**2. Theoretical Background**

Vibration based inspection (VBI) is a research domain, which has found its place not just in rock engineering but also in other branches of engineering. Much research has dealt with direct or inverse solutions [22], that is, with the assessment of the effect of structural damage on its parameters as well as with the problem of detecting, locating, and quantifying the extent of the problem. In order to understand the behavior of longitudinal wave propagation in a bar, a certain explanation must be given. Figure 3 shows a segmen<sup>t</sup> of a bar with cross section A, material density r, and elasticity modulus E, where the infinitesimal element of δx is in equilibrium. If the bar is loaded with dynamical force F(x,t) in the direction of its axis, it will yield a displacement u(x,t).

**Figure 3.** A segmen<sup>t</sup> of a bar with position of infinitesimal element.

By solving the equation:

$$\frac{\partial F}{\partial \mathbf{x}} = \rho A \frac{\partial^2 \mathbf{u}}{\partial t^2} \tag{3}$$

the natural frequencies of the bar can be obtained.

By rearrangemen<sup>t</sup> of the equation, while employing the expression for the velocity *v* = (*E*ρ and considering a solution in the form:

$$u(\mathbf{x},t) = \mathcal{U}(\mathbf{x}) \cdot (A \cos \omega t + B \sin \omega t) \tag{4}$$

where *A* and *B* are constants depending on the initial conditions and ω is the angular frequency, Equation (3) becomes:

$$\frac{\partial^2 U}{\partial x^2} + \frac{\alpha^2}{v^2} U = 0 \tag{5}$$

and it has a solution of the form:

$$
\Delta U = M \cos \frac{a\upsilon}{v} + N \sin \frac{a\upsilon}{v} \tag{6}
$$

where *M* and *N* are constants depending on the bar's boundary conditions.

In the present paper, as will be shown later, it is of interest to determine the natural frequencies of a rock bolt model, where a system with both ends fixed, as well as a system with one end fixed and the other free, is considered. Due to the infinite number of solutions there are an infinite number of natural frequencies, one for each *n* = 1, 2, 3, ... , which can be determined by solving Equation (3):

$$
\omega\_n = \frac{n\pi}{L} \sqrt{\frac{E}{\rho}} \text{ (for both fixed ends)}\tag{7}
$$

$$
\omega\_n = \frac{(2n-1)\pi}{2L} \sqrt{\frac{E}{\rho}} \text{ (for one end fixed, the other free)} \tag{8}
$$

### **3. Experimental Testing of Rock Bolt Models**

### *3.1. Rock Bolt Models*

For the purpose of this research, 51 physical models of grouted bars were made and tested at 1:1 scale. The models were designed to simulate different cases of grouting from the aspect of different grouting percentages and in regard to the position of the grout. Since steel rods of 2100 mm (2000 m bar with 50 mm thread on each side of a bar) were used and the grouting sections had a resolution of 10% of the total length of the rod, the grouted and non-grouted sections were 200 mm long. Therefore, a grouting section is represented with a 200 × 100 × 100 mm element (length × width × height) and the steel bar of 25 mm diameter is centered within this section. Figure 4 shows the schemes of model combinations. The reason for many models is to cover as large as possible a range of grouting percentages and defect positions. Based on the schemes and considering the possibility of generating impulses on both sides of the steel rod (50 mm bar threads are made on each side), a total of 94 testing combinations in respect to different grouting percentages and grou<sup>t</sup> positions were made. In addition, the natural frequencies of the bar alone, i.e., 0% grouted percentage, were determined.


**Figure 4.** Complete scheme of rock bolt laboratory models.

After the wooden framework was filled with a grou<sup>t</sup> based on the grouting scheme from Figure 4, and after smoothing of the upper surface, the framework was removed, and models were left to reach a 28-day strength before being subjected to the tests. Figure 5a shows, as an example, ten models that have a regular increase in the length of the grouted section. Figure 5b refers to, as an example, ten randomly selected rock bolts from each group of different grouting percentages. These two groups will be later used for numerical modelling.

**Figure 5.** Rock bolt laboratory models: (**a**) Group I and (**b**) Group II.

### *3.2. Characteristics of the Models' Materials*

For the construction of the physical models, the reinforcing steel bars B500B with 25 mm diameter were used. Since the parameters of the steel bar can be considered as 'reliable', in the sense that these steel bars were produced under strictly controlled manufacturing conditions, no additional tests for determination of their geometrical and physical and mechanical parameters were performed. In the present study, a cement-based mixture was used as the grout, made from pure Portland cement, water, and filling while no additives were used. During the preparation of the grout, samples were continuously taken in order to determine the physical–mechanical and chemical characteristics. A w/c ratio of 0.42 was used for all models. The laboratory tests to which all samples, were subjected after reaching the 28-day strength are:


**Figure 6.** Conduction of ultrasound testing on grou<sup>t</sup> samples.

All tests were documented, and the results were used as an input for subsequent numerical simulations. Figure 7 shows a wave velocity determined by the ultrasonic tests (a), density (b), and stiffness at a small strain (c). In particular, the figure shows the results for samples taken for the purpose of preparing a rock bolt model with 30% of grouted section, as an example. Since numerical analyses require only one value input, the testing results are averaged. This procedure has been carried out for samples of all rock bolt models.

**Figure 7.** Laboratory test results for 30% grouted model: (**a**) ultrasound wave velocity, (**b**) density,and (**c**) small strain stiffness.

(**c**)

(**a**) 

It could be seen from Figure 7 that the difference in wave velocities for all samples is up to 10%, while the density difference is up to 1.5%. A small strain stiffness difference is up to 18% which is expected taking into consideration that the square of wave velocity value is implemented into the equation. These exemplary results are in line with laboratory test results for all samples taken for preparation of all rock bolt models.
