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Vertical Dynamic Analysis of Rigid Strip Foundation on Layered Unsaturated Media

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Abstract

This paper analytically investigates the vertical dynamic response of a rigid strip foundation on layered unsaturated media. Using the triphasic Biot-type model and extended precise integration method, we derive the flexibility coefficient for layered unsaturated media. On this basis, by introducing the Bessel function series of the first kind, the dual integral equations of the mixed boundary value problem in this study are transformed into a set of linear equations. Finally, we obtain explicit expressions for the contact stress and vertical compliance, which are used to evaluate the soil-structure interaction. After the proposed solution is verified, several parameters are presented to study the impacts of the stratification, soil thickness, saturation degree, air-entry value and dimensionless frequency.

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References

  1. Ai, Z.Y., Ye, Z.K.: Dynamic analysis of eccentrically loaded rigid strip foundation on layered transversely isotropic poroelastic media. Appl. Math. Model. 109, 473–491 (2022)

    Article  MathSciNet  Google Scholar 

  2. He, R.: Dynamic vertical impedance of a submarine strip foundation in ocean engineering: water wave pressure effect. Ocean Eng. 139, 116–126 (2017)

    Article  Google Scholar 

  3. Ba, Z.N., Liang, J.W., Lee, V.W., Kang, Z.Q.: Dynamic impedance functions for a rigid strip footing resting on a multi-layered transversely isotropic saturated half-space. Eng. Anal. Bound. Elem. 86, 31–44 (2018)

    Article  MathSciNet  Google Scholar 

  4. Reissner, E.: Stationäre, axialsymmetrische, durch eine schüttelnde Masse erregte Schwingungen eines homogenen elastischen Halbraumes. Ing.-Arch. 7, 381–396 (1936)

    Article  Google Scholar 

  5. Keer, L.M.: Mixed boundary-value problems for an elastic half-space. Math. Proc. Camb. Philos. Soc. 63, 1379–1386 (1967)

    Article  Google Scholar 

  6. Selvaduari, A.P.S.: The rotation of a rigid elliptical disc inclusion embedded in a transversely isotropic elastic solid. Mech. Res. Commun. 11(1), 41–48 (1984)

    Article  Google Scholar 

  7. Selvaduari, A.P.S.: Fundamental results concerning the settlement of a rigid foundation on an elastic medium due to an adjacent surface load. Int. J. Numer. Anal. Methods Geomech. 7(2), 209–223 (1983)

    Article  Google Scholar 

  8. Selvaduari, A.P.S.: Rotary oscillations of a rigid disc inclusion embedded in an isotropic elastic infinite space. Int. J. Solids Struct. 17(5), 493–498 (1984)

    Article  Google Scholar 

  9. Selvaduari, A.P.S.: The settlement of a rigid circular foundation resting on a half-space exhibiting a near surface elastic non-homogeneity. Int. J. Numer. Anal. Methods Geomech. 20(5), 351–364 (1996)

    Article  Google Scholar 

  10. Selvaduari, A.P.S.: Boussinesq’s problem for an elastic half-space reinforced with a rigid disk inclusion. Math. Mech. Solids 5(4), 483–499 (2000)

    Article  MathSciNet  Google Scholar 

  11. Eskandari-Ghadi, M., Nabizadeh, S.M., Ardeshir-Behrestaghi, A.: Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space. Soil Dyn. Earthq. Eng. 61–62, 135–139 (2014)

    Article  Google Scholar 

  12. Ai, Z.Y., Zhang, Y.F.: Vertical vibration of a rigid strip footing on a transversely isotropic multilayered half-plane. Appl. Math. Model. 40(23–24), 10521–10532 (2016)

    Article  MathSciNet  Google Scholar 

  13. Ai, Z.Y., Liu, C.L., Jiang, J.P.: Dynamic analysis of a vertically loaded rigid disc in a transversely isotropic multilayered half-space. Meccanica 51(8), 1887–1895 (2016)

    Article  MathSciNet  Google Scholar 

  14. Pan, E.N., Liu, H., Zhang, Z.Q.: Vertical and torsional vibrations of a rigid circular disc on a transversely isotropic and layered half-space with imperfect interfaces. Soil Dyn. Earthq. Eng. 113, 442–453 (2018)

    Article  Google Scholar 

  15. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am. 28, 168–178 (1956)

    Article  MathSciNet  Google Scholar 

  16. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher-frequency range. J. Acoust. Soc. Am. 28(2), 179–191 (1956)

    Article  MathSciNet  Google Scholar 

  17. Yue, Z.Q., Selvaduari, A.P.S.: On the mechanics of a rigid disc inclusion embedded in a fluid-saturated poroelastic medium. Int. J. Eng. Sci. 33(11), 1633–1662 (1995)

    Article  Google Scholar 

  18. Keawsawasvong, S., Senjuntichai, T., Plangmal, R., Kaewjuea, W.: Rocking vibrations of rigid foundations on multi-layered poroelastic media. Mar. Georesour. Geotechnol. 38(4), 480–492 (2020)

    Article  Google Scholar 

  19. Zhang, Z.Q., Pan, E.N.: Coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic and layered poroelastic half-space. Appl. Math. Model. 114, 270–290 (2023)

    Article  MathSciNet  Google Scholar 

  20. Senjuntichai, T., Phulsawat, B., Keawsawasvong, S., Kaewjuea, W.: Dynamic impedances of multiple strips on multi-layered transversely isotropic poroelastic soils. Forces Mech. 14, 100260 (2024)

    Article  Google Scholar 

  21. Knight, R., Dvorkin, J., Nur, A.: Acoustic signatures of partial saturation. Geophysics 63(1), 132–138 (1998)

    Article  Google Scholar 

  22. Wang, Y., Gao, G.Y., Yang, J., Song, J.: The influence of the degree of saturation on dynamic response of a cylindrical lined cavity in a nearly saturated medium. Soil Dyn. Earthq. Eng. 71, 27–30 (2015)

    Article  Google Scholar 

  23. Ye, Z., Ai, Z.Y.: Poroelastodynamic response of layered unsaturated media in the vicinity of a moving harmonic load. Comput. Geotech. 138, 104358 (2021)

    Article  Google Scholar 

  24. Degrande, G., De Roeck, G., van den Broeck, P., Smeulders, D.: Wave propagation in layered dry, saturated and unsaturated poroelastic media. Int. J. Solids Struct. 35(34–35), 4753–4778 (1998)

    Article  Google Scholar 

  25. Yang, J.: Rayleigh surface waves in an idealised partially saturated soil. Geotechnique 55(5), 409–414 (2005)

    Article  Google Scholar 

  26. Vardoulakis, I.: Dynamic behavior of nearly saturated porous media. Mech. Mater. 5(1), 87–108 (1986)

    Article  Google Scholar 

  27. Lo, W.C., Sposito, G.: Acoustic waves in unsaturated soils. Water Resour. Res. 49(9), 5674–5684 (2013)

    Article  Google Scholar 

  28. Santos, J.E., Corberó, J.M., Douglas, J. Jr: Static and dynamic behavior of a porous solid saturated by a two-phase fluid. J. Acoust. Soc. Am. 87, 1428–1438 (1990)

    Article  Google Scholar 

  29. Steeb, H., Kurzeja, P.S., Schmalholz, S.M.: Wave propagation in unsaturated porous media. Acta Mech. 225(8), 2435–2448 (2014)

    Article  MathSciNet  Google Scholar 

  30. Xu, M.J., Wei, D.M.: 3D non-axisymmetrical dynamic response of unsaturated soils. Eng. Mech. 28(3), 78–85 (2011)

    Google Scholar 

  31. Lu, Z., Fang, R., Yao, H.L., Dong, C., Xian, S.H.: Dynamic responses of unsaturated half-space soil to a moving harmonic rectangular load. Int. J. Numer. Anal. Methods Geomech. 42(9), 1057–1077 (2018)

    Article  Google Scholar 

  32. Zou, H.Y., Xiong, F., Lu, Y., Ge, Q.: Vertical vibration of a rigid strip footing on a half-space of unsaturated soil. Comput. Geotech. 148, 104840 (2022)

    Article  Google Scholar 

  33. Ye, Z., Su, R.B., Ai, Z.Y., Chen, Y.H., Li, Y., Xue, D.W.: Vertical dynamic analysis of a rigid disc in a layered transversely isotropic unsaturated soil. Comput. Geotech. 171, 106342 (2024)

    Article  Google Scholar 

  34. Elorduy, J., Nieto, J.A., Szekely, E.M.: Dynamic response of bases of arbitrary shape subjected to periodic vertical loading. In: Proceedings of International Symposium on Wave Propagation and Dynamic Properties of Earth Materials, pp. 105–123. University of New Mexico, Albuquerque (1967)

    Google Scholar 

  35. Hryniewicz, Z.: Dynamic-response of a rigid strip on an elastic half-space. Comput. Methods Appl. Mech. Eng. 25(3), 355–364 (1981)

    Article  Google Scholar 

  36. Eskandari-Ghadi, M., Mehdizadeh, D., Morshedifard, A., Rahimian, M.: A family of exponentially-gradient elements for numerical computation of singular boundary value problems. Eng. Anal. Bound. Elem. 80, 184–198 (2017)

    Article  MathSciNet  Google Scholar 

  37. Luco, J.E., Westmann, R.A.: Dynamic response of a rigid footing bonded to an elastic half space. J. Appl. Mech. 39(2), 527–534 (1972)

    Article  Google Scholar 

  38. Bishop, A.: The principle of effective stress. Tekn. Ukebl. 39, 859–863 (1955)

    Google Scholar 

  39. Lu, N., Godt, J.W., Wu, D.T.: A closed-form equation for effective stress in unsaturated soil. Water Resour. Res. 46, W05515 (2010)

    Article  Google Scholar 

  40. Lu, N., Likos, W.J.: Unsaturated Soil Mechanics. Wiley, New Jersey (2004)

    Google Scholar 

  41. Lo, W.C., Sposito, G., Majer, E.: Wave propagation through elastic porous media containing two immiscible fluids. Water Resour. Res. 41(2), W02025 (2005)

    Article  Google Scholar 

  42. Tuncay, K., Corapcioglu, M.Y.: Wave propagation in poroelastic media saturated by two fluids. J. Appl. Mech. 64(2), 313–320 (1997)

    Article  Google Scholar 

  43. Xu, M.J., Wei, D.M.: 3D non-axisymmetrical dynamic response of unsaturated soils. Eng. Mech. 28(03), 78–85 (2011)

    Google Scholar 

  44. Fredlund, D.G.: Soil mechanics for unsaturated soils. Wiley. 286(321) (1993)

  45. Van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980)

    Article  Google Scholar 

  46. Zhong, W.X., Lin, J.H., Gao, Q.: The precise computation for wave propagation in stratified materials. Int. J. Numer. Methods Eng. 60(1), 11–25 (2004)

    Article  Google Scholar 

  47. Ye, Z., Ai, Z.Y.: Elastodynamic analyses of transversely isotropic unsaturated subgrade pavement system under moving loads. Int. J. Numer. Anal. Methods Geomech. 46(11), 2138–2162 (2022)

    Article  Google Scholar 

  48. Wang, P., Cai, Y.Q., Xu, C.J., Sun, H.L., Wang, L.Z.: Rocking vibrations of a rigid circular foundation on poroelastic half-space to elastic waves. Soil Dyn. Earthq. Eng. 31(4), 708–715 (2011)

    Article  Google Scholar 

  49. Jin, B., Liu, H.: Horizontal vibrations of a disk on a poroelastic half-space. Soil Dyn. Earthq. Eng. 19(4), 269–275 (2000)

    Article  Google Scholar 

  50. Japon, B.R., Gallego, R., Dominguez, J.: Dynamic stiffness of foundations on saturated poroelastic soils. J. Eng. Mech. 123(11), 1121–1129 (1997)

    Article  Google Scholar 

  51. Teh, K.L., Leung, C.F., Chow, Y.K., Cassidy, M.J.: Centrifuge model study of spudcan penetration in sand overlying clay. Geotechnique 60(11), 825–842 (2010)

    Article  Google Scholar 

  52. Xie, Y., Leung, C.F., Chow, Y.K.: Centrifuge modelling of spudcan-pile interaction in soft clay overlying sand. Geotechnique 67(1), 69–77 (2017)

    Article  Google Scholar 

  53. Gazetas, G.: Analysis of machine foundation vibrations: state of the art. Int. J. Soil Dyn. Earthqu. Eng. 2(1), 2–42 (1983)

    Google Scholar 

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant No. 41672275).

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Authors and Affiliations

  1. Department of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, 1239 Siping Road, Shanghai, 200092, P.R. China

    Zhi Yong Ai, Li Wei Shi & Lei Sheng

  2. Department of Civil and Environmental Engineering, National University of Singapore, Singapore, 117576, Singapore

    Lei Sheng

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Contributions

Zhi Yong Ai: Conceptualization, Methodology, Investigation, Writing- Reviewing and Editing. Li Wei Shi: Data curation, Validation, Investigation, Writing- Original draft preparation. Lei Sheng: Investigation, Writing- Reviewing and Editing. All authors reviewed the manuscript.

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Correspondence to Zhi Yong Ai.

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Appendix

Appendix

The detailed expressions of \(M_{11} - M_{24}\) are as follows:

$$\begin{aligned} M_{11} &= \frac{\beta \chi - nS_{r}}{K_{s}} + \frac{nS_{r}}{K_{w}},\qquad M_{12} = \frac{\beta \left ( 1 - \chi \right ) - n\left ( 1 - S_{r} \right )}{K_{s}} + \frac{n\left ( 1 - S_{r} \right )}{K_{a}},\qquad \\ M_{13} &= 1 - n - \frac{K_{b}}{K_{s}},\qquad \\ M_{14} &= nS_{r},\qquad n\left ( 1 - S_{r}\right ). \\ M_{21} &= \frac{ - S_{r}\left ( 1 - S_{r} \right )}{K_{w}} + M_{n},\qquad M_{22} = \frac{S_{r}\left ( 1 - S_{r} \right )}{K_{a}} - M_{n},\qquad \\ M_{23} &= - S_{r}\left ( 1 - S_{r} \right ),\qquad \\ M_{24} &= S_{r}\left ( 1 - S_{r} \right ),\qquad M_{n} = - \phi mk\left ( 1 - S_{r\mathit{min}} \right )S_{e}^{1 + \frac{1}{m}}\left ( S_{e}^{ - \frac{1}{m}} - 1 \right )^{1 - \frac{1}{k}}. \end{aligned}$$

The detailed expressions of \(m_{11} - m_{24}\) are as follows:

$$\begin{aligned} m_{11} &= \frac{M_{11}M_{24} - M_{21}M_{15}}{M_{14}M_{24} - M_{15}M_{23}},\qquad m_{12} = \frac{M_{12}M_{24} - M_{22}M_{15}}{M_{14}M_{24} - M_{15}M_{23}},\qquad \\ m_{13} &= \frac{M_{13}M_{24}}{M_{14}M_{24} - M_{15}M_{23}},\qquad \\ m_{21} & = \frac{M_{14}M_{21} - M_{23}M_{11}}{M_{14}M_{24} - M_{15}M_{23}},\qquad m_{22} = \frac{M_{14}M_{22} - M_{23}M_{12}}{M_{14}M_{24} - M_{15}M_{23}},\qquad \\ m_{23} &= \frac{ - M_{13}M_{23}}{M_{14}M_{24} - M_{15}M_{23}}. \end{aligned}$$

The detailed expressions of \(\mathbf{W}_{1}\sim \mathbf{W}_{4}\) are as follows:

$$\begin{aligned} \mathbf{W}_{1} &= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & \xi _{x}\frac{\lambda}{\lambda + 2G} & - \xi _{x}\left ( \frac{2G\beta \chi}{\lambda + 2G} + w^{2}nS_{r}\rho _{w}a_{w} \right ) & - \xi _{x}\left ( \textstyle\begin{array}{l} \frac{2G\beta \left ( 1 - \chi \right )}{\lambda + 2G} + \\ w^{2}n\left ( 1 - S_{r} \right )\rho _{a}a_{a} \end{array}\displaystyle \right ) \\ - \xi _{x} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\displaystyle \right ], \\ \mathbf{W}_{2} &= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \left ( \frac{4\lambda G + 4G^{2}}{\lambda + 2G} \right )\xi _{x}^{2} - o & 0 & 0 & 0 \\ 0 & - o & \frac{\omega ^{2}\rho _{w}^{2}gnS_{r}a_{w}}{k_{w}} & \frac{\omega ^{2}\rho _{a}^{2}gn\left ( 1 - S_{r} \right )a_{a}}{k_{a}} \\ 0 & 0 & \frac{\rho _{w}g}{k_{w}} & 0 \\ 0 & 0 & 0 & \frac{\rho _{a}g}{k_{a}} \end{array}\displaystyle \right ],\qquad \\ \mathbf{W}_{3} &= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \frac{1}{G} & 0 & 0 & 0 \\ 0 & \frac{1}{\lambda + 2G} & \frac{\chi \beta}{\lambda + 2G} & \frac{\left ( 1 - \chi \right )\beta}{\lambda + 2G} \\ 0 & \frac{k_{w}}{\rho _{w}g}\frac{b_{13}}{\lambda + 2G} & \frac{k_{w}}{\rho _{w}g}\psi _{1} & \frac{k_{w}}{\rho _{w}g}\psi _{2} \\ 0 & \frac{k_{a}}{\rho _{a}g}\frac{b_{23}}{\lambda + 2G} & \frac{k_{a}}{\rho _{a}g}\psi _{3} & \frac{k_{a}}{\rho _{a}g}\psi _{4} \end{array}\displaystyle \right ],\qquad \mathbf{W}_{4} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & \xi _{x} & 0 & 0 \\ - \xi _{x}\frac{\lambda}{\lambda + 2G} & 0 & 0 & 0 \\ \xi _{x}\frac{k_{w}}{\rho _{w}g}\frac{2Gb_{13}}{\lambda + 2G} & 0 & 0 & 0 \\ \xi _{x}\frac{k_{a}}{\rho _{a}g}\frac{2Gb_{23}}{\lambda + 2G} & 0 & 0 & 0 \end{array}\displaystyle \right ]. \end{aligned}$$

where \(o = \omega ^{2}\left ( 1 - n \right )\rho _{s} + \mathrm{i}\omega ^{3}nS_{r}\rho _{w}a_{w}c_{w} + \mathrm{i}\omega ^{3}n\left ( 1 - S_{r} \right )\rho _{a}a_{a}c_{a}\), \(\psi _{1} = b_{11} + \frac{b_{13}\beta \chi}{\lambda + 2G} + \xi _{x}^{2}\), \(\psi _{2} = b_{12} + \frac{b_{13}\beta \left ( 1 - \chi \right )}{\lambda + 2G}\), \(\psi _{3} = b_{21} + \frac{b_{23}\beta \chi}{\lambda + 2G}\), \(\psi _{4} = b_{22} + \frac{b_{23}\beta \left ( 1 - \chi \right )}{\lambda + 2G} + \xi _{x}^{2}\).

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Ai, Z.Y., Shi, L.W. & Sheng, L. Vertical Dynamic Analysis of Rigid Strip Foundation on Layered Unsaturated Media. J Elast 157, 6 (2025). https://doi.org/10.1007/s10659-024-10099-0

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