Underground rock engineering has long relied on failure criteria that consider only two of the three principal stresses acting in the Earth’s crust. A new numerical modeling study argues that this simplification can substantially distort stability assessments, and it demonstrates the consequences with a three-dimensional extension of the widely used Hoek–Brown failure criterion implemented in the FLAC3D finite-difference code. According to the simulations, accounting for the intermediate principal stress component reduces predicted plastic-zone volumes around excavations by large margins and raises the calculated factor of safety of a mine pillar from 2.3 to 2.7.
The work, published in GeoEnergy Communications by Duncan Maina, Roy Morgenstern, Fabian Weber, Friederike Tiedtke and Heinz Konietzky, addresses a persistent gap between laboratory rock mechanics and engineering practice. Conventional stability analyses in mining, tunneling and borehole design typically use stress-based failure criteria such as Mohr–Coulomb or the classical Hoek–Brown criterion, both of which involve only the major and minor principal stresses. Yet true triaxial testing of many rock types has documented that the intermediate principal stress, the middle value among the three orthogonal principal stresses, has a significant influence on rock strength.
The classical Hoek–Brown criterion is nonlinear, its parameters are easy to determine, and it can be readily implemented in numerical codes, which explains its popularity. But because it ignores the intermediate principal stress, the authors argue that it underestimates rock mass strength and can produce inaccurate predictions of stability and safety factors, potentially leading to suboptimal designs and an overestimation of the excavation damaged zone, the region of rock around an opening that is plastically deformed or fractured.
To close this gap, the team selected the extended Hoek–Brown constitutive law proposed by Li and colleagues, which incorporates the intermediate principal stress while requiring only one additional parameter beyond the familiar Hoek–Brown inputs. That parameter, called b, acts as a shape factor of the failure envelope, controlling its Lode-angle dependence. When b equals zero, the extended criterion reduces exactly to the classical Hoek–Brown formulation, which allowed the researchers to validate their implementation against the standard model. Many true triaxial tests show that for a constant minor principal stress, rock strength rises with increasing intermediate principal stress until a peak and then gradually declines, a behavior the extended criterion is designed to capture.
The researchers built a full elasto-plastic numerical model around this criterion, combining isotropic elasticity with a yield surface and flow rules that vary with stress state, and programmed it in C++ as a user-defined model compiled into FLAC3D. For their demonstration calculations they used mechanical properties of the Chelungpu siltstone, scaled to rock mass conditions with an assumed Geological Strength Index value of 80. Because scaling laws for the b parameter from intact rock to rock mass do not yet exist, they took the b values directly from the corresponding laboratory rock data.
The pillar simulations produced some of the most striking numbers. In a three-dimensional model of a 7.5 by 7.5 by 5 meter mine pillar with pre-mining stresses of 10 megapascals in the vertical and horizontal directions, the volume of plastic zones was 178.2 cubic meters when the intermediate stress was ignored, equivalent to the classical Hoek–Brown result. With a b value of 0.464, the plastic volume fell to 130.9 cubic meters, and with b equal to 1 it dropped further to 107.24 cubic meters. In other words, disregarding the intermediate principal stress overestimated the extent of yielded rock in the pillar.
Excavation models showed even larger contrasts. For a horseshoe-shaped drift under an anisotropic in-situ stress state of 40, 20 and 10 megapascals in the three directions, the classical model predicted a plastic-zone volume of 9,188 cubic meters versus 1,713 cubic meters for the extended model with b equal to 0.464. Under a rotated stress field the figures were 10,947 versus 2,026 cubic meters. Under isotropic stress conditions of 25 megapascals in each direction, the difference narrowed sharply, to 882 versus 578 cubic meters, showing that the influence of the intermediate principal stress depends strongly on the stress state. Under the anisotropic conditions, plastic zones formed in horizontal and vertical butterfly-shaped patterns whose extent was markedly larger with the classical criterion. In a twin circular tunnel model with 4-meter-radius tunnels separated by 10 meters, the classical model predicted 2,702 cubic meters of plastic zones against 1,844 cubic meters for the extended model, and the pillar of rock between the tunnels remained elastic under the extended criterion but became fully plastified when the intermediate stress was ignored.
The factor of safety calculations used the strength reduction method, in which rock mass strength parameters are progressively reduced by increasing strength reduction factors until the model can no longer reach mechanical equilibrium; the factor at failure is taken as the factor of safety. For the quarter pillar model, stepwise reduction of the uniaxial compressive strength yielded a factor of safety of 2.3 with the classical formulation and 2.7 with the extended model, an improvement of roughly 18 percent when all three principal stress components are considered. The authors interpret this as evidence that classical simulation techniques are conservative, overestimating the excavation damaged zone and underestimating safety factors.
The study’s broader context comes from an extensive survey of published true triaxial test data, which indicates that considering the intermediate principal stress can increase predicted maximum strength by roughly 10 to 30 percent at an optimum level of that stress component, compared with analyses that assume the intermediate and minor stresses are equal. Laboratory borehole breakout tests have likewise shown that true triaxial strength criteria agree much better with stresses at the breakout boundary than the commonly used Mohr–Coulomb criterion. Despite this laboratory evidence, three-dimensional constitutive models of this kind have not been used extensively for in-situ rock engineering projects, in part because earlier work on safety factors focused on mine pillars with laboratory-scale parameters rather than rock mass parameters derived through the Geological Strength Index system.
The authors are careful to note the limitations of their demonstration. The calculations are exemplary simulations rather than validated case studies, and in-situ case studies are still needed to confirm that the model, its parameters and the resulting simulations reflect actual mine conditions. Scaling laws for the b value from intact rock to fractured rock mass remain to be established, although for some undisturbed rock masses such as rock salt or claystone laboratory data can be used directly or with minor adjustment. Future work is planned on improving the flow rules, which will require high-precision deformation measurements, and on extending the model to time-dependent behavior such as creep and damage from subcritical crack growth. The numerical behavior of the extended model mirrors that of the classical criterion, so existing restrictions on mesh dependency and mesh resolution apply equally. Even so, the message for practitioners is direct: designs based on conservative empirical laws that ignore the intermediate principal stress may be safe, but they are not optimal in terms of economy, recovery performance or mine life, and incorporating all three principal stresses offers a path to more realistic and more economical underground design.
Subject of Research: Numerical modeling of how the intermediate principal stress component affects predicted stability, excavation damaged zones and factors of safety in underground rock engineering.
Article Title: How important is the intermediate principal stress component for underground rock engineering?
Article References: Maina, D., Morgenstern, R., Weber, F., Tiedtke, F., & Konietzky, H. (2025). How important is the intermediate principal stress component for underground rock engineering?. GeoEnergy Communications, 1(1), Article 6. https://doi.org/10.1007/s44421-025-00001-6
Image Credits: AI Generated
DOI: 10.1007/s44421-025-00001-6
Keywords: rock mechanics, intermediate principal stress, Hoek–Brown criterion, FLAC3D, excavation damaged zone, factor of safety, true triaxial testing, mine pillars, tunnel stability, strength reduction method, Geological Strength Index, numerical modeling
News Source: Violet Maxwell. (October 30, 2026). Ignoring the Intermediate Principal Stress May Skew Underground Rock Safety Designs. Scienmag.
