Analytical Solutions to the Axisymmetrical Consolidation of Unsaturated Soils
Publication: Journal of Engineering Mechanics
Volume 144, Issue 1
Abstract
In this study, analytical solutions are presented for the axisymmetrical consolidation of an unsaturated soil stratum on the basis of Fredlund and Hasan’s one-dimensional consolidation theory. Two different loading conditions, i.e., instantaneous loading and ramp loading, were considered. The derivation process of the proposed analytical solution is simpler in contrast to the existing analytical solutions and it was verified to be accurate using an analytical solution in the literature. In addition to the analytical solutions under the assumption of free strain, the analytical solutions under the assumption of equal strain were further derived. In the parametric study, the variations in the excess pore-water and pore-air pressures as well as the average degree of consolidation are presented. Graphs are given to show the influence of the different hydraulic conductivity ratios between the air and water phases as well as the construction time (under a ramp loading condition) to the consolidation process. These two factors have significant impact on the consolidation of the air phase but little influence on the consolidation process of the water phase.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work has been completed by the financial support from the National Natural Science Foundation of China (Grant No. 51508585), the Macau Science and Technology Development Grant (FDCT/011/2013/A1), and the University of Macau Research Fund MYRG2015-00112-FST.
References
Barron, R. A. (1948). “Consolidation of fine-grained soils by drain wells.” Trans. ASCE, 113(1), 718–742.
Bergado, D. T., Chai, J. C., Miura, N., and Balasubramaniam, A. S. (1998). “PVD improvement of soft Bangkok clay with combined vacuum and reduced sand embankment preloading.” Geotech. Eng. SE Asian Geotech. Soc., 29(1), 95–122.
Chen, W. Z., Jiang, G. L., Zhao, H. S., Wu, L. J., and Li, A. H. (2014). “Analysis of nonlinear settlement for an unsaturated soil under stage continuous loading.” J. Cent. South Univ., 21(12), 4690–4697.
Cheng, Y., Zhang, L. L., Li, J. H., Zhang, L. M., Wang, J. H., and Wang, D. Y. (2017). “Consolidation in spatially random unsaturated soils based on coupled flow-deformation simulation.” Int. J. Numer. Anal. Methods Geomech., 41(5), 682–706.
Conte, E. (2004). “Consolidation analysis for unsaturated soils.” Can. Geotech. J., 41(4), 599–612.
Conte, E. (2006). “Plane strain and axially symmetric consolidation in unsaturated soils.” Int. J. Geomech., 131–135.
Feng, J., Wu, X. Y., Zhu, B. L., and Yang, Q. X. (2015). “Analytical solution to one-dimensional consolidation in unsaturated soils under sinusoidal cyclic loading.” J. Cent. South Univ., 22(2), 646–653.
Fredlund, D. G., and Hasan, J. U. (1979). “One-dimensional consolidation theory: Unsaturated soils.” Can. Geotech. J., 16(3), 521–531.
Fredlund, D. G., Rahardjo, H., and Fredlund, M. D. (2012). Unsaturated soil mechanics in engineering practice, Wiley, Hoboken, NJ.
Hansbo, S. (1981). “Consolidation of fine-grained soils by prefabricated drains.” Proc., 10th Int. Conf. on Soil Mechanics and Foundation Engineering, Vol. 3, A.A. Balkema, Rotterdam, Netherlands, 677–682.
Ho, L., and Fatahi, B. (2015). “Analytical solution for the two-dimensional plane strain consolidation of an unsaturated soil stratum subjected to time-dependent loading.” Comput. Geotech., 67, 1–16.
Ho, L., and Fatahi, B. (2016). “One-dimensional consolidation analysis of unsaturated soils subjected to time-dependent loading.” Int. J. Geomech., 04015052.
Ho, L., and Fatahi, B. (2017). “Axisymmetric consolidation in unsaturated soil deposit subjected to time-dependent loadings.” Int. J. Geomech., 04016046.
Ho, L., Fatahi, B., and Khabbaz, H. (2014). “Analytical solution for one-dimensional consolidation of unsaturated soils using eigenfunction expansion method.” Int. J. Numer. Anal. Methods Geomech., 38(10), 1058–1077.
Ho, L., Fatahi, B., and Khabbaz, H. (2015). “A closed form analytical solution for two-dimensional plane strain consolidation of unsaturated soil stratum.” Int. J. Numer. Anal. Methods Geomech., 39(15), 1665–1692.
Ho, L., Fatahi, B., and Khabbaz, H. (2016). “Analytical solution to axisymmetric consolidation in unsaturated soils with linearly depth-dependent initial conditions.” Comput. Geotech., 74, 102–121.
Indraratna, B., Rujikiatkamjorn, C., and Sathananthan, I. (2005a). “Radial consolidation of clay using compressibility indices and varying horizontal permeability.” Can. Geotech. J., 42(5), 1330–1341.
Indraratna, B., Sathananthan, I., Rujikiatkamjorn, C., and Balasubramaniam, A. S. (2005b). “Analytical and numerical modeling of soft soil stabilized by prefabricated vertical drains incorporating vacuum preloading.” Int. J. Geomech., 114–124.
Lei, G. H., Zheng, Q., Ng, C. W. W., Chiu, A. C. F., and Xu, B. (2015). “An analytical solution for consolidation with vertical drains under multi-ramp loading.” Géotechnique, 65(7), 531–547.
Leo, C. J. (2004). “Equal strain consolidation by vertical drains.” J. Geotech. Geoenviron. Eng., 316–327.
Lo, W. C., and Lee, J. W. (2015). “Effect of water content and soil texture on consolidation in unsaturated soils.” Adv. Water Resour., 82, 51–69.
Lo, W. C., Sposito, G., Lee, J. W., and Chu, H. (2016). “One-dimensional consolidation in unsaturated soils under cyclic loading.” Adv. Water Resour., 91, 122–137.
Oosterbaan, R. J., and Nijland, H. J. (1994). “Determining the saturated hydraulic conductivity.” Chapter 12, Drainage principles and applications, H. P. Ritzema, ed., International Institute for Land Reclamation and Improvement, Wageningen, Netherlands.
Qin, A., Sun, D., Yang, L., and Weng, Y. (2010). “A semi-analytical solution to consolidation of unsaturated soils with the free drainage well.” Comput. Geotech., 37(7–8), 867–875.
Qin, A., Sun, D., and Zhang, J. (2014). “Semi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soils.” Comput. Geotech., 62, 110–117.
Rujikiatkamjorn, C., Indraratna, B., and Chu, J. (2008). “2D and 3D numerical modeling of combined surcharge and vacuum preloading with vertical drains.” Int. J. Geomech., 144–156.
Shan, Z., Ling, D., and Ding, H. (2013). “Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition.” J. Zhejiang Univ. A, 14(1), 61–70.
Shan, Z., Ling, D., and Ding, H. (2014). “Analytical solution for the 1D consolidation of unsaturated multi-layered soil.” Comput. Geotech., 57, 17–23.
Tan, F., Zhou, W. H., and Yuen, K. V. (2016). “Modeling the soil water retention properties of same-textured soils with different initial void ratios.” J. Hydrol., 542, 731–743.
Tang, X. W., and Onitsuka, K. (2000). “Consolidation by vertical drains under time-dependent loading.” Int. J. Numer. Anal. Methods Geomech., 24(9), 739–751.
Tsiampousi, A., Smith, P. G. C., and Potts, D. M. (2017a). “Coupled consolidation in unsaturated soils: An alternative approach to deriving the governing equations.” Comput. Geotech., 84, 238–255.
Tsiampousi, A., Smith, P. G. C., and Potts, D. M. (2017b). “Coupled consolidation in unsaturated soils: From a conceptual model to applications in boundary value problems.” Comput. Geotech., 84, 256–277.
Walker, R., and Indraratna, B. (2007). “Vertical drain consolidation with overlapping smear zones.” Géotechnique, 57(5), 463–467.
Wang, X. S., and Jiao, J. J. (2004). “Analysis of soil consolidation by vertical drains with double porosity model.” Int. J. Numer. Anal. Methods Geomech., 28(14), 1385–1400.
Yoshikuni, H., and Nakanodo, H. (1974). “Consolidation of fine-grained soils by drain wells with finite permeability.” Japan Soc. Soil Mech. Found., 14(2), 35–46.
Yuan, S., and Zhong, H. (2016). “A weak form quadrature element formulation for coupled analysis of unsaturated soils.” Comput. Geotech., 76, 1–11.
Zeng, G. X., and Xie, K. H. (1989). “New development of the vertical drain theories.” Proc., 12th Int. Conf. on Soil Mechanics and Foundation Engineering, A.A. Balkema, Rotterdam, Netherlands, 1435–1438.
Zhou, W. H. (2013). “Axisymmetric consolidation of unsaturated soils by differential quadrature method.” Math. Probl. Eng., 1–14.
Zhou, W. H., Lok, M. H., Zhao, L. S., Mei, G. X., and Li, X. B. (2017a). “Analytical solutions to the axisymmetric consolidation of a multi-layer soil system under surcharge combined with vacuum preloading.” Geotext. Geomembr., 45(5), 487–498.
Zhou, W. H., and Zhao, L. S. (2014). “One-dimensional consolidation of unsaturated soil subjected to time-dependent loading with various initial and boundary conditions.” Int. J. Geomech., 291–301.
Zhou, W. H., Zhao, L. S., Garg, A., and Yuen, K. V. (2017b). “Generalized analytical solution for the consolidation of unsaturated soil under partially permeable boundary conditions.” Int. J. Geomech., 04017048.
Zhou, W. H., Zhao, L. S., and Li, X. B. (2014). “A simple analytical solution to one-dimensional consolidation for unsaturated soils.” Int. J. Numer. Anal. Methods Geomech., 38(8), 794–810.
Zhou, Y., and Chai, J. C. (2017). “Equivalent ‘smear’ effect due to non-uniform consolidation surrounding a PVD.” Géotechnique, 67(5), 410–419.
Zhu, G., and Yin, J. H. (2004). “Consolidation analysis of soil with vertical and horizontal drainage under ramp loading considering smear effects.” Geotext. Geomembr., 22(1–2), 63–74.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 144 • Issue 1 • January 2018
Copyright
©2017 American Society of Civil Engineers.
History
Received: Feb 17, 2017
Accepted: Jun 28, 2017
Published online: Nov 8, 2017
Published in print: Jan 1, 2018
Discussion open until: Apr 8, 2018
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.