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Introduction
To consider the uncertainties in the evaluation of liquefaction resistance, several studies have been conducted based on probabilistic analysis to improve the existing cyclic resistance ratio(CRR) curves proposed in the literature. Juang etal. (2000) have proposed a new approach for developing a liquefaction limit state function related to the Youd etal. (2001) model, which defines a boundary that separates liquefaction from no liquefaction occurrence. In Juang etal. (2009), a procedure for estimating uncertainty of the Youd etal. (2001) method was developed. Goharzay etal. (2017) used gene expression programming (GEP) to evaluate the occurrence of soil liquefaction in terms of liquefaction field performance and factor of safety in logistic regression by using the liquefaction resistance model of Idriss and Boulanger (2010). Sebaaly and Muhsin (2019) have also proposed a procedure to evaluate the uncertainty of the Idriss and Boulanger (2010) models based on Standard Penetration Test and Cone Penetration Test tests. Bagheripour etal. (2012) have performed a reliability analysis based on advanced first-order second-moment (AFOSM) technique associated with genetic algorithm (GA) to estimate the reliability index and the probability of liquefaction using the CRR model of Youd etal. (2001). Based on a probabilistic analysis, Al-Zoubi (2015) suggested a design method based on a predetermined reliability for selecting the coefficients of active and passive lateral earth pressures and their variations under seismic conditions. A reliability analysis of rock slope using soft computing techniques was conducted by Prithvendra etal. (2020) to show that Extreme Learning Machine and Multivariate Adaptive Regression Splines models are well capable of predicting the reliability of slope in terms of the factor of safety of rock slope, considering statistical predictnds.
After the earthquakes of Alaska (1964) and Nigata in Japan (1964), Seed and Idriss (1971) developed a simplified procedure based on insitu tests to evaluate the liquefaction potential, which is defined by a safety factor calculated by the ratio between CRR and the cyclic stress ratio (CRR/CSR). Thereafter, this procedure was modified and improved, in particular, by Seed (1979), Seed and Idriss (1982), Seed etal. (1985), and Youd etal. (1997, 2001). This procedure is based on simplifying the hypothesis by considering the soil column as a rigid body with the assumption that the actual peak shear stress (τd) induced at depth h is always less than that predicted by the simplified procedure (τr) of Seed and Idriss (1971) (rd= τd/τr<1). All the expressions proposed for rd in the literature are based on a many equivalent linear site response analyses in which each site is submitted to one corresponding earthquake. Filali and Sbartai (2017) conducted the same analysis, but by submitting each site to many earthquakes (38 in their study) in order to show the influence of the variation of the input motions on rd profiles. Thus, Filali and Sbartai (2017), in their study, showed that the dynamic cyclic shear stress (CSRD) can, in many cases, be greater than the simplified shear stress (CSR) according to the used earthquake. Therefore, rd can be greater than 1;this result (rd>1) was found in the study conducted by Farrokhzad (2016) for many sites at a significant depth and in the works reported by Sun etal. (2020), Cetin and Seed (2004), and Dismuke (2014) at a shallow depth for a few sites. In this case, this procedure cannot be considered as conservative; thus, the simplified procedure of Seed and Idriss (1971) cannot be applied because it is based on the assumption that rd<1, and all the modifications and improvements made in the literature are based on this assumption. For this reason, in order to generalize the use of the simplified procedure, Filali and Sbartai (2017) have proposed a corrector factor to adjust the simplified CSR in the range where rd>1, which corresponds to a maximum acceleration of the earthquake less than 0.30g (amax≤0.30g). In this paper, we will present a probabilistic analysis of liquefaction potential based on the proposed correction (Filali and Sbartai, 2017) in order to define the CRR curves used to characterize the boundary between liquefied and non-liquefied regions. For this purpose, we have used the case history database compiled after the Chi-Chi earthquake, which consists of 287 cases including 163 liquefied sets and 124 non-liquefied sets. The liquefaction during this earthquake appeared in several sites such as Nantou, Wufenf, and Yuanlin for soils with low and high fine content (FC), which is the characteristic unique to this earthquake (Hwang and Yang, 2001).
Deterministic Model
The approach of Seed and Idriss (1971) is the most widely used procedure in practice for estimating the liquefaction resistance of sandy soils. To represent the ground motions caused by earthquakes with one single parameter, a simplified procedure has been developed by Seed and Idriss (1971) and updated in Youd etal. (2001). The resistance to liquefaction is evaluated by comparing a property index of the soil to the CSR given by the following equation for a magnitude of the earthquake adjusted to 7.5:
{\rm{CSR}} = {{{\tau _{{\rm{cyc}}}}} \over {\sigma _{{\rm{v0}}}^\prime}} = 0.65 \times \left( {{{{{\rm{a}}_{\max }}} \over g}} \right) \times \left( {{{{\sigma _{\rm{v}}}} \over {\sigma _{\rm{v}}^\prime}}} \right) \times {r_d}
where σv is the vertical total stress of the soil at the depth studied, σ′v the vertical effective stress of the soil at the depth studied, amax the peak horizontal ground surface acceleration, g the acceleration of gravity, and rd is the shear stress reduction factor. The variable rd is calculated in accordance with Youd etal. (2001) as follows:
\matrix{ {{r_d} = 1 - 0.00765zz \le 9.15m} \hfill & {} \hfill \cr {{r_d} = 1.174 - 0.0267z} \hfill & {9.15 \le z \le 23m} \hfill \cr {{r_d} = 0.744 - 0.008z} \hfill & {23 \le z \le 30m} \hfill \cr {{r_d} = 0.5} \hfill & {z > 30m} \hfill \cr }
The various works existing in the literature, such as Cetin and Seed (2004), Lasley etal. (2016), and others, related to rd factor use a suite of equivalent linear site response analyses to adjust the dynamic and simplified results (deformable and rigid body). After Filali and Sbartai (2017), as the assumption rd<1 is verified only when amax>0.30g, in other words, when amax<0.30g, which corresponds to rd>1. Then, the deformable and rigid body is not adjusted in accordance with the assumption on which is based the simplified procedure. Also, in order to generalize the use of the simplified method by adjusting the deformable and rigid body whatever the used earthquake is, the authors have proposed a new earthquake corrector factor, RC, in the range where amax≤0.30g in order to adjust the dynamic and simplified results when rd>1 and ensure the reliability of the simplified method by giving the most conservative case for all earthquakes. The proposed correction (Filali and Sbartai, 2017) is defined by an earthquake corrector factor, RC, which is the ratio between the dynamic and the simplified shear stress, expressed as follows:
\left\{ {\matrix{ {{\rm{RC}}\,{\rm{ = 0}}{\rm{.696}}{{\left( {{{{{\rm{a}}_{\max }}} \over g}} \right)}^{ - 0.577}}} \hfill & {{\rm{if}}\,{a_{\max }} \le 0.30g} \hfill \cr {{\rm{RC}}\, = 1} \hfill & {{\rm{if}}\,{a_{\max }} > 0.30g} \hfill \cr } } \right.
This correction can be applied only when amax≤0.30g; otherwise, Eq. (1) is kept without correction (RC=1).
Then, by applying this correction, the original form of CSR (Eq. 1) can be rewritten in accordance with the following expression (Filali and Sbartai, 2017):
\left\{ {\matrix{ {{\rm{CSR}}\,{\rm{ = 0}}{\rm{.65}} \times \left( {{{{a_{\max }}} \over g}} \right) \times \left( {{{{\sigma _{v0}}} \over {\sigma _{v0}^\prime}}} \right) \times {r_d}} \hfill & {{\rm{if}}\,{a_{\max }} > 0.30g} \hfill \cr {{\rm{CSR}}\,{\rm{ = 0}}{\rm{.65}} \times \left( {{{{a_{\max }}} \over g}} \right) \times \left( {{{{\sigma _{v0}}} \over {\sigma _{v0}^\prime}}} \right) \times {r_d} \times {\rm{RC}}} \hfill & {{\rm{if}}\,{a_{\max }} \le 0.30g} \hfill \cr } } \right.
Cyclic resistance ratio
The empirical graph for evaluating liquefaction resistance based on SPT test developed by Seed etal. (1984) has been in the first term approximated by an equation proposed by Rauch (1997) based on the corrected blow count N160. To consider the effect of FC, Youd etal. (2001) have introduced the corrected blow counts for cleansands and given this equation by the following expression:
{\rm{CR}}{{\rm{R}}_{7.5}} = {1 \over {34 - {N_{160{\rm{cs}}}}}} + {{{N_{160cs}}} \over {135}} + {{50} \over {{{\left( {10{N_{160{\rm{cs}}}} + 45} \right)}^2}}} - 1/200
where N160cs is the corrected blow count for clean sands expressed as:
{N_{160{\rm{cs}}}} = a + b{N_{160}}
where a and b are two constant parameters introduced to account the effect of FC and are both functions of FC. The coefficients a and b are given by the following equations:
\left\{ {\matrix{ {{\rm{a}} = 0} \hfill & {{\rm{for}}\,{\rm{FC}}\, \le \,{\rm{5\% }}} \hfill \cr {{\rm{a}} = \exp \,\left( {1.76 - {{190} \over {{\rm{F}}{{\rm{C}}^2}}}} \right)} \hfill & {{\rm{for}}\,5\% \, < \,{\rm{FC}}\, < 35\% } \hfill \cr {{\rm{a}} = 5} \hfill & {{\rm{for}}\,{\rm{FC}}\, \ge \,35\% } \hfill \cr } } \right. \left\{ {\matrix{ {{\rm{b}} = 1} \hfill & {{\rm{for}}\,{\rm{FC}}\, \le \,{\rm{5\% }}} \hfill \cr {{\rm{b}} = \,0.99 + {{{\rm{F}}{{\rm{C}}^{1.5}}} \over {1000}}} \hfill & {{\rm{for}}\,5\% \, < \,{\rm{FC}}\, < 3\,{\rm{5\% }}} \hfill \cr {{\rm{b}} = 1.2} \hfill & {{\rm{for}}\,{\rm{FC}}\, \ge \,35\% } \hfill \cr } } \right.
Also, N160 is the corrected blow counts expressed as
{N_{160}} = {N_m}{C_N}{C_E}{C_B}{C_R}{C_S}
where Nm is the measured standard penetration resistance, CE the correction for hammer energy ratio, CB the correction factor for borehole diameter, CR the correction factor for rod length, CS the correction for samplers with or without liners, and CN is the factor to normalize Nm to a common reference effective overburden stress expressed as (Youd etal., 2001; Liao and Whitman, 1986a)
{C_N} = {\left( {{{{P_a}} \over {\sigma _{v0}^\prime}}} \right)^{0.5}} \le 1.7
The CRR7.5 should be corrected for the earthquake magnitude, overburden pressure, and static shear stress (Seed and Idriss, 1982, 1983; Boulanger and Idriss, 2004) as follows:
{\rm{CR}}{{\rm{R}}_{{M_w}}} = {\rm{CR}}{{\rm{R}}_{7.5cs}}\left( {{\rm{MSF}}} \right){K_\sigma }{K_\alpha }
where MSF is the magnitude scaling factor and Kσ and Kα are the factors for overburden and initial static stress ratio corrections, respectively. These factors are calculated with the formulae recommended by Boulanger and Idriss (2004).
Magnitude scaling factor
Several equations have been proposed for the assessment of MSF according to the earthquake moment magnitude (Seed and Idriss, 1982; Idriss, 1999). Idriss (1999) proposed the MSF as
{\rm{MSF}} = 6.9\,\exp \left( { - {M_w}/4} \right) - 0.058 \le 1.8
Overburden correction factor Kσ
The overburden correction factor Kσ can be estimated by the relationship proposed by Boulanger and Idriss (2004) as follows:
{K_\sigma } = 1 - {C_\sigma }\ln \left( {\sigma _{v0}^\prime/{P_a }} \right) \le 1.1
where the coefficient Cσ can be expressed in terms of corrected shear wave velocity as follows:
{C_\sigma } = 1/\left( {18.9 - 2.55{{\left( {{N_1}} \right)}_{60}}} \right) \le 0.3
Static shear stress correction factor kα
To take into account the influence of static shear stresses on CRR, Seed etal. (1983) have proposed a correction factor Kα to correct the CRR. Several studies were conducted by Idriss and Boulanger (2003a, 2003b). The author believes that these results can be used. As the soil layers are considered horizontal, the value of Kα in this study is kept as equal to 1.
Bayesian Mapping Function (BMF)
Since the deterministic safety factor (Fs) is the most widely used in geotechnical practice, it is interesting to relate it to the probability of liquefaction in order to facilitate the use of the probabilistic approach for engineers to take a correct decision. Juang etal. (1999) have proposed a mapping function approach which linked the deterministic Fs to the probability of liquefaction; this approach has been refined by Juang etal. (2000a, 2000b). In this approach, the conditional probability of liquefaction for a given site is deduced from the information contained in the case history database (Juang etal., 2000b, 2002) according to the following equation:
{P_L} = {{{f_L}\left( {Fs} \right)} \over {{f_L}\left( {Fs} \right) + {f_{NL}}\left( {Fs} \right)}}
where fL(Fs) and fNL(Fs) are the probability density functions of the calculated FS for the sets of liquefied cases and non-liquefied cases, respectively. Based on the obtained Eq. (14), the probability of liquefaction is calculated for each of the287 cases in the database using the original and the corrected versions of the simplified procedure.
The variation of the probability of liquefaction against the deterministic safety factor (Fs) calculated using the original version of the simplified procedure (Seed and Idriss, 1971) is plotted in Fig. 1. The set of the 287 points can be fitted in terms of mapping function, which linked PL to Fs by the following equation:
{P_L} = {1 \over {1 + {{\left( {{{{F_S}} \over {0.9674}}} \right)}^{7.558}}}}
The deterministic curve model is defined by Fs=1. Thus, the Youd etal. (2001) curve can be characterized with a probability of liquefaction of 45% based on Bayesian mapping model. This result is very close to that obtained by Juang etal. (2000). From Eq. (15), we can plot for a given value of PL the CRR boundary curves presented in Fig. 2. This figure shows that the value of N160CS converges to 32 for high values of CSR.
Figure 1
Relationship between PL and Fs based on Bayesian mapping function using the original version of the simplified procedure. PL = probability of liquefaction.
Figure 2
Bayesian mapping function along the case history data base using the original version of the simplified procedure. CSR = cyclic stress ratio.
The safety factor calculated using Eq. (4) for the CSR and Eq. (5) for the CRR is used to recalculate the probability of liquefaction. By fitting the set of points presented in Fig. 3, the mapping function can be expressed by the relationship below:
{P_L} = {1 \over {1 + {{\left( {{{{F_S}} \over {0.7585}}} \right)}^{5.076}}}}
In this equation, a value of Fs=1 corresponds to the deterministic curve model. Therefore, for this case, the Youd etal. (2001) curves can be characterized with a probability of liquefaction of 20% based on Bayesian mapping model. From Eq. (16), we can plot for a given value of PL, the CRR boundary curves presented in Fig. 4. This figure shows that the value of N160CS converges to 32 for high values of CSR.
Figure 3
Relationship between PL and Fs based on Bayesian mapping function using the corrected version of the simplified procedure. PL = probability of liquefaction.
Figure 4
Bayesian mapping function along the case history data base using the corrected version of the simplified procedure. CSR = cyclic stress ratio.
The figure also shows that the Youd etal. (2001) boundary curve is characterized by a PL=0.20. In accordance with the corrected version of the simplified procedure, it is not conservative because it cannot be considered as a boundary curve, which separates the liquefied and non-liquefied cases and must be adjusted to the curve corresponding to PL=0.40, which is very close to the true boundary between the two zones.
By fitting the true boundary between the liquefied and non-liquefied sets using the same shape of the Youd etal. (2001) model, the CRR, CRR7.5, can be expressed by the following equation:
{\rm{CR}}{{\rm{R}}_{7.5}} = {1 \over {34 - {N_{160cs}}}} + {{{N_{160cs}}} \over {96.83}} + {{344.1} \over {21.43{N_{160cs}} + 87.33}} - 1/100
By comparing this equation with that proposed by Youd etal. (2001), we can say that only the curve-fitting parameters have changed. This result is reasonable because according to the corrected version of the simplified procedure, the values of the CSR have changed; therefore, the boundary between the liquefied and non-liquefied cases may also change and the curve-fitting parameters must be adjusted to the new position of the boundary. Since the mathematical model of the true boundary is defined, we must recalculate the safety factor and the probability of liquefaction for all cases in the database using Eqs (4) and (17). In the same manner, the mapping function deduced by fitting the set of points presented in Fig. 5 can be expressed as follows:
{P_L} = {1 \over {1 + {{\left( {{{{F_S}} \over {0.8976}}} \right)}^{6.271}}}}
This equation shows that the deterministic boundary curve, which correspond to Fs=1 is characterized by a probability of liquefaction of 35% instead of40%. The set of probabilistic boundary curves deduced from Eqs (17) and (18) are plotted in Fig. 6. The deterministic design decision is always made based on the safety factor, which indicates that the liquefaction occurs or not according to a reference value by choosing the most conservative case. The liquefaction boundaries plotted in Fig. 6 show that the Youd etal. (2001) CRR curve is characterized by a probability of 15% using the Bayesian mapping function with the deterministic model given by Eq. (17) based on the corrected simplified method, which corresponds to a deterministic safety factor (Fs) of 1.20, while the adjusted model proposed in this study shown in Eq. (17) is related to a probability of 35%, which corresponds to Fs=1. Then, according to these results, the more conservative case is always given by the corrected simplified method. Thus, the set of curves shown in Fig. 6 indicates liquefaction for the zone above the boundary curve of PL=90% and no liquefaction for the zone below the boundary curve of PL=15%. The zone between PL=15% and PL=90% is an intermediate zone, in which 15% and 35% represent the lower and marginal probabilities, respectively, and above 35%, the risk of liquefaction increases with the probability of liquefaction. To define the severity of the liquefaction potential using a probabilistic analysis, Juang etal. (2001) have proposed a liquefaction likelihood classification that can be used for probabilistic design decision using the corrected version of the simplified method.
Figure 5
Relationship between PL and Fs based on Bayesian mapping function using the corrected version of the simplified procedure. PL = probability of liquefaction.
Figure 6
Bayesian mapping function along the case history data base using the corrected version of the simplified procedure with Eq. (17). CSR = cyclic stress ratio.
Comparison with Other Curves
The liquefaction resistance correlation based on standard penetration test has been studied by several authors. Based on an updated case history database used to develop the Idriss and Boulanger (2008) and the Boulanger and Idriss (2004) liquefaction correlation for cohesionless soils, Idriss and Boulanger (2010) have developed a new liquefaction resistance correlation based on the standard penetration test. Hwang etal. (2012), based on a case history database collected after the Chi-Chi Taiwan earthquake 2001, have proposed a hyperbolic model to express a liquefaction resistance correlation based on an SPT test. Other comprehensive studies have been performed by the geotechnical experts in order to develop liquefaction resistance correlations (Youd etal., 2001; Seed etal., 1984, 1985; Cetin etal., 2016). A comparison between these liquefaction resistance correlations and the adjusted model proposed in this study is presented in Fig. 7. This figure shows clearly that the best fit is given by the corrected version of the simplified method, which materializes the true liquefaction boundary expressed by Eq. (17). The other correlations in Fig. 7 cannot be considered as boundary curves because according to the corrected version of the simplified method, the values of CSR for all cases in the database for which amax≤0.30g are adjusted through a corrector factor, RC, defined by Eq. (3). Therefore, the plotted set of points of the case history database is translated upward, which leads to move the boundary curve to a new position different to that defined by the original simplified method. Then, in this figure, we have kept the original position of the plotted curves in order to show the effect of the proposed correction on these curves.
Figure 7
Comparison of the original and the adjusted CRR-N160cs curves. CRR = cyclic resistance ratio.
Validation
Validation of the obtained results will be performed through two case studies for Yuanlin and Coastal road Skikda sites and by using the case history database based on SPT test of Cetin etal. (2016).
YualinTaiwan site
Nantou site is located approximately 0–5 km from the fault rupture within the Taichung basin. The Chi-Chi, Taiwan earthquake of 09/25/1999 caused significant damage in the village of Yuanlin; for example, liquefaction, landslide, and major faults appeared on the surface. The geological environment of Nantou is in the form of young alluvial sediments with shallow groundwater (within about 0.5–5 m of the surface). The National Center for Research on Earthquake Engineering (NCREE) has conducted several investigative programs based on the in situ CPT, SPT, and shear wave velocity (VS) testing. The soil stratigraphy is generally silty medium to fine sand interspersed with very dense layers of small gravel with a percentage of fines of 7%–45%. The profile of the soil along the SPT boring MAA-BH6 is shown in Fig. 8 (NCREE, 2001).
Figure 8
Profile of soil according to the depth (Nantou site).
In this example, we will evaluate the liquefaction potential with the original and corrected versions of the simplified procedure in order to define which of the two methods gives the more conservative case. Then, the CSR is calculated by using both Eqs (1) and (4); for the estimation of the CRR, we will use Eq. (5) adjusted to FC ≤5% and Eq. (17). The peak ground acceleration value, amax, used for the calculation of CSR is taken to be equal to 0.1687g. The depth of the groundwater table is kept at 1.2m relative to the ground surface. The average value of the unit weight is taken to be equal to 18.3kN/m3 above the water table and 20.35kN/m3 below the water table. In Fig. 9 are shown the profiles of safety factor according to the depth computed by the original and corrected versions of the simplified procedure; the profile of the dynamic Fs is deduced from a dynamic analysis conducted with Shake91_input software (Idriss and Sun, 1992) by using Eq. (17) to estimate the CRR.
Figure 9
Safety factor according to the depth computed by the original and corrected versions of the simplified method (Nantou site).
Fig. 9 shows that the more conservative case is given by the corrected version of the simplified procedure and the profile of the corrected safety factor is very close to the dynamic profile. These results indicate that the maximum shear stress given by the corrected version, which is very close to that computed from a dynamic analysis, is always for this case greater than the shear stress estimated by the original simplified method, which implies that the stress corrector factor, rd, is greater than 1. To confirm this, we have conducted a dynamic analysis using Shake91_input software (Idriss and Sun, 1992), in which the Chi-Chi Taiwan 2001 earthquake is simulated by the TCU075 accelerogram applied at the bottom of the soil profile. In this analysis, we have calculated the maximum shear stress for soil profile using Shake91_input and the simplified method with the original and corrected versions using the maximum acceleration of the TCU075 accelerogram, which is 0.1687g. The results are presented in Fig. 10. This figure shows clearly that the maximum shear stress computed by the original simplified method is less than that given by the dynamic analysis conducted using Shake91_input (rd>1, amax<0.30g), while the corrected version of the simplified method gives values greater than or equal to those of the dynamic method (rd≤1).
Figure 10
Maximum shear stress according to depth with dynamic and simplified analysis (Yuanlin site).
Then, for this site, the liquefaction potential evaluation must be conducted using the corrected version of the simplified method because the original version cannot be applied since rd is greater than 1.
Coastal road Skikda site (Algeria)
Based on the request of the National Petroleum Refining Company of Skikda department (NAFTEC), the laboratory has performed a geophysical investigation with three down hole tests. The study site is located within the industrial zone of Skikda; it has a flat topography. The down hole test SC02 detected the presence of a sandy horizon, reddish to brownish, which extended up to depth 20 m and was saturated with a mean diameter D50 varying between 0.11 and 1 mm. The average value of the unit saturated weight is taken to be between19.6 and 20.5kN/m3. The water table is assumed to be on the ground surface. The magnitude of the earthquake is 6.8, and the maximum acceleration at the surface is equal to 0.122g. The site is classified as zone II according to the Algerian earthquake code RPA 2003. The profile of soil and shear wave velocity chosen in this study are shown in Fig. 11.
Figure 11
Profile of soil and shear wave velocity according to the depth (petrochemical zone site).
In Fig. 12 are shown the profiles of safety factor according to the depth computed by the original and corrected versions of the simplified procedure. The profile of the dynamic FS is deduced from a dynamic analysis performed with Shake_input software (Idriss and Sun, 1992), in which the dynamic cyclic stress ratio (CSRD) was expressed as the ratio of the maximum shear stress and the vertical effective stress.
Figure 12
Safety factor according to the depth computed by the original and corrected versions of the simplified method (Coastal roadSkikda site).
For this site, the conclusion is the same as for the Treasure Island site. To confirm this, we have conducted a dynamic analysis using Shake91_input software (Idriss and Sun, 1992), in which the Boumerdes earthquake of 21/05/2003 is simulated by the Azazga station accelerogram EW component applied at the bottom of the soil profile. In this analysis, we have calculated the maximum shear stress for soil profile using Shake91_input and the simplified method with the original and corrected versions using the maximum acceleration of the used accelerogram, which is 0.122g. The results are presented in Fig. 13. This figure shows clearly that the maximum shear stress computed by the original simplified method is less than that given by the dynamic analysis conducted using Shake91_input (rd>1, amax<0.30g), while the corrected version of the simplified method gives values greater than or equal to those of the dynamic method (rd≤1).
Figure 13
Maximum shear stress according to the depth with dynamic and simplified analysis (Coastal road Skikda site).
Case history database
From a case history database of SPT liquefaction (Cetin etal., 2016) including 210 cases, we have retained20liquefied cases with a maximum acceleration less than 0.30g and a safety factor computed by the original simplified method greater than 1. For each case in the database, we have computed the safety factor using the proposed correction and the original simplified method; the results are presented in Table 1 shown in Appendix A. By examining the results, we can conclude that the proposed correction indicates that all cases are liquefied (FSSMC<1), while the original simplified method indicates otherwise (FSSM>1 for 18 cases). Then, it is clearly visible that the database confirms the results of the proposed correction.
Conclusion
A probabilistic analysis has been conducted in this paper based on the original simplified method (Seed and Idriss, 1971) and the corrected version of this method (Filali and Sbartai, 2017) by using a Bayesian mapping function based on standard penetration test. The results show the following:
The boundary curve is characterized on one hand by PL=0.45, which corresponds to FS=1, by using the original simplified method, and on the other hand by PL=0.40, which corresponds to FS=0.82, by using the corrected version of this method with the Youd etal. (2001) shape of the CRR expressed by Eq. (7).
Then, the proposed model for the CRR curve of Youd etal. (2001) must be adjusted to the new boundary in accordance with the corrected version of the simplified method, which corresponds to PL=0.40, because the boundary curve is obtained by plotting CSR against N160cs from the case history data; also, as the CSR changes for all sites in the database where amax<0.30g, the boundary curve must also change and may be readjusted.
According to the corrected version of the simplified method, the boundary between liquefied and non-liquefied zones is readjusted using the proposed Eq. (17). By using the shape of the proposed CRR curve (Eq. 17), the probability of liquefaction, which corresponds to a deterministic FS=1, becomes 0.35 instead of 0.40.
Then, the proposed model of CRR curve is characterized by a probability of 0.35. This correction is only valid for clean sand (FC<5%). For other sands e (FC>5%) an adjustment to clean sand may be made according to N160cs in order to be able to use the proposed correction.
