Neural Prediction of Tunnels’ Support Pressure in Elasto-Plastic, Strain-Softening Rock Mass

Featured Application

This work can be conjunctly used with the support characteristic curve of circular tunnels to find the optimum time of the installation of the support system in a way to restrict the displacements to a specific value. The approach described in this manuscript facilitates the design of circular tunnels for elasto-plastic, strain-softening rock masses obeying both Mohr–Coulomb and Hoek–Brown strength criteria.

Abstract

The prediction of the support pressure (P_i) and the development of the ground reaction curve (GRC) are crucial elements of the convergence–confinement procedure used to design underground structures. In this paper, two different types of artificial neural networks (ANNs) are used to predict the P_i of circular tunnels in elasto-plastic, strain-softening rock mass. The developed ANNs consider the stress state, the radial displacement of tunnel and the material softening behavior. Among these parameters, strain softening is the parameter of the deterioration of the material’s strength in the plastic zone. The analysis also presents separate solutions for the Mohr–Coulomb and Hoek–Brown strength criteria. In this regard, multi-layer perceptron (MLP) and radial basis function (RBF) ANNs were successfully applied. MLP with the architectures of 15-5-10-1 for the Mohr–Coulomb criteria and 17-5-15-1 for the Hoek–Brown criteria appeared optimum for the prediction of the P_i. On the other hand, the RBF networks with the architectures of 15-5-1 for the Mohr–Coulomb criterion and 17-3-12-1 for the Hoek–Brown criterion were found to be the optimum for the prediction of the P_i.

1. Introduction

Convergence–confinement is a broadly applied method to design underground structures. In this method, a ground reaction curve (GRC) is applied to evaluate the interaction between the rock mass and the support system near the tunnel’s face at the time of construction [1,2,3]. The classical problem of a circular tunnel constructed in the rock mass medium is illustrated in Figure 1b (according to [4]). This problem uses two well-known rock mass strength criteria: Mohr–Coulomb and Hoek–Brown. These criteria are usually adopted to investigate the behavior of materials. As described in references [4,5,6], the governing equilibrium equation of this problem is as presented in Figure 1d.

In Figure 1, ${\sigma }_r$ is the radial stress, and ${\sigma }_{\theta }$ represents the tangential (hoop) stress, while the distance from the center of tunnel is shown using the $r$ parameter. Thus, compatibility equations are simultaneously used to solve the stress–strain around the tunnel. In this regard, a plastic potential function and associated/non-associated flow rules are applied to find the governing relationship between the radial and tangential plastic strains [1,4,5,7,8,9,10,11,12,13]. The stress–strain around the tunnel is also calculated, considering the strength parameters in different zones (the peak, residual, and softening parameters), as presented in Figure 1a,c, according to reference [4]. As explained in reference [4], using different yield and potential functions, the non-associated flow rule is the modeled and adopted as the flow rule in all the cases. Also, starting from the peak strength parameters, the deterioration of the strength parameters of the material (referred to as the softening behavior of the rock mass around the tunnel) begins and continues. For softening parameter values higher than the critical value, the residual parameters are taken into the considerations. As shown in Figure 1a,c, $\omega \left(\eta \right)$ represents each of the strength parameters of the rock mass, and $\eta$ and ${\eta }^{*}$, respectively, are the softening and the critical softening parameters. In this figure, ${\sigma }_0$ is the far-field in situ rock stress. In addition, Mohr–Coulomb and Hoek–Brown strength criteria are commonly used as two appropriate criteria to correctly evaluate the post-elastic behavior of rock mass. These criteria are, also, broadly applied in the literature for the development of the GRC. Hence, in this paper, available solutions with regard to the described problem are gathered, and rigorous datasets for both Mohr–Coulomb and Hoek–Brown strength criteria are obtained. As presented in the following sections, the collected datasets are fed into the different types of artificial neural networks (ANNs). The main reason of ANN modeling of the tunnels’ support pressure (P_i) is to save time and still obtain high accuracy in the predicted results. Previously, the evolutionary polynomial regression technique (EPR) was used by the authors to predict the P_i and to develop the GRC [4]. Nevertheless, there still exist some considerable prediction errors specially in the case of the Hoek–Brown strength criterion. Thus, we suppose that, in this case, ANNs may be more accurate. In this regard, available datasets should be sorted in the way shown in Figure 1e (a typical dataset for the case of the Hoek–Brown strength criterion). As depicted, each data pair stands for a point in the $U_i-P_i$ space. $U_i$ is the tunnel’s radial convergence and $P_i$ represents the corresponding required internal P_i. Each of the described data is a function of some input parameters (the number of input parameters is determined on the basis of the type of the used strength criterion) and presents the P_i based on the all affecting parameters. For instance, in Figure 1e, $P_i$ is a function of 17 independent input parameters.

The challenge is to predict the P_i of the internal support system [7,8,9,14]. The commonly known ways of P_i prediction employ different rock mass quality systems (e.g., rock mass rating, RMR; geological strength index, GSI, etc.) [15]. Some of these methods are also based on numerical codes, theoretical calculations, or coupled semi-analytical solutions [1,4,5,7,10,11,12,16,17,18,19,20]. Most of the commercial numerical packages, usually, use different finite element/difference codes [21]. These packages first define the element types, the material properties, and the geometry, together with the boundary and loading conditions. Discrete element packages are, also, used to model a rock medium containing specific joint sets with pre-defined orientations. All the described approaches are used to derive the governing differential equations and to present the solutions of the developed system [5,7,12]. Hence, they first need to be well validated against rigorous, available solutions and case studies. In addition, some of the existing methods have drawbacks (e.g., some of the available numerical solutions do not consider the deterioration of the strength parameters or do not correctly take the plastic straining into consideration [1]). Also, they require an adequate knowledge and background about the fundamentals and the theory of the convergence–confinement method and numerical and mathematical techniques. In addition to requiring an expert for the analysis, it is usually a time-consuming process to obtain the GRC. There are also other complex characteristics in the original problem which can be taken into the account, for instance, the material softening.

Considering the above, there is still a need to develop new methods for the reliable prediction of P_i in the elasto-plastic, strain-softening rock mass. In this regard, in this paper, the applicability of another new intelligent method of ANN modeling to predict the P_i of circular tunnels constructed in rock masses with different qualities is investigated.

The remainder of the paper is organized as follows. Section 2 presents the description of the research problems with background information, data acquisition, and the principles of different types of ANNs. Section 3 is the performance evaluation and compares the performance of ANNs with that of the previously proposed EPR models. Section 4, summarizes the results of the analyses.

2. Methodology

ANNs are broadly applied in engineering [22,23,24,25,26,27,28,29]. Also, over the last decades, ANNs have appeared as efficient meta-modelling methods applicable to a wide range of sciences, including material science and structural engineering [30,31,32]. An important characteristic of ANNs is that they can be used to build soft sensors, i.e., models with the ability to estimate critical quantities without having to measure them [30]. In particular, such alternative models are built after a training process with only a few available data, selected in a previous phase, to forecast the parameters. This considerably reduces the time and money needed for the experiments. One of the applicable fields of ANNs described in the literature is the prediction of the mechanical properties of concrete materials [33,34]. One of the main benefit of ANNs is that there is no need of any prior knowledge about the nature of the problem [35]. This makes ANNs applicable for a fast and reliable calculation of the P_i by practitioners who are not expert in numerical modeling and programming. In this paper, two different types of ANNs, namely, the multi-layer perceptron (MLP) and the radial basis function (RBF), are applied to predict the P_i.

Data Acquisition

A dataset was compiled from input parameters for the Mohr–Coulomb and the Hoek–Brown criteria.

The datasets for the Mohr–Coulomb criteria were from reference [8]. A total of 168 independent input and output parameters and their range of variation are presented in

Table 1. Input and output parameters and their range of variation used in the prediction of the P_i (Mohr–Coulomb case).

Parameter	Range of Variation	Standard Deviation	Coefficient of Variation (%)
$GS{I}^{peak}$	21.4–64.9	17.48	34.45
$GS{I}^{res}$	15.1–33	6.91	26.20
${\sigma }_{ci}$	23–162	60.09	60.49
$\gamma$	26–26.7	0.33	1.27
$E$	1.1–24	10.57	92.47
$\vartheta$	0.25–0.3	0.02	8.27
$c^{peak}$	0.34–3.7	1.55	83.51
${\phi }^{peak}$	24.81–57.8	14.26	33.24
$c^{res}$	0.27–0.96	0.31	51.67
${\phi }^{res}$	15.69–51	15.42	42.51
$m_i$	10–20	4.47	27.61
$\psi$	0–14	6.14	90.30
${\sigma }_0$	10.4–26	7.27	41.89
${\eta }^{*}$	0.0465–0.1394	0.037	43.43
$U_i$	8.86–1263.69	307.86	147.24
$P_i$	0–2	0.55	71.55

$GS{I}^{peak}$: peak geological strength index, $GS{I}^{res}$: residual geological strength index, ${\sigma }_{ci}$: uni-axial compressive strength, $\gamma$: unit weight, $E$: Young’s modulus, $\vartheta$: Poisson’s ratio, $c^{peak}:$ peak cohesion, ${\phi }^{peak}:$ peak friction angle, $c^{res}:$ residual cohesion, ${\phi }^{res}:$ residual friction angle, $m_i$: mi constant, $\psi :$ dilation angle, ${\sigma }_0:$ in-situ rock stress, ${\eta }^{*}:$ critical softening parameter, $U_i:$ radial displacement, $P_i:$ support pressure.

. Although the datasets are those adopted from reference [8], the presented ANN models have the following superiority: When a new dataset is available, ANN helps to have a prediction based on the relationships they have found between previously used input and output parameters.

It should be noted that, in this table, the unit of $U_i$ is mm, and the remaining parameters follow the units presented in the nomenclature.

Similar to the Mohr–Coulomb criteria, the dataset for the Hoek–Brown criteria was collected from the cases studied in references [5,6,9,36] and [14,37]. In total, 547 datasets were applied.

Table 2. Input and output parameters and their range of variation used in the prediction of the P_i (Hoek–Brown case).

Parameter	Range of Variation	Standard Deviation	Coefficient of Variation (%)
$r_i$	3–5.35	0.95	26.78
$GS{I}^{peak}$	25–100	29.52	41.34
$GS{I}^{res}$	10–100	35.89	67.07
${\sigma }_{ci}$	27.6–75	22.83	42.59
$E$	1.38–36.51	9.03	112.80
$\vartheta$	0.25	0	0
$m^{peak}$	0.5–4.09	0.94	57.06
$s^{peak}$	0.0002–0.0622	0.016	203.39
$a^{peak}$	0.5	0	0
$m^{res}$	0.1–1.173	0.29	41.75
$s^{res}$	0–0.002	0.00077	107.25
$a^{res}$	0.5–0.6	0.03	5.99
${\psi }^{peak}$	0–30	9.38	84.71
${\psi }^{res}$	0–20	5.50	94.45
${\sigma }_0$	3.31–30	6.54	43.53
${\eta }^{*}$	0.004742–100	34.66	216.12
$U_i$	0–0.282765	0.045	150.29
$P_i$	0–29.4	5.63	107.80

$r_i:$ radius of tunnel, $GS{I}^{peak}$: peak geological strength index, $GS{I}^{res}$: residual geological strength index, ${\sigma }_{ci}$: uni-axial compressive strength, $\gamma$: unit weight, $E$: Young’s modulus, $\vartheta$: Poisson’s ratio, $m^{peak}:$ peak m constant, $s^{peak}:$ peak s constant, $a^{peak}:$ peak a constant, $m^{res}:$ residual m constant, $s^{res}:$ residual s constant, $a^{res}:$ residual a constant, ${\psi }^{peak}:$ peak dilation angle, ${\psi }^{res}:$ residual dilation angle, ${\sigma }_0:$ in-situ rock stress, ${\eta }^{*}:$ critical softening parameter, $U_i:$ radial displacement, $P_i:$ support pressure.

shows all the studied input and output parameters together with their range of variation. Differently from Table 1, in Table 2, the unit of $U_i$ is m.

The multi-layer perceptron (MLP) ANN is still used by researchers in engineering [38]. The transfer function applied in this study is the TANSIG transfer function, according to the following formula:

$$TANSIG\left(e_x\right)=\frac{2}{1+e^{\left(-2e_x\right)}}-1,$$

(1)

where $e_x$ is the weighted sum of the inputs for a neuron [39].

The radial basis function (RBF) ANNs are also popular in engineering. In RBF-ANN, by increasing the Euclidean distance between the input vector and its center, the output attained using the activation function will tend to zero. The width of RBF controls the decreasing rate of the output. Equation (2) shows the Gaussian basis function:

$$y_j=\exp \left(\frac{-\Vert \overrightarrow{x}-{\overrightarrow{\mu }}_j{\Vert }^2}{{\sigma }_j^2}\right),$$

(2)

3. Performance Evaluation

This section reports the evaluation of the performance of ANNs. It was conducted on the basis of the commonly known coefficient of determination ($R^2$), root-mean-square error ($RMSE$), and mean absolute error ($MAE$) [22,24,27,28,40,41,42,43,44,45,46,47].

3.1. The Mohr–Coulomb Criterion

To evaluate the ANNs’ performances, ANNs with different architectures and neurons were built. To do this, 70% of the total data fed into an ANN was used as the training dataset. The other remaining 30% was divided into two datasets, with 15% of the data used for cross-validation and 15% for testing. Then, the ANNs showing the best predictions (in terms of maximum $R^2$ and minimum $RMSE$ and $MAE$ values) were selected. As shown in

Table 3. Performance of different multi-layer perceptron (MLP) and radial basis function (RBF) artificial neural networks (ANNs) in the prediction of the P_i for the testing data series (Mohr–Coulomb case).

Architecture	ANN Type
	MLP-ANN			RBF-ANN
	R2 (%)	RMSE (MPa)	MAE (MPa)	R2 (%)	RMSE (MPa)	MAE (MPa)
15-3-12-1	94.47	0.126617	0.09375	79.42	0.22383	0.1494
15-3-15-1	93.61	0.136705	0.100808	96.51	0.09361	0.06803
15-5-1	92.17	0.163929	0.102924	65.12	0.346439	0.256144
15-5-10-1	99.48	0.03883	0.02825	77.20	0.3193	0.2607
15-5-15-1	91.98	0.185252	0.128403	69.26	0.320201	0.180665
15-10-1	92.31	0.153122	0.110948	88.89	0.177232	0.142425
15-10-5-1	98.63	0.066325	0.042416	93.37	0.17276	0.124671
15-12-3-1	95.99	0.118647	0.06867	63.24	0.427169	0.340091
15-15-1	98.22	0.075068	0.058276	99.21	0.050576	0.040918
15-15-3-1	93.72	0.164167	0.107004	87.43	0.182931	0.140225
15-15-5-1	99.41	0.0459213	0.028215	81.90	0.255812	0.190918

RMSE: root mean squared error, MAE: mean absolute error.

, the 15-5-10-1 and 15-15-1 networks were, respectively, the best MLP-ANN and RBF-ANN in the prediction of P_i for the Mohr–Coulomb strength criterion. In this case, the overall performance of MLP-ANN was better than that of the RBF-ANN.

Figure 2 (as a sample representative of the RBF-ANN) depicts the topology of the best RBF-ANN developed for the Mohr–Coulomb case (15-15-1 architecture with $R^2=99.21\%$, $RMSE=0.050576 \mathrm{MPa}$, and $MAE=0.040918 \mathrm{MPa}$). Figure 3a,b show the predicted P_i versus the calculated P_i.

Similar to Figure 3a,b, Figure 4a shows a high correlation between the calculated and the ANN-based prediction for the RBF-ANN in the case of Mohr–Coulomb strength criterion. Figure 4b, which better describes the occurred errors in the predictions for each sample, is another demonstration of the high capability of the proposed ANNs.

3.2. The Hoek–Brown Criterion

Similar to the Mohr–Coulomb case, the evaluation of the performance of ANNs was carried out on the 15% of the whole data series. By investigating different one- and two-layered neural ANNs, the 15-5-15-1 and 17-3-12-1 architectures were found to be the most efficient ANNs in the prediction of the P_i of tunnels in the Hoek–­Brown case. As indicated in

Table 4. Performance of different MLP and RBF ANNs in the prediction of the P_i for the testing data series (Hoek–Brown case).

Architecture	ANN Type
	MLP-ANN			RBF-ANN
	R2 (%)	RMSE (MPa)	MAE (MPa)	R2 (%)	RMSE (MPa)	MAE (MPa)
17-3-12-1	99.72	0.268343	0.204655	93.18	1.558064	1.078099
17-3-15-1	93.37	1.36379	0.965679	84.51	2.515467	1.703367
17-5-1	99.63	0.317273	0.233694	78.35	2.570293	1.65224
17-5-10-1	90.37	1.565927	1.24049	89.05	2.165716	1.561432
17-5-15-1	99.91	0.179285	0.12516	87.72	1.810275	1.213031
17-10-1	99.67	0.300537	0.225826	77.64	2.497819	1.639146
17-10-5-1	99.87	0.252432	0.164251	80.62	2.627902	1.729953
17-12-3-1	99.80	0.239714	0.167833	88.72	2.06676	1.310764
17-15-1	99.65	0.322873	0.233356	90.51	1.877567	1.26464
17-15-3-1	99.85	0.20853	0.125573	77.22	3.019557	2.056784
17-15-5-1	99.72	0.307093	0.162638	72.15	3.060725	2.036629

, the 17-5-15-1 MLP-ANN had a coefficient of determination of 99.91%, an RMSE of 0.179285 MPa, and a MAE of 0.12516 MPa. On the other hand, the most accurate RBF-ANN had an $R^2=93.18\%$, an $RMSE=1.558064 \mathrm{MPa}$, and a $MAE=1.078099 \mathrm{MPa}$. Although the developed ANNs for the Mohr–Coulomb case are more accurate compared to the Hoek–Brown case in terms of occurred errors, the ANNs suggested for the Hoek–Brown modeling were still useful and efficient. Also, it was shown that that MLP-ANN are more rigorous in the prediction of the P_i of circular tunnels excavated in the elasto-plastic, strain-softening, Hoek–Brown rock mass.

The architecture and the topology of the most efficient MLP-ANN proposed for the modeling of the P_i in the Hoek–Brown case 17-5-15-1 is shown in Figure 5.

For the best constructed MLP-ANN, the predicted and calculated P_i, along with the relative differences of predictions and calculations for each independent exemplars are, respectively, shown in Figure 6. As clearly illustrated, the predictions are similar to the exact values of the calculations. This result, which was generally observed also for the RBF networks (Figure 7), confirms that both MLP and RBF ANNs are highly applicable to the determination of the P_i.

3.3. Comparison with Previously Obtained Models

In this section, the performance of the developed ANNs is compared to that of a previous artificial intelligence (AI) technique developed by other authors [4], i.e., the evolutionary polynomial regression (EPR), for the prediction of P_i. In this regard, the testing data series used in each of the ANNs was fed into the predictive EPR models, and the P_i were approximated. The results of the comparison between the EPR and the ANN predictions are presented. On the basis of the results, all the proposed MLP and RBF ANNs predicted the pressures more accurately than the corresponding EPR models. As an example, Figure 8 compares the results obtained from an EPR model to those predicted by the 15-5-10-1 MLP-ANN for the case of Mohr–Coulomb criterion and by the 17-5-15-1 MLP-ANN for the case of Hoek–Brown strength criterion. As shown, all the performance’s evaluation criteria presented higher accuracy in the ANN-based predictions than in the EPR model.

4. Conclusions and Perspectives

The available classical methods require a broad understanding of the governing mechanisms of the convergence–confinement method and programming skills to derive the GRC for the elasto-plastic, strain-softening rock mass. Considering the proven ability of AI techniques in the prediction of the P_i of circular tunnels [4], the applicability of another intelligent method is investigated in this study to obtain even more accurate predictions. To do this, the performance of two different ANN types, namely, MLP and RBF, were evaluated and compared. The described methods were applied to the GRC development in both the Mohr–Coulomb and the Hoek–Brown rock mass cases.

In this regard, available solutions with regard to the problem were collected, and a total of 168 and 547 datasets were compiled for the Mohr–Coulomb and the Hoek–Brown cases, respectively. Elastic rock mass properties, peak and residual strength parameters, softening and geometrical parameters, in situ stress, and the convergence of tunnel were employed as the input parameters to predict the internal P_i.

After building various one- and two-layer neural networks with the TANSIG transfer function and assessing the values of the coefficient of determination, root-mean-square error, and mean absolute error, the following conclusions were obtained:

• The ANN-based method appeared to be a highly performant method, applicable to the the development of GRC and the estimation of the P_i of circular tunnels;
• The 15-5-10-1 ($R^2=99.48\%$, $RMSE=0.03883 \mathrm{MPa}$, and $MAE=0.02825 \mathrm{MPa}$) and 15-15-1 ($R^2=99.21\%$, $RMSE=0.050576 \mathrm{MPa}$, and $MAE=0.040918$ MPa) networks were reported as the best MLP and RBF networks for the Mohr–Coulomb case, respectively;
• For the Hoek–Brown case, the 17-5-15-1 MLP ($R^2=99.91\%$, $RMSE=0.179285 \mathrm{MPa}$, and $MAE=0.12516 \mathrm{MPa}$) and the 17-3-12-1 RBF ($R^2=93.18\%$, $RMSE=1.558064 \mathrm{MPa}$, and $MAE=1.078099 \mathrm{MPa}$) architectures were the most accurate proposed neural networks;
• It was shown that the overall performance of MLP networks was better than that of the RBF networks for both Mohr–Coulomb and Hoek–Brown cases;
• The results obtained from the comparison between neural network and EPR models proved the superiority of ANN to the EPR in the prediction of P_i;
• The proposed networks can be effectively applied by design engineers and practitioners to accurately, time-effectively, and economically obtain the GRC using a new set of data is available;
• The proposed networks can be successfully applied in conjunction with the support characteristic curve to calculate the proper time of the installation of the tunnels’ supports;
• Regarding the successful application of ANNs to the problem and as suggestion for future works, the applicability of other soft computing techniques (e.g., genetic programming, ant or bee colony, etc.) can be investigated;
• As another perspective of the current research, stress–strain and time-dependent behavior of rock masses can be studied on the basis of the implementation of viscose constitutive models;
• The formation of damaged zones around the tunnel’s surface (which is the subject of new work by the authors) and new EPR and ANN methods for the prediction of pressures are other interesting perspectives suggested by the present paper.