Identification of Important Random Field Domain in Foundation Engineering through Reliability Sensitivity Analysis
Publication: Journal of Geotechnical and Geoenvironmental Engineering
Volume 150, Issue 6
Abstract
In the random field model’s consideration of the spatial variability of soil, soil properties at different locations play different roles in the reliability analysis of the foundation. Investigating the importance distribution of the random field through reliability sensitivity analysis (RSA) is beneficial for understanding how the random field affects the reliability of the foundation. However, many existing RSA methods for the random field model are deficient in terms of efficiency, accuracy, and applicability under complex engineering conditions. Consequently, this study proposes an efficient RSA method for the random field model based on the Karhunen–Loève (KL) expansion method and the first-order reliability method (FORM) to identify the important random field domain in foundation engineering. In the proposed method, the mean reliability sensitivity index (MRSI) is extended to a random field model of continuous form to characterize the importance distribution of the random field. The MRSI is analytically derived based on the results of the KL expansion method and the FORM without additional limit state function (LSF) calculations. Subsequently, the important random field domain, in which the variation of the mean of the soil property contributes significantly to the reliability index, is identified based on the MRSI. Last, two foundation engineering examples that consider the cross-correlated random fields of cohesion and friction angle, including strip footing on single-layer soil and pile in multiple-layer soil, were used to verify the proposed method. The results showed that an important random field domain with a small area dominates the variation of the reliability index of a foundation, and important random field domain area increases with autocorrelation distance (ACD). This innovative identification method holds great engineering significance, because it allows geotechnical practitioners to gain a comprehensive understanding of the failure modes and foundation treatment areas of foundations in spatially varying soil.
Practical Applications
In the random field model’s consideration of the spatial variability of soil, soil properties at different locations play different roles in the reliability analysis of the foundation. Investigating the importance distribution of the random field through RSA is beneficial for understanding how the random field affects the reliability of the foundation. However, many existing RSA methods for the random field model are deficient in terms of efficiency, accuracy, and applicability under complex engineering conditions. Consequently, this study proposes an efficient RSA method for the random field model to identify the important random field domain, in which the variation of the mean of the soil property contributes significantly to the reliability index. Two foundation engineering examples that consider the cross-correlated random fields of cohesion and friction angle, including strip footing on single-layer soil and pile in multiple-layer soil, were used to verify the proposed method. The results showed that the innovative identification will allow geotechnical practitioners to gain a comprehensive understanding of the failure modes and foundation treatment areas of foundations in spatially varying soil.
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Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos. 41972278 and 42372302).
References
Betz, W., I. Papaioannou, and D. Straub. 2014. “Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion.” Comput. Methods Appl. Mech. Eng. 271 (Apr): 109–129. https://doi.org/10.1016/j.cma.2013.12.010.
Bowles, J. E. 1996. Foundation analysis and design. 5th ed. New York: McGraw-Hill.
Chan, C. L., and B. K. Low. 2012. “Sensitivity analysis of laterally loaded pile involving correlated non-normal variables.” Int. J. Geotech. Eng. 6 (2): 163–169. https://doi.org/10.3328/IJGE.2012.06.02.163-169.
Cheng, H. Z., J. Chen, R. P. Chen, and G. L. Chen. 2019. “Comparison of modeling soil parameters using random variables and random fields in reliability analysis of tunnel face.” Int. J. Geomech. 19 (1): 04018184. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001330.
Ching, J. Y., Y. G. Hu, and K. K. Phoon. 2018. “Effective Young’s modulus of a spatially variable soil mass under a footing.” Struct. Saf. 73 (Jul): 99–113. https://doi.org/10.1016/j.strusafe.2018.03.004.
Ching, J. Y., and K. K. Phoon. 2013a. “Mobilized shear strength of spatially variable soils under simple stress states.” Struct. Saf. 41 (Mar): 20–28. https://doi.org/10.1016/j.strusafe.2012.10.001.
Ching, J. Y., and K. K. Phoon. 2013b. “Probability distribution for mobilised shear strengths of spatially variable soils under uniform stress states.” Georisk 7 (3): 209–224. https://doi.org/10.1080/17499518.2013.801273.
Ching, J. Y., K. K. Phoon, and P. H. Kao. 2014. “Mean and variance of mobilized shear strength for spatially variable soils under uniform stress states.” J. Eng. Mech. 140 (3): 487–501. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000667.
Cho, S. E., and H. C. Park. 2010. “Effect of spatial variability of cross-correlated soil properties on bearing capacity of strip footing.” Int. J. Numer. Anal. Methods Geomech. 34 (1): 1–26. https://doi.org/10.1002/nag.791.
Deodatis, G. 1991. “Weighted integral method. I: Stochastic stiffness matrix.” J. Eng. Mech. 117 (8): 1851–1864. https://doi.org/10.1061/(ASCE)0733-9399(1991)117:8(1851).
Der Kiureghian, A., and J. B. Ke. 1988. “The stochastic finite element method in structural reliability.” Probab. Eng. Mech. 3 (2): 83–91. https://doi.org/10.1016/0266-8920(88)90019-7.
Fei, S. Z., X. H. Tan, W. P. Gong, X. L. Dong, F. S. Zha, and L. Xu. 2021. “Reliability analysis of strip footing under rainfall using KL–FORM.” Geomech. Eng. 24 (2): 167–178. https://doi.org/10.12989/gae.2021.24.2.167.
Fei, S. Z., X. H. Tan, X. Lin, Y. Xiao, F. S. Zha, and L. Xu. 2022. “Evaluation of the scale of fluctuation based on variance reduction method.” Eng. Geol. 308 (Oct): 106804. https://doi.org/10.1016/j.enggeo.2022.106804.
Fenton, G. A., and D. V. Griffiths. 2002. “Probabilistic foundation settlement on spatially random soil.” J. Geotech. Geoenviron. Eng. 128 (5): 381–390. https://doi.org/10.1061/(ASCE)1090-0241(2002)128:5(381).
Fenton, G. A., and D. V. Griffiths. 2005. “Three-dimensional probabilistic foundation settlement.” J. Geotech. Geoenviron. Eng. 131 (2): 232–239. https://doi.org/10.1061/(ASCE)1090-0241(2005)131:2(232).
Fenton, G. A., and E. H. Vanmarcke. 1990. “Simulation of random fields via local average subdivision.” J. Eng. Mech. 116 (8): 1733–1749. https://doi.org/10.1061/(ASCE)0733-9399(1990)116:8(1733).
Haldar, S., and G. L. Sivakumar Babu. 2012. “Response of vertically loaded pile in clay: A probabilistic study.” Geotech. Geol. Eng. 30 (Feb): 187–196. https://doi.org/10.1007/s10706-011-9461-6.
Hasofer, A. M., and N. C. Lind. 1974. “Exact and invariant second-moment code format.” J. Eng. Mech. Div. 100 (1): 111–121. https://doi.org/10.1061/JMCEA3.0001848.
Hu, Y. G., and J. Y. Ching. 2015. “Impact of spatial variability in undrained shear strength on active lateral force in clay.” Struct. Saf. 52 (Jan): 121–131. https://doi.org/10.1016/j.strusafe.2014.09.004.
Ji, J., and J. K. Kodikara. 2015. “Efficient reliability method for implicit limit state surface with correlated non-Gaussian variables.” Int. J. Numer. Anal. Methods Geomech. 39 (17): 1898–1911. https://doi.org/10.1002/nag.2380.
Ji, J., C. S. Zhang, Y. F. Gao, and J. Kodikara. 2018. “Effect of 2D spatial variability on slope reliability: A simplified FORM analysis.” Geosci. Front. 9 (6): 1631–1638. https://doi.org/10.1016/j.gsf.2017.08.004.
Ji, J., C. S. Zhang, Y. F. Gao, and J. Kodikara. 2019. “Reliability-based design for geotechnical engineering: An inverse FORM approach for practice.” Comput. Geotech. 111 (Jul): 22–29. https://doi.org/10.1016/j.compgeo.2019.02.027.
Jiang, S. H., and J. S. Huang. 2016. “Efficient slope reliability analysis at low-probability levels in spatially variable soils.” Comput. Geotech. 75 (May): 18–27. https://doi.org/10.1016/j.compgeo.2016.01.016.
Jiang, S. H., D. Q. Li, Z. J. Cao, C. B. Zhou, and K. K. Phoon. 2015. “Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation.” J. Geotech. Geoenviron. Eng. 141 (2): 04014096. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001227.
Jiang, S. H., X. Liu, J. S. Huang, and C. B. Zhou. 2022. “Efficient reliability-based design of slope angles in spatially variable soils with field data.” Int. J. Numer. Anal. Methods Geomech. 46 (13): 2461–2490. https://doi.org/10.1002/nag.3414.
Jiang, S. H., I. Papaioannou, and D. Straub. 2018. “Bayesian updating of slope reliability in spatially variable soils with in-situ measurements.” Eng. Geol. 239 (May): 310–320. https://doi.org/10.1016/j.enggeo.2018.03.021.
Jiang, S. H., G. Y. Zhu, Z. Z. Wang, Z. T. Huang, and J. S. Huang. 2023. “Data augmentation for CNN-based probabilistic slope stability analysis in spatially variable soils.” Comput. Geotech. 160 (Aug): 105501. https://doi.org/10.1016/j.compgeo.2023.105501.
Juang, C. H., J. Zhang, M. F. Shen, and J. Z. Hu. 2019. “Probabilistic methods for unified treatment of geotechnical and geological uncertainties in a geotechnical analysis.” Eng. Geol. 249 (Jan): 148–161. https://doi.org/10.1016/j.enggeo.2018.12.010.
Li, C. C., and A. Der Kiureghian. 1993. “Optimal discretization of random fields.” J. Eng. Mech. 119 (6): 1136–1154. https://doi.org/10.1061/(ASCE)0733-9399(1993)119:6(1136).
Li, D. Q., X. H. Qi, Z. J. Cao, X. S. Tang, W. Zhou, K. K. Phoon, and C. B. Zhou. 2015a. “Reliability analysis of strip footing considering spatially variable undrained shear strength that linearly increases with depth.” Soils Found. 55 (4): 866–880. https://doi.org/10.1016/j.sandf.2015.06.017.
Li, J. H., Y. H. Tian, and M. J. Cassidy. 2015b. “Failure mechanism and bearing capacity of footings buried at various depths in spatially random soil.” J. Geotech. Geoenviron. Eng. 141 (2): 04014099. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001219.
Li, Y. J., M. A. Hicks, and P. J. Vardon. 2016. “Uncertainty reduction and sampling efficiency in slope designs using 3D conditional random fields.” Comput. Geotech. 79 (Oct): 159–172. https://doi.org/10.1016/j.compgeo.2016.05.027.
Lin, X., X. H. Tan, S. Z. Fei, Z. H. Sun, and J. Zhang. 2022a. “Regional reliability sensitivity analysis considering spatial variability of soil.” In Proc., 8th Int. Symp. On Geotechnical Safety and Risk (ISGSR 2022), 376–382. Singapore: Research.
Lin, X., X. H. Tan, Y. C. Yao, X. L. Dong, S. Z. Fei, and L. Ma. 2022b. “Realization of multi-dimensional random field based on Jacobi–Lagrange–Galerkin method in geotechnical engineering.” Comput. Geotech. 144 (Apr): 104533. https://doi.org/10.1016/j.compgeo.2021.104533.
Liu, L. L., Y. M. Cheng, and S. H. Zhang. 2017. “Conditional random field reliability analysis of a cohesion-frictional slope.” Comput. Geotech. 82 (Feb): 173–186. https://doi.org/10.1016/j.compgeo.2016.10.014.
Liu, W. K., T. Belytschko, and A. Mani. 1986. “Random field finite elements.” Int. J. Numer. Anal. Methods Geomech. 23 (10): 1831–1845. https://doi.org/10.1002/nme.1620231004.
Lo, M. K., and Y. F. Leung. 2017. “Probabilistic analyses of slopes and footings with spatially variable soils considering cross-correlation and conditioned random field.” J. Geotech. Geoenviron. Eng. 143 (9): 04017044. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001720.
Lo, M. K., and Y. F. Leung. 2018. “Reliability assessment of slopes considering sampling influence and spatial variability by Sobol’ sensitivity index.” J. Geotech. Geoenviron. Eng. 144 (4): 04018010. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001852.
Low, B. K. 2014. “FORM, SORM, and spatial modeling in geotechnical engineering.” Struct. Saf. 49 (Aug): 56–64. https://doi.org/10.1016/j.strusafe.2013.08.008.
Lumb, P. 1966. “The variability of natural soils.” Can. Geotech. J. 3 (2): 74–97. https://doi.org/10.1139/t66-009.
Luo, Z., S. Atamturktur, Y. Q. Cai, and C. H. Juang. 2012. “Reliability analysis of basal-heave in a braced excavation in a 2-D random field.” Comput. Geotech. 39 (Jan): 27–37. https://doi.org/10.1016/j.compgeo.2011.08.005.
Madsen, H. O., S. Krenk, and N. C. Lind. 1986. Methods of structural safety. Englewood Cliffs, NJ: Prentice Hall.
Moore, E. H. 1920. “On the reciprocal of the general algebraic matrix.” Bull. Am. Math. Soc. 26 (9): 412–426. https://doi.org/10.1090/S0002-9904-1920-03332-X.
Penrose, R. 1955. “A generalized inverse for matrices.” Math. Proc. Cambridge Philos. Soc. 51 (3): 406–413. https://doi.org/10.1017/S0305004100030401.
Phoon, K. K., Z. J. Cao, J. Ji, Y. F. Leung, S. Najjar, T. Shuku, C. Tang, Z. Y. Yin, Y. Ikumasa, and J. Y. Ching. 2022. “Geotechnical uncertainty, modeling, and decision making.” Soils Found. 62 (5): 101189. https://doi.org/10.1016/j.sandf.2022.101189.
Phoon, K. K., S. P. Huang, and S. T. Quek. 2002. “Implementation of Karhunen–Loeve expansion for simulation using a wavelet-Galerkin scheme.” Probab. Eng. Mech. 17 (3): 293–303. https://doi.org/10.1016/S0266-8920(02)00013-9.
Phoon, K. K., and F. H. Kulhawy. 1999. “Characterization of geotechnical variability.” Can. Geotech. J. 36 (4): 612–624. https://doi.org/10.1139/t99-038.
Polak, E. 1997. Optimization: Algorithms and consistent approximations. New York: Springer.
Popescu, R., G. Deodatis, and J. H. Prevost. 1998. “Simulation of homogeneous nonGaussian stochastic vector fields.” Probab. Eng. Mech. 13 (1): 1–13. https://doi.org/10.1016/S0266-8920(97)00001-5.
Rackwitz, R. 2000. “Reviewing probabilistic soils modeling.” Comput. Geotech. 26 (3–4): 199–223. https://doi.org/10.1016/S0266-352X(99)00039-7.
Schuëller, G. I., and H. A. Jensen. 2008. “Computational methods in optimization considering uncertainties–An overview.” Comput. Methods Appl. Mech. Eng. 198 (1): 2–13. https://doi.org/10.1016/j.cma.2008.05.004.
Tabarroki, M., J. Y. Ching, C. P. Lin, J. J. Liou, and K. K. Phoon. 2022. “Homogenizing spatially variable Young modulus using pseudo incremental energy method.” Struct. Saf. 97 (Apr): 102226. https://doi.org/10.1016/j.strusafe.2022.102226.
Tan, X. H., P. Li, M. F. Shen, M. Z. Hu, X. L. Hou, and H. C. Ma. 2020. “Evaluation of the spatial variability characteristics of the unsaturated clay in Hefei, China.” Soils Found. 60 (2): 454–465. https://doi.org/10.1016/j.sandf.2020.03.010.
Tang, X. S., D. Q. Li, G. Rong, K. K. Phoon, and C. B. Zhou. 2013. “Impact of copula selection on geotechnical reliability under incomplete probability information.” Comput. Geotech. 49 (Apr): 264–278. https://doi.org/10.1016/j.compgeo.2012.12.002.
Vanmarcke, E. 1983. Random fields: Analysis and synthesis. Cambridge, MA: MIT Press.
Vořechovský, M. 2008. “Simulation of simply cross correlated random fields by series expansion methods.” Struct. Saf. 30 (4): 337–363. https://doi.org/10.1016/j.strusafe.2007.05.002.
Wang, M. X., X. S. Tang, D. Q. Li, and X. H. Qi. 2020. “Subset simulation for efficient slope reliability analysis involving copula-based cross-correlated random fields.” Comput. Geotech. 118 (Feb): 103326. https://doi.org/10.1016/j.compgeo.2019.103326.
Yu, Y., M. F. Shen, and C. H. Juang. 2019. “Assessing initial stiffness models for laterally loaded piles in undrained clay: Robust design perspective.” J. Geotech. Geoenviron. Eng. 145 (10): 04019073. https://doi.org/10.1061/(ASCE)GT.1943-5606.0002074.
Zhang, X. L., B. H. Jiao, Y. Han, S. L. Chen, and X. Y. Li. 2021. “Random field model of soil parameters and the application in reliability analysis of laterally loaded pile.” Soil Dyn. Earthquake Eng. 147 (Aug): 106821. https://doi.org/10.1016/j.soildyn.2021.106821.
Zhao, T. Y., and Y. Wang. 2018. “Simulation of cross-correlated random field samples from sparse measurements using Bayesian compressive sensing.” Mech. Syst. Signal Process. 112 (Nov): 384–400. https://doi.org/10.1016/j.ymssp.2018.04.042.
Zheng, Z. B., and H. Z. Dai. 2017. “Simulation of multi-dimensional random fields by Karhunen–Loève expansion.” Comput. Methods Appl. Mech. Eng. 324 (Sep): 221–247. https://doi.org/10.1016/j.cma.2017.05.022.
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Published In
Journal of Geotechnical and Geoenvironmental Engineering
Volume 150 • Issue 6 • June 2024
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© 2024 American Society of Civil Engineers.
History
Received: Jan 17, 2023
Accepted: Dec 26, 2023
Published online: Mar 26, 2024
Published in print: Jun 1, 2024
Discussion open until: Aug 26, 2024
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