When saturated sand is subjected to an earthquake, because of the rapid vibrations, drainage is prevented and the tendency towards volume reduction, causes the transfer of the effective overburden stress to the pore water until excess pore water pressure becomes equal to the initial effective overburden stress; after which liquefaction happens. The most commonly reported liquefaction manifests in saturated loose or medium sandy soil have been observed during the most massive earthquakes worldwide. The 1964 magnitude 9.2 earthquake in Alaska and the magnitude 7.6 earthquake in Niigata of the same year, prompted extensive research on this phenomenon. Soil liquefaction has also been observed in recent earthquakes in China [1], Japan [2], Indonesia [3] and the USA [4,5].

Three main methods have been employed in relevant studies. The first one is the stress-based method, which was introduced by Seed et al. [6]. And other researchers performed research to develop models using in-situ tests [7–9], laboratory tests [10,11] and numerical simulation [12–19]. Additionally, the strain-based method was first introduced by Dobry et al. [12]. They assumed under cyclic loading approximately 0.01% for threshold strain and initial water pressure (u_w) increasing. After that, the shear strain was compared with 0.01 according to Serikawa et al. [2] or 0.02 [3]. Next, water pressure was estimated through experimental graphs. In the end, this water pressure was compared to confining stress to predict the triggering of liquefaction. In other words, depending on the following condition, liquefaction may or may not occur:

u_w > σ_v0 liquefaction occurs.

u_w < σ_v0 liquefaction does not occur.

where σ_v0 is initial vertical effective stress.

The coupled numerical models, since 1975, have been presented [13–17] based on Biot’s theory [18–20] which was the clarification of effective stress concept and coupled phases interaction between solid porous materials and fluid. Recently, some numerical simulation was also performed by researchers [21–28]. The fourth method includes strain energy-based methods developed by applying seismic energy dissipated in the soil [21–30]. This method has been applied in three main procedures by researchers which are using histories of site exploration liquefied [29–31], and laboratory test results [23,27,29–40] and Arias intensity-based models [32,41]. To evaluate the potential of liquefaction in energy concept method, the capacity strain energy (W) value of the soil is required to be estimated to compare with the energy transferred to the soil by the earthquake loads. Since the energy dissipated by mechanisms (e.g., cohesion and frictional mechanisms) cannot be easily discerned from laboratory and field data, the energy dissipated by frictional mechanisms is estimated by the total dissipated energy as the frictional mechanisms are expected to be dominant growing interest in earthquake engineering. In addition, the total amount of dissipated energy to the liquefaction point should be relatively independent of the sequence for increasing the load. On the contrary, the viscous mechanisms of energy dissipation can be considered for the low increase of strains where the rate of energy dissipated by this mechanism is directly related to the sequence used. In order to enhance the conventional load, the dissipated energy is expected to be greater than the liquefaction point, which is either independent or increases due to the load sequence used.

Based on laboratory test results six input parameters including effective confining pressure (σ’_c) kPa, initial relative density (D_r)%, FC%, coefficient of uniformity (C_u), mean grain size (D₅₀) (mm) and coefficient of curvature (C_c), have been identified and confirmed as the most influential factors in modeling liquefaction to estimate liquefaction resistance of sandy soil based on capacity strain energy concept [23,27,29–40]. Clearly, permeability of the soil is considered implicitly in soil properties parameters of D_r, C_u, D₅₀ and C_c.

These studies, except Cabalar et al. [37], extracted C_c in their final correlation due to its limited range values in the datasets and hence, its limited effect. Furthermore, they included all ranges of the parameters in their models without special consideration to their value.

Further, Maurer et al. [42] analyzed 7,000 case histories from Canterbury Earthquakes in 2010–2011 and concluded when soils contain a high value of FC, assessment of liquefaction is less reliable. Zhang et al. via laboratory tests showed that liquefaction potential is closely related to FC [38]. Liu et al. [43] performed some experimental tests on marine sediments and proposed a critical value for FC to evaluate liquefaction resistance. Additionally, Tao [44] defined the limit value of 28% for estimating the liquefaction resistance, and through laboratory test results showed liquefaction resistance becomes more dependent on D_r when FC is higher than 28%, rather than FC value. While, in all presented models there has not been consideration to different effect of FC in different range.

Regarding the evaluation of liquefaction strength by applying the strain energy approach, several models have been developed using artificial neural network (ANN) [35], genetic programming (GP) [45,46], multi expression programming (MEP) [46], neuro-fuzzy Interface system (ANFIS) [37], and multivariate adaptive regression splines (MARS) [38]. Although the importance of the validating phase has been indicated by many researchers [47–49], in all studies, data division was performed randomly in two groups of testing and training phases, without considering the statistical characteristics of the data. Also, no validating phase has been performed in order to avoid overtraining of the models.

Due to the uncertainty of geotechnical problems, particularly, liquefaction phenomena, some studies, such as Bayesian methods, have been performed to develop probabilistic forms and reliability analysis to evaluate the potential of liquefaction [50–56]. Furthermore, artificial intelligence [57–63] and Monte Carlo simulation (MCS) which is a classic approach to assess risk in quantitative analysis, has recently been used in engineering and sciences [64–71] and also in the evaluation of liquefaction potential [72–74]. Jha et al. [73] presented the probability of liquefaction due to factor of safety using FOSM method, an advanced first-order second-moment (FOSM), Hasofer–Lind reliability method, a point estimation (PEM), and an MCS method. They presented a new combined method using both FOSM and PEM to find the cyclic stress ratio (CSR) and cyclic strength ratio (CRR) statistically. They showed that the factor of safety measured by the combined method is similar to the PEM and MCS methods. They also indicated FOSM, PEM, and MCS methods present nearly the same probabilities of liquefaction by considering input variability. Using the jointly distributed random variables method and using the data from triaxial test results, Johari et al. [74] presented a reliability assessment of liquefaction and compared the results with the Monte Carlo simulation. The results exhibited close probability density functions of the safety factor applying both methods.

In this study, to investigate the complex effect of FC on liquefaction resistance of soil in terms of the unit energy, two datasets were arranged. The first dataset was collected from the literature as the largest and likely most complete dataset employed by researchers covering a large range of parameters. Due to the complicated influence of FC on W and the spares attention to this parameter in developing earlier models, in the second database, according to Tao [44] only samples with FC values less than 28% were selected. Two new ANN models were developed based on these two datasets. A multilayer perceptron network with a backpropagation algorithm was constructed and the samples were divided into three groups, including a validation set to avoid overtraining. These sample groups were formed with similar statistics certificates, and avoided random division, according to Tables 1–4 and Tables 6–9 to enhance the accuracy and capability of trained networks.

This study investigates the effects of all parameters, including C_C while also paying special consideration to the influence of FC in different range according to its critical value in seismic soil liquefaction assessment. To achieve this goal, two different ANN models, one using the entire dataset and the other using the samples with FC value of less than critical value, were developed to compare their predictions for validating and choosing the best one. In development of the models the validation phase was also performed to eliminate overtraining of the models. The data division was performed considering statistics characteristics of the variables instead of performing randomly to increase the accuracy of the trained models. Furthermore, to the best of the author’s knowledge, there has been no previous due attention to the uncertainties of parameters to predict W, which was the motivation behind performing the sensitivity analysis via MCS simulation to investigate the effect of uncertainty and the mean value of parameters on liquefaction resistance. Because of the numerous samples required by MCS and the relative data scarcity, in this study, the MC simulation was performed based on an ANN model developed in this study.

Figueroa et al. [75] developed two equations to evaluate unit energy (E) in a cyclic triaxial test. Alkhatib [76] introduced ER to measure liquefaction resistance. ER is a ratio of the energy computed by area under the stress-strain hysteresis loop to the initial effective confining stress, and through conducting laboratory cyclic triaxial tests. The presented model is as below.

Extensive research has been performed at Case Western Reserve University on energy-based evaluation of liquefaction [31,38–41,75,77–79]. In all procedures, W_u which is the area of the stress-strain hysteresis loops up to the initial liquefaction point was used to define liquefaction resistance. Parameter of δW was introduced first time by Figueroa et al. [31,75,77]. They conducted 27 torsional shear tests on a Reid Bedford sand sample at different shear strain amplitudes and confining pressures and developed a model.

Liang et al. [13,41] conducted 74 liquefaction torsional shear tests on Reid Bedford sand, Lower San Fernando Dam (LSFD) silty sand, and Lapis Luster Dried sand (LSI-30) through random loading. From the test results, they performed a regression analysis and presented an equation to estimate δW.

Kusky [78] developed two equations according to 27 strain-controlled torsional triaxial tests, which were conducted on samples of Reid Bedford.

Rokoff [79] conducted some cyclic torsional shear tests on Nevada sand to investigate the influence of particle size distribution on δW. The regression was limited to special soil properties and geology related to the samples of Nevada region, which contains C_u and C_c as below: (1)log10⁡(δW)=3.6746+0.004877σc′+0.01039Dr+0.21802Cu−2.1444CcR2=0.8195 (2)Cu=D60D10 (3)Cc=D302D10D60

In addition, Figueroa et al. [40] confirmed that Cu and Cc affect δW more than σc′ and D_r. Wallin [80] presented three equations for Nevada sand, LSFD silty sand, and Reid Bedford sand through statistical analysis of tests results which were conducted by other researchers [39,75].

Baziar et al. [35] collected a large dataset from performed shear, cyclic triaxial, and torsional shear laboratory test results, which contained 284 samples from the literature. They developed two Artificial neural network (ANN) models to obtain a correlation between input parameters and Log (W). The first developed ANN model contained six input parameters (σ’_c, D_r%, FC%, C_u, D₅₀, C_c) while the second model was developed by eliminating the parameter C_c. They subsequently demonstrated that FC has the highest effect on W by carrying out a sensitivity analysis. By adding a new dataset to Baziar et al. [35] with the same parameters and applying multigene Genetic Programming, Baziar et al. [45] developed an equation to measure W and then used case histories earthquake data plus laboratory test data to validate and present the accuracy of their model. They concluded that the value of W has a complicated relationship with FC. Alavi et al. [46] presented three equations to estimate Log (W) through applying MEP, GP, and MEP and with the same database and parameters as Baziar et al. [35], as mentioned in the appandix and Table A1. In addition, they conducted sensitivity analysis and confirmed that W is more affected by D_r and σ’_cthan other parameters. Zhang et al. [38] collected 302 samples for their database, which contained six cyclic simple shear, 18 centrifuges, six cyclic simple shear, and 217 cyclic tests. They developed a MARS model with the same five input parameters with Cavallaro et al. [10] to evaluate Log (W). They validated their model using 22 centrifuge test results conducted by Dief [81].

In this study, ANN was used to develop models to estimate the liquefaction resistance of sandy soil based on the capacity strain energy concept and using laboratory test data. The validating phase was performed, in addition to the testing and training phase, to avoid overtraining the model and increasing the model’s capability. An extensive database was collected from literature, including triaxial, simple shear, torsional, and centrifuge test results. ANNs are powerful tools for developing models that can take into account the complexity and non-linearity of the liquefaction issue. To inspect the complicated influence of FC on liquefaction resistance of soil, according to research results presented by Tao [44], two ANN models were developed. The first model was developed using a complete dataset, while the second one was based on the samples by FC less than 28%. The accuracy and capability of the presented models were demonstrated by comparing their predicted values for log (W) with four other available well-known equations. To conduct this comparison, 20 liquefaction test results from Nevada sand and Reid Bedford sand [41], which were independent of the two applied datasets for training the models, were considered. Finally, to investigate the effect of uncertainty in geotechnical parameters, a sensitivity analysis was performed using MCS based on the response surface provided by the second presented ANN model, which showed higher accuracy. The results of sensitivity analysis were illustrated through some graphs to indicate the correlation between the variables and their uncertainties with the liquefaction resistance of the soil in order to capacity energy. The limitation of the present study includes its application in the issue of strain energy, not in the other methods such as stress-based or numerical methods.

In conclusion, this study has demonstrated:

Artificial neural network (ANN) is a powerful tool to assess liquefaction in soil with high non-linearity. Adding validation phase and performing data division by considering the statistical aspects, instead of random division, provides significant precision on the model.