General Approximate Analytical Solutions for One-Dimensional Nonlinear Consolidation of Soft Soils under Time-Dependent Loading
Publication: International Journal of Geomechanics
Volume 22, Issue 5
Abstract
A surcharge preloading method with time-dependent loading is widely used for accelerating the consolidation process of soft soils, and different loading patterns have a great effect on the consolidation characteristics. For one-dimensional nonlinear consolidation of soft soils considering the variable compressibility and permeability, the general approximate analytical solutions for one-dimensional nonlinear consolidation under time-dependent loading are derived. Moreover, two forms of boundary conditions are considered according to engineering practice. With reference to the general analytical solutions presented, the expressions for the solutions under several common loading patterns, such as the multistage instantaneous loading pattern, multistage linear loading pattern, and cyclic loading pattern, are given. The accuracy and correctness of the approximate analytical solutions are proven by comparing the approximate analytical solutions with the finite-difference solutions and two experimental results with different boundary conditions. Based on the approximate analytical solutions proposed, the consolidation behaviors of soft soils under different loading patterns are analyzed. In conclusion, the proposed approximate analytical solutions have a simple and clear form, which is convenient for calculations in engineering applications.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant Nos. 51378469 and 51708497); the Natural Science Foundation of Zhejiang Province, China (Grant Nos. LZ20E080001, LY19E080013, and LQ19E080009); the Natural Science Foundation of Ningbo, China (Grant No. 2019A610444); and the Application Research of Public Welfare Technology of Ningbo, China (Grant No. 2019C50016). In particular, we thank the reviewers and the editor for their valuable comments and suggestions on how to improve the quality of this paper.
Reference
Abbasi, N., H. Rahimi, A. A. Javadi, and A. Fakher. 2007. “Finite difference approach for consolidation with variable compressibility and permeability.” Comput. Geotech. 34 (1): 41–52. https://doi.org/10.1016/j.compgeo.2006.09.003.
Berry, P. L., and W. R. Wilkinson. 1969. “The radial consolidation of clay soils.” Géotechnique 19 (2): 253–284. https://doi.org/10.1680/geot.1969.19.4.534.
Cai, Y. Q., X. Y. Geng, and C. J. Xu. 2007. “Solution of one-dimensional finite-strain consolidation of soil with variable compressibility under cyclic loadings.” Comput. Geotech. 34 (1): 31–40. https://doi.org/10.1016/j.compgeo.2006.08.008.
Chen, Y., H. Xie, H. Ke, and R. Chen. 2009. “An analytical solution for one-dimensional contaminant diffusion through multi-layered system and its applications.” Environ. Geol. 58 (5): 1083–1094. https://doi.org/10.1007/s00254-008-1587-3.
Chen, Y. M., X. W. Tang, and J. Wang. 2004. “An analytical solution of one-dimensional consolidation for soft sensitive soil ground.” Int J. Numer. Anal. Methods Geomech. 28 (9): 919–930. https://doi.org/10.1002/nag.353.
Cheng, D., W. Wang, X. Chen, and Z. Zhang. 2017. “Finite analytic method for one-dimensional nonlinear consolidation under time-dependent loading.” Shock. Vib. 2017: 4071268. https://doi.org/10.1155/2017/4071268.
Conte, E., and A. Troncone. 2006. “One-dimensional consolidation under general time-dependent loading.” Can. Geotech. J. 43 (11): 1107–1116. https://doi.org/10.1139/t06-064.
Conte, E., and A. Troncone. 2007. “Nonlinear consolidation of thin layers subjected to time-dependent loading.” Can. Geotech. J. 44 (6): 717–725. https://doi.org/10.1139/t07-015.
Davis, E. H., and G. P. Raymond. 1965. “A non-linear theory of consolidation.” Géotechnique 15 (2): 161–173. https://doi.org/10.1680/geot.1965.15.2.161.
Geng, X. Y., C. J. Xu, and Y. Q. Cai. 2006. “Non-linear consolidation analysis of soil with variable compressibility and permeability under cyclic loadings.” Int. J. Numer. Anal. Methods Geomech. 30 (8): 803–821. https://doi.org/10.1002/nag.505.
Gray, H. 1945. “Simultaneous consolidation of contiguous layers of unlike compressive soils.” Trans. ASCE 110 (1): 1352–1356. https://doi.org/10.1061/taceat.0005788.
Huang, J., and D. V. Griffiths. 2010. “One-dimensional consolidation theories for layered soil and coupled and uncoupled solutions by the finite-element method.” Géotechnique 60 (9): 709–713. https://doi.org/10.1680/geot.08.p.038.
Kim, P., K. S. Ri, Y. G. Kim, K. N. Sin, H. B. Myong, and C. H. Paek. 2020. “Nonlinear consolidation analysis of a saturated clay layer with variable compressibility and permeability under various cyclic loadings.” Int. J. Geomech. 20 (8): 04020111. https://doi.org/10.1061/(asce)gm.1943-5622.0001730.
Lee, P. K. K., K. H. Xie, and Y. K. Cheung. 1992. “A study on one dimensional consolidation of layered systems.” Int. J. Numer. Anal. Methods Geomech. 16 (11): 815–831. https://doi.org/10.1002/nag.1610161104.
Lekha, K. R., N. R. Krishnaswamy, and P. Basak. 2003. “Consolidation of clays for variable permeability and compressibility.” J. Geotech. Geoenviron. 129 (11): 1001–1009. https://doi.org/10.1061/(asce)1090-0241(2003)129:11(1001).
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, C. X., M. M. Lu, and H. Ma. 2020. “Solutions for one-dimensional rheological consolidation of a clay layer with threshold hydraulic gradient under multistage loading.” Int. J. Geomech. 20 (11): 06020031. https://doi.org/10.1061/(asce)gm.1943-5622.0001846.
Li, L., A. Qin, and L. Jiang. 2021. “Semi-analytical solution for the one-dimensional consolidation of multi-layered unsaturated soils with semi-permeable boundary.” J. Eng. Math. 130 (1): 1–17. https://doi.org/10.1007/s10665-021-10162-y.
Li, C. X., and K. H. Xie. 2013. “One-dimensional nonlinear consolidation of soft clay with the non-Darcian flow.” J. Zhejiang Univ. Sci. A 14 (6): 435–446. https://doi.org/10.1631/jzus.a1200343.
Mei, G. X., and Q. M. Chen. 2013. “Solution of Terzaghi one-dimensional consolidation equation with general boundary conditions.” J. Cent. South. Univ. 20 (8): 2239–2244. https://doi.org/10.1007/s11771-013-1730-5.
Pyrah, I. C. 1996. “One-dimensional consolidation of layered soils.” Géotechnique 46 (3): 555–560. https://doi.org/10.1680/geot.1996.46.3.555.
Qi, T. 2008. Theoretical and experimental studies on one-dimensional nonlinear consolidation of soft soil. [In Chinese.] Hangzhou: Zhejiang Univ.
Savović, S., and A. Djordjevich. 2012. “Finite difference solution of the one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media.” Int. J. Heat Mass Transfer 55 (15–16): 4291–4294. https://doi.org/10.1016/j.ijheatmasstransfer.2012.03.073.
Schiffman, R. L., and J. R. Stein. 1970. “One-dimensional consolidation of layered systems.” J. Soil. Mech. Found. Div. 96 (4): 1499–1504. https://doi.org/10.1061/jsfeaq.0001453.
Terzaghi, K. 1925. Erdbaumechanik auf bodenphysikalischer Grundlage. Vienna, Austria: F. Deuticke.
Wilson, N. E., and N. M. Elgohary. 1974. “Consolidation of soils under cyclic loading.” Can. Geotech. J. 11 (3): 420–423. https://doi.org/10.1139/t74-042.
Wu, W. B., M. F. Zong, M. H. E. I. Naggar, G. X. Mei, and R. Z. Liang. 2018. “Analytical solution for one-dimensional consolidation of double-layered soil with exponentially time-growing drainage boundary.” Int. J. Distrib. Sens. Netw. 14 (10): 155014771880671. https://doi.org/10.1177/1550147718806716.
Xie, K. H., and Q. Y. Pan. 1995. “One-dimensional consolidation of soil stratum of arbitrary layers under time-dependent loading.” [In Chinese.] Chin. J. Geotech. Eng. 17 (5): 82–87.
Xie, K. H., T. Qi, and Q. Y. Dong. 2006. “Nonlinear analytical solution for one-dimensional consolidation of soft soil under cyclic loading.” J. Zhejiang Univ. Sci. A 7 (8): 1358–1364. https://doi.org/10.1631/jzus.2006.a1358.
Xie, K. H., X. Y. Xie, and X. Gao. 1999. “Theory of one-dimensional consolidation of two-layered soil with partially drained boundaries.” Comput. Geotech. 24 (4): 265–278. https://doi.org/10.1016/s0266-352x(99)00012-9.
Xie, K. H., X. Y. Xie, and W. Jiang. 2002. “A study on one-dimensional nonlinear consolidation of double-layered soil.” Comput. Geotech. 29 (2): 151–168. https://doi.org/10.1016/s0266-352x(01)00017-9.
Zhao, X. D., and W. H. Gong. 2019. “Numerical solution of nonlinear large strain consolidation based on non-Darcian flow.” Math. Probl. Eng. 2019: 1–14. https://doi.org/10.1155/2019/5745068.
Zhao, X. D., Y. Liu, and W. H. Gong. 2020. “Analytical solution for one-dimensional electro-osmotic consolidation of double–layered system.” Comput. Geotech. 122: 103496. https://doi.org/10.1016/j.compgeo.2020.103496.
Zheng, G. Y., P. Li, and C. Y. Zhao. 2013. “Analysis of non-linear consolidation of soft clay by differential quadrature method.” Appl. Clay Sci. 79: 2–7. https://doi.org/10.1016/j.clay.2013.02.025.
Zhu, G., and J. H. Yin. 1998. “Consolidation of soil under depth-dependent ramp load.” Can. Geotech. J. 35 (2): 344–350. https://doi.org/10.1139/t97-092.
Information & Authors
Information
Published In
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Aug 20, 2021
Accepted: Dec 27, 2021
Published online: Mar 9, 2022
Published in print: May 1, 2022
Discussion open until: Aug 9, 2022
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.
Cited by
- Shangqi Ge, Wenhao Jiang, Lingwei Zheng, Xinyu Xie, Kanghe Xie, Guohui Feng, Xunli Zhang, Theoretical analysis and experimental verification of large deformation electro-osmosis consolidation treatment of dredged slurry, Engineering Geology, 10.1016/j.enggeo.2022.106924, 312, (106924), (2023).
- Luyu Wang, Zhen-Yu Yin, Gang Han, Meilu Yu, Toward temporal evolution of consolidation in fluid-saturated poroelastic media with various permeable conditions, Computers and Geotechnics, 10.1016/j.compgeo.2023.105273, 156, (105273), (2023).