Theoretical study and application of energy extreme value discrimination method of softening zone for bursting hazard

Abstract

Considering the current situation that it's difficult to quantify the discrimination of rockburst hazard, an analytical solution for the softening zone energy of surrounding rock under rockburst in a circular roadway is obtained theoretically based on the elastoplastic softening model of a circular roadway. The critical softening zone energy, critical softening zone radius, critical roadway shrinkage displacement, and critical roadway shrinkage rate are determined. An energy extreme value discrimination method of softening zone for bursting hazard is proposed. The critical softening zone energy is taken as the evaluation basis of the bursting liability of the coal seam where the roadway (a full coal roadway) is located. The uniform energy index, which is the ratio of the external disturbance energy of the surrounding rock softening zone to the critical softening zone energy, is used as the evaluation basis for bursting hazards. Theoretical analysis shows that the smaller the critical softening zone energy of the roadway surrounding rock is, the stronger the bursting liability of the coal seam is, and the greater the uniform bursting hazard index is, the stronger the bursting hazard of the roadway. Subsequently, based on the critical softening zone energy and the uniform energy index, a classification standard of bursting liability and an evaluation method of bursting hazard are proposed. The reliability of the softening zone energy extremum criterion, the bursting liability classification standard, and the bursting hazard evaluation method was verified through indoor and field tests. The energy extreme value discrimination method of softening zone for bursting hazard considers the internal and external factors of roadway rockburst comprehensively and can judge the rockburst risk through field real-time observation, which is a more scientific and reasonable method to predict the risk of roadway rockburst. The research work has important theoretical guiding significance for the prevention and control of roadway rockbursts.

Introduction

Rockburst is one of the most serious disasters associated with coal mining.^1–3 And for the mine with rockburst risk, rockburst prediction becomes the premise of safety production. At present, bursting liability identification and bursting hazard assessment are the most basic and universal methods for rockburst prediction.^4–9 Therefore, the rationality and reliability of bursting liability identification and bursting hazard assessment are very important for the prediction and prevention of rockbursts. In recent years, scholars have conducted extensive research on the bursting liability and hazard of coal seams.

Bieniawski et al. ¹⁰ proposed two bursting liability indexes: the coal and rock elastic strain energy index and the bursting energy index. Kidybinski ¹¹ proposed that the elastic strain energy index of coal and rock mass should be considered as the bursting liability index. Singh^12–14 and Sears and Heasley ¹⁵ proposed a method to judge the bursting liability of coal and rock based on an energy release index and analyzed the relationship between the energy release index and the physical and mechanical properties of coal and rock. Tang et al. ¹⁶ used the residual energy index, obtained from the difference between the elastic strain energy stored before reaching the peak strength and the energy consumed after reaching the peak strength, to judge the bursting liability of coal and rock. Pan et al. ¹⁷ and Zhu et al. ¹⁸ proposed three new indexes including the bursting energy velocity index, critical softening zone coefficient and critical stress coefficient by considering the time effect. Qi et al. ¹⁹ proposed using the uniaxial compressive strength of coal as a new index to evaluate the bursting liability of coal seams. In China, the current bursting liability indexes for coal seams mainly include the elastic strain energy index, bursting energy index, dynamic destruction time, and uniaxial compressive strength.

The bursting liability can only describe the properties of the coal seam and reflect the degree of bursting liability of the coal seam. However, the bursting hazard reflects the degree of bursting hazard of a rockburst area. An accurate assessment of bursting hazard is the basis of targeted rockburst prevention and control. Bardet ²⁰ and Larsson ²¹ proposed a safety evaluation index method for coal mines. Kornowski ²² proposed a comprehensive assessment method for bursting hazard. Dou and Drzezia ²³ and Dou et al.^24–26 proposed the ground-sound method and the superposition method of the relative stress concentration coefficient for bursting hazard evaluation and established vibration wave computed tomographic detection technology for the dynamic prediction and evaluation of bursting hazard. Jiang et al. ²⁷ proposed a fuzzy evaluation method for the bursting hazard of a coal mass by using the ratio of vertical stress to the uniaxial compressive strength of the coal mass and the elastic strain energy index. The current coal mine safety regulations in China stipulate that “the comprehensive index method is preferred to determine the risk of rockburst.”

To sum up, the existing evaluation methods and indicators of bursting liability and bursting hazard have achieved a lot of research results in practice, but there are still some deficiencies. On the one hand, the current bursting liability index is proposed by considering a single influence factor respectively, which is relatively independent, so the evaluation results are often inconsistent and discrete. On the other hand, the present comprehensive index evaluation method is to calculate a comprehensive index based on the analysis of engineering geological and mining technical conditions, which lacks a certain theoretical basis for the selection of influencing factors, value range and weight. In the view of this, based on the mechanism of roadway rockbursts, a new discriminant method of rockburst hazard will be proposed by considering the influencing factors of the bursting liability and hazard of coal seams in a unified manner. Furthermore, a new bursting liability index and bursting hazard index will be proposed, which will be verified based on actual rockburst cases.

Theoretical study of energy extreme value discrimination method of softening zone for bursting hazard

Limitations of the current bursting liability index

At present, measurement of the uniaxial compressive strength ${\sigma }_{\mathrm{c}}$ , dynamic destruction time DT, elastic strain energy index W_ET, and bursting energy index K_E under the laboratory conditions stipulated in measurement, monitoring and prevention of rockburst (GB/T 25217.2-2010) is the generally applied classification and index measurement method for coal and rock bursting liability in China. However, there are still some problems and limitations associated with this method.

Uniaxial compressive strength

The uniaxial compressive strength, ${\sigma }_{\mathrm{c}}$ , is the ratio of the failure load sustained by a standard coal specimen to the area of its bearing surface under uniaxial compression in the laboratory. The unit of this parameter is megapascals（hereinafter referred to as MPa）, and the schematic of its calculation is presented in Figure 1. In the current classification standard for bursting liability index, the coal is deemed to have no bursting liability when ${\sigma }_{\mathrm{c}}<7$ , weak bursting liability when $7\le {\sigma }_{\mathrm{c}}<14$ , and strong bursting liability when ${\sigma }_{\mathrm{c}}\ge 14$ . In other words, the greater the uniaxial compressive strength is, the stronger the bursting liability is. However, the bursting liability of coal seams with high uniaxial compressive strength is not always strong, and the bursting liability of coal seams with low uniaxial compressive strength may not always be weak. For example, with regard to Figure 1, the relationship between the uniaxial compressive strengths ${\sigma }_{\mathrm{c}1}$ , ${\sigma }_{\mathrm{c}2}$ , and ${\sigma }_{\mathrm{c}3}$ , which correspond to stress–strain curves I, II, and III, is ${\sigma }_{\mathrm{c}1}>{\sigma }_{\mathrm{c}2}>{\sigma }_{\mathrm{c}3}$ . However, curves I, II, and III, correspond to the ideal elastic-plastic material, elastic-plastic material, and ideal elastic–brittle material, respectively, and the corresponding relationship in terms of bursting liability is I ＜ II ＜ III. The curve in Figure 1 shows that the stress drops $\mathrm{\Delta }{\sigma }_1$ , $\mathrm{\Delta }{\sigma }_2$ , and $\mathrm{\Delta }{\sigma }_3$ corresponding to curves, I, II, and III, respectively, increased successively. In other words, the greater the stress drop is, the stronger the bursting liability is. It is evident that the stress drop, rather than uniaxial compressive strength, truly reflects the bursting liability of the coal seam in the stress-strain curve under uniaxial compression in the laboratory.

Figure 1.

Calculation of uniaxial compressive strength.

2. Elastic strain energy index and bursting energy index

The elastic strain energy index, W_ET, is the ratio of the elastic strain energy to the plastic strain energy as the coal specimen unloads when the stress reaches a certain value (before failure) under uniaxial compression. A schematic of the calculation of this parameter is shown in Figure 2. In fact, the coal seam is not loaded from zero, but part of the plastic strain energy is consumed because it has been subjected to in situ stress and tectonic stress. When a rockburst occurs, the load on the coal seam exceeds the value of ${\sigma }_{\mathrm{c}}^{\mathrm{\prime }}$ (75%–85% ${\sigma }_{\mathrm{c}}$ ), reaching the ultimate strength ${\sigma }_{\mathrm{c}}$ . Therefore, the current method for elastic strain energy index measurement is not sufficiently accurate.

Figure 2.

Calculation of elastic strain energy index.

The bursting energy index, K_E, is the ratio of the strain energy accumulated before reaching the peak value to the strain energy dissipated after reaching the peak in the entire stress–strain curve of the coal specimen under uniaxial compression, and the schematic representation of the calculation of this parameter is shown in Figure 3. Here, A_S and A_X represent the strain energy before and after reaching the peak value, respectively. However, in practical scenarios, the coal seam is not loaded from zero, as it is already subjected to in-situ stress and tectonic stress. Figures 2 and 3 show that A_S does not account for all the strain energy accumulated before the peak and that some of this energy has been dissipated. Therefore, the strain energy before reaching the peak value should not be calculated from the beginning of loading, instead, the dissipated energy should be excluded.

Figure 3.

Calculation of bursting energy index.

Therefore, the existing measurement method can be modified to determine the elastic strain energy index and bursting energy index simultaneously. First, the coal specimen is subjected to uniaxial compression. When loaded to 75%–85% of the average failure load, it is unloaded to 1%–5% of the average failure load and then loaded up to any point after the residual strength is reached. Thus, the entire stress–strain curve is obtained. The schematic of the calculation of the modified elastic strain energy index and the bursting energy index is shown in Figure 4, where ${\mathrm{\Phi }}_{\mathrm{SP}}$ , $A_{\mathrm{S}}$ , and $A_X$ represent the plastic strain energy before the peak, strain energy accumulated before the peak, and strain energy dissipated after the peak, respectively. Thus, the elastic strain energy index is $W_{\mathrm{ET}}=A_{\mathrm{S}}/{\mathrm{\Phi }}_{\mathrm{SP}}$ , and the bursting energy index is $K_{\mathrm{E}}=A_{\mathrm{S}}/A_{\mathrm{X}}$ . However, even with this correction, there are still difficulties and shortcomings involved in measuring the elastic strain energy index and bursting energy index by using the method of calculating the area under the stress–strain curve. At initial loading, the curve indicates the densification stage of the coal specimen instead of a linear elastic deformation stage. This leads to the error in ${\mathrm{\Phi }}_{\mathrm{SP}}$ . Even after the complete failure of the coal specimen, the elastic strain energy of the coal body is not released completely. Therefore, the curve tends to be flat. However, the flat section does not represent the energy consumed during rockburst. This makes it difficult to define section DF.

Figure 4.

Calculation of modified elastic strain energy index and bursting energy index.

3. Dynamic destruction time DT

The dynamic destruction time, DT, is the time taken for the complete failure of the coal specimen after its ultimate strength is reached under uniaxial compression. The schematic representation of the calculation of this parameter is depicted in Figure 5. However, the measurement of dynamic destruction time in the laboratory is affected by the stiffness of the testing machine. If an ordinary testing machine is used, the coal specimen directly fails after reaching the ultimate strength, without any appreciable dynamic destruction time. If a rigidity testing machine is used, the dynamic destruction time changes with the loading rate. However, the existing dynamic destruction time measurement method does not account for the stiffness of the testing machine. This also affects the determination of bursting liability.

Figure 5.

Calculation of dynamic destruction time.

Therefore, it is evident that the limitations in the existing measurement and classification approaches have a significant influence on the accurate determination of coal seam bursting liability. Additionally, the existing bursting liability indexes were developed by considering individual influencing factors. This can lead to differences or inconsistent results when judging coal seam bursting liability based on several indexes.

Proposal of the constitutive model

As discussed in the “Limitations of the current bursting liability index” section, a reasonable constitutive model of coal should exclude the influence of the plastic strain energy lost before the peak value is reached and the residual strength after the peak value. Therefore, a bilinear constitutive relation (as shown in Figure 6) is selected in this study. In other words, the maximum slope of the curve during the loading phase after unloading, which is before the peak value, is selected as the elastic modulus E. The slope absolute value (maximum) of the curve removed the residual strength after the peak value is selected as the softening modulus $\lambda$ to ensure that the measurement results of coal bursting liability are safer and closer to actual situations.

Figure 6.

Calculation of bursting liability indexes.

Analytical solution for the softening zone energy of roadway surrounding rock

The development and occurrence of rockbursts are physical processes involving coal and rock mass loading, energy storage, energy dissipation, sudden deformation and failure, and a large amount of energy instantaneous release. Under the action of the far-field stress, stress concentration develops in the area surrounding the roadway, and the coal masses are broken to form a softening zone when the stress exceeds the peak intensity. In contrast, the surrounding rock not reaching its peak value remains in the elastic zone. As the far-field stress increases, the energy in the elastic zone is gradually released into the softening zone and the deformation of the roadway surrounding rock increases. When the softening zone energy reaches an extremely high value, the surrounding rock deformation system of the roadway is in a critical stable equilibrium state. At this time, in the case of any disturbance, the surrounding rock deformation system of the roadway becomes unstable. And the energy exceeding the extreme energy value of the softening zone makes the coal masses in the softening zone be destroyed quickly. Thereafter, these coal masses in the softening zone are driven into the roadway, resulting in the occurrence of a rockburst.

For the convenience of analysis, the roadway is simplified into a circular roadway with radius a, which is subjected to a hydrostatic pressure P in the far field, and the effect of gravity is neglected. A unit width is taken along the roadway axis for subsequent analysis. Thus, the roadway rockburst is simplified into an axisymmetric plane strain problem. This roadway analysis model is shown in Figure 7.

Figure 7.

Roadway analysis model.

The coordinate conversion of the bilinear constitutive relation in Figure 6 is carried out and ${\epsilon }_0$ is set as the origin. The peak value refers to the compressive strength of coal, ${\sigma }_{\mathrm{c}}$ , and the corresponding strain is ${\epsilon }_{\mathrm{c}}$ . The coal is in the stage of linear elastic deformation before the peak value, the slope E is the elasticity modulus of coal, and the constitutive relation of the elastic stage is $\sigma =E\epsilon$ . After the peak strength, the coal exhibits linear softening, and the slope absolute value $\lambda$ refers to the softening modulus of coal, such that $\lambda =\left|\frac{\mathrm{\Delta }\sigma }{\mathrm{\Delta }\epsilon }\right|$ , which reflects the strain softening characteristics of coal after the peak strength is reached. In addition, $\lambda /E$ reflects the bursting liability of coal, ¹ then let

$$k={\displaystyle \frac{\lambda }{E}}$$

(1)

where k is defined as the deformation modulus index. The larger the value of $\lambda$ , the greater the value of k, and the more rapid the impact failure of coal, indicating stronger bursting liability of coal. Moreover, $\lambda =\mathrm{\infty }$ and $k=\mathrm{\infty }$ correspond to ideal elastic–brittle materials, whereas $\lambda =0$ and k = 0 corresponds to ideal elastic-plastic materials.

According to Figure 6, the constitutive relation of the softening stage is

$$\sigma ={\sigma }_{\mathrm{c}}-\lambda (\epsilon -{\epsilon }_{\mathrm{c}})$$

(2)

The yield criterion of coal in the softening zone is the Mohr-Coulomb criterion.

$${\sigma }_{\theta }=m{\sigma }_r+{\sigma }_{\lambda }(r)$$

(3)

where ${\sigma }_{\theta }$ and ${\sigma }_r$ are the tangential stress and radial stress, respectively, $m=(1+\sin \phi )/(1-\sin \phi )$ ; and $\phi$ is the internal friction angle of coal. When the coal enters the softening zone, the internal friction angle only undergoes minor changes (approximately 30°). Hence, it can be assumed that $\phi$ remains constant, with $\phi ={30}^{\circ }$ . Thus, m = 3. Here, ${\sigma }_{\lambda }(r)$ is the bearing capacity of coal in the softening zone, and it is a function of the deformation in coal. As the deformation in coal varies with the depth of the softening zone, the bearing capacity ${\sigma }_{\lambda }(r)$ can be regarded as a function of the depth r of the softening zone.

On substituting m = 3 into equation (3), we obtain

$${\sigma }_{\theta }=3{\sigma }_r+{\sigma }_{\lambda }(r)$$

(4)

The equilibrium equation is

$${\displaystyle \frac{\mathrm{d}{\sigma }_r}{\mathrm{d}r}+{\displaystyle \frac{{\sigma }_r-{\sigma }_{\theta }}{r}=0}}$$

(5)

The geometric equation is

$${\epsilon }_r={\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r},{\epsilon }_{\theta }={\displaystyle \frac{u}{r}}}$$

(6)

where u is the radial displacement; ${\epsilon }_r$ and ${\epsilon }_{\theta }$ are the radial strain and tangential strain, respectively.

As the roadway analysis model is an axisymmetric plane strain problem, ${\epsilon }_z=0$ . If the volume of the softening zone is incompressible, then

$${\epsilon }_r+{\epsilon }_{\theta }+{\epsilon }_z=0$$

(7)

On substituting equation (6) into equiation (7), we obtain

$$u={\displaystyle \frac{B}{r}}$$

(8)

where B is the integral constant. Furthermore, by substituting equation (8) into equation (6), we obtain

$${\epsilon }_r={\displaystyle \frac{-B}{r^2},{\epsilon }_{\theta }={\displaystyle \frac{B}{r^2}}}$$

(9)

Thus, the strain strength of the softening zone is calculated as

$${\epsilon }_i=\sqrt{{\displaystyle \frac{2}{9}[{({\epsilon }_r-{\epsilon }_z)}^2+{({\epsilon }_r-{\epsilon }_{\theta })}^2+{({\epsilon }_{\theta }-{\epsilon }_z)}^2]}}={\displaystyle \frac{2}{\sqrt{3}}{\displaystyle \frac{B}{r^2}}}$$

(10)

If $\epsilon$ is substituted with ${\epsilon }_i$ in equation (2), the bearing capacity of coal in the softening zone can be obtained:

$${\sigma }_{\lambda }(r)={\sigma }_{\mathrm{c}}(1+k)-{\displaystyle \frac{2Ek}{\sqrt{3}}{\displaystyle \frac{B}{r^2}}}$$

(11)

At the junction of the softening and elastic zones, i.e. when $r=\rho$ , ${\sigma }_{\lambda }(r)|{}_{r=\rho }={\sigma }_{\mathrm{c}}$ . On substituting these relations into equation (11), we obtain

$$B={\displaystyle \frac{\sqrt{3}}{2}{\rho }^2{\epsilon }_{\mathrm{c}}}$$

(12)

Then,

$${\epsilon }_i={\displaystyle \frac{{\rho }^2{\epsilon }_{\mathrm{c}}}{r^2}}$$

(13)

Furthermore,

$${\sigma }_{\lambda }(r)={\sigma }_{\mathrm{c}}(1+k)-k{\sigma }_{\mathrm{c}}{\displaystyle \frac{{\rho }^2}{r^2}}$$

(14)

$$u={\displaystyle \frac{\sqrt{3}{\rho }^2{\epsilon }_{\mathrm{c}}}{2r}}$$

(15)

$${\epsilon }_r=-{\displaystyle \frac{\sqrt{3}{\rho }^2{\epsilon }_{\mathrm{c}}}{2r^2},{\epsilon }_{\theta }={\displaystyle \frac{\sqrt{3}{\rho }^2{\epsilon }_{\mathrm{c}}}{2r^2}}}$$

(16)

On substituting equations (4) and (14) into equation (5), we obtain

$$r\left({\displaystyle \frac{\mathrm{d}{\sigma }_r}{\mathrm{d}r}}\right)=2{\sigma }_r-k{\sigma }_{\mathrm{c}}{\displaystyle \frac{{\rho }^2}{r^2}+{\sigma }_{\mathrm{c}}(1+k)}$$

(17)

The radial stress, ${\sigma }_r$ , in the softening zone is obtained. If it is substituted into equation (4), ${\sigma }_{\theta }$ can be determined.

$${\sigma }_r={\displaystyle \frac{k{\sigma }_{\mathrm{c}}}{4}{\displaystyle \frac{{\rho }^2}{r^2}+Q{r}^2-{\displaystyle \frac{1+k}{2}{\sigma }_{\mathrm{c}}}}}$$

(18)

$${\sigma }_{\theta }=-{\displaystyle \frac{k{\sigma }_{\mathrm{c}}}{4}{\displaystyle \frac{{\rho }^2}{r^2}+3Q{r}^2-{\displaystyle \frac{1+k}{2}{\sigma }_{\mathrm{c}}}}}$$

(19)

where Q is the integration constant.

At the roadway boundary, i.e. when r = a, ${\sigma }_r(a)=0$ . On substituting these relations into equation (18), we obtain

$$Q={\displaystyle \frac{1}{a^2}\left({\displaystyle \frac{1+k}{2}{\sigma }_{\mathrm{c}}-{\displaystyle \frac{k{\sigma }_{\mathrm{c}}}{4}{\displaystyle \frac{{\rho }^2}{a^2}}}}\right)}$$

(20)

If equation (20) is substituted into equations (18) and (19), we obtain

$${\sigma }_r={\displaystyle \frac{{\sigma }_c}{2}\{\left[-{\displaystyle \frac{k}{2}{\displaystyle \frac{{\rho }^2}{a^2}+(1+k)}}\right]{\left({\displaystyle \frac{r}{a}}\right)}^2+{\displaystyle \frac{k}{2}{\displaystyle \frac{{\rho }^2}{r^2}-(1+k)}}\}}$$

(21)

$${\sigma }_{\theta }={\displaystyle \frac{3{\sigma }_c}{2}\left\{\left[-{\displaystyle \frac{k}{2}{\displaystyle \frac{{\rho }^2}{a^2}+(1+k)}}\right]{\left({\displaystyle \frac{r}{a}}\right)}^2+{\displaystyle \frac{k}{2}{\displaystyle \frac{{\rho }^2}{r^2}-(1+k)}}\right\}+{\sigma }_{\mathrm{c}}\left[1-k\left({\displaystyle \frac{{\rho }^2}{r^2}-1}\right)\right]}$$

(22)

Thus, the strain energy of the softening zone is

$${\mathrm{\Pi }}^{\mathrm{p}}={\int }_0^{2\pi }{\int }_a^{\rho }\left[{\displaystyle \frac{1}{2}{\sigma }_{\theta }{\epsilon }_{\theta }+{\displaystyle \frac{1}{2}{\sigma }_r{\epsilon }_r}}\right]r\mathrm{d}r\mathrm{d}\theta =-{\displaystyle \frac{\sqrt{3}\pi a^2{\sigma }_{\mathrm{c}}^2}{8E}[k{\left({\displaystyle \frac{\rho }{a}}\right)}^6\text{-}2(k+1){\left({\displaystyle \frac{\rho }{a}}\right)}^4+(k+2){\left({\displaystyle \frac{\rho }{a}}\right)}^2]}$$

(23)

Assuming that the roadway shrinkage displacement corresponds to the radial displacement at r = a, it is considered a basic quantity for describing the deformation state of the roadway. On substituting this into equation (15), the roadway shrinkage rate is obtained as follows:

$${\displaystyle \frac{u_a}{a}={\displaystyle \frac{\sqrt{3}{\sigma }_{\mathrm{c}}}{2E}{\displaystyle \frac{{\rho }^2}{a^2}}}}$$

(24)

Therefore,

$${\displaystyle \frac{{\rho }^2}{a^2}={\displaystyle \frac{2E}{\sqrt{3}{\sigma }_{\mathrm{c}}}{\displaystyle \frac{u_a}{a}}}}$$

(25)

On substituting equation (25) into equation (23), we obtain

$${\mathrm{\Pi }}^{\mathrm{p}}=-\pi {\sigma }_{\mathrm{c}}{a}^2\left[{\displaystyle \frac{k{E}^2}{3{\sigma }_{\mathrm{c}}^2}{\left({\displaystyle \frac{u_a}{a}}\right)}^3-{\displaystyle \frac{(k+1)E}{\sqrt{3}{\sigma }_{\mathrm{c}}}{\left({\displaystyle \frac{u_a}{a}}\right)}^2+{\displaystyle \frac{k+2}{4}{\displaystyle \frac{u_a}{a}}}}}\right]$$

(26)

Analysis of critical softening zone energy

On taking the derivative of equation (26) with respect to $u_a/a$ ,

$${\displaystyle \frac{\mathrm{\partial }{\mathrm{\Pi }}^{\mathrm{p}}}{\mathrm{\partial }(u_a/a)}=-\pi {\sigma }_{\mathrm{c}}{a}^2\left[{\displaystyle \frac{k{E}^2}{{\sigma }_{\mathrm{c}}^2}{\left({\displaystyle \frac{u_a}{a}}\right)}^2-{\displaystyle \frac{2(k+1)E}{\sqrt{3}{\sigma }_{\mathrm{c}}}{\displaystyle \frac{u_a}{a}+{\displaystyle \frac{k+2}{4}}}}}\right]}$$

(27)

When $\frac{\mathrm{\partial }{\mathrm{\Pi }}^{\mathrm{p}}}{\mathrm{\partial }(u_a/a)}=0$ ${\mathrm{\Pi }}^{\mathrm{p}}$ has a maximum value, and the surrounding rock deformation system of the roadway is in a critical stable equilibrium state. In this case, any disturbance causes the surrounding rock deformation system of the roadway to lose its stability, which leads to roadway rockbursts. Thus, when $\frac{\mathrm{\partial }{\mathrm{\Pi }}^{\mathrm{p}}}{\mathrm{\partial }(u_a/a)}=0$ , the critical tunnel shrinkage rate can be obtained, which is expressed by $\frac{u_{a\mathrm{cr}}}{a}$ . On substituting it into equation (24), as $\frac{u_{a\mathrm{cr}}}{a}$ and $\frac{{\rho }^2}{a^2}$ always satisfy $\frac{u_{a\mathrm{cr}}}{a}\ge 0$ and $\frac{{\rho }^2}{a^2}\ge 0$ , then

$${\displaystyle \frac{u_{a\mathrm{cr}}}{a}={\displaystyle \frac{\sqrt{3}{\sigma }_{\mathrm{c}}}{2E}(1+{\displaystyle \frac{1}{k})\left[{\displaystyle \frac{2}{3}+{\displaystyle \frac{1}{3}\sqrt{1+{\displaystyle \frac{3}{{(k+1)}^2}}}}}\right]}}}$$

(28)

Furthermore,

$${\left({\displaystyle \frac{{\rho }_{\mathrm{cr}}}{a}}\right)}^2=(1+{\displaystyle \frac{1}{k})\left[{\displaystyle \frac{2}{3}+{\displaystyle \frac{1}{3}\sqrt{1+{\displaystyle \frac{3}{{(k+1)}^2}}}}}\right]}$$

(29)

where ${\rho }_{\mathrm{cr}}$ is the critical softening zone radius (also known as the “critical softening zone depth”) of the roadway rockburst. On substituting equations (2) and (29) into equation (23), the critical softening zone energy of the roadway rockburst can be obtained:

$$\begin{array}{rl}{\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}& =-{\displaystyle \frac{\sqrt{3}\pi a^2{\sigma }_{\mathrm{c}}^2}{8E}\{k{\left[(1+{\displaystyle \frac{1}{k})\left({\displaystyle \frac{2}{3}+{\displaystyle \frac{1}{3}\sqrt{1+{\displaystyle \frac{3}{{(k+1)}^2}}}}}\right)}\right]}^3}\\ & -2(k+1){\left[(1+{\displaystyle \frac{1}{k})\left({\displaystyle \frac{2}{3}+{\displaystyle \frac{1}{3}\sqrt{1+{\displaystyle \frac{3}{{(k+1)}^2}}}}}\right)}\right]}^2\\ & +(k+2)\left[(1+{\displaystyle \frac{1}{k})\left({\displaystyle \frac{2}{3}+{\displaystyle \frac{1}{3}\sqrt{1+{\displaystyle \frac{3}{{(k+1)}^2}}}}}\right)}\right]\}\end{array}$$

(30)

where $\frac{\pi a^2}{2E}$ is the energy density, and $\pi a^2$ is the volume per unit length of the roadway. When the softening zone energy reaches the critical value ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ , the surrounding rock system of the roadway is in the critical stable equilibrium state, and it loses stability even under the slightest disturbance. Furthermore, the excess energy causes the coal in the softening zone to fail rapidly and rush into the roadway, forming a roadway rockburst.

Let

$$\begin{array}{rl}{C}_k& =-{\displaystyle \frac{\sqrt{3}}{4}\{k{\left[(1+{\displaystyle \frac{1}{k})\left({\displaystyle \frac{2}{3}+{\displaystyle \frac{1}{3}\sqrt{1+{\displaystyle \frac{3}{{(k+1)}^2}}}}}\right)}\right]}^3}\\ & -2(k+1){\left[(1+{\displaystyle \frac{1}{k})\left({\displaystyle \frac{2}{3}+{\displaystyle \frac{1}{3}\sqrt{1+{\displaystyle \frac{3}{{(k+1)}^2}}}}}\right)}\right]}^2\\ & +(k+2)\left[(1+{\displaystyle \frac{1}{k})\left({\displaystyle \frac{2}{3}+{\displaystyle \frac{1}{3}\sqrt{1+{\displaystyle \frac{3}{{(k+1)}^2}}}}}\right)}\right]\}\end{array},$$

then equation (29) can be simplified as

$${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}=\pi a^2{\displaystyle \frac{{\sigma }_{\mathrm{c}}^2}{2E}{C}_k}$$

(31)

In order to analyze the influence of roadway radius a, elasticity modulus E, uniaxial compressive strength ${\sigma }_{\mathrm{c}}$ and deformation modulus index k on the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ in equation (31), the four parameters are assigned values, and three of them are kept constant while one parameter is changed. The relation curves of a, E, ${\sigma }_{\mathrm{c}}$ k and ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ are drawn respectively, as shown in Figure 8.

Figure 8.

Relation curves of roadway radius, elasticity modulus, uniaxial compressive strength, deformation modulus index and critical softening zone energy, respectively.

According to equation (31) and Figure 8, the critical softening zone energy reflects the rockburst tendency. The lower the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is, the higher the probability of the occurrence of a rockburst. The value ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is affected by the deformation modulus index k, uniaxial compressive strength ${\sigma }_{\mathrm{c}}$ , elasticity modulus E, and roadway radius a. The larger the value of k is, the lower the value of ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is; the larger the value of ${\sigma }_{\mathrm{c}}$ is, the higher the value of ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is; and the larger the value of E is, the lower the value of ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is; the larger the value of a is, the higher the value of ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is. Among these parameters, k plays the main role; the larger the value of k is, the lower the value of ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is, and the stronger the bursting liability is. When k = ∞, ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ corresponds to ideal elastic-brittle materials. Moreover, when k = 0, ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ corresponds to ideal elastic-plastic materials.

Analysis of uniform energy index

The uniform energy index for the occurrence of roadway rockbursts is defined as

$$K_{\mathrm{\Pi }}={\displaystyle \frac{{\mathrm{\Pi }}^{\mathrm{e}}}{{\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}}}$$

(32)

where ${\mathrm{\Pi }}^{\mathrm{e}}$ is the energy of disturbance outside the softening zone on its outer boundary. When the energy of disturbance acting on the outer boundary is greater than the critical softening zone energy, a rockburst starts. The energy exceeding ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ makes the coal in the softening zone break rapidly and forms a crushing zone around the roadway. Supposing that the width of the crushing zone is $\mathrm{\Delta }a$ , which is equivalent to the rapid expansion $\mathrm{\Delta }a$ of the radius, then equation (31) becomes

$${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}=\pi (a+\mathrm{\Delta }a{)}^2{\displaystyle \frac{{\sigma }_{\mathrm{c}}^2}{2E}{C}_k}$$

(33)

Under the action of the rockburst, the coal in the crushing zone expands rapidly and breaks into the roadway.

Assuming that the dilatancy coefficient is $C_f$ after the coal completely fails and that the rapid shrinkage displacement of the roadway is $\mathrm{\Delta }d$ , then

$$\pi \left({(a-u_{\mathrm{cr}})}^2-{\left(a-u_{\mathrm{cr}}-{\displaystyle \frac{\mathrm{\Delta }d}{2}}\right)}^2\right)=\pi [{(a-u_{\mathrm{cr}}+\mathrm{\Delta }a)}^2-{(a-u_{\mathrm{cr}})}^2](C_f-1)$$

(34)

where the critical roadway shrinkage displacement, $u_{\mathrm{cr}}$ , is considerably small compared with the roadway radius a. Hence, it can be neglected. Thus, the rapid shrinkage displacement $\mathrm{\Delta }d$ is

$$\mathrm{\Delta }d=2\left(a-\sqrt{a^2{C}_f-{(a+\mathrm{\Delta }a)}^2(C_f-1)}\right)$$

(35)

Equations (32)–(35) show that the uniform energy index represents the hazard of rockbursts. The larger the uniform energy index $K_{\mathrm{\Pi }}$ is, the greater the roadway rapid shrinkage displacement $\mathrm{\Delta }d$ is when rockbursts occur, the more serious the roadway failure is, and the more dangerous the rockburst is (that is, the stronger the rockburst hazard is).

In summary, the critical softening zone energy is proposed by comprehensively considering the relationship among the stress, elasticity modulus, and softening modulus of surrounding rock. This can be used to characterize the bursting liability of surrounding rock. The uniform energy index combines the critical softening zone energy, external load, and structural parameters of the roadway to reflect the bursting hazard of the roadway. In actual mining engineering, the mining conditions, geological structure, coal and rock properties, and mining methods are various. Thus, the bursting hazard of the coal seam does not remain the same; it varies with the mining process. Therefore, from equations (31) and (32), the advantage of bursting hazard evaluation based on energy is that this approach prevents contradictory results between bursting liability classification and bursting hazard evaluation under a single influencing factor, and also guides on-site, real-time tracking, monitoring, and evaluation to ensure the safety of roadways.

Classification of bursting liability and bursting hazard

In equation (31), $\pi a^2$ is the volume per unit length of the roadway. When the roadway is determined, the radius a is fixed (it is generally considered as 2.5 m); hence, this remains constant. ${\sigma }_{\mathrm{c}}^2/(2E)$ is the energy density, $C_k$ is a function of k ( $k=\lambda /E$ ), and k is equivalent to the bursting energy index. Therefore, in this study, the existing classification standards of the bursting liability indexes are unified into the critical softening zone energy, and two demarcation points ( $1.01\times {10}^6\mathrm{J}$ and $2.05\times {10}^5\mathrm{J}$ ) for bursting liability are obtained. In other words, there is no bursting liability when ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}>1.01\times {10}^6\mathrm{J}$ , there exists a weak bursting liability when $2.05\times {10}^5\mathrm{J}<{\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}\le 1.01\times {10}^6\mathrm{J}$ , and there exists a strong bursting liability when ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}<2.05\times {10}^5\mathrm{J}$ . The rationality and accuracy need to be revised continuously. The classification results of coal seam bursting liability are listed in Table 1.

Table 1.

Classification of bursting liability.

No bursting liability	Weak bursting liability	Strong bursting liability
${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}>1.01\times {10}^6\mathrm{J}$	$2.05\times {10}^5\mathrm{J}<{\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}\le 1.01\times {10}^6\mathrm{J}$	${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}<2.05\times {10}^5\mathrm{J}$

The classification of bursting hazard uses the bursting liability classification results in Table 1 as the reference point, combined with the energy transmitted from the source to the outer boundary of the softening zone of the roadway surrounding rock and the roadway failure caused by the residual energy released into the roadway. When ${\mathrm{\Pi }}^{\mathrm{e}}<{\mathrm{\Pi }}_{{}^{\mathrm{cr}}}^{\mathrm{p}}$ (that is, $K_{\mathrm{\Pi }}<1$ ), there is no residual energy released to the roadway, and consequently, a rockburst does not occur. In this case, the roadway has no bursting hazard. When ${\mathrm{\Pi }}^{\mathrm{e}}\ge {\mathrm{\Pi }}_{{}^{\mathrm{cr}}}^{\mathrm{p}}$ (that is, $K_{\mathrm{\Pi }}\ge 1$ ), residual energy is released into the roadway, resulting in a rockburst and failure of the roadway. Based on actual rockburst cases, when the roadway failure ratio (that is, the ratio of rapidly shrinking displacement to the roadway diameter) is less than 4%, the roadway does not undergo serious failure. In contrast, the roadway failure ratio exceeding 8% results in serious consequences. Therefore, the bursting hazard indexes corresponding to roadway failure ratios of 4% and 8%, i.e. 1.08 and 1.16, respectively, are selected as the demarcation points for the bursting level. Thus, $1\le K_{\mathrm{\Pi }}<1.08$ , $1.08\le K_{\mathrm{\Pi }}<1.16$ , and $K_{\mathrm{\Pi }}\ge 1.16$ indicate weak, medium, and strong bursting hazards, respectively. It should be noted that the reliability of this approach needs to be evaluated via tests and engineering inspections in order to realize further improvements. The classification results for the coal seam bursting hazard are shown in Table 2.

Table 2.

Classification of bursting hazard.

No bursting hazard	Weak bursting hazard	Medium bursting hazard	Strong bursting hazard
$K_{\mathrm{\Pi }}<1$	$1\le K_{\mathrm{\Pi }}<1.08$	$1.08\le K_{\mathrm{\Pi }}<1.16$	$K_{\mathrm{\Pi }}\ge 1.16$

Analysis on influencing factors of critical softening zone energy

According to the “Analysis of critical softening zone energy” section, the parameters that affect the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ of a roadway rockburst are the deformation modulus index k, uniaxial compressive strength ${\sigma }_{\mathrm{c}}$ , and elasticity modulus E. In this study, the rationality of the criterion and the critical softening zone energy were verified using a similarity simulation test under varying values of k, ${\sigma }_{\mathrm{c}}$ , and E. According to the relationship between roadway shrinkage rate and softening zone energy and the relationship between critical roadway shrinkage rate and critical softening zone energy, as expressed in equations (26), (28), and (30), the roadway shrinkage displacement in the similarity simulation test could be monitored directly. Furthermore, it was converted into the actual value under working conditions based on the similarity ratio. Thus, the roadway shrinkage rate was calculated, and then, the critical softening zone energy of the roadway surrounding rock under actual working conditions was obtained.

The main aggregate materials selected in the similarity simulation test include sand and gypsum. By adding a certain proportion of rosin to a mixture of sand, gypsum, and water, composite materials with low material strength and certain brittle failure characteristics are prepared. These materials are suitable for simulating roadway rockbursts. Seven groups of similar material models were created. The rosin was dissolved in alcohol, and then added to the mixture of sand, gypsum, and water, according to pre-set proportions. The material proportions and mechanical parameters of the similarity model are shown in Table 3, and the results retain three valid digits. The similarity ratio between the model and the actual roadway surrounding rock was 1:100; the dimensions of the model were 400 × 400 × 200 mm, and the roadway was excavated relative to the center of the model. The section of the roadway was $\mathrm{\Phi }$ 50 mm. A schematic representation of the roadway rockburst simulation test system is shown in Figure 9.

Figure 9.

Schematic representation of roadway rockburst simulation test system.

Table 3.

Material proportions and mechanical parameters of similarity model.

Serial number	Proportions	Mechanical parameters
	Sand:gypsum:water: rosin:alcohol	$\gamma$ (g/cm3)	${\sigma }_{\mathrm{c}}$ (MPa)	E (MPa)	$\lambda$ (MPa)	k
1	84:25:5:8:4	2.40	0.170	14.0	80.0	5.70
2	78:20:4:5:3	2.40	0.170	14.0	42.0	3.00
3	74:12:3:4:2	2.40	0.170	14.0	28.0	2.00
4	82:24:6:7:4	2.40	0.150	14.0	80.0	5.70
5	81:22:6:6:3	2.40	0.120	14.0	80.0	5.70
6	83:24:5:7:4	2.40	0.170	12.0	68.4	5.70
7	85:26:6:9:5	2.40	0.170	16.0	91.2	5.70

Figure 10 shows a diagram of roadway deformation and failure. Figure 10 (a) is the model diagram of the roadway surrounding rock at the initial loading stage. During the loading process of the model, cracks are generated around the roadway, and a softening zone is formed, as shown in Figure 10 (b). With an increase in the loading, these cracks further expand, and the range of the softening zone expands. When the range of the softening zone expands to a certain limit, the roadway is destroyed suddenly, and a roadway rockburst occurs. Figure 10 depicts the roadway failure after impact.

Figure 10.

Deformation and failure of roadway. (a) initial loading (b) roadway deformation diagram during loading (c) sudden shrinkage failure of roadway.

Figures 11–13 show the roadway shrinkage rate–time curves under various k, ${\sigma }_{\mathrm{c}}$ , and E values, respectively. Table 4 shows a comparison between the theoretical calculation and test results. And the results retain three valid digits.

Figure 11.

Roadway shrinkage rate–time curves under various deformation modulus indexes.

Figure 12.

Roadway shrinkage rate–time curves under various uniaxial compressive strengths.

Figure 13.

Roadway shrinkage rate–time curves under various elasticity moduli.

Table 4.

Comparison of theoretical and experimental results.

Serial number	Theoretical results		Experimental results		Error/%
	uacr/a	${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ /104 J）	uacr/a	${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ /104 J）
1	0.0125	18.2	0.0127	18.0	1.60
2	0.0144	39.6	0.0146	39.5	1.40
3	0.0166	67.0	0.0166	67.6	0
4	0.0110	14.2	0.0113	13.7	2.70
5	0.00880	9.10	0.00910	8.60	3.40
6	0.0146	21.2	0.0146	21.2	0
7	0.0109	15.9	0.0113	15.0	3.70

As indicated by the roadway shrinkage rate–time curves in Figures 11–13, the roadway shrinkage rate increases almost linearly with the loading. A sudden increase is observed when the roadway shrinkage rate reaches a certain value, and a significant catastrophe point appears on the curve. This catastrophe point denotes the rockburst failure of the roadway, and the roadway shrinkage rate at this point is the critical roadway shrinkage rate. Figure 11 depicts the results of the tests on Groups 1–3. According to the results of Groups 1–3, as shown in Figure 11 and Table 4, when ${\sigma }_{\mathrm{c}}$ and E are fixed, the theoretical calculation results show that the critical roadway shrinkage rates are 0.0125, 0.0144, and 0.0166 when k = 5.70, 3.00, and 2.00, respectively, and the corresponding critical softening zone energies are 1.82 × 10⁵, 3.96 × 10⁵, and 6.70 × 10⁵ J, respectively. On the other hand, the experimental results show that the critical roadway shrinkage rates are 0.0127, 0.0146, and 0.0166 when k = 5.70, 3.00, and 2.00, respectively, and the corresponding critical softening zone energies are 1.80 × 10⁵, 3.95 × 10⁵, and 6.76 × 10⁵ J, respectively. Therefore, the larger the deformation modulus index k, the lower the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ , and the more easily the roadway fails. This is consistent with the theoretical results.

Figure 12 depicts the results of the tests on Groups 1, 4, and 5. According to the results of Groups 1, 4, and 5, as shown in Figure 12 and Table 4, when k and E are fixed, the theoretical calculation results show that the critical roadway shrinkage rates are 0.0125, 0.0110, and 0.00880 when ${\sigma }_{\mathrm{c}}$  = 17.0, 15.0, and 12.0 MPa, respectively, and the corresponding critical softening zone energies are 1.82 × 10⁵, 1.42 × 10⁵, and 9.10 × 10⁴ J, respectively. On the other hand, the experimental results show that the critical roadway shrinkage rates are 0.0127, 0.0113, and 0.00910 when ${\sigma }_{\mathrm{c}}$  = 17.0, 15.0, and 12.0 MPa, respectively, and the corresponding critical softening zone energies are 1.80 × 10⁵, 1.37 × 10⁵, and 8.60 × 10⁴ J, respectively. Therefore, it can be seen that the lower the uniaxial compressive strength ${\sigma }_{\mathrm{c}}$ , the lower the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ , and the more easily the roadway fails. This is consistent with the theoretical results.

Figure 13 presents the results of the tests on Groups 1, 6, and 7. According to the results of Groups 1, 6, and 7, as shown in Figure 13 and Table 4, when k and ${\sigma }_{\mathrm{c}}$ are fixed, the theoretical calculation results show that the critical roadway shrinkage rates are 0.0125, 0.0146, and 0.0109 when E = 1.40 × 10³, 1.20 × 10³, and 1.60 × 10³ MPa, respectively, and the corresponding critical softening zone energies are 1.82 × 10⁵, 2.12 × 10⁵, and 1.59 × 10⁵ J, respectively. On the other hand, the experimental results show that the critical roadway shrinkage rates are 0.0127, 0.0146, and 0.0113 when E = 1.40 × 10³, 1.20 × 10³, and 1.60 × 10³ MPa, respectively, and the corresponding critical softening zone energies are 1.80 × 10⁵, 2.12 × 10⁵, and 1.50 × 10⁵ J, respectively. Therefore, the larger the elasticity modulus E, the lower the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ , and the more easily the roadway fails. This is consistent with the theoretical results.

In summary, the roadway failure and the trends in the roadway shrinkage rate–time curve observed in the similarity simulation tests are consistent with the theoretical results, proving that a critical value of roadway rockburst exists. When $k$ , ${\sigma }_{\mathrm{c}}$ , and E vary, the error between the critical roadway shrinkage rates determined via tests and theoretical calculations is less than 5%, which is within the error range. When the roadway radius a is determined, the larger the deformation modulus index $k$ is, the lower the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is. Furthermore, the higher the uniaxial compressive strength ${\sigma }_{\mathrm{c}}$ is, the higher the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is. The larger the elasticity modulus E is, the lower the critical softening zone energy ${\mathrm{\Pi }}_{\mathrm{cr}}^{\mathrm{p}}$ is. These results are consistent with the theoretical analyses.

Tests on bursting liability identification and bursting hazard evaluation

A case study of a rockburst mine in Henan province was adopted to verify the rationality of the theoretical analyses.

Bursting liability identification

Coal samples from the 13190, 13210, and 13230 working faces of a coal mine in Henan province were selected for the uniaxial compression test, in order to determine the critical softening zone energy. Raw coal retrieved from the field was cut into coal specimens with dimensions of 50 mm× 50 mm× 100 mm. The uniaxial compressive strength ${\sigma }_{\mathrm{c}}$ , elasticity modulus E, and softening modulus λ of the coal samples were measured according to the bursting liability measurement method in Figure 6, and the deformation modulus index $k$ was calculated. According to equation (31), the critical softening zone energies of the roadway surrounding rock, where the sampling points of the three working faces are located, were calculated. The results are shown in Table 5, which retains three valid digits.

Table 5.

Test and calculated results.

Coal samples	Working face		13190 working face			13210 working face			13230 working face
	Serial number		M1	M2	M3	M4	M5	M6	M7	M8	M9
Roadway radius/m			2.50	2.50	2.50	2.50	2.50	2.50	2.50	2.50	2.50
Uniaxial compressive strength/MPa			16.0	17.1	17.1	16.9	17.2	16.9	15.8	16.1	16.1
Elasticity modulus/(103 MPa)			1.43	1.45	1.47	1.43	1.46	1.42	1.38	1.40	1.42
Softening modulus/(103 MPa)			7.80	8.08	8.12	8.17	8.21	8.05	7.18	7.30	7.37
Deformation modulus index			5.45	5.57	5.52	5.71	5.62	5.67	5.20	5.21	5.19
Critical softening Zone energy/(105 J)		Trial value	1.84	1.81	1.82	1.76	1.81	1.77	1.77	1.81	1.80
		Average value	1.82			1.78			1.79

Table 5 shows that the critical softening zone energies of the roadway surrounding rock at the 13190, 13210, and 13230 working faces are 1.82 × 10⁵, 1.78 × 10⁵, and 1.79 × 10⁵ J, respectively. Additionally, Table 1 indicates that all three working faces have strong bursting liability.

Bursting hazard evaluation

According to the event with the largest residual energy (that is, the event causing the most serious damage to the roadway) among the three working surface rockburst events recorded by a microseismic monitoring system, the mine earthquake energy ${\mathrm{\Pi }}^{\mathrm{e}}$ transmitted from the source to the outer boundary of softening zone was recorded through microseismic sensors placed at the softening zone outer boundary of the roadway surrounding rock. The uniform energy index $K_{\mathrm{\Pi }}$ of the roadway was calculated according to equation (32). The theoretical value of the roadway rapid shrinkage displacement was calculated using equation (35). The measured roadway rapid shrinkage displacement was obtained through real-time monitoring using a displacement sensor. The error between the theoretical and measured values was also obtained. According to the classification standard of a bursting hazard, as listed in Table 2, the bursting hazard of the sampling points at the three working faces could be determined. The evaluation results are shown in Table 6, which retains three valid digits.

Table 6.

Results of bursting hazard assessments.

Working face		13190 working face	13210 working face	13230 working face
Rockburst events		Event 1	Event 2	Event 3
Energy of mine earthquake	At the source/(106)	60.0	9.00	6.40
	At the softening zone outer boundary/(105)	2.90	2.02	2.72
Bulking factor		1.50	1.50	1.50
Critical softening zone energy/(105)		1.82	1.78	1.79
Roadway rapid shrinkage displacement	theoretical value/m	0.800	0.170	0.690
	measured value/m	0.780	0.180	0.660
	error/%	2.50	5.90	4.30
Uniform energy index		1.59	1.13	1.52
Bursting hazard classes		Strong	Medium	Strong

As shown in Table 6, the maximum source energies of the rockburst events at the 13190, 13210, and 13230 working faces were 6.00 × 10⁷, 9.00 × 10⁶, and 6.40 × 10⁶ J, respectively. The energies propagated to the outer boundary of the roadway softening zone reduced to 2.90 × 10⁵, 2.02 × 10⁵, and 2.72 × 10⁵ J, respectively. Furthermore, they were divided by the corresponding critical softening zone energies, and the unified energy indexes obtained were 1.59, 1.13, and 1.52, respectively. According to the classification standard of a bursting hazard, as shown in Table 2, the sampling points of the 13190 and 13230 working faces exhibited a strong bursting hazard, whereas the sampling points of the 13210 working face exhibited a medium bursting hazard.

Therefore, greater source energy in the far field does not necessarily indicate more severe roadway damage, and stronger bursting liability does not necessarily indicate a stronger bursting hazard. The bursting hazard and degree of roadway damage are determined by the critical softening zone energy and the energy of the softening zone outer boundary of the roadway surrounding rock that is transmitted from the far-field source. In other words, even if the source energy is considerably high, a rockburst will not occur if the energy of softening zone outer boundary that is transmitted from the far-field source does not reach the critical softening zone energy. However, if the energy of softening zone outer boundary that is transmitted from the far-field source is considerably larger than the critical softening zone energy, a severe rockburst will occur. In addition, a uniform energy index is obtained based on historical rockburst events and the critical softening zone energy. Hence, the bursting hazard varies for different regions of the same coal seam or even for different mining stages in the same region. This is also in line with the actual conditions of underground mining. Therefore, it is necessary to perform real-time monitoring during mining and provide the dynamic evaluation results of bursting hazards, which can serve as the basis for rockburst predictions.

In addition, the theoretical values of the roadway rapid shrinkage displacement for the three rockburst events are 0.780, 0.170, and 0.690 m, while the measured values are 0.780, 0.180, and 0.660 m, respectively. Hence, the corresponding errors are 2.50%, 5.90%, and 4.30%, respectively. As is evident, the errors between the theoretical and measured values are less than 10%, which are within the allowable range of errors. Therefore, the calculation results obtained using the critical softening zone energy and uniform energy indexes are deemed correct and reasonable.

Conclusions

1. Based on the mechanism of roadway rockbursts and by considering the influencing factors of the bursting liability and hazard of a coal seam, an elastoplastic softening mechanical model was established. In addition, a softening zone energy extremum criterion for roadway rockbursts was proposed. The analytical solution of the softening zone energy for a rockburst in a circular roadway was obtained. The critical roadway shrinkage displacement, the critical roadway shrinkage rate, and the critical softening zone radius of the roadway rockburst were determined. In addition, an energy extreme value discrimination method of the softening zone for bursting hazards was proposed.

2. It was proposed that the critical softening zone energy can be used as a judgment basis for bursting liability, and the analytical solution was determined. The analyses show that the smaller the critical softening zone energy is, the stronger the bursting liability is. Thereafter, the bursting liability of the coal seam was classified according to the corresponding critical softening zone energy. The advantage of using the critical softening zone energy is that all the influencing factors of bursting liability are considered in a unified manner, which helps prevent contradictory results compared to bursting liability index classification using a single influencing factor.

3. It was proposed that the uniform energy index can be used as a bursting hazard index, and an analytical solution was determined. The analyses show that the larger the uniform energy index is, the greater the roadway rapid shrinkage displacement when a rockburst occurs, i.e. the stronger the bursting hazard is. The classification of bursting hazards was conducted according to the uniform energy index. The uniform energy index was obtained based on previous rockburst events and the corresponding critical softening zone energy. This approach considers all the influencing factors of the bursting hazard, conforms with the actual conditions of the field, and affords dynamic evaluation results for the bursting hazard.

4. Through a similarity simulation test of the rockbursts in circular roadways, it was proved that a critical value of the rockburst in a circular roadway exists; the obtained law was found to be consistent with theoretical analyses. Through a field test, the bursting liabilities of three working faces in a mine were classified, and a uniform energy index was obtained based on the critical softening zone energy and the previous rockburst events that occurred at each working face. Subsequently, a bursting hazard evaluation of the working faces was conducted, and the results were proved to be reasonable and reliable.