## 3 Theoretical Background for the Convergence Confinement Method  
The convergence confinement method (CCM) serves as a straightforward analytical tool capable of estimating the deformational load acting on an excavation support. Initially, the CCM method was based on the stress state solution of anelastic disc of a uniform thickness proposed by [PERSON] [23]. Over subsequent years, the method underwent further refinement and enhancement, including considerations such as the development of the plastic zone around the excavation [24; 25; 26; 27; 28]. Regardless of the specific yield criterion used, these adaptations were based on the fundamental assumption of rock mass continuity [8]. [PERSON] [29] conducted research on potential disruptions in this continuity, leading to the formulation of the hypothesis that appropriate radial deformability and the timing of support installation are critical factors in ensuring both safety and cost-effectiveness in excavation design. CCM is a graphical method consisting of three basic components [22; 30]:  
* Ground reaction curve (GRC)--illustrates the correlation between fictitious support pressure and the radial displacement of the excavation boundary. In essence, fictitious pressure refers to the internal support pressure required to prevent further ground displacement. * Support characteristics curve (SCC)--depicts the relationship between increases in the radial displacement of the support and the external radial stress applied to the support (deformational pressure). * Longitudinal displacement profile (LDP)--depicts the correlation between the radial displacement of the excavation boundary and the longitudinal distance from the excavation face. In comparison to the design method advocated by Polish Standards, the convergence-confinement method is capable of:  
* More accurately considering the three-dimensional character of an underground roadway or chamber excavation. * Considering rigid support systems, which enable the quick suppression of rock mass displacement. * Considering the correlation between support system stiffness and the value of the excavation boundary displacement. * Considering the deformational load acting on the designed support system. Algorithm for the Establishment of a Ground Reaction Curve for an Elasto-Plasto-Fractured Rock Mass  
An important issue connected with the use of the CCM method is the difficulty in evaluating the real actions of rock masses within inelastic zones [31]. The simplest procedure for determining a GRC curve is based on the model of an elasto-perfectly-plastic medium [2]. [PERSON] and [PERSON] claim that such a model can only be used for rock masses of low quality (GSI! 30) [32]. Therefore, for a rock mass of medium (30! GSI! 70) or high (GSI? 70) quality, a number of alternative equations have been introduced over the years, considering such issues as post-failure acting, including rock mass softening, stiffness degradation, and dilatancy [31; 33]. The development of a three-phase rock mass model, based on the model of elasto-plasto-fractured medium, was considered an important achievement of Polish mining engineering [18]. This three-phase model was introduced into Polish Standards [14; 15; 16; 17] and successfully applied in the design of underground roadways and chambers located at significant depths in the vicinity of low and medium-quality rock masses. The properties of an elasto-plasto-fractured medium may by depicted by polyline (Figure 2) [18]:  
* Within the elastic, the zone action of the rock mass is depicted by a straight line inclined by the angle \(arctg\big{(}E_{g}\big{)}\), where \(E_{g}\) is a rock mass deformation modulus. * Within the plastic zone a following condition, depicting rock mass action, is met\[\frac{R_{cg}}{E_{g}}<\varepsilon_{t}<\varepsilon_{ng}, \tag{8}\]  
where:  
\(R_{cg}\)--rock mass compressive strength in the elastic zone, MPa,  
\(E_{g}\)--rock mass Young modulus, MPa,  
\(\varepsilon_{t}\)--longitudinal strain of the rock mass.