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 (τ_{d}) induced at depth h is always less than that predicted by the simplified procedure (τ_{r}) of Seed and Idriss (1971) (rd= τ_{d}/τ_{r}<1). All the expressions proposed for _{d}_{d}_{d}_{d}>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 _{d}<1_{d}_{max}≤0.30

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:
_{v} 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, _{max} the peak horizontal ground surface acceleration, _{d}_{d}_{d}_{d}_{max}>0.30_{max}<0.30_{d}_{max}≤0.30_{d}

This correction can be applied only when _{max}≤0.30

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):

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 _{160}. 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:
_{160cs} is the corrected blow count for clean sands expressed as:
_{160} is the corrected blow counts expressed as
_{m}_{E}_{B}_{R}_{S}_{N}_{m}_{7.5} should be corrected for the earthquake magnitude, overburden pressure, and static shear stress (Seed and Idriss, 1982, 1983; Boulanger and Idriss, 2004) as follows:
_{σ}_{α}

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

_{σ}

The overburden correction factor _{σ}_{σ}

_{α}

To take into account the influence of static shear stresses on CRR, Seed etal. (1983) have proposed a correction factor _{α}_{α}

Since the deterministic safety factor (_{L}_{NL}_{S}

The variation of the probability of liquefaction against the deterministic safety factor (_{L}_{L}_{160CS} converges to 32 for high values of CSR.

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:
_{L}_{160CS} converges to 32 for high values of CSR.

The figure also shows that the Youd etal. (2001) boundary curve is characterized by a _{L}_{L}

By fitting the true boundary between the liquefied and non-liquefied sets using the same shape of the Youd etal. (2001) model, the CRR, CRR_{7.5}, can be expressed by the following equation:

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 _{max}≤0.30

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).

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).

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, _{max}, used for the calculation of CSR is taken to be equal to 0.1687^{3} above the water table and 20.35kN/m^{3} 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

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, r_{d}, 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 0.1687_{d}>1, _{max}<0.30_{d}≤1).

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 _{d}

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 D_{50} varying between 0.11 and 1 mm. The average value of the unit saturated weight is taken to be between19.6 and 20.5kN/m^{3}. 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 0.122

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.

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 0.122_{d}>1, _{max}<0.30_{d}≤1).

From a case history database of SPT liquefaction (Cetin etal., 2016) including 210 cases, we have retained20liquefied cases with a maximum acceleration less than 0.30_{SMC}<1), while the original simplified method indicates otherwise (FS_{SM}>1 for 18 cases). Then, it is clearly visible that the database confirms the results of the proposed correction.

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 _{L}_{L}

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 _{L}_{160cs} from the case history data; also, as the CSR changes for all sites in the database where _{max}<0.30

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

#### Summary of updated Cetin etal.’s (2016) field performance case history parameters (20 cases retained for validation).

_{v} (kPa) |
_{v} (kPa) |
_{max} ( |
_{w} |
_{σ} |
_{s} |
_{R} |
_{B} |
_{E} |
_{N} |
_{160CS} |
_{SM} |
_{SM} |
_{SMC} |
_{SMC} |
||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

6 | 1964 Niigata | Arayamo-tomachi | Yes | 3.3 | 1 | 56 | 34 | 0.09 | 0.09 | 7.6 | 0.967 | 1.31 | 5 | 4.4 | 1 | 0.86 | 1 | 1.22 | 1.72 | 8.4 | 0.099 | 0.126 | 0.121 | 0.153 | ||

24 | 1975 Haicheng | Panjin Ch. F. P. | Yes | 8 | 1.5 | 148 | 85 | 0.13 | 0.13 | 7 | 1.193 | 1.041 | 67 | 8.1 | 1 | 1 | 1 | 0.83 | 1.09 | 10.9 | 0.121 | 0.151 | 0.149 | 0.185 | ||

25 | 1975 Haicheng | Ying Kou G. F. P. | Yes | 7 | 1.5 | 130 | 76 | 0.2 | 0.2 | 7 | 1.193 | 1.071 | 48 | 12.4 | 1 | 0.98 | 1 | 1 | 1.16 | 17.9 | 0.191 | 0.244 | 0.239 | 0.305 | ||

26 | 1975 Haicheng | Ying Kou P. P. | Yes | 7.5 | 1.5 | 139 | 80 | 0.2 | 0.18 | 7 | 1.193 | 1.057 | 20 | 10.3 | 1 | 0.99 | 1 | 1 | 1.12 | 13.6 | 0.146 | 0.185 | 0.182 | 0.229 | ||

30 | 1976 Tangshan | Coastal Region | Yes | 4.5 | 1.1 | 83 | 50 | 0.13 | 0.13 | 7.6 | 0.967 | 1.189 | 12 | 9.45 | 1 | 0.9 | 1 | 1 | 1.43 | 13.5 | 0.145 | 0.167 | 0.181 | 0.208 | ||

47 | 1978 Miyagiken-Oki | Nakamura 4 | Yes | 4 | 0.5 | 75 | 40 | 0.12 | 0.14 | 6.5 | 1.442 | 1.257 | 5 | 5.6 | 1 | 0.89 | 1 | 1 | 1.59 | 8.4 | 0.099 | 0.180 | 0.121 | 0.219 | ||

58 | 1978 Miyagiken-Oki | Hiyori-18 | Yes | 3.3 | 2.4 | 57 | 49 | 0.24 | 0.18 | 7.7 | 0.935 | 1.195 | 20 | 9.1 | 1 | 0.86 | 1 | 1.09 | 1.43 | 14.4 | 0.154 | 0.172 | 0.192 | 0.214 | ||

70 | 1978 Miyagiken-Oki | Yuriage Br-2 | Yes | 2.4 | 1.3 | 43 | 32 | 0.24 | 0.21 | 7.7 | 0.935 | 1.33 | 7 | 11.4 | 1 | 0.82 | 1 | 1.12 | 1.79 | 19.5 | 0.209 | 0.260 | 0.262 | 0.325 | ||

81 | 1979 Imperia’ Valley | Radio Tower B1 | Yes | 4.3 | 2 | 72 | 50 | 0.18 | 0.16 | 6.53 | 1.426 | 1.189 | 43.5 | 4.45 | 1 | 0.86 | 1 | 1.13 | 1.42 | 9.7 | 0.110 | 0.187 | 0.135 | 0.229 | ||

83 | 1979 Imperia’ Valley | River Park A | Yes | 1.1 | 0.3 | 18 | 10 | 0.16 | 0.18 | 6.53 | 1.426 | 1.778 | 91 | 2.65 | 1 | 0.66 | 1 | 1.13 | 2 | 7.3 | 0.090 | 0.228 | 0.109 | 0.275 | ||

95 | 1983 Nihonkai-Chubu | Takeda Ele.Sch. | Yes | 4.5 | 0.4 | 80 | 40 | 0.12 | 0.13 | 7.1 | 1.151 | 1.257 | 0 | 7.85 | 1 | 0.9 | 1 | 1.22 | 1.6 | 14.3 | 0.153 | 0.222 | 0.191 | 0.276 | ||

97 | 1983 Nihonkai-Chubu | Aomori Station | Yes | 5.8 | 0 | 108 | 52 | 0.12 | 0.14 | 7.7 | 0.935 | 1.178 | 3 | 9 | 1 | 0.94 | 1 | 1.22 | 1.4 | 15 | 0.160 | 0.176 | 0.200 | 0.220 | ||

122 | 1987 Superstition Hills | Wildlife B | Yes | 4.7 | 0.9 | 86 | 49 | 0.2 | 0.19 | 6.54 | 1.42 | 1.195 | 26.2 | 7.85 | 1 | 0.88 | 1 | 1.13 | 1.44 | 14.1 | 0.151 | 0.257 | 0.188 | 0.319 | ||

132 | 1989 Loma Prieta | P007-2 | Yes | 6.2 | 3 | 118 | 88 | 0.22 | 0.17 | 6.93 | 1.224 | 1.032 | 3 | 12.7 | 1 | 0.93 | 1 | 0.92 | 1.08 | 13.3 | 0.143 | 0.181 | 0.178 | 0.225 | ||

134 | 1989 Loma Prieta | POR-2&3&4 | Yes | 4.9 | 3.5 | 79 | 65 | 0.15 | 0.09 | 6.93 | 1.224 | 1.114 | 50 | 3.15 | 1 | 0.89 | 1 | 0.92 | 1.24 | 6.8 | 0.086 | 0.117 | 0.103 | 0.141 | ||

135 | 1989 Lama Prieta | Sandholdt UC-B10 | Yes | 3.2 | 1.7 | 58 | 43 | 0.26 | 0.22 | 6.93 | 1.224 | 1.235 | 2 | 9.15 | 1 | 0.81 | 1 | 1.25 | 1.53 | 14.8 | 0.158 | 0.239 | 0.197 | 0.298 | ||

139 | 1989 Lama Prieta | Treasure Island | Yes | 5.3 | 1.5 | 91 | 55 | 0.18 | 0.17 | 6.93 | 1.224 | 1.161 | 20 | 5.05 | 1 | 0.9 | 1 | 1.13 | 1.36 | 9.8 | 0.111 | 0.158 | 0.136 | 0.194 | ||

140 | 1989 Lama Prieta | Wood Marine UC-B4 | Yes | 1.8 | 1 | 31 | 24 | 0.25 | 0.21 | 6.93 | 1.224 | 1.429 | 35 | 5.75 | 1 | 0.72 | 1 | 1 | 2 | 11.9 | 0.130 | 0.228 | 0.161 | 0.282 | ||

143 | 1989 Lama Prieta | Marine Laboratory UC-B2 Yes | Yes | 3.5 | 2.5 | 63 | 53 | 0.26 | 0.2 | 6.93 | 1.224 | 1.172 | 3 | 13 | 1 | 0.83 | 1 | 1 | 1.38 | 15.4 | 0.164 | 0.235 | 0.205 | 0.294 | ||

210 | 1995 Hyogo ken-Na nbu | Torishima Dike | Yes | 4.8 | 0 | 86 | 39 | 0.25 | 0.29 | 6.9 | 1.238 | 1.265 | 20 | 8.5 | 1 | 0.91 | 1 | 1.22 | 1.61 | 17.5 | 0.186 | 0.292 | 0.233 | 0.365 |

Reviews on Finite Element Modeling Practices of Stone Columns for Soft Soil Stabilization Beneath an Embankment Dam Proposal of concept for structural modelling of hybrid beams Resonance of a structure with soil elastic waves released in non-linear hysteretic soil upon unloading Settlement Analysis of a Sandy Clay Soil Reinforced with Stone Columns The evolution of the shape of composite dowels Parametric study of the earth dam's behaviour subjected to earthquake Impact of longwall mining on slope stability – A case study