGoogle Scholar
  5. 5.
    Gao L, Wang K, Wu J et al (2017) Analytical solution for the dynamic response of a pile with a variable-section interface in low-strain integrity testing. J Sound Vib 395:328–340CrossRefGoogle Scholar
  6. 6.
    Gao L, Wang K, Xiao S et al (2016) An analytical solution for excited pile vibrations with variable section impedance in the time domain and its engineering application. Comput Geotech 73(2):170–178CrossRefGoogle Scholar
  7. 7.
    Grabe J, Heins E (2017) Coupled deformation–seepage analysis of dynamic capacity tests on open-ended piles in saturated sand. Acta Geotech 12(1):211–223CrossRefGoogle Scholar
  8. 8.
    Jardine RJ, Thomson NV, Mygind M, Liingaard MA, Thilsted CL (2015) Axial capacity design practice for North European wind-turbine projects. In: Meyer V (ed) Proceedings of 3rd international symposium on frontiers in Offshore Geotechnics 2015 in Oslo (Norway), vol 2. Taylor & Francis Group, London, pp 581–586Google Scholar
  9. 9.
    Kolymbas D (1991) Longitudinal impacts on piles. Soil Dyn Earthq Eng 10(5):264–270CrossRefGoogle Scholar
  10. 10.
    Koten HV, Middendorp P, Brederode PV (1980) An analysis of dissipative wave Propagation in a Pile. In: International seminar on the application of stress wave theory on pilesGoogle Scholar
  11. 11.
    Kuhlemeyer RL (1979) Vertical vibration of piles. J Geotech Geoenviron Eng 105:14393Google Scholar
  12. 12.
    Kuhlemeyer RL (1979) Static and dynamic laterally loaded floating piles. J Geotech Geoenviron Eng 106(2):289–304Google Scholar
  13. 13.
    Kuo KA, Hunt HEM (2013) An efficient model for the dynamic behaviour of a single pile in viscoelastic soil. J Sound Vib 332(10):2549–2561CrossRefGoogle Scholar
  14. 14.
    Liu C, Soltani H, Muraleetharan KK et al (2016) Cyclic and seismic response of single piles in improved and unimproved soft clays. Acta Geotech 11(6):1431–1444CrossRefGoogle Scholar
  15. 15.
    Militano G, Rajapakse R (1999) Dynamic response of a pile in a multi-layered soil to transient torsional and axial loading. Geotechnique 49(1):91–109CrossRefGoogle Scholar
  16. 16.
    Novak M, Aboul-Ella F (1978) Impedance functions of piles in layered media. J Eng Mech Div 104(3):643–661Google Scholar
  17. 17.
    Novak M, Sachs K (1973) Torsional and coupled vibrations of embedded footings. Earthq Eng Struct Dyn 2(1):11–33CrossRefGoogle Scholar
  18. 18.
    Novák M (1977) Vertical vibration of floating piles. J Eng Mech Division 103(1):153–168Google Scholar
  19. 19.
    Paikowsky S, Chernauskaus LR (2008) Dynamic analysis of open ended pipe piles. In: Proceedings of 8th international conference on the application of the stress wave theory to piles, Lisbon, Portugal, pp 59–76Google Scholar
  20. 20.
    Shi L, Xu C, Cai Y et al (2014) Dynamic impedances and free-field vibration analysis of pile groups in saturated ground. J Sound Vib 333(16):3709–3731CrossRefGoogle Scholar
  21. 21.
    Taylor RN (ed) (1995) Geotechnical centrifuge technology. Chapman & Hall, GlasgowGoogle Scholar
  22. 22.
    Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill Book Company, New YorkzbMATHGoogle Scholar
  23. 23.
    Wang K, Wenbing W, Zhang Z, Leo CJ (2010) Vertical dynamic response of an inhomogeneous viscoelastic pile. Comput Geotech 37(4):536–544CrossRefGoogle Scholar
  24. 24.
    Wang K, Zhang Z, Leo CJ et al (2009) Dynamic torsional response of an end bearing pile in transversely isotropic saturated soil. J Sound Vib 327(3):440–453CrossRefGoogle Scholar
  25. 25.
    Wu WB, Wang KH, Zhang ZQ, Leo CJ (2013) Soil-pile interaction in the pile vertical vibration considering true three-dimensional wave effect of soil. Int J Numer Anal Meth Geomech 37(17):2860–2876CrossRefGoogle Scholar
  26. 26.
    Wu W, Jiang G, Huang S et al (2014) Vertical dynamic response of pile embedded in layered transversely isotropic soil. Math Probl EngGoogle Scholar
  27. 27.
    Zheng C, Kouretzis G, Ding X, Liu H, Poulos H (2016) Three-dimensional effects in low strain integrity testing of piles: analytical solution. Can Geotech J 53(2):225–235CrossRefGoogle Scholar
  28. 28.
    Zheng C, Liu H, Ding X et al (2016) Non-axisymmetric response of piles in low-strain integrity testing. Géotechnique 67(2):181–186CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.MOE Key Laboratory of Soft Soils and Geoenvironmental EngineeringZhejiang UniversityHangzhouChina
  2. 2.Research Center of Coastal Urban Geotechnical EngineeringZhejiang UniversityHangzhouChina

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