Prediction of vertical displacement for a buried pipeline subjected to normal fault using a hybrid FEM-ANN approach

Abstract

Fault movement during earthquakes is a geotechnical phenomenon threatening buried pipelines and with the potential to cause severe damage to critical infrastructures. Therefore, effective prediction of pipe displacement is crucial for preventive management strategies. This study aims to develop a fast, hybrid model for predicting vertical displacement of pipe networks when they experience faulting. In this study, the complex behavior of soil and a buried pipeline system subjected to a normal fault is analyzed by using an artificial neural network (ANN) to generate predictions the behavior of the soil when different parameters of it are changed. For this purpose, a finite element model is developed for a pipeline subjected to normal fault displacements. The data bank used for training the ANN includes all the critical soil parameters (cohesion, internal friction angle, Young’s modulus, and faulting). Furthermore, a mathematical formula is presented, based on biases and weights of the ANN model. Experimental results show that the maximum error of the presented formula is 2.03%, which makes the proposed technique efficiently predict the vertical displacement of buried pipelines and hence, helps to optimize the upcoming pipeline projects.

Graphical abstract

Keywords

buried pipelines / normal Fault / finite element method / multilayer perceptron neural network / formulation

1 Introduction

Geotechnical phenomena play a critical role in infrastructure engineering, and can cause tremendous damage to underground structures and utilities such as buried pipelines. Such utilities transmit vital fluids such as oil, gas, water, or wastewater. Therefore, damage to these pipelines can lead to disruption of human life. The intensity of such potential damages depends on factors like loading type, soil native characteristics, trench specification, pipe stiffness, and hunching [1]. The environment of a buried pipeline can vary in characteristics, such as soil type and seismicity conditions. Ground rupture from seismic wave propagation and permanent ground deformation (PGD) are two crucial factors that could threaten such infrastructures [2–5]. Surface faulting has been the main cause of damage to pipelines during past earthquakes, as exampled by the 1971 San Fernando, 1994 Los Angeles, and 1995 Japan earthquakes when numerous breaks in the water, natural gas, and sewer pipelines occurred [6–8].

The importance of safety in pipelines has convinced many researchers working in this field. Consequently, several experimental, numerical, and analytical research studies have been done to assess the reliability of pipelines during earthquakes. Newmark and Hall [9] focused on the effect of PGD on buried pipelines, using analytical modeling. By considering lateral interaction and large axial strain effects, Kennedy et al. [10] extended the idea of Karamitros et al. [11] presented an analytical method to analyze stress in buried pipelines subjected to a strike-slip fault event. Later, Karamitros et al. [12] extended this study by considering a normal fault. Development of the analytical methodology proposed by Karamitros et al. [11] was continued by Trifonov and Cherniy [13], where the aim was to analyze pipeline response when subjected to normal faults, without symmetric conditions, at the fault-pipeline intersection. Trifunac [14] studied wave propagation and its effect on structures near earthquake faults, while the semi-analytical method proposed by Takada et al. [15] aimed to determine the maximum strain of steel pipelines crossing faults. Trifonov and Cherniy [16] used the plane strain plasticity theory to develop an analytical model for stress-strain investigation of buried pipelines subjected to active faults. In addition, some research focused on underground pipes subjected to compressive transverse loading. With the help of a mechanical performance evaluation, Rafiee and Habibagahi conducted a study in 2018. They studied Glass-Fiber Reinforced-Plastic pipes subjected to compressive, transverse loading and studied damage progression and failure mechanism. They achieved the extent to which pipes can withstand diametric deflection without experiencing any failure mode [17]. In another work, Rafiee and Ghorbanhosseini developed a multi-stage test scenario for analyzing the creep response of Glass Fiber Reinforced Polymer pipes experiencing compressive, transverse loading. Their study validated the efficiency of their proposed procedure, a short-term multi-stage loading test [18]. These two researchers performed another study in which the impact of structural parameters, including the thickness of the liner, lay-up orientation, and the number of layers, was evaluated [19].

Among various scientific tools, the finite element method (FEM) is a common, convenient approach that has helped researchers to study diverse topics in material engineering [20–22] and structures [23], including buried pipelines. This method can consider the nonlinearity of pipe, soil, and their interaction. FEM has been recently applied in various studies, and researchers have used it to develop and verify analytical methods or to evaluate the factors that affect the response of pipes subjected to different types of PGD [24–29]. Despite the accuracy and popularity of FEM for analyzing the response of buried pipes subjected to permanent fault displacements, it is considerably time-consuming.

Recent advances in machine learning (ML) techniques have motivated researchers to use data-driven models (DDMs) in diverse application fields. In contrast to numerical and analytical models, a DDM is constructed based on the experimental data acquired from real systems [30,31]. A wide range of linear and nonlinear models can be selected as DDM, including state-of-the-art ML models, like XGBoost [32], LightGBM [33] and CatBoost [34], and classical ML models, like support vector machine (SVM) [35], artificial neural network (ANN) [36], and K-nearest neighbors (KNN) [37]. One of the most popular ML methods is ANN, either with simple architecture or deep architecture, which has been employed for solving complex and nonlinear problems in various branches of science [38–41]. Civil engineering is not exempted from this approach, and different branches of this area of science have been involved, especially in structural engineering [42–45], transportations engineering (rail profile measurement) [46,47], material engineering [48,49], and geotechnical engineering [50–54].

One limitation in the current literature is that most studies have focused on either experimental or numerical simulations, which can limit the generalizability of the findings. Additionally, there is a lack of consensus on the appropriate analytical models and methods for predicting vertical displacement under normal faulting, which can lead to inconsistencies in the results.

Another gap relates to the limited understanding of the effects of various parameters, such as soil properties, pipeline characteristics, and fault geometry, on the prediction of vertical displacement. Hence, more research is needed to investigate the role of these parameters in predicting vertical displacement and to develop reliable and accurate prediction models that can be applied to different scenarios.

The main contribution of the current work is to address these limitations and gaps in the literature to improve the accuracy and reliability of practical predictions for pipelines by the adoption of FEM with ANN, executing jointly on Abaqus and Matlab platforms. The soil-pipeline system simulated using Abaqus was subjected to the normal fault. After parametric analysis and training of the ANN, a mathematical formula was extracted from the ANN to facilitate the analysis of soil displacement.

The rest of this paper is organized as follows. Section 2 presents a description of FEM implementation in Abaqus. In Section 3, the procedure of data bank creation is described. Section 4 explains the ANN development. In Section 5, the study results are presented. The paper is concluded in Section 6.

2 Numerical modeling of the pipeline subjected to the normal fault

The first step of this work is to model the soil, the pipeline, and their interaction. The basis of the modeling procedure in this study is the physical model introduced by Yeganeh Khaksar et al. [55–57]. To overcome the geometrical limitation of small-scale physical modeling, the authors in Ref. [55] used an innovative combination of a centrifuge and numerical modeling methods, which helped them to calibrate their model. For simulation, they utilized a powerful numerical software for modeling, Abaqus, which is widely used to simulate pipes’ response to fault displacements. Their model, in Ref. [55], considers soil, pipeline, the interaction between them, and the impact of faulting upon PGD.

The interaction of soil and pipe can be considered in terms of frictional contact. The frictional contact angle is the maximum value of the frictional static angle, which cannot be determined easily. Different estimations exist based on the properties of the contact surface, soil density, burial depth, and the values and relative displacement surface [58]. Generally, the values of the frictional angle between contact surfaces are between 20° and the value of the soil frictional angle (ϕ_soil) [59]. In this research, the frictional contact angle is taken to be ϕ_soil /2, and the internal friction angle of adjacent soil in denser regions is considered to be 35.5° (based on the direct shear test on the 191 sand). Hence, the friction coefficient of the contact surface is 0.32, which can provide an estimate of soil and pipe nonlinearity interaction.

According to the guidelines of ASCE, Eurocode 1998-4, IITK-GSDMA, and the research done by Ref. [9], a conservative limit value of 3% strain can be used as the permissible tensile strain in the analysis and design of pipelines [55].

An important point that must be considered in a modeling procedure is the determination of the loading type. Generally, there are three loading types: static, quasi-static, and dynamic. Since fault movement occurs during an earthquake, it seems to be a purely dynamic problem. However, by focusing on the fault phenomenon, it is considered by the authors that the velocity of loading/displacement is not large enough to assume the problem is purely dynamic. Therefore, the faulting model is mainly categorized as static.

Fault movement is a geotechnical phenomenon in which large deformation and displacement occur. In problems with large deformations, strain cannot be defined as a variable of displacement in the initial situation. In this situation, establishing special equations for strain and stress is needed. In this research, considering that the contact surface is changing, using arrangements to determining where the relevant meshes contact can model the contact with the interface element. Moreover, the contact can be modeled by using contact surfaces that gradually deform and displace, by using Abaqus software.

Continuous pipelines mainly break in tension and buckle in compression as a result of permanent deformations of the ground. Due to these deformations, the response of the pipe at the place of occurrence experiences bending and axial tensile forces, and in this case the mechanism of pipe failure is of the tensile type.

To prevent the lack of convergence, an explicit solver is used in finite element modeling. The current problem is categorized as “dynamic problems with large deformations,” and the studied materials have linear behavior. Moreover, the contact phenomenon related to the interaction of soil and pipe is considered here. Hence, an explicit solver is a great choice for addressing the convergence issue, since it has proven capabilities in solving both linear and nonlinear problems.

It is important to specify the type of soil and pipeline before creating the model. Yeganeh Khaksar et al. used the standard Firozkoh 191 sand and TP304 stainless steel in their study. Tab.1 shows the properties of the soil and pipe used for the proposed modeling. Since the physical model in this study is identical to the model presented in Ref. [55], we use the same properties for the soil and the pipe as used in Ref. [55] and consider this work as the baseline research. For this study, we rely on the data extracted from the baseline research. Hence, we do not consider the data as being raw data and no further optimization is needed.

The model geometry involves two parts, pipeline and soil. In the first part (pipeline), the length, diameter, thickness, and depth from the surface of the earth are 38.4, 1, 0.02, and 1.6 m, respectively. In the second part (soil), the length, width, and height are 38.4, 28, and 9.2 m, respectively. Fig.1 shows the top and detailed views of the meshed soil. As shown, the meshing should be done in a way that closer approach to the sensitive points (faulting locations), correlates with the meshing becoming more fine-grained. Since the purpose of this research is to study the response of the soil-pipeline system to the normal fault movement, vertical displacement is selected as a decision criterion. According to the specifications of the soil failure plane in the present study, it is assumed that the shear band is formed in the direction of an element column with a width of 0.16 mm [60]. The expressed width in this research for the gravity of 40 g is equal to 102 mm (standard sand of Firoozkouh). Considering this mesh width for all model elements makes the numerical analysis time consuming and practically impossible. Therefore, this width is only considered for the shear band elements and the mesh becomes coarser from the band to the sides with a specific pattern.

Displacement of various parts of the soil can be different, and therefore, two paths are selected for soil displacement analysis: the top and bottom regions of the pipe (Fig.2). For better and more realistic analysis, 30 different points along the pipe are examined for each path (Tab.2).

In this study, the modeling results simulated using Abaqus are compared with the experimental and numerical results of the baseline research [55]. The model is calibrated by using the experimental model in Ref. [55], and it is developed based on the geometry and characteristics of the soil and pipeline in the physical model of the baseline research and normal-fault simulation. To calibrate the model, we use the characteristics of standard sand 191 of Firuzkoh, which are used in the baseline research [55]. Soil parameters that are investigated include Young’s modulus, internal friction angle, and cohesion, and the values for the calibrated model are shown in Tab.1. The displacement of the fault is also considered to be 1 m. After structuring the calibrated model with Abaqus, the outputs (vertical displacement) are compared with the physical model. Finally, the displacement of the pipe in the calibrated model is compared with the baseline research model, and the results are shown in Fig.3. This figure verifies that the soil displacements in all models are similar, with an error value close to 10%.

3 Data bank production

As mentioned earlier, this study aims to analyze the response of the soil-pipeline system subjected to normal fault and develop a numerical model. Based on the direction of relative movements on two sides of a fault, there are different fault types [50]: normal, reverse, strike-slip, and oblique. Hence, it is decided to simulate the different conditions for a modeled environment in Abaqus while performing a parametric analysis. Different characteristics can affect the response of buried pipelines subjected to fault, and result in pipeline displacement when an earthquake happens [61]. Since cohesion, internal friction angle, and Young’s modulus are three crucial parameters that have the most effect on soil behavior, the authors consider these parameters, along with faulting, as variables in this study. Based on specified ranges (Tab.3), these four parameters are automatically changed based on Eqs. (1)−(4).

To create a data bank, first, we develop a tool in Matlab to invoke Abaqus and automatically change the parameters. Then MATLAB gets the outputs from Abaqus and stores them in a file. The parameters are changed based on an equation written for each parameter. This innovation increases the speed of the process. Finally, for all 4 parameters, there are 4 groups of output to be used in the neural network structure.

(1)C=(j\times 999.995)+0.1,j\in \text{[0,20],}

(2)E=(j+9)\times 1000000,j\in [0,40],

(3)U_{\text{f}}=(-j/20)-0.05,j\in [0,19],

(4)\phi =j+0.5,j\in [30,39],

where C is cohesion, E is Young’s modulus, U_f is the displacement of fault (faulting), \phi is internal friction angle, and j is a number in the interval. The automated modeling process is continued until the chosen interval for parameters is satisfied. The data bank is then ready for the analysis step.

4 Artificial neural network development for the data bank analysis

Pattern recognition, automatic control, and predictions are only few of the common applications of ANN [62,63]. Studies have demonstrated that ANNs are capable of extracting complex nonlinear relationships between dependent and independent variables after training [64,65]. The main motivation for using ANN is the nonlinear nature of the problem in this work. Other ML techniques that are based on linear programming cannot be used for this problem. Moreover, ANN can work with incomplete knowledge. With a proper data bank (which is demonstrated in Section 5, below), ANN creates a safe margin assuring the accuracy of the results. Another advantage of the ANN is that it works deterministically and does not need to be reprogrammed, making it a perfect candidate for this study.

4.1 Multi-layer perceptron

Multilayer perceptron neural network (MLPNN) is a type of ANN that uses a feedforward architecture. Generally, an MLPNN has three layers: input, hidden, and output. The input layer is associated with the input data, while the output layer deals with the target. Between the input and output layers, hidden layers are located. In general, one hidden layer is sufficient for most analyses. However, two or more hidden layers can be used in highly complex modeling. For each input, there exists one neuron in the input layer. For the output layer, the number of neurons is equal to the number of outputs. For the hidden layer, the number of neurons is determined experimentally [47,66,67].

An MLPNN employs supervised learning to adjust connection weights within the network through iterative training. Neurons of each layer depend on neuron(s), bias, and weights of the previous layer. Neurons are multiplied by their weight to calculate their value. After that, weighted values of neurons are added to the bias, and the sum is calculated. Following this, the obtained value is transferred to the next layer by passing it through an activation function. Finally, the output (h_j) can be calculated using Eq. (5).

(5)h_j=f\left(\sum _i{x}_i{w}_{ij}+b\right),

where f denotes the activation function, x_i is the activation of the ith hidden layer node, w_ij is the connection weight between the jth neuron in a layer with the ith neuron in the previous layer, and b indicates the neuron bias.

The input parameters for the neural network are the soil parameters. Four different inputs are introduced for the neural network that is fed in terms of a 4 × 92 matrix, such that each column represents a modeling step in Abaqus and each row shows a parameter. The outputs of Abaqus which form a 30 × 92 matrix, are considered the target of the neural network. Each column of this matrix represents the output of the corresponding element in the input matrix. 30 rows are defined since there are 30 different points along the pipe.

Perceptron neural networks can have different architectures based on the project specifications and data quality. Although some researchers, such as Ref. [68], made some efforts in this field and tried to use the available tools to reach an optimum architecture, researchers did not follow the same strategy for finding the optimized architecture. There is large room for development in this area; however, some general rules can help researchers to choose a proper architecture. Increasing the layers leads to making the network more complex and imposes significant overheads on the network. The number of layers is defined based on the problem, and in general, one or two layers can provide a desirable prediction. In the current study, two layers (including one output layer) are considered. The number of neurons in each layer depends on the number of inputs and outputs. The low number of neurons can result in poor prediction by the network. Hence, 12 neurons are considered for each layer in this work (Fig.4).

Sigmoid and tangent hyperbolic are the commonly-used activation functions. The sigmoid function applies a biased and evaluated function on the neurons of each layer which originate from a linear equation. This function determines the inputs in any desirable range, and generates the relative outputs in a range between 0 and 1. The tangent hyperbolic function is similar to sigmoid, but with this difference: it generates outputs in a range between −1 and 1. In this work, the tangent hyperbolic function is used in the first layer, while a linear one is used as an activation function in the second (output) layer. Tab.4 shows the hyperparameter of the employed neural network. The normalization process is conducted using Eq. (6).

(6)X_{\mathrm{n}}=-1+2\times \frac{{X_i-X}_{i,\mathrm{m}\mathrm{i}\mathrm{n}}}{X_{i,\mathrm{m}\mathrm{a}\mathrm{x}}-X_{i,\mathrm{m}\mathrm{i}\mathrm{n}}},

where X_{\mathrm{n}} are normalized values, and X_{i,\mathrm{m}\mathrm{i}\mathrm{n}} and X_{i,\mathrm{m}\mathrm{a}\mathrm{x}} are minimum and maximum of X_i values, respectively.

However, there are some challenges such as overfitting while working with ANNs. Overfitting can be meant as mimicry from target data rather than learning. A model that suffers from overfitting has likely poor predictive success. To prevent this problem, the authors use the K-fold cross validation technique. K-fold cross-validation is a statistical technique used to evaluate the performance of an ML algorithm. The purpose of this technique is to estimate the performance of a model on unseen data by splitting the available data set into multiple subsets or folds. In K-fold cross-validation, the data are divided into K equal-sized subsets. The model is then trained on K-1 subsets and validated on the remaining subset. This process is repeated K times, with each of the K subsets used exactly once as the validation data. The results from each validation run are then averaged to give a final performance estimate. The advantage of K-fold cross-validation is that it uses all the available data for training and validation, which can lead to a more accurate estimate of model performance. Additionally, it helps to reduce overfitting by providing a more robust estimate of model performance on unseen data. In this study, K is considered equal to 5 and data points are considered equal to 35.

4.2 Extraction of the mathematical formula from artificial neural network

ANNs can solve many complicated and time-consuming mathematical problems with high accuracy. Hence, the mathematical operation of an ANN can be extracted as a formula. The effectiveness of the formula depends on the following factors: 1) number of layers, 2) specifications of layers.

ANN models can be converted into mathematical formulas using Eq. (7). This equation presents the relation between an output parameter and input parameters (vertical displacement) [69]:

(7)h=f_{\text{OH}}({bias}_{\text{h}}+\sum _{k=1}^h{V}_k{f}_{\text{IH}}({bias}_{\text{h}k}+\sum _{i=1}^m{w}_{ik}{x}_i)),

where {bias}_{\mathrm{h}} denotes the hidden layer bias, V_k is the weight between hidden and output layer neurons, {bias}_{\mathrm{h}k} indicates the bias of neurons in the hidden layer, w_{ik} is the weight between inputs and hidden layer neurons, x_i are inputs parameters, f_{\mathrm{O}\mathrm{H}} is the activation function between the hidden and output layer, and f_{\mathrm{I}\mathrm{H}} is the activation function between inputs and the hidden layer.

5 Results

5.1 Parametric study

A calibrated model in Abaqus is used to analyze the displacement in a pipeline subjected to normal faulting. For a more comprehensive and realistic analysis, three key parameters of soil are chosen that change within the ranges shown in Tab.3. Fig.5 depicts the vertical displacements of various parts of the pipeline under the maximum and minimum values of soil parameters. In each part of Fig.5, A to D, one desired parameter takes its maximum and minimum values, while three other parameters remain constant. These values are 10 kPa, 1 m, 10 MPa, and 31° for cohesion, faulting, Young’s modulus, and internal friction angle, respectively. Vertical displacement is determined for both paths, top (t) and bottom (b). Fig.6 presents the soil mass and pipeline condition when the maximum value is applied to one soil parameter, while the other three parameters are fixed. In this figure, the blue color represents the parts of the pipeline that experienced the lowest displacement, where the highest displacement is shown with red. Other colors indicate gradual change between these two abovementioned conditions.

5.2 Evaluation of multilayer perceptron neural network performance

An MLPNN with a single hidden layer is used for predicting the vertical displacement of the pipeline subjected to normal faulting. Neural network training is continued until the best performance (by obtaining the minimum error) is achieved. In other words, training is stopped once the mean squared error (MSE) reaches its minimum value. Fig.7 shows the graph of MSE versus epoch. It can be seen that the best performance is achieved in Epochs 166 and 179 for the top and bottom paths since MSE is fixed in the six subsequent Epochs. Here, the MLPNN architecture consists of one hidden layer with 12 neurons. Fig.8 and Fig.9 present the actual vertical displacement versus predicted vertical displacement by the MLPNN for the top and bottom paths through the pipeline. As seen in these figures, the MLPNN is successful in forecasting vertical displacement.

For analyzing the performance of the neural network, the correlation coefficient (R) is one of the most important parameters to determine. The correlation of the model is higher when R has a value close to 1. In general, values higher than 0.8 for R are categorized in an acceptable range [70]. Despite being helpful in results interpretation, this measure alone is not a reasonable index for evaluating the model’s accuracy [71]. Consequently, three other performance criteria in this study are chosen: MSE, mean absolute error (MAE), and performance index (PI). For MSE and MAE, we are looking for values close to 0 to get more accurate models. PI, proposed by Gandomi et al. [72], can evaluate the performance of trained models. The PI index considers the changes in both correlation and error functions. PI can be in the range of [0 to +∞], and lower values mean higher precision of the model. Equations (8) to (11) help readers understand these criteria in detail, while Tab.5 shows the values of R, MSE, MAE, and PI for the MLPNN modeled in this study.

(8)R=\frac{\sum _{i=1}^n(h_i-{\overline{h}}_i)(t_i-{\overline{t}}_i)}{\sqrt{{\sum }_{i=1}^n{(h_i-{\overline{h}}_i)}^2{\sum }_{i=1}^n{(t_i-{\overline{t}}_i)}^2}},

(9)MSE=\frac{{\sum }_{i=1}^n{(h_i-t_i)}^2}{n},

(10)MAE=\frac{1}{n}{\sum }_{i=1}^n\left|h_i-t_i\right|,

(11)PI=\frac{{\displaystyle \frac{1}{\left|{\overline{h}}_i\right|}}\sqrt{MSE}}{R+1},

where h_i and t_i present the real and estimated values for ith output, respectively. {\overline{h}}_i is the average of h_i, and the total number of samples is represented by n.

Observing the error values can help to predict the process evaluation. Any representative of the error values (i.e., error histogram) can reveal the discrepancies between calculated values after training a feedforward neural network and the target values. These error values show how values predicted by an MLPNN differ from the target values from Abaqus. Fig.10 and Fig.11 display the error histogram for the top and bottom paths, respectively. The X-axis represents the error, which is the difference between the target and the output. For instance, the error value for the top path ranges between −0.3553 and 0.5414, which is divided into 30 parts (bins). Y-axis denotes the number of samples from the data set which fall in a particular bin number. Fig.10 and Fig.11 confirm that the prediction process for the majority of data was successful and with low error values, nearly 0. The maximum error value for the top and bottom paths are 0.5414 and 0.07984, respectively.

5.3 Sensitivity analysis

Sensitivity analysis helps to find how the input parameters contribute to the output. A simple method for sensitivity analysis is sensitivity percentage. By following the formulas described in Ref. [73], the sensitivity (S_i) of the displacement is determined for four inputs: cohesion, friction angle, Young’s modulus, and faulting.

(12)N_i=f_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(x_i\right)-f_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(x_i\right),

(13)S_i=\frac{N_i}{\sum _{j=1}^n{N}_j}\times 100,

where f_max(x_i) and f_min(x_i) denote the maximum and minimum of the estimated output over the ith input domain, and other variables are equal to their mean values.

The sensitivity percentage for all 30 points along the pipe is presented in Tab.6. The results confirm that the highest sensitivity level of pipe displacement is related to Young’s modulus.

5.4 Mathematical formulation of MLPNN

After proving that MLPNN is successfully developed, we can extract a mathematical formula from the MLPNN based on Eq. (7). If A and F_k are described as Eqs. (14) and (15), Eq. (7) can be transformed into Eq. (16).

(14)A={bias}_k+\sum w_{ik}{x}_i,

(15)F_k={bias}_j+\sum V_{jk}{f}_{\mathrm{I}\mathrm{H}}(A_j)

(16)h=f_{\text{OH}}(F_k),

where w_ik is the assigned weights to the parameters in the first layer neurons, x_i is the input parameter, and bias_k is the bias in the first (hidden) layer. V_jk is the assigned weight to the inputs in the output layer, f_IH denotes the activation function of the first layer, and bias_j is the output layer bias.

As mentioned in previous sections, the neural network constructed in this work has four inputs, one hidden layer with 12 neurons, and 30 outputs for the displacement of 30 points on the pipeline. The activation function for the hidden layer (f_IH) and output layer (f_OH) are Tanh and Linear, respectively. Therefore, F_k and A_j for the hidden layer can be described by Eqs. (17) and (18), respectively.

(17)F_k={bias}_j+{\sum }_{j=1}^{12}{V}_{jk}\left(\frac{e^{2A_j}-1}{e^{2A_j}+1}\right),

(18)A_j={(C}_{\text{n}}\times w_{1k})+({Uf}_{\text{n}}\times w_{2k})+(E_{\text{n}}\times w_{3k})+({\phi }_{\text{n}}\times w_{4k})+{bias}_k,

where C_n, Uf_n, E_n, and φ_n are normalized cohesion, faulting, Young’s modulus, and internal friction angle, respectively. The numbers of hidden layer neurons (k) and of outputs (h) are 12 and 30, respectively. All weights and biases of the MLPNN are presented in the Electronic Supplementary Materials.

After computing the weights and biases, the last step is to return the mathematical formula from normalized form to a standard form. Equation (19) shows how this transformation is performed. For this purpose, we have to obtain the maximum and minimum values of outputs. Since there are 30 different outputs (points on the pipeline), we have to calculate the highest and the lowest values of vertical displacement for these 30 points at the top and bottom paths of the pipe.

Tab.7 present the values of vertical displacement for the bottom path of the pipe, while Tab.8 shows the same values for the top path. Consequently, for any desired point around the pipeline, first, the maximum and minimum values are chosen from Tab.7 and Tab.8, and then, the vertical displacement can be determined using Eq. (19).

(19)U_j\left(\mathrm{c}\mathrm{m}\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left(U_j\right)+(\mathrm{m}\mathrm{a}\mathrm{x}\left(U_j\right)-\mathrm{m}\mathrm{i}\mathrm{n}\left(U_j\right))\frac{(F_k+1)}{2},

where U_j is soil displacement and F_k is the value calculated using Eq. (17).

Fig.12 shows the difference between the denormalized output of the mathematical formula and the target values (in Abaqus). In both paths, the error value is less than 2%, which confirms the high accuracy of the extracted mathematical formula.

6 Conclusions

In this study, the complex behavior of soil and buried pipeline system subjected to a normal fault is analyzed using an ANN to predict the behavior of the soil-pipeline system when different soil parameters (cohesion, Young’s modulus, internal friction angle, and movement of fault) are changed. A reliable experimental model is used as baseline research and a calibrated model is created using the FEM in Abaqus. The outputs of Abaqus (vertical displacement of pipeline in top and bottom paths) are utilized as a data bank to build the neural network. The results of the neural network for the bottom (R = 0.99986, MAE = 0.001110, MSE = 1.13 × 10^–4, PI = 0.01177) and top (R = 0.99894, MAE = 0.00135, MSE = 1.69 × 10^–4, PI = 0.01869) paths show the high accuracy of neural network. Based on sensitivity analysis of selected parameters for 30 different points along the pipe, it is confirmed that the highest sensitivity level of pipe displacement is related to Young’s modulus.

Making use of gathered data, an MLPNN is conducted to provide a high correlation coefficient and low error, and a comparison is performed for the Abaqus and mathematical models. Due to the satisfactory and good convergence obtained from this comparison, a mathematical formula is presented using biases and weights of the MLPNN model. Experimental results show that the maximum error of the presented formula is 2.03%, which proves that the proposed technique can efficiently predict the vertical displacement of buried pipelines. Such a model can be helpful for the initial studies of buried pipeline projects and resulting in saving time and budget.

Both proposed techniques (MLPNN model and the mathematical formula) can efficiently estimate the condition that had not previously been evaluated by field and experimental tests. It should be noted that these methods are only valid for the specified range of the pipe and soil specifications mentioned in this study. For future research, investigation of the following topics is planned.

1) Investigating other outputs of Abaqus, such as tension and strain.

2) Considering different types of faulting.

3) Changing the type of soil and material and size of the pipe.

4) Using other artificial intelligence methods to increase the accuracy of the method.

References

[1] Rafiee R, Habibagahi M R. On the stiffness prediction of GFRP pipes subjected to transverse loading. KSCE Journal of Civil Engineering, 2018, 22(11): 4564–4572

[2] Datta T K. Seismic response of buried pipelines: a state-of-the-art review. Nuclear Engineering and Design, 1999, 192(2−3): 271–284

[3] Izadi M, Bargi K. Improvement of mechanical behavior of buried pipelines subjected to strike-slip faulting using textured pipeline. Frontiers of Structural and Civil Engineering, 2019, 13(5): 1105–1119

[4] KhaksarRMoradiMGhalandarzadehA. A Novel Experimental-Numerical Approach to Model Buried Pipes Subjected to Reverse Faulting, in Physical Modelling in Geotechnics. Boca Raton: CRC Press, 2018: 565–569

[5] KhaksarR YMoradiMKhorasaniM. Numerical investigation of factors influencing oil and gas lifelines behavior due to normal faulting geo-hazard. Current World Environment, 2015, 10 (Special Issue): 806

[6] MaCaffreyM A. Buried pipeline response to reverse faulting during the 1971 San Fernando Earthquake. ASME, PVP, 1983, 77: 151–159

[7] SchiffA J. Northridge Earthquake: Lifeline Performance and Post-Earthquake Response. New York: American Society of Civil Engineers, 1995

[8] Scawthorn C, Yanev P I. 17 January 1995, Hyogo-ken Nambu, Japanese earthquake. Engineering Structures, 1995, 17(3): 146–157

[9] NewmarkN MHallW J. Pipeline design to resist large fault displacement. In: Proceedings of US national conference on earthquake engineering. Ann Arbor, US: Earthquake Engineering Research Institute, 1975

[10] Kennedy R P, Williamson R A, Chow A M. Fault movement effects on buried oil pipeline. Transportation Engineering Journal, 1977, 103(5): 617–633

[11] Karamitros D K, Bouckovalas G D, Kouretzis G P. Stress analysis of buried steel pipelines at strike-slip fault crossings. Soil Dynamics and Earthquake Engineering, 2007, 27(3): 200–211

[12] Karamitros D K, Bouckovalas G D, Kouretzis G P, Gkesouli V. An analytical method for strength verification of buried steel pipelines at normal fault crossings. Soil Dynamics and Earthquake Engineering, 2011, 31(11): 1452–1464

[13] Trifonov O V, Cherniy V P. A semi-analytical approach to a nonlinear stress–strain analysis of buried steel pipelines crossing active faults. Soil Dynamics and Earthquake Engineering, 2010, 30(11): 1298–1308

[14] Trifunac M D. The role of strong motion rotations in the response of structures near earthquake faults. Soil Dynamics and Earthquake Engineering, 2009, 29(2): 382–393

[15] Takada S, Hassani N, Fukuda K. A new proposal for simplified design of buried steel pipes crossing active faults. Earthquake Engineering & Structural Dynamics, 2001, 30(8): 1243–1257

[16] Trifonov O V, Cherniy V P. Elastoplastic stress–strain analysis of buried steel pipelines subjected to fault displacements with account for service loads. Soil Dynamics and Earthquake Engineering, 2012, 33(1): 54–62

[17] Rafiee R, Habibagahi M R. Evaluating mechanical performance of GFRP pipes subjected to transverse loading. Thin-walled Structures, 2018, 131: 347–359

[18] Rafiee R, Ghorbanhosseini A. Analyzing the long-term creep behavior of composite pipes: Developing an alternative scenario of short-term multi-stage loading test. Composite Structures, 2020, 254: 112868

[19] Rafiee R, Ghorbanhosseini A. Experimental and theoretical investigations of creep on a composite pipe under compressive transverse loading. Fibers and Polymers, 2021, 22(1): 222–230

[20] Yue T, Wahab M A. Finite element analysis of fretting wear under variable coefficient of friction and different contact regimes. Tribology International, 2017, 107: 274–282

[21] Ling Y, Ni J, Antonissen J, Ben Hamouda H, Vande Voorde J, Abdel Wahab M. Numerical prediction of microstructure and hardness for low carbon steel wire Arc additive manufacturing components. Simulation Modelling Practice and Theory, 2023, 122: 102664

[22] Nguyen K D, Thanh C L, Vogel F, Nguyen-Xuan H, Abdel-Wahab M. Crack propagation in quasi-brittle materials by fourth-order phase-field cohesive zone model. Theoretical and Applied Fracture Mechanics, 2022, 118: 103236

[23] Tran-NgocHKhatirSLe-XuanTDe RoeckGBui-TienTAbdel WahabM. Finite element model updating of a multispan bridge with a hybrid metaheuristic search algorithm using experimental data from wireless triaxial sensors. Engineering with Computers, 2021: 1–19

[24] Vazouras P, Dakoulas P, Karamanos S A. Pipe–soil interaction and pipeline performance under strike-slip fault movements. Soil Dynamics and Earthquake Engineering, 2015, 72: 48–65

[25] Zhang L, Zhao X, Yan X, Yang X. A new finite element model of buried steel pipelines crossing strike-slip faults considering equivalent boundary springs. Engineering Structures, 2016, 123: 30–44

[26] Vazouras P, Karamanos S A, Dakoulas P. Mechanical behavior of buried steel pipes crossing active strike-slip faults. Soil Dynamics and Earthquake Engineering, 2012, 41: 164–180

[27] Trifonov O V. Numerical stress-strain analysis of buried steel pipelines crossing active strike-slip faults with an emphasis on fault modeling aspects. Journal of Pipeline Systems Engineering and Practice, 2015, 6(1): 04014008

[28] LimY MKimM KKimT WJaeW J. The behavior analysis of buried pipeline: Considering longitudinal permanent ground deformation. Advances in Pipelines Engineering and Construction, 2001: 1–11

[29] Vazouras P, Karamanos S A, Dakoulas P. Finite element analysis of buried steel pipelines under strike-slip fault displacements. Soil Dynamics and Earthquake Engineering, 2010, 30(11): 1361–1376

[30] HabibM KAyankosoS ANagataF. Data-driven modeling: Concept, techniques, challenges and a case study. In: 2021 IEEE International Conference on Mechatronics and Automation (ICMA). Takamatsu: IEEE, 2021

[31] SolomatineDSeeL MAbrahartR J. Data-driven modelling: Concepts, approaches and experiences. Practical Hydroinformatics, 2008: 17–30

[32] ChenTGuestrinC. Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining. New York: Association for Computing Machinery, 2016

[33] KeGMengQFinleyTWangTChenWMaWYeQLiuT. Lightgbm: A highly efficient gradient boosting decision tree. Advances in Neural Information Processing Systems, 2017, 30

[34] DorogushA VErshovVGulinA. CatBoost: Gradient boosting with categorical features support. ArXiv preprint ArXiv, 2018: 1810.11363

[35] SuthaharanS. Support Vector Machine, In Machine Learning Models and Algorithms for Big Data Classification. Berlin: Springer, 2016

[36] Dongare A, Kharde R, Kachare A D. Introduction to artificial neural network. International Journal of Engineering and Innovative Technology, 2012, 2(1): 189–194

[37] Peterson L E. K-nearest neighbor. Scholarpedia Journal, 2009, 4(2): 1883

[38] Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790

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

[40] GuoHZhuangXRabczukT. A deep collocation method for the bending analysis of Kirchhoff plate. ArXiv preprint ArXiv, 2021: 2102.02617

[41] Zhuang X, Guo H, Alajlan N, Zhu H, Rabczuk T. Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. European Journal of Mechanics. A, Solids, 2021, 87: 104225

[42] Khatir S, Tiachacht S, Le Thanh C, Ghandourah E, Mirjalili S, Abdel Wahab M. An improved Artificial Neural Network using Arithmetic Optimization Algorithm for damage assessment in FGM composite plates. Composite Structures, 2021, 273: 114287

[43] Ho L V, Trinh T T, De Roeck G, Bui-Tien T, Nguyen-Ngoc L, Abdel Wahab M. An efficient stochastic-based coupled model for damage identification in plate structures. Engineering Failure Analysis, 2022, 131: 105866

[44] Ho L V, Nguyen D H, Mousavi M, De Roeck G, Bui-Tien T, Gandomi A H, Wahab M A. A hybrid computational intelligence approach for structural damage detection using marine predator algorithm and feedforward neural networks. Computers & Structures, 2021, 252: 106568

[45] Hamidian P, Soofi Y J, Bitaraf M. A comparative machine learning approach for entropy-based damage detection using output-only correlation signal. Journal of Civil Structural Health Monitoring, 2022, 12(5): 975–990

[46] Wang S, Wang H, Zhou Y, Liu J, Dai P, Du X, Abdel Wahab M. Automatic laser profile recognition and fast tracking for structured light measurement using deep learning and template matching. Measurement, 2021, 169: 108362

[47] Karballaeezadeh N, Zaremotekhases F, Shamshirband S, Mosavi A, Nabipour N, Csiba P, Várkonyi-Kóczy A R. Intelligent road inspection with advanced machine learning; hybrid prediction models for smart mobility and transportation maintenance systems. Energies, 2020, 13(7): 1718

[48] Guo H, Zhuang X, Chen P, Alajlan N, Rabczuk T. Stochastic deep collocation method based on neural architecture search and transfer learning for heterogeneous porous media. Engineering with Computers, 2022, 38(6): 1–26

[49] Guo H, Zhuang X, Chen P, Alajlan N, Rabczuk T. Analysis of three-dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis. Engineering with Computers, 2022, 38(6): 1–22

[50] Liu X, Zheng Q, Wu K, Yang Y, Zhao Z, Zhang H. Development of a novel approach for strain demand prediction of pipes at fault crossings on the basis of multi-layer neural network driven by strain data. Engineering Structures, 2020, 214: 110685

[51] Fan X, Wang X, Zhang X, Yu X B. Machine learning based water pipe failure prediction: The effects of engineering, geology, climate and socio-economic factors. Reliability Engineering & System Safety, 2022, 219: 108185

[52] Zheng Q, Liu X, Zhang H, Gu X, Fang M, Wang L, Adeeb S. Reliability evaluation method for pipes buried in fault areas based on the probabilistic fault displacement hazard analysis. Journal of Natural Gas Science and Engineering, 2021, 85: 103698

[53] Sadat Shokouhi S K, Dolatshah A, Ghobakhloo E. Seismic strain analysis of buried pipelines in a fault zone using hybrid FEM-ANN approach. Earthquakes and Structures, 2013, 5(4): 417–438

[54] Moghadas Tafreshi S N, Tavakoli Mehrjardi G, Moghadas Tafreshi S M. Analysis of buried plastic pipes in reinforced sand under repeated-load using neural network and regression model. International Journal of Civil Engineering, 2007, 5(2): 118–133

[55] Yeganeh Khaksar R, Moradi M, Ghalandarzadeh A. Response of buried oil and gas pipelines subjected to reverse faulting: A novel centrifuge-finite element approach. Scientia Iranica, 2018, 25(5): 2501–2516

[56] JalaliHYeganeh KhaksarR. Development of a numerical model using artificial intelligence for investigation of the response of soil and buried pipe system under the effect of normal faulting. In: 12th International Congress on Civil Engineering. Mashhad: Ferdowsi University of Mashhad, 2021

[57] Yeganeh KhaksarRMoradiM. A numerical model developed to investigate the response of buried pipelines subjected to the normal faulting. In: 10th International Congress on Civil Engineering. Tabriz: Springer Science and Business Media Deutschland Gmbh, 2015

[58] Yimsiri S, Soga K, Yoshizaki K, Dasari G R, O’Rourke T D. Lateral and upward soil-pipeline interactions in sand for deep embedment conditions. Journal of Geotechnical and Geoenvironmental Engineering, 2004, 130(8): 830–842

[59] Yoshimi Y, Kishida T. A ring torsion apparatus for evaluating friction between soil and metal surfaces. Geotechnical Testing Journal, 1981, 4(4): 145–152

[60] Potts D, Dounias G, Vaughan P. Finite element analysis of progressive failure of Carsington embankment. Geotechnique, 1990, 40(1): 79–101

[61] Liu X, Zhang H, Han Y, Xia M, Zheng W. A semi-empirical model for peak strain prediction of buried X80 steel pipelines under compression and bending at strike-slip fault crossings. Journal of Natural Gas Science and Engineering, 2016, 32: 465–475

[62] Naderpour H, Rafiean A H, Fakharian P. Compressive strength prediction of environmentally friendly concrete using artificial neural networks. Journal of Building Engineering, 2018, 16: 213–219

[63] Ismail A. ANN-based empirical modelling of pile behaviour under static compressive loading. Frontiers of Structural and Civil Engineering, 2018, 12(4): 594–608

[64] El-Abbasy M S, Senouci A, Zayed T, Mirahadi F, Parvizsedghy L. Artificial neural network models for predicting condition of offshore oil and gas pipelines. Automation in Construction, 2014, 45: 50–65

[65] Ziaee S A, Sadrossadat E, Alavi A H, Mohammadzadeh Shadmehri D. Explicit formulation of bearing capacity of shallow foundations on rock masses using artificial neural networks: Application and supplementary studies. Environmental Earth Sciences, 2015, 73(7): 3417–3431

[66] Soofi Y J, Bitaraf M. Output-only entropy-based damage detection using transmissibility function. Journal of Civil Structural Health Monitoring, 2022, 12(1): 191–205

[67] Moghadas NejadFMehrabiAZakeriH. Prediction of asphalt mixture resistance using neural network via laboratorial X-ray images. Journal of Industrial and Intelligent Information, 2015, 3(1): 48−53

[68] Hamdia K M, Zhuang X, Rabczuk T. An efficient optimization approach for designing machine learning models based on genetic algorithm. Neural Computing & Applications, 2021, 33(6): 1923–1933

[69] Yilmaz I. The effect of the sampling strategies on the landslide susceptibility mapping by conditional probability and artificial neural networks. Environmental Earth Sciences, 2010, 60(3): 505–519

[70] Karballaeezadeh N, Ghasemzadeh Tehrani H, Mohammadzadeh Shadmehri D, Shamshirband S. Estimation of flexible pavement structural capacity using machine learning techniques. Frontiers of Structural and Civil Engineering, 2020, 14(5): 1083–1096

[71] Gandomi A H, Roke D A. Assessment of artificial neural network and genetic programming as predictive tools. Advances in Engineering Software, 2015, 88: 63–72

[72] Gandomi A H, Alavi A H, Mousavi M, Tabatabaei S M. A hybrid computational approach to derive new ground-motion prediction equations. Engineering Applications of Artificial Intelligence, 2011, 24(4): 717–732

[73] Gandomi A H, Yun G J, Alavi A H. An evolutionary approach for modeling of shear strength of RC deep beams. Materials and Structures, 2013, 46(12): 2109–2119

RIGHTS & PERMISSIONS

Higher Education Press