1 Introduction

A karst cavity is a type of underground cavities in rock masses that can be formed from the dissolution of soluble rocks such as limestone, dolomite, and gypsum (Huang et al. 2017). Collapse of a karst cavity is a serious threat to human lives as well as an economic or environmental disaster. Therefore, obtaining information about karst cavity sizes and locations as well as the stability assessment of a spherical opening is of paramount importance to assure the safety of the existence of underground cavities.

The failure patterns of underground cavities in soils were investigated by Craig (1990) and Abdulla (1995) using centrifuge model tests as well as numerical and analytical techniques (e.g., Drumm et al. 1990; Abdulla and Goodings 1996; Tharp 1999; Vaziri et al. 2001; Augarde et al. 2003; Keawsawasvong and Ukritchon 2019; Shiau and Al-Asadi 2020; Shiau et al. 2021; Keawsawasvong 2021). The finite element limit analysis (FELA) is a powerful numerical technique that has been widely used by many researchers to determine the upper bound (UB) or the lower bound (LB) solutions of plastic collapse load of various stability problems based on the plastic bound theorems (Sloan 2013; Drucker et al. 1952). By using FELA, exact collapse loads can be accurately bracketed from UB and LB solutions. For the stability analysis of underground cavities, several researchers employed FELA to numerically solve the solutions to this problem (Augarde et al. 2003; Shiau and Smith 2006, Keawsawasvong and Ukritchon 2019; Shiau et al. 2016a, b; Keawsawasvong 2021). However, their solutions are limited to the cases of underground cavities in soils obeying the Mohr–Coulomb failure criterion. It is well known that rock masses are discontinuous materials with joints and fractures. The curved shape of the true failure envelope of rock masses cannot be replaced by a linear expression of the Mohr–Coulomb failure criterion. Consequently, it is seldom used to predict the stability of underground cavities in rock masses, in spite that the Mohr–Coulomb failure criterion constitutes a good approximation of the true failure envelope for some weak rock masses (Hoek et al. 2002).

The Hoek–Brown (HB) failure criterion (Hoek et al. 2002) has been widely accepted and adopted by many geotechnical engineers to predict the failure of intact rocks. It is an empirical failure criterion and the basic idea of the Hoek–Brown criterion was to start with the properties of intact rock, and then the factors to reduce those properties are added to the equation because of the existence of joints in the rock (Hoek et al. 2002). By adopting the HB failure criterion into FELA, many researchers have investigated the stability of tunnels and underground openings in HB rock masses under plane strain conditions (e.g., Fraldi and Guarracino 2009; Ukritchon and Keawsawasvong 2019a, b; Keawsawasvong and Ukritchon 2020; Rahaman and Kumar 2020; Xiao et al. 2018, 2019, 2021; Wu et al. 2020; Zhang et al. 2019).

Currently, there is no solution or stability criterion published for stability assessment of cavities in HB rock masses under axisymmetric conditions in the literature. This technical note aims to develop and propose a stability criterion for stability assessment of cavities in HB rock masses based on UB and LB solutions obtained from axisymmetric FELA. The considered parameters include the cover-depth ratio of cavities and the HB material parameters that have significant influences on the normalized collapse pressure applied at the rock surface above a cavity. Nonlinear regression analysis is employed to develop a closed-form approximate equation of this problem. This developed equation is valuable for engineers in practice to estimate the stability of underground cavities in rock masses.

2 Problem Statement and FELA Modelling

Figure 1 shows the problem definition of a spherical cavity in a rock mass. The cavity has a diameter (D) and a cover depth (C). A uniform surcharge (σs) at the collapse is applied over the surface area. The stability problem is investigated under 2D axisymmetric conditions. The work stated herein used the Hoek–Brown (HB) failure criterion (Hoek et al. 2002) to investigate the failure of a cavity in a rock mass. The Hoek–Brown (HB) parameters for a rock mass include σci, GSI, and mi, and a unit weight of γ.

Fig. 1
figure 1

The stability problem under axisymmetry

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The expression of the HB failure criterion is in the form of a power-law relationship between the effective major and minor principal stresses (σ1 and σ3) as shown in Eq. (1).

$$ - \sigma_{3} = \, - \sigma_{1} + \sigma_{ci} \left( { - m_{b} \frac{{\sigma_{1} }}{{\sigma_{ci} }} + s} \right)^{\alpha } $$
(1)

It is to be noted that the compression negative sign convention applies to Eq. 1 and σci denotes the uniaxial compressive strength of intact rock masses. Other parameters such as mb, s, and α are expressed in Eqs. (2)–(4).

$$ m_{b} = \,m_{i} \exp \left( {\frac{GSI - 100}{{28 - 14DF}}} \right) $$
(2)
$$ s = \,\exp \left( {\frac{{{\text{GSI}} - 100}}{{9 - 3{\text{DF}}}}} \right) $$
(3)
$$ \alpha = \frac{1}{2}\, + \frac{1}{6}\left( {e^{{ - \frac{{{\text{GSI}}}}{15}}} - e^{{ - \frac{20}{3}}} } \right) $$
(4)

In the HB failure criterion, GSI is the geological strength index describing the quality of an in-situ rock mass. A GSI of 10 represents an extremely poor rock mass whilst 100 is used for intact rock. mi is the parameter used to describe the frictional strength of the intact rock mass. Noting that DF is the disturbance factor that has the range of 0–1, an undisturbed in-situ rock mass with DF = 0 is studied throughout the paper.

Using the concept of dimensionless ratios for practical design purposes, the stability solutions are determined through the use of five dimensionless variables as shown in Eq. (5).

$$ \frac{{\sigma_{s} }}{{\sigma_{ci} }} = f\left( {\frac{C}{D},\frac{{\sigma_{ci} }}{\gamma D},m_{i} ,{\text{GSI}}} \right) $$
(5)

where σs/σci is the normalized collapse surcharge; C/D is the cover depth ratio; σci/γD is the normalized uniaxial compressive strength.

The computer program OptumG2 (FELA, OptumCE 2019) is employed to perform the numerical analyses of the upper bound (UB) and lower bound (LB) finite element limit analysis (FELA). The FELA is based on the plastic bound theorems for a perfectly plastic material with an associated flow rule in conjunction with the finite element discretization and the mathematical optimization (Sloan 2013). The results from FELA include the UB and LB solutions that can bracket the true limit load from above and below.

In UB FELA, the rock mass is discretized by using six-noded quadratic triangular elements to describe the overall velocity fields. In LB FELA, the rock mass is modelled using three-noded triangular elements to describe the linear stress field. The UB and LB solutions of this problem are computed by solving the optimization problem that minimizes (for the UB method) or maximizes (for LB method) the active surface pressure (σs) i.e., the collapse pressure at the ground surface. The mesh adaptivity technique (e.g., Ciria et al. 2008) is a powerful feature for improving LB and UB solutions. By activating this feature, more elements are added to the sensitive regions with large shear strain gradients at any iteration step, aiming to bridge the differences between UB and LB solutions. Five iterations of mesh adaptivity were used for all UB and LB simulations in the study, with 5000–10,000 elements in all analyses. It is interesting to note that the current technique reveals the location of a possible failure mechanism at the final stage of mesh adaptivity.

Figure 2 shows a typical domain for the analysis of a spherical cavity. The left boundary is the plane of axisymmetry where only vertical movements can take place. The same condition applies to the right boundary. Nevertheless, velocities are fixed in both vertical and horizontal directions at the bottom boundary. At the rock surface, there is a uniform surcharge σs applied over the surface area. The size of the domain is chosen to be large enough to avoid any interferences due to boundary effects. The current model does not allow internal pressure inside the cavity.

Fig. 2
figure 2

A spherical cavity under axisymmetry

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3 Results and Discussions

The chosen ranges of dimensionless parameters for the study are for σci/γD = 100–∞, GSI = 40 − 100, mi = 5–30 and C/D = 1–5. A total of 320 computed LB and UB solutions are obtained. Table 1 summarizes sixteen examples of computed LB and UB solutions of the normalized collapse surcharge σs/σci. The differences between UB and LB solutions with respect to their averages for all cases are also reported. These results show that the rigorous UB and LB results can bracket the “exact” solution to within 5% for all cases. Other numerical results are not shown here for the sake of limited space. They are available upon request. Graphically, selected LB and UB solutions of σs/σci are presented in Figs. 3, 4, 5, and 6 for demonstrating the effects of C/D, GSI, mi, and σci/γD respectively.

Table 1 Selected numerical results of σs/σci for cavities in rock masses
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Figure 3a–d show that σs/σci increases as the cover depth C/D increases. This trend is for σci/γD = 100 and all GSI values presented. In general, the larger the value of mi is, the greater the increase rate. The effect of GSI on σs/σci is shown in Fig. 4, where σci/γD = 100 and mi = 5, 10, 20, and 30. The figure shows a nonlinear relationship between GSI and σs/σci. An increase in GSI results in an exponential increase in σs/ci, as reflected in the mathematic equations of HB failure criterion in Eqs. (2)–(4).

Fig. 3
figure 3

Effect of C/D on the normalized collapse pressure σs/σci a GSI = 40, b GSI = 60, c GSI = 80 and d GSI = 100

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Fig. 4
figure 4

Effect of GSI on the normalized collapse pressure σs/σci a mi = 5, b mi = 10, c mi = 20 and d mi = 30

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The effect of mi on σs/σci is illustrated in Fig. 5a–d for four different values of GSI = 40, 60, 80, and 100. The study is for σci/γD = ∞ and five cover depth ratios C/D = 1–5. In general, σs/σci increases linearly as mi increases for all C/D. Note that as C/D increases, the gradient of the linear line also increases. Figure 6 shows the effect of σci/γD on σs/σci. This is for the cases of mi = 20. It is clear that the effect σci/γD on σs/σci is insignificant for all C/D, given all the horizontal lines in the figure. Although it is physically impossible to have a weightless rock, the obtained results in Figs. 5 and 6 for σci/γD = ∞ are simply representatives of a very “strong” rock (i.e., very large strength ratio). Numerically, we can either put a very large value of σci or a very small γ to achieve the numerical results, which are needed to develop the stability criterion in the next section.

Fig. 5
figure 5

Effect of mi on the normalized collapse pressure σs/σci a GSI = 40, b GSI = 60, c GSI = 80 and d GSI = 100

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Fig. 6
figure 6

Effect of σci/γD on the normalized collapse pressure σs/σci a C/D = 1, b C/D = 2, c C/D = 4 and d C/D = 5

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Figure 7 presents upper bound adaptive meshes of four various cover depth ratios C/D. The chosen rock mass is for GSI = 80, mi = 10, and σci/γD = 100. It is important to note that the automatic adaptive meshing technique utilizes shear power dissipation as the control variable for re-meshing estimation. The number of elements in areas with very high shear power dissipation is automatically increased through successive iterations using the adaptive technique. Although the final adaptive mesh so produced resembles a so-called failure mechanism, it is common to use the contour plots of shear power dissipation to depict possible failure mechanisms of a soil structure, as it provides a good indicator of the intensity of non-zero plastic strains. Technically speaking, the actual values of the contour are not important in a perfectly plastic material model using FELA, and therefore the contour bars for these plots are not normally shown in technical documents. Figure 8 demonstrates such a contour plot obtained from the kinematic solutions (or UB solutions). Both Figs. 7 and 8 have shown that the larger the cover depth ratio C/D, the larger the size of the plastic shear zone, and the greater the rock stability (σs/σci). For C/D = 5, the failure zone appears to initiate from the invert, whilst for C/D = 1, a more chimney type of vertical slippage is observed. The information on surface failure extent may be useful in decision making in relation to the development of evacuation zone.

Fig. 7
figure 7

Upper bound axisymmetric mesh for various cover depth ratios C/D (GSI = 80, mi = 10, σci/γD = 100) a C/D = 1, b C/D = 2, c C/D = 4 and d C/D = 5

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Fig. 8
figure 8

Shear power dissipation contours for various cover depth ratios C/D (GSI = 80, mi = 10, σci/γD = 100) a C/D = 1, b C/D = 2, c C/D = 4 and d C/D = 5

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4 Stability Criterion

Design tables and charts of σs/σci have been presented in the earlier sections. Nevertheless, in most cases, the limit pressure σs is greater than the unconfined compressive strength of the intact rock σci. To obtain solutions with small values of σs/σci, an approximate expression for calculating collapse pressures at the rock surface above a cavity is developed by using a curve fitting method. The average values of UB and LB are employed to determine an appropriate mathematical expression. The proposed stability criterion is presented in Eq. (6).

$$ \sigma_{s} = N_{c} \sigma_{ci} - N_{\gamma } \gamma D $$
(6)

Both Nc and Nγ are to be determined using Eqs. (7)–(11) with known values of C/D, GSI, mi, and σci/γD

$$ N_{c} = F_{1} + F_{2} m_{i} $$
(7)
$$ N_{\gamma } = F_{3} $$
(8)
$$ F_{1} = {\text{GSI}}\left[ {b_{1} + b_{2} \frac{C}{D} + b_{3} \left( \frac{C}{D} \right)^{2} } \right] + {\text{GSI}}^{2} \left[ {c_{1} + c_{2} \frac{C}{D} + c_{3} \left( \frac{C}{D} \right)^{2} } \right] $$
(9)
$$ F_{2} = e_{1} + e_{2} \frac{C}{D} + {\text{GSI}}\left[ {f_{1} + f_{2} \frac{C}{D} + f_{3} \left( \frac{C}{D} \right)^{2} } \right] + {\text{GSI}}^{2} \left[ {g_{1} + g_{2} \frac{C}{D}} \right] + {\text{GSI}}^{3} \left( {d_{2} \frac{C}{D}} \right) $$
(10)
$$ F_{3} = - a_{1} - a_{2} \frac{C}{D} $$
(11)

where ai, bi, ci, di, ei, fi, and gi are constant coefficients that were determined by performing a least square method (Sauer 2014). The optimum values of these constant coefficients are shown in Table 2.

Table 2 Constants for the design equation
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The value of R2 of the proposed new stability criterion is about 99.98%, meaning that the approximation from Eq. (6) fits the FELA results very well. Figure 9 shows that Nγ is a function of C/D only and the relationship between the two is linear. Note that Nγ is negative for C/D < 2.5. This means that for cavities at relatively small depths, the self-weight of the ground has a positive effect on the limit load (i.e., σs increases with γ). Intuitively, this is not right. Nevertheless, by taking another look at Eqs. (7, 9, and 10) would have explained this. Noting that the values of (F1, F2, and Nc) become small when C/D < 2.5, it may result in a total decrease of σs (see Eq. 6). On the other note, Fig. 10 shows that Nc is a function of both C/D and GSI for the given values of mi = 5, 10, 20, and 30. In general, Nc increases nonlinearly as the cover depth C/D increases for all GSI values selected in the figures. The rate of increase is different for each GSI value. The curve becomes flattered (gradient decrease) as GSI decreases.

Fig. 9
figure 9

The factor Nγ

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Fig. 10
figure 10

The factor Nc a mi = 5; b mi = 10; c mi = 20; and d mi = 30

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5 Conclusions

This short technical note has successfully studied the stability of spherical cavity in axisymmetric Hoek–Brown rock mass using the rigorous upper and lower bound finite element limit analysis. The solution was formulated to find the limit normalized surface pressures σs/σci that is a function of four dimensionless parameters; namely the cover-depth ratio C/D, the Geological Strength Index GSI, the Hoek–Brown mi parameter, and the normalized uniaxial compressive strength σci/γD. A new stability criterion for predicting the stability of cavities in rock masses is developed by using a least square method of the computed solutions. The main findings of the present study are summarized as follows.

  • The limit normalized surface pressures σs/σci increases as the cover depth ratio C/D increases. The greater the values of GSI and mi, the larger the σs/σci. In addition, the effect of σci/γD on σs/σci is insignificant for all considered depth ratios in this study.

  • The failure mechanism of a cavity resembles a chimney type of vertical slippage when C/D is small. The lateral size of the failure mechanism extends when C/D increases.

  • The new cavity stability factors Nc and Nγ for the stability of cavities in rock masses are proposed in this paper, where Nc is a function of C/D only while Nγ is a function of C/D, GSI, and mi.