# Energy Evaluation of Triggering Soil Liquefaction Based on the Response Surface Method

^{*}

## Abstract

**:**

## 1. Introduction

_{max}). Then, the cyclic stress ratio (CSR) and the cyclic shear strength (CRR) are determined, and by comparing them the potential of liquefaction is analyzed [2,3,4,5,6,7,8,9,10,11]. Strain-based methods have also been conducted by supposing that pore water pressure grows by control of the cyclic shear strain during dynamic loads [12,13].

_{h}) [17,18], and laboratory test results [19,20]. Shafee et al. [21] performed some uniaxial shaking tests and demonstrated that the difference between strain energy generated in the soil caused by biaxial and uniaxial shaking tests is negligible. Zheghal et al. [22] studied the effect of non-proportionality and the phase angle of the induced shear stresses on rising pore water pressure. Moreover, researchers investigated the influence of five parameters, including the initial effective mean confining pressure (${\sigma}_{c}^{\prime}$), initial relative density (D

_{r})%, fine content (FC)%, coefficient of uniformity (C

_{u}), and mean grain size (D

_{50}), on the capacity energy of soils (W) [13,23,24,25,26,27,28,29,30,31] by considering laboratory test results. Baziar et al. [26] collected a large number of datasets with a wide range of test results, including six parameters, and they divided them randomly into a testing and training phase in order to present artificial neural networks (ANNs). Furthermore, they eliminated the coefficient of curvature (C

_{c}) due to no increase in the model’s accuracy, and they presented new artificial neural network (ANN) and multi-linear regression (MLR) models, including five parameters containing ${\sigma}_{c}^{\prime}$, D

_{r}%, FC%, C

_{u}, and (D

_{50}) in mm. With the same dataset collected by Baziar et al. [26], Alavi et al. [28] developed three new models. By adding new data to the dataset of Baziar et al. [26] and applying a neuro-fuzzy interface system (ANFIS), Cabalar et al. [32] developed a model that included six parameters containing C

_{c}and demonstrated its influence. However, data division was conducted randomly in all studies, without considering the statistical aspects of parameters. Furthermore, data was divided into two groups, testing and training, without performing a validating phase to prevent overtraining of the ANN model. A validating phase is applied to minimize overfitting of the trained model [33,34]. In addition, Tao [35] investigated the complicated influence of FC and illustrated that liquefaction resistance, in terms of the unit energy, starts to increase with an increase in FC above 28%. They also indicated that the liquefaction resistance becomes less dependent on relative density when FC is less than 28%. Maurer et al. [36] investigated 7000 dataset case histories from the 2010–2011 Canterbury Earthquakes and indicated that the evaluation of liquefaction is less accurate when soils have a high FC value. Although these studies have indicated an altered influence of a high FC value on liquefaction assessment, it has not yet been taken into account to propose a model.

_{r}%, FC%, C

_{u}, D

_{50}, and C

_{c}. To analyze the complicated influence of FC, the first ANN model, without any constraints in the range of input parameters, is derived similarly to other studies that have been performed and explained herein, and the second ANN model is derived through a database with FC values less than 28%, as in Tao [35]. To increase the accuracy and capability of the ANN models, the dataset is divided into three groups by considering the statistical aspects of parameters with similar mean as well as mean coefficient of variation (COV) values, instead of random division. The first group is for the training phase, the second is for testing, and the third is for the validating phase, to prevent overtraining in the training phase. In the second step, six visualized equations are captured by using the response surface method (RSM), which was demonstrated to be a capable method for evaluating liquefaction in sandy soil by Pirhadi et al. [37]. To the best of our knowledge, no other studies have been conducted on liquefaction using RSM. In Section 5, the dataset and two derived ANN models and their characteristics are described. According to any ANN model, three different design of experiments (DOEs) are performed. Therefore, three equations are obtained to illustrate and calculate correlation between six defined parameters and W as a target. During this step, the meaningfulness of all terms of the equations are analyzed through hypothesis testing, and to obtain more capable and reliable equations, some equation terms that do not provide a meaningful correlation with the target are eliminated, instead of performing an overall elimination of parameters such as C

_{c}. The final equations thus contain all six parameters and are presented in Section 6. Furthermore, by applying three different DOEs, their influence and capability are studied to determine the best DOE that can be applied for similar issues. Finally, to demonstrate the accuracy and capability of the presented equations, their predicted results are compared with four existing, well-known, and highly rated models that are currently used. Section 7 presents this comparison, using 20 samples from Dief [38] that are not included in the database to develop the ANN and RSM models for this study. Figure 1 illustrates the flowchart of the process applied in this study to develop the RSM equations.

## 2. Approaches Based on Laboratory Test Results

_{r}, a

_{max}, and initial effective confining stress.

^{3}), Γ is the shear strain amplitude, and R

^{2}is the coefficient of determination.

_{u}and the C

_{c}, and the author presented the equations expressed as:

_{i}is the particle diameter, which is given by a grain-size distribution for a given percent finer, denoted by the subscript i.

_{r}%, FC%, C

_{u}, D

_{50}(in mm), and C

_{c}—and the output (the target) was W (J/m

^{3}). In the second model, they eliminated C

_{c}and developed the model with the five extra input parameters.

## 3. Artificial Neural Network

## 4. Response Surface Method

#### 4.1. Selection of Regression Model Function

#### 4.2. Design of Experiment

#### 4.3. Coding of the Input Variables

_{i}is the coded value, X

_{i}is the real value, Z

_{i}is the middle of the real value range, and L is the major coded limit value in the matrix for each variable.

#### 4.4. Hypothesis Test

## 5. Databases and Artificial Neural Network Models

_{r}%, FC%, C

_{u}, D

_{50}(in mm), and C

_{c}—and the target was log W. Also, the criterion for the triggering of liquefaction is r

_{u}= 1 for the initiate of liquefaction or strain equal to 5% (ε

_{DA}= 5%). The database that was used to develop the first group of equations includes 217 cyclic, triaxial laboratory test results [56]; 61 cyclic, torsional laboratory test results [22,57]; six cyclic simple tests [58], and 22 centrifuge test results [36]. In addition, new data were added from 22 samples from the VErification of Liquefaction Analysis by Centrifuge Studies (VELACS) program [25,35,58], along with 48 cyclic, triaxial laboratory test results [59], and 27 cyclic, torsional laboratory test results [35]. In total, the database was composed of 403 samples.

#### 5.1. First Artificial Neural Network Model

#### 5.2. Second Artificial Neural Network Model

## 6. The RSM Equations

^{2}demonstrates the power of the regression, taking into account the number of predictors. In other words, it is a modified version of the R

^{2}, and it is always lower than the R

^{2}; however, when it is closer to the R

^{2}, this demonstrates a greater accuracy and ability to predict. It must be considered that to use these equations to predict W, the real values of six input parameters must first be transferred to the coded value in Equation (16) below; then, the value of W can be estimated by substituting the coded value in the RSM equations presented in Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10.

- (1)
- Both RSM equations require soil properties and laboratory test results to estimate D
_{r}%, FC%, C_{u}, D_{50}(in mm), and C_{c}. - (2)
- (3)
- The second RSM equation is only applicable for samples with an FC value of less than 28%.
- (4)
- It is necessary to transfer the real value of the parameters to the coded value as in Equation (16), then input this into the equations to estimate the results.

## 7. Comparison of the Predicted Capacity Energy of Liquefaction between the RSM Equations and Existing Models

## 8. Summary and Conclusions

_{r})%, FC, coefficient of uniformity (Cu), coefficient of curvature (Cc) and mean grain size (D

_{50}), with no limitation on the range of the parameters, whereas the second dataset was built by eliminating all samples with FCs higher than 28%. To establish the RSM equations, three common DOEs, the Box-Behnken (BB), central composite (CC), and half central composite (HCC), were applied to assess the best DOE. Then, after performing a hypothesis test based on P-values, some terms of the original equations were eliminated instead of eliminating a parameter such as C

_{c}. The RSM procedure was then repeatedly rebuilt to obtain the most accurate and capable final equations.

- Applying a validation phase provides a significant increase in the accuracy of the model in predicting W. Furthermore, performing data division considering statistical factors instead of random division raises the performance of the model.
- The second group of models containing three equations demonstrate higher capability and accuracy for measuring W. It should be considered that the second group of models were derived on the basis of a smaller dataset, 309 samples, due to eliminated samples with FCs higher than 28%. Therefore, FCs in varying amounts that are higher than 28% are confirmed to have different effects on W.
- In general, the RSM is a capable tool for predicting the potential of liquefaction, and it can be used by researchers.
- Of all the DOEs inspected in the present study, the CC and BB designs demonstrated the highest capability and accuracy in predicting W; they both displayed lower RMSEs and MAEs, and they both had a higher R
^{2}.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

BF | Equation | BF | Equation |
---|---|---|---|

BF1 | $\mathrm{max}\left(0,{D}_{50}-0.12\right)$ | BF8 | $\mathrm{max}\left(0,{D}_{r}-17\right)\ast \mathrm{max}\left(0,35-FC\right)$ |

BF2 | $\mathrm{BF}1\ast \mathrm{max}\left(0,{D}_{r}-69.2\right)$ | BF9 | $BF1\ast \mathrm{max}\left(0,{C}_{u}-1.68\right)$ |

BF3 | $\mathrm{max}\left(0,{\sigma}_{c}^{\prime}-100.5\right)$ | BF10 | $\mathrm{max}\left(0,FC-20\right)$ |

BF4 | $\mathrm{max}\left(0,100.5-{\sigma}_{c}^{\prime}\right)$ | BF11 | $\mathrm{max}\left(0,20-FC\right)$ |

BF5 | $\mathrm{max}\left(0,17-{D}_{r}\right)$ | BF12 | $\mathrm{max}\left(0,{C}_{u}-2.63\right)$ |

BF6 | $BF1\ast \mathrm{max}\left(0,{D}_{r}-17\right)$ | BF13 | $\mathrm{max}\left(0,2.63-{C}_{u}\right)$ |

BF7 | $\mathrm{max}\left(0,{D}_{r}-17\right)\ast \mathrm{max}\left(0,FC-35\right)$ | BF14 | $BF1\ast \mathrm{max}\left(0,{C}_{u}-1.66\right)$ |

## References

- Seed, H.B.; Idriss, I.M. Simplified procedure for evaluating soil liquefaction potential. J. Geotech. Eng. Div.
**1971**, 97, 1171. [Google Scholar] - Robertson, P.K.; Wride, C.E. Evaluating cyclic liquefaction potential using the cone penetration test. Can. Geotech. J.
**1998**, 35, 442–459. [Google Scholar] [CrossRef] - Andrus, R.D.; Kenneth, S.H., II. Liquefaction Resistance of Soils from Shear-Wave Velocity. J. Geotech. Geoenviron. Eng.
**2000**, 126, 1015–1025. [Google Scholar] [CrossRef] - Youd, T.L. Liquefaction resistance of soils: Summary report from the 1996 NCEER and 1998 NCEER/NSF workshops on evaluation of liquefaction resistance of soils. J. Geotech. Geoenviron. Eng.
**2001**, 127, 297–313. [Google Scholar] [CrossRef] - Juang, C.H.; Yuan, H.; Lee, D.-H.; Lin, P.-S. Simplified Cone Penetration Test-based Method for Evaluating Liquefaction Resistance of Soils. J. Geotech. Geoenviron. Eng.
**2003**, 129, 66–80. [Google Scholar] [CrossRef] - Andrus, R.D.; Stokoe, K.H.; Juang, C.H. Guide for Shear-Wave-Based Liquefaction Potential Evaluation. Earthq. Spectra
**2004**, 20, 285–308. [Google Scholar] [CrossRef] - Idriss, I.M.; Boulanger, R.W. Semi-empirical procedures for evaluating liquefaction potential during earthquakes. Soil Dyn. Earthq. Eng.
**2006**, 26, 115–130. [Google Scholar] [CrossRef] - Moss, R.E.; Seed, R.B.; Kayen, R.E.; Stewart, J.P. CPT-Based Probabilistic and Deterministic Assessment of In Situ Seismic Soil Liquefaction Potential. J. Geotech. Geoenviron. Eng.
**2006**, 132, 1032–1051. [Google Scholar] [CrossRef] [Green Version] - Baxter, C.D.P.; Bradshaw, A.S.; Green, R.A.; Wang, J.H. Correlation between Cyclic Resistance and Shear-Wave Velocity for Providence Silts. J. Geotech. Geoenviron. Eng.
**2008**, 134, 37–46. [Google Scholar] [CrossRef] - Idriss and Boulanger. CPT and SPT Based Liquefaction Triggering Procedures Center for Geotechnical Modeling; Report UCD/CGM-10/02; Department of Civil and Environmental Engineering, University of California: Davis, CA, USA, 2010; p. 77. [Google Scholar]
- Ghafghazi, M.; DeJong, J.; Wilson, D. Evaluation of Becker Penetration Test Interpretation Methods for Liquefaction Assessment in Gravelly Soils. Can. Geotech. J.
**2017**, 54, 1272–1283. [Google Scholar] [CrossRef] - Dobry, R.; Ladd, R.S.; Yokel, F.Y.; Chung, R.M.; Powell, D. Prediction of Pore Water Pressure Buildup and Liquefaction of Sands during Earthquakes by the Cyclic Strain Method; National Bureau of Standards Building Science Series; U.S. Department of Commerce: Washington, DC, USA, 1982; Volume 138.
- Liang, L. Development of an Energy Method for Evaluating the Liquefaction Potential of a Soil Deposit. Ph.D. Thesis, Department of Civil Engineering, Case Western Reserve University, Cleveland, OH, USA, 1995. [Google Scholar]
- Davis, R.O.; Berrill, J.B. Energy dissipation and seismic liquefaction in sands. Earthq. Eng. Struct. Dyn.
**1982**, 10, 59–68. [Google Scholar] - Law, K.T.; Cao, Y.L.; He, G.N. An energy approach for assessing seismic liquefaction potential. Can. Geotech. J.
**1990**, 27, 320–329. [Google Scholar] [CrossRef] [Green Version] - Trifunac, M.D. Empirical criteria for liquefaction in sands via standard penetration tests and seismic wave energy. Soil Dyn. Earthq. Eng.
**1995**, 14, 419–426. [Google Scholar] [CrossRef] - Running, D.L. An energy-based Model for soil Liquefaction. Ph.D. Thesis, Washington State University, Pullman, WA, USA, 1996; 267p. [Google Scholar]
- Kayen, R.E.; Mitchell, J.K. Assessment of Liquefaction Potential during Earthquakes by Arias Intensity. J. Geotech. Geoenviron. Eng.
**1997**, 123, 1162–1174. [Google Scholar] [CrossRef] - Azeiteiro, R.J.N.; Coelho, P.A.; Taborda, D.M.; Grazina, J.C. Energy-based evaluation of liquefaction potential under non-uniform cyclic loading. Soil Dyn. Earthq. Eng.
**2017**, 92, 650–665. [Google Scholar] [CrossRef] [Green Version] - Kokusho, T. Liquefaction potential evaluations by energy-based method and stress-based method for various ground motions: Supplement. Soil Dyn. Earthq. Eng.
**2017**, 95, 40–47. [Google Scholar] [CrossRef] - Shafee, O.E.; Abdoun, T.; Zeghal, M. Centrifuge modelling and analysis of site liquefaction subjected to biaxial dynamic excitations. Géotechnique
**2017**, 67, 260–271. [Google Scholar] [CrossRef] - Zeghal, M.; El-Shafee, O.; Abdoun, T. Analysis of soil liquefaction using centrifuge tests of a site subjected to biaxial shaking. Soil Dyn. Earthq. Eng.
**2018**, 114, 229–241. [Google Scholar] [CrossRef] - Kusky, P.J. Influence of Loading Rate on the Unit Energy Required for Liquefaction. Master’s Thesis, Department of Civil Engineering, Case Western Reserve University, Cleveland, OH, USA, 1996. [Google Scholar]
- Figueroa, J.L.; Saada, A.S.; Rokoff, M.D.; Liang, L. Influence of Grain-Size Characteristics in Determining the Liquefaction Potential o f a Soil Deposit by the Energy Method. In Proceedings of the International Workshop on the Physics and Mechanics of Soil Liquefaction, Baltimore, MD, USA, 10–11 September 1998; pp. 237–245. [Google Scholar]
- Rokoff, M.D. The Influence of Grain-Size Characteristics in Determining the Liquefaction Potential o f a Soil Deposit by the Energy Method. Master’s Thesis, Department of Civil Engineering, Case Western Reserve University, Cleveland, OH, USA, 1999. [Google Scholar]
- Baziar, M.H.; Jafarian, Y. Assessment of liquefaction triggering using strain energy concept and ANN model: Capacity Energy. Soil Dyn. Earthq. Eng.
**2007**, 27, 1056–1072. [Google Scholar] [CrossRef] - Baziar, M.H.; Jafarian, Y.; Shahnazari, H.; Movahed, V.; Tutunchian, M.A. Prediction of strain energy-based liquefaction resistance of sand–silt mixtures: An evolutionary approach. Comput. Geosci.
**2011**, 37, 1883–1893. [Google Scholar] [CrossRef] - Alavi, A.H.; Gandomi, A.H. Energy-based numerical models for assessment of soil liquefaction. Geosci. Front.
**2012**, 3, 541–555. [Google Scholar] [CrossRef] [Green Version] - Zhang, W.; Goh, A.T.; Zhang, Y.; Chen, Y.; Xiao, Y. Assessment of soil liquefaction based on capacity energy concept and multivariate adaptive regression splines. Eng. Geol.
**2015**, 188, 29–37. [Google Scholar] [CrossRef] - Jin, J.-X.; Cui, H.-Z.; Liang, L.; Li, S.-W.; Zhang, P.-Y. Variation of Pore Water Pressure in Tailing Sand under Dynamic Loading. Shock Vib.
**2018**, 2018, 1921057. [Google Scholar] [CrossRef] - Qu, D.; Cai, X.; Chang, W. Evaluating the Effects of Steel Fibers on Mechanical Properties of Ultra-High Performance Concrete Using Artificial Neural Networks. Appl. Sci.
**2018**, 8, 1120. [Google Scholar] [CrossRef] - Cabalar, A.F.; Cevik, A.; Gokceoglu, C. Some applications of Adaptive Neuro-Fuzzy Inference System (ANFIS) in geotechnical engineering. Comput. Geotech.
**2012**, 40, 14–33. [Google Scholar] [CrossRef] - Zeng, X.; Martinez, T.R. Distribution-balanced stratified cross-validation for accuracy estimation. J. Exp. Theor. Artif. Intell.
**2000**, 12, 1–12. [Google Scholar] [CrossRef] - Kohavi, R. A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection. Presented at the 14th International Joint Conference on Artificial Intelligence (IJCAI’95), Montreal, QC, Canada, 20–25 August 1995; Volume 14. [Google Scholar]
- Tao, M. Case History Verification of the Energy Method to Determine the Liquefaction Potential of Soil Deposits. Ph.D. Thesis, Department of Civil Engineering, Case Western Reserve University, Cleveland, OH, USA, 2013; p. 173. [Google Scholar]
- 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 Dyn. Earthq. Eng.
**2015**, 76, 58–68. [Google Scholar] [CrossRef] - 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, 112. [Google Scholar] [CrossRef] - Dief, H.M. Evaluating the Liquefaction Potential of Soils by the Energy Method in the Centrifuge. Ph.D. Thesis, Reserve University, Cleveland, OH, USA, 2000. [Google Scholar]
- Alkahatib, M. Liquefaction Assessment by Strain Energy Aprroach. Ph.D. Thesis, Wayne State University, Detroit, MI, USA, 1994; p. 212. [Google Scholar]
- Liang, L.; Figueroa, J.L.; Saada, A.S. Liquefaction under random loading: Unit energy approach. J. Geotech. Eng.
**1995**, 121, 776–781. [Google Scholar] [CrossRef] - Wallin, M.S. Evaluation of Normalized Pore Water Pressure vs. Accumulated Unit Energy Relationships for Determining Liquefaction Potential in Soils. Ph.D. Thesis, Department of Civil Engineering, Case Western Reserve University, Cleveland, OH, USA, 2000. [Google Scholar]
- de Julián-Ortiz, J.; Pogliani, L.; Besalú, E. Modeling Properties with Artificial Neural Networks and Multilinear Least-Squares Regression: Advantages and Drawbacks of the Two Methods. Appl. Sci.
**2018**, 8, 1094. [Google Scholar] [CrossRef] - Gherman, A.; Kovács, K.; Cristea, M.; Toșa, V. Artificial Neural Network Trained to Predict High-Harmonic Flux. Appl. Sci.
**2018**, 8, 2106. [Google Scholar] [CrossRef] - Kose, U. An Ant-Lion Optimizer-Trained Artificial Neural Network System for Chaotic Electroencephalogram (EEG) Prediction. Appl. Sci.
**2018**, 8, 1613. [Google Scholar] [CrossRef] - Lee, H.; Oh, J. Establishing an ANN-Based Risk Model for Ground Subsidence Along Railways. Appl. Sci.
**2018**, 8, 1936. [Google Scholar] [CrossRef] - Mato-Abad, V.; Jiménez, I.; García-Vázquez, R.; Aldrey, J.M.; Rivero, D.; Cacabelos, P.; Andrade-Garda, J.; Pías-Peleteiro, J.M.; Yánez, S.R. Using Artificial Neural Networks for Identifying Patients with Mild Cognitive Impairment Associated with Depression Using Neuropsychological Test Features. Appl. Sci.
**2018**, 8, 1629. [Google Scholar] [CrossRef] - Zhou, P.; Zhou, G.; Zhu, Z.; Tang, C.; He, Z.; Li, W.; Jiang, F. Health Monitoring for Balancing Tail Ropes of a Hoisting System Using a Convolutional Neural Network. Appl. Sci.
**2018**, 8, 1346. [Google Scholar] [CrossRef] - Haykin, S. Neural Networks: A Comprehensive Foundation, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, USA, 1998. [Google Scholar]
- Coulibaly, P.; Anctil, F.; Bobée, B. Daily reservoir inflow forecasting using artificial neural networks with stopped training approach. J. Hydrol.
**2000**, 230, 244–257. [Google Scholar] [CrossRef] - Box, G.E.P.; Wilson, K.B. On the Experimental Attainment of Optimum Conditions (with discussion). J. R. Stat. Soc. Ser. B
**1951**, 13, 1–45. [Google Scholar] - Park, S.; Kang, H. Multivariate Analysis of Laser-Induced Tissue Ablation: Ex Vivo Liver Testing. Appl. Sci.
**2017**, 7, 974. [Google Scholar] [CrossRef] - Chu, Z.; Zheng, F.; Liang, L.; Yan, H.; Kang, R. Parameter Determination of a Minimal Model for Brake Squeal. Appl. Sci.
**2018**, 8, 37. [Google Scholar] [CrossRef] - Takahashi, H.; Kurita, M.; Iijima, H.; Sasamori, M. Interpolation of Turbulent Boundary Layer Profiles Measured in Flight Using Response Surface Methodology. Appl. Sci.
**2018**, 8, 2320. [Google Scholar] [CrossRef] - Box, G.E.P.; Draper, N.R. Empirical Model-Building and Response Surfaces; Wiley: New York, NY, USA, 1987. [Google Scholar]
- Box, G.E.P.; Behnken, D.W. Some New Three Level Designs for the Study of Quantitative Variables. Technometrics
**1960**, 2, 455–475. [Google Scholar] [CrossRef] - Green, R.A. Energy-Based Evaluation and Remediation of Liquefiable Soils. Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, 2001. [Google Scholar]
- Towhata, I.; Ishihara, K. Shear work and pore water pressure in undrained shear. Soils Found.
**1985**, 25, 73–84. [Google Scholar] [CrossRef] - Arulanandan, K.; Scott, R.F. Project VELACS-Control Test Results. J. Geotech. Eng.
**1993**, 119, 1276–1292. [Google Scholar] [CrossRef] - Kanagalingam, T. Liquefaction Resistance of Granular Mixes Based on Contact Densityand Energy Considerations. Ph.D. Thesis, Department of Civil, Structural, and Environmental Engineering, and Environmental Engineering, The State University of New York at Buffalo, Buffalo, NY, USA, 2006; p. 386. [Google Scholar]

**Figure 1.**Flowchart of the process to derive the response surface method (RSM) equations in this study; BB: Box-Behnken; CC: central composite; HCC: half central composite.

**Figure 2.**Capacity energy predicted by RSM equation on the basis of the Box-Behnken DOE and the first ANN model versus the measured values of laboratory tests.

**Figure 3.**Capacity energy predicted by RSM equation on the basis of the central composite (CC) design of experiment (DOE) and the first ANN model versus the measured values of laboratory tests.

**Figure 4.**Capacity energy predicted by RSM equation on the basis of the half central composite (HCC) design of experiment (DOE) and the first ANN model versus the measured values of laboratory tests.

**Figure 5.**Capacity energy predicted by RSM equation on the basis of the Box-Behnken (BB) design of experiment (DOE) and the second ANN model versus the measured values of laboratory tests.

**Figure 6.**Capacity energy predicted by RSM equation on the basis of the central composite (CC) and design of experiment (DOE) and the second ANN model versus the measured values of laboratory tests.

**Figure 7.**Capacity energy predicted by RSM equation on the basis of half central composite (HCC) and design of experiment (DOE) and the second ANN model versus the measured values of laboratory tests.

**Figure 8.**Capacity energy predicted by genetic programming (LGP) versus the measured values of laboratory tests.

**Figure 9.**Capacity energy predicted by multi expression programming (MEP) versus the measured values of laboratory tests.

**Figure 10.**Capacity energy predicted by genetic programming (GP) versus the measured values of laboratory tests.

**Figure 11.**Capacity energy predicted by Zhang’s equation versus the measured values of laboratory tests.

**Table 1.**Characteristics of complete input parameters used for the first ANN model and RSM equation.

Parameters | Min. Value | Max. Value | Mean Value | Average Value |
---|---|---|---|---|

${\sigma}_{c}^{\prime}$ (kpa) | 40.00 | 294.00 | 98.87 | 167.00 |

D_{r} (%) | −44.50 | 105.10 | 48.13 | 30.30 |

FC (%) | 0.00 | 100.00 | 18.97 | 50.00 |

C_{u} | 1.57 | 5.88 | 2.38 | 3.73 |

D_{50} (mm) | 0.03 | 0.46 | 0.23 | 0.25 |

C_{c} | 0.74 | 1.61 | 0.95 | 1.18 |

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

R^{2} | 0.9544 | 0.93967 | 0.91301 | 0.94797 |

**Table 3.**Characteristics of the complete input parameters used for the second ANN model and RSM equation.

Parameters | Min. Value | Max. Value | Mean Value | Average Value |
---|---|---|---|---|

${\sigma}_{c}^{\prime}$ (kpa) | 40 | 400 | 106.00 | 220 |

D_{r} (%) | −44.5 | 98.3 | 46.89 | 26.9 |

FC (%) | 0 | 26 | 7.83 | 13 |

C_{u} | 1.52 | 28.12 | 3.72 | 14.82 |

D_{50} (mm) | 0.13 | 0.46 | 0.25 | 0.295 |

C_{c} | 0.74 | 10.89 | 1.68 | 5.815 |

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

R^{2} | 0.94811 | 0.92569 | 0.91192 | 0.94079 |

**Table 5.**RSM equation with the DOE of the BB design, based on the first ANN model (R

^{2}= 79.83%, R

^{2}[adjust] = 69.46%).

Term | Constant | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r} | FC | C_{u} | D_{50} | C_{c} | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r}*D_{r} | FC*FC |

Coef | 2.066 | 0.306 | 0.487 | −0.247 | −0.008 | −0.176 | 0.399 | 0.509 | 0.388 | −0.297 |

Term | C_{u}*C_{u} | D_{50}*D_{50} | C_{c}*C_{c} | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{D}}_{\mathit{r}}$ | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}\mathit{F}\mathit{C}$ | D_{r}*D_{50} | D_{r}*C_{c} | FC*D_{50} | FC*C_{c} | - |

Coef | 0.512 | 0.821 | 0.385 | −0.318 | −0.323 | 0.368 | −0.373 | −0.867 | 0.370 | - |

**Table 6.**RSM equation with the DOE of the CC design, based on the first ANN model (R

^{2}= 73.89%, R

^{2}[adjust] = 66.81%).

Terms | Constant | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r} | FC | C_{u} | D_{50} | C_{c} | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r}*D_{r} | FC*FC |

Coef | 2.13752 | 0.21066 | 0.5284 | −0.04392 | 0.0966 | −0.0777 | −0.059 | 0.14882 | 0.31044 | 0.155 |

Terms | D_{50}*D_{50} | C_{c}*C_{c} | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{D}}_{\mathit{r}}$ | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}\mathit{F}\mathit{C}$ | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{C}}_{\mathit{c}}$ | D_{r}*FC | D_{r}*D_{50} | D_{r}*C_{c} | FC*C_{u} | FC*D_{50} |

Coef | 0.253135 | 0.27647 | 0.1039 | −0.15012 | −0.1627 | −0.2512 | 0.3619 | −0.1242 | 0.12629 | −0.644 |

**Table 7.**RSM equation with the DOE of the HCC design, based on the first ANN model (R

^{2}= 72.74%, R

^{2}[adjust] = 60.62%).

Terms | Constant | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r} | FC | C_{u} | D_{50} | C_{c} | D_{r}*D_{r} | FC*FC |

Coef | 2.160 | 0.115 | 0.513 | −0.007 | 0.173 | −0.091 | −0.109 | 0.402 | 0.144 |

Terms | D_{50}*D_{50} | C_{c}*C_{c} | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}\mathit{F}\mathit{C}$ | D_{r}*FC | D_{r}*C_{u} | D_{r}*D_{50} | FC*C_{u} | FC*D_{50} | - |

Coef | 0.347 | 0.273 | −0.211 | −0.217 | −0.192 | 0.382 | 0.189 | −0.620 | - |

**Table 8.**RSM equation with the DOE of the BB design, based on the second ANN model (R

^{2}= 96.49%, R

^{2}[adjust] = 94.52%).

Terms | Constant | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r} | FC | C_{u} | D_{50} | C_{c} | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r}*D_{r} |

Coef | 1.922 | 0.526 | 0.939 | 0.229 | −1.682 | −0.134 | −0.76 | 0.233 | 0.157 |

Terms | C_{u}*C_{u} | D_{50}*D_{50} | C_{c}*C_{c} | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}\mathit{F}\mathit{C}$ | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{C}}_{\mathit{u}}$ | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{C}}_{\mathit{c}}$ | D_{r}*C_{u} | D_{r}*D_{50} | D_{r}*C_{c} |

Coef | 0.633 | 0.134 | 0.346 | −0.202 | −0.1333 | −0.461 | −0.15 | −0.234 | −0.224 |

Terms | C_{u}*D_{50} | C_{u}*C_{c} | - | - | - | - | - | - | - |

Coef | −0.481 | −0.308 | - | - | - | - | - | - | - |

**Table 9.**RSM equation with the DOE of the CC design, based on the second ANN model (R

^{2}= 93.37%, R

^{2}[adjust] = 91.80%).

Terms | Constant | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r} | FC | C_{u} | D_{50} | C_{c} | D_{r}*D_{r} | |

Coef | 2.038 | 0.45 | 0.769 | 0.264 | −1.514 | −0.25 | 0.619 | 0.158 | 0.133 |

Terms | C_{u}*C_{u} | D_{50}*D_{50} | C_{c}*C_{c} | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}\mathbf{\ast}{\mathit{C}}_{\mathit{u}}$ | D_{r}*C_{u} | D_{r}*D_{50} | D_{r}*C_{c} | C_{u}*D_{50} | C_{u}*C_{c} |

Coef | 0.224 | 0.122 | 0.277 | 0.113 | 0.369 | −0.182 | −0.11 | −0.271 | −0.283 |

**Table 10.**RSM equation with the DOE of the HCC based on the second ANN model (R

^{2}= 94.63%, R

^{2}[adjust] = 92.02%).

Terms | Constant | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ | D_{r} | FC | C_{u} | D_{50} | C_{c} | D_{r}*D_{r} | |

Coef | 1.8996 | 0.46 | 0.8 | 0.239 | −1.539 | −0.22 | 0.507 | 0.183 | 0.154 |

Terms | C_{u}*C_{u} | D_{50}*D_{50} | C_{c}*C_{c} | D_{r}*FC | D_{r}*C_{u} | D_{r}*D_{50} | FC*D_{50} | C_{u}*D_{50} | C_{u}*C_{c} |

Coef | 0.317 | 0.145 | 0.304 | −0.171 | 0.389 | −0.156 | 0.178 | −0.278 | −0.316 |

**Table 11.**The 20 samples from Dief [38] that were used for a comparison of models’ performance.

Test No. | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{\prime}}$ (kpa) | D_{r} | FC | C_{u} | D_{50} | C_{c} |
---|---|---|---|---|---|---|

1 | 27.8 | 49.7 | 0 | 2.27 | 0.15 | 0.95 |

2 | 34.6 | 51.7 | 0 | 2.27 | 0.15 | 0.95 |

3 | 28.7 | 60.7 | 0 | 2.27 | 0.15 | 0.95 |

4 | 33.9 | 58.5 | 0 | 2.27 | 0.15 | 0.95 |

5 | 28.4 | 64.7 | 0 | 2.27 | 0.15 | 0.95 |

6 | 34.4 | 66.5 | 0 | 2.27 | 0.15 | 0.95 |

7 | 29.8 | 72 | 0 | 2.27 | 0.15 | 0.95 |

8 | 34.7 | 72.1 | 0 | 2.27 | 0.15 | 0.95 |

9 | 28.8 | 76.3 | 0 | 2.27 | 0.15 | 0.95 |

10 | 34.9 | 74 | 0 | 2.27 | 0.15 | 0.95 |

11 | 28.8 | 51 | 0 | 1.67 | 0.26 | 0.97 |

12 | 33.6 | 51 | 0 | 1.67 | 0.26 | 0.97 |

13 | 28.65 | 60.5 | 0 | 1.67 | 0.26 | 0.97 |

14 | 34.07 | 60.2 | 0 | 1.67 | 0.26 | 0.97 |

15 | 28.9 | 64.3 | 0 | 1.67 | 0.26 | 0.97 |

16 | 34.3 | 65.5 | 0 | 1.67 | 0.26 | 0.97 |

17 | 29.1 | 71.8 | 0 | 1.67 | 0.26 | 0.97 |

18 | 34.54 | 72.5 | 0 | 1.67 | 0.26 | 0.97 |

19 | 29.3 | 78.7 | 0 | 1.67 | 0.26 | 0.97 |

20 | 34.6 | 80.4 | 0 | 1.67 | 0.26 | 0.97 |

Test No | Log W | BB | CC | HCC | BB28 | CC28 | HCC28 |
---|---|---|---|---|---|---|---|

1 | 2.568 | 2.370 | 2.167 | 2.133 | 2.456 | 2.943 | 3.102 |

2 | 2.690 | 2.369 | 2.205 | 2.175 | 2.503 | 2.965 | 3.129 |

3 | 2.771 | 2.531 | 2.350 | 2.276 | 2.612 | 3.065 | 3.236 |

4 | 2.778 | 2.473 | 2.317 | 2.260 | 2.596 | 3.040 | 3.211 |

5 | 2.895 | 2.596 | 2.426 | 2.331 | 2.669 | 3.111 | 3.286 |

6 | 2.971 | 2.597 | 2.471 | 2.372 | 2.713 | 3.132 | 3.312 |

7 | 2.968 | 2.712 | 2.583 | 2.442 | 2.781 | 3.197 | 3.381 |

8 | 3.035 | 2.689 | 2.592 | 2.456 | 2.797 | 3.198 | 3.384 |

9 | 3.148 | 2.793 | 2.682 | 2.506 | 2.843 | 3.249 | 3.437 |

10 | 3.241 | 2.721 | 2.636 | 2.486 | 2.826 | 3.221 | 3.410 |

11 | 2.740 | 2.847 | 2.479 | 2.412 | 2.657 | 2.904 | 2.962 |

12 | 2.851 | 2.826 | 2.485 | 2.424 | 2.670 | 2.905 | 2.964 |

13 | 2.940 | 3.013 | 2.661 | 2.565 | 2.765 | 2.992 | 3.061 |

14 | 2.948 | 2.982 | 2.662 | 2.574 | 2.777 | 2.989 | 3.061 |

15 | 3.035 | 3.081 | 2.743 | 2.631 | 2.811 | 3.028 | 3.103 |

16 | 3.111 | 3.078 | 2.777 | 2.666 | 2.841 | 3.040 | 3.118 |

17 | 3.049 | 3.223 | 2.920 | 2.767 | 2.903 | 3.101 | 3.187 |

18 | 3.207 | 3.209 | 2.945 | 2.794 | 2.928 | 3.109 | 3.197 |

19 | 3.064 | 3.360 | 3.100 | 2.898 | 2.991 | 3.172 | 3.267 |

20 | 3.225 | 3.367 | 3.155 | 2.945 | 3.029 | 3.190 | 3.290 |

Test No. | Log W | LGP [27] | MEP [27] | GP [27] | Zhang [28] |
---|---|---|---|---|---|

1 | 2.568 | 3.044 | 2.841 | 3.046 | 2.282 |

2 | 2.690 | 3.090 | 2.880 | 3.067 | 2.306 |

3 | 2.771 | 3.115 | 2.887 | 3.085 | 2.308 |

4 | 2.778 | 3.130 | 2.905 | 3.089 | 2.281 |

5 | 2.895 | 3.138 | 2.901 | 3.098 | 2.282 |

6 | 2.971 | 3.184 | 2.941 | 3.119 | 2.309 |

7 | 2.968 | 3.191 | 2.937 | 3.128 | 2.304 |

8 | 3.035 | 3.222 | 2.965 | 3.140 | 2.337 |

9 | 3.148 | 3.211 | 2.948 | 3.141 | 2.360 |

10 | 3.241 | 3.235 | 2.975 | 3.147 | 2.400 |

11 | 2.740 | 3.347 | 3.093 | 3.188 | 2.385 |

12 | 2.851 | 3.378 | 3.111 | 3.199 | 2.419 |

13 | 2.940 | 3.435 | 3.154 | 3.224 | 2.416 |

14 | 2.948 | 3.471 | 3.174 | 3.236 | 2.390 |

15 | 3.035 | 3.472 | 3.180 | 3.239 | 2.388 |

16 | 3.111 | 3.524 | 3.211 | 3.257 | 2.418 |

17 | 3.049 | 3.544 | 3.229 | 3.268 | 2.400 |

18 | 3.207 | 3.593 | 3.259 | 3.286 | 2.425 |

19 | 3.064 | 3.610 | 3.275 | 3.296 | 2.429 |

20 | 3.225 | 3.671 | 3.313 | 3.318 | 2.402 |

Model | R | RMSE | MAE |
---|---|---|---|

LGP | 0.627 | 0.402 | 0.369 |

MEP | 0.584 | 0.182 | 0.157 |

GP | 0.664 | 0.259 | 0.227 |

Zhang | 0.614 | 0.620 | 0.600 |

BB | 0.693 | 0.251 | 0.207 |

CC | 0.830 | 0.376 | 0.348 |

HCC | 0.792 | 0.476 | 0.456 |

BB28 | 0.911 | 0.218 | 0.203 |

CC28 | 0.722 | 0.173 | 0.139 |

HCC28 | 0.508 | 0.293 | 0.244 |

© 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.
Energy Evaluation of Triggering Soil Liquefaction Based on the Response Surface Method. *Appl. Sci.* **2019**, *9*, 694.
https://doi.org/10.3390/app9040694

**AMA Style**

Pirhadi N, Tang X, Yang Q.
Energy Evaluation of Triggering Soil Liquefaction Based on the Response Surface Method. *Applied Sciences*. 2019; 9(4):694.
https://doi.org/10.3390/app9040694

**Chicago/Turabian Style**

Pirhadi, Nima, Xiaowei Tang, and Qing Yang.
2019. "Energy Evaluation of Triggering Soil Liquefaction Based on the Response Surface Method" *Applied Sciences* 9, no. 4: 694.
https://doi.org/10.3390/app9040694