Introduction

With the increase of coal mining depth and strength, coal and rock dynamic disasters such as rock burst are becoming more and more serious, which seriously affects the safe and efficient production of mines1,2,3. The occurrence of coal-rock dynamic disasters is closely related to the activity law of overlying strata4,5. Therefore, it is of great theoretical and engineering significance to reveal the evolution characteristics of overlying strata structure and the induced mechanical response of stope for the prevention and control of coal-rock dynamic disasters.

In recent years, many scholars national and international have carried out in-depth research on the occurrence mechanism and early warning prevention of coal-rock dynamic disasters. Among them, in terms of the evolution characteristics of overlying strata structure and the mechanism of instability and disaster, the overburden structure model represented by masonry beam6 and transfer rock beam7 is proposed. On this basis, the large space stope structure model and its influence on mine pressure behavior8, the ‘three-zone structure loading model’ structure model and the mechanism of rock burst9, the macro ‘large-small’ structure model of overlying strata and the mechanism of instability and disaster10, the arch structure of loose layer and its mechanism of action on the stability of mining overlying strata11 are analyzed. In the study of the mechanism of overlying strata breaking and rock burst, the characteristics of hard rock strata breaking and instability and its dynamic disturbance to the stope are mainly studied12.It is considered that the hard rock layer instantaneously breaks and releases strain energy, and transmits strain energy outward in the form of stress wave, causing damage to the overburden structure of the stope13,14,15,16. Based on existing research on the fracture-induced rock burst mechanism of overlying strata, this study investigates the control mechanisms of overlying strata spatial structure evolution on ground pressure and surface fractures17. The proposed stress arch theory of overlying strata provides a mechanical basis for coal pillar recovery18,19. At the specimen scale, the study revealed the dynamic mechanical degradation mechanisms of red sandstone under high temperatures20. This study provides a direct theoretical basis for optimizing experimental parameters, selecting monitoring indicators, and designing protective measures for dynamic impact testing apparatus.

The existing research determines the static load distribution characteristics of the stope through the research and analysis of the structural characteristics and load characteristics of the overlying strata in the stope. Through on-site monitoring, theoretical estimation and other methods, the characteristics of strain energy release in the stope and its dynamic disturbance to the working face are obtained, and then the formation conditions and occurrence mechanism of dynamic disasters in the stope are determined, which provides scientific guidance for the prevention and control of coal-rock dynamic disasters.

The mechanical response of coal rock in the stope is the result of the combined action of internal and external factors21,22. When the internal and external factors are transient, the mechanical response will inevitably change. In particular, the instability and fracture of the hard rock stratum causes the change of the geometric properties and structural characteristics of the spatial structure of the overlying strata within the influence range. The coal and rock in the stope transits from the previous equilibrium state to the next equilibrium state, accompanied by the release of strain energy and gravitational potential energy. The existing research focuses on the influence of near-field factors such as the main roof on the mechanical response of surrounding rock in the two equilibrium states, without considering the influence of far-field surrounding rock23. It is considered that the instantaneous breaking of hard rock only forms the source and disturbs the working face, ignoring the interaction between the front and back equilibrium hard rock and its adjacent rock, and the transient characteristics of mechanical response.

In fact, it can be known from the theory of roof cutting and pressure relief24,25 that after artificial roof cutting, the peak value of abutment pressure of coal body in gob-side entry decreases and the peak position migrates to deep coal body, and the mechanical response of coal rock in stope will change within its influence range. On the basis of this cognition, and based on the concept of dynamics, the author has proposed that the transient spatial structure of overlying strata induced by the fracture of hard rock strata will lead to the occurrence of dynamic response of coal and rock in the stope and the step change of static mechanical response, and the dissipated energy in the dynamic evolution process is composed of two parts : strain energy and gravitational potential energy released by overlying strata26. However, it is mainly a discussion of the concept and lacks systematic research. Under the transient excitation of overlying strata spatial structure, the characteristics of stress field and displacement field of coal and rock in the stope will respond accordingly. The vertical displacement of different areas of hard rock strata has positive and negative step change characteristics, but the reason is that the stress transfer path of overlying strata changes. Therefore, it is of great significance to reveal the response characteristics of coal-rock stress field under structural transient excitation for the prevention and control of coal-rock dynamic disasters.

Based on the existing research, this paper uses the physical similarity simulation experiment method to analyze the transient characteristics of the spatial structure of the overlying strata and the mechanical response of the coal rock. Furthermore, through theoretical analysis and numerical calculation method, the mechanism of overburden stress field step change under the influence of mechanical response when hard rock stratum is broken is studied. The research results have certain theoretical reference value for the prevention and control of coal and rock dynamic disasters.

Transient characteristics of spatial structure of overlying strata

In order to study the evolution characteristics of the mechanical response of coal and rock in the stope, this paper takes the longwall working face of B4-1 coal seam with strong impact tendency of W1143 in a mine as the background, and builds a physical similarity simulation experiment for analysis. The main coal seam B4-1 and the B4-2 coal seam roof with an average spacing of 9 m have strong impact tendency, which is a typical hard thick roof working face. The coal and rock information is shown in Table 1. The physical similar material simulation experiment scheme is shown in Reference27.

Table 1 Characteristics of coal and rock strata27.
Full size table

In reference27, the general law of dynamic and static response of stope coal under structural transient excitation is expounded through the case of the third cycle of pressure in the model experiment. For this reason, the following mainly analyzes the transient characteristics of the spatial structure of the overburden rock through the first weighting of the model experiment. At the same time, based on the stress sensor and acceleration sensor laid inside the model, the dynamic and static responses of coal and rock in different areas of stope space are monitored, and the internal relationship between the dynamic and static response characteristics of coal and rock and the transient characteristics of overlying strata spatial structure is clarified.

Figure 1 shows the transient characteristics of the spatial structure of overlying strata during the first weighting. It can be seen from the figure that the transient characteristics of the spatial structure of overlying strata caused by the fracture of hard rock strata are symmetrical. After the formation of the working face, the immediate roof collapses with the mining, and the coarse-grained sandstone (the lower hard rock layer) above the B4-1 coal seam is a fixed beam structure at both ends. When the working face advances 60 cm, in the process of model standing, the fixed beam structure of the lower hard rock layer breaks instantaneously and loses stability, forming the first weighting phenomenon. The height of the fracture surface of the overlying rock suddenly increased from 7.6 to 11.7 cm, and the overlying rock fell into the goaf in the transient area, and it was symmetrical along the strike.

Fig. 1
figure 1

Transient characteristics of spatial structure of overlying strata before and after the first weighting.

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Figure 2 shows the monitoring results of abutment pressure of coal and lower hard rock before and after the first weighting and periodic weighting. It can be seen from the figure that:

Fig. 2
figure 2

The step change characteristics of abutment pressure of coal seam and hard rock layer before and after the first weighting.

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Before and after the pressure, the abutment pressure of coal body shows obvious step change characteristics, and it is symmetrical along the strike. Compared with before pressure, the abutment pressure in the range of 0–12.5 cm in front of the working face decreases sharply after weighting, forming a negative amount of step change zone, and the maximum amount of step change is − 1.47 MPa. The abutment pressure increases sharply in the range of 12.5–40 cm in front of the working face, forming a positive amount of step change area, and the maximum amount of step change is 1.96 MPa. The peak value of abutment pressure decreases and the action position jumps to the deep ahead of coal body. The peak value of abutment pressure of coal body decreases from 9.8 to 9.45 MPa, and the peak position jumps to the deep ahead of coal body by 5 cm. The abutment pressure of the coal body on the side of the cutting hole also shows similar step-change characteristics, and the step-change characteristics are slightly different, as shown in Fig. 2a.

Similar to the case of coal body, the abutment pressure on the lower hard rock strata also shows obvious step change characteristics before and after the first weighting and periodic weighting. However, the difference is that the peak value of abutment pressure on hard rock increases and the peak position jumps to the side of goaf. Moreover, compared with the first weighting, the step change characteristics before and after the periodic weighting are more obvious. Compared with before the first weighting, the peak value of the abutment pressure on the hard rock layer after the first weighting increased from 7.01 to 7.80 MPa, and the peak position jumped 2.5 cm to the goaf side. Compared with that before periodic weighting, the peak value of abutment pressure on hard rock increases from 7.33 to 8.28 MPa after periodic weighting, and the peak position jumps to the side of goaf by 10 cm.

It can be seen that because the hard rock stratum has the mechanical characteristics of instantaneous fracture, the fracture of the lower hard rock stratum in different pressure stages will cause the static mechanical response of the coal and rock in the stope to change. Due to the symmetry of the transient spatial structure of the overlying strata during the first weighting and the asymmetry of the periodic weighting, the dynamic and static response characteristics induced by the periodic weighting are also symmetrical and asymmetrical. Moreover, due to the influence of the mechanical properties of coal rock, there are significant differences in the static mechanical response between coal and lower hard rock. Therefore, the following theoretical analysis method is used to further reveal the step change mechanism of static mechanical response of coal rock in different areas of stope space.

Mechanical response mechanism of hard rock fracture

Based on the simulation results shown in Figs. 1 and 2, a schematic diagram of the step change mechanism of the static mechanical response of coal and rock in the stope under the structural transient excitation shown in Fig. 3 is constructed.

Fig. 3
figure 3

Schematic diagram of step change mechanism of coal rock static mechanical response under structure transient excitation.

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In Fig. 3, σm and σmf are the abutment pressure of coal before and after structural transient, MPa ; σrd and σrdf are the abutment pressure of the lower hard rock layer outside the transient region before and after the structural transient, MPa; σrdz is the abutment pressure of the lower hard rock layer in the transient region before the structural transient, MPa; σrdx and σrdf are the support load of the gangue in the goaf before and after the structural transient, respectively, MPa; Δσmf and Δσrdf are the amount of step changes of the peak abutment pressure of coal and lower hard rock under structural transient excitation, MPa, respectively. dmf and drdf are the transition distance of abutment pressure peak position of coal and lower hard rock under structural transient excitation, respectively, m; the σrux is the support load of the middle load layer to the upper hard rock layer, MPa; ldf is the first weighting step, m; hs1, hd, hs2 and hu are the thickness of the lower load layer, the lower hard rock layer, the middle load layer and the upper hard rock layer, respectively, m. In the following analysis, the middle of the goaf is taken as the coordinate origin O, the advancing direction of the working face is taken as the x-axis, and the vertical direction is taken as the y-axis.

Occurrence mechanism of loading step change of lower hard rock stratum

The bearing capacity of lower hard rock strata before transient

It can be seen from Fig. 3a that before the fracture of the lower hard rock stratum (structural transient), due to the influence of mining, the load characteristics of the lower hard rock stratum in different regions are different, but the total load it bears is still the same as that in the original rock stress state. Therefore, the bearing capacity Qd of the lower hard rock stratum in the initial state is:

$$Q_{d} = \sigma_{d0} \left( {2l_{rdfh} + l_{df} } \right) = 2\int_{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}}^{{l_{rdfh} }} {\sigma_{rd} } dx + \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\sigma_{rdz} } dx$$
(1)

In the formula, lrdfh is the advanced mining influence range of the lower hard rock stratum (the maximum value before and after the breaking), m; σd0 is the stress acting on the lower hard rock stratum under the original rock stress state (including the self-weight of the lower hard rock stratum), MPa.

The bearing capacity of lower hard rock after transient

From the previous simulation results and the internal and external mechanical responses, it can be seen that after the structural transient, the mechanical response of the coal rock in the influence range will inevitably change. The abutment pressure distribution state of the lower hard rock layer changes from σrd changes to σrdf; moreover, because the transient area falls into the goaf, the total load it bears is reduced, as shown in Fig. 3b. According to the principle of equal stope load before and after hard rock breaking, the bearing capacity of lower hard rock Qrdf after breaking is as follows:

$$\begin{gathered} Q_{rdf} = \sigma_{d0} \left( {2l_{rdfh} + l_{df} } \right) - \left( {\gamma_{d} h_{d} + \gamma_{s2} h_{s2} } \right)l_{df} \hfill \\ \, = 2\int_{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}}^{{l_{rdfh} }} {\sigma_{rdf} } dx + \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\sigma_{ruxf} } dx \hfill \\ \end{gathered}$$
(2)

In the formula, γd and γs2 are the bulk density of the lower hard rock layer and the middle load layer, kN/m3, respectively.

The step change mechanism of bearing capacity of lower hard rock stratum

From Eq. (1) and Eq. (2), it can be obtained that the amount of step change ΔQd of the total load borne by the lower hard rock layer outside the transient region (part of the rock layer) before and after the structural transient is

$$\begin{aligned} \Delta Q_{d} & = Q_{rdf} - Q_{d} = 2\int_{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}}^{{l_{rdfh} }} {\left( {\sigma_{rd} - \sigma_{rdf} } \right)} dx = - \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\sigma_{rdz} } dx \\ & \quad + \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\sigma_{ruxf} } dx + \left( {\gamma_{d} h_{d} + \gamma_{s2} h_{s2} } \right)l_{df} \\ \end{aligned}$$
(3)

For the abutment pressure σrdz of the hard rock stratum under the transient region in Eq. ( 3 ), considering the effect of overburden structure, it is expressed as

$$\int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\sigma_{rdz} } dx \le \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\sigma_{rux} } dx + \left( {\gamma_{d} h_{d} + \gamma_{s2} h_{s2} } \right)l_{df}$$
(4)

For the sake of simplification, let the left and right sides of formula ( 4 ) be equal. Substitute it into formula (3) and sort it out.

$$\Delta Q_{d} = 2\int_{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}}^{{l_{rdfh} }} {\left( {\sigma_{rd} - \sigma_{rdf} } \right)} dx = - \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\left( {\sigma_{rux} - \sigma_{ruxf} } \right)} dx$$
(5)

It can be seen from the formula (5) that the amount of step change of the bearing capacity of the lower hard rock layer before and after the structural transient is negative because the interlayer load σrux is much larger than the support load σruxf of the gangue in the goaf. That is, compared with before the structural transient, the total load of the lower hard rock layer after the transient increases instantaneously, and the bearing pressure of the lower hard rock layer increases, forming a positive step change zone of the bearing pressure, which is consistent with the experimental results shown in 0.1. The reason is that after the instantaneous breaking of the upper hard rock layer, the supporting load σrux of the middle load layer to the upper hard rock layer is transferred to both sides of the goaf, and the bearing capacity of the lower hard rock layer has a step-by-step evolution phenomenon, which leads to the instantaneous loading state.

The occurrence mechanism of coal seam load step change

The bearing capacity of coal before transient

Similarly, it can be seen from Fig. 3a that before the structural transient, the relationship between the total load Qm on the coal body and its abutment pressure is

$$\begin{aligned} Q_{m} & = \sigma_{m0} \left( {2l_{mfh} + l_{df} } \right) - \gamma_{s1} h_{s1} l_{df} \\ & = 2\int_{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}}^{{l_{mfh} }} {\sigma_{m} } dx + \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\sigma_{rdx} } dx \\ \end{aligned}$$
(6)

In the formula, lmfh is the influence range of advanced mining of coal body (the maximum value before and after breaking), m; γs1 is the bulk density of the lower load layer, kN/m3; σm0 is the stress acting on the coal body under the stress state of the original rock, MPa.

The bearing capacity of coal after transient

After the structure is transient, the transient area falls into the goaf, and the total load on the coal body decreases instantaneously, as shown in Fig. 3b. Similar to the solution method of the bearing capacity of the lower hard rock stratum, the relationship between the total load Qmf of the coal body after the structural transient and its supporting pressure σmf is

$$\begin{aligned} Q_{mf} & = \sigma_{m0} \left( {2l_{mfh} + l_{df} } \right) - \left( {\gamma_{s1} h_{s1} + \gamma_{d} h_{d} + \gamma_{s2} h_{s2} } \right)l_{df} \\ & = 2\int_{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}}^{{l_{mfh} }} {\sigma_{mf} } dx + \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\sigma_{ruxf} } dx \\ \end{aligned}$$
(7)

The step change mechanism of coal bearing capacity

It can be obtained from Eq. (7) and Eq. (6) that the amount of step change ΔQd of the total load borne by the coal body before and after the transient is:

$$\begin{aligned} \Delta Q_{d} & = 2\int_{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}}^{{l_{mfh} }} {\left( {\sigma_{m} - \sigma_{mf} } \right)} dx \\ & = \int_{{{{ - l_{df} } \mathord{\left/ {\vphantom {{ - l_{df} } 2}} \right. \kern-0pt} 2}}}^{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}} {\left( {\sigma_{ruxf} - \sigma_{rdx} } \right)} dx{ + }\left( {\gamma_{d} h_{d} + \gamma_{s2} h_{s2} } \right)l_{df} \\ \end{aligned}$$
(8)

Due to the small change of the supporting load of the overlying strata caused by the gangue in the goaf before and after the pressure, it can be approximately considered that the two are equal, that is,

$$\sigma_{rdx} = \sigma_{ruxf}$$
(9)

Substituting Eq. (9) into Eq. (8) and sorting out, we can get:

$$\Delta Q_{d} = 2\int_{{{{l_{df} } \mathord{\left/ {\vphantom {{l_{df} } 2}} \right. \kern-0pt} 2}}}^{{l_{mfh} }} {\left( {\sigma_{m} - \sigma_{mf} } \right)} dx = \left( {\gamma_{d} h_{d} + \gamma_{s2} h_{s2} } \right)l_{df}$$
(10)

It can be seen from Eq. (10) that the amount of step change ΔQd of the total load on the coal body is positive, that is, after the structural transient, the total load on the coal body decreases instantaneously, which is consistent with the experimental results shown in Fig. 1. The reason is that after the lower hard rock layer is broken, the overburden rock in the transient area falls to the goaf, and the load in this area is borne by the floor of the goaf, resulting in a decrease in the total load of the coal body on the working face, which is in the instantaneous unloading state.

It can be seen that the transient spatial structure of overlying strata induced by the fracture of hard rock strata will form the instantaneous unloading and load transfer at the fracture surface, which will lead to the instantaneous loading and unloading of coal and rock mass in the stope, and then cause different degrees of mechanical response of coal and rock in the stope. The above theoretical analysis preliminarily reveals the internal mechanism of the step change phenomenon of abutment pressure of coal and lower hard rock under the transient excitation of structure, and the mechanical mechanism of different step change characteristics, but it has not revealed the transition mechanism of the peak position of abutment pressure. Therefore, based on the theory of elastic foundation28, the step change mechanism and its influencing factors of static mechanical response of coal and rock in different areas of stope space under structural transient excitation are further studied.

In practical engineering, due to the existence of plastic zone, the distribution of bearing pressure has a peak value. In order to simplify the theoretical solution, it is simplified as a piecewise linear load, as shown in the figure. Where k and ks are the elastic foundation coefficients in the elastic zone and the plastic zone, respectively, q0, q1, q2 and qL are the load values at their respective positions. According to the above mechanical analysis, it can be seen that after the structure is transient, instantaneous unloading and load transfer will be formed, and different instantaneous loading and unloading will be formed for coal and rock in different areas of the stope space. For the key strata, instantaneous unloading will be formed in the transient region; for the basic roof, in addition to the instantaneous unloading on its fracture surface, the transferred load will also form instantaneous loading on its upper surface. Therefore, the breaking modes of the basic roof and the key layer belong to similar situations in the mechanical model, which can be deduced by the graphical unified model and distinguished according to the different substitution values of different situations. According to this, the unit width is taken along the tendency of the working face, and the mechanical model of the hard rock stratum (key stratum and basic roof) is shown in Fig. 4 when the equilibrium state before and after the structural transient is taken.

Fig. 4
figure 4

Mechanical model of hard rock under structure transient excitation.

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For the hard rock fracture mechanics model shown in the above diagram, in order to facilitate the solution, it can be divided into two parts as shown in the diagram to solve separately. Among them, M0 and ML are the bending moment at the section, N/m; Q0 and QL are the shear force at this section, N.

For the rock beam part of the transient region shown in Fig. 4, the corresponding differential equation of the deflection curve is:

$$EI\frac{{d^{2} y_{0} (x)}}{{dx^{2} }} = M_{0} + Q_{0} x + \frac{1}{2}Q_{x0} x^{2} + \frac{1}{3}(q_{0} - Q_{x0} )x^{2}$$
(11)

In the formula, Qx0 is the distributed load equation of rock beam in transient region, which can be expressed as:

$$Q_{x0} = \frac{{q_{0} - q_{L} }}{{L_{0} }}x + q_{0}$$
(12)

From Eqs. (11) and (12), the corresponding deflection curve equation z11(x) is:

$$\begin{aligned} y_{0} (x) & = A_{0} x + B_{0} + \frac{1}{{6EIL_{0} }}\left[ {\frac{1}{20}(q_{0} - q_{L} )x^{5} + \frac{1}{4}L_{0} q_{0} {\mkern 1mu} x^{4} + \frac{1}{12}(6L_{0}^{2} q_{0} + 6q_{L} {\mkern 1mu} L_{0}^{2} - 12Q_{L} L_{0} )x^{3} } \right. \\ & \quad \left. { + \frac{1}{8}(4L_{0}^{3} q_{0} + 8q_{L} {\mkern 1mu} L_{0}^{3} - 24Q_{L} {\mkern 1mu} L_{0}^{2} - 24M_{L} L_{0} )x^{2} } \right] \\ \end{aligned}$$
(13)

In the formula, A0 and B0 are unknown constants.

For the rock beam part in front of the working face shown in Fig. 4b, the differential equation of the deflection curve can be obtained from the theory of elastic foundation beam.

$$\left\{ \begin{gathered} \frac{{d^{4} y_{1} \left( x \right)}}{{dx^{4} }}{ + 4}\beta_{1}^{{4}} y_{1} \left( x \right) = \frac{{Q_{x1} }}{EI} \hfill \\ \frac{{d^{4} y_{2} \left( x \right)}}{{dx^{4} }}{ + 4}\beta_{2}^{{4}} y_{2} \left( x \right) = \frac{{Q_{x2} }}{EI} \hfill \\ \frac{{d^{4} y_{3} \left( x \right)}}{{dx^{4} }}{ + 4}\beta_{2}^{{4}} y_{3} \left( x \right) = \frac{{q_{2} }}{EI} \hfill \\ \end{gathered} \right.$$
(14)

In the formula, β1 = [ks/(4EI)]^(1/4), β2 = [k/(4EI)]^(1/4) are constants, 1/m; E is elastic modulus, N/m2; I is the inertia moment, m4. The corresponding deflection curve equation can be expressed as

$$\left\{ \begin{gathered} y_{1} \left( x \right) = e^{{\beta_{1} x}} \left( {A_{1} \cos \beta_{1} x + B_{1} \sin \beta_{1} x} \right) + e^{{ - \beta_{1} x}} \left( {C_{1} \cos \beta_{1} x + D_{1} \sin \beta_{1} x} \right) + \frac{{Q_{x1} }}{{k_{s} }} \hfill \\ y_{2} \left( x \right) = e^{{\beta_{2} x}} \left( {A_{2} \cos \beta_{2} x + B_{2} \sin \beta_{2} x} \right) + e^{{ - \beta_{2} x}} \left( {C_{2} \cos \beta_{2} x + D_{2} \sin \beta_{2} x} \right) + \frac{{Q_{x2} }}{k} \hfill \\ y_{3} \left( x \right) = e^{{\beta_{2} x}} \left( {A_{3} \cos \beta_{2} x + B_{3} \sin \beta_{2} x} \right) + e^{{ - \beta_{2} x}} \left( {C_{3} \cos \beta_{2} x + D_{3} \sin \beta_{2} x} \right) + \frac{{q_{2} }}{k} \hfill \\ \end{gathered} \right.$$
(15)

In the formula, C1, C2, C3, D1, D2 and D3 are unknown constants, and the distributed load equation is:

$$\left\{ \begin{gathered} Q_{x1} = \frac{{q_{1} - q_{0} }}{{L_{s} }}x + q_{0} \hfill \\ Q_{x2} = \frac{{q_{2} - q_{1} }}{{L_{1} }}(x - L_{s} ) + q_{1} \hfill \\ \end{gathered} \right.$$
(16)

At the same time, from the bending moment and shear force conditions at the 0 section, it can be obtained that

$$\left\{ \begin{gathered} M_{0} = \frac{1}{2}L_{0}^{2} (q_{L} - \frac{1}{3}q_{0} ) - M_{L} \hfill \\ Q_{0} = q_{L} L_{0} + \frac{1}{2}q_{0} L_{0} - Q_{L} \hfill \\ \end{gathered} \right.$$
(17)

For the position of x = − L0, when the mechanical model represents the key strata, the position satisfies that the rotation angle and shear force are 0, that is:

$$\left\{ \begin{gathered} \frac{{dy_{0} (x)}}{dx}\left| {_{{x = - L_{0} }} } \right. = 0 \hfill \\ Q_{L} = 0 \hfill \\ \end{gathered} \right.$$
(18)

When the mechanical model represents the Main roof, this position is the free end, and the values of bending moment and shear force are 0, that is:

$$\left\{ \begin{gathered} M_{L} = 0 \\ Q_{L} = 0 \\ \end{gathered} \right.$$
(19)

The boundary continuity conditions for the remaining positions are as follows:

$$\left\{ \begin{gathered} \frac{{d^{2} y_{1} (x)}}{{dx^{2} }}\left| {_{x = 0} } \right. = \frac{{M_{0} }}{EI} \hfill \\ \frac{{d^{3} y_{1} (x)}}{{dx^{3} }}\left| {_{x = 0} } \right. = \frac{{Q_{0} }}{EI} \hfill \\ \end{gathered} \right.$$
(20)
$$\left\{ \begin{gathered} y_{1} (x)\left| {_{{x = L_{s} }} } \right. = y_{2} (x)\left| {_{{x = L_{s} }} } \right. \hfill \\ \frac{{dy_{1} (x)}}{dx}\left| {_{{x = L_{s} }} } \right. = \frac{{dy_{2} (x)}}{dx}\left| {_{{x = L_{s} }} } \right. \hfill \\ \frac{{d^{2} y_{1} (x)}}{{dx^{2} }}\left| {_{{x = L_{s} }} } \right. = \frac{{d^{2} y_{2} (x)}}{{dx^{2} }}\left| {_{{x = L_{s} }} } \right. \hfill \\ \frac{{d^{3} y_{1} (x)}}{{dx^{3} }}\left| {_{{x = L_{s} }} } \right. = \frac{{d^{3} y_{2} (x)}}{{dx^{3} }}\left| {_{{x = L_{s} }} } \right. \hfill \\ \end{gathered} \right.$$
(21)
$$\left\{ \begin{gathered} y_{2} (x)\left| {_{{x = L_{s} + L_{1} }} } \right. = y_{3} (x)\left| {_{{x = L_{s} + L_{1} }} } \right. \hfill \\ \frac{{dy_{2} (x)}}{dx}\left| {_{{x = L_{s} + L_{1} }} } \right. = \frac{{dy_{3} (x)}}{dx}\left| {_{{x = L_{s} + L_{1} }} } \right. \hfill \\ \frac{{d^{2} y_{2} (x)}}{{dx^{2} }}\left| {_{{x = L_{s} + L_{1} }} } \right. = \frac{{d^{2} y_{3} (x)}}{{dx^{2} }}\left| {_{{x = L_{s} + L_{1} }} } \right. \hfill \\ \frac{{d^{3} y_{2} (x)}}{{dx^{3} }}\left| {_{{x = L_{s} + L_{1} }} } \right. = \frac{{d^{3} y_{3} (x)}}{{dx^{3} }}\left| {_{{x = L_{s} + L_{1} }} } \right. \hfill \\ \end{gathered} \right.$$
(22)
$$y_{3} (x)\left| {_{x \to + \infty } } \right. = \frac{{q_{2} }}{k}$$
(23)

The Ai, Bi, Ci and Di coefficients can be calculated and solved by simultaneous boundary conditions. However, due to the limitation of space and the length of the calculation results, only the numerical solution is calculated here to analyze the deflection, bending moment, rotation angle and other results.

From the mechanics of materials, the formula 24 can be used to calculate the bending moment and rotation angle of the beam based on the deflection expression of rock beam, which can further calculate and analyze the distribution characteristics of bending moment and rotation angle before and after the fracture of hard rock layer, where y(x) can represent one of y1(x), y2(x) and y3(x).

$$\left\{ \begin{gathered} M\left( x \right) = EI\frac{{d^{2} y\left( x \right)}}{{dx^{2} }} \hfill \\ \varphi \left( x \right) = \frac{dy\left( x \right)}{{dx}} \hfill \\ \end{gathered} \right.$$
(24)

Example and analysis

The values of the basic parameters in the model are shown in Table 2. By substituting the calculation parameters into Eqs. (15) and (24), the static mechanical response of the hard rock strata before and after the structural transient and its corresponding step change characteristics can be obtained.

Table 2 Related physical parameters in the model.
Full size table

From the perspective of material mechanics, different loading conditions will cause different mechanical responses of the rock beam, and in essence, the principal stress state at each position inside the rock beam has step characteristics.

Figure 5 shows the characteristics of flexural deformation of hard rock before and after structural transient. It can be seen from the figure:

  1. (1)

    There are differences in the deflection curves of hard rock strata before and after structural transients, resulting in certain positive and negative step change characteristics. Compared with before the transient, the key stratum sinks and deforms. The distribution law of the deflection curve before and after the transient is symmetrical along the left and right sides of the goaf, and the maximum subsidence occurs in the center of the goaf. The deflection deformation of the main roof has obvious regional characteristics. The deflection deformation near the goaf side is large and rebounds after the transient. The deflection deformation in the deep area far away from the working face is small but produces small subsidence.

  2. (2)

    Relative to the structure before the transient, the key layer in the goaf after the transient and in front of the working face within 75 m range of subsidence deformation, the maximum subsidence in the middle of the goaf is 0.034 m. Due to the instantaneous unloading of shear force and bending moment at the fracture position after transient, the main roof rebounds within 73 m in front of the working face, and the maximum rebound amount is 0.106 m. The instantaneous loading of the transfer load causes the instantaneous subsidence of the main roof in the deep range, and the maximum subsidence is 0.006 m.

  3. (3)

    Due to the influence of mining, the main roof is broken, and the internal force (shear force and bending moment) at the fracture position and the support reaction force at the lower part of the key stratum are instantaneously released. The stress and displacement boundary conditions of the hard rock strata (main roof and key stratum) in the mining area of the stope are changed, resulting in a certain degree of flexural deformation. Due to the influence of the transient characteristics of the spatial structure of the overlying strata, the deformation of main roof and the hard rock strata produce regional positive and negative step change characteristics.

Fig. 5
figure 5

Characteristics of deflection step change of rock beam in hard rock strata before and after weighting.

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Figure 6 shows the bending moment distribution characteristics of hard rock strata before and after structural transient. It can be seen from the figure that:

  1. (1)

    Due to the regional positive and negative step change characteristics of rock beam deflection before and after structural transient, the bending moment distribution of hard rock strata also has some differences. The bending moment distribution of the key strata is symmetrical along the left and right sides of the goaf, and its peak value is generated in the center of the goaf and the coal and rock mass on both sides. Because the key stratum bends and sinks after the structural transient, the degree of bending deformation increases and the degree of bending becomes more obvious. Before the structural transient, the main roof has two bending moment extremums at the working face position and the depth of the coal wall. After the structural transient, the bending moment is instantaneously released at the breaking position, and the bending degree of the rock beam is reduced.

  2. (2)

    The bending moment distribution of the main roof and the key stratum is more obvious than that before the structural transient, and the peak bending moment is concentrated in the middle of the goaf and the position of 50 m deep on both sides of the coal wall. Due to the increase of the bending degree of the key layer in the middle of the goaf, the peak value increases from 5.54 to 6.42 G N m; the peak bending moment of the key strata on both sides of the goaf increases from 2.41 to 2.89 G N m. After the transient, the bending degree of the main roof decreases, and its peak value decreases from 3.57 to 1.28 G N m.

  3. (3)

    Under the influence of the transient spatial structure of the overburden rock, the hard rock stratum has a certain degree of positive and negative flexural step change characteristics. The bending degree of the key stratum is more obvious when it bends and sinks, so that the bending moment distribution increases as a whole. However, due to the release of internal forces after self rupture, the main roof undergoes flexural deformation and rebound, while the overall bending moment value decreases.

Fig. 6
figure 6

Characteristics of bending moment step change of rock beam in hard rock strata before and after weighting.

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Figure 7 shows the distribution characteristics of the rotation angle of hard rock before and after the structural transient. It can be seen from the diagram:

  1. (1)

    Due to the change of stress and displacement constraints of hard rock strata, it produces regional flexural deformation characteristics, and the distribution characteristics of rotation angle at each position of rock beam are also significantly different. The rotation angle curve of the key stratum is symmetrically distributed in the center. Some rock beams on the left and right sides of the goaf rotate clockwise and counterclockwise respectively. After the transient, the rotation angle increases due to the increase of the bending degree. The basic roof in front of the working face mainly rotates counterclockwise before and after the structural transient, and the extreme value of the rotation angle decreases with the decrease of the bending degree after the transient.

  2. (2)

    After the transient, the bending degree of the key stratum increases, and the extreme value of the rotation angle increases from 0.12° to 0.15°, and the extreme position is located at 13 m deep of the coal wall. The difference is that the deflection angle of the key layer increases as a whole after the transient, while the deflection angle of the basic top decreases as a whole, and the extreme value of the basic top angle decreases from 0.19° to 0.06°.

  3. (3)

    When the hard rock layer produces different deflection deformation before and after the structural transient, the deflection characteristics at each position of the rock beam will change accordingly. Although the bending degree of the key layer in the middle of the goaf increases, its tangential rotation angle always remains horizontal, and the deflection degree at the extreme value position of the bending moment on both sides of the goaf is the largest. Due to the difference between the deflection deformation characteristics of the basic top and the key layer, the bending degree at each position of the key layer increases and the deflection degree increases, while the deflection angle of the basic top decreases due to the decrease of its bending degree.

Fig. 7
figure 7

The hard rock strata rock beam angle step change characteristics before and after weighting.

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It can be seen that under the influence of the transient spatial structure of the overlying rock, the hard rock layer has a certain regional mechanical response of the step change characteristics, which is essentially the change of the stress state at each position inside the rock beam. According to the definition of material mechanics, the bending stress will be generated after the bending deformation of the rock beam, and its direction is perpendicular to the cross section of the rock beam. It is tensioned on one side of the neutral layer and compressed on the other side. The tension and compression state depends on the positive and negative of the bending moment. In this plane problem, the first and third principal stresses are the main basic parameters of the in-plane stress field characteristics, and the intermediate principal stress is perpendicular to the plane direction. When the spatial structure of the overlying rock changes, the stress transfer path of the coal rock in the stope changes. Under the influence of the change of the rotation angle of the rock beam, the deflection state of the principal stress in different regions evolves, forming the step change characteristics of the stress field of the overlying rock. Therefore, the magnitude and direction of the principal stress of the hard rock layer will change after the transient of the spatial structure of the overlying rock, and the step change characteristics of the mechanical response such as the deflection before and after the breaking of the hard rock layer affect the step change law of the internal principal stress state. In the following, the numerical model is established to extract the principal stress distribution curve of hard rock strata and analyze the principal stress step change law of hard rock strata.

Numerical simulations

Model establishment

According to the mine data of Reference27 and the physical and mechanical parameters of rock strata, the 3DEC discrete element plane numerical calculation model shown in Fig. 8 is established. The model is 400 m long, 1 m wide and 200 m high. The z-direction displacement constraint is applied at the bottom of the model, the x-direction displacement constraint is applied at the left and right boundaries of the model, and the y-direction displacement of the front and rear boundaries of the model is limited. The vertical load of 5 MPa is applied at the top of the model to simulate the formation depth of 200 m. The number of blocks in the model is 14,410, and the number of elements is 528,553. The constitutive model adopts the Mohr–Coulomb constitutive model.

Fig. 8
figure 8

Numerical calculation model.

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The 3DEC discrete element numerical calculation software can better simulate the failure and collapse characteristics of overlying strata. However, due to the limitation of the software’s own calculation method, the calculation of the instability and failure of hard rock strata is still a quasi-static progressive evolution process. It is difficult to achieve the short and severe failure process of brittle materials such as hard rock strata. Therefore, this paper attempts to carry out secondary development of the pre-processing module of the model. The flowchart is shown in Fig. 9.

Fig. 9
figure 9

Calculation flow of transient process of spatial structure of overlying strata.

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The key idea is that through the FISH language built in 3DEC, after each advance of the working face, the position of the joint with the largest tensile stress value in the hard rock stratum is searched globally to determine whether the maximum tensile stress of the joint surface of the hard rock stratum reaches the preset limit value of the mechanical strength of the coal rock. When the conditions are met, the relevant parameters of the joint are cleared, and then the balance is calculated to realize the first or periodic weighting simulation of the model. If it does not exceed the preset value, the next excavation calculation will continue until the end of the calculation (Fig. 9).

In order to more intuitively analyze the transient characteristics of static mechanical response of coal rock under structural transient excitation, the post-processing module of the model is redeveloped by FISH language. Combined with the elastic–plastic theory, by extracting and processing the result information of each unit node in the target area, the data with low value and weak correlation can be transformed into data with high value and strong correlation.

The step change characteristics of overlying strata principal stress value

Figures 10 and 11 show the step change characteristics of the principal stress of the overlying rock after the hard rock layer is broken. Among them, the red area of the cloud image is a positive-amount of step change area (increased area), while the blue part is a negative-amount of step change area (reduced area); in each rock survey line, the principal stress value per unit length is extracted and the distribution curve is calculated. The corresponding values before and after the pressure are expressed in blue and red respectively, and the amount of step changes of the corresponding values after the pressure are expressed in black. As can be seen from Fig. 10:

  1. (1)

    After the main roof is broken, the failure mode of overburden rock changes from rectangle to isosceles trapezoid, and the geometric characteristics of fracture surface are symmetrical. Because the hard rock has the mechanical properties of transient breaking, the transient area falls into the goaf after the pressure, and the supporting effect on the key layer is instantaneously contacted. The constraint at the breaking position of the main roof is changed from the shear force and bending moment constraint to the mutual extrusion constraint between the rock blocks. Therefore, under the influence of the symmetrical failure mode of the overburden rock, the transient characteristics of the coal-rock stress field outside the fracture surface are also symmetrical. The positive step change zone of the first principal stress of overlying strata is concentrated in the middle of the key stratum and the area near the coal wall on both sides of the goaf, as well as the deep area where the main roof and coal seam are far away from the goaf. The negative step change area is mainly concentrated in the area near the basic roof and coal seam near the goaf on both sides.

  2. (2)

    The distribution of the first principal stress of the key strata is symmetrically distributed along both sides of the goaf. The first principal stress is opposite on both sides of the neutral layer. After the bending deformation of the key strata in the upper part of the goaf, the first principal stress is in the state of tension in the lower part and compression in the upper part, and the tension area is larger. The first principal stress of the key strata on both sides is mainly in the state of tension in the upper part and compression in the lower part, and the compression area is larger. After the transient structure, the key stratum further bends and sinks, and its bending degree increases, resulting in an increase in the stress on both sides of the neutral layer of the rock beam. The overall first principal stress shows a positive step change distribution. The first principal stress in the middle of the goaf increases by 1.57 MPa, and the concentration area of principal stress step change in front of the working face increases by 0.63 MPa at 37 m. In the area of 20 m in front of the working face, the first principal stress changes from the tensile state before fracture to the compressive state, and the peak stress decreases from 5.99 MPa tensile stress at 5.5 m in front of the working face to the compressive stress at 2.5 m. The first principal stress of the key stratum is in the compression state from 20 m to the deep area, and the first principal stress increases after the structural transient. The first principal stress in the coal seam is a compressive stress state, and its stress peak is reduced from 3.09 MPa at the position of 13 m from the coal wall to 2.78 MPa at the position of 16 m before pressure. Under the influence of the reduction of the bending degree of the main roof, the first principal stress peak decreases and migrates to the deep ahead away from the goaf.

  3. (3)

    When the main roof is broken, it can be approximately regarded as the deformation and failure of the fixed beam at both ends under uniform load. However, due to the instantaneous unloading of the reaction force of the lower support, the transient characteristics of the boundary constraint conditions are symmetrically distributed, so the distribution of the first principal stress of the overburden is approximately symmetrical. The step change characteristics of mechanical response of hard rock strata are deeply related to the distribution law of internal principal stress. After structural transient, the first principal stress of key strata increases with the increase of bending degree, while the first principal stress of basic roof does not decrease with the decrease of bending degree. Under this influence, the principal stress of coal and rock mass near the working face has a release reduction zone, the peak value of the first principal stress in the coal seam decreases, and the peak position shifts towards the front of the coal seam.

Fig. 10
figure 10

Step change characteristics of the first principal stress of overlying strata before and after the weighting.

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Fig. 11
figure 11

Step change characteristics of the third principal stress of overlying strata before and after the weighting.

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According to the definition of elastic mechanics, the value of the third principal stress is the smallest, and the stress characteristics of the overburden are mainly compressive stress, so the third principal stress represents the stress transfer characteristics of the overburden. It can be seen from the step change characteristics of third principal stress cloud diagram in Fig. 11:

  1. (1)

    Due to the symmetrical distribution characteristics of overburden failure modes, the distribution law of the third principal stress step change is also symmetrically distributed. The negative step change area of the third principal stress is mainly concentrated in the overburden above the goaf and the area near the goaf on both sides, while the positive step change area is the deep coal and rock mass area on both sides of the goaf. After the main roof is broken, the broken rock mass rotates and falls to the floor under the action of self-weight, and the other side is squeezed with the coal wall and the unbroken section to form a hinged structure. Therefore, the third principal stress in the area near the coal wall increases under the extrusion effect compared with that before the fracture, and increases-decreases-increases along the advancing direction.

  2. (2)

    The first principal stress of the key strata is symmetrically distributed, and the peak stress is formed on both sides of the goaf, and the peak stress jumps away from the goaf after the transient. Compared with before pressure, the peak value of the third principal stress increases from 13.77 MPa at the position of 9 m deep from the coal wall to 13.80 MP at the position of 19 m under the influence of structural transient. The central position of the goaf is the minimum value of the third principal stress, which is reduced by 0.49 MPa after weighting. The distribution law and step change characteristics of the third principal stress of the main roof and the coal seam are similar, showing a step change trend of increasing–decreasing-increasing along the strike, with a stress peak and a transition to the depth ; compared with before pressure, the peak value of the third principal stress of the basic roof decreased from 19.14 MPa at the position of 5.5 m from the coal wall to 17.96 MPa at the position of 11.5 m, and the peak value of the third principal stress of the coal seam decreased from 23.46 MPa at the position of 6.5 m from the working face to 20.35 MPa at the position of 10.5 m.

  3. (3)

    With the breaking of hard rock strata, the overlying strata above the goaf sinks with the deformation of the key strata, which increases the bending degree and releases the overlying strata stress to concentrate on both sides of the goaf. There are obvious positive and negative step change regions of the third principal stress in the key stratum and the main roof, and the step change characteristics of the coal seam are controlled by the mechanical response of the main roof, which leads to the obvious migration characteristics of the concentrated position after the structural transient. However, the decrease of the bending degree of the main roof after the structural transient leads to the migration of the principal stress concentration area and the decrease of the peak value, while the increase of the bending degree of the key stratum after the transient of the overburden structure makes the migration of the principal stress concentration area and the increase of the peak value.

Step change characteristics of principal stress direction of overburden rock

With the breaking of hard rock strata, due to the transient influence of the principal stress, the basic roof and key strata in different regions have a certain degree of rotation and deformation, and the principal stress deflection state will change at the same time. From the Fig. 12 principal stress deflection characteristic amount of step change vector diagram, it can be seen that:

  1. (1)

    The fracture of hard rock strata has a significant effect on the principal stress deflection state of overlying strata. Before pressure, the principal stress direction of overlying strata deflects under the influence of mining. The principal stress of overlying strata on the right side of goaf deflects counterclockwise, while the left side deflects clockwise. The principal stress deflection angle of the upper part of the two hard rock strata is the largest. After weighting, the broken rock mass of the main roof falls to the goaf, and the gravity of the overburden rock is mainly borne by the key stratum and further deforms and sinks. Under this influence, the deflection degree of the principal stress of the overburden rock in the upper area of the goaf increases significantly on the basis of before pressure.

  2. (2)

    The principal stress deflection characteristics of the overlying strata before and after the main roof breaking are symmetrically distributed, and the principal stress deflection step characteristics of the overlying strata and the bottom plate above the goaf and the near-field coal and rock are more obvious. Compared with before pressure, the deflection range of the principal stress of the overlying strata after weighting becomes larger under the influence of the subsidence of the key strata, and the maximum step change amounts of the deflection is 35.92°, so that the transmission direction of part of the self-weight points to the coal and rock mass on both sides of the stope. The significant position of the principal stress deflection characteristics of the key stratum after the pressure is concentrated in the upper area on both sides of the goaf, and the maximum deflection value is 8.96°. The main roof near the working face rebounds after weighting, which makes the main stress direction of the coal rock near it rotate clockwise, and the maximum rotation is 32.89°.

  3. (3)

    Before the breaking of the hard rock, the main roof and the key stratum are bent and deformed respectively under the action of the self-weight load of the overlying rock, which makes the rock mass in the upper part of the two hard rock layers show a large principal stress deflection characteristic. Combined with the analysis of the principal stress, it can be seen that there is a stress release zone in the overburden, and the basic roof plays a major role before the fracture. After the fracture, the supporting reaction force of the lower part of the key stratum is released instantaneously, so that it can bear more overburden weight and further bend and sink, and transfer the overburden gravity to the coal and rock on both sides. The basic roof rebounds and the bending degree decreases under the influence of the release of the fracture section constraint conditions and the stress transfer characteristics of the overburden. Under the influence of the transient characteristics of the structure, the principal stress deflection characteristics of the key layer are more significant after the pressure, while the principal stress deflection degree of the basic roof is weakened, which is corresponding to the rotation angle step change characteristics of the hard rock layer in Fig. 7. Because the concentration area of the principal stress deflection step change characteristics in the key layer and the basic roof is the same as the rotation angle step change characteristics, which are concentrated near the top of the coal wall on both sides of the goaf. Therefore, under the transient excitation of the structure, the structure that plays a major role in controlling the overburden load is transformed from the stress arch I formed by the basic roof before the pressure to the stress arch II formed by the key layer after the pressure.

Fig. 12
figure 12

Step change characteristics of principal stress deflection of overlying strata before and after the weighting.

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It can be seen from this that when the hard rock layer is broken, the load state and constraint conditions of the basic roof and the key layer change, resulting in a certain flexural mechanical response, and the principal stress size and direction deflection state at different positions of itself will change accordingly. However, because the overburden stress field is mainly controlled by the mechanical state of the two hard rock layers in the model, the regional positive and negative step change characteristics of the overburden stress field are finally made.

Discussion

In this paper, combined with theoretical modeling, similar material simulation experiments and numerical simulation methods, the step change characteristics of stress field under the transient influence of overburden spatial structure are analyzed from two aspects: mechanical response of beam and unit principal stress. Compared with the previous studies that only focused on the position and size of the stress concentration area of coal and rock in the stope before the transient of the structure, this study considers the stress field after the transient, and explains the mechanical relationship and evolution law between the strata within the mining influence range from the perspective of the spatial structure of the overlying strata.

Mine earthquake is a typical dynamic problem of coal rock, which is a dynamic process of evolution to a new space–time structure after the original space–time structure of coal rock in the stope is broken. In the current research, the theoretical analysis and numerical calculation still use the static model. Under the guidance of the concept of dynamics, the mechanical models of the two equilibrium states before and after the structural transient are constructed respectively, and the evolution mechanism of the overburden stress field under the structural transient excitation is compared and studied. However, in the process of hard rock breaking and caving, the stress field of overlying strata has a dynamic evolution process from the equilibrium state before caving to the backward equilibrium state. In order to accurately reveal the complex dynamic response mechanism of the mechanical behavior of coal and rock mass in the evolution process, it is very important to clarify the dynamic evolution process of the spatial structure system of overlying strata. In addition, although the research process can qualitatively reveal the dynamic mechanical response process of overburden induced by hard rock breaking, the actual mine earthquake process is not only the release of stress or energy from a single point source. There are many factors that affect and excite each other, but it is essentially a dynamic process caused by hard rock breaking. Therefore, it is necessary to establish targeted prevention and control measures based on the concept of "adjusting structure and controlling response" by analyzing the internal relationship between structural transients and coal-rock dynamic and static responses, and combining with the refined research of coal-rock dynamics at the actual engineering scale.

Conclusion

In this study, the evolution law of overburden stress field is analyzed by combining similar material simula-tion experiments and sensor-based coal and rock stress monitoring results, and the theoretical modeling analysis is further used to reveal the flexural mechanical response under the influence of hard rock fracture. Finally, through numerical calculation and secondary development of post-processing, the step change law of overburden stress field under structural transient excitation is studied, and the influence mechanism of mechanical response on the step change of overburden stress field during hard rock fracture is analyzed and verified. The main conclusions are as follows:

  1. (1)

    The step change law of overburden stress field is closely related to the mechanical response characteristics of hard rock strata (main roof and key strata). When the main roof breaks, the mechanical constraints at the fracture surface change instantaneously, so that the hard rock strata have different flexural mechanical responses before and after weighting. The hard rock layer has a controlling effect on the stress state of coal and rock in the influence range. The combined deformation law of the main roof and the key stratum under the transient excitation of the structure causes the stress state of the coal and rock in the stope to have certain positive and negative step change.

  2. (2)

    Under the influence of hard rock breaking, the spatial structure of overlying strata is transient, and the step change characteristics of key strata under this influence are symmetrical. The increase of bending degree after transient makes the deflection, bending moment and rotation angle produce positive step change. Due to the decrease of the bending degree of the basic roof after the transient, the deflection, bending moment and rotation angle of the basic roof have negative step changes.

  3. (3)

    After weighting, the mechanical response of the hard rock changes the principal stress state inside the rock beam, and further controls the step change law of the coal rock stress in the stope. The symmetrical deformation mechanical response of the key strata makes the step change characteristics of the overburden stress field symmetrical, and the overburden stress above the goaf is released and transferred to both sides. The flexural deformation response of the main roof leads to the regional positive and negative step change characteristics of the stress state of the coal and rock mass within its influence range, which reduces the degree of stress concentration and migrates the peak position.

  4. (4)

    Under the structural transient excitation, the positive step change of the bending deformation of the key strata and the negative step change of the bending deformation of the basic roof cause the step change response of the stress state of the coal and rock in the stope. During the dynamic evolution process of the pressure, the stress arch that plays a major role in the overburden load is instantaneously transformed from the basic roof to the mechanical structure formed by the key layer. In this process, the mechanical response process of different areas of coal rock in the stope is an important factor for reverse analysis of the possible dangerous state in its evolution process.