Semianalytical Solution for Long-Term Settlement of a Single Pile Embedded in Fractional Derivative Viscoelastic Soils
Publication: International Journal of Geomechanics
Volume 21, Issue 2
Abstract
This paper proposes a semianalytical solution to analyze the long-term settlement of a single pile embedded in fractional derivative viscoelastic soils under time-dependent loading. In this paper, a three-dimensional (3D) fractional derivative viscoelastic model will be introduced to describe the rheological behavior of soils around the pile. Based on the shear deformation compatibility between the pile and soils, the semianalytical solution will be strictly derived using the correspondence principle and the Laplace transform technique. Three well-documented cases will be used to verify the correctness and reliability of the proposed semianalytical solution, all of which show a very close agreement. Furthermore, worked examples that include instantaneous and ramp loadings will be carried out to capture the influence of six dimensionless parameters, that is, a, b, c, d, κ, and α, on the long-term settlement of a single pile. The results indicate that: the dimensionless settlement for fractional derivative viscoelastic models is larger in the early stage and smaller in the later stage than those for conventional viscoelastic models; the dimensionless settlement capacity of the pile is controlled by the parameters a and d, and the dimensionless settlement rate is controlled by the parameters b and α, which means the selection of these four parameters has a great influence on the long-term settlement; and the parameter c causes little variation in the results.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, or code generated or used during the study are available from the corresponding author by request.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 51778345), the Shandong Provincial Natural Science Foundation for Distinguished Young Scholars (No. JQ201811), the Key Research and Development Foundation of Shandong Province of China (No. 2019GSF109006), the program of Qilu Young Scholars of Shandong University, and the Young Scholars Program of Shandong University (No. 2017WLJH32). Great appreciation goes to the editorial board and the reviewers of this paper.
References
Abate, J., and P. P. Valkó. 2004. “Multi-precision Laplace transform inversion.” Int. J. Numer. Methods Eng. 60 (5): 979–993. https://doi.org/10.1002/nme.995.
Abbas, J. M., Z. H. Chik, M. R. Taha, and Q. S. M. Shafiqu. 2010. “Time-dependent lateral response of pile embedded in elasto-plastic soil.” J. Central South Univ. Technol. 17 (2): 372–380. https://doi.org/10.1007/s11771-010-0055-x.
Bagley, R. L., and P. J. Torvik. 1986. “On the fractional calculus model of viscoelastic behavior.” J. Rheol. 30 (1): 133–155. https://doi.org/10.1122/1.549887.
Booker, J. R., and H. G. Poulos. 1975. “Analysis of creep settlement of pile foundations.” J. Geotech. Eng. Div. 102 (1): 1–14.
Butterfield, R., and P. K. Bannerjee. 1970. “A note on the problem of a pile reinforced half space.” Géotechnique 20 (1): 100–103. https://doi.org/10.1680/geot.1970.20.1.100.
Chen, R. P., W. H. Zhou, and Y. M. Chen. 2009. “Influences of soil consolidation and pile load on the development of negative skin friction of a pile.” Comput. Geotech. 36 (8): 1265–1271. https://doi.org/10.1016/j.compgeo.2009.05.011.
Christensen, R. M. 1982. Theory of viscoelasticity. New York: Academic Press.
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.
Danno, K., and M. Kimura. 2009. “Evaluation of long-term displacements of pile foundation using coupled FEM and centrifuge model test.” Soils Found. 49 (6): 941–958. https://doi.org/10.3208/sandf.49.941.
Durbin, F. 1974. “Numerical inversion of Laplace transforms: An efficient improvement to Dubner and Abate’s method.” Comput. J. 17 (4): 371–376. https://doi.org/10.1093/comjnl/17.4.371.
Edil, T. B., and I. B. Mochtar. 1988. “Creep response of model pile in clay.” J. Geotech. Eng. 114 (11): 1245–1260. https://doi.org/10.1061/(ASCE)0733-9410(1988)114:11(1245).
Feng, S., X. Li, F. Jiang, L. Lei, and Z. Chen. 2017. “A nonlinear approach for time-dependent settlement analysis of a single pile and pile groups.” Soil Mech. Found. Eng. 54 (1): 7–16. https://doi.org/10.1007/s11204-017-9426-8.
Feng, S. Y., L. M. Wei, C. Y. He, and Q. He. 2014. “A computational method for post-construction settlement of high-speed railway bridge pile foundation considering soil creep effect.” J. Central South Univ. 21 (7): 2921–2927. https://doi.org/10.1007/s11771-014-2258-z.
Flaate, K. 1972. “Effects of pile driving in clays.” Can. Geotech. J. 9 (1): 81–88. https://doi.org/10.1139/t72-006.
Gaver, D. P. 1966. “Observing stochastic processes, and approximate transform inversion.” Oper. Res. 14 (3): 444–459. https://doi.org/10.1287/opre.14.3.444.
Gemant, A. 1936. “A method of analyzing experimental results obtained from elasto-viscous bodies.” Physics 7 (8): 311–317. https://doi.org/10.1063/1.1745400.
Guo, W. D. 2000a. “Visco-elastic consolidation subsequent to pile installation.” Comput. Geotech. 26 (2): 113–144. https://doi.org/10.1016/S0266-352X(99)00028-2.
Guo, W. D. 2000b. “Visco-elastic load transfer models for axially loaded piles.” Int. J. Numer. Anal. Methods Geomech. 24 (2): 135–163. https://doi.org/10.1002/(SICI)1096-9853(200002)24:2%3C135::AID-NAG56%26gt;3.0.CO;2-8.
Guo, W. D., and M. F. Randolph. 1997. “Vertically loaded piles in non-homogeneous media.” Int. J. Numer. Anal. Methods Geomech. 21 (8): 507–532. https://doi.org/10.1002/(SICI)1096-9853(199708)21:8%3C507::AID-NAG888%26gt;3.0.CO;2-V.
Guo, W. D., and M. F. Randolph. 1998. “Rationality of load transfer approach for pile analysis.” Comput. Geotech. 23 (1–2): 85–112. https://doi.org/10.1016/S0266-352X(98)00010-X.
Huang, M. H., and J. C. Li. 2019. “Consolidation of viscoelastic soil by vertical drains incorporating fractional-derivative model and time-dependent loading.” Int. J. Numer. Anal. Methods Geomech. 43 (1): 239–256. https://doi.org/10.1002/nag.2861.
Jeong, S., D. Seo, and Y. Kim. 2009. “Numerical analysis of passive pile groups in offshore soft deposits.” Comput. Geotech. 36 (7): 1164–1175. https://doi.org/10.1016/j.compgeo.2009.05.003.
Kawada, Y., T. Yajima, and H. Nagahama. 2013. “Fractional-order derivative and time-dependent viscoelastic behaviour of rocks and minerals.” Acta Geophys. 61 (6): 1690–1702. https://doi.org/10.2478/s11600-013-0153-x.
Kelesoglu, M. K., and S. M. Springman. 2011. “Analytical and 3D numerical modelling of full-height bridge abutments constructed on pile foundations through soft soils.” Comput. Geotech. 38 (8): 934–948. https://doi.org/10.1016/j.compgeo.2011.07.011.
Koeller, R. C. 1984. “Applications of fractional calculus to the theory of viscoelasticity.” J. Appl. Mech. 51 (2): 299–307. https://doi.org/10.1115/1.3167616.
Konrad, J.-M., and M. Roy. 1987. “Bearing capacity of friction piles in marine clay.” Géotechnique 37 (2): 163–175. https://doi.org/10.1680/geot.1987.37.2.163.
Kraft, L. M., R. P. Ray, and T. Kagawa. 1981. “Theoretical t-z curves.” J. Geotech. Eng. Div. 107 (11): 1543–1561.
Li, X. M., Q. Q. Zhang, R. F. Feng, J. G. Qian, and H. W. Wei. 2020. “Long-term deformation analysis for a vertical concentrated force acting in the interior of fractional derivative viscoelastic soils.” Int. J. Geomech. 20 (5): 04020040. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001649.
Li, Z. Y., K. H. Wang, S. H. Lv, and W. B. Wu. 2015. “A new approach for time effect analysis in the settlement of single pile in nonlinear viscoelastic soil deposits.” J. Zhejiang Univ.-Sci. A 16 (8): 630–643. https://doi.org/10.1631/jzus.A1400329.
Meral, F. C., T. J. Royston, and R. Magin. 2010. “Fractional calculus in viscoelasticity: An experimental study.” Commun. Nonlinear Sci. Numer. Simul. 15 (4): 939–945. https://doi.org/10.1016/j.cnsns.2009.05.004.
Mishra, A., and N. R. Patra. 2018. “Time-dependent settlement of pile foundations using five-parameter viscoelastic soil models.” Int. J. Geomech. 18 (5): 04018020. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001122.
Mishra, A., and N. R. Patra. 2019. “Analysis of creep settlement of pile groups in linear viscoelastic soil.” Int. J. Numer. Anal. Methods Geomech. 43 (14): 2288–2304. https://doi.org/10.1002/nag.2976.
Nguyen, V. T., A. M. Tang, and J. M. Pereira. 2017. “Long-term thermo-mechanical behavior of energy pile in dry sand.” Acta Geotech. 12 (4): 729–737. https://doi.org/10.1007/s11440-017-0539-z.
Olgun, C. G., T. Y. Ozudogru, S. L. Abdelaziz, and A. Senol. 2015. “Long-term performance of heat exchanger piles.” Acta Geotech. 10 (5): 553–569. https://doi.org/10.1007/s11440-014-0334-z.
Pasten, C., and J. C. Santamarina. 2014. “Thermally induced long-term displacement of thermoactive piles.” J. Geotech. Geoenviron. Eng. 140 (5): 06014003. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001092.
Podlubny, I. 1999. Fractional differential equations. Millbrae, CA: Academic Press.
Randolph, M. F., and C. P. Wroth. 1978. “Analysis of deformation of vertically loaded piles.” J. Geotech. Eng. Div. 104 (12): 1465–1488.
Small, J. C., and H. L. S. Liu. 2008. “Time-settlement behaviour of piled raft foundations using infinite elements.” Comput. Geotech. 35 (2): 187–195. https://doi.org/10.1016/j.compgeo.2007.04.004.
Stehfest, H. 1970. “Algorithm 368: Numerical inversion of Laplace transforms [D5].” Commun. ACM 13 (1): 47–49. https://doi.org/10.1145/361953.361969.
Timoshenko, S. P., and J. N. Goodier. 1970. Theory of elasticity. 3rd ed. New York: McGraw-Hill.
Valkó, P. P., and J. Abate. 2004. “Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion.” Comput. Math. Appl. 48 (3–4): 629–636. https://doi.org/10.1016/j.camwa.2002.10.017.
Wang, L., D. A. Sun, P. 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.
Wimp, J. 1981. Sequence transformations and their applications. New York: Academic Press.
Wu, W. B., K. H. Wang, Z. Q. Zhang, and J. L. Chin. 2012. “A new approach for time effect analysis of settlement for single pile based on virtual soil-pile model.” J. Central South Univ. 19 (9): 2656–2662. https://doi.org/10.1007/s11771-012-1324-7.
Zeng, Y. Q., J. Zhou, and J. T. Qu. 2005. “Method for long-term settlement prediction of pile-foundation in consideration of time effect of stress-strain relationship.” [In Chinese.] Rock Soil Mech. 26 (8): 1283–1287.
Zhang, C. C., H. H. Zhu, B. Shi, and L. C. Liu. 2014. “Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation.” J. Rock Mech. Geotech. Eng. 6 (4): 373–379. https://doi.org/10.1016/j.jrmge.2014.04.007.
Zhang, Q. Q., S. W. Liu, R. F. Feng, and X. M. Li. 2019. “Analytical method for prediction of progressive deformation mechanism of existing piles due to excavation beneath a pile-supported building.” Int. J. Civ. Eng. 17 (6): 751–763. https://doi.org/10.1007/s40999-018-0309-9.
Zhao, C. Y., W. M. Leng, and G. Y. Zheng. 2013. “Calculation and analysis for the time-dependency of settlement of the single-driven pile in double-layered soft clay.” Appl. Clay Sci. 79 (7): 8–12. https://doi.org/10.1016/j.clay.2013.02.022.
Zhu, H. H., C. C. Zhang, G. X. Mei, B. Shi, and L. Gao. 2017. “Prediction of one-dimensional compression behavior of Nansha clay using fractional derivatives.” Mar. Georesour. Geotechnol. 35 (5): 688–697. https://doi.org/10.1080/1064119X.2016.1217958.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 21 • Issue 2 • February 2021
Copyright
© 2020 American Society of Civil Engineers.
History
Received: May 3, 2020
Accepted: Sep 4, 2020
Published online: Nov 19, 2020
Published in print: Feb 1, 2021
Discussion open until: Apr 19, 2021
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.