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

^{1}

^{2}

^{*}

## Abstract

**:**

## 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

_{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

_{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.

_{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.

_{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.

## 2. Methodology

_{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

## 3. Performance Evaluation

#### 3.1. The Mohr–Coulomb Criterion

_{i}for the Mohr–Coulomb strength criterion. In this case, the overall performance of MLP-ANN was better than that of the RBF-ANN.

_{i}versus the calculated P

_{i}.

#### 3.2. The Hoek–Brown Criterion

_{i}of tunnels in the Hoek–Brown case. As indicated in Table 4, 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.

_{i}in the Hoek–Brown case 17-5-15-1 is shown in Figure 5.

_{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

_{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

_{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.

_{i}.

- 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.

## Author Contributions

## Conflicts of Interest

## Nomenclature

Symbol | Description | Unit | Symbol | Description | Unit |

${a}^{res}$ | Residual a constant | [-] | ${w}_{j}$ | Weight between neurons | [-] |

${a}^{peak}$ | Peak a constant | [-] | $\overrightarrow{x}$ | Input | [-] |

${c}_{i}$ | Calculated value | [-] | ${y}_{j}$ | Gaussian basis function | [-] |

${c}^{peak}$ | Peak cohesion | MPa | $\gamma $ | Unit weight | kN/m^{3} |

${c}^{res}$ | Residual cohesion | MPa | $\eta $ | Softening parameter | [-] |

${e}_{x}$ | Weighted sum of the inputs | [-] | ${\eta}^{*}$ | Critical softening parameter | [-] |

$E$ | Young’s modulus | GPa | $\vartheta $ | Poisson’s ratio | [-] |

$GS{I}^{peak}$ | Peak geological strength index | [-] | ${\overrightarrow{\mu}}_{j}$ | Center of the Gaussian basis function | [-] |

$GS{I}^{res}$ | Residual geological strength index | [-] | ${\sigma}_{0}$ | In-situ stress | MPa |

$MAE$ | Mean absolute error | [-] | ${\sigma}_{ci}$ | Uni-axial compressive strength | MPa |

${m}_{i}$ | mi constant | [-] | ${\sigma}_{j}$ | Spread of the Gaussian basis function | [-] |

${m}^{peak}$ | Peak m constant | [-] | ${\sigma}_{r}$ | Radial stress | MPa |

${m}^{res}$ | Residual m constant | [-] | ${\sigma}_{\theta}$ | Tangential stress | MPa |

$n$ | Number of datasets | [-] | ${\phi}^{peak}$ | Peak friction angle | ° |

${p}_{i}$ | Predicted value | [-] | ${\phi}^{res}$ | Residual friction angle | ° |

${P}_{i}$ | Support pressure | MPa | $\psi $ | Dilation angle | ° |

$r$ | Distance from the tunnel center | m | ${\psi}^{peak}$ | Peak dilation angle | ° |

${R}^{2}$ | Coefficient of determination | [-] | ${\psi}^{res}$ | Residual dilation angle | ° |

$RMSE$ | Root-mean-square error | [-] | ${\omega}^{peak}$ | Peak strength parameters | [-] |

${s}^{peak}$ | Peak s constant | [-] | ${\omega}^{res}$ | Residual strength parameters | [-] |

${s}^{res}$ | Residual s constant | [-] | ${U}_{i}$ | Radial displacement | m & mm |

$TANSIG$ | Tangent hyperbolic function | [-] |

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**Figure 1.**Description of: (

**a**) different zones and corresponding strength parameters; (

**b**) a circular tunnel in an elastic-plastic, strain-softening rock mass; (

**c**) corresponding relationships for strength parameters; (

**d**) governing equilibrium equation; (

**e**) schematic view of a typical ground reaction curve (GRC) and parameters (Hoek–Brown case) affecting on the curve.

**Figure 2.**The topology of the optimum RBF-ANN (15-15-1) developed for the prediction of the P

_{i}(Mohr–Coulomb criterion).

**Figure 3.**(

**a**) ANN-based predicted P

_{i}versus calculated P

_{i}; (

**b**) differences between ANN-based predicted P

_{i}and calculated P

_{i}for the best MLP-ANN (15-5-10-1) applied to the testing data series for the Mohr–Coulomb criterion.

**Figure 4.**(

**a**) ANN-based predicted P

_{i}versus calculated P

_{i}; (

**b**) differences between ANN-based predicted P

_{i}and calculated P

_{i}for the best RBF-ANN (15-15-1) for the Mohr–Coulomb criterion.

**Figure 5.**The topology of the optimum MLP-ANN (17-5-15-1) developed for the prediction of the P

_{i}(Hoek–Brown criterion).

**Figure 6.**(

**a**) Neural predicted P

_{i}versus calculated P

_{i}; (

**b**) differences between neural predicted P

_{i}and calculated P

_{i}for the best MLP-ANN (17-5-15-1) applied to the testing data series for the Hoek–Brown criterion.

**Figure 7.**(

**a**) Neural predicted P

_{i}versus calculated P

_{i}; (

**b**) differences between neural predicted P

_{i}and calculated P

_{i}for the best RBF-ANN (17-3-12-1) applied to the testing data series for the Hoek–Brown criterion.

**Figure 8.**Comparison between the performance of (

**a**) 15-5-10-1 MLP-ANN and evolutionary polynomial regression technique (EPR) model (Mohr–Coulomb case); (

**b**) 17-5-15-1 MLP-ANN and EPR model (Hoek–Brown case) in the prediction of the P

_{i}.

**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 |

**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 |

**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 | |||||

R^{2} (%) | RMSE (MPa) | MAE (MPa) | R^{2} (%) | 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 |

**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 | |||||

R^{2} (%) | RMSE (MPa) | MAE (MPa) | R^{2} (%) | 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 |

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Ghorbani, A.; Hasanzadehshooiili, H.; Sadowski, Ł.
Neural Prediction of Tunnels’ Support Pressure in Elasto-Plastic, Strain-Softening Rock Mass. *Appl. Sci.* **2018**, *8*, 841.
https://doi.org/10.3390/app8050841

**AMA Style**

Ghorbani A, Hasanzadehshooiili H, Sadowski Ł.
Neural Prediction of Tunnels’ Support Pressure in Elasto-Plastic, Strain-Softening Rock Mass. *Applied Sciences*. 2018; 8(5):841.
https://doi.org/10.3390/app8050841

**Chicago/Turabian Style**

Ghorbani, Ali, Hadi Hasanzadehshooiili, and Łukasz Sadowski.
2018. "Neural Prediction of Tunnels’ Support Pressure in Elasto-Plastic, Strain-Softening Rock Mass" *Applied Sciences* 8, no. 5: 841.
https://doi.org/10.3390/app8050841