Dynamic response based empirical liquefaction model

Dynamic response-based methodology, wherein integrated effect of dynamic soil properties and ground motion parameters proposed by authors, has been found to detect liquefaction susceptibility. The present work necessarily deals with the formulation of a comprehensive empirical liquefaction model (ELM) using this methodology. The absolute form of the ELM is dimensionally homogeneous and yields a correlation between proposed “liquefaction potential term” and “normalized standard penetration blow count corrected for fines content, (N 1 ) 60cs .” The developed ELM demonstrates unbiased performance when verified over a wide range of significant parameters. One of the prominent features of the present ELM is accurate prediction of possibility of liquefaction. The proposed ELM has proven to work well on varied data-sets of more than 1000 case records within the given range of model parameters. Moreover, the dynamic response-based ELM proves its ability when compared with other liquefaction evaluation procedures. Thus, a generalized and optimistic ELM simulating realistic field conditions is formulated. It is anticipated that for accurate prediction of liquefaction occurrence, it would be more appropriate to employ the proposed ELM which will minimize the enormous losses caused due to liquefaction. Subjects: Civil, Environmental and Geotechnical Engineering; Georisk & Hazards; Soil Mechanics

ABOUT THE AUTHOR Pathak is working in this field for more than a decade; the various areas dealt so far are 1-g shake table testing, Static and Cyclic Triaxial testing, and analytical modeling to assess susceptibility of liquefaction by varying soil and seismic parameters. The publications include 20 plus international and national journal papers in the field of seismic soil liquefaction; to name a few: ASCE, Natural Hazards, Geomechanics and Geoengineering, and equivalent papers presented and published in international and national conferences on the research theme.

PUBLIC INTEREST STATEMENT
Liquefaction is a phenomenon wherein a loose sandy soil loses its strength and flows like a fluid and is a topic of interest around the world. It was more thoroughly brought to the attention of engineers after 1964 Niigata and 1964 Alaska earthquakes which have witnessed major damage due to seismic soil liquefaction. Accordingly, prediction of liquefaction potential is the key step which will minimize the enormous losses caused due to liquefaction. The focus of the present work is thus formulation of a comprehensive and cogent empirical liquefaction model (ELM) and validating the same to ensure realistic dynamic soil response to earthquake-induced liquefaction. The final form of the ELM thus developed, engenders realistic yet optimistic prediction of liquefaction at a particular site within a given range of model parameters. ELM shows its versatility by performing on over 1000 case records and accurately predicts liquefaction occurrence.

Introduction
The use of empirical relationships based on correlation of observed field behavior with various in situ "index" tests is the dominant approach in common engineering practice to assess liquefaction susceptibility. Based on extensive literature review of empirical liquefaction procedures, Pathak and Dalvi (2011) inferred that inclusion of dynamic soil properties in conjunction with ground motion parameters would enhance the predictability of the model. Furthermore, it is observed that although there exist several studies to assess the liquefaction potential (LP), it is well understood that a large number of seismic as well as soil parameters affect the occurrence of liquefaction during an earthquake, making it necessary to identify the significant parameters based on their contribution to liquefaction phenomenon to expedite the assessment of liquefaction. Accordingly, Dalvi, Pathak, and Rajhans (2014) extracted significant parameters responsible for the phenomenon of liquefaction by employing multi-criteria decision-making tools (MCDM), namely: AHP and Entropy analysis. Such an attempt of applying MCDM tools in the field of earthquake-induced liquefaction has been firstly introduced by the authors.
Subsequently, Pathak and Dalvi (2012) presented a preliminary approach referred as, "dynamic response based methodology" to evaluate LP by incorporating thus extracted significant parameters to represent realistic dynamic response of soils. Based on this dynamic response-based methodology, Pathak and Dalvi (2013) established an elementary empirical liquefaction model (ELM) to identify susceptibility of liquefaction which is given by the following Equation. 1 as Where (N 1 ) 60 = normalized SPT blow count and, where, "v max " is the peak ground velocity (m/s) , "G max " is the small strain shear modulus (kPa), "dur" is duration of strong ground motion (sec), and " ′ v " is effective overburden pressure (kPa).
The performance of thus developed elementary model ascertained the potential of using the functional form LP term in assessing liquefaction susceptibility (Pathak & Dalvi, 2013). However, in order to predict liquefaction in practical situations, a dimensionless relationship might be required. Moreover, it is also required to take into account all types of soils ranging from sandy soils to silty sands and also site conditions to simulate actual field conditions. Thus, in the present work, an endeavor has been made to establish an exhaustive generalized ELM. The main motive behind the present work is to detect the occurrence of liquefaction at a particular site based on dynamic response. A sequential procedure as elaborated in preceding sections is thus adopted to arrive at the final form of ELM.

Development of ELM
In order to develop an exhaustive ELM to replicate pragmatic dynamic response which will take into account these issues to be applicable in realistic situations, the fundamental relationship as in Equation. 1 is employed. Hence, initially, the elementary model is transformed into a dimensionally homogeneous form using Fourier's principle of dimensional homogeneity, expressing the dimensions of the LP term in terms of primary quantities M-L-T, i.e. mass, length and time as Thus, the term remains with the units of length; accordingly, the epicentral distance is chosen as the dimensional homogeneity factor to convert it into a homogeneous form. Now, in order to formulate the model, a total of 314 case records (Appendix I) have been utilized which are extracted from Boulanger, Wilson, and Idriss (2012), Cetin et al. (2004), Davis and Berrill (1982), Hamada and Wakamatsu (1996), Hanna, Ural, and Saygili (2007), and Zhang (1998). Out of the total parameters involved in formulating the present model, the values of earthquake magnitude (M w ), peak ground acceleration (a max ), epicentral distance (r), normalized SPT blow count (N 1 ) 60 , and effective overburden pressure ( ′ v ) are used directly as available in the database, whereas the parameters, namely, peak ground velocity(v max ),small strain shear modulus (G max ), and duration of strong ground motion (dur) have been computed as detailed in Pathak and Dalvi (2013). Sampling bias and class balance have been maintained to ensure realistic predictability as illustrated by Oommen, Baise, and Vogel (2011) and accordingly, 220 cases have been employed for formulating the model and 94 cases have been used to validate the ELM. Thus, regression analysis has been performed in MATLAB using the training data-set of 220 cases out of the total 314 cases using the functional form as in Equation 1 and regression coefficients are obtained after applying dimensional homogeneity factor which resulted in Equation 2 as: where LP m term = modified LP term which is dimensionally homogeneous The boundary curve representing Equation 2 thus obtained is found to accurately discriminate the cases of liquefaction and non-liquefaction when validated on remaining 94 cases. All the 47 liquefaction cases are correctly predicted by Equation 2, hence signifies the theme underlying the present work. However, this proposed model is observed to predict "no" liquefaction conservatively, as some of the non-liquefied cases have been identified as liquefied by the model. Few such misclassified non-liquefaction case records indicate the normalized SPT values < 5 which actually illustrate high risk of liquefaction. Further, it is observed that for many of the remaining wrongly predicted "NO" cases, the value of normalized SPT blow count is within the range of 10-20, indicating intermediate risk of liquefaction. Thus, though the cases are of "no" liquefaction, the model rightly predicts them as "yes" cases. The model thereby achieves an overall success rate of 78% with accurate prediction for liquefied cases. Such a dimensionally homogeneous proposed model ensures it with physical significance in liquefaction assessment studies.
Further to simulate different soil conditions on site, range of soils is then incorporated through the compositional factor "fines content" which is known to affect both cyclic shear strength and penetration resistance of soils. To do so, initially, the effect of fines content on the model performance is verified by categorizing the whole data-set of 314 case records as: Class (A) Low FC, clean sand (FC ≤ 5%), Class (B) Intermediate FC, silty sand (6 ≤ FC ≤ 35%), and Class (C) High FC, silty sand to sandy silt (FC > 35%), to distinguish the soil types. Consequently, three separate equations each representing a particular type of soil have been obtained. The regression coefficients as evaluated for Classes A, B, and C are as tabulated below.
These three categories are also associated with corresponding relative risk of liquefaction (Dalvi et al., 2014) as stated in Table 1. It can also be observed from Table 1 that the regression coefficients corresponding to intermediate fines content are the same as those proposed through Equation 2. Interestingly, the other two ranges of fines content define the upper and lower bounds of boundary curves as illustrated in Figure 1.
It is worth mentioning that despite considering a different form to represent LP, the trend of boundary curves as proposed in present work ( Figure 1) indicates similarity with those established by previous researchers such as (Andrus & Stokoe II, 2000;Bolton Seed, Tokimatsu, Harder, & Chung, 1985;Rezania, Javadi, & Giustolisi, 2010). The trend of boundary curves as depicted in Figure 1 clearly indicates that fines content is an important parameter which affects the performance of the where C FINES is as given by Seed et al. (2003); thus, the final form of the model culminates into Equation 4: This liquefaction triggering correlation obtained is in terms of dimensionally homogeneous "liquefaction potential term" (LP m term) and the normalized SPT blow count corrected for fines content (N 1 ) 60cs . After verifying the predictive performance of the developed Equation. 4, on remaining 94 cases, it is found that by inclusion of fines content as model parameter, the predictive performance improves by over 4%, giving an overall success rate of 82%. Most importantly, all the 47 liquefaction cases are correctly predicted and 30 cases out of the 47 non-liquefaction cases have been identified correctly as shown in Figure 2.
From Figure 2, it can be seen that the scatter of non-liquefaction cases lying above the boundary curve is lesser when correction for fines content is applied. It is also noticed that the wrongly pre-

Performance of ELM
Now in order to justify the comprehensiveness of the proposed ELM, it is projected to assess its performance over diversified applications such as: (3.1) Performance of ELM over the range of significant parameters,

Performance of ELM over the range of significant parameters
Based on MCDM analysis, the effect of range of significant parameters, namely: a max , M w, and effective overburden pressure, is studied in present work. It has been demonstrated that these parameters are not only statistically significant but also possess physical relevance with the actual phenomenon (Dalvi et al., 2014). As a wide range of these parameters is included in the proposed ELM, the ranges are classified such that a particular range of respective parameter reflects the actual field behavior, indicating possible relative risk of liquefaction. Accordingly, the entire data-set of 314 cases is classified into three categories, namely: Class (A) High risk of liquefaction, Class (B) Intermediate risk of liquefaction, and Class (C) Low risk of liquefaction. These ranges of the respective parameters and their corresponding potential risk of liquefaction are summarized in Table 2.

Effect of peak ground acceleration (a max )
a max has been commonly used to describe the ground motion because of its inherent relationship with inertial forces; indeed, the largest dynamic forces are assumed to be closely related to the a max (Kramer, 1996). The scatter of data points relative to each class corresponding to ranges of acceleration as stated in Table 2 is represented graphically through Figure 3(a-c). Figure 3(a) is indicating low range of a max (a max < 0. 2 g) categorized as Class C. Figure 3(b) demonstrates variation of ELM over  From these Figures (3 (a-c), it can be noted that the overall distribution of liquefaction and nonliquefaction points relative to the proposed ELM across these ranges as specified above appears to be fairly balanced, indicating efficiency of the developed model. Moreover, for intermediate and high a max ranges, (Figure 3(b) and (c), respectively), more than 60% of correct prediction of non-liquefied sites is observed. It is worth mentioning that the positive predictability of the proposed ELM through Equation (4) remains unaltered, although a particular range of a max is considered.

Effect of earthquake magnitude (M w)
It is a known fact that for liquefaction to occur, there must be ground shaking and the potential for liquefaction increases with the increase in earthquake shaking represented by M w . A similar observation as above is found when the model is verified relative to ranges of M w as: Class C (M w < 6.5), Class B (M w = 6.5 to 7.5), and Class A (M w > 7.5) as depicted in Figure 4(a-c), respectively. The scatter of data points shown in Figure 4 (a) indicates that most of the misclassified non-liquefaction cases bear the (N 1 ) 60cs value less than 10 which is indicative of high risk of liquefaction.  Figure 4 (c) clearly indicates that all the liquefaction cases of Class A are correctly predicted by the ELM. It can be stated that the model performance remains impartial at defined ranges of magnitude as well as acceleration of an earthquake as the overall success rate of accurate prediction remains unchanged. The functional form of LP term as employed in the present study includes these two parameters through v max . This ensures the realistic seismic soil response as frequency content gets included.

Effect of vertical effective overburden pressure
Further, representing geological setting of soil strata, overburden stress effects for liquefaction analysis procedures have been investigated by Boulanger (2003),  and Seed et al. (2003). Moreover, it is known that soil behaves as a deformable body with increasing depth; hence, depth is a site condition parameter and has a huge impact on soil response against its capacity to resist liquefaction. In addition to this, for liquefaction to occur, groundwater table should be at sufficient depth to create saturated soil conditions. Thus, the depth to groundwater table is an another important consideration in identifying soils that are susceptible to liquefaction. The vertical effective overburden pressure obviously takes into account the effect of depth as well as groundwater table level. For this purpose, data have been categorized w.r.t ′ v in three different classes to represent the pertinent degree of risk of liquefaction ( Table 2). The variation of ELM w.r.t these classes is illustrated through Figure 5 (a) to (c). (4) is unbiased, relative to the variation in vertical effective overburden pressure. Moreover, it rightly indicates that the susceptibility of liquefaction decreases (LP m term) with increase in vertical effective overburden pressure. Thus, it is inferred that the observed behavior on field is replicated through the proposed model. In summary, the deterministic ELM not only yields accurate prediction of liquefied sites within a given range of model parameters but also replicates actual field behavior, thus an optimistic yet realistic ELM is developed.

Performance of proposed ELM using varied data-sets
Based on the investigation of predictive performance of recently developed empirical models, it is inferred that the success rate of accurate prediction reduces if the developed model is tested upon a totally different data-set and thus in general, the models are data specific (Pathak & Dalvi, 2011).
In this work, the predictability of the present ELM is ascertained on a varied data-set of around reported 740 cases. Among these, 386 case records are obtained from Hanna et al. (2007) and . Based on verification over these data points, it is prominently observed that the developed ELM succeeds to accurately detect the occurrence of liquefaction for all the 227 "yes" cases out of the total 386 cases under consideration.
As the remaining 354 case records extracted from Kayen et al. (2013) include database of corrected shear wave velocity (V s1 ), the (N 1 ) 60 value has been computed using the established correlation of Andrus and Stokoe II (2000) as given by: where V s1 = overburden stress-corrected shear wave velocity, B 1 = 93.2 ± 6.5 and B 2 = 0.231 ± 0.022. For this data-set, the predictability of ELM is verified against Equation (2) which gives the relation between LP m term and (N 1 ) 60 . The scatter of liquefied cases is as depicted in Figure 6 below.

Figure 4. Variation of ELM with earthquake magnitude (M w ).
It is interesting to note that most of the 238 "liquefied" cases are accurately predicted as could be seen lying within the zone of liquefaction as shown in Figure 6. Although few of the data points could Thus, overall, the ELM performance is verified on 740 case records apart from the 314 cases employed in formulating and validating the model. It is prominently observed that the developed ELM succeeds to accurately predict the occurrence of liquefaction for all the 465 "yes" cases out of the total 740 cases under consideration. Although conservative results are observed for non-liquefaction cases, it is to be noted that such wrongly predicted "no" liquefaction cases bear the properties which are prone to liquefaction. Finally, the ELM has performed well on over 1,054 case history records within the given range of model parameters, covering worldwide 46 number of earthquakes occurred during 1906-2011. Thus, the proposed final form ELM (Equation 4) certainly bears remarkable potential in detecting liquefaction by correlating the field conditions with the dynamic soil properties and ground motion parameters. The performance of the developed ELM to predict liquefaction is then compared with existing empirical models to demonstrate its applicability in the field as discussed in the next section. The comparison is purely meant to focus the inclusion of dynamic response in liquefaction evaluation procedures as proposed by the authors.

Comparison of proposed ELM with other approaches
The proposed ELM being fundamental in its nature has to be verified for its performance with SPTand V s -based simplified procedures. The SPT-based approach as originally proposed by Bolton Seed et al. (1985) is based on empirical evaluations of field observations. Although this method has been modified and improved periodically since that time, it presents the fundamental concept of inclusion of fines content into liquefaction evaluation procedures. Thus, the model performance when  compared with these SPT-based approaches indicates that the developed ELM is efficient in correctly identifying the liquefied sites which are misclassified by these approaches. Further, it is known that Stokoe (1997, 2000) pioneered in the use of V s , a dynamic soil property as a field index of liquefaction resistance. Similar to the simplified procedure, they proposed CRR-V s1 curves corresponding to the range of fines content using compiled case histories. As stated in Andrus and Stokoe II (2000), only two evidences from 1989, Loma Prieta event (M w = 6.93), lie incorrectly in "No" liquefaction zone; however, these two misclassifications lie very near to the demarcation curve; interestingly, the proposed ELM rightly predicts liquefaction occurrence for these two evidences which were actually reported as liquefied. This also indicates that success rate of correct prediction is more when dynamic soil properties are taken into account.
The performance of ELM is verified using the approaches by Youd et al. (2001): (Approach I) and Cetin et al. (2004): (Approach II) and also two recently developed approaches, namely: SPT-based approach by Oommen, Baise, and Vogel (2010): (Approach III) and V s -based approach by Zhang (2010): (Approach IV). The research carried out by various researchers is undeniably exemplary. It is to be noted that the basis for this comparison is to verify the performance of various ELMs over a common data-set which in present work is as extracted from Cetin et al. (2004). The values of overall accuracy (OA) and success rate for liquefaction (SRL) by approaches I, II, and III have been mentioned as reported in Oommen et al. (2010), whereas the values of the same for approach IV and ELM are computed. The predictive performance in terms of OA and SRL is then summarized in Table 3.
From Table 3 it is observed that the OA in correct prediction can be seen to be at par with other four approaches. Moreover, the proposed ELM succeeds to detect liquefaction occurrence more accurately. It is to be noted that the comparison of ELM with other models is cited purely to symbolize the effectiveness of using the proposed methodology, although the basic framework of liquefaction assessment for each of these models differs significantly. Overall, the above discussion justifies the extensiveness of the developed ELM and thus for accurate prediction of liquefaction occurrence, it would be more appropriate to employ the proposed ELM which will minimize the enormous losses caused due to liquefaction.

Conclusions
Formulation, validation, and verification of a fundamental yet comprehensive ELM for detection of liquefaction occurrence is presented. Although it follows the general format of simplified procedure based on SPT, the liquefaction potential is characterized by integrating the effect of dynamic soil properties and ground motion parameters. The final form of the model is not only dimensionally homogeneous but also takes into account the fines content correction. Realistic predictability is achieved as sampling bias and class balance is maintained while formulating the ELM. Based on the performance of the ELM relative to soil and seismic parameters, it bears the potential to replicate the actual field behavior. A final correlation thus established is found to perform well for the diversified 1,000 case history records. Accurate prediction of liquefied cases for the specified range of model parameters is the prominent feature of the developed ELM.