Consolidation Theory of Unsaturated Soils with Vertical Drains Considering Well Resistance and Smear Effect under Time-Dependent Loading
Publication: Journal of Engineering Mechanics
Volume 147, Issue 9
Abstract
In this investigation, an approximate analytical solution to the axisymmetric consolidation problem of unsaturated soils improved with vertical drain is presented under the equal-strain condition. The smear effect of soil, the well resistance of vertical drain, and the application of time-dependent loading are considered in the solution. Matrix analysis is used to solve the solution under constant loading for comparison against the available solution. The influence of soil parameters, smear parameters, and well resistance parameters on the consolidation behavior is investigated under ramp loading. Moreover, two types of loading which are noncyclic forms (i.e., instantaneous, multistage ramp, and exponential loading) and cyclic forms (i.e., triangular, sinusoidal, and cosine cyclic loading) are incorporated in the analyses. It is found that a higher hydraulic conductivity ratio in the undisturbed zone and a larger radius of the influence zone can speed up the consolidation process. The consolidation process is slowed down by increasing the radius of the disturbed zone and reducing its hydraulic conductivity, and the influence of hydraulic conductivity is more significant. The effect of radius and hydraulic conductivity of vertical drain on the consolidation behavior is similar. The consolidation behavior generally follows the pattern of the applied loading apparently.
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Data Availability Statement
All data, models, and code generated or used during the study appear in the published article.
Acknowledgments
This work had been supported by the National Natural Science Foundation of China (41672296, 41867034, 51878185, 52008124, and 52078506), the Natural Science Foundation of Guangxi Province (2016GXNSFGA380008 and 2019GXNSFBA185028), the Bagui Scholars Program (2016A31), the China Scholarship Council (CSC) (201808455023 and 201906660001), and the Systematic Project of Guangxi Key Laboratory of Disaster Prevention and Engineering Safety (2019ZDK041).
References
Barron, R. A. 1948. “Consolidation of fine-grained soils by drain wells.” Trans. Am. Soc. Civ. Eng. 113 (1): 718–742. https://doi.org/10.1061/TACEAT.0006098.
Chen, R. P., W. H. Zhou, H. Z. Wang, and Y. M. Chen. 2005. “One-dimensional nonlinear consolidation of multi-layered soil by differential quadrature method.” Comput. Geotech. 32 (5): 358–369. https://doi.org/10.1016/j.compgeo.2005.05.003.
Chen, Z., P. P. Ni, G. X. Mei, and Y. F. Chen. 2020. “Semi-analytical solution for consolidation of ground with partially penetrating PVDs under the free-strain condition.” J. Eng. Mech. 147 (2): 04020148. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001884.
Conte, E., and A. Troncone. 2009. “Radial consolidation with vertical drains and general time-dependent loading.” Can. Geotech. J. 46 (1): 25–36. https://doi.org/10.1139/T08-101.
Conte, E., A. Troncone, and M. Vena. 2017. “A method for the design of embedded cantilever retaining walls under static and seismic loading.” Géotechnique 67 (12): 1–9. https://doi.org/10.1680/jgeot.17.D.013.
Fredlund, D. G., and G. Delwyn. 2006. “Unsaturated soil mechanics in engineering practice.” J. Geotech. Geoenviron. Eng. 132 (3): 286–321. https://doi.org/10.1061/(ASCE)1090-0241(2006)132:3(286).
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.
Geng, X. Y., B. Indraratna, and C. Rujikiatkamjorn. 2012. “Analytical solutions for a single vertical drain with vacuum and time-dependent surcharge preloading in membrane and membraneless systems.” Int. J. Geomech. 39 (1): 51–62. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000106.
Hansbo, S. 1981. “Consolidation of fine-grained soils by prefabricated drains.” In Proc., 10th Int. Conf. on Soil Mechanics and Foundation Engineering, 677–682. Stockholm, Sweden: International Society of Soil Mechanics and Foundation Engineering.
Ho, L., and B. Fatahi. 2015. “Analytical solution for the two-dimensional plane strain consolidation of an unsaturated soil stratum subjected to time-dependent loading.” Comput. Geotech. 67 (Jun): 1–16. https://doi.org/10.1016/j.compgeo.2015.02.011.
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., and B. Fatahi. 2017. “Axisymmetric consolidation in unsaturated soil deposit subjected to time-dependent loadings.” Int. J. Geomech. 17 (2): 04016046. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000686.
Ho, L., and B. Fatahi. 2018. “Analytical solution to axisymmetric consolidation of unsaturated soil stratum under equal strain condition incorporating smear effects.” Int. J. Numer. Anal. Methods Geomech. 42 (15): 1890–1913. https://doi.org/10.1002/nag.2838.
Ho, L., B. Fatahi, and H. Khabbaz. 2016. “Analytical solution to axisymmetric consolidation in unsaturated soils with linearly depth-dependent initial conditions.” Comput. Geotech. 74 (Apr): 102–121. https://doi.org/10.1016/j.compgeo.2015.12.019.
Horne, M. R. 1964. “The consolidation of a stratified soil with vertical and horizontal drainage.” Int. J. Mech. Sci. 6 (2): 187–197. https://doi.org/10.1016/0020-7403(64)90015-3.
Hu, Y. Y., W. H. Zhou, and Y. Q. Cai. 2014. “Large-strain elastic viscoplastic consolidation analysis of very soft clay layers with vertical drains under preloading.” Can. Geotech. J. 51 (2): 144–157. https://doi.org/10.1139/cgj-2013-0200.
Leo, C. J. 2004. “Equal strain consolidation by vertical drains.” J. Geotech. Geoenviron. Eng. 130 (4): 316–327. https://doi.org/10.1061/(ASCE)1090-0241(2004)130:3(316).
Li, C. X., J. S. Huang, L. Z. Wu, J. F. Lu, and C. Q. Xia. 2018. “Approximate analytical solutions for one-dimensional consolidation of a clay layer with variable compressibility and permeability under a ramp loading.” Int. J. Geomech. 18 (11): 06018032. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001296.
Li, L., H. H. Chen, and J. P. Li. 2021. “An elastoplastic solution to undrained expansion of a cylindrical cavity in SANICLAY under plane stress condition.” Comput. Geotech. 132 (Apr): 103990. https://doi.org/10.1016/j.compgeo.2020.103990.
Lu, M. M., S. Y. Wang, S. W. Sloan, B. Indraratna, and K. H. Xie. 2015. “Nonlinear radial consolidation of vertical drains under a general time-variable loading.” Int. J. Numer. Anal. Methods Geomech. 39 (1): 51–62. https://doi.org/10.1002/nag.2295.
Lu, M. M., K. H. Xie, and S. Y. Wang. 2011. “Consolidation of vertical drain with depth-varying stress induced by multi-stage loading.” Comput. Geotech. 38 (8): 1096–1101. https://doi.org/10.1016/j.compgeo.2011.06.007.
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 (Mar): 45–54. https://doi.org/10.1016/j.compgeo.2018.11.020.
Ni, P. P., K. Xu, G. X. Mei, and Y. L. Zhao. 2019. “Effect of vacuum removal on consolidation settlement under a combined vacuum and surcharge preloading.” Geotext. Geomembr. 47 (1): 12–22. https://doi.org/10.1016/j.geotexmem.2018.09.004.
Onoue, A. 1988. “Consolidation of multilayered anisotropic soils by vertical drains with well resistance.” Soils Found. 28 (3): 75–90. https://doi.org/10.3208/sandf1972.28.3_75.
Qin, A. F., D. A. Sun, L. P. Yang, and Y. F. Weng. 2010. “A semi-analytical solution to consolidation of unsaturated soils with the free drainage well.” Comput. Geotech. 37 (7–8): 867–875. https://doi.org/10.1016/j.compgeo.2010.07.006.
Tang, X. W., and K. Onitsuka. 1998. “Consolidation of ground with partially vertical drains.” Geotech. Eng. J. 29 (2): 209–231.
Tang, X. W., and K. Onitsuka. 2001. “Consolidation of double-layered ground with vertical drains.” Int. J. Numer. Anal. Methods Geomech. 25 (14): 1449–1465. https://doi.org/10.1002/nag.191.
Tang, X. W., and K. Onitsuka. 2015. “Consolidation by vertical drains under time-dependent loading.” Int. J. Numer. Anal. Methods Geomech. 24 (9): 739–751. https://doi.org/10.1002/1096-9853(20000810)24:9%3C739::AID-NAG94%3E3.0.CO;2-B.
Walker, R., and B. Indraratna. 2006. “Vertical drain consolidation with overlapping smear zones.” Géotechnique 57 (5): 463–467. https://doi.org/10.1680/geot.2007.57.5.463.
Wang, L., D. A. Sun, A. F. Qin, and Y. F. Xu. 2017. “Semi-analytical solution to one-dimensional consolidation for unsaturated soils with semi-permeable drainage boundary under time-dependent loading.” Int. J. Numer. Anal. Methods Geomech. 41 (16): 1636–1655. https://doi.org/10.1002/nag.2694.
Wang, L., Y. F. Xu, X. H. Xia, Y. L. He, and T. Y. Li. 2019. “Semi-analytical solutions to two-dimensional plane strain consolidation for unsaturated soils under time-dependent loading.” Comput. Geotech. 109 (May): 144–165. https://doi.org/10.1016/j.compgeo.2019.01.002.
Wang, L., Y. F. Xu, X. H. Xia, and A. N. Zhou. 2020a. “Semi-analytical solutions to the two-dimensional plane strain consolidation for unsaturated soil with the lateral semi-permeable drainage boundary under time-dependent loading.” Comput. Geotech. 124 (Aug): 103562. https://doi.org/10.1016/j.compgeo.2020.103562.
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 (Sep): 103706. https://doi.org/10.1016/j.compgeo.2020.103706.
Xie, K. H., and G. X. Zeng. 1989. “Consolidation theories for drain wells under equal strain condition.” [In Chinese.] Chin. J. Geotech. Eng. 11 (2): 3–17.
Zhou, F., Z. Chen, and X. D. Wang. 2018a. “An equal-strain analytical solution for the radial consolidation of unsaturated soils by vertical drains considering drain resistance.” Adv. Civ. Eng. 2018: 5069159.
Zhou, W. H. 2013. “Axisymmetric consolidation of unsaturated soils by differential quadrature method.” Math. Probl. Eng. 2013 (1): 497161.
Zhou, W. H., and S. Tu. 2012. “Unsaturated consolidation in a sand drain foundation by differential quadrature method.” Procedia Earth Planet. Sci. 5 (2012): 52–57. https://doi.org/10.1016/j.proeps.2012.01.009.
Zhou, W. H., L. S. Zhao, M. H. Lok, G. X. Mei, and X. B. Li. 2018b. “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. 2004. “Consolidation analysis of soil with vertical and horizontal drainage under ramp loading considering smear effects.” Geotext. Geomembr. 22 (1): 63–74. https://doi.org/10.1016/S0266-1144(03)00052-9.
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Published In
Journal of Engineering Mechanics
Volume 147 • Issue 9 • September 2021
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© 2021 American Society of Civil Engineers.
History
Received: Jun 4, 2020
Accepted: Mar 30, 2021
Published online: Jul 7, 2021
Published in print: Sep 1, 2021
Discussion open until: Dec 7, 2021
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