Impact of Autocorrelation Function Model on the Probability of Failure
Publication: Journal of Engineering Mechanics
Volume 145, Issue 1
Abstract
The scale of fluctuation (SOF) of a spatially variable soil property has been known to be the most important parameter that characterizes the effect of spatial averaging, whereas the type of the autocorrelation model (e.g., single exponential versus squared exponential model) is thought to be of limited impact. This paper shows that when extending this statement (SOF is the most important parameter) to the probability of failure, one must be cautious regarding whether the limit-state function is completely governed by spatial averaging. Spatial averaging is a function of the input random field in its classical form—it is not related to the limit-state function. For a limit state that happens to be completely governed by spatial averaging, e.g., a friction pile under axial compression, the statement is indeed true, but for a limit state that is not completely governed by spatial averaging, the statement may not be true and the type of autocorrelation model can have significant impact. This paper shows that the second type of limit-state functions are not uncommon. In particular, this paper shows that the sample path smoothness can be another important feature that signifcantly affects the probability of failure for this type of limit-state function. An autocorrelation model that can control the sample path smoothness using a smoothness parameter is adopted in this paper. Five practical examples are presented to illustrate the effect of . It is observed that , which is another characteristic of the autocorrelation model that is distinctive from the scale of fluctuation, can have significant impact on the probability of failure for these examples.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors would like to thank Dr. Yu-Gang Hu and Miss Tzu-Ting Lin for their efforts in producing the results in some plots.
References
Abramowitz, M., and I. Stegun. 1970. Handbook of mathematical functions. New York: Dover.
Akbas, S. O., and F. H. Kulhawy. 2009. “Reliability-based design approach for differential settlement of footings on cohesionless soils.” J. Geotech. Geoenviron. Eng. 135 (12): 1779–1788. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000127.
CEN (European Committee for Standardization). 1994. Geotechnical design, general rules. Part 1: Eurocode 7. Brussels, Belgium: CEN.
Ching, J., S. W. Lee, and K. K. Phoon. 2016. “Undrained strength for a 3D spatially variable clay column subjected to compression or shear.” Probab. Eng. Mech. 45: 127–139. https://doi.org/10.1016/j.probengmech.2016.03.002.
Ching, J., and K. K. Phoon. 2013a. “Mobilized shear strength of spatially variable soils under simple stress states.” Struct. Saf. 41: 20–28. https://doi.org/10.1016/j.strusafe.2012.10.001.
Ching, J., and K. K. Phoon. 2013b. “Probability distribution for mobilized 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., K. K. Phoon, J. L. Beck, and Y. Huang. 2018a. “On the identification of geotechnical site-specific trend function.” ASCE-ASME J. Risk Uncertainty Eng. Syst. Part A: Civ. Eng. 3 (4): 04017021. https://doi.org/10.1061/AJRUA6.0000926.
Ching, J., K. K. Phoon, and P. H. Kao. 2014. “Mean and variance of the mobilized shear strengths 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.
Ching, J., K. K. Phoon, and S. P. Sung. 2017. “Worst case scale of fluctuation in basal heave analysis involving spatially variable clays.” Struct. Saf. 68: 28–42. https://doi.org/10.1016/j.strusafe.2017.05.008.
Ching, J., and S. P. Sung. 2016. “Spatial averaging of stationary random fields along curves—Simulation and variance reduction.” J. GeoEng. 11 (1): 33–43. http://dx.doi.org/10.6310%2fjog.2016.11(1).4.
Ching, J., T. J. Wu, A. W. Stuedlein, and T. Bong. 2018b. “Estimating horizontal scale of fluctuation with limited CPT soundings.” Geosci. Front.9 (6): 1597–1608. https://doi.org/10.1016/j.gsf.2017.11.008.
Christian, J. T., and W. D. Carrier. 1978. “Janbu, Bjerrum, and Kjaernsli’s chart reinterpreted.” Can. Geotech. J. 15 (1): 124–128. https://doi.org/10.1139/t78-010.
DeGroot, D. J., and G. B. Baecher. 1993. “Estimating autocovariances of in-situ soil properties.” J. Geotech. Eng. 119 (1): 147–166. https://doi.org/10.1061/(ASCE)0733-9410(1993)119:1(147).
Fenton, G. A., and D. V. Griffiths. 2008. Risk assessment in geotechnical engineering. New York: Wiley.
Griffiths, D. V., J. Huang, and G. A. Fenton. 2011. “Probabilistic infinite slope analysis.” Comput. Geotech. 38 (4): 577–584. https://doi.org/10.1016/j.compgeo.2011.03.006.
Guttorp, P., and T. Gneiting. 2006. “Studies in the history of probability and statistics XLIX on the Matérn correlation family.” Biometrika 93 (4): 989–995. https://doi.org/10.1093/biomet/93.4.989.
Hicks, M. A. 2013. “An explanation of characteristic values of soil properties in Eurocode 7.” In Modern geotechnical design code of practice, edited by P. Arnold, G. A. Fenton, M. A. Hicks, T. Schweckendiek, and B. Simpson. Amsterdam, Netherlands: IOS Press.
Hicks, M. A., and J. D. Nuttall. 2012. “Influence of soil heterogeneity on geotechnical performance and uncertainty: A stochastic view on EC7.” In Proc., 10th Int. Probabilistic Workshop. Stuttgart, Germany: Institut für Geotechnik.
Hristopulos, D. T., and M. Žukovič. 2011. “Relationships between correlation lengths and integral scales for covariance models with more than two parameters.” Stochastic Environ. Res. Risk Assess. 25 (1): 11–19. https://doi.org/10.1007/s00477-010-0407-y.
Hu, Y. G., and J. Ching. 2015. “Impact of spatial variability in soil shear strength on active lateral forces.” Struct. Saf. 52: 121–131. https://doi.org/10.1016/j.strusafe.2014.09.004.
Jaksa, M. B., W. S. Kaggwa, and P. I. Brooker. 1999. “Experimental evaluation of the scale of fluctuation of a stiff clay.” In Proc., 8th Int. Conf. on Application of Statistics and Probability, 415–422. Rotterdam, Netherlands: A.A. Balkema.
Janbu, N., L. Bjerrum, and B. Kjaernsli. 1956. Veiledning vedlosning av fundamentering–soppgaver. [In Norwegian.]. Oslo, Norway: Norwegian Geotechnical Institute.
Jha, S. K., and J. Ching. 2013. “Simulating spatial averages of stationary random field using Fourier series method.” J. Eng. Mech. 139 (5): 594–605. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000517.
Lacasse, S., and F. Nadim. 1996. “Uncertainties in characterizing soil properties.” In Proc., Uncertainty in the Geologic Environment: From Theory to Practice, 49–75. Reston, VA: ASCE.
Liu, W. F., Y. F. Leung, and M. K. Lo. 2017. “Integrated framework for characterization of spatial variability of geological profiles.” Can. Geotech. J. 54 (1): 47–58. https://doi.org/10.1139/cgj-2016-0189.
Marchant, B., and R. Lark. 2007. “The Matérn variogram model: Implications for uncertainty propagation and sampling in geostatistical surveys.” Geoderma 140 (4): 337–345. https://doi.org/10.1016/j.geoderma.2007.04.016.
Minasny, B., and A. B. McBratney. 2005. “The Matérn function as a general model for soil variograms.” Geoderma 128 (3–4): 192–207. https://doi.org/10.1016/j.geoderma.2005.04.003.
Phoon, K. K., S. T. Quek, and P. An. 2003. “Identification of statistically homogeneous soil layers using modified Bartlett statistics.” J. Geotech. Geoenviron. Eng. 129 (7): 649–659. https://doi.org/10.1061/(ASCE)1090-0241(2003)129:7(649).
Rasmussen, C. E., and C. K. I. Williams. 2006. Gaussian processes for machine learning. Cambridge, MA: MIT Press.
Santoso, A., K. K. Phoon, and S. T. Quek. 2011. “Probability models for SWCC and hydraulic conductivity.” In Proc., 14th Asian Regional Conf. on Soil Mechanics and Geotechnical Engineering. Hong Kong: Hong Kong Polytechnic Univ.
Stein, M. L. 1999. Interpolation of spatial data: Some theory for kriging. New York: Springer.
Timoshenko, S., and J. N. Goodier. 1970. Theory of elasticity. 3rd ed. New York: McGraw-Hill.
Uzielli, M., G. Vannucchi, and K. K. Phoon. 2005. “Random field characterisation of stress-normalised cone penetration testing parameters.” Geotechnique 55 (1): 3–20. https://doi.org/10.1680/geot.2005.55.1.3.
Vanmarcke, E. H. 1977. “Probabilistic modeling of soil profiles.” J. Geotech. Eng. Div. ASCE 103 (11): 1227–1246.
Vanmarcke, E. H. 1983. Random fields: Analysis and synthesis. Cambridge, MA: MIT Press.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 145 • Issue 1 • January 2019
Copyright
©2018 American Society of Civil Engineers.
History
Received: Jun 1, 2017
Accepted: Jun 28, 2018
Published online: Oct 26, 2018
Published in print: Jan 1, 2019
Discussion open until: Mar 26, 2019
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.