Stability Analysis of Rib Pillars in Highwall Mining Under Dynamic and Static Loads in Open-Pit Coal Mine

The highwall miner can be used to mine the retained coal in the end slope of an open-pit mine. However, the instability mechanism of the reserved rib pillar under dynamic and static loads is not clear, which restricts the safe and efficient application of the highwall mining system. In this study, the load-bearing model of the rib pillar in highwall mining was established, the cusp catastrophe theory and the safety coefficient of the rib pillar were considered, and the criterion equations of the rib pillar stability were proposed. Based on the limit equilibrium theory, the limit stress of the rib pillar was analyzed, and the calculation equations of plastic zone width of the rib pillar in highwall mining were obtained. Based on the Winkler foundation beam theory, the elastic foundation beam model composed of the rib pillar and roof under the highwall mining was established, and the calculation equations for the compression of the rib pillar under dynamic and static loads were developed. The results show that with the increase of the rib pillar width, the total compression of the rib pillar under dynamic and static loads approximately decreases in an inverse function, and the compression of the rib pillar caused by static loads of the overlying strata and trucks has a decisive role. Numerical simulation and theoretical calculation were performed in this study. In the Numerical simulation, the coal seam with a buried depth of 122 m and a thickness of 3 m was mined by the highwall miner. According to the established rib pillar instability model of the highwall mining system, it is found that when the mining tunnel width is 3 m, the reasonable width of the rib pillar is at least 1.3 m, and the safety factor of the rib pillar is 1.3. The numerical simulation results are in good agreement with the results of theoretical calculation, which verifies the feasibility of the theoretical analysis of the rib pillar stability. The research results can provide an important reference for the stability analysis of rib pillars under highwall mining. stress of the rib pillar is analyzed, and the calculation equation of the plastic zone width of the rib pillar on both sides of the mining tunnel is obtained. The results show that the plastic


Introduction
At present, near-horizontal coal seams are mostly encountered in the large-scale open-pit coal mines under development in China, and the mining procedure of internal pressure relief in zones is widely adopted (Chen and Wu 2016;Wu et al. 2020;Chen et al. 2020). According to the traditional open-pit mining design theory, the transportation lines discharged by trucks in these large-scale open-pit mines are set on the two ends of the wall, and the formed end slope angle is less than the stable slope angle. As a result, a large number of coal resources are occupied by the end slope (Wang and Zhang 2019;Chen et al. 2020). With the continuous advancement of the inner dump and working slope, the coal occupied by the end slope will be re-buried by the inner dump.
Due to the limitations of the current mining technology and mining equipment in China, it is difficult to conduct secondary mining in these mines Pan et al. 2021).
Generally, a double lane inner row line of heavy-duty and no-load large mining trucks is set on the haulage bench of the end slope of the open-pit mine. For heavy-duty trucks, the truck weight plus load can reach 600 t, which brings a strong dynamic load and a great impact on the stability of the lower rib pillar Chen et al. 2020). Therefore, when the highwall miner (Hood et al. 2020;Sasaoka et al. 2016;Chang et al. 2021) is used to mine the retained coal in the end slope, the load on the reserved rib pillar is the composite load superimposed by the strong dynamic load generated by mining trucks and the non-uniform static load generated by end slopes.
Wilson's two-zone constraint theory has been widely used in the load calculation of rib pillars in strip mining in China (Wilson 1973). By this method, the ultimate bearing capacity and actual load of rib pillars are calculated, and the safety coefficient of the rib pillar is calculated to evaluate the stability of the rib pillar. Lin'Kov (2001) calculated the strength at different positions in the core area by combing the strength of the core area in a rib pillar with the actual stress, and put forward a general equation for calculating the failure envelope of the long rib pillar. Pietruszczak and Mroz (1981) regarded the strike section of the goaf as a flat elliptical orifice in an infinite plate with uniformly distributed load acting on the boundary, and derived the stress calculation equation of any distance point between the rib pillar at the end of the orifice and the coal wall using the elastic fracture theory.  Mo et al. (2018) quantitatively studied the influence of backfilling on rib pillar strength in highwall mining by using the FLAC2D software. Zhong and Ma (2020), He et al. (2021) researched the slope stability by using the ANSYS/LSDYNA software. From the 1950s to 1960s, the ideal elastic-plastic analytical solution of surrounding rock of axisymmetric circular roadway, namely Kastner equation, was derived (Obert and Duvall 1967;Jaeger and Cook 1978). Based on the stress balance theory of loose media and the stress differential equilibrium equation, Hou and Ma (1989) obtained the coal seam interface stress and the width of the stress limit equilibrium zone (plastic zone) of the coal body. On the basis of the elastic-plastic mechanics, Li et al. (2004) deduced the width of stress limit equilibrium zone of retained strip rib pillar using the stress balance differential equation and Coulomb criterion. Besides, the plastic zone width of the strip rib pillar was also deduced by using the complex function model of elastic theory, and then different theoretical equations were obtained. Tang et al. The external load of the strip-supported rib pillar mainly comes from the static load of overlying strata and the dynamic load caused by the driving of mining trucks. Due to the action of external load, a yield zone is formed on both sides of the rib pillar. According to the two-zone constraint theory proposed by Wilson (1973), from the peak value of the rib pillar stress to the boundary of the rib pillar (i.e. in the plastic zone), the stress of the rib pillar exceeds the yield stress and flows to the goaf. This area is surrounded by the plastic zone and constrained by the plastic zone, which is called the elastic core zone of the rib pillar. According to the effective area theory, the elastic core zone of the rib pillar carries three kinds of load, including the static load of rock strata above the upper elastic core zone of the rib pillar, the load of the overlying strata evenly divided by adjacent rib pillars and the dynamic load caused by trucks.
For the dynamic loads generated by mining trucks, the following four factors should be considered: (1) The running speed of the mining truck on the haulage bench of an open-pit mine is generally not more than 30 km/h. Therefore, the dynamic loads caused by the horizontal movement of the mining truck is ignored, and only the impact load caused by the uneven road surface of the mining truck is considered; (2) Because the wheel diameter of a large mining truck is larger than the width of the rib pillar, the front and rear wheels of the mining truck will not act on the same rib pillar at the same time, that is, each rib pillar can only carry the front or rear wheels of the mining truck; (3) Commonly used mining trucks generally have 2 front wheels and 4 rear wheels. As shown in Fig. 2, under the full load, the load distribution of front and rear axles is approximately 1:2, that is, the specific pressure of wheel-toground is approximately equal and the wheel-to-ground contact area is approximately the same; (4) When the rear wheel of a fully-loaded mining truck acts directly above the rib pillar, the rib pillar bears the maximum external load. At this time, the wheel-to-ground contact area of the mining truck is calculated by a×b, where a is the length of the wheel-to-ground contact area and b is the wheel width.
Combined with factor (1), the wheel-to-ground load can be simplified as a uniformly distributed static load, with a width of a, the load value of q0 and the dynamic load coefficient of kd.
When the cutting width of the highwall miner is 2.9-3.7 m (Chen et al. 2013;Yin and Dong 2015), no caving occurs in the roof of the rib pillar roof. As shown in Fig. 1, it is assumed that the average thickness of the overlying strata is H and the unit weight is γ, the width of the rib pillar is Wp and the width of the mining tunnel is Wm, then the load borne by the rib pillar (in plane) is: The relationship of the rib pillar stress σ, strain ε and damage parameter D can be expressed as follows (Xie et al. 2009;Xie 1990): where ε0 is the strain of the rib pillar under load; E is the initial elastic modulus of the rib pillar.
Assuming that the width of the plastic zone on one side of the rib pillar is Y and the thickness of the coal seam is Hp; in the plastic zone of the rib pillar with a total width of 2Y or in the elastic core zone of the rib pillar with a width of Wp-2Y, the relationship between plastic zone load Ps, elastic core zone load Pe and deformation u can be expressed by the following equation (2): where u0 is the compression of the rib pillar under load.

The rib pillar failure and instability model based on the cusp catastrophe theory
The strain energy V1 of the plastic zone of the stripsupported rib pillar, the elastic potential energy V2 of the elastic core zone of the strip-supported rib pillar and the gravity potential energy V3 of the overlying strata and the truck can be obtained from equations (1) respectively: The total potential energy of the roof-rib pillar system composed of rib pillar, overlying strata and truck can be expressed as: Taking u as the state variable, the catastrophe theory is used for analysis. The first derivative of V(u) is taken and V(u)'=0, then the equation of the equilibrium surface can be obtained: Equation (8) is derived further to obtain the equation: After solving equation (9), then: The dimensionless quantity ζ is introduced as the state variable, p and q as the control variables, and let: where k0 is the stiffness ratio, k0=ke/ks; ke is the stiffness of medium in the elastic core zone of the rib pillar, ke=E(Wp- By substituting equation (13) and equation (14) into the bifurcation set equation (16), the bifurcation set equation (17) of the cusp catastrophe (Chen et al. 2021;Dong et al. 2021) of the system can be obtained:

Criterion equation of the rib pillar stability
The stress of the rib pillar σs can be obtained as: According to the ultimate strength theory, the safety factor f of the rib pillar can be obtained from the ultimate stress σp of the rib pillar and stress σs borne by the rib pillar.

Calculation of plastic zone width and ultimate stress of the rib pillar
Based on the limit equilibrium theory, it is assumed that the roof and floor of coal seams are consistent, and the compressive strength is greater than that of coal. Without considering the physical strength, the mechanical analysis model of the lower mining tunnel and surrounding rock in highwall mining is established (Gu et al. 2014). As shown in where c0 and φ0 are the cohesion and internal friction angle of the interface between coal seams and the roof and floor.  The equilibrium equation in the x-axis direction of the micro element is established and simplified as follows: According to the Mohr-Coulomb yield criterion, the plastic zone of coal is in the limit equilibrium state, and the following equation can be obtained: 1 3 1 sin 2 cos 1 sin 1 sin The vertical stress and horizontal stress of micro element are the main stress. In equation (23), c and φ are cohesion and internal friction angle of coal body, and σ1=σy, σ3=σx.
In equation (23), X is differentiated and A=(1+sinφ)/(1sinφ), then boundary conditions The vertical ultimate stress σp of the rib pillar can be obtained from the equation (24): Combined with the principle of stress balance, after the mining tunnel is excavated, the load originally borne by the coal body in the mining tunnel section is transferred to the rib pillars on both sides of the mining tunnel. Assuming that the vertical stress and horizontal stress of micro elements are equal to the average stress distributed along with the coal seam thickness, there are: The equation (26) is simplified, and let X=(2YAtanφ0)/Hp, then equation (27) can be obtained: 1 sin tan 2 cos 2 cos 1 e 2 1 sin 1 sin tan tan 1 sin tan The expression of plastic zone width Y of the rib pillar on both sides of the mining tunnel is finally obtained by solving the equation (27):

Theoretical model of the beam on Winkler elastic foundation
To establish the differential equation of the deflection curve satisfied by the beam (in Fig. 5), a section of the beam with a length of l, a width of b1 and a height of h is taken, and the foundation coefficient of the elastic foundation is presented as k and the width as b2. Under the load q(x), the compression of the foundation is y(x), the reaction of the foundation to the beam is qf(x), a micro section dx is intercepted in the beam, the bending moment is M and the shear force is Q. According to Winkler's assumption (Soon et al. 2021;Li et al. 2021), the base reaction force is: If the influence of shear force on beam deflection is not considered, according to the balance condition of the force, let α=[kb2/4E1I] 1/4 and αx is used to replace x, then: The homogeneous solution of the differential equation (30) of the deflection curve can be obtained by introducing hyperbolic function: According to the boundary conditions of the beam on the elastic foundation, Equation (32) is substituted into equation (31) to obtain: and As shown in Fig. 6, the additional term caused by

Rib pillar compression under dynamic loads of the truck
Since the compression of the rib pillar under the dynamic loads of a fully-load truck is greater than that of a no-load truck, only the compression of the rib pillar under the dynamic load of a fully-loaded truck is discussed. As shown in Fig. 7, the rib pillar group between two permanent rib pillars is taken as the research object along the direction of the haulage bench in the end slope. Assuming that the compression generated on the rib pillar in the middle of the rib pillar group (near y=0) does not decay when the truck is driving, the compression generated by dynamic loads of a truck on a certain rib pillar can be simplified as the sum of the compression generated by the static load between the truck and the rib pillar at different distances, and this compression is defined as u2.
As shown in Fig. 8 The obtained z(y) curve is integrated on the interval of y [-l1,l1]. Finally, the compression u2 of the rib pillar caused by the dynamic loads of the fully-loaded truck is obtained:   9 shows the established coordinate system xoz. As shown in Fig. 9, assuming that the width of the stope is Wm, the height is Hp, the width of the supporting rib pillar is Wp, the elastic modulus is E, the width of the plastic zone is Y, the height of the roof is h, the length is l2, the elastic modulus is E1, then the width of the roof is b1=Wm+Wp, and the width of the elastic core area of the rib pillar is b2=Wp-2Y.
The foundation coefficient k is defined as: The elastic index value of roof (beam) α is defined as: As shown in Fig. 9, the boundary conditions of the model are: By substituting the boundary conditions (41) Table 1. Fig. 11a shows the schematic diagram of the wheel-toground contact area when the mining truck is fully loaded (600 t). According to the calculation of design parameters, the bearing capacity of each wheel is 100 t. Figure 11b shows the schematic diagram of the wheel-to-ground contact area when the mining truck is unloaded (237 t). Each wheel of the front wheel carries 54.5 t, and each wheel of the rear wheel carries 32 t.

Calculation of plastic zone width of the rib pillar
The plastic zone width of the rib pillar includes the sum of the plastic zone width caused by static load (slope) and the plastic zone width caused by dynamic load (truck). The No. 1 haulage bench is taken as an example. According to Table 1 (27) and equation (28), as shown in Fig. 12.   Fig. 12 The plastic zone width of the rib pillar with different pillar widths beneath the No. 1 haulage bench.
As shown in Fig. 12, for the rib pillar directly below the No. 1 haulage bench, the plastic zone width Y on both sides of the rib pillar gradually decreases with the increase of the width Wp of the rib pillar. When the pillar width Wp increases from 0.54 m to 3.00 m, the pillar plastic zone width Y decreases from 0.269 m to 0.260 m, decreasing by 3.3%. The plastic zone width Y of the rib pillar decreases linearly.

Compression of the rib pillar under dynamic loads of the truck
Along the direction of a mining tunnel, the rib pillar is equivalent to the elastic foundation. As the result of the highwall mining, the load above the elastic foundation is redistributed. When the fully-loaded truck and unloaded truck meet on each two-lane haulage bench at the same time, then the maximum static load is generated by trucks.
As shown in Fig. 8, one permanent isolated rib pillar is reserved in every ten rib pillars, and the width of the isolated rib pillar is 12 m; then half the length of the model  [-l1,l1], and then the compression u2 of the rib pillar due to the dynamic load generated by the fully-loaded truck is obtained, as shown in Fig. 13.   Fig. 13 The compression of the rib pillar under the dynamic load of No. 1 transport flat truck at full load.

Compression of the rib pillar under the combined action of static loads of overlying strata and truck
The loads of overlying strata that deform the elastic foundation are q1, q2, …, q7，and the loads of the truck that deform the elastic foundation are q8, q9, …, q23, therefore, n=7， m=16 in the equation (42). The overlying strata load at all levels can be expressed by the following equation (45)  The parameter setting is as follow: the roof length is l2=500 m, the roof height is h=12 m, the roof elastic modulus is E1=4 GPa, the mining tunnel height is Hp=3 m, the mining width is Wm=3 m, and the foundation coefficient is k=1/3 GPa.
As shown in Fig. 9   As shown in Fig. 17, the criterion value Δ sharply increases with the gradual increase of the rib pillar width Wp. When the rib pillar width Wp increases from 0.54 m to 3.00 m, the criterion value Δ increases from 7 to 80065 in an approximate power function.

Ultimate stress and safety factor of the rib pillar
When the rib pillar width takes different values, the calculated plastic zone width Y of the rib pillar and the total compression u0 of the rib pillar can be substituted into equation (18) and equation (19). The ultimate stress σp and safety factor f under different rib pillar widths can be obtained, and the calculation results are shown in Fig. 18 and Fig. 19.  As shown in Fig. 18 and Fig. 19, with the increase of the rib pillar width, the ultimate stress of the rib pillar under No.
1 haulage bench approximately decreases linearly, and the safety factor of the rib pillar approximately increases linearly.
When the rib pillar width Wp increases from 0.54 m to 3.00 m, the ultimate stress of the rib pillar gradually decreases from 11.965 MPa to 11.733 MPa, and the safety factor of the rib pillar increases from 0.66 to 2.21. The ultimate stress of the rib pillar decreases slightly, with a reduced rate of 1.9%, indicating that the ultimate stress is less affected by the change of the rib pillar width. The safety factor of the rib pillar increases greatly, with a growth rate of 234.8%, which is greatly affected by the change of the rib pillar width.

Analysis on reasonable reserved width of the rib pillar
Combined with Fig. 17 and Fig. 19 (20), equation (25), equation (27) and equation (28) haulage benches are checked, and the results are shown in Table 2.

Verification of numerical simulation by the ANSYS/LS-DYNA
Referring to the dimensions and physical and mechanical parameters of the end slope model shown in Fig. 10 and Table   1, the end slope model in In order to eliminate the influence of boundary conditions and reduce the calculation amount of the model, two rib pillars in the middle of the rib pillar group are taken as the main research objects and the mesh refinement is carried out, as shown in Fig. 21. The grid division of other strata is shown in Fig. 22. After the initial in-situ stress is balanced, the lifeand-death unit command is used to delete the grid of the mining tunnel (Fig. 23).
As shown in Fig. 24, a moving load on each haulage bench is applied, and the load of a truck is simplified into four-point loads. The load parameters are set referring to Chapter 4, and the load moving speed is 30 km/h.

Conclusions
In this study, the difficulty in mining the retained coal in the end slope of an open-pit mine was taken as the research object; the stress distribution, energy evolution mechanism and instability disaster mechanism of the rib pillar under the superposition of the dynamic loads of the truck in the largescale open-pit and the non-uniform static load of the end slopes were comprehensively considered. Based on the catastrophe theory, energy theory and rock mass mechanics, the mechanical analysis and numerical simulation were performed, the instability disaster mechanism of the rib pillar under dynamic and static loads in highwall mining system was comprehensively studied, and the design method of the rib pillar width in highwall mining was put forward. The main research results are as follows: (1) According to the stress characteristics of the rib pillar in highwall mining, the cusp catastrophe theory and the requirements of rib pillar safety coefficient are considered, a comprehensive criterion for predicting the rib pillar stability is established. The comprehensive criterion is mainly determined by three factors: the plastic zone width of the rib pillar, the ultimate stress of the rib pillar and the corresponding compression of the rib pillar under external load.
(2) According to the limit equilibrium theory, the limit stress of the rib pillar is analyzed, and the calculation equation

Conflicts of interest
The authors declare no conflict of interest.
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