1 Introduction

After the underground coal seam is mined, a large-scale underground goaf will be formed. The stability of goaf is the key to ensure the safety of stope and surface (Liu et al. 2023; Pan et al. 2020). For ensuring the stability of underground goaf, a large number of coal pillars are set in and around the stope to support the overlying load (Han et al. 2022; Song et al. 2022). However, the residual coal pillars will be destroyed under plenty of factors over time (Zhu et al. 2019). To date, many researches have reflected that coal pillar is a part of coal–rock system (He et al. 2023). Further, the instability of coal pillar is caused by the combined action of coal and rock (Prassetyo et al. 2019). Therefore, it is of great significance to study the overall failure mode and mechanism of coal–rock system. Through numerical simulation and theoretical analysis, Zhou et al. (2021) showed that coal pillar is subjected to non-uniform load due to non-uniform deformation of roof, and the failure of coal pillar at different positions present different characteristics. Qin et al. (2006) studied the failure mechanism of coal pillar–roof system through catastrophe theory and concluded that the instability of coal pillar mainly depends on the stiffness ratio of the whole system. Yin et al. (2021) compared the failure of coal–roof combined bodies with different rocks, it was found that the mechanical characteristics of combined bodies mainly depend on coal chiefly. Yang et al. (2015) theoretically analyzed the influence of roof structure on the bearing capacity and internal stress distribution in coal pillar. These results play an important guiding role for ensuring the stability of underground goaf.

However, as presented in Fig. 1, the residual coal pillars are set in the form of groups in pillar goaf. In this instance, the single coal pillar and its roof form single coal pillar–roof system, while coal pillars and roof form coal pillars–roof system. In the coal pillars–roof system, the interrelation and integration not only exist in among coal pillars, but also between coal pillars and roof (Li et al. 2020; Zhou et al. 2018). After a coal pillar becomes unstable, the overlying load it bore originally will transfer to the adjacent coal pillars, which may lead to the instability of adjacent coal pillars due to excessive load as well. By analogy, it is possible to trigger large-scale instability of coal pillars (Zhang et al. 2017). Thenceforth, the roof will break after reaching its ultimate bearing strength due to the loss of effective support from large-scale coal pillars. Meanwhile, the broken roof reacts to coal pillars and exacerbates their instability. Eventually, there will be a large area of instability in pillar goaf based upon the domino effect of coal pillars–roof system. However, the existing researches only consider the coal pillars as a whole to study its instability (Tian et al. 2023). Liu et al. (2021) demonstrated the gradual instability process of coal pillars and put forward that coal pillar reinforcement technologies can prevent the overall instability of goaf. Song et al. (2023) studied the failure characteristics of multiple coal pillars with different strengths, it was found that different coal pillars show imbalanced deformation characteristics. Feng et al. (2021) defined the “key pillars” in goaf and took it as the starting point to evaluate the instability of coal pillars.

Fig. 1
figure 1

Engineering model of coal pillars–roof structure in pillar goaf

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In summary, there are few researches on instability of coal pillars–roof system that consider not only the interaction among coal pillars but also between coal pillars and roof based on domino effect. Therefore, the coal pillars–roof system is studied as a collaborative whole in this paper. The engineering model of coal pillars–roof system is difficult to be studied directly due to many factors (Bao et al. 2021; Zhukova et al. 2020). Thus, drawing on existing research methods, the coal–rock system in engineering was simplified as the coal–rock combined body (Li et al. 2022; Gao et al. 2020). The single coal pillar–roof combined body was designed firstly to characterize single coal pillar–roof system. Based on it, the double coal pillar–roof combined bodies with different combinations were also designed to characterize the coal pillars–roof system. The uniaxial compression tests were carried out by PFC. To point out the source inducement and load transfer characteristics of coal pillars -roof system, the damage process of single body and double bodies were recorded from the mesoscopic perspective. On this foundation, the instability criterions of double coal pillar–roof combined bodies with different combinations were established. The research results of this paper can provide theoretical and technical support for the prevention and control of domino—type disaster in pillar goaf.

2 Numerical model and procedure

2.1 Model setup

As a discontinuous medium programming software based on the discrete element method, PFC has significant advantages in simulating the mechanical behavior of coal and rock (Roos et al. 2019). In PFC program, the coal and rock models are composed of abundant tiny particles. By assigning geometric parameters and mechanical parameters, models with identical mechanical properties or different mechanical properties can be established (Yin et al. 2019; Chen et al. 2020). The damage of coal and rock are realized by particle movement and interaction between particles, which can reflect the mechanical behavior of coal and rock more comprehensively from the mesoscopic perspective. Therefore, this study used PFC to simulate uniaxial compression of coal and rock. As coal and rock are aggregates composed of mineral particles, the contact between particles mostly adopts parallel bonding model in the PFC model. The parallel bonding model can transmit force and moment at the same time, which is more realistic (Poulsen et al. 2018).

Under the same roof in pillar goaf, the mechanical properties of coal pillars may be the same or similar, or there may be obvious differences. Therefore, to construct combined bodies with different combinations, two types of coal with different mechanical properties were selected, and the same sandstone was selected for the rock. As presented in Fig. 2a, b, single coal pillar–roof body A (SCRA) and single coal pillar–roof body B (SCRB) were designed. Factors such as height and interface were not considered (Liu et al. 2018; Yu et al. 2022; Sun et al. 2021). The overall sizes of models are 50 mm × 100 mm, the height ratio of coal and rock is 1:1, the lower part is coal and the upper is rock. Furthermore, two series of double coal pillar–roof bodies composed of two single bodies with the same mechanical properties were designed, which are named double coal pillar–roof bodies A (DCRAA) and double coal pillar-roof bodies B (DCRBB) respectively, as presented in Fig. 2c, d. Meanwhile, double bodies composed of two single bodies with different mechanical properties were designed as well, which is named double coal pillar–roof bodies AB (DCRAB), as presented in Fig. 2e. To avoid the influence of geometric parameters on the results, the geometric parameters in all models are kept consistent. When modeling, a single coal pillar–roof model was established and preserved after preloading firstly. In subsequent simulations, the corresponding mechanical parameters were given after calling this model. For the modeling of double coal pillar–roof bodies, the saved single coal pillar–roof model was first called to generate one side of the double bodies, and then it was copied and moved by FISH language to get the other side of double bodies. Moreover, based on the existing experimental design methods of coal pillars in coal mines and pillars in non-coal mines (Bai 2019; Zhou et al. 2020), the influence of distances among coal pillars were also not considered during the establishment of the double bodies, but the distance between the two component bodies remains consistent all the time.

Fig. 2
figure 2

Numerical model

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2.2 Mesoscopic parameters confirmation

To make the PFC model consistent with the macroscopic mechanical properties of actual coal and rock, the "trial-and-error method" was applied to calibrate the mesoscopic parameters in combination with laboratory tests, which was carried out before uniaxial compression simulation of combined bodies. When the stress–strain curve and failure mode of the PFC simulation are in good agreement with that of the laboratory test, it is considered that the microscopic parameters meet the simulation requirements (Sarfarazi et al. 2023; Ozturk et al. 2017). The PFC model of each coal or rock was set as a two-dimensional rectangle of 50 mm × 100 mm, and the corresponding laboratory prototype is a standard specimen of φ 50 × 100 mm. Displacement loading method was applied in both laboratory test and numerical simulation, the loading rates were both 0.05 mm/s.

The comparisons of laboratory tests and numerical simulations for each specimen are presented in Fig. 3. The experimental peak strengths of coal A, coal B and sandstone are 19.21 MPa, 10.02 MPa and 39.35 MPa respectively, while the numerical peak strengths are 19.44 MPa, 10.14 MPa and 39.89 MPa respectively. The experimental peak strength strains of coal A, coal B and sandstone are 0.74%, 0.87% and 1.17% respectively, while the numerical peak strength strains are 0.71%, 0.84% and 1.14% respectively. There is a difference between the numerical simulations and laboratory tests, especially the difference in strains. That is because the numerical models are composed of uniformly distributed rigid particles with the same mechanical parameters, while the mechanical parameters of the mineral particles that make up the laboratory bodies are discrete. Additionally, there are many randomly distributed original pores and cracks in the laboratory bodies (Chen et al. 2023). Therefore, the experimental and numerical stress–strain curves cannot completely coincide, especially the numerical stress–strain curve does not have the stage of original pore compaction, but the experimental stress–strain curve does, which leads to the numerical strain is greater than the experimental one. However, for all that, the stress–strain curves and the final failure models of laboratory tests and numerical simulations are close to each other relatively. To sum up, the calibrated mesoscopic parameters can be applied to simulate the mechanical behavior of experimental coal and rock. Some specific mesoscopic parameters are presented in Table 1.

Fig. 3
figure 3

Comparisons between numerical simulations and laboratory tests

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Table 1 Some basic mesoscopic parameters
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2.3 Simulation procedure

The uniaxial compression simulation of each single combined body and double combined bodies were carried out in sequence. Uniaxial compression was realized by displacement loading, that is, vertical downward displacement was applied to the upper wall. Meanwhile, vertical upward displacement was applied to the lower wall. The loading rate was fixed at 0.05 mm/s all the time. The uniaxial compression was stopped when the model was destroyed (Zhao et al. 2021). During the compression of single combined body, the load was obtained by monitoring the counterforce of boundary walls. For the double coal pillar–roof combined bodies, the stress of each component body was monitored separately during the compression, and the load was obtained by multiplying the stress by the cross-sectional area. Similarly, the total load borne by the double combined bodies was also obtained by monitoring the counterforce of the boundary walls. Meanwhile, FISH language was adopted to monitor the average strain of each body in real time.

3 Results analysis

3.1 Single combined body

The bearing and deformation laws of SCRA and SCRB are presented in Figs. 4 and 5 respectively. Different characteristic points were selected on the load–step curves to analyze the failure process, as presented in Table 2 and Table 3 respectively. Explanatorily, the black symbols in the bodies indicate tension cracks and the red symbols indicate shear cracks. Although the loads of the two types of bodies are different, the failure modes are the same. The bodies both go through the process of “cracks initiation-cracks propagation-cracks penetration”, which finally leads to the macroscopic failure. According to the laws of cracks development in different locations and types, the failure can be subdivided into five stages.

  • Stage I (a–a1, b–b1): Stage of coal tension cracks initiation. At the initial stage of axial compression, external load cannot break the bonds among model particles yet. Subsequently, the increase of interparticle force induces tension stress concentration at coal with loading. The bonds among the particles at the stress concentration exhibit tension fractures, leading to the initiation of tension cracks at the coal.

  • Stage II (a1–a2, b1–b2): Stage of coal shear cracks initiation. The tension cracks propagate slowly and penetrate partially. At the same time, shear cracks initiate due to the shear fractures of bonds among particles at coal.

  • Stage III (a2–a3, b2–b3): Stage of slow coal cracks propagation and penetration. The number and penetration of coal cracks continued to increase before the peak load, but the increase rate is still slow.

  • Stage IV (a3–a4, b3–b4): Stage of rock tension cracks initiation. After the peak load, the number and penetration of coal cracks increase significantly at a high rate, and the rock tension cracks initiate near the interface of coal and rock as well. Meanwhile, the bearing capacity of whole body decreases gradually.

  • Stage V (a4–a5, b4–b5): Stage of rapid cracks penetration. Although rock cracks continue to increase and propagate, the overall number is small relatively, while the speed and degree of coal failure increase significantly, especially the shear cracks. Microcracks propagate and penetrate to form a “V” shaped macroscopic crack at coal, leading to the overall instability of combined body ultimately.

Fig. 4
figure 4

Bearing capacity and deformation laws of SCRA

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Fig. 5
figure 5

Bearing capacity and deformation laws of SCRB

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Table 2 Failure process of SCRA
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Table 3 Failure process of SCRB
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As presented in Figs. 4b and 5b, the strains of coal and rock present different evolutions. Meanwhile, the strain of coal is far greater than that of rock. The strain of coal increases continuously, and the rate increases obviously after the peak load, while the strain evolution of rock is consistent with the load. That is, the strain of rock increases linearly before the peak load, but decreases parabolically at a relatively small rate after the peak load.

Generally speaking, coal dominates the destruction of single combined body. The coal is broken firstly and induces rock damage and the rock damage aggravates the destruction of coal in turn. Finally, the overall body loses bearing capacity based upon domino effect. It is inferred that there is domino-type instability in the “small system” composed of single coal pillar and its roof. If a single coal pillar is destroyed, it will lead to the deformation of the roof easily. Afterwards, the roof deformation will aggravate the destruction of coal pillar, which lead to the overall instability of single coal pillar–roof system.

3.2 Double combined bodies with same properties

The bearing and deformation laws of DCRAA and DCRBB are presented in Figs. 6 and 7. Similarly, different characteristic points were selected on the load–step curves to analyze the failure characteristics. The failure processes of double bodies are presented in Tables 4 and 5. Explanatorily, taking DCRAA as an example, “LCRA” represents the left component body and “RCRA” represents the right one; “LCA” represents the left coal and “RCA” represents the right one; “LR” represents the left rock and “RCA” represents the right one. The same representation is applied in subsequent simulation.

Fig. 6
figure 6

Bearing capacity and deformation laws of DCRAA

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Fig. 7
figure 7

Bearing capacity and deformation laws of DCRBB

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Table 4 Failure process of DCRAA
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Table 5 Failure process of DCRBB
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Although the loads are different, the bearing and failure characteristics of the two series are similar. In each series, the load–step curves of two component bodies are highly coincident with each other. Concomitantly, two component bodies divide the external load uniformly during uniaxial compression. Meanwhile, the peak loads of two component bodies are equal basically. Synchronously, not only the strain laws of component bodies are consistent, but also the cracks initiation, propagation and penetration occur simultaneously with the highly consistent number and mode. In addition, the evolution law of the total load–step curve of the whole system is consistent with that of any component body, but the total load is much greater than either of them. With the synchronous failure of the two component bodies, the whole system loses its bearing capacity immediately as well.

To sum up, the bearing and deformation characteristics of double bodies with same mechanical properties are similar to those of single combined bodies mentioned above. According to the development of different positions and types about cracks, the failure of the whole double bodies system can be also subdivided into the five stages mentioned above. Likewise, the damage of component bodies still occurs in coal mainly, while only a few cracks in rock appear near the interface. However, different from the single body mentioned above, the double bodies with the same mechanical properties not only have domino—type damage in the “small system” for each component body, but also has domino—type damage in the whole “big system” composed of component bodies. Specifically, the mechanical properties of the left and right bodies are the same, if any one of them is destroyed, the other body will destroy synchronously due to the domino effect, which will lead to the rapid instability of the whole double bodies system. It can be inferred that under the same roof condition in pillar goaf, for coal pillars with the same or similar mechanical properties, all the coal pillars are distributed evenly and deformed synchronously. If a coal pillar is damaged and unstable, it will cause large-scale coal pillars instantly. After that, wide roof may collapse synchronously without support, resulting in goaf disaster finally.

3.3 Double combined bodies with different properties

Uniaxial compression simulation was carried out on the double coal pillar–roof bodies composed of single body A and single body B with different mechanical properties. Figure 8 presents the bearing capacity and deformation laws of DCRAB, and Table 6 presents the failure process. The load–step curve of any component body in DCRAB shows the same evolution as that of the single body and the double bodies with same properties mentioned above, but the total load–step curve shows obvious differences with them. According to the law of total load–step curve, the instability of DCRAB can be divided into four stages.

  • Stage I (ab–ab1): Stage of load linear increase. At the initial stage of loading, the total load and the load of each component body increase linearly. Meanwhile, the strain of each coal and rock increases linearly as well. At the same time, combined with Table 6, it can be seen that RCB first initiates tension and shear microcracks in turn. With loading, these microcracks propagate, penetrate and form macrocracks eventually, while only a few tension microcracks distributed randomly and appear at RCA evenly. As the load of RCRB reaches its peak, the total load also reaches the peak synchronously.

  • Stage II (ab1–ab2): Stage of gradually weakening load. After the peak load of RCRB, its bearing capacity begins to decrease, while the bearing capacity of LCRA is still in the stage of linear increase. The decrease in bearing capacity of RCRB is greater than the increase of LRCA, so the total load decreases slowly. The strain rate of RCB increases with its failure, and the RR strain rebounds after reaching its peak. Additionally, tension microcracks appear at RR near the interface. On the contrary, the strains of LCA and LR still increase linearly, and the tension microcracks at LCA are still distributed randomly and evenly with few numbers.

  • Stage III (ab2–ab3): Stage of secondary load increase. The load of LCRA still increases and reaches the peak gradually, while RCRB gradually loses its bearing capacity completely. During this period, the load RCRB borne originally transfers to LCRA slowly, which leads to LCRA gradually becoming the main bearing body. Because the increase in bearing capacity of LRCA is greater than the decrease of RCRB, although the total load evolution oscillates slightly, it shows an overall increasing trend. Concomitantly, the second total load peak appears when LCRA reaches its peak load. Additionally, the strains of LCA and LR still increase linearly. The tension microcracks in LCA continue to develop and macrocracks begin to form locally.

  • Stage IV (ab3–ab4): Stage of rapid load decline. After the peak load of LCRA, the rapid increase of cracks leads to the destruction of LCA. Meanwhile, tension microcracks also appear at LR. Subsequently, the failure of LCRA leads to the rapid decline of the total load. Finally, the whole system of DCRAB loses bearing capacity synchronously with the instability of LCRA.

Fig. 8
figure 8

Bearing capacity and deformation laws of DCRAB

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Table 6 Failure process of DCRAB
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From the above analysis, it can be seen that double bodies with different mechanical properties also have domino—type instability in the whole "big system" during uniaxial compression. Specifically, after the component body with low strength becomes instability firstly, the external load originally borne by it transfers to another component body with high strength. The component body with high strength is destroyed due to the increase of external load, which leads to instability of the whole system eventually. Therefore, it is considered that if the mechanical properties of coal pillars are different under the same roof in pillar goaf, the single coal pillar with the weakest bearing capacity loses stability firstly, which leads to load transfer and domino—type failure of other coal pillars. After losing the effective support of the coal pillars in a large area, the roof will collapse in a wide range.

Overall, whether it is for double combined bodies with same properties or different properties, the overall failure of the whole system is caused by the instability of one of the component bodies. Meanwhile, the instability of the component body is caused and dominated by the destruction of coal. Therefore, the instability of a single coal pillar is the initial cause of the instability of the whole coal pillars–roof system. That is, the instability of a single coal pillar is equivalent to triggering the first card in the “Domino” of coal pillars–roof system, which leads to chain disasters in the “big system” composed of coal pillars and roof easily.

4 Establishment of instability criterion

4.1 Single combined body bearing capacity

As presented in Fig. 9, the uniaxial loading model of the single combined body is established without considering the influence of the interface between coal and rock (Huang et al. 2013; Liu et al. 2015). The coal and rock of the combined body both meet the elastic-strain softening constitutive equation, as presented in Fig. 10 (Yang 2019).

Fig. 9
figure 9

Schematic diagram of the single body under uniaxial loading

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Fig. 10
figure 10

Elastic-strain softening constitutive relationship

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During the early stage of compression, both the coal and rock of the combined body are in the elastic stage, which is the section oa in Fig. 10. During this stage, both the coal and rock meet the following requirement:

$$\sigma = E\varepsilon$$
(1)

where σ represents the axial stress; E represents the elastic modulus of coal or rock; ε represents the strain of coal or rock.

As the force is transmitted equally in the combined body, it can be seen that:

$$E_{c} \varepsilon_{c} S_{c} = E_{r} \varepsilon_{r} S_{r}$$
(2)

where Ec represents the elastic modulus of coal; εc represents the strain of coal; Sc represents the cross-sectional area of coal; Er represents the elastic modulus of rock; εr represents the strain of rock; Sr represents the cross-sectional area of rock.

The total axial deformation of the combined body is:

$$\varepsilon_{{\text{c}}} h_{c} + \varepsilon_{r} h_{r} = u$$
(3)

where hc represents the height of coal; hr represents the height of rock; u represents the total axial deformation of the single combined body.

By combining Eqs. (2) and (3), it can be concluded that:

$$\varepsilon_{{\text{c}}} = \frac{{E_{r} S_{r} }}{{E_{r} S_{r} h_{c} + E_{c} S_{c} h_{r} }}u$$
(4)

The overall bearing capacity of the combined body in the elastic stage can be expressed as:

$$F_{1} = \frac{{E_{c} E_{r} S_{r} S_{c} }}{{E_{r} S_{r} h_{c} + E_{c} S_{c} h_{r} }}u$$
(5)

With the compression, the coal in the combined body reaches the peak bearing capacity firstly. At the same time, the whole body reaches the peak bearing capacity as well. And then the coal enters the strain softening stage, which is the section ab in Fig. 10, while the rock is still in the elastic stage.

After coal enters the strain softening stage, it meets the following requirement:

$$\sigma^{\prime } = \sigma_{a} - E_{c}^{\prime } \left( {\varepsilon_{c} - \varepsilon_{ca} } \right)$$
(6)

where σ′ represents the axial stress of coal in the descending section of ab; σa represents the peak strength of coal; Ec′ represents the slope of coal in the descending section of ab; εca represents the peak strength strain of coal.

By equally transmitting force within the combined body, it can be concluded that:

$$\left[ {\sigma_{a} - E_{c}^{\prime } \left( {\varepsilon_{c} - \varepsilon_{ca} } \right)} \right]S_{c} = E_{r} \varepsilon_{r} S_{r}$$
(7)

By combining Eqs. (3) and (7) and substituting εca = σa/Ec into them, it can be concluded that:

$$\varepsilon_{r} = \frac{1}{{E_{c} }}\frac{{E_{c} S_{c} \sigma_{a} h_{c} + E_{c}^{\prime } S_{c} \sigma_{a} h_{c} - E_{c} S_{c} E_{c}^{\prime } u}}{{E_{r} S_{r} h_{c} - E_{c}^{\prime } S_{c} h_{r} }}$$
(8)

The overall bearing capacity of the combined body can be represented as:

$$F_{2} = \frac{{E_{r} S_{r} }}{{E_{c} }}\left( {\frac{{E_{c} S_{c} \sigma_{a} h_{c} + E_{c}^{\prime } S_{c} \sigma_{a} h_{c} }}{{E_{r} S_{r} h_{c} - E_{c}^{\prime } S_{c} h_{r} }} - \frac{{E_{c} S_{c} E_{c}^{\prime } }}{{E_{r} S_{r} h_{c} - E_{c}^{\prime } S_{c} h_{r} }}u} \right)$$
(9)

In summary, the expression for the bearing capacity of the single combined body is:

$$F = \left\{ {\begin{array}{*{20}l} {\frac{{E_{c} E_{r} S_{r} S_{c} }}{{E_{r} S_{r} h_{c} + E_{c} S_{c} h_{r} }}u,} \hfill & {0 < u_{c} \le u_{ca} } \hfill \\ {\frac{{E_{r} S_{r} }}{{E_{c} }}\left( {\frac{{E_{c} S_{c} \sigma_{a} h_{c} + E_{c}^{\prime } S_{c} \sigma_{a} h_{c} }}{{E_{r} S_{r} h_{c} - E_{c}^{\prime } S_{c} h_{r} }} - \frac{{E_{c} S_{c} E_{c}^{\prime } }}{{E_{r} S_{r} h_{c} - E_{c}^{\prime } S_{c} h_{r} }}u} \right),} \hfill & {u_{ca} < u_{c}^{\prime } < \infty } \hfill \\ \end{array} } \right.$$
(10)

where uc represents the displacement of coal in section oa; uca represents the displacement of coal at the peak bearing capacity; uc′ represents the displacement of coal in section ab.

4.2 Instability criterion of double bodies with same mechanical properties

Combined with the numerical simulation results, the load–displacement curves of double bodies with the same mechanical properties are presented in Fig. 11, which is consistent with the load–displacement curve of single body under load. The whole system is unstable synchronously after any component body loses its stability, so the instability criterion of any component body can be used as the instability criterion of the whole system. This paper takes the left component body as an example to analyze.

Fig. 11
figure 11

Load–displacement curves of double combined bodies with same mechanical properties

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The overall bearing capacity of the double bodies with same mechanical properties can be expressed as:

$$F_{s} = \left\{ {\begin{array}{*{20}l} {2\frac{{E_{cl} E_{rl} S_{rl} S_{cl} }}{{E_{rl} S_{rl} h_{cl} + E_{cl} S_{cl} h_{rl} }}u_{l} ,} \hfill & {0 < u_{cl} \le u_{cal} } \hfill \\ {2\frac{{E_{rl} S_{rl} }}{{E_{cl} }}\left( {\frac{{E_{cl} S_{cl} \sigma_{al} h_{cl} + E_{cl}^{\prime } S_{cl} \sigma_{al} h_{cl} }}{{E_{r} S_{r} h_{c} - E_{c}^{\prime } S_{c} h_{r} }} - \frac{{E_{cl} S_{cl} E_{cl}^{\prime } }}{{E_{rl} S_{rl} h_{cl} - E_{cl}^{\prime } S_{cl} h_{rl} }}u_{l} } \right),} \hfill & { u_{cal} < u_{cl}^{\prime } < \infty } \hfill \\ \end{array} } \right.$$
(11)

where Fs represents the total load of double bodies with same mechanical properties; the other parameters are all marked with a subscript l added to the corresponding parameters mentioned above, these parameters represent the same meaning, but the subscript l represents the left component body.

The section oa in Fig. 11 corresponds to the first expression in Eq. (11). During this stage, the component bodies and the whole system are all in the elastic stage. Therefore, the whole system is stable. Section ab corresponds to the second expression in Eq. (11). During this stage, the bearing capacities of the component bodies and the whole system all begin to decline.

For the second expression in Eq. (11), if:

$$E_{rl} S_{rl} h_{cl} - E_{cl}^{\prime } S_{cl} h_{rl} > 0$$
(12)

Then the coefficient of displacement ul is negative, indicating that the body still moves downward (when the displacement is vertically downward, it is in the negative direction). Although the bearing capacity of the body gradually decreases, the energy generated by external load can still be accumulated in the body, and the system still has bearing capacity at this time.

If:

$$E_{rl} S_{rl} h_{cl} - E_{cl}^{\prime } S_{cl} h_{rl} < 0$$
(13)

Then the coefficient of displacement ul is positive, it means that the body moves upward. In this way, the body is broken seriously, which leads to the energy accumulated in it release and act externally. At this time, the body is unstable and no longer has the bearing capacity.

If:

$$E_{rl} S_{rl} h_{cl} - E_{cl}^{\prime } S_{cl} h_{rl} = 0$$
(14)

Then the coefficient of displacement ul is negative infinity, the bearing capacity of the body is infinitely close to zero, which indicates that the body is in a critical instability state at this time and will be unstable once it is disturbed slightly.

To sum up, the stable state of the double bodies with same mechanical properties can be determined by Eqs. (12–14), which are the criterion for determining whether the double bodies with same mechanical properties are unstable. At the same time, it is specially explained that these three Equations are applicable to any left or right component body, the applicable conditions and results are consistent.

4.3 Instability criterion of double bodies with different mechanical properties

The load–displacement curves of the double bodies with different mechanical properties are presented in Fig. 12. The load–displacement curves of the left and right bodies are consistent with that of the single body, but the total load–displacement curve is different.

Fig. 12
figure 12

Load–displacement curves of double combined bodies with different mechanical properties

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The overall bearing capacity of double bodies with different mechanical properties can be expressed by the following equation:

$$F_{d} = \left\{ {\begin{array}{*{20}l} {\frac{{E_{cl} E_{rl} S_{rl} S_{cl} }}{{E_{rl} S_{rl} h_{cl} + E_{cl} S_{cl} h_{rl} }}u_{l} + \frac{{E_{cr} E_{rr} S_{rr} S_{cr} }}{{E_{rr} S_{rr} h_{cr} + E_{cr} S_{cr} h_{rr} }}u_{r} ,} \hfill & {0 < u_{cr} \le u_{car} } \hfill \\ {\frac{{E_{cl} E_{rl} S_{rl} S_{cl} }}{{E_{rl} S_{rl} h_{cl} + E_{cl} S_{cl} h_{rl} }}u_{l} + \frac{{E_{rr} S_{rr} }}{{E_{cr} }}\left( {\frac{{E_{cr} S_{cr} \sigma_{ar} h_{cr} + E_{cr}^{\prime } S_{cr} \sigma_{ar} h_{cr} }}{{E_{rr} S_{rr} h_{cr} - E_{cr}^{\prime } S_{cr} h_{rr} }} - \frac{{E_{cr} S_{cr} E_{cr}^{\prime } }}{{E_{rr} S_{rr} h_{cr} - E_{cr}^{\prime } S_{cr} h_{rr} }}u_{r} } \right),} \hfill & {u_{car} < u_{cl} \le u_{cal} } \hfill \\ {\frac{{E_{rl} S_{rl} }}{{E_{cl} }}\left( {\frac{{E_{cl} S_{cl} \sigma_{al} h_{cl} + E_{cl}^{\prime } S_{cl} \sigma_{al} h_{cl} }}{{E_{rl} S_{rl} h_{cl} - E_{cl}^{\prime } S_{cl} h_{rl} }} - \frac{{E_{cl} S_{cl} E_{cl}^{\prime } }}{{E_{rl} S_{rl} h_{cl} - E_{cl}^{\prime } S_{cl} h_{rl} }}u_{l} } \right) + \frac{{E_{rr} S_{rr} }}{{E_{cr} }}\left( {\frac{{E_{cr} S_{cr} \sigma_{ar} h_{cr} + E_{cr}^{\prime } S_{cr} \sigma_{ar} h_{cr} }}{{E_{rr} S_{rr} h_{cr} - E_{cr}^{\prime } S_{cr} h_{rr} }} - \frac{{E_{cr} S_{cr} E_{cr}^{\prime } }}{{E_{rr} S_{rr} h_{cr} - E_{cr}^{\prime } S_{cr} h_{rr} }}u_{r} } \right),} \hfill & { u_{cal} < u_{cl}^{\prime } < \infty } \hfill \\ \end{array} } \right.$$
(15)

where Fd is the total load of double bodies with different mechanical properties; the subscript l and r represent the left and right component body respectively.

The section oa in Fig. 12 corresponds to the first expression in Eq. (15). At this stage, both the component bodies and whole system are in the elastic stage, the whole system is stable. The section ac corresponds to the second expression in Eq. (15). At this stage, the bearing capacity of the right body with low strength gradually decreases, while the bearing capacity of the left body with high strength still increases linearly and becomes the main bearing body gradually. The whole system is still stable due to the bearing capacity of the left body. The section cd corresponds to the third expression in Eq. (15). During this stage, the bearing capacity of the left body decreases gradually, and the whole system is stable before the left body becomes unstable. It can be seen that the high-strength component body determines whether the whole system is stable or not. By analyzing the section of the high-strength body of the third expression in Eq. (15) in the same way as that of double bodies with the same mechanical properties, the stable state of double bodies with different mechanical properties can be obtained. That is to say, applying Eqs. (12–14) to the strain softening stage of high-strength component body can determine the stable state of double bodies with different mechanical properties.

5 Conclusions

Single coal pillar–roof combined body and double coal pillar–roof combined bodies were designed to characterize single coal pillar–roof system and coal pillars–roof system in pillar goaf, respectively. A series of numerical simulations of uniaxial compression were conducted using PFC, the source inducement, failure modes and load transfer characteristics of coal pillars–roof system were studied. On this basis, the instability criterions of double combined bodies with different combinations were deduced theoretically. Compared with existing researches, this study highlights four important conclusions:

  1. (1)

    Coal dominates the destruction during the single combined body is compressed. That is to say, coal is not only the first to be destroyed, but also induces rock damage. Meanwhile, the rock damage aggravates the destruction of coal in turn. Finally, the overall body loses bearing capacity based upon domino effect. The instability of a single coal pillar is the initial cause of the instability of the whole coal pillars–roof system in pillar goaf.

  2. (2)

    For double combined bodies with same mechanical properties, the two component bodies bear external load evenly and deform in coordination during compression. The whole system and its component bodies reach the bearing limits synchronously, and the loss of bearing capacity of any component body will lead to the instability of the whole system. When evaluating the stability of pillar goaf, the large-scale roof collapse disaster caused by the synchronous instability of coal pillars with the same or similar mechanical properties should be pay special attention to.

  3. (3)

    When double combined bodies with different mechanical properties are compressed, the low-strength body is damaged and reaches its bearing limit firstly. Concomitantly, the whole system reaches the first bearing peak as well. After that, as the external load originally borne by low-strength body transfers to the high -strength component body, the high-strength body reaches the bearing peak. Synchronously, the whole system reaches the second bearing peak. Thereafter, the high-strength body loses its stability gradually, resulting in the whole system no longer has bearing capacity simultaneously. For coal pillars with different mechanical properties in pillar goaf, it is necessary to accurately locate the coal pillars with weak bearing capacities and take reinforcement measures to prevent them from losing stability first, so as to prevent more coal pillars from losing stability due to load transfer.

  4. (4)

    The instability of any single component body can be regarded as the whole instability criterion of double bodies with same mechanical properties, while the instability of the single component body with high strength should be regarded as the instability criterion of the double bodies with different mechanical properties.

In the future, the failure and load transfer mechanism of multi coal pillar–roof combined bodies under more factors should be studied. Incidentally, the instability criterions should be applied to engineering pillar goaf to improve its practicability and accuracy.