**1. Introduction**

Ensuring the stability of rock mass excavation is a challenging task for all persons involved in a construction process. Any kind of instability, which is a result of an inadequate support system can lead to undesirable temporal, financial and, what is most important, safety consequences. For decades, there has been a practice of using cement-based grouted rock bolts for rock mass reinforcement and much experience, both positive and negative, has been gained through their use. Rock bolts are an inevitable part of a large number of projects, such as rock slopes (Figure 1), where different levels of rock bolt design are used, ranging from empirical, analytical, and numerical, the latter having the most expanding applications.

Regardless of the design method used and the level of detail of the rock bolt parameter assignment, the usual practice is to take into consideration fully grouted rock bolts, without any grou<sup>t</sup> defects along their length. Although some authors [1] have stated that grouted rock bolts present the optimal reinforcement method for strengthening weak or fractured rocks, a number of authors point to the fact that such rock masses may cause issues with grouting quality. For example, application of grouted rock bolts in karstic rock mass, characterized by the presence of voids and discontinuities, can lead to the severe grou<sup>t</sup> loss [2,3], limiting their applicability in these types of rock masses [4,5]. The presence of karstic phenomena can lead to poor grouting, as shown in Figure 2. However, most of rock mass discontinuities are non-persistent, even though their visible traces can lead to conclusions as if they were open fractures. Assumption of full persistence gives incorrect predictions of hydraulic conductivity, as well as of grou<sup>t</sup> flow. Since the visible trace length of a discontinuity can be a poor indicator of true persistence, some innovative methods, such as FERM given by Shang et al. [6] may help to detect the internal connections of rock bridges that hinder the flow of grout.

**Figure 1.** An example of the installed rock bolt system.

**Figure 2.** Good and poor grouting quality of a rock bolt.

Although the grouted rock bolts can produce a higher degree of load transfer in comparison to the other types of bolts, the grouting procedures and characteristics of the bonds between the bar and the grou<sup>t</sup> and with the rock mass are the least controlled in practice [7], while the rock mass grouting procedures remain a predominantly empirical practice [8]. The importance of grouting quality can be seen through the potential rock bolt failure mechanisms, where most significant mechanisms include pull-out failures at bar–grout (B–G) contact, as well on grout–rock mass contact (G–RM). The main role of the grou<sup>t</sup> is to provide a mechanism for transferring the load between the rock mass and the reinforcement element, and grouting quality has an important role in this task, along with the interface strength between bolt, grout, and rock, influenced by the adhesion, friction, and mechanical interlocking. Ren et al. [9] conclude that, based on many laboratory and in situ tests, the most common failure type of fully grouted rock bolts is at the bar–grout or grout–rock mass interface. It can be understood that if the grou<sup>t</sup> and rock have similar strengths and if the required grouted rock bolt length is inadequate, then failure could occur at the bolt–grout interface [10]. If the surrounding rock mass is softer, as is the often case with karst, then the failure could happen at the grout–rock interface. There are other pull-out mechanisms which are not directly linked with grouting quality but depend on rock mass characteristics. However, these are out of scope of this paper. The relevant equations for the rock bolt pull-out capacities are:

$$R\_{B \cdot G} = \, d\_{BR} \cdot \pi \cdot l \cdot \mu \cdot \tau\_{B \cdot G} \tag{1}$$

$$R\_{G\cdot RM} = d\_{BH} \cdot \pi \cdot l \cdot \mu \cdot \pi\_{G\cdot RM} \tag{2}$$

The product of bar diameter *dBR* (m) or borehole diameter *dBH* (m), installed rock bolt length l (m), π and unit shear strength of bar–grout contact τ*B-G* or unit shear strength of grout–rock mass contact τ*G-RM* (kN/m2), theoretically give the values of pull-out capacity at bar–grout contact *RB-G* (kN) and pull-out capacity at grout–rock mass contact *RG-RM* (kN), respectively. Since the pull-out resistance can be achieved only on the grouted portion of a rock bolt, it is of importance to properly determine the actual grouting percentage of the installed rock bolt, μ (—). It is a common designer's assumption of a fully grouted rock bolt (μ = 1), but due to the above-mentioned reasons, a μ < 1 value is more appropriate for the rock mass characterized by the presence of voids and discontinuities.

Much research has dealt, with greater or lesser success, with problems of determining the rock bolt grouting quality taking in consideration its importance for the pull-out capacity. Baˇci´c et al. [11] and Song et al. [12] give the overview of the testing methods. Among many acoustic methods, the first one developed for determining grouting quality is the Boltometer method, which has several limitations when applied to soft and highly fractured rock masses. By using this method, the assumption of a lower sound velocity in the grou<sup>t</sup> in comparison to the velocities in the bar and rock mass is made [13], but this may not be the case with karstic rock masses characterized by low stiffness [14]. If the rock mass has similar stiffness as the grout, a large portion of the energy will dissipate in the rock mass, making the reflections more difficult to recognize [15]. Some progress in this field was made through analysis of ultrasound guided waves [16], where the upgraded version of the Boltometer is currently in development under the name of rock bolt tester [17]. When it comes to vibration-based methods, the GRANIT system [18] stands out and it is getting frequently implemented in practice. This non-destructive system is however dominantly oriented towards determining the force in an anchor. Still, many findings of the GRANIT development process have been significant for the implementation of vibration based non-destructive testing of these types of structural elements. Kovaˇcevi´c et al. [3] had the basic objective of determining the dominant frequency, determined from the power spectrum, of rock bolt models and developing a correlation between the dominant frequency and the grouting percentage. A laboratory testing was conducted on 16 models representing different grouting defects and constructed of 25 mm bar embedded in square concrete blocks of 25 cm side length, having the total length of 3 m. Additionally, 30 field models of 3 m length were tested on location of a tunnel. Overall, the research was not continued and no clear link between the dynamic response and the grouting quality was established, but the authors stress that the procedure is very attractive for further research. Other methods based on an analysis of the frequency response are in development, but need further improvement. The potential of using electrical (such as time-domain reflectometry, electrical resistance, or the Mise-a-la-Masse method) and electromagnetic methods (such as ground penetrating radar) for determining grouting quality have been considered by many authors [19–21], but these methods have not found wider application in this domain.

This paper presents a research on the possibility of determining grouting percentage of rock bolts based on an analysis of the dynamical response by considering its natural frequencies after the generation of an artificial impulse to its head. By considering three natural frequencies instead of analyzing only the dominant frequency in the spectrum, as was done by Kovaˇcevi´c et al. [3], a more concrete correlation could be established so that the value of grouting percentage factor (μ) used for verification of designed grouting assumptions is determined in a more reliable manner.
