*3.6. Regression Analysis*

Figure 10 shows the first three frequencies of each testing combination, set in the corresponding diagrams. Here, the distribution scatter plots show the pairs of values of natural frequency vs. grouting percentage. The presented scatter plots are a starting point in the correlation and regression analysis for establishing a connection between the grouting percentage and the first three natural frequencies for the whole dataset for each frequency (average trendline marked as 'A'). As an optimal regression function, the second order polynomial function was applied. The fitting uses the least squares method to minimize the squares of the residual deviations. The value of the coe fficient of correlation implies no correlation or minor correlation when 0 ≤ R < 0.2, mild correlation for 0.2 ≤ R < 0.4, significant correlation for 0.4 ≤ R < 0.7, and high or very high correlation for 0.7 ≤ R ≤ 1.0. In this case, the correlation coe fficients R of the regression polynomial functions describing the overall dependence of the natural frequencies on the grouting percentage have the values of 0.79 (first natural frequency), 0.83 (second natural frequency), and 0.86 (third natural frequency). These values of correlation coe fficients clearly demonstrate benefits of analyzing first, second, and third natural frequency for determination of the grouting percentage along the rock bolt. Therefore, better understanding of rock bolt dynamic response was achieved when compared to the Kovaˇcevi´c et al. [3] study where it was shown that there is no clear correlation when only dominant frequency is observed. The diagrams on Figure 10 also show the theoretical function (T curves), obtained by the means of Equations (10) and (11). The trend and values of a T-curve from Figure 10, are influenced by the grouting percentage and grou<sup>t</sup> characteristics. The overall trend of natural frequency increase with the decrease of grouting percentage is evident for both theoretical and experimental curves. However, certain deviations of experimental results are noticed in comparison to theoretical values and these can be attributed to additional influence of grou<sup>t</sup> position on natural frequencies values. Additionally, minimum and maximum boundaries for each natural frequency are stressed out.

**Figure 10.** *Cont*.

**Figure 10.** Regression functions for natural frequencies vs. grouting percentage for: (**a**) first, (**b**) second and (**c**) third natural frequency.

### **4. Numerical Modelling of Grouting Defects**

In order to verify the experimental results and to further analyze the applicability for rock bolts installed in rock mass, a numerical modelling was conducted. Within the scope of this paper, two groups of rock bolt models were selected among the experimental models, as shown in Figure 5:


### *4.1. Determination of Numerical Input Parameters*

The laboratory test presented in Section 3.2 were conducted for all samples taken for the preparation of all the rock bolt models. An overview of material characteristics used as input for all numerical simulations is given in Table 2.


**Table 2.** Material properties of steel bar and grou<sup>t</sup> mixtures.

### *4.2. Composition of Numerical Models*

Numerical analyses were carried out using a finite element method Ansys Workbench computer program, which is widely accepted in a wide range of human activities and is very often applied for scientific research purposes. Of the many modules offered by Ansys Workbench, in this case, the Harmonic Response module was used, since it enables the acquisition of a full frequency spectrum for the result of the force applied to the rock bolt head.

To carry out the analysis, the following steps were followed:


Geometrical models were created using the Autocad 3D 2019 computer program, which is a sophisticated universal utility that supports three-dimensional modelling of complex objects. The advantage of its application is its simplicity and compatibility with the selected Ansys Workbench program. For the purposes of the analysis, the two groups of rock bolts that were analyzed through the experimental setup, Group I and Group II, were modelled. They are shown in Figure 11.

**Figure 11.** Rock bolt geometrical models: (**a**) Group I and (**b**) Group II.

Since the numerical analyses in the presented research are dynamic analyses, which imply small strains, the constituent model chosen for all materials is linear-elastic, which assumes that the stresses are directly proportional to deformation. Once the constitutive model and the material characteristics of the material are defined, it is necessary to define the bonds between the individual materials. Since this numerical modelling employs a linear dynamic analysis, it is recommended to avoid nonlinear contacts. Therefore, the "no separation" or "bonded" contacts from a range of types of contact between the steel and the grou<sup>t</sup> could be chosen. The "no separation" contact configuration can be applied to the sides of the region or to its edges, and by its application the separation of the sides that are in contact is not allowed, but a little slip of the sides may appear without friction. However, in case the so-called "bonded" type of contact is chosen, the 3D contact surface–surface element (contact 174) is selected for simulation of contacts between the target surfaces (steel–grout). A bonded contact, used in this research, can be applied to all regions that are in contact and in that case sliding or separation is not allowed, and the bodies tied to this contact act as mutually glued. As a method for simulating contacts, the so-called MPC (multi-point constraint) method is chosen where the contact is directly and efficiently formulated due to the internal addition of edge shift equations to match the displacement between the contact points. Since the magnitude of the force applied to the rock bolt head was not considered in experimental testing, it did not have a significant impact on the numerical modelling results. Namely, by varying the force magnitude, frequency spectra with equal values of natural frequencies were obtained which, depending on the magnitude of the force applied, have di fferent amplitudes and these are of no interest for the subject of this research. Since the load magnitude has no e ffect on the value of the natural frequencies but only on their amplitudes, an anchorage load of 1 N has been imposed on the rock bolt head in all numerical analyses, in the direction of the anchor axis. The model discretization was performed in such a way that the steel bar was divided into 50 mm length elements while the grou<sup>t</sup> was divided into 50 × 50 × 50 mm bodies. With mesh refinement, the numerical results are closer to the experimental, but the calculation time is prolonged. Figure 12a shows the discretization of the fully grouted model (S100\_1), while Figure 12b shows the discretization of the rock bolt model S60\_7 from Group II.

**Figure 12.** Discretization of numerical models: (**a**) S100\_1 and (**b**) S60\_7.

### *4.3. Comparison of Numerical Modelling and Experimental Results*

The results of the numerical analysis are presented as a comparison of the first three natural frequencies with the results of the laboratory model testing of models from Group I and Group II. Figure 13 shows the comparison between calculated and measured values of natural frequencies for the Group I and Group II, respectively.

The difference between the numerically calculated first three frequencies and the experimentally measured first three frequencies for both groups of selected groups of models, Group I and Group II, are up to 12%, which can be considered acceptable taking into account possible irregularities in the production of laboratory models (despite the strictly controlled conditions), as well as variations in the measuring environment. Accepting the above, the further numerical calculations include the consideration of grou<sup>t</sup> stiffness variation influence on the dynamic response of models.

**Figure 13.** Comparison of the values of first, second, and third calculated natural frequency and experimentally obtained values for: (**a**) Group I and (**b**) Group II.

### *4.4. Analysis of Grout Sti*ff*ness Influence*

Following the comparison of numerical and experimental results, the next phase of numerical modelling implies variation of small strain sti ffness of grouts for all models within Groups I and II, in order to evaluate the influence of sti ffness on the natural frequencies. Considering that the numerical input, besides values of elasticity modulus, requires a density value input, it is selected in all analysis as constant reference value of 2000 kg/m3. Used values of elasticity modulus, density, and (calculated) wave velocities are shown in Table 3.

A range of elasticity modulus from 20 to 40 GPa is varied since these values represent commonly achieved values for rock bolt systems. Figure 14 shows the influence of the modulus of elasticity of the grouting mixture on the values of the natural frequencies for Group I, while in Figure 15 shows the same for Group II. A clear reduction of the natural frequency values is apparent from Figures 14 and 15 with a decrease of the modulus of elasticity of the grout. This reduction is least pronounced for the first natural frequency and is most clearly expressed for the third natural frequency. In addition, the reduction of the value of the natural frequencies is generally more significant in the models with a higher percentage of grouting, although locally there are values that do not meet the specified. The variation of the elastic modulus does not, however, a ffect the general trend of increasing natural frequency with the reduction in grouting percentage, whereby the shape of the natural frequency—the grouting percentage curve is independent of the value of modulus of elasticity.


**Table 3.** Parameters for analysis of grou<sup>t</sup> sti ffness influence.

 **Figure 14.** *Cont*.

 **Figure 14.** Influence of grou<sup>t</sup> stiffness on (**a**) first, (**b**) second, and (**c**) third natural frequency of Group I models.

(**c**)

**Figure 15.** Influence of grou<sup>t</sup> stiffness on (**a**) first, (**b**) second, and (**c**) third natural frequency of Group II models.
