Predicting lateral displacement caused by seismic liquefaction and performing parametric sensitivity analysis: Considering cumulative absolute velocity and fine content

Abstract

Lateral displacement due to liquefaction (D_H) is the most destructive effect of earthquakes in saturated loose or semi-loose sandy soil. Among all earthquake parameters, the standardized cumulative absolute velocity (CAV₅) exhibits the largest correlation with increasing pore water pressure and liquefaction. Furthermore, the complex effect of fine content (FC) at different values has been studied and demonstrated. Nevertheless, these two contexts have not been entered into empirical and semi-empirical models to predict D_H. This study bridges this gap by adding CAV₅ to the data set and developing two artificial neural network (ANN) models. The first model is based on the entire range of the parameters, whereas the second model is based on the samples with FC values that are less than the 28% critical value. The results demonstrate the higher accuracy of the second model that is developed even with less data. Additionally, according to the uncertainties in the geotechnical and earthquake parameters, sensitivity analysis was performed via Monte Carlo simulation (MCS) using the second developed ANN model that exhibited higher accuracy. The results demonstrated the significant influence of the uncertainties of earthquake parameters on predicting D_H.

Graphical abstract

Keywords

lateral spreading displacement / cumulative absolute velocity / fine content / artificial neural network / sensitivity analysis / Monte Carlo simulation

Introduction

Liquefaction-induced lateral movement is one of the most destructive effects of earthquakes in saturated loose or semi-loose sandy soil. The liquefaction or pore water pressure accumulation induced by an earthquake leads to the soil losing its cohesive strength and the relative movement of its mass to the surrounding soil. Observations from previous earthquakes (e.g., Tohoku and Lushan earthquakes) show that liquefaction occurred during several earthquakes and was the root cause of severe damage. Extensive damage to constructions caused by liquefaction has been reported owing to the 2011 Tohoku earthquake [1]. Furthermore, sand boiling caused by liquefaction was observed during the 2013 Lushan earthquake [2].

Franke [3] categorized the models for the estimation of lateral displacement due to liquefaction (D_H) into three groups: empirical, semi-empirical, and analytical models. Empirical models are based on measured field data [4–9], semi-empirical models are theoretical and computational approaches using field data [8,10–12] while analytical models are mathematical closed-form solutions [13–19].

Among these models, analytical models contain the most complex approaches and require a large number of complicated parameters, whereas empirical and semi-empirical models are based on data case histories that are simple to use.

Several research works have studied the complex influence of fine content (FC) in saturated sand during dynamic loads [20–22]. Most of these studies introduced a soil sample with FC values between 20% and 30% as a transition behavior. Derakhshandi et al. [21] performed some strain-control laboratory tests, which indicated that sand samples with FC values less than 20% exhibited sand-like responses, but displayed clay-like responses when its FC values were larger than 30% during dynamic loads. A large case history data set, containing 7 000 samples of Canterbury earthquakes in 2010 and 2011, was investigated by Maurer et al. [23]. They inferred that the presented models are less accurate in evaluating liquefaction at high FC values. Furthermore, Tao conducted a few laboratory tests and demonstrated that the liquefaction resistance in a soil sample of FC larger than 28% exhibits a higher dependency on relative density (D_r) than in a soil sample of FC less than 28%. Therefore, there is a significant dependency between FC and the potential of liquefaction in a soil sample with less than 28% of FC [20]. Although the complex effect of FC on the resistance of soil against liquefaction at different FC values has been demonstrated, to our knowledge, except for Ref. [24], no attempt has been made to harness this relationship and assess liquefaction. All the available empirical and semi-empirical models for estimating D_H contain the entire range of the parameters for FC without any limitations.

Furthermore, machine learning and intelligent algorithms have been applied to develop models and predict D_H [25–29]. Using the data set of Bartlett and Youd [30], Wang and Rahman [25] developed a back-propagation learning neural network algorithm with one hidden layer to develop an artificial neural network (ANN) model and estimate D_H. Additionally, Baziar and Ghorbani [26] created a back-propagation algorithm and presented an ANN model based on the data set collected by Youd et al. [31]. They also carried out sensitivity analysis to evaluate the potency of the input parameters. Using the data set of Youd et al. [31], Javadi et al. developed an ANN model. They presented three models for free face, gentle slope, and both conditions. Then, they derived three equations based on the three ANN models to predict D_H. García et al. [28] applied a neuro-fuzzy system using the data set of Youd et al. [31]. The major limitation in all presented models of Refs. [25–28] was the division of the data set into two subsets for the testing and training phase, which lacked a validation phase to prevent the over training of the models. Furthermore, data division was performed randomly without considering the statistical features of the parameters.

Kramer and Mitchell [32] examined the intensities of more than 300 earthquakes and inferred that standardized cumulative absolute velocity(CAV₅) provides more efficiency and sufficiency to the prediction of excess pore water pressure owing to the motion of earthquakes in liquefiable sandy soil. To present a model that estimates CAV₅, they collected a large database including 284 records of 40 earthquakes with causative fault types, such as normal, strike-slip, reverse, and reverse-oblique fault types. However, the available models have not included the parameters proposed in this study to predict D_H.

According to the study on the significance of the input parameters in estimating D_H, Baziar and Ghorbani [26] carried out sensitivity analysis to evaluate the potency of the parameters. However, they did not consider the uncertainties of the parameters. Ganji et al. [33] investigated the effect of parameters’ uncertainties on a geotechnical construction and demonstrated its significant influence; however, there are no studies on the uncertainties of D_H parameters.

Two ANN models were proposed in this study to investigate the complex influence of FC. First, a data set was created by estimating CAV₅ using the attenuation equation of Kramer and Mitchel [32] and adding it to the data set of Youd et al. [31]. Therefore, the main data set was capable of covering all aspects of earthquakes, including near fault zone, frequency load motion, and causative fault types. After that, by calculating the correlation coefficient (R), the statistical analysis was applied to discover meaningful parameters and create the main data set. Next, the first ANN model was derived via the main data set, which covers the entire range of the parameters. Then, to investigate the complex influence of FC on the D_H, the second data set was arranged by eliminating all samples with FC values larger than 28%, which was defined by Tao [20] as a critical FC value in the evaluation of excess pore water pressure in liquefaction. Finally, the second ANN model was developed based on the second data set. To achieve higher model accuracy, data was divided into three subsets with the same statistical features, instead of a random division. Further, the validation phase was conducted alongside the training and testing phases to prevent the overtraining of the models. Finally, owing to the uncertainties in the characterization of soil properties and the prediction of earthquake intensities, to evaluate the effect of the parameters and their uncertainties, parametric sensitivity analysis was conducted via Monte Carlo simulation (MCS) based on the second developed ANN model, which provides more accuracy than the first one.

Theoretical bases of methods

In this study, the validation phase was added to the most common constructed network to demonstrate its influence on terminating the training criteria. Then, to validate their capability and determine the best ANN model, the created ANN models were compared with additional available models. Furthermore, to illustrate the effect of earthquake-parameter uncertainties on D_H, a sensitivity analysis was performed based on the best ANN.

Artificial neural network

A neural network is a computing system that works similar to the human brain information processing model by connecting simple nodes (networks of nodes). Consequently, it is called a “neural network”. Different types of ANNs have been introduced by scientists. Among them, multi-layer perceptron [34] is more popular in the literature. Hornik et al. [35] showed that multilayer perceptrons (MLPs), which are supervised networks, possess the best capacity and ability to approximate any function with high accuracy. These include three types of layers: an input layer, which distributes the input data and contains one neuron for each input variable, one or more hidden layers, which perform nonlinear transformations, additions, and multiplications, and an output layer for estimated final results, which contains a number of neurons equal to the number of targets that is to be approximated by the ANN model.

A back propagation network has been commonly applied by researchers as a supervised learning technique. The function of a neural network is its ability to learn the representation in the training data, which enables it to predict the desired target value. During the training process, the object minimizes the total error for the predicted output, as defined below:

(1)E={\displaystyle \sum _{n=\mathrm{1}}^N{\displaystyle \sum _{i=\mathrm{1}}^I{(t-p)}^{\mathrm{2}}}},

where N, I, t, and P represent the number of samples, number of outputs (targets), target value, and the value predicted by the ANN for the target, respectively.

The Levenberg–Marquardt back-propagation algorithm, which is a well-known network for solving nonlinear systems, was first introduced by Levenberg [36]. Here, the connection weights are updated by estimating the error and distributing it through the layers of neurons. This entails two steps, which are iterated to obtain a pre-specified tolerance range for the output. In the first step, the network produces an output, and in the second step, the estimated error at the output layer is distributed to the hidden layers and then to the input layer to modify the weights. The Levenberg-Marquardt algorithm (LVM) is a combination of the gradient descent algorithm and the Gauss-Newton-Method and is expressed as:

(2)\left(J^{T}J+\lambda I\right)\delta =J^T[y-f(\beta )],

where J, I, and l represent the Jacobian, identity matrix, and dumping factor, respectively.

The three main indices used to inspect the accuracy of the ANN models are the root mean squared error (RMSE), coefficient of correlation (R), and mean absolute error (MAE). However, R is the most common index and is expressed as:

(3)R={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^{ns}\left(x_i-\overline{x}\right)}\left(y_i-\overline{y}\right)}{\sqrt{{\displaystyle \sum _{i=\mathrm{1}}^{ns}{\left(x_i-\overline{x}\right)}^{\mathrm{2}}}}\sqrt{{\displaystyle \sum _{i=\mathrm{1}}^{ns}{\left(y_i-\overline{y}\right)}^{\mathrm{2}}}}}},

where ns is the number of the samples; x_i and y_i are the sample points of \overline{x}{\displaystyle \frac{\mathrm{1}}{ns}}{\displaystyle \sum _{i=\mathrm{1}}^{ns}{x}_i} (mean value of the sample) and\bar{y}, respectively.

Network samples are usually separated randomly into two subsets. One subset is used for training by adjusting its weights while the other, which does not contribute to the training phase, is applied to testing the network. Additionally, an additional subset, called the validating set, can be considered to prevent network overtraining. During training, the correlation coefficient increases, conversely, with a decrease in the correlation coefficient of the validating subset, it exhibits overfitting.

Recently, the practicability of using ANN in geotechnical construction has been confirmed [37,38], particularly in nonlinear situations that are similar to the liquefaction phenomenon.

Monte Carlo simulation

MCS is a statistical assessment of mathematical functions by applying random samples. It requires a large number of samples to provide more accurate results [39]. In contrast with deterministic variables or parameters, it estimates variables that depend on one or more random factors. MCS is used to investigate the effect of uncertainties in model predictions. Furthermore, it obtains numerical solutions for complex problems that are impossible to solve analytically [40]. Generally, MCS can be performed in four major steps. First, outline a domain for inputs, then define a distribution function for random variables. Next, generate input samples by applying a distribution function to the domain of the variables. Finally, conduct simulation on the input samples.

Monte Carlo simulation using ANN for parametric sensitivity analysis

As earlier mentioned, MCS requires large number of samples. Additionally, for any given project, it is expensive and time consuming to prepare samples, perform laboratory tests, in situ tests, or computation calculations. To address this limitation, the first ANN model was developed to generate a large number of samples. Figure 1 illustrates the process applied in this study to perform the MCS-based ANN model. The Latin hypercube sampling method [41] was applied to generate samples and estimate output by ANN to perform MCS.

Cumulative absolute velocity

In the electric power research institute (EPRI), Eed et al. [42] introduced cumulative absolute velocity (CAV) as a criterion for shutting down nuclear power plants. After that, O’Hara et al. [43] presented an equation in EPRI to predict CAV, which is expressed as:

(4)CAV={\displaystyle {\int }_{\mathrm{0}}^{t_{\max }}|}a(t)|\mathrm{d}t,

where a(t), t, and t_max are the acceleration of ground motion graph, time, and duration of the earthquake, respectively.

Although the momentum magnitude of an earthquake (Mw) and its peak acceleration (a_max) are the most common earthquake motion input in the empirical, semi-empirical, and analytical models to assess liquefaction hazards, Kramer and Mitchell [32] evaluated the effect of 300 earthquake parameters on liquefaction. They investigated these parameters to determine which one of them correlated closely with the accumulated pore water pressure in sand during earthquakes. Their collected database included 450 ground motions from 22 earthquakes with a broad range of faulting types, rupture distance (r_rup), and Mw. Furthermore, the authors evaluated the efficiency and sufficiency of the parameters according to Luco and Cornell [44]. Finally, it was inferred that the CAV₅ parameter provides the highest efficiency and sufficiency with liquefaction assessment, as expressed in Eq. (5). Additionally, they utilized the pacific earthquake engineering research center (PEER) strong motion database containing 282 records from 40 earthquakes to present an attenuation equation to predict CAV₅, as expressed in Eq. (6).

CA{V}_{\mathrm{5}}={\displaystyle {\int }_{\mathrm{0}}^{\infty }\langle \chi \rangle }\left|a\right(t\left)\right|,

where

(5)

(6)\begin{array}{l}\ln CA{V}_5\hfill \\ =3.495+2.764(Mw-6)\hfill \\ +8.539\ln (M/6)+1.008\ln \left(\sqrt{r_{\mathrm{rup}}^2+6.155}\right)+0.464F_{\mathrm{N}}+0.165F_{\mathrm{R}}\hfill \end{array},

where CAV₅ is a form of CAV excluding small amplitude accelerations less than 5 cm/s². F_N = F_R = 0 for strike slip faults F_N = 1 and F_R = 0, for normal faults .and F_N = 0 and F_R = 1 for reverse or reverse-oblique faults (F_N and F_R) are the expressions.

Case history database

Bartlett and Youd [45] collected case histories of lateral spread for four US and four Japanese earthquakes including 476 displacement vectors considering 267 standard penetration test (SPT) boreholes. Consequently, they developed an empirical equation using a multiple linear regression (MLR) approach. The main part of the database was provided by investigating a large number of lateral spreads in Nigata, Japan [4]. Furthermore, they selected influential parameters in three categories: seismic, topographic, and geotechnical parameters. Among earthquake intensities, they considered the moment magnitude (M_w) and horizontal distance from the site to the rupture surface (r_rup) in km. They also inspected the free-face ratio (W) and ground slope (S), both in percentage, as influential topographic parameters. Additionally, the average fines content in the T₁₅ layer (F₁₅) (in percentage) within thickness of the layer with corrected blow counts (N1)₆₀<15 (T₁₅) and average mean grain size in the T₁₅ layer (D50₁₅) calculated in meters (m) and millimeters (mm), respectively. Finally, they presented two separate MLR equations for free face and ground slope conditions. Youd and Bartlett revised their models by adding data from three earthquake case histories to their data set and eliminated eight samples due to their boundary effects [31].

First, in this study, the authors entered CAV₅ into the data set to extend it to cover all aspects of the earthquakes, such as causative fault type, frequency of earthquake loads, and near-fault zone effects. Therefore, causative fault types of the earthquakes in the data set were discovered, and by using Kramer and Mitchel’s [32] attenuation equation (Eq. (6)), this parameter was estimated for all samples in the data set. In the second step, considering the applicable range of Eq. (6), the data obtained from the Alaska-1964 site with a magnitude of 9.2 was deleted from the data set. The equation provided by Kramer and Mitchell [32] is applicable for a M_w range of 4 to 8. Consequently, the data set provides a magnitude range from 6.4 to 7.9. In the third step, statistical analysis was conducted for all parameters by evaluating the correlation coefficient (R) of all parameters with D_H successively. Contrary to previous assumptions, the results of the statistical analysis exhibit a positive value of 0.104 for R between r_rup and D_H, and a negative value of -0.98 for R between S and D_H. Consequently, by excluding these two parameters, the main data set created with 215 samples included only the free face condition with 215 samples.

Development of ANN models

In the fourth step, to assess the complex influence of F15, two ANN models were presented. The first one was based on the main data set, which was called ANN1 while the second one (ANN2) was based on the second data set.

Data sets

The first model (ANN1) was based on the entire range of parameters, whereas the second model (ANN2) was based on the samples with F₁₅ values less than the critical value of 28% [20]. Therefore, the main data set defined in Section 6 was the first data set, which was void of alterations and includes 215 case histories (samples) with six parameters. The second database, containing 182 samples, was arranged by excluding all samples with F₁₅ values larger than 28%. Tables 1 and 2 illustrate the features of these two data sets.

Data division

In this study, a validation phase was applied to prevent the model from being over-trained. To achieve this goal, the data were divided into three subsets for the training, testing, and validation phases. Furthermore, to obtain more effective and precise models, the data were divided by attempting to provide all three subsets with similar statistical features, such as minimum, maximum, and mean values, as opposed to random divisions. Generally, 70% of the samples were considered for the training phase while the rest were considered for the testing and validation phases. Therefore, the data set was divided into 151, 32, and 32 samples for training, testing, and validating, respectively. Additionally, the second data set with the same process was separated into 128, 27, and 27 samples for the three phases of the ANN model to estimate D_H. The statistical features of the testing and validation subsets of both data sets are presented in Tables 3–8. As considered in any data set division, we tried to provide all three subsets with approximately equal features.

ANN Models

In this study, a single hidden layer MLP with a back-propagation algorithm and sigmoid activation function was utilized. As the most common network, MLP has been demonstrated to have superior capability in assessing the liquefaction phenomenon [25–27,46,47]. The three-layer MLP applied includes only one hidden layer, as illustrated in Fig. 2. For the MLP with an input vector x = (x₁, x₂,…,x_m) and an output vector y = (y₁, y₂,…,y_p), a correlation exists between its inputs and outputs, which can be expressed as:

(8)y_j=f_{\mathrm{output}}\left[{\displaystyle \sum _{h=\mathrm{1}}^k{w}_{jh}}{f}_{\mathrm{hidden}}\left({\displaystyle \sum _{i=\mathrm{1}}^m{w}_{hi}}{x}_i+w_{h\mathrm{0}}\right)+w_{j\mathrm{0}}\right],

where m is the number of input units, k is the number of neurons in the hidden layer, p is the number of output units, x_i is the ith input unit, w_hi is the weight between input i and hidden neuron h, w_jh is the weight between hidden neuron h and output neuron j, w_ho is the threshold (or bias) for neuron h, w_jo is the threshold for neuron j, f_hidden is the transfer function of the hidden layer, and f_output is the transfer function of the output layer.

The network developed in this study presents two models: the first model for the entire range of the parameters (ANN1) and the second one for the entire range of the parameters except those with F₁₅ values larger than 28% (ANN2). Tables 3 and 4 present the features of both models. The value of R for the three subsets of both models were approximately 90%, indicating high accuracy of the models.

It should be mentioned that the main objective of this research was to investigate the effect of the validation phase, CAV₅, and critical value of F₁₅. Consequently, the most common network suitable for the liquefaction analysis, as confirmed by researchers [25,26,34,46,47], has been considered herein. Therefore, the development of more complex networks with more than one hidden layer [48] have been avoided.

Comparison with existing models

Evidence of liquefaction, such as sand boiling and lateral displacement, was observed after the Chi-Chi earthquake with a magnitude of 7.7 in central Taiwan, China, in 1999. The epicenter of the earthquake was 23.86° N and 120.81° E, with a focal depth of 11 km. MAA [49,50] conducted CPT shortly after the earthquake. Chu et al. [51] analyzed D_H at five sites in Wufeng and Nantou cities

Among all reported samples, 28 of them contain all defined data fields and necessary parameters for applying the three models obtained from Youd et al. [31], Javadi et al. [26], and Rezania et al. [9]. Table 11 defines these samples according to their parameter values. The F₁₅ values of the samples ranged from 13 to 48.55%.

To verify the models presented in this study, the 28 samples and estimated values of three additional models were compared with ANN1. Furthermore, 16 samples (samples 1 to 16) with F₁₅ values less than 28% were applied to compare ANN1 and ANN2 results with three additional models. The degree of accuracy of the predicted value for D_H was determined by estimating the error difference between their results and measured values at the sites. To summarize this comparison, the MAE, RMSE, and R were estimated. Tables 12 and 13 present the summary of this comparison.

(8)RSME=\sqrt{\frac{{\displaystyle \sum _N{\left(X_{\mathrm{m}}{X}_{\mathrm{P}}\right)}^2}}{N}},

(9)MAE=\frac{{\displaystyle \sum _N\left|X_{\mathrm{m}}{X}_{\mathrm{P}}\right|}}{N},

where X_m, N, and X_P represent the measured value at the sites, number of samples (28 and 16, herein), and predicted value obtained by the models, respectively.

Figures 3 and 4 illustrate the comparison between the proposed ANN models and the other three models considering the independent case histories from the Chi-Chi earthquake. Figure 3 plots the predicted value of D_H by the first model and the other three models against the 28 measured values for D_H at the sites. Figure 4 shows the D_H predicted by both the developed ANN models and the other three models against the 16 measured values for D_H, including those of F₁₅ values less than 28%.

Sensitivity analysis

Most geotechnical engineering parameters are uncertain due to measurement, estimation, and statistical (related to shortage of data) errors in laboratory or field conditions. To address these uncertainties, some probabilistic approaches have been developed and carried out recently by researchers to create a plot form for decision making and risk analysis [48,52–55]. In this study, MSC was applied to perform sensitivity analysis and demonstrate the effect of changing the mean value and coefficient of variation (COV) on D_H. A few methods have been introduced by researchers to quantify the uncertainties of the parameters and their correlations [56–58]. As mentioned earlier, MCS is a technique that relies on a large number of samplings, which is expensive and time consuming. To address this shortage, the second ANN model, which exhibits the highest accuracy and capability, was applied to 500 000 samples to achieve convergence results by MCS. The parameters were assumed to be uncertain and were considered as variables with statistical factors and distribution functions. The COV (ratio between the standard deviation (ʋ) and the mean value) and distribution function were defined according to Table 14. Lumb [59] and Tan et al. [60] proposed a normal distribution function for soil properties via laboratory tests. Furthermore, when ʋ is insignificant, the error in normal distribution function is negligible, and other variables were also supposed to have this distribution. Additionally, to construct a correlation matrix, it is necessary to define the dependency between random variables via the correlation coefficient (r). The authors determined r between CAV₅ and M_w to be 62.5% by applying standard statistical methods to the main data set. Additional variables were assumed to be independent, with a r value equal to zero.

To perform the sensitivity analysis of each parameter, the other five parameters were fixed in their mean value with their mean COV value according to Table 14, whereas the target variable changes according to its range to demonstrate its influence. Owing to its close proximity to the mean value of D_H in the main database, 1 m was considered for performing sensitivity analysis. Figure 5 illustrates the influence of parameters and their uncertainties on lateral displacements larger than 1 m.

Results and discussion

Tables 9 and 10 present the certificate of both ANN models presented in this study. All models provide R values of approximately 90% for testing, training, and validation, which indicates the high precision of the model. Twenty-eight data points from the Chi-Chi earthquake possessed the parameters necessary for applying the available models. They were selected to compare the performance of ANN1 and three other available models. It should be noted that larger R values and lower MAE and RSME values indicate higher accuracy of predicted results and therefore superior performance. Table 12 shows that, among all models, ANN1 provides a larger R value of 0.731 compared to the 0.433, -0.74, and 0.514 values presented in the models of Rezania et al. [9], Javadi et al. [27], and Youd et al. [31], respectively. Additionally, compared to others, ANN1 provides lower MAE and RSME values of 0.28 and 0.32. Rezania et al.’s model [9] exhibits better performance with MAE and RSME values of 0.49 and 0.7, respectively. Considering these results, ANN1 predicts D_H with higher accuracy.

Furthermore, data points with F₁₅ higher than 28% were eliminated from the 28 samples of the Chi-Chi earthquake. Therefore, 16 samples were kept to evaluate the performance of ANN2 presented in this study. We predicted the D_H values of both ANN models, including that of the additional available models. These values were compared with the measured values of the 16 samples to estimate the R, MAE, and RSME values for all models. This comparison is presented in Table 13. As can be observed, the R value presented in Youd et al. [31] was the largest value (0.934), which is closely followed by the second model with an R value of 0.923. Furthermore, ANN1 provides an R value of 0.846 in comparison with Rezania et al. [9] and Javadi et al. [27], who reported values of 0.233 and -0.813, respectively. Regarding MAE and RSME, ANN2 exhibited higher accuracy with values equal to 0.23 and 0.29. Rezania et al.’s model with MAE and RSME values of 0.42 and 0.57 as well as Javadi et al.’s [27] model with 1.2 and 1.3, respectively, exhibited a lower error in predicting D_H. Consequently, we show that for the entire range of the data points, ANN1 in this study provides a prediction power higher than other available models. Furthermore, for data points with F₁₅ less than the critical value of 28%, the ANN2 proposed in this study predicts D_H with a satisfactory accuracy. The ANN1 and the model presented by Rezania et al. [9] exhibited the highest performance.

As can be observed from Figs. 3 and 4, among all models, the models proposed by Youd et al. [31] and Javadi et al. [27] provide an overestimated D_H, both with and without any limitations to the parameters’ values. Furthermore, ANN1 and the model proposed by Rezania et al. [9] underestimated D_H. The second ANN model developed in this study estimated D_H with the highest accuracy.

Figure 5 illustrates the results of parametric sensitivity analysis by plotting the parameters against the probability of D_H, which is greater than 1 m. We observed that by increasing Mw in the range of the second data set from 6.5 to 7.9, the probability of D_H>1 m increases to approximately 28%. In addition, probability increases by 4.5% and 7% when the COV of Mw falls from 0.05 to 0.1 then to 0.15. The probability of D_H>1 m increases abruptly from 61% to 75% in the W ranged from 1.64 to approximately 8 then experiences a slight increase to 87% until W reaches 56.8. The increase in uncertainties does not indicate significant influence. As T₁₅ increases from 0.5 to 16.7, the probability of D_H>1 m steadily increases from 36 to 60%. Furthermore, the increase in COV influences the probability of D_H>1 m alone in T₁₅ values higher than approximately 15. This significant influence on F₁₅ is illustrated by moving from 0 to 92% in the range of F₁₅ value (zero to 28%). Furthermore, by increasing F₁₅, the changes in COV provide higher effect on D_H>1 m. The maximum increase by 5% and 3.5% can be observed, with COV rising by 0.1, from 0.1 to 0.2 then 0.3. The probability of D_H>1 m indicates a uniform decrease between 56% and 16% in the D50₁₅ ranged from 0.086 to 1.98. Additionally, in the critical range of 1.2 to 1.98, by any 0.1 rise in COV from 0.1 to 0.2 then 0.3, the parameter experiences a 2.5% increase in the probability. Furthermore, the probability of D_H>1 m rises from 13% to 79% within the considered CAV₅ range in this study. As COV increases from 0.1 to 0.2 and then 0.3 in the critical values between 8 and 14, the probability increases by approximately 2.5% every time.

Summary and conclusions

Among all liquefaction effects, the prediction of lateral displacement is the most vital issue in the hazard analysis of the liquefaction phenomenon based on it being the destructive effect. Semi-empirical and empirical models, which are the most common models in the evaluation of liquefaction, have been developed by researchers mostly on the basis of the data set collected by Youd and Bartlett [31,45]. Furthermore, the ability of machine learning and intelligent algorithms have been validated to present models for estimating D_H. Furthermore, it has been confirmed that among earthquake parameters, CAV₅ provides the highest sufficiency and efficiency for estimating the increasing pore water pressure and liquefaction in saturated sandy soil caused by the motion of earthquakes. In this study, the CAV₅ parameter was first estimated and added to the data set. Therefore, the data set became more capable in covering all aspects of the earthquake conditions, such as causative fault types, frequency of the earthquake’s motion, and effect of near-fault regions. Furthermore, by conducting statistical analysis, we can safely remove r_rup and S from the database owing to their poor statistical results. Additionally, the complex influence of FC at different values (critical value) has been demonstrated by researchers. Hence, to evaluate this influence, two data sets were created and two ANN models were developed on the base of the data sets. The first model for the entire range of parameters in the main data set, and the second model belonging to the data set with samples possessing F₁₅ values less than the critical value of 28%, were derived. To demonstrate the performance of the ANN models, some data points from near fault zones of the Chi-Chi earthquake in Wufeng and Nantu districts were selected and were not included in the data sets of the developed ANN models in this study. After that, the predicted values of two ANN models, as well as additional available three models were compared to the measured values for D_H in the sites. Although the results validate the high accuracy of the two ANN models, the second ANN model trained on a smaller set exhibits a higher capability. Hence, the results reveal that considering the critical and complex effects of FC, a positive match with other studies, as well as conducting a validation phase, enhances capability of the model. Finally, the sensitivity analysis was carried out by MCS using the second ANN model to examine the influence of the parameters and their uncertainties on the D_H. The results highlight that among all geotechnical properties, F₁₅ possesses the most significant influence on D_H. However, CAV₅ exhibits a higher effect on D_H among other earthquake parameters. Summing up the results of the sensitivity analysis, it is evident that the uncertainties of the parameters, particularly in some critical values, provide significant effects. For example, the increase in COV by 10% can cause an approximately 5% alteration in D_H. These findings suggest that the uncertainties of the parameters and the probabilistic approach should be considered instead of deterministic approaches in the hazard analysis of liquefaction.

References

[1] Bhattacharya S, Hyodo M, Goda K, Tazoh T, Taylor C A. Liquefaction of soil in the Tokyo Bay area from the 2011 Tohoku (Japan) earthquake. Soil Dynamics and Earthquake Engineering, 2011, 31(11): 1618–1628

[2] Zhang Y, Dong S, Hou C, Guo C,Yao X, Li B, Du J, Zhang J.Geohazards induced by the Lushan Ms7.0 Earthquake in Sichuan Province, Southwest China: Typical examples, types and distributional characteristics. Acta Geologica Sinica (English Edition), 2013, 87(3): 646–657

[3] Franke K W. Development of a Performance-Based Model for Prediction of Lateral Spreading Displacements. Seattle: University of Washingtone, 2005

[4] Hamada M, Towhata I, Yasuda S, Isoyama R. Study on permanent ground displacement induced by seismic liquefaction. Computers and Geotechnics, 1987, 4(4): 197–220

[5] Shamoto Y, Zhang J M, Tokimatsu K. Methods for evaluating residual post-liquefaction ground settlement and horizontal displacement. Soils and Foundations, 1998, 38(Special): 69–83

[6] Kanıbir A, Ulusay R, Aydan Ö. Assessment of liquefaction and lateral spreading on the shore of Lake Sapanca during the Kocaeli (Turkey) earthquake. Engineering Geology, 2006, 83(4): 307–331

[7] Franke K W, Kramer S L. Procedure for the empirical evaluation of lateral spread displacement hazard curves. Journal of Geotechnical and Geoenvironmental Engineering, 2014, 140(1): 110–120

[8] Gillins D, Bartlett S. Multilinear regression equations for predicting lateral spread displacement from soil type and cone penetration test data. Journal of Geotechnical and Geoenvironmental Engineering, 2014, 140(4): 04013047

[9] Rezania M, Faramarzi A, Javadi A A. An evolutionary based approach for assessment of earthquake-induced soil liquefaction and lateral displacement. Engineering Applications of Artificial Intelligence, 2011, 24(1): 142–153

[10] Idriss I M, Boulange R W. Soil Liquefaction during Earthquakes. Monograph MNO-12. Oakland, CA: Earthquake Engineering Research Institute, 2008

[11] Zhang G, Robertson P K, Brachman R W I. Estimating liquefaction-induced lateral displacements using the standard penetration test or cone penetration test. Journal of Geotechnical and Geoenvironmental Engineering, 2004, 130(8): 861–871

[12] Faris A T, Seed R B, Kayen R E, Wu J. A semi-empirical model for the estimation of maximum horizontal displacement due to liquefaction-induced lateral spreading. In: The 8th U.S. National Conference of Earthquake Engineering. San Francisco, CA: Earthquake Engineering Research Institute, 2006

[13] Bray D J, Asce F, Travasarou T. Simplified procedure for estimating earthquake-induced deviatoric slope displacements. Journal of Geotechnical and Geoenvironmental Engineering, 2007, 133(4): 381–392

[14] Byrne P M, Seid-Karbasi M. Seismic liquefaction, lateral spreading, and flow slides: A numerical investigation into void redistribution.Canadian Geotechnical Journal, 2007, 44: 873–890

[15] Olson S M, Johnson C I. Analyzing liquefaction-induced lateral spreads using strength ratios. Journal of Geotechnical and Geoenvironmental Engineering, 2008, 134(8): 1035–1049

[16] Saygili G, Rathje E. Empirical predictive models for earthquake-induced sliding displacements of slopes. Journal of Geotechnical and Geoenvironmental Engineering, 2008, 134: 790–803

[17] Lam I, Arduino P, Mackenzie-Helnwein P. OPENSEES Soil-Pile Interaction Study under Lateral Spread Loading. Orlando, FL, 2009

[18] Arulanandan K, Li X S, Sivathasan K S. Numerical simulation of liquefaction-induced deformations. Journal of Geotechnical and Geoenvironmental Engineering, 2000, 126(7): 657–666

[19] Lopez-Caballero F, Modaressi Farahmand-Razavi A. Numerical simulation of liquefaction effects on seismic SSI. Soil Dynamics and Earthquake Engineering, 2008, 28(2): 85–98

[20] Tao M. Case History Verification of the Energy Method to Determine the Liquefaction Potential of Soil Deposits. Cleveland, OH: Case Western Reserve University, 2003, 173

[21] Derakhshandi M, Rathje E M, Hazirbaba K, Mirhosseini S M. The effect of plastic fines on the pore pressure generation characteristics of saturated sands. Soil Dynamics and Earthquake Engineering, 2008, 28(5): 376–386

[22] Phan V T A, Hsiao D H, Nguyen P T L. Effects of fines contents on engineering properties of sand-fines mixtures. Procedia Engineering, 2016, 142: 213–220

[23] Maurer B W, Green R A, Cubrinovski M, Bradley B A. Fines-content effects on liquefaction hazard evaluation for infrastructure in Christchurch, New Zealand. Soil Dynamics and Earthquake Engineering, 2015, 76: 58–68

[24] Pirhadi N, Tang X, Yang Q, Kang F. A new equation to evaluate liquefaction triggering using the response surface method and parametric sensitivity analysis. Sustainability, 2018, 11(1): 112–136

[25] Wang J, Rahman M S. A neural network model for liquefaction-induced horizontal ground displacement. Soil Dynamics and Earthquake Engineering, 1999, 18(8): 555–568

[26] Baziar M H, Ghorbani A. Evaluation of lateral spreading using artificial neural networks. Soil Dynamics and Earthquake Engineering, 2005, 25(1): 1–9

[27] Javadi A A, Rezania M, Nezhad M M. Evaluation of liquefaction induced lateral displacements using genetic programming. Computers and Geotechnics, 2006, 33(4–5): 222–233

[28] García S R, Romo M P, Botero E. A neurofuzzy system to analyze liquefaction-induced lateral spread. Soil Dynamics and Earthquake Engineering, 2008, 28(3): 169–180

[29] Hassan B M, Alireza S A. Evaluation of lateral spreading utilizing artificial neural network and genetic programming. International Journal of Civil Engineering, 2013, 11: 100–111

[30] Bartlett S F, Youd T L. Empirical Analysis of Horizontal Ground Displacement Generated by Liquefaction-Induced Lateral Spread. Tech. Rep. No. NCEER-92–0021. Buffalo, NY: National Center for Earthquake Engineering Research, 1992

[31] Youd T L, Hansen C M, Bartlett S F. Revised multilinear regression equations for prediction of lateral spread displacement. Journal of Geotechnical and Geoenvironmental Engineering, 2002, 128(12): 1007–1017

[32] Kramer S L, Mitchell R A. Ground motion intensity measures for liquefaction hazard evaluation. Earthquake Spectra, 2006, 22(2): 413–438

[33] Ganji H T, Alembagheri M, Khaneghahi M H. Evaluation of seismic reliability of gravity dam-reservoir in homogeneous foundation coupled system. Frontiers of Structural and Civil Engineering, 2019, 13(3): 701–715

[34] Haykin S. Neural Networks: A Comprehensive Foundation. 2nd ed.Prentice Hall, 1998

[35] Hornik K, Stinchcombe M, White H. Multilayer feed forward networks are universal approximators. Neural Networks, 1989, 2(5): 359–366

[36] Levenberg K. A method for the solution of certain non-linear problems in least squares. Quarterly of Applied Mathematics, 1944, 2(2): 164–168

[37] Rezaei S, Choobbasti A J. Liquefaction assessment using microtremor measurement, conventional method and artificial neural network (Case study: Babol, Iran). Frontiers of Structural and Civil Engineering, 2014, 8(3): 292–307

[38] Singh T, Pal M, Arora V K. Modeling oblique load carrying capacity of batter pile groups using neural network, random forest regression and M5 model tree. Frontiers of Structural and Civil Engineering, 2019, 13(3): 674–685

[39] Metropolis N, Ulam S. The Monte Carlo Method. Journal of the American Statistical Association, 1949, 44(247): 335–341

[40] Harr M E. Reliability-Based Design in Civil Engineering. New York: McGraw-Hill, 1987

[41] Levy S, Steinberg D M. Computer experiments: A review. Advances in Statistical Analysis, 2010, 94(4): 311–324

[42] EPRI. A Criterion for Determining Exceedance of the Operating Basis Earthquake. Report No. EPRI NP-5930. Palo Alto, CA, 1988

[43] EPRI. Standardization of the Cumulative Absolute Velocity. EPRI TR-100082–T2. Palo Alto, CA, 1991

[44] Luco N, Cornell C A. Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions. Earthquake Spectra, 2007, 23(2): 357–392

[45] Bartlett S F, Youd T L. Empirical prediction of liquefaction-induced lateral spread. Journal of Geotechnical Engineering, 1995, 121(4): 316–329

[46] Baziar M H, Nilipour N. Evaluation of liquefaction potential using neural-networks and CPT results. Soil Dynamics and Earthquake Engineering, 2003, 23(7): 631–636

[47] Hanna A M, Ural D, Saygili G. Neural network model for liquefaction potential in soil deposits using Turkey and Taiwan, China earthquake data. Soil Dynamics and Earthquake Engineering, 2007, 27(6): 521–540

[48] Hamdia K, Ghasemi H, Bazi Y, AlHichri H, Alajlan N, Rabczuk T. A novel deep learning based method for the computational material design of flexoelectric nanostructures with topology optimization. Finite Elements in Analysis and Design, 2019, 165: 21–30

[49] MAA. Soil Liquefaction Assessment and Remediation Study, Phase I (Yuanlin, Dachun, and Shetou), Summary Report and Appendixes. Taipei, Taiwan, China: Moh and Associates (MAA), Inc., 2000 (in Chinese)

[50] MAA. Soil Liquefaction Investigation in Nantou and Wufeng Areas. Taipei, Taiwan, China: Moh and Associates (MAA), Inc., 2000

[51] Chu D B, Stewart J P, Youd T L, Chu B L. Liquefaction-induced lateral spreading in near-fault regions during the 1999 Chi-Chi, Taiwan, China Earthquake. Journal of Geotechnical and Geoenvironmental Engineering, 2006, 132(12): 1549–1565

[52] Guo H, Zhuang X, Rabczuk T. A Deep collocation method for the bending analysis of Kirchhoff plate. Computers, Materials & Continua, 2019, 59(2): 433–456

[53] Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials & Continua, 2019, 59(1): 345–359

[54] Hamdia K M, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

[55] Hamdia K M, Ghasemi H, Zhuang X, Alajlan N, Rabczuk T. Sensitivity and uncertainty analysis for flexoelectric nanostructures. Computer Methods in Applied Mechanics and Engineering, 2018, 337: 95–109

[56] Vu-Bac N, Rafiee R, Zhuang X, Lahmer T, Rabczuk T. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464

[57] Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535

[58] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100(C): 19–31

[59] Lumb P. The variability of natural soils. Canadian Geotechnical Journal, 1966, 3(2): 74–97

[60] Tan C P, Donald I B, Melchers R E. Probabilistic Slope Stability Analysis-State of Play. In: Proceedings of the Conference on Probabilistic Methods in Geotechnical Engineering. Canberra: CRC Press 1993

[61] Juang C H, Rosowsky D V, Tang W H. Reliability-Based Method for Assessing Liquefaction Potential of Soils. Journal of Geotechnical and Geoenvironmental Engineering, 1999, 125(8): 684–689

RIGHTS & PERMISSIONS

Higher Education Press