Semianalytical Solution for One-Dimensional Consolidation in a Multilayered Unsaturated Soil System with Exponentially Time-Growing Permeable Boundary
Publication: Journal of Engineering Mechanics
Volume 147, Issue 5
Abstract
This paper proposes new semianalytical solutions to one-dimensional consolidation of unsaturated soils with a multilayered system under the partially permeable boundary condition, in which a time-decaying exponential function is applied to represent the variations of excess pore pressures for both water and air at the boundary. The general solutions are explicitly presented by the aid of the Laplace transform. Based on the general solutions, interfacial continuity between layers, and boundary conditions, the transfer matrix approach is adopted to deduce the analytical solutions of the multilayered unsaturated soil consolidation in the Laplace space, and the numerical inverse computations of Laplace transform in terms of Crump’s technique are carried out to obtain the final analytical solutions. Then, the existing solution for single-layered soils and the numerical solutions for two-layered unsaturated soils performed by finite-difference methods are utilized to demonstrate the validity of the present methodology as well as semianalytical solutions. Finally, a case study considering two distinct soil profiles is implemented to illustrate the influence of the impeded boundary parameters on the settlement of two-layered unsaturated soils.
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Data Availability Statement
All data, models, and code generated or used during the study appear in the published article.
Acknowledgments
The research described in this paper was financially supported by the National Natural Science Foundation of China (Grant Nos. 42072292 and 41372279).
References
Ai, Z. Y., Z. Z. Zhao, and W. J. Liu. 2020. “Fractional derivative modeling for axisymmetric consolidation of multilayered cross-anisotropic viscoelastic porous media.” Comput. Math. Appl. 79 (5): 1321–1334. https://doi.org/10.1016/j.camwa.2019.08.033.
Ausilio, E., and E. Conte. 1999. “Settlement rate of foundations on unsaturated soils.” Can. Geotech. J. 36 (5): 940–946. https://doi.org/10.1139/t99-055.
Booker, J. R., and J. P. Carter. 1987. “Elastic consolidation around a point sink embedded in a half-space with anisotropic permeability.” Int. J. Numer. Anal. Methods Geomech. 11 (1): 61–77. https://doi.org/10.1002/nag.1610110106.
Chiou, Y. J., and S. Y. Chi. 1994. “Boundary element analysis of BIOT consolidation in layered elastic soils.” Int. J. Numer. Anal. Methods Geomech. 18 (6): 377–396. https://doi.org/10.1002/nag.1610180603.
Conte, E. 2004. “Consolidation analysis for unsaturated soils.” Can. Geotech. J. 41 (4): 599–612. https://doi.org/10.1139/t04-017.
Crump, K. S. 1976. “Numerical inversion of Laplace transforms using a Fourier series approximation.” J. ACM 23 (1): 89–96. https://doi.org/10.1145/321921.321931.
Desai, C. S., and J. H. Christian. 1977. Numerical methods in geotechnical engineering. New York: McGraw-Hill.
Fazeli, A., A. Keshavarz, and M. Moradi. 2018. “Coupled consolidation of layered unsaturated soil under general time-dependent loading.” Int. J. Geomech. 18 (8): 04018088. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001193.
Feng, W. Q., and J. H. Yin. 2017. “A new simplified Hypothesis B method for calculating consolidation settlements of double soil layers exhibiting creep.” Int. J. Numer. Anal. Methods Geomech. 41 (6): 899–917. https://doi.org/10.1002/nag.2635.
Fredlund, D. G., and J. U. Hasan. 1979. “One-dimensional consolidation theory: Unsaturated soils.” Can. Geotech. J. 16 (3): 521–531. https://doi.org/10.1139/t79-058.
Fredlund, D. G., and H. Rahardjo. 1993. Soil mechanics for unsaturated soils. New York: Wiley.
Fredlund, D. G., H. Rahardjo, and M. D. Fredlund. 2012. Unsaturated soil mechanics in engineering practice. Hoboken, NJ: Wiley.
Gao, Y., D. A. Sun, Z. C. Zhu, and Y. F. Xu. 2019. “Hydromechanical behavior of unsaturated soil with different initial densities over a wide suction range.” Acta Geotech. 14 (9): 417–428. https://doi.org/10.1007/s11440-018-0662-5.
Gray, H. 1945. “Simultaneous consolidation of contiguous layers of unlike compressible soils.” Trans ASCE 110: 1327–1356.
Ho, L., and B. Fatahi. 2016. “One-dimensional consolidation analysis of unsaturated soils subjected to time-dependent loading.” Int. J. Geomech. 16 (2): 04015052. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000504.
Ho, L., B. Fatahi, and H. Khabbaz. 2018. “Analytical solution to one-dimensional consolidation in unsaturated soil deposit incorporating time-dependent diurnal temperature variation.” Int. J. Geomech. 18 (5): 04018029. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001153.
Kodikara, J., C. Jayasundara, and A. N. Zhou. 2020. “A generalised constitutive model for unsaturated compacted soils considering wetting/drying cycles and environmentally-stabilised line.” Comput. Geotech. 118: 103332. https://doi.org/10.1016/j.compgeo.2019.103332.
Li, P. C., L. Z. Li, and M. M. Lu. 2020. “Semi-analytical solution to one-dimensional consolidation of viscoelastic saturated soils subject to an inner sink.” Eng. Comput. 37 (5): 1787–1804. https://doi.org/10.1108/EC-05-2019-0196.
Li, P. C., K. Y. Wang, G. K. Fang, and D. T. Lu. 2017. “Steady-state analytical solutions of flow and deformation coupling due to a point sink within a finite fluid-saturated poroelastic layer.” Int. J. Numer. Anal. Methods Geomech. 41 (8): 1093–1107. https://doi.org/10.1002/nag.2663.
Liu, J. C., and G. H. Lei. 2013. “One-dimensional consolidation of layered soils with exponentially time-growing drainage boundaries.” Comput. Geotech. 54: 202–209. https://doi.org/10.1016/j.compgeo.2013.07.009.
Lo, W. C., R. I. Borja, J. H. Deng, and J. W. Lee. 2020. “Analytical solution of soil deformation and fluid pressure change for a two-layer system with an upper unsaturated soil and a lower saturated soil under external loading.” J. Hydrol. 588: 124997. https://doi.org/10.1016/j.jhydrol.2020.124997.
Mei, G. X., T. M. H. Lok, J. Xia, and S. S. Wu. 2014. “One-dimensional consolidation with asymmetrical exponential drainage boundary.” Geomech. Eng. 6 (1): 47–63. https://doi.org/10.12989/gae.2014.6.1.047.
Moradi, M., A. Keshavarz, and A. Fazeli. 2019. “One dimensional consolidation of multi-layered unsaturated soil under partially permeable boundary conditions and time-dependent loading.” Comput. Geotech. 107: 45–54. https://doi.org/10.1016/j.compgeo.2018.11.020.
Qin, A. F., G. J. Chen, Y. W. Tan, and D. A. Sun. 2008. “Analytical solution to one dimensional consolidation in unsaturated soils.” Appl. Math. Mech. 29 (10): 1329–1340. https://doi.org/10.1007/s10483-008-1008-x.
Qin, A. F., D. A. Sun, and J. L. Zhang. 2014. “Semi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soils.” Comput. Geotech. 62 (10): 110–117. https://doi.org/10.1016/j.compgeo.2014.06.014.
Senjuntichai, T., and R. K. N. D. Rajapakse. 1995. “Exact stiffness method for quasi-statics of a multi-layered poroelastic medium.” Int. J. Solids Struct. 32 (11): 1535–1553. https://doi.org/10.1016/0020-7683(94)00190-8.
Shan, Z. D., D. S. Ling, and H. J. Ding. 2012. “Exact solutions for one-dimensional consolidation of single-layer unsaturated soil.” Int. J. Numer. Anal. Methods Geomech. 36 (6): 708–722. https://doi.org/10.1002/nag.1026.
Shan, Z. D., D. S. Ling, and H. J. Ding. 2014. “Analytical solution for the 1d consolidation of unsaturated multi-layered soil.” Comput. Geotech. 57: 17–23. https://doi.org/10.1016/j.compgeo.2013.11.009.
Talbot, A. 1979. “The accurate numerical inversion of Laplace transform.” J. Inst. Math. Appl. 23 (1): 97–120. https://doi.org/10.1093/imamat/23.1.97.
Terzaghi, K. 1925. Erdbaumechanik aud bodenphysikalischer grundlage. Leipzig, Germany: Deuticke.
Tsiampousi, A., P. G. C. Smith, and D. M. Potts. 2017. “Coupled consolidation in unsaturated soils: An alternative approach to deriving the governing equations.” Comput. Geotech. 84: 238–255. https://doi.org/10.1016/j.compgeo.2016.10.007.
Wang, L., D. A. Sun, P. C. Li, and Y. Xie. 2017. “Semi-analytical solution for one-dimensional consolidation of fractional derivative viscoelastic saturated soils.” Comput. Geotech. 83: 30–39. https://doi.org/10.1016/j.compgeo.2016.10.020.
Wang, L., D. A. Sun, and A. F. Qin. 2018. “Semi-analytical solution to one-dimensional consolidation for unsaturated soils with exponentially time-growing drainage boundary conditions.” Int. J. Geomech. 18 (2): 04017144. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001056.
Wang, L., Y. F. Xu, X. H. Xia, L. Z. Li, and Y. L. He. 2020a. “A series of semianalytical solutions of one-dimensional consolidation in unsaturated soils.” Int. J. Geomech. 20 (6): 06020005. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001661.
Wang, L., A. N. Zhou, Y. F. Xu, and X. H. Xia. 2020b. “Consolidation of unsaturated composite ground reinforced by permeable columns.” Comput. Geotech. 125: 103706. https://doi.org/10.1016/j.compgeo.2020.103706.
Wang, L. J., B. Zhu, Y. M. Chen, R. P. Chen, and X. S. Shi. 2019. “Precise model for predicting excess pore-water pressure of layered soils induced by thermal-mechanical loads.” J. Eng. Mech. 145 (1): 04018114. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001544.
Yin, J. H., and F. Tong. 2011. “Constitutive modeling of time-dependent stress–strain behaviour of saturated soils exhibiting both creep and swelling.” Can. Geotech. J. 48 (12): 1870–1885. https://doi.org/10.1139/t11-076.
Zhou, A. N., D. C. Sheng, S. W. Sloan, and A. Gens. 2012a. “Interpretation of unsaturated soil behaviour in the stress–saturation space. I: Volume change and water retention behaviour.” Comput. Geotech. 43: 178–187. https://doi.org/10.1016/j.compgeo.2012.04.010.
Zhou, A. N., D. C. Sheng, S. W. Sloan, and A. Gens. 2012b. “Interpretation of unsaturated soil behaviour in the stress–saturation space. II: Constitutive relationships and validations.” Comput. Geotech. 43: 111–123. https://doi.org/10.1016/j.compgeo.2012.02.009.
Zhou, W. H., and L. S. Zhao. 2014. “One-dimensional consolidation of unsaturated soil subjected to time-dependent loading with various initial and boundary conditions.” Int. J. Geomech. 14 (2): 291–301. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000314.
Zhou, W. H., L. S. Zhao, A. Garg, and K. V. Yuen. 2017. “Generalized analytical solution for the consolidation of unsaturated soil under partially permeable boundary conditions.” Int. J. Geomech. 17 (9): 04017048. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000942.
Zhou, W. H., L. S. Zhao, T. M. H. Lok, G.-X. Mei, and X. B. Li. 2018. “Analytical solutions to the axisymmetrical consolidation of unsaturated soils.” J. Eng. Mech. 144 (1): 04017152. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001382.
Zhu, G. F., and J. H. Yin. 1999. “Consolidation of double soil layers under depth-dependent ramp load.” Géotechnique 49 (3): 415–421. https://doi.org/10.1680/geot.1999.49.3.415.
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Published In
Journal of Engineering Mechanics
Volume 147 • Issue 5 • May 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Sep 2, 2020
Accepted: Jan 20, 2021
Published online: Mar 11, 2021
Published in print: May 1, 2021
Discussion open until: Aug 11, 2021
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