1
R=0.95 ; E=0.89 ; RMSE = 0.17
Fig. 6.23 Comparision of notional probabilities of liquefaction obtained for 144 cases based on β2 (using µcmf=2.08 and COV=0.2) with PL obtained from mapping function based on β1
Another example of a liquefied case from 1979, Imperial Valley, California earthquake at Mckim Ranch A as presented in Moss (2003) has been analyzed to find PL. There was
occurrence of liquefaction at critical depth, d =2.75m; qc = 2.69 MPa; fs = 30.56 kPa; σv =
47.75 kPa; σ’v = 35.49 kPa; amax = 0.51 g and Mw = 6.5. The COV of the parameters: qc, fs,
σv, σ’v, amax and Mw are 32.3, 14.2, 17, 12.3, 9.8 and 2%, respectively. The CSR model
(Eq.6.6) and the CRR (Eq.6.44) are used to form the limit state of liquefaction and considering the model uncertainty (µcmf = 2.08 and COV = 0.2), FORM analysis was made
using the developed code in MATLAB. The reliability index, β2 and corresponding notional
probability of liquefaction PL using Eq. (6.18) are found out to be -2.271 and 0.99
respectively, and the result confirms the case as liquefied one. These two examples illustrate the procedure for evaluation of PL of a site in a future seismic event using the proposed
reliability based analysis if the uncertainties of soil and seismic parameters of the site are known.
168 6.4 CONCLUSIONS
The conclusions drawn from probabilistic method for both SPT and CPT-based models are presented separately as follows.
6.4.1 Conclusions based on SPT- based reliability analysis
The following conclusions are drawn based on the results and discussion of SPT-based reliability analysis as presented above.
i. The developed MGGP-based CRR model has been characterized with an uncertainty of mean value 0.98 and COV of 0.1 on the basis rigorous FORM analysis of 94 cases of the database. As the mean value of model uncertainty is very close to 1 the CRR model, which represents the boundary surface separating the liquefied cases from non-liquefied cases, can be considered as un-biased one. This is also evident from the proposed PL-Fs mapping function that yields PL=0.503, when Fs=1. Thus, the
degree of conservatism of the limit state boundary surface is quantified in terms of PL as 50.3%.
ii. The probability of liquefaction, PL can be estimated from the developed PL-β1
mapping function using a reliability index, β1 that is calculated by FORM
considering only parameter uncertainties. Alternatively, the reliability index β2 can
be determined by FORM considering both model and parameter uncertainties, and then, the PL can be obtained with the notional probability concept (using Eq.
6.18).The notional concept to estimate the PL of a future case is preferred as the
model uncertainty of the adopted limit state has been determined and also it is a well- accepted approach in the reliability theory.
iii. In absence of parameter uncertainties the proposed PL-Fs mapping function as
defined by Eq.(6.33) can be used to estimate probability of liquefaction, where the Fs
is calculated based on the CSR and CRR models as presented by Eq. (6.6) and Eq. (6.32), respectively.
iv. Using the developed code for FORM two examples, one liquefied case and the other non-liquefied case, are analyzed and corresponding probability of liquefaction on the
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basis of notional probability concept, using the obtained “true” uncertainty of the limit state model were found out to be 0.91 and 0.49, respectively. The PL of the
above liquefied case was also found out to be 0.91 as per the available regression- based reliability method. This indicates accuracy of the proposed MGGP-based reliability method for evaluation liquefaction potential on the basis of the above example.It is pertinent to mention here that the proposed MGGP-based reliability method is developed on basis of the most recent CSR formulation whereas the available reliability method is based on an older CSR model.
6.4.2 Conclusions based on CPT- based reliability analysis
The following conclusions are drawn based on the results and discussion of CPT-based reliability analysis as presented above.
i. The developed MGGP-based CRR model has been characterized with an uncertainty of mean value 2.08 and COV of 0.2 on the basis rigorous FORM analysis of 144 cases of the post liquefaction CPT database.
ii. As discussed for the SPT database, the probability of liquefaction, PL can be
estimated from the developed PL-β1 mapping function using a reliability index, β1
that is calculated by FORM considering only parameter uncertainties. Alternatively, the reliability index β2 can be determined by FORM considering both model and
parameter uncertainties, and then, the PL can be obtained with the notional
probability concept (using Eq. 6.18).The notional concept to estimate the PL of a
future case is preferred as the model uncertainty of the adopted limit state has been determined and also it is a well-accepted approach in the reliability theory.
iii. The characterization of the developed MGGP-based limit state model uncertainty using the proposed reliability analysis was based on all the 144 cases of the database with maximum COV of input parameters is 1.26, whereas the estimation of uncertainty of available ANN-based limit state model was made using only 64 cases of the same database with maximum COV of the input parameters is 0.30. Thus, the proposed MGGP-based reliability method can be used for wider range of COV of
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soil parameters compared to the available ANN-based reliability method of Juang et al. (2006).
iv. In absence of parameter uncertainties the proposed PL-Fs mapping function as
defined by Eq.(6.45) can be used to estimate probability of liquefaction, where the Fs
is calculated based on the CSR and CRR model as presented by Eq. (6.6) and Eq. (6.44), respectively.
v. Using the developed code for FORM two examples, one non-liquefied case and the other liquefied case, are analyzed and the corresponding probability of liquefaction on the basis of notional probability concept, using the obtained “true” uncertainty of the limit state model, are found out to be 0.17 and 0.99, respectively. But, the PL of
the non-liquefied case was found out to be 0.27 as per the available ANN-based reliability method, which is slightly more than the present finding. This indicates accuracy of the proposed MGGP-based reliability method for evaluation liquefaction potential is more than that of the available ANN-based reliability method on the basis of the above example.
171 7.1 SUMMARY
Natural hazards like earthquake, tsunami, flood, cyclone and landslide pose severe threat to human life and its environment. But, now days natural hazards are no longer considered as a rare act of God, but as a recurrent natural phenomenon whose disastrous effects can and should be mitigated. Out of the various seismic hazards, soil liquefaction is a major cause of both loss of life and damage to infrastructures and lifeline systems. Soil liquefaction phenomena have been noticed in many historical earthquakes after first large scale observations of damage caused by liquefaction in the 1964 Niigata, Japan and 1964 Alaska, USA, earthquakes. Accurate evaluation of liquefaction potential of soil is one of the most important steps towards mitigating liquefaction hazard. Though, different approaches like cyclic strain-based, energy-based and cyclic stress-based approach are in use, the stress- based approach is the most widely used method for evaluation of liquefaction potential of soil. Seed and Idriss (1971) first developed a stress-based simplified semi-empirical model, using laboratory tests and post liquefaction SPT-based field manifestations in earthquakes, which presents a limit state function that separates liquefied cases from the non-liquefied cases. Due to difficulty in obtaining high quality undisturbed samples and cost involved therein, further development of this simplified method was made using SPT- based field test data (Seed et al. 1983 and Seed et al. 1985). Though, SPT is most widely used soil exploration method now a days cone penetration test (CPT) is also preferred by geotechnical engineers for liquefaction potential evaluation as it is consistent, repeatable and also able to identify continuous soil profile.