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Neural network and support vector machine models for the prediction of the liquefaction-induced uplift displacement of tunnels
Journal Pre-proofs Neural network and support vector machine models for the prediction of the liquefaction-induced uplift displacement of tunnels Gang Zheng, Wenbin Zhang, Wengang Zhang, Haizuo Zhou, Pengbo Yang PII: DOI: Reference:
S2467-9674(19)30101-1 https://doi.org/10.1016/j.undsp.2019.12.002 UNDSP 127
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Underground Space
Received Date: Revised Date: Accepted Date:
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Please cite this article as: G. Zheng, W. Zhang, W. Zhang, H. Zhou, P. Yang, Neural network and support vector machine models for the prediction of the liquefaction-induced uplift displacement of tunnels, Underground Space (2020), doi: https://doi.org/10.1016/j.undsp.2019.12.002
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Neural network and support vector machine models for the prediction of the liquefaction-induced uplift displacement of tunnels Gang Zhenga,b,c, Wenbin Zhanga,b, Wengang Zhangd, Haizuo Zhoua,b,c*, Pengbo Yanga,b a
School of Civil Engineering, Tianjin University, Tianjin 300072, China Key Laboratory of Coast Civil Structure Safety, Tianjin University, Ministry of Education, Tianjin 300072, China c State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China d National Joint Engineering Research Center of Geohazards Prevention in Reservoir Area Environment, School of Civil Engineering, Chongqing University, Chongqing 400045, China b
*Corresponding
author (Haizuo Zhou): Tel./fax: 022 27402341 E-mail address: [email protected]
Abstract Tunnels buried in liquefiable deposits are vulnerable to liquefaction-induced uplift damage during earthquakes. This paper presents support vector machine (SVM) and artificial neural network (ANN) models for predicting the liquefaction-induced uplift displacement of tunnels based on artificial databases generated by the finite difference method (FDM). The performance of the SVM and ANN models are assessed using statistical parameters, including the coefficient of determination (R2), the mean absolute error (MAE), and the root mean squared error (RMSE). The applications for the above-mentioned approach for predicting the uplift displacement of tunnels are compared and discussed. A relative importance analysis is adopted to quantify the sensitivity of each input variable. The precision of the presented models is demonstrated using the centrifuge test results available from previous studies. 1
Keywords: artificial neural network, support vector machine, liquefaction, uplift displacement
2
1. Introduction Underground structures buried in liquefiable deposits may suffer considerable uplift damage during major earthquakes. In the 2011 Great East Japan Earthquake, the underground pipeline was subjected to a significant uplift displacement in reclaimed liquefiable deposits, and over 112 manholes were uplifted [1]. The phenomenon was also observed in the 1964 Niigata Earthquake, the 1989 Loma Prieta Earthquake and the 1995 Kobe Earthquake [2-4]. The submerged unit weight of underground structures is lower than the surrounding soil; thus, the underground structure tends to uplift, and under static conditions, the weight and shear strength of the soil resist the uplift. However, when an earthquake occurs, the liquefiable soil around the underground structures loses most of its shear strength, and the buoyant force may be larger than the resistant force; the unbalanced force thus leads to the floating of underground structures. For the purpose of evaluating the uplift damage, centrifuge tests were conducted to verify the uplift behaviour of underground structures (e.g., buried tunnels, pipelines and manholes) [5-7]. Numerical simulations were performed to study the potential uplift damage of real tunnels such as the BART Tunnel [8] and George Masset Tunnel [2]. Furthermore, a parametric analysis was conducted to study potential factors that affect the uplift of the tunnels. Azadi et al. [9, 10] numerically analysed the effect of the friction angle and the relative density of soil, the diameter and the buried depth of the tunnel and the frequency and amplitude of the input motions. Chian et al. [11] analysed the effect of the magnitude of the input motions, soil conditions, such as the relative density, and the structural configuration, such as the diameter. Zheng et al. [12] established a multivariate adaptive regression splines (MARS) model based on numerical artificial data and analysed the relative importance of nine parameters, including the characteristics of the ground motion, structure and soil condition. These previous studies showed that the liquefaction-induced uplift displacement is greatly influenced by the relative density and the friction angle of the soil, the diameter and buried depth of the tunnel and the amplitude, frequency and duration of the input 3
motion. Therefore, these parameters are considered in this study. Koseki et al. [13] proposed a factor of safety against uplift to evaluate the potential of tunnel floating [5, 6, 14]. The factor of safety can qualitatively evaluate the uplift behaviour of tunnels, but a quantitative evaluation method has rarely been reported. The artificial intelligence methods can reveal the relationship between the complex input parameters and target output and has been applied in geotechnical engineering problems [15, 16], such as the prediction of liquefaction-induced lateral spreading [17], lateral capacity [18], drivability [19, 20] friction capacity of piles [16, 21], tunnelling-induced displacement [22] and excavation-induced displacement [23-27]. This paper presents the prediction model for the liquefaction-induced uplift of underground structures by means of the support vector machine (SVM) and artificial neural network (ANN) models. The training and testing of the model are based on a numerical database, and the model validation is conducted through the centrifuge test results from previous studies.
2. Methodology 2.1. Artificial neural network (ANN) The ANN is a common method for nonlinear regression/classification. A feed-forward propagation neural network is adopted in this study. The units are generally placed as three layers, including the input layer, hidden layer/layers and output layer in a neural network. In the generation of the ANN model, the model is first trained using a training data set. The model training aims to map the input variables to the output value by determining the optimal connection weights and biases through the back-propagation procedure. Two phases of data flow are included in the back-propagation algorithm. In the first phase, the input data propagate forward from the input layer to the output layer, and an actual output value is produced. In the second phase, the errors between the target values and actual outputs are propagated backwards from the output layer to the previous layers, and the connection weights 4
are updated to reduce the errors between the actual output values and the target output values [19]. Generally, the number of hidden neurons is typically determined by selecting the smallest number of neurons that acquire better network performance (i.e., judged by the coefficient of determination (R2) of the testing data set). The neural network established in this study is used with the Levenberg-Marquardt back-propagation algorithm based on MATLAB software.
2.2. Support vector machine (SVM) The SVM model is a statistical learning algorithm where the prediction error and model complexity are simultaneously minimized [21]; this method has been extended to regression problems. The basic idea of the SVM method is to transform the input features into a higher-dimensional space to linearly separate the data into two classes by a hyper-plane [28] which maximized the margin and minimized the error. The regression model can be expressed as follows:
f X WT X b
(1)
where X is the training dataset; ϕ is a nonlinear function to map the low-dimensional input features into a high-dimensional space; W is the unknown adjustable vector that is normal to the hyper-plane; and b is the bias of the hyper-plane. Some misclassification errors may allow for the SVM model [29], and the following constrained conditions are adopted in the SVM model.
yi (W x i ) b i ,
i 1, 2,..., k,
(W x i ) b yi i* ,
i 1, 2,..., k,
i 0 and i* 0,
i 1, 2,..., k,
(2)
where ε is the maximum error allowed, and ξi and ξi* are the slack variables that 5
quantified the deviation of the output. The optimization problem of the model can be solved using the Lagrangian multipliers method. Considering that the linear regression may not be appropriate for the input data, the input features have to be mapped into a high-dimensional space by ϕ. The kernel function (defined as the dot product of the transformed input vectors (ϕ(xi) ∙ ϕ(xj)) is introduced to reduce the computational demand [30]. The previous model can be rewritten as follows:
N
f (x) ( i i* )K (x i x j ) b
(3)
i 1
where the αi and αi* are the Lagrangian multipliers and K (xi, xj) is the kernel function. Some common kernel functions, such as the Gaussian function, polynomial function or radial basis function, are typically considered in the SVM model. The Gaussian kernel is applied in this study. The equation of the Gaussian kernel is as follows:
2
K (x i , x j ) exp( x i x j )
(4)
In the present study, the SVM analysis is performed using MATLAB software. 3. Numerical analyses 3.1. Model geometry and boundary conditions Numerical analyses are performed using Fast Lagrangian Analysis of Continua (FLAC version 8.0) software [31]. A series of analyses is performed to study the effect of structural configurations, soil conditions and ground motion characteristics on the performance of the liquefaction-induced uplift displacement of tunnels. As shown in Fig. 1, the height of the model soil is 30 m, and the water table is located at the model surface. The width of the model is 50 m, and the free field boundary 6
condition is adopted in the model to minimize the influence of the boundary effects. The element size is set to 1 m to allow the wave of the frequencies considered in this study to transmit through the model. The tunnel with diameter D buried at a depth of H (from the centre of the tunnel to the ground surface). In this study, the tunnel is modelled as a linearly elastic material using a beam element, the elastic modulus is 2.2 × 107 kPa, the unit weight of the tunnel is 24 kN/m3, and the thickness of the tunnel is 0.3 m. The interface is set between the tunnel and the soil, and the friction angle and shear stiffness are set to 15° and 63 MPa, respectively [9]. A Rayleigh damping ratio of 5% is used in this study.
Fig. 1. Model configuration
3.2. Constitutive model and input motions The behaviour of the liquefiable deposits is simulated using the FLAC build-in Finn model. The relationship between the cyclic shear-strain amplitude (γ) and the increment in the volume decrease ( ΔεVd ) is determined by the following empirical equation [32]:
2 C3εvd Δεvd C1 ( -C2 εvd ) C4 εvd
(5)
7
where C1, C2, C3 and C4 are model parameters. Byrne [33] proposed a modified two-parameter model based on equation (1); the two-parameter model is expressed as:
vd
c1 exp( c2
vd )
(6)
where c1 and c2 are constants related to the relative density (Dr) of soil, and the expression of c1 and c2 is as follows:
c1 7600(Dr )2.5 c2
(7)
0.4 c1
(8)
The properties of the soil include the unit weight soil = 15 kN/m3, shear modulus G = 2.0 × 104 kPa, and Poisson’s ratio 𝜐 = 0.23 [12]. The cohesion and dilation angle of the soil are set to 0. A series of sinusoidal accelerations with different peak acceleration (PA), frequency (f) and duration (T) values are selected as the input motions, and these input motions are applied at the base of the model. In this study, a total of 1300 analyses are performed to study the effect of seven parameters: the relative density (Dr) and friction angle (φ) of soil, frequency (f), amplitude (PA) and duration (T) of shaking motion, embedded depth (H) and diameter (D) of tunnel. Table 1 shows the statistical parameters of the input variables and their ranges considered in the model for the generation of the numerical database. A total of 960 data points in the numerical dataset are randomly selected as the training dataset, while the remaining data are the testing dataset.
Table 1 Statistical parameters of the data Dr (%)
φ (°)
f (Hz)
PA (g)
8
H (m)
D (m)
T (s)
Min.
30.000
20.000
0.500
0.050
3.330
3.330
2.000
Max.
60.000
50.000
2.000
0.200
20.000
10.000
10.000
Training
Mean.
38.419
25.343
1.045
0.130
8.523
6.272
5.863
set
Std. Dev.
11.863
6.457
0.296
0.047
3.599
2.226
2.737
Min.
30.000
20.000
0.500
0.050
3.330
3.330
2.000
Max.
60.000
50.000
2.000
0.200
20.000
10.000
10.000
Testing
Mean.
38.922
25.183
1.055
0.131
8.679
6.302
5.862
set
Std. Dev.
11.963
6.753
0.324
0.047
3.773
2.221
2.756
3.3. Validation of the finite difference method model The results of the centrifuge test and numerical simulation conducted by Chian et al. [11] are selected to demonstrate the accuracy of the established numerical model. The diameter of the prototype tunnel is 5 m, and the buried depth is 7.5 m. The properties of the lining include the elastic modulus E = 3.0 × 107 kPa, the unit weight str = 26.5 kN/m3, and thickness t = 0.35 m. The soil is characterized by a unit weight of 18.6 kN/m3, a shear modulus of 5.5 × 103 kPa, a Poisson’s ratio of 0.34, an internal friction angle of 33°, a permeability of 1.0 × 10-3 m/s and a cohesion of 0 kPa. The sine wave is applied at the base of the model, and the frequency, peak acceleration and duration of the sine wave is 0.75 Hz, 0.1 g and 27 s, respectively. A comparison between the numerical simulation results from this study and the results from the literature is shown in Fig. 2. The error between the centrifuge result and numerical simulation is approximately 6%, indicating that the numerical model can accurately capture the liquefaction-induced uplift displacement.
9
Uplift displacement (cm)
80 Present Study (Numerical) Chian et al. (Numerical) Chian et al. (Experimental)
60 40 20 0 0
5
10
15
20
25
30
35
Time (s) Fig. 2. Comparison between the results of the simulation and centrifuge measurements
4. Results and discussion 4.1. Results The coefficient of determination (R2) is selected as the criterion to describe the performance of the model in this study. To evaluate the capability of the presented models, the numerical data are separated into a training dataset and a testing dataset. Fig. 3 and Fig. 4 depict the performance of the SVM model and the ANN model, respectively. For the SVM model, the prediction of the training data is better than that of the testing dataset. Specifically, the R2 values for the training data and testing data are 0.9512 and 0.9072, respectively. For the ANN model, the prediction for training data (i.e., R2 = 0.9560) exhibits slightly better performance than that for the testing data (i.e., R2 = 0.9379). Although the performance of testing data for ANN model is better than that for SVM model, but a series of parameters can affect the prediction in ANN model, such as the number of hidden nodes and the number of training epochs. It is difficult to determine these parameters for the ANN model. However, the SVM model can be optimized through linearly constrained quadratic programming [30]; thus, the SVM model can interpret the input-output relationship.
10
200
200
Predicted value (cm)
Predicted value (cm)
SVM - Testing: R2 = 0.9072 Reference line 150
100
50
150
100
50
SVM - Training: R2 = 0.9512 Reference line 0
0
50
100
150
0
200
0
50
100
150
Training value (cm)
Testing value (cm)
(a) Training dataset
(b) Testing dataset
200
Fig. 3. Performance of the support vector machine (SVM) model
200
200
Predicted value (cm)
Predicted value (cm)
ANN - Testing: R2 = 0.9379 Reference line 150
100
50
150
100
50
ANN - Training: R2 = 0.956 Reference line 0
0
50
100
150
0
200
0
50
100
150
Training value (cm)
Testing value (cm)
(a) Training dataset
(b) Testing dataset
200
Fig. 4. Performance of the artificial neural network (ANN) model
For the purpose of evaluating the performance of the established SVM and ANN models, 37 well-documented centrifuge test results [7, 11] are used to assess the applicability of the previous models. As shown in the Table 2, the centrifuge tests include a liquefiable Huston sand layer with a relative density of 45% and an internal friction angle of soil of 33°. The prototype buried depth and diameter of the tunnel are in the range of 1.425 m – 7.5 m and 0.95 m – 5 m, respectively. The input motion is represented by sine waves with a frequency of 0.75 Hz and a peak acceleration of 0.1 g – 0.22 g. The duration of the earthquake is between 5 s and 50 s. The measured uplift displacement (Uc) is between 12 cm and 89.4 cm. Fig. 5 presents comparisons 11
between the model predictions and the uplift displacement measured in the centrifuge tests. The R2 values for the SVM and ANN models are 0.8983 and 0.8662, respectively, indicating that the SVM model shows better performance than the ANN model. The SVM model performs better with respect to the prediction of the centrifuge results. This improved performance may because the SVM model is able to capture the underlying relationship between the inputs and output rather than just fitting data, and the SVM model is optimized to allow better generalization.
Table 2 Centrifuge data used for validation Dr
φ
f
PA
D
H
T
Uc
(%)
(°)
(Hz)
(g)
(m)
(m)
(s)
(cm)
45
33
0.75
0.22
5.5
5
5
22.1
45
33
0.75
0.22
5.5
5
10
37.3
45
33
0.75
0.22
5.5
5
15
51.7
45
33
0.75
0.22
5.5
5
20
65.3
45
33
0.75
0.22
5.5
5
25
77.2
45
33
0.75
0.22
5.5
5
30
82
45
33
0.75
0.22
5.5
5
35
82
45
33
0.75
0.22
5.5
5
12
45
45
33
0.75
0.1
7.5
5
10
12
45
33
0.75
0.1
7.5
5
15
19.8
45
33
0.75
0.1
7.5
5
20
31.8
Chian and
45
33
0.75
0.1
7.5
5
25
41
Madabhushi
45
33
0.75
0.1
7.5
5
30
43
[7]
45
33
0.75
0.1
7.5
5
35
43
45
33
0.75
0.1
7.5
5
17
25
45
33
0.75
0.22
7.5
5
5
13.2
45
33
0.75
0.22
7.5
5
10
33.8
45
33
0.75
0.22
7.5
5
15
45
45
33
0.75
0.22
7.5
5
20
53
45
33
0.75
0.22
7.5
5
25
59
45
33
0.75
0.22
7.5
5
30
62
45
33
0.75
0.22
7.5
5
35
63
45
33
0.75
0.22
7.5
5
7
26
45
33
0.75
0.22
5.5
5
25
81.6
45
33
0.75
0.22
7.5
5
25
67.6
Chian and
45
32
0.75
0.22
5.5
5
25
43.6
Madabhushi
45
32
0.75
0.22
7.5
5
25
33.7
[34]
45
33
0.75
0.22
5.5
5
25
89.4
12
Reference
45
33
0.75
0.22
7.5
5
25
55.9
45
33
0.75
0.23
7.5
5
40
75.4
45
32
0.75
0.23
5.5
5
40
44.7
45
33
0.75
0.08
2.13
1.42
25
15.4
45
33
0.75
0.09
2.13
1.42
50
35.7
Chian and
45
33
0.75
0.09
2.84
1.42
25
26.9
Madabhushi
45
33
0.75
0.17
4.995
3.33
18
18.2
[34]
45
33
0.75
0.17
1.425
0.95
18
19.3
45
33
0.75
0.18
1.425
0.95
25
23.6
45
33
0.75
0.18
1.425
0.95
36
28.9
100 SVM - Prediction: R2 = 0.8983 Reference line
80
Predicted value (cm)
Predicted value (cm)
100
60 40 20 0
0
20
40
60
80
80 60 40 20 0
100
Centrifuge result (cm)
ANN - Prediction: R2 = 0.8662 Reference line
0
20
40
60
80
100
Centrifuge result (cm)
(a) Prediction of SVM
(b) Prediction of ANN
Fig. 5. Comparation between the model predicted values and centrifuge results
4.2. Discussion Fig. 6 shows the bar graphs comparing the mean absolute error (MAE) and root mean squared error (RMSE) for the testing dataset of both models. The MAE measures the variation in the error term by term and eliminates the importance of large errors; the RMSE value focuses more on large errors than on small errors. The ANN model has lower MAE and RMSE values, demonstrating that the ANN model is preferable for estimating the uplift displacement of the tunnel for the numerical testing data.
13
MAE / RMSE Value
10 8
Testing data ANN SVM
6 4 2 0
MAE
RMSE
Fig. 6. Comparison of MAE and RMSE values from the ANN and SVM models
Fig. 7 illustrates the cumulative probability of the ratio between the predicted uplift displacement (Up) and the numerical uplift displacement (Un) of the testing dataset. The model has an adequate prediction when Up /Un is equal to 1, while Up /Un > 1.0 indicates overprediction and underprediction when Up /Un < 1.0. The cumulative probability P50 and P90 values for the SVM model are 0.995 and 1.519, respectively, while those values for the ANN model are 0.961 and 1.231, respectively. The P50 values for both models are close to 1.0, indicating that both models exhibit good performance at a cumulative probability of 50%. The P90 values for both models are larger than 1.0, indicating the overprediction of these models. The P90 value for the ANN model is lower than that for the SVM model, demonstrating that the ANN model generally demonstrates better performance than the SVM model for the testing data.
14
2.0
Testing data ANN SVM
1.8 1.6
Up / Un
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
0
10
20
30
40
50
60
70
80
90 100
Cumulative probability (%)
Fig. 7. Cumulative probability of Up/Un for the SVM and ANN models
Fig. 8 shows the relative importance of the input parameters. The sensitivity analysis was carried out by partitioning the hidden-output connection weights into components connected with each input parameter [35, 36] in the ANN model. As the figure depicts, the characteristics of the ground motions (duration, frequency and peak acceleration) are the most significant factors affecting the uplift displacement of the tunnel, and the earthquake duration is the most critical parameter. In the case of the tunnel geometry, the diameter and embedded depth of the tunnel also have an important effect on the uplift displacement, and the influence of tunnel diameter is slightly larger than the embedded depth. From the perspective of engineering design, more attention should be paid to these two features of tunnels in liquefiable deposits. The properties of the liquefiable layer have a moderate effect on the uplift displacement in this study. This finding agrees with a previous study [12].
15
30
Sensitivity (%)
25 20 15 10 5 0
Dr
φ
f
PA
T
H
D
Input Parameters Fig. 8. Feature sensitivity analysis by ANN
5. Conclusions This paper presents an SVM model and an ANN model for predicting the liquefaction-induced uplift displacement of a tunnel. The training and testing processes of both models use the artificial dataset based on a numerical simulation. The performance of the presented models is evaluated via statistical parameters, such as the coefficient of determination (R2), the MAE and the RMSE. The ANN model exhibits a better performance regarding the numerical data, but the SVM model better generalizes the centrifuge results. A relative importance analysis is performed to assess the sensitivity of the input variables. The characteristics of the ground motions are the most significant factors, especially the duration of shaking. The tunnel diameter and embedded depth clearly affect the uplift displacement (attention should be placed on these two features of tunnels in liquefiable deposits), and the relative density and friction angle of the liquefiable layer have a moderate effect. The predictions of the established ANN and SVM models are generally consistent with the 16
centrifuge results, indicating that these models can be used as an alternative approach to preliminarily estimate the liquefaction-induced uplift of tunnels.
17
Acknowledgements This research was funded by the National Natural Science Foundation of China (No. 51708405 and 41630641). The authors appreciate the financial support.
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earthquake induced uplift displacement of tunnels using multivariate adaptive regression splines, Comput Geotech, 113 (2019). [13] J. Koseki, Matsuo, O., and Koga, Y., Uplift behavior of underground structures caused by liquefaction of surrounding soil during earthquake, Soils and Foundations, 37(1) (1997) 97-108. [14] T. Tobita, G.-C. Kang, S. Iai, Estimation of Liquefaction-Induced Manhole Uplift Displacements and Trench-Backfill Settlements, Journal of Geotechnical and Geoenvironmental Engineering, 138 (2012) 491-499. [15] W. Zhang, A.T.C. Goh, Multivariate adaptive regression splines for analysis of geotechnical engineering systems, Comput Geotech, 48 (2013) 82-95. [16] S. Suman, S.K. Das, R. Mohanty, Prediction of friction capacity of driven piles in clay using artificial intelligence techniques, International Journal of Geotechnical Engineering, 10 (2016) 469-475. [17] A.T.C. Goh, W.G. Zhang, An improvement to MLR model for predicting liquefaction-induced lateral spread using multivariate adaptive regression splines, Engineering Geology, 170 (2014) 1-10. [18] S.K. Das, S. Suman, Prediction of Lateral Load Capacity of Pile in Clay Using Multivariate Adaptive Regression Spline and Functional Network, Arabian Journal for Science and Engineering, 40 (2015) 1565-1578. [19] W. Zhang, A.T.C. Goh, Multivariate adaptive regression splines and neural network models for prediction of pile drivability, Geoscience Frontiers, 7 (2016) 45-52. [20] W. Zhang, C. Wu, Y. Li, L. Wang, P. Samui, Assessment of pile drivability using random forest regression and multivariate adaptive regression splines, Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, (2019) 1-14. [21] P. Samui, Prediction of friction capacity of driven piles in clay using the support vector machine, Can Geotech J, 45 (2008) 288-295. [22] A.T.C. Goh, W. Zhang, Y. Zhang, Y. Xiao, Y. Xiang, Determination of earth pressure balance tunnel-related maximum surface settlement: a multivariate adaptive regression splines approach, B Eng Geol Environ, 77 (2018) 489-500. [23] G. Zheng, X.Y. Yang, H.Z. Zhou, Y.M. Du, J.Y. Sun, X.X. Yu, A simplified prediction method for evaluating tunnel displacement induced by laterally adjacent excavations, Comput Geotech, 95 (2018) 119-128. [24] W. Zhang, Y. Zhang, A.T.C. Goh, Multivariate adaptive regression splines for inverse analysis of soil and wall properties in braced excavation, Tunn Undergr Sp Tech, 64 (2017) 24-33. [25] W.G. Zhang, R.H. Zhang, W. Wang, F. Zhang, A.T.C. Goh, A Multivariate Adaptive Regression Splines model for determining horizontal wall deflection envelope for braced excavations in clays, Tunn Undergr Sp Tech, 84 (2019) 461-471. [26] A.T.C. Goh, Y.M. Zhang, R.H. Zhang, W.G. Zhang, Y. Xiao, Evaluating stability of underground entry-type excavations using multivariate adaptive regression splines and logistic regression, Tunn Undergr Sp Tech, 70 (2017) 148-154. [27] Wengang Zhang, Runhong Zhang, Chongzhi Wu, Anthony Teck Chee Goh, Suzanne Lacasse, Zhongqiang Liu, H. Liu, State-of-the-art review of soft computing applications in underground excavations, Geoscience Frontiers, Accepted (2019).
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[28] V. Rodriguez-Galiano, M. Sanchez-Castillo, M. Chica-Olmo, M. Chica-Rivas, Machine learning predictive models for mineral prospectivity: An evaluation of neural networks, random forest, regression trees and support vector machines, Ore Geology Reviews, 71 (2015) 804-818. [29] K.P. Bennett, O.L. Mangasarian, Robust linear programming discrimination of two linearly inseparable sets, Optimization Methods and Software, 1 (1992) 23-34. [30] P. Samui, Predicted ultimate capacity of laterally loaded piles in clay using support vector machine, Geomechanics and Geoengineering, 3 (2008) 113-120. [31] FLAC—Fast Lagrangian analysis of continua, version 8.0, in, Itasca Consulting Group, Inc, Minneapolis. [32] W.D.L. Finn, G.R. Martin, K.W. Lee, An Effective Stress Model for Liquefaction, Journal of the Geotechnical Engineering Division, 103 (1977) 517-533. [33] B.P. M, A cyclic shear-volume coupling and pore pressure model for sand, in: Second Int. Conf. Recent Advances in Geotechnical Earthquake Engg. and Soil Dynamics, St. Louis, Missouri, 1991, pp. 47-55. [34] S.C. Chian, K. Tokimatsu, S.P.G. Madabhushi, Soil Liquefaction Induced Uplift of Underground Structures: Physical and Numerical Modeling, Journal of Geotechnical and Geoenvironmental Engineering, 140 (2014) 04014057. [35] G.D. Garson, Interpreting neural-network connection weights, Artif. Intell. Expert, 6(7) (1991) 47-51. [36] A.T.C. Goh, Seismic Liquefaction Potential Assessed by Neural Networks, Journal of Geotechnical and Geoenvironmental Engineering, 120(9) (1994) 1467-1480.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
21
S2467-9674(19)30101-1 https://doi.org/10.1016/j.undsp.2019.12.002 UNDSP 127
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Please cite this article as: G. Zheng, W. Zhang, W. Zhang, H. Zhou, P. Yang, Neural network and support vector machine models for the prediction of the liquefaction-induced uplift displacement of tunnels, Underground Space (2020), doi: https://doi.org/10.1016/j.undsp.2019.12.002
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Neural network and support vector machine models for the prediction of the liquefaction-induced uplift displacement of tunnels Gang Zhenga,b,c, Wenbin Zhanga,b, Wengang Zhangd, Haizuo Zhoua,b,c*, Pengbo Yanga,b a
School of Civil Engineering, Tianjin University, Tianjin 300072, China Key Laboratory of Coast Civil Structure Safety, Tianjin University, Ministry of Education, Tianjin 300072, China c State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China d National Joint Engineering Research Center of Geohazards Prevention in Reservoir Area Environment, School of Civil Engineering, Chongqing University, Chongqing 400045, China b
*Corresponding
author (Haizuo Zhou): Tel./fax: 022 27402341 E-mail address: [email protected]
Abstract Tunnels buried in liquefiable deposits are vulnerable to liquefaction-induced uplift damage during earthquakes. This paper presents support vector machine (SVM) and artificial neural network (ANN) models for predicting the liquefaction-induced uplift displacement of tunnels based on artificial databases generated by the finite difference method (FDM). The performance of the SVM and ANN models are assessed using statistical parameters, including the coefficient of determination (R2), the mean absolute error (MAE), and the root mean squared error (RMSE). The applications for the above-mentioned approach for predicting the uplift displacement of tunnels are compared and discussed. A relative importance analysis is adopted to quantify the sensitivity of each input variable. The precision of the presented models is demonstrated using the centrifuge test results available from previous studies. 1
Keywords: artificial neural network, support vector machine, liquefaction, uplift displacement
2
1. Introduction Underground structures buried in liquefiable deposits may suffer considerable uplift damage during major earthquakes. In the 2011 Great East Japan Earthquake, the underground pipeline was subjected to a significant uplift displacement in reclaimed liquefiable deposits, and over 112 manholes were uplifted [1]. The phenomenon was also observed in the 1964 Niigata Earthquake, the 1989 Loma Prieta Earthquake and the 1995 Kobe Earthquake [2-4]. The submerged unit weight of underground structures is lower than the surrounding soil; thus, the underground structure tends to uplift, and under static conditions, the weight and shear strength of the soil resist the uplift. However, when an earthquake occurs, the liquefiable soil around the underground structures loses most of its shear strength, and the buoyant force may be larger than the resistant force; the unbalanced force thus leads to the floating of underground structures. For the purpose of evaluating the uplift damage, centrifuge tests were conducted to verify the uplift behaviour of underground structures (e.g., buried tunnels, pipelines and manholes) [5-7]. Numerical simulations were performed to study the potential uplift damage of real tunnels such as the BART Tunnel [8] and George Masset Tunnel [2]. Furthermore, a parametric analysis was conducted to study potential factors that affect the uplift of the tunnels. Azadi et al. [9, 10] numerically analysed the effect of the friction angle and the relative density of soil, the diameter and the buried depth of the tunnel and the frequency and amplitude of the input motions. Chian et al. [11] analysed the effect of the magnitude of the input motions, soil conditions, such as the relative density, and the structural configuration, such as the diameter. Zheng et al. [12] established a multivariate adaptive regression splines (MARS) model based on numerical artificial data and analysed the relative importance of nine parameters, including the characteristics of the ground motion, structure and soil condition. These previous studies showed that the liquefaction-induced uplift displacement is greatly influenced by the relative density and the friction angle of the soil, the diameter and buried depth of the tunnel and the amplitude, frequency and duration of the input 3
motion. Therefore, these parameters are considered in this study. Koseki et al. [13] proposed a factor of safety against uplift to evaluate the potential of tunnel floating [5, 6, 14]. The factor of safety can qualitatively evaluate the uplift behaviour of tunnels, but a quantitative evaluation method has rarely been reported. The artificial intelligence methods can reveal the relationship between the complex input parameters and target output and has been applied in geotechnical engineering problems [15, 16], such as the prediction of liquefaction-induced lateral spreading [17], lateral capacity [18], drivability [19, 20] friction capacity of piles [16, 21], tunnelling-induced displacement [22] and excavation-induced displacement [23-27]. This paper presents the prediction model for the liquefaction-induced uplift of underground structures by means of the support vector machine (SVM) and artificial neural network (ANN) models. The training and testing of the model are based on a numerical database, and the model validation is conducted through the centrifuge test results from previous studies.
2. Methodology 2.1. Artificial neural network (ANN) The ANN is a common method for nonlinear regression/classification. A feed-forward propagation neural network is adopted in this study. The units are generally placed as three layers, including the input layer, hidden layer/layers and output layer in a neural network. In the generation of the ANN model, the model is first trained using a training data set. The model training aims to map the input variables to the output value by determining the optimal connection weights and biases through the back-propagation procedure. Two phases of data flow are included in the back-propagation algorithm. In the first phase, the input data propagate forward from the input layer to the output layer, and an actual output value is produced. In the second phase, the errors between the target values and actual outputs are propagated backwards from the output layer to the previous layers, and the connection weights 4
are updated to reduce the errors between the actual output values and the target output values [19]. Generally, the number of hidden neurons is typically determined by selecting the smallest number of neurons that acquire better network performance (i.e., judged by the coefficient of determination (R2) of the testing data set). The neural network established in this study is used with the Levenberg-Marquardt back-propagation algorithm based on MATLAB software.
2.2. Support vector machine (SVM) The SVM model is a statistical learning algorithm where the prediction error and model complexity are simultaneously minimized [21]; this method has been extended to regression problems. The basic idea of the SVM method is to transform the input features into a higher-dimensional space to linearly separate the data into two classes by a hyper-plane [28] which maximized the margin and minimized the error. The regression model can be expressed as follows:
f X WT X b
(1)
where X is the training dataset; ϕ is a nonlinear function to map the low-dimensional input features into a high-dimensional space; W is the unknown adjustable vector that is normal to the hyper-plane; and b is the bias of the hyper-plane. Some misclassification errors may allow for the SVM model [29], and the following constrained conditions are adopted in the SVM model.
yi (W x i ) b i ,
i 1, 2,..., k,
(W x i ) b yi i* ,
i 1, 2,..., k,
i 0 and i* 0,
i 1, 2,..., k,
(2)
where ε is the maximum error allowed, and ξi and ξi* are the slack variables that 5
quantified the deviation of the output. The optimization problem of the model can be solved using the Lagrangian multipliers method. Considering that the linear regression may not be appropriate for the input data, the input features have to be mapped into a high-dimensional space by ϕ. The kernel function (defined as the dot product of the transformed input vectors (ϕ(xi) ∙ ϕ(xj)) is introduced to reduce the computational demand [30]. The previous model can be rewritten as follows:
N
f (x) ( i i* )K (x i x j ) b
(3)
i 1
where the αi and αi* are the Lagrangian multipliers and K (xi, xj) is the kernel function. Some common kernel functions, such as the Gaussian function, polynomial function or radial basis function, are typically considered in the SVM model. The Gaussian kernel is applied in this study. The equation of the Gaussian kernel is as follows:
2
K (x i , x j ) exp( x i x j )
(4)
In the present study, the SVM analysis is performed using MATLAB software. 3. Numerical analyses 3.1. Model geometry and boundary conditions Numerical analyses are performed using Fast Lagrangian Analysis of Continua (FLAC version 8.0) software [31]. A series of analyses is performed to study the effect of structural configurations, soil conditions and ground motion characteristics on the performance of the liquefaction-induced uplift displacement of tunnels. As shown in Fig. 1, the height of the model soil is 30 m, and the water table is located at the model surface. The width of the model is 50 m, and the free field boundary 6
condition is adopted in the model to minimize the influence of the boundary effects. The element size is set to 1 m to allow the wave of the frequencies considered in this study to transmit through the model. The tunnel with diameter D buried at a depth of H (from the centre of the tunnel to the ground surface). In this study, the tunnel is modelled as a linearly elastic material using a beam element, the elastic modulus is 2.2 × 107 kPa, the unit weight of the tunnel is 24 kN/m3, and the thickness of the tunnel is 0.3 m. The interface is set between the tunnel and the soil, and the friction angle and shear stiffness are set to 15° and 63 MPa, respectively [9]. A Rayleigh damping ratio of 5% is used in this study.
Fig. 1. Model configuration
3.2. Constitutive model and input motions The behaviour of the liquefiable deposits is simulated using the FLAC build-in Finn model. The relationship between the cyclic shear-strain amplitude (γ) and the increment in the volume decrease ( ΔεVd ) is determined by the following empirical equation [32]:
2 C3εvd Δεvd C1 ( -C2 εvd ) C4 εvd
(5)
7
where C1, C2, C3 and C4 are model parameters. Byrne [33] proposed a modified two-parameter model based on equation (1); the two-parameter model is expressed as:
vd
c1 exp( c2
vd )
(6)
where c1 and c2 are constants related to the relative density (Dr) of soil, and the expression of c1 and c2 is as follows:
c1 7600(Dr )2.5 c2
(7)
0.4 c1
(8)
The properties of the soil include the unit weight soil = 15 kN/m3, shear modulus G = 2.0 × 104 kPa, and Poisson’s ratio 𝜐 = 0.23 [12]. The cohesion and dilation angle of the soil are set to 0. A series of sinusoidal accelerations with different peak acceleration (PA), frequency (f) and duration (T) values are selected as the input motions, and these input motions are applied at the base of the model. In this study, a total of 1300 analyses are performed to study the effect of seven parameters: the relative density (Dr) and friction angle (φ) of soil, frequency (f), amplitude (PA) and duration (T) of shaking motion, embedded depth (H) and diameter (D) of tunnel. Table 1 shows the statistical parameters of the input variables and their ranges considered in the model for the generation of the numerical database. A total of 960 data points in the numerical dataset are randomly selected as the training dataset, while the remaining data are the testing dataset.
Table 1 Statistical parameters of the data Dr (%)
φ (°)
f (Hz)
PA (g)
8
H (m)
D (m)
T (s)
Min.
30.000
20.000
0.500
0.050
3.330
3.330
2.000
Max.
60.000
50.000
2.000
0.200
20.000
10.000
10.000
Training
Mean.
38.419
25.343
1.045
0.130
8.523
6.272
5.863
set
Std. Dev.
11.863
6.457
0.296
0.047
3.599
2.226
2.737
Min.
30.000
20.000
0.500
0.050
3.330
3.330
2.000
Max.
60.000
50.000
2.000
0.200
20.000
10.000
10.000
Testing
Mean.
38.922
25.183
1.055
0.131
8.679
6.302
5.862
set
Std. Dev.
11.963
6.753
0.324
0.047
3.773
2.221
2.756
3.3. Validation of the finite difference method model The results of the centrifuge test and numerical simulation conducted by Chian et al. [11] are selected to demonstrate the accuracy of the established numerical model. The diameter of the prototype tunnel is 5 m, and the buried depth is 7.5 m. The properties of the lining include the elastic modulus E = 3.0 × 107 kPa, the unit weight str = 26.5 kN/m3, and thickness t = 0.35 m. The soil is characterized by a unit weight of 18.6 kN/m3, a shear modulus of 5.5 × 103 kPa, a Poisson’s ratio of 0.34, an internal friction angle of 33°, a permeability of 1.0 × 10-3 m/s and a cohesion of 0 kPa. The sine wave is applied at the base of the model, and the frequency, peak acceleration and duration of the sine wave is 0.75 Hz, 0.1 g and 27 s, respectively. A comparison between the numerical simulation results from this study and the results from the literature is shown in Fig. 2. The error between the centrifuge result and numerical simulation is approximately 6%, indicating that the numerical model can accurately capture the liquefaction-induced uplift displacement.
9
Uplift displacement (cm)
80 Present Study (Numerical) Chian et al. (Numerical) Chian et al. (Experimental)
60 40 20 0 0
5
10
15
20
25
30
35
Time (s) Fig. 2. Comparison between the results of the simulation and centrifuge measurements
4. Results and discussion 4.1. Results The coefficient of determination (R2) is selected as the criterion to describe the performance of the model in this study. To evaluate the capability of the presented models, the numerical data are separated into a training dataset and a testing dataset. Fig. 3 and Fig. 4 depict the performance of the SVM model and the ANN model, respectively. For the SVM model, the prediction of the training data is better than that of the testing dataset. Specifically, the R2 values for the training data and testing data are 0.9512 and 0.9072, respectively. For the ANN model, the prediction for training data (i.e., R2 = 0.9560) exhibits slightly better performance than that for the testing data (i.e., R2 = 0.9379). Although the performance of testing data for ANN model is better than that for SVM model, but a series of parameters can affect the prediction in ANN model, such as the number of hidden nodes and the number of training epochs. It is difficult to determine these parameters for the ANN model. However, the SVM model can be optimized through linearly constrained quadratic programming [30]; thus, the SVM model can interpret the input-output relationship.
10
200
200
Predicted value (cm)
Predicted value (cm)
SVM - Testing: R2 = 0.9072 Reference line 150
100
50
150
100
50
SVM - Training: R2 = 0.9512 Reference line 0
0
50
100
150
0
200
0
50
100
150
Training value (cm)
Testing value (cm)
(a) Training dataset
(b) Testing dataset
200
Fig. 3. Performance of the support vector machine (SVM) model
200
200
Predicted value (cm)
Predicted value (cm)
ANN - Testing: R2 = 0.9379 Reference line 150
100
50
150
100
50
ANN - Training: R2 = 0.956 Reference line 0
0
50
100
150
0
200
0
50
100
150
Training value (cm)
Testing value (cm)
(a) Training dataset
(b) Testing dataset
200
Fig. 4. Performance of the artificial neural network (ANN) model
For the purpose of evaluating the performance of the established SVM and ANN models, 37 well-documented centrifuge test results [7, 11] are used to assess the applicability of the previous models. As shown in the Table 2, the centrifuge tests include a liquefiable Huston sand layer with a relative density of 45% and an internal friction angle of soil of 33°. The prototype buried depth and diameter of the tunnel are in the range of 1.425 m – 7.5 m and 0.95 m – 5 m, respectively. The input motion is represented by sine waves with a frequency of 0.75 Hz and a peak acceleration of 0.1 g – 0.22 g. The duration of the earthquake is between 5 s and 50 s. The measured uplift displacement (Uc) is between 12 cm and 89.4 cm. Fig. 5 presents comparisons 11
between the model predictions and the uplift displacement measured in the centrifuge tests. The R2 values for the SVM and ANN models are 0.8983 and 0.8662, respectively, indicating that the SVM model shows better performance than the ANN model. The SVM model performs better with respect to the prediction of the centrifuge results. This improved performance may because the SVM model is able to capture the underlying relationship between the inputs and output rather than just fitting data, and the SVM model is optimized to allow better generalization.
Table 2 Centrifuge data used for validation Dr
φ
f
PA
D
H
T
Uc
(%)
(°)
(Hz)
(g)
(m)
(m)
(s)
(cm)
45
33
0.75
0.22
5.5
5
5
22.1
45
33
0.75
0.22
5.5
5
10
37.3
45
33
0.75
0.22
5.5
5
15
51.7
45
33
0.75
0.22
5.5
5
20
65.3
45
33
0.75
0.22
5.5
5
25
77.2
45
33
0.75
0.22
5.5
5
30
82
45
33
0.75
0.22
5.5
5
35
82
45
33
0.75
0.22
5.5
5
12
45
45
33
0.75
0.1
7.5
5
10
12
45
33
0.75
0.1
7.5
5
15
19.8
45
33
0.75
0.1
7.5
5
20
31.8
Chian and
45
33
0.75
0.1
7.5
5
25
41
Madabhushi
45
33
0.75
0.1
7.5
5
30
43
[7]
45
33
0.75
0.1
7.5
5
35
43
45
33
0.75
0.1
7.5
5
17
25
45
33
0.75
0.22
7.5
5
5
13.2
45
33
0.75
0.22
7.5
5
10
33.8
45
33
0.75
0.22
7.5
5
15
45
45
33
0.75
0.22
7.5
5
20
53
45
33
0.75
0.22
7.5
5
25
59
45
33
0.75
0.22
7.5
5
30
62
45
33
0.75
0.22
7.5
5
35
63
45
33
0.75
0.22
7.5
5
7
26
45
33
0.75
0.22
5.5
5
25
81.6
45
33
0.75
0.22
7.5
5
25
67.6
Chian and
45
32
0.75
0.22
5.5
5
25
43.6
Madabhushi
45
32
0.75
0.22
7.5
5
25
33.7
[34]
45
33
0.75
0.22
5.5
5
25
89.4
12
Reference
45
33
0.75
0.22
7.5
5
25
55.9
45
33
0.75
0.23
7.5
5
40
75.4
45
32
0.75
0.23
5.5
5
40
44.7
45
33
0.75
0.08
2.13
1.42
25
15.4
45
33
0.75
0.09
2.13
1.42
50
35.7
Chian and
45
33
0.75
0.09
2.84
1.42
25
26.9
Madabhushi
45
33
0.75
0.17
4.995
3.33
18
18.2
[34]
45
33
0.75
0.17
1.425
0.95
18
19.3
45
33
0.75
0.18
1.425
0.95
25
23.6
45
33
0.75
0.18
1.425
0.95
36
28.9
100 SVM - Prediction: R2 = 0.8983 Reference line
80
Predicted value (cm)
Predicted value (cm)
100
60 40 20 0
0
20
40
60
80
80 60 40 20 0
100
Centrifuge result (cm)
ANN - Prediction: R2 = 0.8662 Reference line
0
20
40
60
80
100
Centrifuge result (cm)
(a) Prediction of SVM
(b) Prediction of ANN
Fig. 5. Comparation between the model predicted values and centrifuge results
4.2. Discussion Fig. 6 shows the bar graphs comparing the mean absolute error (MAE) and root mean squared error (RMSE) for the testing dataset of both models. The MAE measures the variation in the error term by term and eliminates the importance of large errors; the RMSE value focuses more on large errors than on small errors. The ANN model has lower MAE and RMSE values, demonstrating that the ANN model is preferable for estimating the uplift displacement of the tunnel for the numerical testing data.
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MAE / RMSE Value
10 8
Testing data ANN SVM
6 4 2 0
MAE
RMSE
Fig. 6. Comparison of MAE and RMSE values from the ANN and SVM models
Fig. 7 illustrates the cumulative probability of the ratio between the predicted uplift displacement (Up) and the numerical uplift displacement (Un) of the testing dataset. The model has an adequate prediction when Up /Un is equal to 1, while Up /Un > 1.0 indicates overprediction and underprediction when Up /Un < 1.0. The cumulative probability P50 and P90 values for the SVM model are 0.995 and 1.519, respectively, while those values for the ANN model are 0.961 and 1.231, respectively. The P50 values for both models are close to 1.0, indicating that both models exhibit good performance at a cumulative probability of 50%. The P90 values for both models are larger than 1.0, indicating the overprediction of these models. The P90 value for the ANN model is lower than that for the SVM model, demonstrating that the ANN model generally demonstrates better performance than the SVM model for the testing data.
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2.0
Testing data ANN SVM
1.8 1.6
Up / Un
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
0
10
20
30
40
50
60
70
80
90 100
Cumulative probability (%)
Fig. 7. Cumulative probability of Up/Un for the SVM and ANN models
Fig. 8 shows the relative importance of the input parameters. The sensitivity analysis was carried out by partitioning the hidden-output connection weights into components connected with each input parameter [35, 36] in the ANN model. As the figure depicts, the characteristics of the ground motions (duration, frequency and peak acceleration) are the most significant factors affecting the uplift displacement of the tunnel, and the earthquake duration is the most critical parameter. In the case of the tunnel geometry, the diameter and embedded depth of the tunnel also have an important effect on the uplift displacement, and the influence of tunnel diameter is slightly larger than the embedded depth. From the perspective of engineering design, more attention should be paid to these two features of tunnels in liquefiable deposits. The properties of the liquefiable layer have a moderate effect on the uplift displacement in this study. This finding agrees with a previous study [12].
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30
Sensitivity (%)
25 20 15 10 5 0
Dr
φ
f
PA
T
H
D
Input Parameters Fig. 8. Feature sensitivity analysis by ANN
5. Conclusions This paper presents an SVM model and an ANN model for predicting the liquefaction-induced uplift displacement of a tunnel. The training and testing processes of both models use the artificial dataset based on a numerical simulation. The performance of the presented models is evaluated via statistical parameters, such as the coefficient of determination (R2), the MAE and the RMSE. The ANN model exhibits a better performance regarding the numerical data, but the SVM model better generalizes the centrifuge results. A relative importance analysis is performed to assess the sensitivity of the input variables. The characteristics of the ground motions are the most significant factors, especially the duration of shaking. The tunnel diameter and embedded depth clearly affect the uplift displacement (attention should be placed on these two features of tunnels in liquefiable deposits), and the relative density and friction angle of the liquefiable layer have a moderate effect. The predictions of the established ANN and SVM models are generally consistent with the 16
centrifuge results, indicating that these models can be used as an alternative approach to preliminarily estimate the liquefaction-induced uplift of tunnels.
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Acknowledgements This research was funded by the National Natural Science Foundation of China (No. 51708405 and 41630641). The authors appreciate the financial support.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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