Abstract
The stability of surrounding rock in underground excavations is fundamentally governed by the stress evolution induced by excavation unloading and subsequent reloading processes, such as support installation, lining, or internal pressure recovery. However, existing studies have mainly focused on excavation-induced unloading, while the mechanical response of surrounding rock subjected to subsequent reloading remains insufficiently understood. To address this issue, an analytical framework based on the Hoek–Brown strength criterion is proposed to investigate the elastoplastic behavior of surrounding rock during the excavation unloading–reloading process. The proposed framework consists of two complementary components: a limit equilibrium analysis of the overlying rock mass to evaluate the global stability of underground openings, and an elastoplastic analytical solution for a circular cavern to characterize local stress redistribution, plastic zone evolution, and failure mode transition. Based on these solutions, the upper and lower limit pressures associated with tensile and shear failure mechanisms are derived. The results demonstrate that the stress path induced by unloading and reloading plays a critical role in controlling the mechanical response of surrounding rock. Subsequent reloading modifies the stress state around the excavation boundary and may lead to a transition from tensile-dominated instability to shear-controlled failure. Parametric analyses further reveal that burial depth, lateral pressure coefficient, and rock mass quality significantly affect the stability limits and failure characteristics. The proposed method provides an analytical basis for determining allowable pressure ranges and evaluating the stability of underground excavations.
1 Introduction
The mechanical essence of engineering excavation lies in the redistribution of in-situ stresses, which inevitably triggers stress unloading in one or more principal directions. Consequently, all excavation activities induce varying degrees of unloading-related damage within the rock mass. While conventional research has primarily addressed the monotonic loading or unloading behaviors of intact rock (). Real-world geotechnical structures—such as mine tunnels, transportation tunnels, and shafts—frequently encounter a complex combination of initial excavation unloading followed by cyclic loading-unloading disturbances during construction and operation. The deformation characteristics under these coupled conditions diverge significantly from traditional mechanical responses, emerging as a critical research frontier. Therefore, a comprehensive understanding of the cyclic mechanical properties of rock pre-damaged by unloading is essential for rigorous stability analysis and safety assessment. (; , ; ). But the stability analysis in the early stage cannot be ignored.
The elastoplastic response of underground cavities under various loading conditions has long been a pivotal research focus in geomechanics and underground engineering. Extensive theoretical investigations have been conducted regarding cavity expansion and contraction, plastic zone evolution, and ultimate bearing capacity.
For soil media, several scholars have systematically explored the elastoplastic behavior of underground cavities using cavity expansion theory and limit analysis, establishing a robust theoretical framework for stability evaluation (; ; , ; ; ). However, as these studies primarily target soil materials—whose strength criteria, plastic flow characteristics, and deformation patterns differ significantly from those of rock masses—their results cannot be directly applied to engineering structures such as shafts, roadways, and caverns excavated in rock media.
Compared to soil-based cavity problems, research on the excavation response of rock masses focuses more on stress redistribution, plastic zone development, and instability mechanisms following cavity formation. developed an elastoplastic analytical solution for underground excavations and established a fundamental framework for evaluating stress redistribution and plastic zone development around tunnels. Subsequently, extended the solution to the generalized Hoek–Brown criterion, providing a more practical approach for analyzing the elastoplastic response of rock masses. further investigated the stress and displacement fields around circular openings in Hoek–Brown media, improving the prediction of excavation-induced deformation. considered elastic–brittle–plastic behavior and analyzed the influence of brittle failure characteristics on the stability of circular openings. Further developments have focused on improving the applicability of analytical solutions under more complex geological conditions. incorporated strain-softening behavior to describe the progressive evolution of plastic deformation after yielding. extended the analysis to circular tunnels subjected to non-hydrostatic in-situ stresses and investigated the variation of the elastoplastic interface under anisotropic stress conditions. More recently, considered blast-induced damaged zones and strain-softening behavior, proposing a semi-analytical approach for tunnels excavated in Hoek–Brown rock masses. established a unified rock strength theory incorporating the coupled effects of macro-, meso-, and micro-scale defects, providing a quantitative framework for characterizing rock strength and failure in underground engineering. These studies have significantly advanced the understanding of elastoplastic responses around underground openings; however, most existing analytical solutions mainly focus on the stress redistribution induced by excavation or the final equilibrium state after excavation. The influence of excavation unloading history and subsequent reloading processes on the mechanical response and stability limits of surrounding rock remains insufficiently investigated.
Nevertheless, most of these studies confine the problem to a single excavation-induced unloading process. Insufficient attention has been paid to the subsequent loading processes that the surrounding rock may undergo after excavation. These include the confining effects of primary support and secondary linings, the reloading effects induced by backfill or internal liners, and the additional internal pressures exerted when abandoned mines or caverns are repurposed as gas or energy storage facilities (). In practical engineering, surrounding-rock stability can be improved through bolt reinforcement and evaluated using non-destructive testing and early warning of anchorage damage, while the effects of multi-scale rock defects should also be considered (; , ). These measures are generally implemented as subsequent reinforcement and monitoring strategies after the initial stability assessment of the surrounding rock.
Existing stability analyses of the surrounding rock can be broadly categorized into two levels: the first is global stability analysis, which requires that the loads acting on the boundary of the surrounding rock do not exceed the ultimate bearing capacity provided by the overburden (). Various geometric models (as shown in Figure 1)—including conical, logarithmic spiral, and vertical failure models—have been employed for stability analysis (; ). Specifically, researchers have developed analytical methods based on linear failure (), stress analytical models using the Tresca criterion ()), and limit equilibrium methods utilizing rigid cone models (; ). Furthermore, upper bound theorem applications based on the Hoek-Brown criterion have been used to derive uplift failure functions (). Despite these advancements, most existing studies focus exclusively on the upper limit pressure and often rely on the Mohr-Coulomb criterion, resulting in a lack of analytical solutions for the lower limit pressure and a neglect of factors like crack initiation and rupture angles.
FIGURE 1
Beyond limit equilibrium methods, elastoplastic analysis offers a framework for determining both upper and lower limit pressures (). While numerical approaches, such as 3D finite element simulations () and plastic zone distribution analyses (), have successfully modeled in-situ tests and failure patterns, they lack closed-form analytical solutions. Progress in analytical modeling includes studies based on Melan’s lower bound shakedown theory () and solutions for ellipsoidal caverns considering combined shear and tensile strengths (). However, the latter is inapplicable to tunnel-type structures. Recent elastoplastic derivations for upper and lower limit pressures have integrated stress distribution with shear-tensile strengths (), yet these still largely depend on the Mohr-Coulomb criterion.
The Mohr-Coulomb criterion cannot adequately capture the nonlinear failure characteristics of rock materials. The Hoek-Brown strength criterion was first proposed in 1980 and has been continuously improved (; ; ). It can effectively represent the nonlinear failure characteristics of rock masses and the influence of structural planes and stress states on strength (). However, studies considering the coupling effects of the crack initiation angle, rupture angle, and Hoek-Brown strength criterion in surrounding rock stability analysis remain limited.
Consequently, this study evaluates surrounding rock stability through an integrated framework of limit equilibrium and elastoplastic analyses, incorporating the Hoek-Brown strength criterion to better reflect the non-linear failure characteristics of rock masses. A refined conical failure model, which accounts for both crack initiation and rupture angles, is established to derive analytical expressions for the upper limit pressure required to prevent overburden uplift. An integrated Hoek–Brown-based framework combining limit-equilibrium and elastoplastic analyses is proposed to evaluate surrounding-rock stability. The former assesses global overburden uplift limits, and the latter defines local tensile/shear failure pressure bounds. Sensitivity to burial depth, lateral pressure coefficient, and rock mass quality is investigated, offering theoretical guidance for allowable pressure design in underground caverns.
2 Hoek-Brown strength criterion
The Hoek-Brown strength criterion is an empirical strength criterion widely used in rock mechanics. Combined with the Geological Strength Index (), it fully considers the quality characteristics of rock masses and can effectively describe the nonlinear failure behavior of rock materials. The criterion can be expressed in terms of the principal stresses as ():where:where, and represent the absolute maximum and minimum principal stresses, respectively; denotes the uniaxial compressive strength of the intact rock material; is the Geological Strength Index; , and are empirical parameters characterizing the rock mass properties; and is a disturbance parameter.
The Hoek-Brown criterion can also be expressed in terms of normal stress and shear stress on the failure plane(; ):where and represent the normal stress and shear stress on the failure plane, respectively; denotes the tensile strength parameter of the rock mass; and are empirical constants within the range from 0 to 1. By substituting , into Equation 1 the value of can be obtained.
3 Analytical methods for evaluating cavern stability
In underground engineering, the excavation of structures such as roadways, and caverns trigger a redistribution of in-situ stresses. This process leads to unloading, stress concentration, and localized failure within the surrounding rock mass near the cavity walls. As support structures are installed or the load-bearing capacity of backfill material is established or transform abandoned tunnels into gas storage chambers, the boundary conditions of the surrounding rock are further adjusted, potentially subjecting localized zones to reloading. Consequently, the stability of underground rock masses can be characterized as a mechanical response under the coupled effects of excavation-induced unloading and subsequent support- or backfill-induced reloading.
Despite differences in geometric morphology and engineering applications among shafts, roadways, and cavities, their stability analyses consistently involve stress redistribution, plastic zone evolution, and changes in bearing capacity following excavation disturbances. To highlight the primary mechanical characteristics and facilitate analytical research, this study simplifies these underground structures into a deep-buried circular opening model. This model serves as the basis for evaluating the stress state and stability of deep-seated surrounding rock.
3.1 Limit equilibrium analysis of overlying rock mass stability
3.1.1 Establishment of the conical failure model
The conical failure mechanism adopted in this study is based on the limit equilibrium theory for evaluating the global stability of underground caverns under internal pressure. At the ultimate limit state, the increasing internal pressure may induce upward movement of the overlying rock mass, and the failure surface can be idealized as a conical or axisymmetric failure body for relatively homogeneous rock masses without dominant discontinuities (; ; ). It should be noted that this assumption represents a simplified failure mechanism. In practical engineering conditions, geological structures, anisotropy, and heterogeneity may result in more complex failure patterns. Therefore, the proposed model is mainly applicable for stability evaluation of relatively intact rock masses, while further studies considering discontinuities and three-dimensional effects are required for highly fractured rock masses, as shown in Figure 2. The rupture surface ( is the rupture angle) starts from point B (or D) on the cavern wall and propagates along an inclined line to the ground surface point C (or F). The rupture surfaces BC and DF are symmetrical about the vertical axis of the cavern.
FIGURE 2
The rupture surface location is governed by the crack initiation angle . is the burial depth; and are the self-weights of the failed rock mass. and denote the shear and supporting forces on the rupture surface, respectively. is the vertical distance from the crack initiation point to the ground surface, and h is its distance to the cavern crown. is the cavern radius, and is the transmitted pressure.
Figure 2b illustrates the free-body diagram of block ABC. The block is subjected to its gravitational self-weight, while the AC surface, representing the ground level, is considered a traction-free boundary. Both the BC and AB surfaces are subjected to the combined effects of normal pressure and shear forces.
Under axisymmetric loading conditions, the horizontal components of the internal pressure acting on the cavern wall cancel each other, and the remaining resultant force is vertically oriented:
The following assumptions are made: The shear force on the rupture surface follows the Hoek-Brown strength criterion; Surface AB is subjected to earth pressure at rest and shear force.
Based on the vertical force analysis of the overlying rock mass in Figure 2a, and vertical force equilibrium of the overlying rock mass gives:
A mechanical analysis is conducted on block ABC in Figure 2b, where represents the total shear force on the AB plane; while represents the lateral earth pressure at rest exerted by the adjacent rock mass:where, is lateral pressure coefficient; is rock mass gravity.
The shear force on rupture surface BC is obtained from Equation 2:
Horizontal force equilibrium gives:
By combining Equations 3–7 the expression is obtained:In Equation 8, is the positive root of equation .
3.1.2 Identification of crack initiation angle and rupture angle
Due to variations in internal pressure, geostress, and rock mass mechanical properties, the crack initiation position on the cavern wall may vary, resulting in different crack initiation angles. used the Element Free Galerkin (EFG) method to investigate crack initiation points under different internal pressures and lateral pressure coefficients. Their results indicated that the crack initiation position is mainly related to the lateral pressure coefficient.
In this study, the crack initiation angle is determined using the formula proposed by :
When solving Equations 4–9, three unknowns exist: , , and . For a given burial depth and rock mass parameters, each rupture angle corresponds to a specific . The true rupture angle corresponds to the minimum , as shown in Figure 3. Therefore, an iterative method using numerical computation software is employed to simultaneously determine the and .
FIGURE 3
3.1.3 Selection criteria for the safety factor
In engineering design, safety factors are introduced to account for uncertainties during operation.
Currently, no unified safety factor standard exists for underground caverns. In a commercialization study of LRC technology conducted by Itasca in 2001, safety factors were used to verify the required burial depth of large storage caverns(). suggested adopting a safety factor when evaluating cavern uplift safety using limit equilibrium analysis.Therefore, this study adopts a safety factor of for determining the safe burial depth of CAES caverns, consistent with engineering practices in slope stability and tunnel settlement analyses ().
3.2 Elastoplastic analytical solutions for cavern wall stability
3.2.1 Stress analysis of surrounding rock
Although global uplift determines the macroscopic safe burial depth, the microscopic stress evolution in the surrounding rock mass triggers the instability process. Therefore, elastoplastic stress analysis of the cavern wall must be conducted to define the critical pressure for local failure.
This study uses the formation of a continuous plastic zone as a key criterion for evaluating the cavern stability (). For a circular cavern, instability is defined as the point when the surrounding rock mass completely enters the plastic phase around the cavern wall. The failure is assumed to occur when the plastic zone at the cavern wall becomes continuous and extends deeper.
Figure 4a shows the stress model of a cavern in an abandoned roadway, where represents the vertical stress, equal to the self-weight of the overlying rock mass. For simplification, the outer boundary is treated as an infinite boundary, neglecting the effects of temperature and seepage. The surrounding rock is assumed to be an ideal elastoplastic material that conforms to the Hoek-Brown strength criterion. Due to symmetry in the stress distribution, subsequent analysis will only take the range of for calculation simplification.
FIGURE 4
When the surrounding rock does not undergo plastic failure under the pressure , based on the Kirsch equations () and adopting the convention that compressive stress is positive, the expressions for the stress components are obtained as follows:where, .
The relationship between radial and tangential stress on the cavern wall, derived from Equations 12–13, is expressed as:where, .
As can be seen from Equation 12, is a function of and , so the stress state at the cavern wall forms a stress region in the space, defined as the stress transition zone, as shown in Figures 4b–d. Simultaneously, the maximum and minimum values of (obtained from Equations 12–13) can be determined:
Since the pressure acts directly on the surrounding rock, we have ; from Equation 12, it can be derived that .
3.2.2 Analytical solutions for tensile failure
Figures 4b,c present schematic diagrams of the stress paths when tensile failure occurs in the surrounding rock during the unloading and loading phases of the cavern.
By setting the minor principal stress in the Hoek–Brown criterion, the uniaxial compressive strength of the rock mass can be obtained as: . In the principal stress space, represents the intersection point between the Hoek–Brown failure envelope and the -axis. When , as shown in Figure 4b, the stress path within the stress transition zone has no intersection with the strength criterion, indicating that no shear failure will occur; thus, the lower limit pressure is not considered (complete excavation). The upper limit pressure needs to be determined based on the tensile strength. Since the tensile strength of the surrounding rock is very small, it is assumed to be (). When the pressure reaches point A, the tangential stress at the cavern wall is (); if the pressure continues to increase, tensile stress will be generated, leading to tensile failure in the surrounding rock, which is detrimental to the stability of the cavern. Therefore, the pressure at point A is defined as the upper limit pressure, expressed as Equation 15:
When , as shown in Figure 4c, part of the stress path intersects with the Hoek–Brown strength criterion, indicating the occurrence of local shear failure. When the internal pressure decreases below point B, the extent of the shear-yielded region increases; however, the shear failure zone remains discontinuous due to the non-uniform stress distribution induced by the lateral pressure coefficient, and no continuous shear failure zone can be formed. Therefore, although local shear failure occurs, it does not lead to overall shear instability, and the lower limit pressure is not required in this case. The stability of the surrounding rock is still governed by the tensile failure criterion. When the internal pressure reaches point A, the tangential stress at the cavern wall becomes zero; further increase in internal pressure induces tensile stress and causes tensile instability. Therefore, point A is defined as the upper limit pressure.
3.2.3 Analytical solutions for shear failure
When , as shown in Figure 4d, all stress paths within the stress transition zone intersect with the strength criterion, indicating that if the pressure reaches a certain value, shear failure will gradually appear at the cavern wall, eventually forming a continuous shear failure zone, culminating in global shear failure.
By setting , the critical depth for the transition between tensile failure and shear failure can be determined; combining this with Equation 14, the calculation expression for is obtained as:
As shown in Figure 4d, when the pressure is relatively low (i.e., during the excavation phase of the cavern), ; the tangential stress acts as the first principal stress, and the radial stress acts as the third principal stress. As the pressure gradually decreases to point B and below, a failure zone begins to form at the cavern wall. When the pressure drops to point C, the cavern wall fails entirely, forming a continuous shear failure zone, and the pressure at point C is defined as the lower limit pressure . Conversely, when the pressure is relatively high (i.e., during the loading phase), ; the radial stress becomes the first principal stress, and the tangential stress becomes the third principal stress. When the magnitude of the pressure exceeds the radial pressure corresponding to point A, local areas of the surrounding rock begin to undergo shear failure, which will connect with the shear failure zone generated during excavation, ultimately leading to structural failure. Therefore, the radial pressure corresponding to point A is defined as the upper limit pressure.
The stress state at point C is:
By combining Equations 1, 17 the expression for the upper limit pressure under the condition of is derived as:In Equation 18,
The stress state at point A is:
By combining Equations 1, 19 the expression for the upper limit pressure under the condition of is derived as:
If , the high non-linearity of the equation precludes an explicit solution; thus, the upper and lower limit pressures can be solved through numerical methods, such as the Newton-Raphson method.
4 Parametric analysis of ultimate pressures and failure patterns
4.1 Stability analysis of overlying rock mass
4.1.1 Impact of lateral pressure coefficient on overlying rock mass stability
Regarding the limit pressure of the cavern, given the material parameters and cavern geometry, the obtained limit pressure calculation method can be used to conduct stability calculations and analyze the variation laws of the limit pressure and rupture angle with the cavern burial depth. The relationship between the limit pressure, rupture angle, and burial depth under different lateral pressure coefficients is analyzed herein. The parameters are taken as , , , , , , and , with the calculated burial depth ranging from 0 to 250 m. Ultimately, the relationship curves of the limit pressure and rupture angle with burial depth under different lateral pressure coefficients are obtained, as shown in Figure 5.
FIGURE 5
As illustrated in Figure 5a, with the increase in burial depth , does not increase linearly but exhibits a distinct accelerated upward trend. This indicates that under deep burial conditions, the self-weight stress of the overlying rock mass significantly enhances the maximum internal pressure that the cavern can withstand. The impact of on the limit pressure demonstrates a negative correlation. At the same burial depth, the smaller the (i.e., horizontal stress is much less than vertical stress), the higher the limit pressure. It is worth noting that when , the slope of the curve is the steepest, indicating that in a low lateral pressure environment, the stability gain brought by the increased burial depth is the most significant. Conversely, when , the curves tend to overlap, indicating that in a high lateral pressure environment, horizontal tectonic forces gradually dominate the failure mode, and the gain effect of burial depth exhibits a diminishing return. This also implies that within this range, blindly increasing the burial depth may not effectively enhance the limit internal pressure, and the influence of horizontal tectonic stress needs to be additionally considered during design.
Figure 5b displays the variation curves of the rupture angle . It is observed that when , increases with the increase of and gradually levels off. This indicates that the deeper the burial depth, the steeper the shear surface when shear failure occurs in the rock mass. When , changes approximately linearly with . A smaller corresponds to a higher rupture angle curve; at the same burial depth, is inversely proportional to the rupture angle. This is because a smaller lateral pressure coefficient increases the principal stress difference within the rock mass, making it easier to form a high-angle shear rupture plane.
Notably, when , initially decreases and then increases as the burial depth increases. This may be because, at shallow burial depths, the horizontal stress is stronger than the vertical stress, and the strong lateral confinement effect dominates, forcing the initial failure surface to be “flattened”, leading to a decrease in with a slight increase in depth. As the burial depth further increases, surges exponentially. The massive upward thrust becomes absolutely dominant, forcing the rock mass to rupture towards the surface along a steeper and shorter rupture plane (i.e., a large value).
4.1.2 Impact of rock mass grades on overlying rock mass stability
The surrounding rock conditions are selected from Grade I to Grade IV rock masses, and the unit weight, internal friction angle, and cohesion of the surrounding rock are selected in accordance with the Standard for Engineering Classification of Rock Masses (), as shown in Table 1. Because the Hoek–Brown strength criterion is adopted in this study, the equivalent Hoek–Brown parameters are obtained from the Mohr–Coulomb parameters according to the method proposed by , as expressed in Equations 21–23. In these equations, . The lateral pressure coefficient is set to , and the remaining parameters are the same as those in Section 4.1.1.
TABLE 1
| Grade | |||
|---|---|---|---|
| I | 28.0 | 62 | 2.4 |
| II | 26.5 | 56 | 1.5 |
| III | 24.5 | 44 | 0.7 |
| IV | 22.5 | 30 | 0.2 |
Parameters of surrounding rock.
Figure 6 illustrates the variation laws of the upper limit pressure and rupture angle with depth under different rock mass grades. As the surrounding rock mass grade improves from Grade V to Grade I, the unit weight, internal friction angle, and cohesion of the rock mass all increase significantly, and the corresponding upper limit pressure also becomes higher. Moreover, the higher the grade of the surrounding rock (e.g., Grade I), the steeper the slope of its curve, indicating that the bearing potential of high-quality surrounding rock in a deep, high-pressure environment is much higher than that of poor-quality surrounding rock. Meanwhile, the rupture angle is positively correlated with both the rock mass grade and the burial depth; at the same burial depth, a higher rock mass grade corresponds to a larger rupture angle; as the burial depth increases, the rupture surfaces of all grades of surrounding rock tend to become steeper. This suggests that when high-strength rock masses fail, their failure surfaces are more inclined to break directly towards the surface, and their excellent shear strength can effectively restrict the failure scope.
FIGURE 6
At a burial depth of , the upper limit pressure of the Grade I rock mass is approximately , and the rupture angle is about . Due to its extremely high levels of internal friction angle and cohesion, the rock mass itself is of good quality and can support extremely high internal pressures; when a high-strength rock mass fails, the shear surface is steeper, and the failure zone is strictly confined to a narrower area directly above the cavern. In contrast, for the Grade IV rock mass, the upper limit pressure is only about , and the rupture angle is roughly . Compared to the Grade I rock mass; its bearing capacity drops by approximately . This is primarily limited by its lower shear strength, making the rock mass highly susceptible to shear slip under internal pressure.
4.2 Analysis of elastoplastic response of cavern wall
4.2.1 Lateral pressure coefficients effects on elastoplastic response
Based on the parameters provided in Section 4.1.1, elastoplastic analysis is conducted on the surrounding rock under different lateral pressure coefficients to determine the upper and lower limit pressures of the cavern, as shown in Figure 7.
FIGURE 7
It is evident that the upper limit pressure exhibits a significant linear growth trend with the increase in burial depth . Under the same burial depth conditions, the variation of upper limit pressure with the lateral pressure coefficient exhibits a non-monotonic trend. As increases from 0.6 to 1.0, the upper limit pressure increases because the enhanced lateral confinement improves the resistance of surrounding rock against tensile instability. However, when further increases beyond 1.0, excessive horizontal stress causes stress concentration around the cavern and reduces the upper limit pressure. Therefore, the hydrostatic stress state () provides the most favorable condition for resisting high internal pressure.
During the shallow burial stage, the cavern is mainly controlled by tensile failure, with no lower limit pressure, and is relatively low at this time. As depth increases, it gradually transitions to shear failure. Notably, the smaller the , the deeper the critical depth for the failure mode transition (for instance, when , the transition to shear failure occurs at approximately ), whereas conditions near enter the shear failure-dominated stage earlier. Furthermore, merely changing the magnitude of does not affect the upper limit pressure at the depth where the failure mode transition occurs, which can be seen from Equations 14, 16 and 20 indicating that the upper limit pressure at the critical point between tensile and shear failure is independent of .
has a tremendous impact on how early or late appears. The condition corresponds to the shallowest critical depth (about ), while corresponds to the deepest critical depth (about ). This implies that under a non-uniform stress field, when lateral confinement is lower, the cavern requires a minimum internal pressure to maintain stability only at greater depths.
4.2.2 Rock mass grade effects on elastoplastic response
Utilizing the rock parameters for different grades provided in Section 4.1.1, under the condition of a lateral pressure coefficient , the relationship curves between upper limit pressure and burial depth in Grade I-IV surrounding rock masses are obtained based on the calculation formulas. Furthermore, the results were compared with those obtained by using the Mohr-Coulomb criterion, as illustrated in Figure 8.
FIGURE 8
Obviously, the higher the grade of the surrounding rock (the better the quality), the greater the upper limit pressure at the same depth, and the later the lower limit pressure appears (the deeper the critical burial depth for tensile and shear failure). Simultaneously, exhibits linear growth with burial depth , and the slope of the curve increases as the rock mass grade improves. The Grade I rock mass shows the highest gain efficiency of with depth. This indicates that higher-quality rock masses generally exhibit more stable responses to variations in the lateral pressure coefficient under the investigated burial depths. This indicates that high-strength rock masses can maintain stability against tensile failure even at large burial depths, whereas Grade IV rock masses enter the global shear failure stage extremely early.
Under the conditions of a burial depth of and a lateral pressure coefficient , the pressure parameters for various rock mass grades are obtained based on Figure 8, as listed in Table 2. Defining , represents the operating pressure window of the cavern. It is observed that the window reduction from Grade I to Grade II is only , indicating that within the range of high-quality rock masses, slight fluctuations in surrounding rock performance cause minimal disturbance to capacity. However, once entering Grade III and below, the reduction rate climbs rapidly (from to ), exhibiting a distinct non-linear accelerated decay characteristic. This suggests the existence of a threshold for rock mass quality; below this threshold, the self-bearing capacity of the rock mass will suffer a cliff-like loss. At a depth of , the peak-shaving efficiency of the Grade IV rock mass is less than half that of Grade I.
TABLE 2
| Grade | Decreasing rate (%) | |||
|---|---|---|---|---|
| I | 13.5 | 0.05 | 13.45 | ∼ |
| II | 12.8 | 0.40 | 12.40 | 7.8% |
| III | 11.2 | 1.25 | 9.95 | 26.0% |
| IV | 8.8 | 2.85 | 5.95 | 55.8% |
Pressure parameters under different rock grade at a depth of .
Compared with existing analytical models for underground caverns, the proposed framework provides an improved description of the surrounding rock response by incorporating the excavation unloading process and subsequent reloading path. Conventional cavity expansion solutions mainly focus on the stress redistribution caused by boundary pressure changes, while the influence of excavation-induced stress release and irreversible deformation history is generally neglected. In contrast, the present framework establishes a continuous stress evolution process from excavation unloading to pressure recovery, allowing the stability limits and failure mode transition of surrounding rock to be evaluated within a unified analytical framework.
The calculated results in this study show good agreement with those of . As shown in Figure 8a, the predicted upper limit pressures exhibit similar increasing trends with burial depth. The maximum relative differences between the two methods for rock mass classes I–IV are , , , and , respectively, indicating that the proposed model provides comparable predictions of the maximum allowable internal pressure.
In contrast, larger discrepancies are observed in the predicted lower limit pressures (Figure 8b). The maximum relative differences between the two methods reach , , and for rock mass classes II–IV, respectively. These deviations are mainly attributed to the different descriptions of rock strength under low confinement conditions. Furthermore, the predicted critical depths for the transition from tensile-controlled to shear-controlled failure are also comparable, with differences of , , and for rock mass classes II–IV, respectively. Overall, the comparison demonstrates that the proposed model can reasonably capture the stability limits and failure mode transition of underground caverns.
In practical engineering applications, the small difference in has a limited impact on the determination of the maximum operating pressure. However, the deviation in may influence the estimation of the minimum safe pressure during operation, particularly under low confinement conditions. Therefore, additional attention should be paid to the lower pressure limit when designing pressure operation strategies for underground caverns. It should be noted that the proposed framework is not intended to replace detailed numerical simulations, but rather to provide an efficient analytical approach for understanding the mechanical mechanisms, conducting parameter analyses, and providing preliminary guidance for engineering design.
5 Discussion
5.1 Coupling effect of lateral pressure coefficient and rock mass
In Section 4, this study analyzed the upper and lower limit pressures of caverns under different lateral pressure coefficients and rock mass grades, concluding that a high lateral pressure coefficient is unfavorable for the stability of the overlying rock mass; however, the elastoplastic analysis of the surrounding rock indicates exactly the opposite, where a high lateral pressure coefficient corresponds to a higher upper limit pressure. Furthermore, for rock masses of better quality (e.g., Grade I), the overlying rock mass is more stable, and the upper and lower limit pressures obtained from elastoplastic analysis are larger and smaller, respectively. This section discusses the comprehensive impact of different rock mass grades and lateral pressure coefficients on the upper and lower limit pressures of the cavern. The rock mass parameters in Table 1 are adopted, the lateral pressure coefficients are , and the remaining parameters are the same as those in Section 4.1.1.
The Variable Change Percentage () for a specific variable at the same depth under different lateral pressure coefficients is defined as follows:where, is the percentage change of variable ; represents any variable, such as , , etc.,; and is the variable calculated when .
Figure 9 illustrates the sensitivity impact of geostress anisotropy () on the overlying rock mass stability for different rock grades, utilizing the hydrostatic stress state as a baseline. Regardless of the rock grade, the influence of the lateral pressure coefficient on is most pronounced under shallow burial conditions (approximately ), where the deviation of each curve is the largest, especially for the case of . This behavior is attributed to the relatively low initial confinement under shallow burial conditions. In this stage, the variation in horizontal stress directly affects the confinement around the cavern boundary and consequently alters the tensile failure resistance. As the burial depth increases, the increase in geostress enhances the overall confinement of the surrounding rock, and the failure mechanism gradually becomes dominated by shear behavior. Therefore, the relative influence of lateral stress variation on decreases with increasing burial depth.
FIGURE 9
As the burial depth increases, the impact caused by changes in either rock mass grade or lateral pressure coefficient approaches stability. When , as the burial depth gradually increases, and vary by approximately to and to , respectively, compared to the baseline values; when , they vary by approximately to and to , respectively, and the sensitivity of (particularly when ) is significantly higher than that of . This indicates that the influence of lateral pressure coefficient is closely related to the stress confinement level. A low lateral pressure coefficient () produces a more significant reduction in lateral confinement, leading to a larger decrease in , particularly under shallow burial conditions. In addition, the rock mass grade exerts a substantial regulatory effect on the sensitivity to the lateral pressure coefficient; the poorer the rock mass quality (e.g., Grade IV), the larger the amplitude of the affected variations, whereas the Grade I rock mass exhibits the most stable performance.
In Figure 10b, it is observed that under a high lateral pressure coefficient is greater than that under a low lateral pressure coefficient, but both are smaller than that under the hydrostatic pressure state. For instance, at , the for Grade IV rock mass under approaches , while under , it is only about , yet both are smaller than the under . Furthermore, although can improve the magnitude of , given the rock parameters used in this study and the range of , the obtained for poorer-quality rock masses is always lower than that for better-quality ones.
FIGURE 10
From the above analysis, it can be concluded that the hydrostatic stress state () represents the optimal condition for the elastoplastic bearing capacity of the surrounding rock, and any anisotropic geostress deviating from this state will lead to a reduction in the upper limit pressure . However, regarding the global stability of the overlying rock mass, a low lateral pressure environment (e.g., ) conversely significantly enhances its anti-uplift capacity. When the burial depth reaches the critical point transitioning from tensile failure to shear failure, the sensitivity of the upper limit pressure experiences violent fluctuations, and only after entering shear failure do the mechanical response characteristics of different rock mass grades begin to show significant divergence.
5.2 Extreme value analysis of the influence of lateral pressure coefficient
As shown in Figure 7, the lateral pressure coefficient significantly influences the evolution of at a specific burial depth, displaying a trend of an initial increase followed by a decrease. To further explore its extreme-value characteristics near , a burial depth of is selected as the computational condition, with varying from to . The other calculation parameters are identical to those in Section 4.1.1. The response curves of and under different values of are illustrated in Figure 11.
FIGURE 11
The analysis demonstrates that at a burial depth of , the derived from the overlying rock mass stability analysis is consistently higher than that obtained from the elastoplastic analysis. With the increase in , the from the overlying rock mass analysis drops sharply before recovering slowly, reaching its minimum at (the hydrostatic stress state). Conversely, both and from the elastoplastic analysis peak at . A comparison reveals that is considerably more sensitive to changes in than , especially in the transition zone between tensile and shear failures, where undergoes drastic numerical fluctuations.
In addition, non-convergence of the upper-bound pressure occurs as approaches the boundary values ( or ). The physical and mathematical causes are as follows: at , the corresponding angle leads to in Equation 3, causing the system to fail to converge due to the lack of driving force; at , the numerator in Equation 16 approaches zero, rendering the calculation logic invalid.
5.3 Influence of thermal and seepage effects on cavern stability
It should be noted that the present model only considers the mechanical response of the surrounding rock and neglects thermal effects during CAES operation. During the air storage stage, the compressed air is usually injected at a relatively high temperature. According to the ideal gas law, the cavern pressure may temporarily increase with increasing gas temperature under constant volume conditions. However, heat exchange between the compressed air and the surrounding rock gradually reduces the air temperature during the storage period, resulting in pressure decay (; ). Therefore, the actual stable storage pressure is expected to be lower than the upper pressure limit predicted by the present mechanical model, and the corresponding effective storage capacity may be reduced. For practical engineering applications, the thermal effects caused by cyclic compression, injection, and heat transfer should be considered through thermo-mechanical coupled analyses.
In addition to thermal effects, seepage may also influence the stability of underground gas storage caverns. Gas leakage through fractures or the excavation damaged zone can generate additional pore pressure in the surrounding rock, thereby reducing the effective confining stress according to the effective stress principle (). The reduction in confinement weakens the shear resistance of the rock mass and may accelerate plastic zone development. Consequently, the actual allowable upper pressure may be lower than that predicted by the present mechanical model, while the required lower operating pressure may increase. For intact hard rock with low permeability and effective sealing systems, this influence may be limited; however, it should be considered for fractured rock masses or long-term cyclic operation conditions.
6 Conclusion
Based on the analytical framework developed for the excavation unloading–reloading process of surrounding rock, the following conclusions can be drawn:
The mechanical behavior of surrounding rock is fundamentally governed by the stress path associated with excavation unloading and subsequent reloading. The transition from stress release to boundary reloading significantly modifies the stress distribution and deformation characteristics of the surrounding rock.
The proposed analytical model, which integrates global stability analysis of the overlying rock mass with the local elastoplastic response at the excavation boundary based on the Hoek–Brown criterion, establishes a unified analytical framework for describing surrounding rock behavior during the unloading–reloading process.
A distinct transition from tensile-dominated damage to shear-controlled plastic failure is observed with increasing reloading level and in-situ stress. This transition plays a critical role in governing the stability and long-term performance of underground openings.
A minimum support pressure and a stability-controlled loading range are identified. Under the representative condition of a burial depth of and , the operating pressure window decreases from for Grade I rock mass to for Grade IV rock mass, corresponding to a reduction of approximately , indicating a significant deterioration in stability control capacity with decreasing rock mass quality.
Burial depth, lateral pressure coefficient, and rock mass quality significantly affect the ultimate pressure, rupture angle, and failure characteristics of underground caverns. The upper limit pressure exhibits a pronounced nonlinear increase with burial depth, particularly for high-quality surrounding rock. For Grade I rock mass, the upper limit pressure at a burial depth of reaches approximately, whereas that of Grade IV rock mass is only about . Meanwhile, the rupture angle gradually increases with burial depth, indicating that deeper caverns tend to develop steeper shear-dominated failure surfaces.
The proposed analytical results provide a theoretical basis for determining support pressure, optimizing excavation and support strategies, and improving stability control in underground engineering applications. The proposed model is a static analytical framework based on the Hoek-Brown criterion, which does not analyze the dynamic impact of the unloading-loading process itself.
Future work should focus on extending the proposed framework to more complex conditions, including anisotropic rock masses, structural discontinuities, and time-dependent behaviors, as well as integrating analytical solutions with numerical simulations and field observations.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
X-HF: Conceptualization, Project administration, Resources, Writing – review and editing. Z-LG: Data curation, Formal Analysis, Software, Validation, Writing – original draft. H-WS: Data curation, Formal Analysis, Writing – original draft. Z-QL: Formal Analysis, Supervision, Writing – review and editing. H-YH: Project administration, Visualization, Writing – review and editing. P-QM: Investigation, Methodology, Writing – review and editing.
Funding
The author(s) declared that financial support was received for this work and/or its publication. Supported by Major Science and Technology Special Project of China National Coal Group (20231BY001).
Conflict of interest
Authors X-HF and Z-LG were employed by China National Coal Group Corporation. Authors H-WS, H-YH, P-QM were employed by China Coal No.5 Construction Co., Ltd. Authors Z-QL was employed by China Coal Construction Group Co. Ltd.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1
BrandshaugT.ChristiansonM.DamianacB. (2001). Technical Review of the Lined Rock Cavern Concept and Design Methodology: Mechanical Response of Rock Mass. Minneapolis, MN: Itasca Consulting Group, Inc.
2
Carranza-TorresC. (2004). Elasto-plastic solution of tunnel problems using the generalized form of the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci.41, 629–639. 10.1016/j.ijrmms.2004.03.111
3
Carranza-TorresC.FairhurstC. (1999). The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci.36 (6), 777–809. 10.1016/S0148-9062(99)00047-9
4
Carranza-TorresC.FosnachtD.HudakG. (2017). Geomechanical analysis of the stability conditions of shallow cavities for compressed air energy storage (CAES) applications. Geomechanics Geophys. Geo-Energy Geo-Resources3 (2), 131–174. 10.1007/s40948-017-0049-3
5
CarterJ. P.BookerJ. R.YeungS. K. (1986). Cavity expansion in cohesive frictional soils. Géotechnique36 (3), 349–358. 10.1680/geot.1986.36.3.349
6
ChadwickP. (1959). The quasi-static expansion of a spherical cavity in metals and ideal soils. Q. J. Mech. Appl. Math.12 (1), 52–71. 10.1093/qjmam/12.1.52
7
China (2014). Standard for Engineering Classification of Rock Masses (GB/T 50218-2014). China Planning Press.
8
GhalyA.HannaA. (1994). Ultimate pullout resistance of single vertical anchors. Can. Geotechnical J.31 (5), 661–672. 10.1139/t94-078
9
HoekE.BrownE. T. (1980). Empirical strength criterion for rock masses. J. Geotechnical Eng. Div.106 (9), 1013–1035. 10.1061/AJGEB6.0001029
10
HoekE.BrownE. T. (1997). Practical estimates of rock mass strength. Int. J. Rock Mech. Min. Sci.34 (8), 1165–1186. 10.1016/S1365-1609(97)80069-X
11
HoekE.BrownE. T. (2019). The Hoek–Brown failure criterion and GSI – 2018 edition. J. Rock Mech. Geotechnical Eng.11 (3), 445–463. 10.1016/j.jrmge.2018.08.001
12
HoekE.Carranza-TorresC.CorkumB. (2002). Hoek–Brown failure criterion – 2002 edition. Proc. NARMS-TAC Conf.1, 267–273.
13
HouZ.YuW.LiuD.LiC.LiuH. (2020). Experimental study of mechanical properties of marble under triaxial unloading confining pressure after fatigue loading. Rock Soil Mech.41 (5), 1510–1520. 10.16285/j.rsm.2019.0958
14
HuB.YuL.MiX.XuF.LiS.LiW.et al (2024). Comparative analysis of thermodynamic and mechanical responses between underground hydrogen storage and compressed air energy storage in lined rock caverns. Int. J. Min. Sci. Technol.34, 531–543. 10.1016/j.ijmst.2024.04.005
15
KimH.-M. (2014). Stability analysis for ground uplift in underground storage caverns for high pressurized gas using Hoek–Brown strength criterion and geological strength index (GSI). Tunn. Undergr. Space24 (4), 289–296. 10.7474/TUS.2014.24.4.289
16
KirschC. (1898). Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. Z. Des. Vereines Dtsch. Ingenieure42, 797–807.
17
KushnirR.UllmannA.DayanA. (2012). Thermodynamic and hydrodynamic response of compressed air energy storage reservoirs: a review. Rev. Chem. Eng.28 (2–3), 123–148. 10.1515/revce-2012-0006
18
LiG.-Y.MoP.-Q.LuZ.YuanR.YangH.YuH.-S. (2024). Drained cavity expansion–contraction in CASM and its application for pressuremeter tests in sands. J. Geotechnical Geoenvironmental Eng.150 (9), 04024073. 10.1061/JGGEFK.GTENG-12417
19
LiG.-Y.MoP.YangH.YuH.-S.QinY.-J. (2025). Undrained cavity expansion–contraction analysis in CASM and its application for pressuremeter tests. Can. Geotechnical J.62, 1–19. 10.1139/cgj-2024-0011
20
LiaoJ.JiangZ.KonietzkyH.HuangX.TianX. (2025). A modified model considering air mixed convection for analyzing the thermodynamic and heat transfer processes in CAES caverns. J. Energy Storage136, 118428. 10.1016/j.est.2025.118428
21
LuM. (1998). Finite element analysis of a pilot gas storage in rock cavern under high pressure. Eng. Geol.49 (3), 353–361. 10.1016/S0013-7952(97)00067-7
22
LuA.WangS.ZhangX.ZhangN. (2018). Solution of the elasto-plastic interface of circular tunnels in Hoek–Brown media subjected to non-hydrostatic stress. Int. J. Rock Mech. Min. Sci.106, 124–132. 10.1016/j.ijrmms.2018.04.013
23
MoP.WangJ. (2020). Shakedown analysis of cavities in cohesive-frictional materials and its application to underground energy storage caverns. Soils Found.60 (1), 77–89. 10.1016/j.sandf.2020.01.003
24
MoP.-Q.YuH.-S. (2018). Drained cavity expansion analysis with a unified state parameter model for clay and sand. Can. Geotechnical J.55 (7), 1029–1040. 10.1139/cgj-2016-0695
25
MoP.-Q.FangY.YuH.-S. (2020). Benchmark solutions of large-strain cavity contraction for deep tunnel convergence in geomaterials. J. Rock Mech. Geotechnical Eng.12 (3), 596–607. 10.1016/j.jrmge.2019.07.015
26
ParkK.-H.KimY.-J. (2006). Analytical solution for a circular opening in an elastic–brittle–plastic rock. Int. J. Rock Mech. Min. Sci.43 (4), 616–622. 10.1016/j.ijrmms.2005.11.004
27
SharanS. K. (2008). Analytical solutions for stresses and displacements around a circular opening in a generalized Hoek–Brown rock. Int. J. Rock Mech. Min. Sci.45 (1), 78–85. 10.1016/j.ijrmms.2007.03.002
28
ShiH.SongL.ZhangH.ChenW.LinH.LiD.et al (2022). Experimental and numerical studies on progressive debonding of grouted rock bolts. Int. J. Min. Sci. Technol.32, 63–74. 10.1016/j.ijmst.2021.10.002
29
ShiH.ChenW.WuJ.RongC.WangZ.LeiS.et al (2025). Unified rock strength theory incorporating holistic consideration of macro-meso-micro defect coupling effects. Eng. Fract. Mech.328, 111534. 10.1016/j.engfracmech.2025.111534
30
SunG. H.WangZ. X.WangJ.YiQ.MaH. L. (2023). Limit equilibrium method for calculating safe burial depth of underground caverns in compressed air energy storage plants. China Civ. Eng. J.56 (S2), 67–77. 10.15951/j.tmgcxb.2023.s2.s04
31
TunsakulJ.JongpradistP.SoparatP.KongkitkulW.NanakornP. (2014). Analysis of fracture propagation in a rock mass surrounding a tunnel under high internal pressure by the element-free Galerkin method. Comput. Geotechnics55, 78–90. 10.1016/j.compgeo.2013.08.003
32
TunsakulJ.JongpradistP.KimH.-M.NanakornP. (2018). Evaluation of rock fracture patterns based on the element-free Galerkin method for stability assessment of a highly pressurized gas storage cavern. Acta Geotech.13 (4), 817–832. 10.1007/s11440-017-0594-5
33
WangS.YinX.TangH.GeX. (2010). A new approach for analyzing circular tunnel in strain-softening rock masses. Int. J. Rock Mech. Min. Sci.47 (1), 170–178. 10.1016/j.ijrmms.2009.02.011
34
WangZ.JiaW.FengX.WangJ. (2023). Analytical solution of limit storage pressures for tunnel-type lined gas storage caverns. Chin. J. Theor. Appl. Mech.55 (3), 710–718. 10.6052/0459-1879-22-474
35
WuD.WangJ. G.HuB.YangS.-Q. (2020). A coupled thermo-hydro-mechanical model for evaluating air leakage from an unlined compressed air energy storage cavern. Renew. Energy146, 907–920. 10.1016/j.renene.2019.07.034
36
WuJ.WongH. S.ZhangH.YinQ.JingH.MaD. (2024). Improvement of cemented rockfill by premixing low-alkalinity activator and fly ash for recycling gangue and partially replacing cement. Cem. Concr. Compos.145, 105345. 10.1016/j.cemconcomp.2023.105345
37
WuJ.YangS.WilliamsonM.WongH. S.BhudiaT.PuH.et al (2025a). Microscopic mechanism of cellulose nanofibers modified cemented gangue backfill materials. Adv. Compos. Hybrid. Mater.8, 177. 10.1007/s42114-025-01270-9
38
WuJ.ZhangW.WangY.JuF.PuH.RiabokonE.et al (2025b). Effect of composite alkali activator proportion on macroscopic and microscopic properties of gangue cemented rockfill: experiments and molecular dynamic modelling. Int. J. Min. Metall. Mater.32, 1813–1825. 10.1007/s12613-025-3140-8
39
XuY. J.XiaC. C.ZhouS. W.ZhaoH. O.XueX. D. (2022). Anti-uplift failure criterion of caverns for compressed air energy storage based on the upper bound theorem of limit analysis. Chin. J. Rock Mech. Eng.41 (10), 1971–1980. 10.13722/j.cnki.jrme.2022.0018
40
YuJ.LiuZ.LinL.HuangJ.RenW.ZhouL. (2021). Characteristics of dilatancy of marble under variable amplitude cyclic loading and unloading. Rock Soil Mech.42 (11), 2934–2942. 10.16285/j.rsm.2021.0256
41
ZareifardM. R. (2020). A new semi-numerical method for elastoplastic analysis of a circular tunnel excavated in a Hoek–Brown strain-softening rock mass considering the blast-induced damaged zone. Comput. Geotechnics122, 103476. 10.1016/j.compgeo.2020.103476
42
ZhangH.MiaoX.ZhangG.WuY.ChenY. (2017). Non-destructive testing and pre-warning analysis on the quality of bolt support in deep roadways of mining districts. Int. J. Min. Sci. Technol.27, 989–998. 10.1016/j.ijmst.2017.06.028
43
ZhangJ.Hosseini ZadehA.KimS. (2021). Geomechanical and energy analysis on small- and medium-scale CAES in salt domes. Energy221, 119861. 10.1016/j.energy.2021.119861
44
ZhangH.WangL.LiJ.DuH.XuX. (2023). Study on the mechanical properties of unloading damaged sandstone under cyclic loading and unloading. Sci. Rep.13, 7370. 10.1038/s41598-023-33721-y
45
ZhangH.ShiH.ZhaoH.SongL. (2025). Characterization of transverse vibration response of resin-anchored bolt under axial tension and life cycle health assessment of anchorage debonding. Measurement256, 118273. 10.1016/j.measurement.2025.118273
46
ZhengY. R.ZhaoS. Y.DengC. J.LiuM. W.TangS. X.ZhangL. M. (2006). Development of finite element limit analysis method and its applications in geotechnical engineering. Eng. Sci.8 (12), 39–61. (in Chinese).
47
ZhuH. H.ZhangQ.ZhangL. Y. (2013). Review of research progresses and applications of Hoek-Brown strength criterion. Chin. J. Rock Mech. Eng.32 (10), 1945–1963. (in Chinese).
48
ZimmelsY.KirzhnerF.KrasovitskiB. (2002). Design criteria for compressed air storage in hard rock. Energy & Environ.13 (6), 851–872. 10.1260/095830502762231313
Summary
Keywords
elastoplastic analysis, excavation unloading-reloading, Hoek-brown criterion, lateral pressure coefficient, surrounding rock stability
Citation
Feng X-H, Guan Z-L, Sun H-W, Liu Z-Q, Huang H-Y and Mo P-Q (2026) Stability analysis of surrounding rock during excavation unloading-reloading based on the Hoek-Brown criterion. Front. Mater. 13:1927109. doi: 10.3389/fmats.2026.1927109
Received
03 July 2026
Revised
02 September 2026
Accepted
04 September 2026
Published
02 October 2026
Volume
13 - 2026
Edited by
Hao Shi, Anhui University of Science and Technology, China
Reviewed by
Deshan Cui, China University of Geosciences Wuhan, China
Jiawei Xu, Southeast University, China
Duncan Maina, Freiberg University of Mining and Technology, Germany
Updates
Copyright
© 2026 Feng, Guan, Sun, Liu, Huang and Mo.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Hai-Yang Huang, ts24030165p31@cumt.edu.cn
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.