## 2 Deformational Pressure Model According to Polish Standards  
In the late 1970s, the Industrial Standards BN-78/0434-07 [14] and BN-79/0434-04 [15] were introduced into Polish engineering practice. Their aim was to regulate and organize the various methods for calculating loads acting on the support systems of underground roadways and chambers. Previously used rheological models were replaced with the so-called deformational pressure model, which was later implemented in the National Standards PN-G-05600 [16] and PN-G-05020 [17], which remain in use up to today. The equations used to calculate the radii of the inelastic zones and the values of deformational and static pressure acting on excavation supports were based on the solution of a disk of uniform thickness initially subjected to a hydrostatic far-field (primary) stress. Calculations were performed under the following assumptions [18]:* the plain strain condition,
* the hydrostatic far-field (primary) stress condition,
* the circular shape of the excavation section,
* the homogeneity, isotropy and weightlessness of the rock mass,
* that rock mass destruction occurs in cases of exceeding its compressive strength (based on the Mohr-Coulomb yield criterion). The Polish Standards PN-G-05600 and PN-G-05020 distinguish between the following rock mass models [16; 17], depending on the characteristics of the rock mass within the inelastic zone:  
* the elasto-plastic model with softening (referred to simply as elasto-plastic model),
* the elasto-plasto-fractured model. A schematic representation of the rock mass models as provided by the Polish Standards is depicted in Figure 1.  
The choice of the appropriate rock mass model is dependent on the value of the critical longitudinal strain of the rock mass. This strain value marks the point at which the rock mass transitions from a plastic to a fractured medium in terms of its strength properties [18]. The critical longitudinal strain of the rock mass can be calculated using the equation provided by the PN-G-05020 standard [17]:  
\[\varepsilon_{ng}=1.5\cdot\varepsilon_{ns}, \tag{4}\]  
where:  
\(\varepsilon_{ns}\)--the value of critical longitudinal strain of the intact rock, evaluated by loading the rock sample to a value of around 95% of its compressive strength. The PN-G-05020 Standard recommends employing the elasto-plasto-fractured model when the following condition is satisfied [17]:  
\[u_{w}>r_{w}\cdot\varepsilon_{ng}, \tag{5}\]  
where:  
\(u_{w}\)--value of ground displacement, m,  
\(r_{w}\)--excavation radius, m.  
Similarly, the PN-G-05600 Standard advises using the elasto-plasto-fractured model under the condition outlined as follows [16]:  
Figure 1: Deformational pressure model according to Polish Standards. \[0<p_{o}\leq p_{g} \tag{6}\]  
where:  
\(p_{g}\)--radial stress on the boundary of plastic and elastic zone, MPa,  
\(p_{o}\)--radial stress on the boundary of fractured and plastic zone, MPa. In accordance with the PN-G-05600 Standard, the dimensioning of the so-called shell lining (comprising a thin layer of shotcrete with supplementary steel arches and rockbolts) is based solely on the value of the static roof pressure. Deformational pressure can be disregarded since the entire displacement of the rock mass must be transferred by the supporting structure. Therefore, the following condition must be satisfied [16]:  
\[u^{ob}\geq k\cdot u_{w} \tag{7}\]  
where:  
\(u^{ob}\)--radial deformability of the support, m.  
To incorporate a portion of the displacement of the rock mass that occurs before support installation, a coefficient k is introduced into Formula (7). This introduction can be viewed as an attempt to crudely include the three-dimensional effect of road excavation in an otherwise two-dimensional analysis. The PN-G-05600 standard assumes a fixed value of the k factor, set at 0.9. However, the authors of this article regard such an assumption as oversimplified, as it may lead to either overestimation or underestimation of the required deformability of the roadway support.