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Horizontal transient response of a pile group partially embedded in multilayered transversely isotropic soils

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Abstract

This study investigates the horizontal transient response of partially embedded pile groups in multilayered transversely isotropic soils. The dynamic equation of pile groups is derived using the FEM by considering pile–pile interaction. The flexibility matrix is presented by applying the fundamental transient solutions of multilayered transversely isotropic soils. Then, the interaction solution between the piles and soils is obtained by using a FEM–BEM coupled method. The correctness of the present solution is validated by comparing the results with those in existing literature. Numerical examples are given to explore how free-standing length, pile–soil stiffness ratio, pile spacing ratio, soil’s transverse isotropy and stratification affect the horizontal transient response of partially embedded pile groups.

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Abbreviations

\(c\) :

Center distance between adjacent two piles

\(d\), \(r\) :

Diameter and radius of the pile

\(E_{\text{p}}\) :

Elastic modulus of the pile

\(E_{\text{h}}\), \(E_{\text{v}}\) :

Elastic moduli of the soil in the horizontal plane and vertical direction, respectively

\(G_{\text{v}}\) :

Shear modulus of the soil in the vertical direction

\({\mathbf{f}}(t)\) :

Total load vector at the pile node with time

\(f_{0}\) :

Intensity of the horizontal transient loading

\(\{ f\}\), \(\{ f\}^{{\text{ext}}}\) :

Total load vector and external load vector acting at a pile node in the transformed domain, respectively

\(\{ F\}^{m}\), \(\{ F\}^{{\text{ext}}}\) :

External load vectors acting at the \(m\)th pile and the pile group in the transformed domain, respectively

\(h\) :

Thickness of the soil layer

\(i\), \(j\) :

Numbering of the pile

\(J_{1}\) :

Bessel function of the first order

\([K]\), \([K]_{S}^{m}\), \([K]_{GS}\) :

Stiffness matrices for a pile element, the \(m\)th pile, and the pile group, respectively

\([\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{K} ]_{GS}\), \([\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{K} } ]_{GS}\) :

Total stiffness matrices and extended stiffness matrix of a partially embedded pile group, respectively

\(l\) :

Depth of the stratum

\(L\) :

Length of the pile

\({\text{L}}\) :

Symbol of Laplace integral

\(L_{\text{S}}\) :

Length of the pile embedded in the soil

\(L_{\text{F}}\) :

Length of the pile free-standing above the soil surface

\([M]\), \([M]_{S}^{m}\), \([M]_{GS}\) :

Mass matrices for a pile element, the \(m\)th pile and the pile group, respectively

\(M_{h}\) :

Dimensionless moment

\(n_{F}\), \(n_{S}\) :

Number of pile nodes above the soil surface and embedded in the soil, respectively

\(N\) :

Number of piles in a pile group

\(\rho_{\text{p}}\), \(\rho_{\text{s}}\) :

Densities of the pile and the soil, respectively

\(\{ q\}\), \(\{ q\}_{S}^{m}\), \(\{ q\}^{p}\) :

Reaction load vectors acting at a pile node, the \(m\)th pile and the pile group, respectively

\(\{ q\}_{GS}\) :

Load vector acting in the soil

\([Q]\), \([Q]_{S}^{m}\), \([Q]_{GS}\) :

Load transfer matrices for a pile node, the \(m\)th pile and the pile group, respectively

\([R]_{GF}\), \([R]_{S}^{i}\), \([R]_{GS}\) :

Soil flexibility matrices for a fully embedded pile group, a partially embedded single pile and the partially embedded pile group, respectively

\(s\) :

Laplace parameter

\(t\) :

Real time

\([T]_{GS}\), \([\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{T} ]_{GS}\) :

Combined matrix and extended combined matrix with load transfer matrix and soil flexibility matrix

\({\mathbf{u}}(t)\) :

Displacement and rotation vector at the pile node with time

\({\ddot{\mathbf{u}}}(t)\) :

Second derivative of \({\mathbf{u}}(t)\) with time

\(\{ u\}\), \(\{ U\}^{m}\), \(\{ U\}_{GF}\), \(\{ U\}_{GS}\) :

Displacement and rotation vectors in the transformed domain for a pile node, the \(m\)th pile, a fully embedded pile group and a partially embedded pile group, respectively

\(u_{r}\), \(u_{\theta }\), \(U_{ij}\) :

Displacement components and the total displacement of the \(j\)th pile in cylindrical coordinates, respectively

\(U_{0}\), \(U_{2}\) :

Coupled displacement components of the \(j\)th pile in the Laplace–Hankel domain

\(U_{h}\) :

Dimensionless horizontal displacement

\(\mu_{\text{h}}\), \(\mu_{\text{vh}}\) :

Poisson’s ratios of the soil in the horizontal and vertical plane, respectively

\(V_{h}\) :

Dimensionless shear force

\(Z\) :

Depth of the pile

\(\tau\) :

Dimensionless time

\(\xi\) :

Hankel integral parameter

\(\{ \theta \}^{m}\) :

Rotation of the \(m\)th pile in a partially embedded pile group

References

  1. Ai ZY, Li ZX (2015) Dynamic analysis of a laterally loaded pile in a transversely isotropic multilayered half-space. Eng Anal Bound Elem 54:68–75

    Article  MathSciNet  Google Scholar 

  2. Ai ZY, Li ZX (2016) Horizontal vibration of a pile group in transversely isotropic layered soils under souring conditions. Chin J Geotech Eng 38(4):613–618 (in Chinese)

    MathSciNet  Google Scholar 

  3. 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 385:171–183

    Article  Google Scholar 

  4. Ai ZY, Liu CL, Wang LJ, Wang LH (2016) Vertical vibration of a partially embedded pile group in transversely isotropic soils. Comput Geotech 80:107–114

    Article  Google Scholar 

  5. Ai ZY, Li Y, Liu CL (2018) Behavior of a multilayered transversely isotropic half space due to horizontal transient loadings. Comput Geotech 97:217–221

    Article  Google Scholar 

  6. Cairo R, Conte E, Dente G (2005) Analysis of pile groups under vertical harmonic vibration. Comput Geotech 32(7):545–554

    Article  Google Scholar 

  7. Catal HH (2002) Free vibration of partially supported piles with the effects of bending moment, axial and shear force. Eng Struct 24(12):1615–1622

    Article  Google Scholar 

  8. Catal HH (2006) Free vibration of semi-rigid connected and partially embedded piles with the effects of the bending moment, axial and shear force. Eng Struct 28(14):1911–1918

    Article  Google Scholar 

  9. Fan XX, Li Y, Wu WB, Chen YC, Wang XG, Liu H, Liang RZ (2020) Horizontal vibration response of defective large-diameter piles embedded in saturated soils. Chin J Rock Mech Eng 39(2):413–423 (in Chinese)

    Google Scholar 

  10. Heins E, Grabe J (2019) FE-based identification of pile–soil interactions from dynamic load tests to predict the axial bearing capacity. Acta Geotech 14:1821–1841

    Article  Google Scholar 

  11. Henke S, Grabe J (2008) Numerical investigation of soil plugging inside open-ended piles with respect to the installation method. Acta Geotech 3(3):215–223

    Article  Google Scholar 

  12. Hu AF, Fu P, Xia CQ, Xie KH (2016) Lateral dynamic response of a partially embedded pile subjected to combined loads in saturated soil. Mar Georesour Geotechnol 35(6):1–11

    Google Scholar 

  13. Kaynia AM, Kausel E (1982) Dynamic stiffness and seismic response of pile groups. Massachusetts Institute of Technology Technical report, pp 83–103

  14. Kaynia AM, Kausel E (1991) Dynamics of piles and pile groups in layered soil media. Soil Dyn Earthq Eng 10(8):386–401

    Article  Google Scholar 

  15. Kücükarslan S (2008) Transient analysis of piles and pile groups in non-homogeneous soil. Arch Appl Mech 78(1):21–31

    Article  Google Scholar 

  16. Lei ZX, Cheung YK, Tham LG (1993) Vertical response of single piles: transient analysis by time-domain BEM. Soil Dyn Earthq Eng 12(1):37–49

    Article  Google Scholar 

  17. Liu WM, Novak M (1994) Dynamic response of single piles embedded in transversely isotropic layered media. Earthq Eng Struct Dyn 23(11):1239–1257

    Article  Google Scholar 

  18. Luan LB, Ding XM, Cao GW, Deng X (2020) Development of a coupled pile-to-pile interaction model for the dynamic analysis of pile groups subjected to vertical loads. Acta Geotech. https://doi.org/10.1007/s11440-020-00972-2

    Article  Google Scholar 

  19. Luan LB, Zheng CJ, Kouretzis G, Ding XM, Poulos H (2020) A new dynamic interaction factor for the analysis of pile groups subjected to vertical dynamic loads. Acta Geotech. https://doi.org/10.1007/s11440-020-00989-7

    Article  Google Scholar 

  20. Luan LB, Zheng CJ, Kouretzis G, Ding XM (2020) Dynamic analysis of pile groups subjected to horizontal loads considering coupled pile-to-pile interaction. Comput Geotech 117:103276

    Article  Google Scholar 

  21. Meng K, Cui C, Li H (2020) An ontology framework for pile integrity evaluation based on analytical methodology. IEEE Access 8:72158–72168. https://doi.org/10.1109/access.2020.2986229

    Article  Google Scholar 

  22. Millán MA, Domínguez J (2009) Simplified BEM/FEM model for dynamic analysis of structures on piles and pile groups in viscoelastic and poroelastic soils. Eng Anal Bound Elem 33(1):25–34

    Article  MathSciNet  Google Scholar 

  23. Muki R, Sternberg E (1970) Elastostatic load-transfer to a half-space from a partially embedded axially loaded rod. Int J Solids Struct 6(1):69–90

    Article  Google Scholar 

  24. Nogami T, Novak M (1977) Resistance of soil to a horizontally vibrating pile. Earthq Eng Struct Dyn 5(3):247–261

    Article  Google Scholar 

  25. Nogami T, Paulson SK (1985) Transfer matrix approach for nonlinear pile group response analysis. Int J Numer Anal Methods Geomech 9(4):299–316

    Article  Google Scholar 

  26. Novak M, Aboul-Ella F (1978) Impedance functions of piles in layered media. J Eng Mech ASCE 104(3):643–661

    Google Scholar 

  27. Novak M, Nogami T (1977) Soil-pile interaction in horizontal vibration. Earthq Eng Struct Dyn 5(3):263–281

    Article  Google Scholar 

  28. Padrón LA, Aznárez JJ, Maeso O (2007) BEM–FEM coupling model for the dynamic analysis of piles and pile groups. Eng Anal Bound Elem 31(6):473–484

    Article  Google Scholar 

  29. Pak RYS, Jennings PC (1987) Elastodynamic response of pile under transverse excitations. J Eng Mech ASCE 113(7):1101–1116

    Article  Google Scholar 

  30. Rajapakse RKND, Shah AH (1987) On the longitudinal harmonic motion of an elastic bar embedded in an elastic half-space. Int J Solids Struct 23(2):267–285

    Article  Google Scholar 

  31. Rajapakse RKND, Shah AH (1987) On the lateral harmonic motion of an elastic bar embedded in an elastic half-space. Int J Solids Struct 23(2):287–303

    Article  Google Scholar 

  32. Schapery RA (1962) Approximate methods of transform inversion for viscoelastic stress analysis. In: Proceedings of the 4th US national congress on applied mechanics, vol 2, pp 1075–1085

  33. Sneddon IN (1972) The use of integral transforms. McGraw-Hill, New York

    MATH  Google Scholar 

  34. Tajimi H (1969) Dynamic analysis of a structure embedded in an elastic stratum. In: Proceedings of the fourth world conference on earthquake engineering, Santiago, vol 2, pp 53–69

  35. Tham LG, Chu CK, Lei ZX (1996) Analysis of the transient response of vertically loaded single piles by time-domain BEM. Comput Geotech 19(2):117–136

    Article  Google Scholar 

  36. Wu WB, Liu H, Yang XY, Jiang GS, El Naggar M, Mei GX, Liang RZ (2020) New method to calculate apparent phase velocity of open-ended pipe pile. Can Geotech J 57(1):127–138

    Article  Google Scholar 

  37. Zheng T (1997) Dynamic behavior of piles embedded in transversely isotropic layered media. Acta Mech Sin 13(3):241–252

    Article  Google Scholar 

  38. Zhou XL, Wang JH (2009) Analysis of pile groups in a poroelastic medium subjected to horizontal vibration. Comput Geotech 36(3):406–418

    Article  Google Scholar 

  39. Zhu B, Ren PC, Chen YM (2003) Transient response of piles-bridge under horizontal excitation. J Zhejiang Univ Sci A 4(1):28–34

    Article  Google Scholar 

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant No. 41672275). The authors would like to thank the editor Prof. Jian Chu and the reviewers for their valuable comments that contribute to improving the quality of this paper.

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  1. Department of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, College of Civil Engineering, Tongji University, Shanghai, China

    Yang Li & Zhi Yong Ai

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Li, Y., Ai, Z.Y. Horizontal transient response of a pile group partially embedded in multilayered transversely isotropic soils. Acta Geotech. 16, 335–346 (2021). https://doi.org/10.1007/s11440-020-01023-6

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