# New Equations to Evaluate Lateral Displacement Caused by Liquefaction Using the Response Surface Method

^{*}

## Abstract

**:**

_{H}). This paper aims to address this gap in the literature by adding an earthquake parameter of the standardized cumulative absolute velocity (CAV

_{5}) to the original dataset for analyzing. Furthermore, the complex influence of fine content in the liquefiable layer (F

_{15}) is analyzed by deriving two different equations: the first one is for the whole range of parameters, and the second one is for a limited range of F

_{15}values under 28% in order to the F

_{15}’s critical value presented in literature. The new response surface method (RSM) approach is applied on the basis of the artificial neural network (ANN) model to develop two new equations. Moreover, to illustrate the capability and efficiency of the developed models, the results of the RSM models are examined by comparing them with an additional three available models using data from the Chi-Chi earthquake sites that were not used for developing the models in this study. In conclusion, the RSM provides a capable tool to evaluate the liquefaction phenomenon, and the results fully justify the complex effect of different values of F

_{15}.

## 1. Introduction

_{H}).

_{H}caused by liquefaction for some decades. Some of them have proposed numerical approaches [2,3,4,5,6] such as the finite element method (FEM) and the finite difference method (FDM). Next to that, analytical approaches have been developed, for example, minimum potential energy [7] and the sliding block model [8,9,10,11,12].

^{2}and is defined as CAV

_{5}. Sufficiency defines which parameter is independent to estimate the target (increasing pore water pressure herein), and efficiency expresses which parameter is able to predict the target with lower uncertainties [22]. This parameter quantifies aspects of applied frequency load, which can be affected by the near-fault region aspect and causative fault type of earthquakes. Hui et al. [23] proposed an index of PGV to peak ground acceleration (PGA) to characterize the effect of liquefaction on the piles in near-fault zones. Further, Kwang et al. [24], through performing some uniform cyclic simple shear laboratory tests, demonstrated that CAV

_{5}provides the highest correlation with D

_{H}among ground motion parameters. While the significant correlation between CAV

_{5}and the evaluation of liquefaction have been characterized, no attempt has been made to take it into the account when developing empirical and semi-empirical models.

_{H}using databases that were collected from sites [25,26,27,28]. Training is organized to minimize the mean square error (MSE) function. Wang et al. [25] used a back-propagation neural network to develop a model for the prediction of lateral ground displacements caused by liquefaction. They applied the same records used by Bartlet et al. [1], along with 19 datasets of Ambraseys et al. [29]. Among all datasets, 367 data points were used for the training phase, and the extra 99 datasets were used for the testing phase, while no validating phase was conducted. The model was developed using the same parameters suggested by Youd et al. [14].

_{H}, and a validating phase, to prevent overtraining of the artificial neural network (ANN) model. Then, they presented an ANN model using STATISTICA software (version of Statistica 5.1, Dell Software, Round Rock, TX, USA) to estimate D

_{H}. They inspected the performance of their model using validating subset data without considering extra available models. Furthermore, a new model was presented by Javadi et al. on the basis of genetic programming (GP) [27]. They divided the dataset randomly, without paying attention to the statistical properties of the input parameters, into two subsets for the validating and training phase. Garcia et al. established a neuro-fuzzy model to use the advantages of both systems. They randomly separated their dataset into two subsets for training and testing; however, they did not take statistical aspects into account. They also compared the value predicted by their model with extra models to evaluate its performance. Baziar et al. [28] then applied ANN and GP to propose a new model. They divided their dataset randomly into two subsets for the testing and training phase; a validating process was not performed, and the statistical factors of the parameters were not considered.

_{H}. Most of the studies reveal a range of 20% to 30% for the transition of behavior of the response of sand to earthquake and liquefaction occurrence. Maurer et al. [35] investigated the Canterbury earthquakes in 2010 and 2011 through 7000 case history datasets and illustrated that a high value of Fc caused more inaccuracy in liquefaction assessments. Tao performed some laboratory tests and demonstrated that the potential of liquefaction has a significant dependency on initial relative density (D

_{r}) when the Fc value is larger than 28% [32].

_{5}, which is CAV with a 5-cm/sec

^{2}threshold acceleration, through the attenuation equation presented by Kramer et al. [21]. By adding CAV

_{5}, the dataset was expanded and became more capable of and efficient in considering aspects of earthquakes and geology site situations, such as earthquake motion frequency, near-fault effects and the causative fault type of an earthquake. The second dataset was created by eliminating samples with an average Fc in a liquefiable soil layer (F

_{15}) less than 28%. The response surface method (RSM) is used for the first time as a novel method to develop two equations to predict lateral displacement due to liquefaction (D

_{H}) in order to two created datasets herein. Furthermore, the meaningful and effective terms of the equations are discovered through hypothesis testing of the p-value. In this study, two ANNs with back-propagation analysis were developed to measure the coding input data of the RSM. To develop each ANN model, the main dataset is first divided into three subsets for the training, testing, and validating stages, considering statistical properties—instead of random division—to increase the capability and accuracy of the model. To achieve this goal, an attempt is made to create all three subsets with close statistical factors. Finally, the results are compared with data measured from the Chi-Chi earthquake’s near fault zone of Wufeng district (Figure 1) and Nantou district (Figure 2), as well as with the predicted D

_{H}through three extra models [20,27,36] to demonstrate the accuracy and capability of the RSM models.

## 2. Review of Empirical and Semi-Empirical Models

_{H}due to liquefaction; they supposed that earthquake, topographical, geological, and soil factors are the most influential parameters on D

_{H}. They studied 467 displacement vectors from the case history database. Among those vectors, 337 were from the 1964 Niigata and 1983 Nihonkai-Chubu, Japan, earthquakes; 111 were from earthquakes in the United States; and the other 19 cases were selected from Ambraseys’ [29] database. In the end, they developed a new model by using multiple linear regression (MLR) for free-face and ground slope conditions [37], but they did not separate earthquakes according to their region because of a database shortage. Youd et al. revised their MLR by adding case history data from three earthquakes (1983 Borah Peak, Idaho; 1989 Loma Prieta; and 1995 Hyogoken-Nanbu (Kobe)), and they considered coarser-grained materials. They removed eight displacement sites with prevented free lateral movement and developed two equations with more accuracy given as follows:

_{H}is the predicted lateral ground displacement (m), M

_{w}is the moment magnitude of the earthquake, and T

_{15}is the cumulative thickness of saturated granular layers (m) with corrected blow counts ((N

_{1})

_{60}) less than 15. Moreover, F

_{15}is the average fines content of sediment within T

_{15}(%); D50

_{15}is the average mean grain size for granular materials within T

_{15}(mm); S is the ground slope (%); and W is the free-face ratio (H/L), where H is the height of the free face and L is the distance from the base of the free face to the liquefied point. Finally, r is the nearest horizontal or map distance from the site to the seismic energy source (Km).

## 3. Artificial Neural Network

^{2}}

## 4. Response Surface Method

- To present an approximate relationship between input variables and an output variable or response to be able to predict the response variable.
- To discover significant factors or terms of the presented equation using RSM through hypothesis testing such as the p-value.
- To assess the optimization model to obtain a response as a maximum or minimum over a certain range of interest.

#### 4.1. Design of Experiments

#### 4.2. Hypothesis Test

- Defining an initial assumption (null hypothesis).
- Analyzing and assessing sample data by following a formal process.
- Based on the second step, accepting or rejecting the initial assumption in the first step.

## 5. Model Proposed

_{H}in this study. Two models are presented: first one considered the whole range of the parameters and the second one was on the basis of the F

_{15}value being less than 18%.

#### 5.1. Dataset

_{H}from the following eight earthquakes in the United States and Japan: San Francisco 1906, Alaska 1964, Niigata 1964, San Fernando 1971, Imperial Valley 1979, Borah Peak 1983, Nihonkai-Chubu 1983, and Superstition Hills 1987. The parameters of the case histories that they collected to analyze were divided into three groups:

- Seismic parameters—moment magnitude (M
_{W}) and horizontal distance from site to seismic energy source (r) in km. - Topographic parameters—free-face ratio (W) and ground slope (S), both in percent.
- Geotechnical parameters—thickness of layer with corrected blow counts (N1)
_{60}< 15 (T_{15}) in meters, average fines content in the T_{15}layer (F_{15}) in percent, and average mean grain size in the T_{15}layer (D50_{15}) in mm.

_{5}, which is defined in Section 5.1.1. Kramer et al. [21] stated that CAV

_{5}is the most efficient and sufficient earthquake intensity to evaluate liquefaction in sandy soil. Therefore, CAV

_{5}was estimated using the attenuation equation presented by them.. In this way, the causative earthquake fault types of all earthquakes in the dataset were discovered. Then, the sites with a moment magnitude range from 6.4 to 7.9 were selected in order to find the applicable magnitude range for Equation (11); the Alaska 1964 site, with a magnitude of 9.2, was thus deleted from the dataset.

_{H}) one by one. The estimated values of R for r-D

_{H}and S-D

_{H}were a positive value of 0.104 and a negative value of −0.98, respectively, contrary to the supposition for them. This was possibly due to the scarcity of sites that were explored, and consequently, the shortage measured values for r and S in the main dataset. Therefore, in this study, after eliminating r and S from dataset, only the free-face condition was considered, and samples of the ground slope condition were deleted from the main dataset. Furthermore, CAV

_{5}was added to the dataset instead of r. Figure 4 and Figure 5 plot r versus CAV

_{5}for range of M

_{w}in the main dataset from 6.4 to 7 and 7 to 7.9, respectively. It should be mentioned that the bold points show more than one point coincided together.

_{15}value larger than 28%. Therefore, the second dataset included 182 samples.

#### 5.1.1. Cumulative Absolute Velocity

_{max}is the duration of the earthquake.

_{5}, as can be estimated through Equation (11), has better efficiency and sufficiency than other earthquake intensity parameters for liquefaction evaluation. They also utilized the strong Pacific Earthquake Engineering Research (PEER) database, consisting of 282 ground motions from 40 earthquakes to present an equation to calculate CAV

_{5}for shallow crustal events [21].

_{5}is a form of CAV based on Equation (10) (m/sec), M is the moment magnitude, and r is the closest distance to the rupture (km). 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.

_{5}from the dataset by considering the causative fault type.

#### 5.2. Artificial Neural Network Models

_{H}. There are six inputs, including six parameters, which were considered by Youd et al. [20] and Bartlett et al. [37]. In addition, there is the new measured parameter of CAV

_{5}as well as one output as a D

_{H}.

_{i}and d

_{i}are the individual sample points indexed with i, and $\overline{x}$ and $\overline{d}$ are the mean sample sizes.

_{15}, the new dataset was constructed by selecting data samples with an F

_{15}less than critical value of 28%, which was demonstrated by Tao [32], and its characteristics are listed in Table 3. The new dataset was divided again into three groups with the same portion of each site and with similar statistical factors. Therefore, data from each earthquake contributed to the training, testing, and validating phase. In this step, around 15%, 15%, and 70% of the dataset equated to 27, 27, and 129 samples used for the testing, validating, and training processes, respectively. The characteristics of this new ANN model are presented in Table 3. Also, Table 4 presents the certificate of the second model. It can be seen in Table 4 that the R values for all groups of datasets were around 90%.

#### 5.3. The RSM Equations for Predicting D_{H}

_{H}) in coded form was calculated. Thereafter, the RSM equation with 28 terms according to the second-degree polynomial with cross terms was derived; however, through hypothesis testing considering the p-value, some terms were eliminated. Then, the RSM was applied repeatedly to achieve the final equation. The following equation was consequently developed with 22 terms to correlate the D

_{H}caused by liquefaction to the six input parameters for the whole range of parameters in this study, without any limitations on the range of the F

_{15}value:

_{H}= a

_{0}+ a

_{1}M

_{w}+ a

_{2}W + a

_{3}T

_{15}+ a

_{4}F

_{15}+ a

_{5}(D50

_{15}) + a

_{6}(CAV

_{5}) + a

_{7}M

_{w}

^{2}+ a

_{8}W

^{2}+ a

_{9}T

_{15}

^{2}+ a

_{10}(F

_{15})

^{2}+ a

_{11}(D50

_{15})

^{2}+ a

_{12}(CAV

_{5})

^{2}+ a

_{13}M

_{w}T

_{15}+ a

_{14}M

_{w}F

_{15}+ a

_{15}M

_{w}(D50

_{15}) + a

_{16}WT

_{15}+ a

_{17}WF

_{15}+ a

_{18}W(D50

_{15}) + a

_{19}T

_{15}F

_{15}+ a

_{20}T

_{15}D50

_{15}+ a

_{21}F

_{15}CAV

_{5}

^{2}= 87.22%, R

^{2}(predicted) = 78.73%, and R

^{2}(adjust) = 83.99%.

^{2}) illustrates how well the curves fit on the data points. In addition, the R

^{2}(adjust) demonstrates the percentage of variation defined by the independent variables that affect the dependent variable, herein referred to as D

_{H}. Also, the R

^{2}(predicted) defines how well a correlation is able to predict the target for a new observation. The values of coefficients a

_{0}to a

_{21}are listed in Table 5.

_{15}less than 28% (Equation (14)). Table 6 presents the coefficients of the second RSM equation.:

_{H}= a

_{0}+ a

_{1}M

_{w}+ a

_{2}W + a

_{3}T

_{15}+ a

_{4}F

_{15}+ a

_{5}(D50

_{15}) + a

_{6}(CAV

_{5}) + a

_{7}M

_{w}

^{2}+ a

_{8}W

^{2}+ a

_{9}T

_{15}

^{2}+ a

_{10}(F

_{15})

^{2}+ a

_{11}(D50

_{15})

^{2}+ a

_{12}(CAV

_{5})

^{2}+ a

_{13}M

_{w}W + a

_{14}M

_{w}F

_{15}+ a

_{15}M

_{w}(CAV

_{5}) + a

_{16}W(D50

_{15}) + a

_{17}W(CAV

_{5}) + a

_{18}T

_{15}F

_{15}+ a

_{19}T

_{15}(D50

_{15}) + a

_{20}F

_{15}(D50

_{15})

^{2}= 88.51%, R

^{2}(predicted) = 50.95%, and R

^{2}(adjust) = 78.09%.

## 6. Comparison of RSM Equations with Extra Models

_{15}from 13% to 48.5%.

_{m}is the measured value, and X

_{P}is the predicted value.

_{15}greater than 28% were eliminated from the Chi-Chi earthquake cases, and 16 samples consequently remained (samples number 1 to 14 as well as numbers 27 and 28, as can be seen in Table 7). Then, the second RSM equation was validated by applying it to these samples in comparison with the extra three models. Table 9 summarizes the results of all models.

_{15}values of less than 28% at 16 data points.

## 7. Results and Discussion

_{15}values were less than 28%, with three extra well-known models. The models were examined using new data from the Chi-Chi earthquake, which were not included in the two datasets to establish the two RSM models. As can be seen in Table 8, the RSM equation of the whole range of parameters indicated a higher R value of 0.683, in comparison with the extra models whose values were 0.433, −0.74, and 0.514. Furthermore, the RSM model comprises lower MAE and RSME values of 0.3 and 0.37, respectively, compared to 0.49 and 0.7 for Rezania et al., 1.04 and 1.19 for Javadi et al., and 3.77 and 4.37 for Youd et al. Therefore, among all of them, the RSM model provided prediction with higher accuracy.

_{15}less than 28%, the model of Youd et al. provided the highest R value of 0.934, closely followed by the second and the first RSM models with R values equal to 0.891 and 0.846 respectively. Table 9 also illustrates the MAE and RSME criteria values for all models for samples with a limited value for F

_{15}less than 28%. The values of the MAE and RSME in the second RSM model—0.29 and 0.39, respectively—indicates the highest accuracy and performance in comparison with the others. In addition, the model of Rezania et al., with 0.42 and 0.57, illustrated lower values for the MAE and RSME, respectively. Further, Javadi et al. with 1.2 and 1.3, and Youd et al. with 4.84 and 5.34 provide less accuracy for predicting D

_{H}.

_{15}larger than 28% and even by decreasing the number of samples in the dataset.

_{5}parameter to the dataset. As can be seen from Figure 6 and Figure 7, among all models that were considered in the present study to calculate D

_{H}without any limitation on the parameters’ value, the model of Youd et al. was overpredicted. Meanwhile, Youd et al.’s model provided poor and overpredicted results for samples with a limited value of F

_{15}less than 28%. Additionally, considering samples with a limited F

_{15}value shows the first RSM and the model of Javadi et al. present an overpredicted value for D

_{H}. Furthermore, second RSM equation and model of Rezania et al. underpredicted D

_{H}in their predictions.

- (1)
- Both RSM models require standard penetration test SPT and laboratory tests to determine geotechnical properties parameters of T
_{15}, F_{15}, and D50_{15}_{.} - (2)
- Both of the RSM models are valid for free-face conditions but not ground-slope conditions.
- (3)
- Second RSM model is valid only for F
_{15}< 28%. - (4)
- Models are only valid for earthquakes with M
_{w}between 6.4 and 8.0. - (5)
- Specify accuracy limits for each model.
- (6)
- It is necessary to transfer all six input models’ parameters measured value to a coded value using Equation (15) and then put the coded value in the RSM equations to predict D
_{H}.

## 8. Summary and Conclusions

_{H}) is the most important aspect of liquefaction hazard analysis. There are two main types of conditions according to the topography of the sites: free-face and sloping ground conditions. First, the parameter of corrected absolute velocity (CAV

_{5}) of sites was calculated due to it being the most efficient and sufficient parameter for the assessment of liquefaction caused by earthquakes [21], and it was added to develop the dataset to cover all aspects of earthquakes, including the frequency content of earthquake motions and the causative fault type of earthquakes. Then, a statistical parametric analysis was performed by estimating the correlation coefficient (R) between all input parameters and output as D

_{H}. To achieve a more capable and accurate model, based on the estimated values for R, the horizontal distance from a site to the seismic energy source (r) and ground slope (S) was eliminated from the original dataset due to poor correlations to the target. Therefore, the final dataset was created for free-face condition sites.

_{5}into account. To investigate the complex effect of fine content, the main dataset was divided into two subsets. The first dataset included the whole range of parameters, and in the second one, all samples with average fine content in the liquefiable layer (F

_{15}) larger than 28% were removed from the dataset, in line with Tao [32]. Furthermore, the RSM was applied to develop two equations in order to the first and the second dataset to examine its performance to assess liquefaction. In the end, the two presented models in this study were compared to three available models to demonstrate their capability and accuracy with regard to predicting D

_{H}in free-face conditions in a near fault zone case history of the Chi-Chi earthquake.

_{5}as the most sufficient and efficient intensity to liquefaction hazard assessments. In addition, the RSM is a strong tool for the evaluation of complex non-linear phenomena such as liquefaction.

_{15}on the whole range, and they provide significant enhancements to the performance of the model by considering samples with an F

_{15}less than 28% as a critical value defined by Tao [32]. One of the most remarkable results, which shows the complex influence of fine content on evaluation of D

_{H}, is that the second model demonstrated higher accuracy and capability, even though it was developed using a database with fewer samples than the first model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of lateral displacement and ground failure due to liquefaction during the Chi-Chi earthquake in the Wufeng district.

**Figure 2.**Location of lateral displacement and ground failure due to liquefaction during the Chi-Chi earthquake in the Nantou district.

**Figure 3.**Flowchart of the approach applied in this study to present an equation to predict lateral displacement caused by liquefaction.

**Figure 6.**Comparison between the first RSM equation and three extra models with 28 data points measured from sites of Chi-Chi earthquake.

**Figure 7.**Comparison between both RSM equations and three extra models with 16 data points including F

_{15}of less than 28% measured from sites of the Chi-Chi earthquake.

**Table 1.**Characteristics of whole case histories’ input parameters that were used to develop the ANN model and RSM equation.

Parameter | Min Value | Mean Value | Max Value |
---|---|---|---|

M_{w} | 6.4 | 7.18 | 7.9 |

W (%) | 1.64 | 10.25 | 56.8 |

T_{15} (m) | 0.2 | 8.78 | 16.7 |

F_{15} (%) | 1 | 16.57 | 70 |

D50_{15} (mm) | 0.036 | 0.35 | 1.98 |

CAV_{5} (m/sec) | 3.7 | 14.58 | 27.85 |

Data | Training | Testing | Validating | All |
---|---|---|---|---|

R | 0.89 | 0.92 | 0.90 | 0.90 |

**Table 3.**Characteristics of database with F

_{15}≤ 28% used for the second developed ANN model and RSM equation.

Parameter | Min Value | Mean Value | Max Value |
---|---|---|---|

M_{w} | 6.5 | 7.27 | 7.9 |

W (%) | 1.64 | 9.84 | 56.8 |

T_{15} (m) | 0.5 | 8.78 | 16.7 |

F_{15} (%) | 1 | 11.83 | 27 |

D50_{15} (mm) | 0.086 | 0.4 | 1.98 |

CAV_{5} (m/sec) | 3.7 | 15.02 | 16.28 |

Data | Training | Testing | Validating | All |
---|---|---|---|---|

R | 0.92 | 0.95 | 0.89 | 0.91 |

Coefficient | a_{0} | a_{1} | a_{2} | a_{3} | a_{4} | a_{5} | a_{6} | a_{7} |

Value | 0.9174 | −1.6737 | 2.6172 | 0.7685 | −1.0865 | −1.8952 | 1.3425 | −0.36369 |

Coefficient | a_{8} | a_{9} | a_{10} | a_{11} | a_{12} | a_{13} | a_{14} | a_{15} |

Value | −0.3733 | −0.0678 | −0.7474 | −0.4060 | 0.0258 | −0.3766 | 0.2579 | −0.59428 |

Coefficient | a_{16} | a_{17} | a_{18} | a_{19} | a_{20} | a_{21} | ||

Value | 0.3566 | −0.4549 | 0.4603 | −0.6531 | 0.6011 | −0.5063 |

Coefficient | a_{0} | a_{1} | a_{2} | a_{3} | a_{4} | a_{5} | a_{6} | a_{7} |

Value | 3.1271 | 1.1700 | 0.4711 | −0.02313 | −0.6786 | 0.7715 | −0.0208 | 0.5489 |

Coefficient | a_{8} | a_{9} | a_{10} | a_{11} | a_{12} | a_{13} | a_{14} | a_{15} |

Value | −0.0871 | −0.6520 | 0.3773 | 0.3225 | −0.4646 | 0.6225 | −0.7350 | −0.7364 |

Coefficient | a_{16} | a_{17} | a_{18} | a_{19} | a_{20} | |||

Value | −0.7855 | 0.9542 | 0.9622 | −0.8165 | −1.2668 |

Sample No | M_{w} | r | W | S | T_{15} | F_{15} | D50_{15} | PGA | CAV_{5} | D_{H} (m) |
---|---|---|---|---|---|---|---|---|---|---|

1 | 7.6 | 5 | 7.4 | 0 | 0.5 | 20.8 | 0.11 | 0.67 | 45.226 | 0 |

2 | 7.6 | 5 | 13.7 | 0 | 0.8 | 20.8 | 0.11 | 0.67 | 45.226 | 0.45 |

3 | 7.6 | 5 | 18.4 | 0 | 0.8 | 20.8 | 0.11 | 0.67 | 45.226 | 0.55 |

4 | 7.6 | 5 | 25.2 | 0 | 0.8 | 20.8 | 0.11 | 0.67 | 45.226 | 0.8 |

5 | 7.6 | 5 | 37.3 | 0 | 0.8 | 20.8 | 0.11 | 0.67 | 45.226 | 1.05 |

6 | 7.6 | 5 | 49.9 | 0 | 0.8 | 20.8 | 0.11 | 0.67 | 45.226 | 2.05 |

7 | 7.6 | 5 | 5.7 | 0 | 0.5 | 13 | 0.18 | 0.67 | 45.226 | 0 |

8 | 7.6 | 5 | 6.6 | 0 | 0.75 | 13 | 0.18 | 0.67 | 45.226 | 0.1 |

9 | 7.6 | 5 | 7.9 | 0 | 0.75 | 13 | 0.18 | 0.67 | 45.226 | 0.17 |

10 | 7.6 | 5 | 9 | 0 | 0.75 | 13 | 0.18 | 0.67 | 45.226 | 0.23 |

11 | 7.6 | 5 | 15 | 0 | 0.75 | 13 | 0.18 | 0.67 | 45.226 | 0.29 |

12 | 7.6 | 5 | 21.2 | 0 | 0.75 | 13 | 0.18 | 0.67 | 45.226 | 0.49 |

13 | 7.6 | 5 | 11.9 | 0 | 1.1 | 20.8 | 0.11 | 0.67 | 45.226 | 0 |

14 | 7.6 | 5 | 26.3 | 0 | 1.1 | 20.8 | 0.11 | 0.67 | 45.226 | 0 |

15 | 7.6 | 5 | 12.2 | 0 | 0.45 | 30 | 0.13 | 0.67 | 45.226 | 0.4 |

16 | 7.6 | 5 | 14.3 | 0 | 0.45 | 30 | 0.13 | 0.67 | 45.226 | 0.65 |

17 | 7.6 | 5 | 24.6 | 0 | 0.45 | 30 | 0.13 | 0.67 | 45.226 | 1 |

18 | 7.6 | 5 | 57.7 | 0 | 0.45 | 30 | 0.13 | 0.67 | 45.226 | 1.24 |

19 | 7.6 | 5 | 8 | 0 | 1 | 31.4 | 0.1 | 0.67 | 45.226 | 0.35 |

20 | 7.6 | 5 | 10.5 | 0 | 1 | 31.4 | 0.1 | 0.67 | 45.226 | 0.61 |

21 | 7.6 | 5 | 19 | 0 | 1 | 31.4 | 0.1 | 0.67 | 45.226 | 0.96 |

22 | 7.6 | 5 | 31.3 | 0 | 1 | 31.4 | 0.1 | 0.67 | 45.226 | 2.96 |

23 | 7.6 | 5 | 9.6 | 0 | 1.8 | 48.5 | 0.1 | 0.67 | 45.226 | 0.35 |

24 | 7.6 | 5 | 11.7 | 0 | 1.8 | 48.5 | 0.1 | 0.67 | 45.226 | 0.52 |

25 | 7.6 | 5 | 13.3 | 0 | 1.8 | 48.5 | 0.1 | 0.67 | 45.226 | 0.62 |

26 | 7.6 | 5 | 23.7 | 0 | 1.8 | 48.5 | 0.1 | 0.67 | 45.226 | 1.62 |

27 | 7.6 | 13 | 5.9 | 3.8 | 1.7 | 22.3 | 0.12 | 0.39 | 24.816 | 0.05 |

28 | 7.6 | 13 | 16.2 | 3.8 | 1.7 | 22.3 | 0.12 | 0.39 | 24.816 | 0.25 |

Performance Criteria | Models Used to Predict D_{H} | |||
---|---|---|---|---|

Youd et al. [20] | Javadi et al. [26] | Rezania et al. [36] | First RSM | |

R | 0.514 | −0.74 | 0.433 | 0.683 |

MAE | 3.77 | 1.04 | 0.49 | 0.3 |

RSME | 4.37 | 1.19 | 0.7 | 0.37 |

**Table 9.**Performance certificate of second RSM equation, on the basis of samples with F

_{15}≤ 28% in comparison with extra available models.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pirhadi, N.; Tang, X.; Yang, Q.
New Equations to Evaluate Lateral Displacement Caused by Liquefaction Using the Response Surface Method. *J. Mar. Sci. Eng.* **2019**, *7*, 35.
https://doi.org/10.3390/jmse7020035

**AMA Style**

Pirhadi N, Tang X, Yang Q.
New Equations to Evaluate Lateral Displacement Caused by Liquefaction Using the Response Surface Method. *Journal of Marine Science and Engineering*. 2019; 7(2):35.
https://doi.org/10.3390/jmse7020035

**Chicago/Turabian Style**

Pirhadi, Nima, Xiaowei Tang, and Qing Yang.
2019. "New Equations to Evaluate Lateral Displacement Caused by Liquefaction Using the Response Surface Method" *Journal of Marine Science and Engineering* 7, no. 2: 35.
https://doi.org/10.3390/jmse7020035