Room and pillar mining is a common underground mining method that utilizes natural pillar support to stabilize the underground workings and deformation control. A well-designed pillar system ensures safety of miners and equipment, minimize ore dilution, maximizes productivity, extends mine life, maintains stability of mine structure as well as cost-effective and ecological rehabilitation (Potvin et al. 1990). The most important parameters for room-and-pillar design are depth, orebody strength properties, in-situ stresses, pillar geometry, presence of joint sets or faults, weathering of the rock mass and other prevailing environmental conditions (Sheorey et al. 1986). Empirical methods for pillar design are preferred due to their convenience to apply (Kidega et al. 2022) (Hedley, D.G.F., Grant 1972) (Von Kimmelmann et al. 1984) (Krauland, N., Soder 1987) (Sjoberg 1992) (Potvin et al. 1990) (Lunder, P.J., Pakalnis 1997). However, these empirical pillar design formulas have some inherent simplistic assumptions and weaknesses such as inability to consider complex situations, characterized by various factors such as complex in-situ stresses, existence of faults and fractures or other weak inclusions, irregular shape of ore bodies, topography and complex pillar geometry (Wagner 2003). Numerical simulations are a good alternative for mine design and they are already used on a routine basis using classical failure criteria based solely on maximum and minimum principal stresses, like the Mohr–Coulomb or Hoek–Brown (HB) failure criteria. However, researches have demonstrated at the laboratory scale that the intermediate principal stress effects the failure strength of rocks, for instance:(Kwaśniewski 2013) (Feng et al. 2016),(Mogi 1972),(Feng et al. 2019),(Feng et al. 2020) (Zhang et al. 2013) (Jiang and Zhao 2015). The extended and classical Hoek Brown criteria can estimate the strength of intact rock as well as jointed rock masses by reduction of strength parameters on the basis of rock mass classifications or engineering geological data to suit the rock mass behavior under consideration, like demonstrate for instance by (Li et al. 2021) (Hoek and Brown 2019) (Hoek et al. 2002). At the field scale, geotechnical engineering focus on pillar design under the assumption that the intermediate principal stress component, has insignificant contribution to the strength (C.D Martin, W.G Maybee 2000) (Esterhuizen 2007) (Maleki 2017). However, like shown recently by (Chen et al. 2023) for slope stability analysis, the intermediate principle stress component can have some influence on the stability and failure behavior.

This article documents exemplary that the intermediate principal stress component can have a significant influence on the strength and consequently on the load bearing capacity of mine pillars.

The failure criterion proposed by (Li et al. 2021), which is an extended version of the classical HB-criterion considering the intermediate principal stress component, was selected because it can be expressed analytically by two equations if the stress space is subdived in two parts as shown in Fig. 1 and given by Eqs. (1) and (2).

where \({\sigma }_{1}\), \({\sigma }_{2}\) and \({\sigma }_{3}\) are the major,intermediate and minor principal stress respectively. \({\sigma }_{\text{c}}\) is the unconfined compressive strength. \({m}_{\text{b}}\), \(s\) and \(a\) are rock mass material constant.

Extended HB-criterion for different values of parameter b (Li et al. 2021)

All the parameters of the extended HB-criterion can be determined from uniaxial and triaxial tests. Laboratory testing has documented that peak triaxial strength can be increased by up to about 20% for certain types of rock and optimum values of \({\upsigma }_{2}\) (Haimson and Chang 2000) (Hu et al. 2022) (Feng et al. 2016) (Walton 2021) (Mogi 1971) (Ingraham et al. 2013). Equation (1) shows that for low \({\upsigma }_{2}\), \({\upsigma }_{3}\) of the classical HB-criterion can be replaced by \(\frac{{b\sigma_{2} + \sigma_{3} }}{b + 1}\) and Eq. (2) shows that for high \({\upsigma }_{2}\), \({\upsigma }_{1}\) of the classical HB-criterion can be replaced by \(\frac{{b\sigma_{2} + \sigma_{1} }}{b + 1}\). The value b controls the shape of the failure envelope. The intermediate principal stress component increases the strength and a value of b = 0 (ignoring the intermediate principal stress component) signifies the lower bound of the strength envelop or with other words, the classical HB-criterion is always conservative. A simple test of Eqs. (1) and (2) can be used to decide which equation should be applied for the actual stress state: the equation giving the lower values for \({\sigma }_{1}\) has to be used for the extended HB-criterion (see also flowchart in Fig. 3).

3.1 Numerical model set-up and calculation procedure

To investigate the potential effect of the intermediate principal stress component in general, a 3-dimensional numerical pillar model was set-up on the basis of the explicit finite difference code FLAC^3D. In this study, we consider a pillar situated in a large field of pillars (regular room and pillar mining scheme) under assumption of quasi-isotropic and quasi-homogeneous material behavior, which allows to reduce the model to a ¼-model. The model set-up is illustrated in Fig. 2 representing a ¼-model of a pillar with adjacent room.

Numerical model set-up (left: virgin rock mass, right: after excavation)

The outer model dimensions are 8 m × 7 m × 20 m. The pillar height is 5 m, and width and length are of the pillar are 3.5 m and 4 m, respectively. The virgin stress state is characterized by a vertical stress component of 25 MPa and horizontal stresses of 12.5 MPa. At model bottom and vertical outer boundaries, the normal displacements are fixed. At the top of the model, a vertical pressure of 25 MPa is applied. The zones are densely populated at the area of excavation to increase accuracy, with each zone occupying a volume of 0.0047 m³. In the floor and roof regions, each zone occupies 0.125 m³. There are 67,200 zones in the model before excavation with total volume of 910 m³. The material parameters for the classical HB-model used for this exemplary study are given in Table 1.

Stress-strength analysis under pure elastic conditions is conducted to evaluate the violation of failure envelope by verifying the actual stress state violates the yield surface of the classical and extended HB-criteria. The purpose is to identify zones of potential failure around the mine pillar. For this evaluation the parameters \({k}_{1}\), \({k}_{2}\) and \({k}_{3}\) are determined for each zone. Equations 3 and 4 specify the extended HB-criterion and Eq. (5) the classical HB-criterion adopted from (Li et al. 2021). The flowchart shown in Fig. 3 is used to determine the stability numbers \({k}_{1}\), \({k}_{2}\) and \({k}_{3}\) to identify zones, where the failure criterion is violated for the classical as well as the extended HB-criterion. \({k}_{1}\), is valid for the region with lower intermediate principal stress (\({\sigma }_{2}\le {\sigma }_{2}^{*}\)) and k₂ is valid for the region with higher intermediate principal stress (\({\sigma }_{2}\ge {\sigma }_{2}^{*}\)) according to Fig. 1.

Flowchart to detect zones violating the failure envelope

K-values equal or greater than 1 indicate a violation of the strength criterion. For b-value of zero the classical HB-criterion coincides with the extended HB-criterion as illustrated by Fig. 4. Therefore, region of strength exceedance is 24.21m³ for all three cases (\({k}_{1}\), \({k}_{2}\) and \({k}_{3}\)).

Values a \({k}_{1}\) and b \({k}_{2}\) for the extended HB-criterion with b-value of zero and c \({k}_{3}\) for the classical HB-criterion

The selected rock (Chelungpu siltstone I) according to Li et al. (Li et al. 2021) has a b-value of 0.464. Figure 5 shows the simulation results for the k-values. Strength violation according to \({\text{k}}_{1}\) is concentrated at the middle of the pillar height, whereas \({\text{k}}_{2}\) is concentrated at the transition from the pillar to the roof and floor area. Considering the realistic b-value of 0.464, the volume of strength exceedance (8.02 and 6.09 m³, respectively, in total 14.11 m³, see Figs. 5a and b) is much smaller than predicted by the classical HB-criterion with 24.21 m³.

Values a \({k}_{1}\) and b \({k}_{2}\) considering a b-value of 0.464 and c for the classical HB-criterion

The Fig. 6a and b shows the results when parameter b is zero (ignoring the influence of the intermediate principal stress) and 0.464, respectively. The volume of plastic zones indicated by pink colored zones (> 1.0) are 24.21m³ and 14.11m³ for classical and extended Hoek Brown criteria, respectively. This implies that the intermediate principal stress component has strengthening effect on pillar stability.

Regions of strength exceedance zones for a classical HB-criterion and b extended HB-criterion with b-value of 0.464

Figure 7 illustrates the effect of different b-values on the extent of the area with strength violation. According to (Li et al. 2021) realistic b-values can range from about 0.2 to nearly 1.0 depending on the type of rock. Table 2 shows the total volume of zones with strength exceedance by applying different b-values.

Effect of b-values on extent of strength violation area for a b = 0, b b = 0.5, c b = 0.7 and d b = 1.0

Although laboratory tests have documented the influence of the intermediate principal stress component, this knowledge is yet to be fully incorporated and integrated into the engineering practice, where still strength criteria ignoring the influence of the intermediate principal stress component like the classical Mohr–Coulomb or the classical HB-criterion are dominating. Applying these classical approaches is always conservative in terms of safety and stability considerations. However, consideration of the intermediate principal stress component leads to a more realistic assessment of rock and rock mass strength and allows consequently a more economic mine design as presented exemplary for the pillar design. Depending on rock type and stress situation this effect can be quite significant. Therefore, recognizing the great practical importance, incorporation of the intermediate principal stress component for pillar design and safety considerations is highly recommended.

The authors acknowledge that direct application of laboratory results (parameters) for in-situ dimensioning might not be appropriate due to the scale effect, Nevertheless the stabilizing effect of the intermediate principal stress component remains effective in an in-situ scenario.