# Estimation of Modulus of Deformation Using Rock Mass Rating—A Review and Validation Using 3D Numerical Modelling

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## Abstract

**:**

_{d}) of a rock mass is an essential input parameter required in the design of underground excavations. This study involves collecting the results of extensive in situ tested values for various hydroelectric projects in the Himalayan regions, along with the rock mass rating (RMR) values at 35 test sites. E

_{d}is estimated empirically based on statistical analysis. Comparisons were made with the empirical equations already available in the literature, using RMR and the proposed equation for estimating E

_{d}. Although different researchers have proposed many equations for estimating the value of E

_{d}using RMR, a gap exists in validating such equations. In this regard, the proposed equation for E

_{d}was verified by carrying out 3D numerical-modelling studies using FLAC3D, an explicit finite-difference software for an underground powerhouse cavern and comparing the displacement values with the field instrumentation data.

## 1. Introduction

_{d}) of a rock mass typically provides information about the deformation characteristics, i.e., elastic and plastic behavior when the rock mass is subjected to loading and unloading conditions. Joint friction parameters and rock strength play an essential role in the deformation mechanics of rock mass in addition to the E

_{d}value [7]. In recent years, there has been an advancement in numerical tools for analyzing the support system for underground excavations. The output from these numerical tools, however, depends on the reliability of the input data. The E

_{d}value is one of the critical design parameters required for numerical modelling [8] in the design of dam structures and underground excavations.

_{d}is defined as the ratio of stress to strain (elastic and plastic) during the loading of a rock mass, whereas the modulus of elasticity (E

_{e}) is defined as the ratio of stress to strain (elastic) during the unloading of a rock mass. Hence, while carrying out any in situ testing, estimating the E

_{e}value along with E

_{d}is a general practice.

_{d}are the plate-jacking test (PJT) or uniaxial-jacking test, plate-loading test (PLT), and flat jack test (FJT) carried out in drifts or small tunnels. In contrast, the dilatometer test (DT) and goodman jack test (GJT) are conducted in boreholes of NX size [10]. The size of the drift or gallery required for carrying out the in situ testing needs to be as small as required for carrying out the test. During loading and unloading of the rock mass area, the deformations are measured using a multipoint borehole extensometer (MPBX) in boreholes and a linear variable differential transformer (LVDT) case for measuring surface displacements, which are used in determining the in situ value of E

_{d}. However, in situ tests are complicated, expensive, and time-consuming [6]. In addition, each type of in situ test will result in different values due to differences in test procedures and rock mass damage due to blasting [11,12].

_{d}based on rock mass classification systems, such as rock mass rating (RMR), tunneling-quality index (Q), geological strength index (GSI); and intact rock properties, such as uniaxial compressive strength (UCS), Young’s modulus (Ei), disturbance factor (D), and weathering degree (WD). These empirical equations are developed based on the data collected for a particular location and rock type. Using these equations to estimate the deformation modulus value at other sites may not yield correct values.

_{d}based on intact properties (UCS and Ei) gave less reliable results when compared with those of rock mass classification systems. Although many empirical relations are available in the literature for estimating E

_{d}, only those equations with an RMR value as the input parameter are considered in this study [12,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31], since this is the most widely accepted method of rock mass characterization. The RMR classification system was developed by Bieniawski (1974, 1989) [32], updating the charts and tables for the six parameters, i.e., intact rock UCS, rock quality designation (RQD), spacing of joint or discontinuity, joint condition, condition of groundwater, and orientation of joint set. The rock masses at the in situ test locations were classified based on the RMR. Drillability studies conducted on rocks also provided insight into the petrophysicomechanical properties that indicated the influence of various petrographic, physical, and mechanical properties of rock [33,34,35]. From the numerical modelling studies [36], it was observed that when the in situ tested value of E

_{d}is in the range of 1 to 3 GPa, the predicted displacements were almost thrice the measured values. However, suppose the rock mass is too competent, as studied in [37], it can be noted that the in situ value of E

_{d}is higher when compared with that of the back-calculated value from numerical modelling. Depth also was found to influence the E

_{d}values in discontinuum models compared to that of continuum models. At shallow depths, the discontinuities deformed significantly in comparison with that of deeper depths [38]. In addition, studies were carried out for understanding the variations in joint set sizes and orientations on the directional deformation modulus for rock mass [39].

_{d}is the critical design parameter for the design of large underground excavations, which needs to be determined correctly, and which otherwise has the potential to result in time and cost over-runs. Determining E

_{d}values by in situ testing will have huge financial implications for the project. Thus, this study aims to develop a predictor model for estimating the E

_{d}value using the RMR, which can be useful to the designers or project authorities for design of underground excavations if there is a lack of in situ tested data for projects in the Himalayan region.

## 2. Methodology

_{d}values based on the existing empirical relations using the values of the rock mass rating. A new empirical equation is proposed to be developed considering the available in situ tested data from the projects constructed in the Himalayan region. A comparison is made for the value of E

_{d}concerning the existing equations and the newly proposed equation. Finally, 3D numerical modelling studies are carried out considering the value of E

_{d}determined in situ and the value obtained from the proposed equation and comparing the model displacement values with that of the measured values. The empirical equations considered in the study, along with their limitations, coefficient of regression (R

^{2}), number of data sets considered by worldwide researchers, range of RMR values, country of origin, and the lithology considered while developing the relations, are given in Table 1.

_{d}and E

_{e}for 35 test locations in the Himalayan region spanning over India, Bhutan, and Nepal from the published literature [40,41,42,43,44,45] and the National Institute of Rock Mechanics (NIRM) reports [46,47,48,49,50]. In situ tests conducted at the study locations are PLT (deformations measured at the surface), PJT (deformation measured inside the boreholes), carried out in drifts, and the Goodman jack test, carried out in boreholes. The in situ test locations from where data are collected are shown in Figure 1. The in situ tested values of E

_{d}, E

_{e}, and the corresponding RMR values for the identified 35 site locations are shown in Figure 2 (a) and (b), respectively.

_{d}, and an equation to predict E

_{d}from the RMR was proposed. The reliability and predictability of the proposed and the available equations were compared using statistical tools, and the reliable equation for the Himalayan region was presented. The equation was validated using the tested and estimated values in the 3D numerical model developed for Tala Hydroelectric Project, Bhutan. The instrumentation data were utilized for making comparisons with those of the modelling results.

## 3. Statistical Analysis

_{d}values are from 0.118 to 11.591 GPa. It is also observed that the cubic function given in Equation (19) has the highest value of the coefficient of regression (R

^{2}), i.e., R

^{2}= 0.75 when compared to other functions, as shown in Figure 3.

_{d}and RMR values was made for rock types and the cubical function is shown in Figure 4.

_{d}values were calculated for the RMR values at in situ tested locations based on the empirical relations proposed by different authors and are shown in Figure 5. It is noticeable from Figure 5 that Equation (19) closely matches with that of the in situ tested-values curve. The empirical equations proposed by [15,21,22,30] are also in good comparison with that of the in situ tested value. The empirical equations proposed by [12,24,26,28,29] overestimated, and the remaining equations underestimated the E

_{d}values.

## 4. Case Study—Tala Powerhouse Complex

#### 4.1. Geology

#### 4.2. 3D Numerical Modelling

#### 4.3. Excavation Sequence and Support System

#### 4.4. Material Properties

_{d}value of 2.89 GPa (Model A), based on the empirical Equation (19) for an RMR value of 55, and another model with an in situ tested (PLT), E

_{d}value of 6.793 GPa [35,59] (Model B). Other material properties considered in the present analysis for both models are a density of 2650 kg/m

^{3}, cohesion of 2.28 MPa, and friction angle of 28.3° [53].

#### 4.5. Comparison of Modelling Results with Instrumentation Data

_{d}value of 2.89 GPa and Model B with an E

_{d}value of 6.793 GPa, are shown in Figure 8.

_{d}value is on the higher side, enhancing the rock mass properties. Measured convergence matched well in Model A compared to Model B. Hence, the relation proposed in Equation (19) can be utilized to estimate the value of E

_{d}. The displacement contours (in m) at RD 65 m after the complete excavation of the powerhouse complex for Models A and B are shown in Figure 9 (a) and (b), respectively.

## 5. Conclusions

- The review of various empirical models available for estimating E
_{d}values indicates a considerable variation in the value of the deformation modulus for the Himalayan region. The empirical equations proposed by [14,20,21,29] are also in good comparison with the in situ tested value of E_{d}, while equations proposed by [11,23,25,27,28] overestimate, and the remaining equations underestimate E_{d}values. - Based on the data obtained from 35 test locations, a predictive cubic equation (Equation (19)) could be developed, with R
^{2}, RMSE, and VAF values of 0.75, 1.70, and 74.33, respectively. These values indicate higher predictability and maximum accounted-for variance in E_{d}compared with other available correlations available in the literature. - The 3D numerical modelling results show that the E
_{d}value adopted based on the proposed Equation (19) (Model A) correlated well with that of the measured instrumentation data when compared with the value of E_{d}based on the in situ testing (Model B). Model B underpredicts the deformations in the powerhouse complex at all locations, indicating that the in situ tested E_{d}value is higher, enhancing the rock mass properties. Measured convergence matched well in Model A compared to Model B. Hence, the relation proposed in Equation (19) can be utilized to estimate the value of E_{d}. - From the in situ tested data, the average ratio of E
_{e}/ E_{d}for the Himalayan region is 1.5. - The proposed equation validates rock masses from the Himalayan region, with RMR values ranging from 15 to 70.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Google Earth image showing the location of the in situ tests carried out in different hydroelectric projects situated in the Himalayan region.

Equation No. | Ref. | Year | Equation | Type of Equation | R^{2} | Limitations | Data Sets Used | RMR Range | Country of Origin | Lithology |
---|---|---|---|---|---|---|---|---|---|---|

(1) | [12] | 1978 | ${E}_{d}=2RMR-100$ | Linear | - | RMR > 50 | 3 Sites | 51–85 | South Africa | Shale, siltstone, dolerite, mudstone, and sandstone (hard rocks). |

(2) | [14] | 1983 | ${E}_{d}={10}^{(RMR-10)/40}$ | Power | - | RMR ≤ 50 | 15 | 26–83 | - | Dolerite, sandstone, mudstone, shale, siltstone, gneiss, and granite (soft rocks). |

(3) | [24] | 1992 | ${E}_{d}={10}^{(RMR-20)/38}$ | Power | 0.91 | - | 120 | - | India | |

(4) | [25] | 1993 | ${E}_{d}={0.03e}^{0.07RMR}$ | Exponential | - | - | - | - | - | |

(5) | [26] | 1996 | ${E}_{d}={e}^{(4.407+0.081RMR)}$ | Exponential | - | - | - | - | Croatia | Limestone |

(6) | [27] | 1997 | ${E}_{d}={0.0000097RMR}^{3.54}$ | Power | - | - | - | - | - | Gneiss, granite, and sandstone. |

(7) | [28] | 1999 | ${E}_{d}={0.1\left(\frac{RMR}{10}\right)}^{3}$ | Power | - | - | 15 | 26–83 | New Zealand | Graywacke, sandstones, and mudstones. |

(8) | [29] | 1999 | ${E}_{d}={{(7\pm 3)(10}^{(RMR-44)/21})}^{0.5}$ | Non-linear | - | - | - | - | Various | |

(9) | [30] | 2003 | ${E}_{d}={0.0736e}^{\left(0.0755RMR\right)}$ | Exponential | 0.62 | - | 115 | 20–85 | Various | Quartzdiorite, limestone, and shale. |

(10) | [31] | 2003 | ${E}_{d}=19.43\mathrm{ln}RMR-69.03$ | Logarithm | - | - | 57 | 38–84 | Turkey | Grey and pinky quartzdiorite. |

(11) | [15] | 2006 | ${E}_{d}=0.3228{e}^{\left(0.0485RMR\right)}$ | Exponential | 0.36 | - | 8 Sites | - | Korea | |

(12) | [16] | 2008 | ${E}_{rm}=6.7RMR-103.06$ | Linear | 0.94 | RMR ≥ 27 | 9 | 27–61 | Turkey | Graywacke |

(13) | [17] | 2010 | ${E}_{d}=0.0003{RMR}^{3}-0.0193{RMR}^{2}+0.315RMR+3.4065$ | Polynomial | 0.8446 | - | 42 | 10–85 | Iran | Limestone and marble |

(14) | [18] | 2012 | ${E}_{d}=110{e}^{{-\left(\frac{RMR-110}{37}\right)}^{2}}$ | Gaussian function | 0.932 | - | 43 | - | Various | Mudstone, siltstone, sandstone, shale, dolerite (hard rocks), granite, gneiss, mudstone, siltstone, sandstone, shale, and dolerite (soft rocks). |

(15) | [19] | 2013 | ${E}_{d}={10}^{(RMR-16)/50}$ | Power | 0.64 | - | 420 | 7–92 | Korea | Gneiss |

(16) | [20] | 2014 | ${E}_{d}=0.1627RMR-5.0165$ | Linear | 0.6709 | - | 52 | 30–76 | Iran | Sandy siltstone, mudstone, conglomerate, sandstone, dislocated rock mass, faulted rock mass, and shear zone. |

(17) | [21] | 2015 | ${E}_{d}={0.058e}^{\left(0.0785RMR\right)}$ | Exponential | 0.97 | - | 4 Sites | - | Turkey | Basalt, tuffites, and diabases. |

(18) | [22] | 2013 | ${E}_{d}=9E-{7RMR}^{3.868}$ | Power | 0.89 | - | 82 | 39–85 | Iran | Grey-green schist, phyllite, dark grey to black limestone, and limy dolomite. |

S.No. | Type of Equation | Equation | Coefficient of Regression, R^{2} |
---|---|---|---|

1 | Linear | 0.183RMR − 5.81 | 0.53 |

2 | Logarithmic | 5.8log(RMR) − 19.17 | 0.37 |

3 | Cubic | $0.00011{RMR}^{3}-{0.0083RMR}^{2}+0.2RMR-1.3$ | 0.75 |

4 | Exponential | $0.0352{e}^{0.0798RMR}$ | 0.708 |

Cavern | Support System |
---|---|

MHC–Crown | 32 mm diameter, 8 m and 6 m long rock bolts at 1.5 m × 1.5 m pattern Steel-fiber-reinforced shotcrete (SFRS) of 100 mm thickness Steel ribs of ISMB 300 at 0.6 m spacing 32 mm/26.5 mm diameter, 12 m long Dywidag rock bolts at 1.5 m spacing |

MHC–Walls | 32 mm/26.5 mm diameter, 12 m long Dywidag rock bolts at 1.5 m spacing |

THC–Crown | 32 mm diameter, 8 m and 6 m long rock bolts at 3 m × 1.5 m pattern Steel-fiber-reinforced shotcrete (SFRS) of 100 mm thickness Steel ribs of ISMB 350 at 0.6 m spacing |

THC–Walls | 32 mm/26.5 mm diameter, 8 m long Dywidag rock bolts at 1.5 m spacing |

MHC and THC Walls | Initial layer of shotcrete of 50 mm thickness Welded-wire mesh of 100 mm × 100 mm × 5 mm Final two shotcrete layers of 50 mm each |

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**MDPI and ACS Style**

Bellapu, H.V.S.; Sinha, R.K.; Naik, S.R.
Estimation of Modulus of Deformation Using Rock Mass Rating—A Review and Validation Using 3D Numerical Modelling. *Sustainability* **2023**, *15*, 5721.
https://doi.org/10.3390/su15075721

**AMA Style**

Bellapu HVS, Sinha RK, Naik SR.
Estimation of Modulus of Deformation Using Rock Mass Rating—A Review and Validation Using 3D Numerical Modelling. *Sustainability*. 2023; 15(7):5721.
https://doi.org/10.3390/su15075721

**Chicago/Turabian Style**

Bellapu, Hema Vijay Sekar, Rabindra Kumar Sinha, and Sripad Ramchandra Naik.
2023. "Estimation of Modulus of Deformation Using Rock Mass Rating—A Review and Validation Using 3D Numerical Modelling" *Sustainability* 15, no. 7: 5721.
https://doi.org/10.3390/su15075721