Estimation of the Uncertainty of the Robertson and Wride Model for Reliability Analysis of Soil Liquefaction C. Hsein Juang and Susan H. Yang Clemson University

This study was sponsored by the National Science Foundation through Grants No. CMS ACKNOWLEDGMENT

Objectives of the Research Develop a procedure for estimating model uncertainty of limit state models Explore the use of FORM for calculating reliability index and probability of liquefaction Examine the robustness of Bayesian mapping approach

Outline of the Presentation Robertson and Wride Model First Order Reliability Method Parameter and Model Uncertainties Bayesian Mapping Approach Estimation of the Model Uncertainty Conclusions

Review of the Robertson and Wride Model Liquefaction loading: cyclic stress ratio (CSR) Liquefaction resistance: cyclic resistance ratio (CRR) CRR 7.5 = 93(q c1N,cs /1000) , if 50  q c1N,cs <160 CRR 7.5 = 0.833(q c1N,cs /1000)+0.05, if q c1N,cs < 50 CRR = f(q c, f s,  v,  v ) See Robertson and Wride (1998) for detail

Overview of the First Order Reliability Method First order second moment (FOSM) Advanced first order second moment (AFOSM) First order reliability method (FORM)

Reliability Index in the Reduced Variable Space  is defined as the shortest distance between the limit state surface and the origin in the reduced variable space

Limit State Function for Reliability Analysis of Soil Liquefaction When model uncertainty is not considered: g ( ) = CRR  CSR = g (q c, f s,  v,  v, a max, M w ) When model uncertainty is considered: g ( ) = c 1 CRR  CSR = g (c 1, q c, f s,  v,  v, a max, M w )

Model Uncertainty: the uncertainty in the limit state function CSR is used as a reference in the development of the CRR model through calibration with field cases; Whatever uncertainty there is in the CSR model is eventually passed along to the uncertainty in the CRR model. The effect of the uncertainty associated with the CSR model is realized in the CRR model.

Parameter uncertainties in the first baseline reliability analysis Mean to nominal = 1.0 Uncertainty of a parameter is characterized with a coefficient of variation (COV) COV_q c = 0.08COV_f s = 0.12 COV_  v = 0.10COV_  v = 0.10 COV_a max = 0.10COV_M w = 0.05

Coefficient of correlation of the input parameters

Probabilities of Liquefaction By means of notional probability concept By means of a Bayesian mapping function obtained through calibration of the calculated  using a database of field observations

Results of the reliability analysis of the CPT-based case histories

Estimation of Model Uncertainty: Premise The premise: A fully calibrated Bayesian mapping function is a true probability or at least a close approximation of the true probability. This premise stands as the calibration of the calculated reliability index is carried out with a sufficiently large database of case histories.

Estimation of Model Uncertainty: Methodology Model uncertainty is obtained through a trial and error process. The “correct” model uncertainty: The one that produces the probabilities matching best with those produced by the Bayesian mapping function that has been calibrated with a database of field observations.

Effect of the COV component of model uncertainty on β (a)  c1 = 1.0, COV = 0%(b)  c1 = 1.0, COV = 10% (c)  c1 = 1.0, COV = 15%(d)  c1 = 1.0, COV = 20%

Effect of the  c1 component of model uncertainty on β (c)  c1 = 1.0, COV = 10%(d)  c1 = 1.2, COV = 10% (a)  c1 = 0.8, COV = 10%(b)  c1 = 0.9, COV = 10%

Effect of the COV component of model uncertainty on P L (a)  c1 = 1.0, COV = 0%(b)  c1 = 1.0, COV = 10% (c)  c1 = 1.0, COV = 15%(d)  c1 = 1.0, COV = 20%

Effect of the  c1 component of model uncertainty on P L (a)  c1 = 0.8, COV = 10%(b)  c1 = 0.9, COV = 10% (c)  c1 = 1.0, COV = 10%(d)  c1 = 1.2, COV = 15%

Notional probability versus Bayesian probability The model uncertainty is characterized by  c1 = 0.94 and COV = 15% The parameter uncertainty is at the same level as in the first baseline analysis

Notional probability versus Bayesian probability (con’d) The model uncertainty is characterized by  c1 = 0.94 and COV = 15%, The parameter uncertainty is at the same level as in the second baseline analysis.

Conclusions The model uncertainty c 1 can be characterized with two statistical parameters, the mean (  c1 ), and the coefficient of variation (COV). The results show that the  c1 component has a more profound impact than does the COV component.

Conclusions (con’d) The first order reliability method (FORM) is shown to be able to estimate accurately  and P L, provided that the correct parameter and model uncertainties are incorporated in the analysis.

Conclusions (con’d) Robustness of the Bayesian mapping approach is demonstrated in this study. In the situation where the fully calibrated Bayesian mapping function is available, the model uncertainty may be estimated using the Bayesian mapping function.

Conclusions (con’d) The Robertson and Wride model is estimated to have a model uncertainty that is characterized with a  c1 of 0.94 and a COV of 15%.