Study on the Bearing Structure of Key Strata and the Linkage Evolution Mechanism of Surface Subsidence in Shallow Coal Seam Mining

Abstract

Shallow coal seam mining results in the formation of various bearing structures in key strata, leading to varying degrees of surface subsidence and severe disruption to the surface ecological environment. To investigate the coupled evolution characteristics of key strata fracture-bearing structures and surface subsidence in shallow coal seam mining, with a focus on the 1–2 coal seam mining at Longhua Coal Mine in northern Shaanxi as the research background, this study employed physical similarity simulation to establish the correlation between key strata fracture-bearing structures and surface subsidence. The study also utilized theoretical calculations to develop models for the trapezoidal hinged arch structure and the coupling between key strata-bearing structures and surface subsidence. Mechanical properties of bearing structures and the coupled evolution characteristics of surface subsidence were examined, and the scientific validity of the models was verified through field monitoring. The research reveals that the inclined section of the working face in shallow coal seam mining forms a trapezoidal hinged arch structure, where stress transmission actually resembles an arch shape. Based on the fracture characteristics of rock strata, this structure can be categorized into three types: a full-trapezoidal hinged arch structure, a semi-trapezoidal hinged arch structure, and a trapezoidal-like hinged arch structure. A mechanical calculation model for the trapezoidal hinged arch structure was constructed, and the mechanical calculation formula for this structure was derived based on mechanical equilibrium conditions. Using a masonry beam mechanical model, the formula for calculating the subsidence of key blocks in the key strata fracture was obtained. Based on the “masonry beam” mechanical model, a formula was derived to calculate the subsidence of key blocks in fractured key strata. The relationship between key strata-bearing structures and surface subsidence curves was analyzed, leading to the development of a calculation model for both. This model reveals the coupled evolution between rock movement and surface subsidence. Field measurements indicate a maximum surface subsidence of 1.93 m, with a subsidence coefficient of 0.65, showing that the surface helps suppress and reduce the overall subsidence.

1. Introduction

The extensive mining of coal resources has led to the fracturing of key strata within the overlying strata, resulting in surface subsidence [1,2]. This, in turn, has caused significant degradation of the ecological environment. The conflict between economic development and the imperative for green and sustainable environmental practices has become an urgent challenge in need of strategic planning and resolution [3,4,5]. This contradiction is particularly pronounced in the case of shallow coal seams, where surface subsidence is an inherent outcome of mining activities. The manifestation of fractured bearing structures in the key strata of the overlying layers on the surface represents the primary issue left unresolved by mining operations [6,7]. This situation has the potential to trigger a chain reaction, posing unpredictable threats to field production, and has consequently become a central focus for both researchers and industry operators [8,9,10]. At the same time, some scholars, focusing on mining subsidence as their research background, have studied its associated hazards and preventive measures, obtaining relevant results on surface subsidence caused by coal seam mining [11,12,13]. Building on this foundation, their in-depth research on key strata and surface subsidence has become crucial for assessing and managing the ecological environment. This research offers effective methods to address certain mining disasters and the chain reactions they may trigger.

At present, a substantial amount of research has been conducted on key strata and surface subsidence in shallow coal seam mining. Numerous scholars have investigated the mechanical structure of overlying strata and surface subsidence under various mining conditions, primarily analyzing stress characteristics, rock fracture patterns, and subsidence parameters, providing valuable guidance for subsequent research [14,15,16]. Based on the geological characteristics of shallow coal seams, it has been found that after coal seam extraction, the overlying strata can fracture to form hinged rock beam structures and step rock beam structures. Hinged rock beam structures exhibit stable load-bearing capabilities, while step rock beam structures have reduced load-bearing capacities [17,18,19]. The fractured overlying strata play a role in inhibiting and weakening surface subsidence [20]. Step rock beam structures lead to surface subsidence in a step manner and the formation of vertical fault fissures on the surface [4,21,22]. When fractured overlying strata form hinged rock beam structures, the surface subsides uniformly, creating tensional fissures [23,24]. Surface subsidence, as a representation of the fracture pattern of key strata-bearing structures, to some extent, reflects the type of load-bearing structure. Scholars have made significant progress in this area, forming a system of surface subsidence studies [25,26,27]. Through comprehensive research on surface subsidence, factors such as mining thickness, burial depth, and dip angle have been identified as influencing surface subsidence [28,29,30]. Currently, researchers have predominantly conducted separate studies on key strata fracture-bearing structures and surface subsidence [31,32]. While the results of these studies provide guidance for field production, they are constrained by the complexity and variability of geological strata [33,34,35]. The organic correlation between key strata-bearing structures and surface subsidence, as influenced by mine extraction, has not been thoroughly explored and requires continuous research. Therefore, research on the mechanisms governing the rock displacement of key strata-bearing structures and the evolution of surface subsidence in shallow coal seam mining should focus on the interactive mechanisms of key strata fracture-bearing structures and surface subsidence.

This paper, based on the context of shallow coal seam mining, conducts research on the interactive evolution characteristics of overlying strata-bearing structures and surface subsidence through similarity simulation experiments. Building upon the foundation of these experiments, it elucidates the coupled evolution characteristics of these two aspects. The concept of a trapezoidal hinged arch structure is introduced, along with the derivation of its mechanical characteristics. Furthermore, calculation models for both the bearing structures and rock displacement associated with surface subsidence are established. The formula for calculating surface subsidence controlled by key strata-bearing structures is derived, providing a theoretical basis for the study of the rock displacement of key strata-bearing structures and surface subsidence in shallow coal seam mining.

2. Simulation Design for Rock Movement and Surface Subsidence in Key Strata

2.1. Engineering Background

Longhua Coal Mine, as a typical shallow coal seam mine in the Shenfu mining area, primarily extracts 1⁻², 2⁻², and 3⁻² coal seams. The average thickness of these seams is 2.95 m, 3.3 m, and 2.68 m, respectively, with burial depths of 71.08 m, 107.61 m, and 149.11 m. The coal seams have dip angles ranging from 0 to 3°, and the strike longwall mining method is used for all the main coal seam working faces, as shown in Figure 1.

In this study, the 1–2 coal seam serves as the research background, with the working face being nearly horizontal. Fully mechanized mining was ezzmployed. The 1–2 coal seam is overlaid by two key strata, consisting of coarse sandstone (sub-key strata) located 12 m above the coal seam and siltstone (main key strata) located 37 m above the seam. These key strata are the primary load-bearing rock layers. The surface is covered by a loose loess layer approximately 18 m thick, which reflects the bearing structure and movement characteristics of the fractured key strata.

2.2. Similar Material Simulation Experiment

Based on the theory of physical similarity simulation, the mechanical characteristics of coal and rock, and the geological conditions of the study area, a physical similarity simulation model with a geometric similarity constant of 1/100 was established [36]. The dimensions of the experimental model are 300 cm (length) × 20 cm (width) × 155.5 cm (height), with a total research height of 71.5 cm. The mechanical parameters of the coal and rock are listed in

Table 1. Main mechanical properties of model materials.

Number	Lithology	Thickness (m)	Model Thickness (m)	Tensile Strength (MPa)	Compressive Strength (MPa)	Cohesion (MPa)	Bulk Modulus (MPa)	Volumetric Weight (kN·m−3)
14	Loess	18.1	18	0.089	0.42	0.44	1134	16.3
13	Siltstone	16.63	16.5	2.1	31.2	0.72	738	23.4
12	Medium-grained sandstone	5.08	5	2.42	28.4	1.56	1487	23.8
11	Mudstone	2.66	2.5	1.33	13.2	0.73	733	27.6
10	Fine-grained sandstone	1.6	1.5	1.95	28.6	2.43	1536	23.2
9	Siltstone	4.93	5	2.3	32.1	0.56	756	24.1
8	Coarse-grained sandstone	10.77	11	2.54	26.4	1.53	1433	22.8
7	Mudstone	1.2	1.5	1.29	12.9	0.69	752	28.1
6	Fine-grained sandstone	1.23	1.5	1.94	27.9	2.50	1531	23.0
5	Medium-grained sandstone	0.53	0.5	2.33	27.3	165	1494	23.4
4	Fine-grained sandstone	3.7	4	1.87	28.3	2.46	1548	23.5
3	Siltstone	4.65	4.5	2.2	29.8	0.64	746	23.1
2	No. 1–2 coal seam	2.95	3	0.37	12.4	1.23	614	13.2
1	Siltstone	3.92	4	2.4	28.9	0.72	782	24.3

.

In order to establish the relationship between the rock displacement of key strata-bearing structures and surface subsidence, five displacement monitoring lines were set up in the upper part of the 1–2 coal seam. These monitoring lines were positioned at distances of 12 cm, 23 cm, 37 cm, 53.5 cm, and 63.5 cm from the coal seam roof, and they were labeled as A to E accordingly. Lines A and B were located in the lower and upper sections of the sub-key stratum, while lines C and D were positioned in the lower and upper sections of the main key stratum. Line E was situated at the surface. The spacing between each monitoring point was 15 cm. Additionally, a leveling instrument was installed on the surface to monitor surface subsidence characteristics.

3. Analysis of Key Strata Failure Structure and Movement Characteristics in Shallow Coal Seam Mining

3.1. Characteristics of Key Strata Fracture in Shallow Coal Seam Mining

In order to obtain the relationship between the fracture-bearing characteristics of the key strata in the working face and the surface subsidence, Longhua Coal Mine, with two key strata in the overlying strata, was selected as the research background. Taking the left side of the model as the cutting point of the working face, the research on the fracture-bearing structure of the key strata was carried out, and the fracture-bearing structure and bearing structure movement and surface subsidence characteristics of each key strata were obtained.

(1) Breakage characteristics of sub-key strata

Physical similarity simulation experiments reveal that after coal seam mining, the immediate roof layer fractures and collapses, and the overlying strata undergo flexural deformation. In the central part of the goaf, the sub-key stratum experiences inverted arch-shaped subsidence. After the initial fracture of the sub-key stratum, a three-hinged arch structure is formed, with hinged points located in the upper part of the solid coal. The sub-key stratum exhibits mechanical characteristics of upper tension and lower compression in relation to the intact coal, and the distance from the solid coal is proportional to the tangent function of the fracture angle and the height of the sub-key stratum. The hinged points of the three-hinged arch structure in the sub-key stratum are situated in the collapsed rock layer, and the load-bearing strata exhibit mechanical characteristics of upper compression and lower tension from the point of fracture in the rock layers. After the periodic fracture of the sub-key stratum, the interlayer rock layers fracture upward in a layered manner, at which point the load-bearing rock strata and the interlayer rock fracture blocks jointly form the structure controlling the overlying strata movement. Due to the relatively regular fracture blocks of the key strata, a supporting structure resembling a trapezoid but effectively transmitting stress as an arch is formed, which, in this paper, is referred to as the trapezoidal hinged arch structure. This structure exerts hinged load-bearing effects on both ends of the working face and shares the load with the fractured rock mass above the load-bearing strata, controlling the movement of the overlying strata. When load-bearing structures on both ends of the working face form hinged beam structures, it is referred to as the full-trapezoidal hinged arch structure. Due to mining activities causing the fracture of the sub-key stratum, factors such as coal body support, high-strength mining, and increased free movement space make the load-bearing rock strata prone to forming stepped beam structures. When stepped beam structures are formed at both ends of the working face, the structural control of the overlying strata subsidence in the inclined section is referred to as the semi-trapezoidal hinged arch structure. If load-bearing structures at both ends of the working face are different, with one side forming a hinged beam structure and the other side forming a stepped beam structure, it is known as the trapezoidal-like hinged arch structure, which is commonly encountered in shallow coal seam mining, as shown in Figure 2 illustrating the trapezoidal-like hinged arch structure;

(2) Breakage characteristics of main key strata

After the fracture of the sub-key stratum in the working face recovery, it promotes the formation of free movement space in the lower part of the main key stratum, which, under the influence of the load from the overlying rock and soil layers, undergoes fracture to form a bearing structure. In comparison to the fracture-bearing structure of the sub-key stratum, the main key stratum exhibits a lagging behavior, meaning that the main key stratum fractures simultaneously with or after a certain delay relative to the sub-key stratum fracture. During this period, the main key stratum is supported by the sub-key stratum, resulting in a reduction in subsidence space. The fracture blocks of the bearing structure in the main key stratum are longer than those in the sub-key stratum, making it more likely to form a hinged beam structure. Detailed research on this phenomenon has been conducted in the literature [19]. When the sub-key stratum forms a hinged beam structure, the bearing structures in the main key stratum are often the same. If the former forms a stepped beam structure, the latter may develop both hinged beam and stepped beam structures. Due to differences in load bearing, free movement space, and lithology between the main key stratum and the sub-key stratum, periodic fractures of varying magnitudes can occur [37], resulting in smaller fracture block sizes in the main key stratum. This can lead to the formation of a hinged beam structure in the main key stratum, creating a bearing structure similar to that in the sub-key stratum, forming an inclined section full-trapezoidal hinged arch structure as shown in Figure 3.

3.2. Key Strata-Bearing Structure and Surface Movement Characteristics

(1) Trapezoidal-like hinged arch structure movement characteristics

The key strata in the working face form a fracture-bearing structure, which undergoes periodic load movement as the working face advances. Based on the results of experimental simulations, it is evident that after the fracture of the sub-key stratum, a trapezoidal-like hinged arch structure is formed on the inclined section, and this structure’s hinged support reduces the subsidence of the overlying strata. Calculations based on the working face inclined length and the sub-key stratum’s position reveal that the sub-key stratum reaches a state of over-mining conditions. Monitoring lines A and B for sub-key stratum displacement show that the rock layers on the left side of the cutting section exhibit more significant subsidence compared to the right side; with the maximum subsidence occurring in the range of 0~22 m and 75~120 m on the left side of the working face, the load-bearing action of the sub-key stratum causes line B to move toward both sides of the working face. As line A has no load-bearing rock layer support at its position, the rock layers exhibit smaller horizontal displacement after compaction, confirming the controlling effect of the trapezoidal-like hinged arch structure on the subsidence of the overlying rock and soil layers. Analysis of the fitted curves for lines A and B reveals that the fitted curve for line B has a larger slope, indicating the presence of inflection points and stretching zones at both ends of the working face. The fitted curve for line A exhibits a funnel-like shape with no inflection points, demonstrating that the key strata-bearing structure has a certain control over the subsidence of the overlying rock layers, effectively reducing the amount of subsidence of the overlying strata, as shown in Figure 4 in the subsidence curve of the trapezoidal-like hinged arch structure;

(2) Full-trapezoidal hinged arch structure and surface movement characteristics

The fracture of the sub-key stratum results in an increased free movement space in the lower part of the main key stratum, forming a full-trapezoidal hinged arch structure on the inclined section. Based on the rock fracture angle and coal pillar support, the inclined length of the full-trapezoidal hinged arch structure is smaller than that of the trapezoidal-like hinged arch structure, with the difference being twice the thickness of the interlayer rock. Due to the supporting effect of the full-trapezoidal hinged arch structure, the subsidence of rock and soil layers in the inclined section ranges from 0 to 10 m and 94 to 120 m and is relatively small. In the range of 10~32 m and 69~94 m, the subsidence gradient of rock and soil layers is higher, leading to an intensified subsidence trend. Within the range of 32~69 m, the subsidence values for various monitoring lines are essentially the same, indicating that the maximum subsidence point has been reached. Comparing the fitted curves of rock and soil layers, only monitoring line C has an inflection point on the left side of the working face. Fitted curves for lines D and E exhibit a high degree of similarity, with inflection points and stretching zones on both sides of the working face. The subsidence gradient for the latter is gentler, indicating a higher degree of fit. The fitted curve for line E closely resembles the surface subsidence curve in mining subsidence theory and is symmetric with the maximum subsidence point, indicating that the fracture-bearing structure of the key strata can effectively control and mitigate surface subsidence. It possesses self-repairing properties, resulting in a smoother surface subsidence curve and reducing the impact of mining-induced damage on the surface. Therefore, it can be observed that the trapezoidal hinged arch structure effectively controls the movement of overlying rock and soil layers, reducing the detrimental effects on the surface soil layer, as shown in Figure 5 in the diagram of the full-trapezoidal hinged arch structure and surface subsidence curve.

4. Mechanical Calculation and Analysis of Key Strata-Inclined-Bearing Structure

4.1. Building a Mechanical Model

Based on the mechanical load-bearing structural characteristics of the key strata along the strike section of the working face, the key strata are considered as a whole for analysis of the fractured rock mass. According to the results of physical similarity simulation, the sub-key stratum is regarded as a trapezoidal-like hinged arch structure, while the main key stratum is considered a full-trapezoidal hinged arch structure. The mechanical model of the key strata fractured structure is shown in Figure 6.

In Figure 5, h is the thickness of the coal seam (m); h₁ is the thickness of the immediate roof (m); h₂ is the thickness of the sub-key strata (m); h₃ is the thickness of the interlayer rock (m); h₄ is the thickness of the main key strata (m); h₅ is the thickness of the loess layer, (m); ${\beta }_{\mathrm{Z}1}$, ${\beta }_{\mathrm{Z}2}$, ${\beta }_{\mathrm{Y}1}$, and ${\beta }_{\mathrm{Y}2}$ are the key strata breaking angles (°); ${\theta }_{\mathrm{Z}1}$, ${\theta }_{\mathrm{Z}2}$, ${\theta }_{\mathrm{Y}1}$, ${\theta }_{\mathrm{Y}2}$, and ${\theta }_{\mathrm{Y}3}$ are the key strata to break the key block rotation angle (°); L_Z1, L_Z2, L_Z3, L_Y1, L_Y2, L_Y3, and L_Y4 are the key block lengths of key strata breaking, (m); P_Z1 and P_Z2 are the loads transferred from the loess layer to key blocks B and B₁ (kN); T_A1 and T_A2 are full-trapezoidal hinged arch structure horizontal extrusion forces (kN); P_ZJ1, P_ZJ2, and P_ZJ3 are the self-weight loads of key blocks B, B₁, and C (kN); W_C1 and W_C2 are the acting forces of interlayer strata on key blocks B₂ and B₃ (kN); T_B1 and T_B2 are horizontal extrusion forces of the trapezoidal-like hinged arch structure (kN); P_Y1, P_Y2, P_Y3, and P_Y4 are the key blocks of B₂, B₃, C₃, and C₂ weight loads (kN); and Q_Y1 and Q_Y2 are the shear forces on the contact hinge of the trapezoidal-like hinged arch structure (kN).

4.2. Mechanical Analysis of Full-Trapezoidal Hinged Arch Load-Bearing Structure

The stability and mechanical characteristics of the key strata, which serve as critical load-bearing rock layers controlling surface subsidence, have a significant impact on mining operations. According to the results of physical similarity simulation experiments, the fracture-bearing structure of the key strata has a favorable control effect on rock layer subsidence. Therefore, the stability of the load-bearing structure determines the subsidence of the overlying rock and soil layers. The mechanical characteristics of the key strata fracture-bearing structure were simplified into a mechanical calculation model of a full-trapezoidal hinged arch structure, as shown in Figure 7.

According to the action relationship of force, the key blocks B, C, and B₁ form a unified force system with hinge points F₁, F₂, F₃, and F₄. Based on the stress balance condition, the following can be obtained:

$$\left\{\begin{array}{l}{\displaystyle \sum M_{\mathrm{F}1}=0}\\ {\displaystyle \sum M_{\mathrm{F}2}=0}\\ {\displaystyle \sum F_{\mathrm{Zy}}=0}\\ {\displaystyle \sum F_{\mathrm{Zx}}=0}\end{array}\right.$$

(1)

Considering the difficulty of calculation, the uniform load of the mechanical structure is simplified as the point load acting on the middle of the key block, and then, P_Z1 and P_Z2 are:

$$\left\{\begin{array}{l}{P}_{\mathrm{Z}1}={\gamma }_5{h}_5{L}_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}\\ P_{\mathrm{Z}2}={\gamma }_5{h}_5{L}_{\mathrm{Z}2}\cos {\theta }_{\mathrm{Z}2}\end{array}\right.$$

(2)

where ${\gamma }_5$ is the bulk density of the loess layer (kN/m³).

In the full-trapezoidal hinged arch structure, all the rock masses compacted and bearing the fractured strata are considered key block C; thus the point load P_Z3 is given by:

$$P_{\mathrm{Z}3}={\gamma }_5{h}_5{L}_{\mathrm{Z}3}$$

(3)

From the fracture angle of the main key stratum rock layers and the fracture length of the key block, L_Z3 can be determined as follows:

$$L_{\mathrm{Z}3}=L-{\displaystyle \sum _{n=t}^4{h}_{\mathrm{t}}}\left(\cot {\displaystyle \frac{{\beta }_{\mathrm{Y}1}+{\beta }_{\mathrm{Z}1}}{2}}+\cot {\displaystyle \frac{{\beta }_{\mathrm{Y}2}+{\beta }_{\mathrm{Z}2}}{2}}\right)-\left(L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}+L_{\mathrm{Z}2}\cos {\theta }_{\mathrm{Z}2}\right)$$

(4)

where L is the inclined length of the working face (m).

Based on the theory of masonry beams, it can be determined that the contact area between key blocks is 0.5a. From the key block rotation and thickness, the value of “a” can be obtained as follows:

$$a=0.5\left(h_4-L_{\mathrm{Z}1}\sin {\theta }_{\mathrm{Z}1}\right)$$

(5)

According to Formula (1), ${\displaystyle \sum M_{\mathrm{F}1}}=0$ can be derived as follows:

$$Q_{\mathrm{Z}2}={\displaystyle \frac{M_{\mathrm{Z}1\text{-}1}+M_{\mathrm{Z}1\text{-}2}+M_{\mathrm{Z}1\text{-}3}}{L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}+h_4\cot {\beta }_{\mathrm{Z}1}+L_{\mathrm{Z}3}+L_{\mathrm{Z}2}\cos {\theta }_{\mathrm{Z}2}+h_4\cot {\beta }_{\mathrm{Z}2}}}$$

(6)

wherein:

$$\left\{\begin{array}{l}{M}_{\mathrm{Z}1\text{-}1}=\left(P_{\mathrm{ZJ}1}+P_{\mathrm{Z}1}\right)\left({\displaystyle \frac{L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}}{2}}+h_4\cot {\beta }_{\mathrm{Z}1}\right)\\ M_{\mathrm{Z}1\text{-}2}=\left(P_{\mathrm{Z}3}+P_{\mathrm{ZJ}3}+P_{\mathrm{Z}2}+P_{\mathrm{ZJ}2}-P_{\mathrm{Z}4}\right)\left(L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}+h_4\cot {\beta }_{\mathrm{Z}1}+{\displaystyle \frac{L_{\mathrm{Z}3}}{2}}\right)\\ M_{\mathrm{Z}1\text{-}3}=-{\displaystyle \frac{\left(P_{\mathrm{Z}2}+P_{\mathrm{ZJ}2}\right)L_{\mathrm{Z}2}\cos {\theta }_{\mathrm{Z}2}}{2}}\end{array}\right.$$

(7)

where P_Z3 is the load transferred from the loess layer to the key block C (kN); and P_Z4 is the supporting load of the interlayer rock stratum after key block C is broken (kN).

According to Formula (1), ${\displaystyle \sum M_{\mathrm{F}2}}=0$ can be derived as follows:

$$M_{\mathrm{Z}2\text{-}1}=M_{\mathrm{Z}2\text{-}2}+M_{\mathrm{Z}2\text{-}3}+M_{\mathrm{Z}2\text{-}4}+M_{\mathrm{Z}2\text{-}5}$$

(8)

wherein:

$$\left\{\begin{array}{l}{M}_{\mathrm{Z}2\text{-}1}=T_{\mathrm{A}2}\left(h_4-L_{\mathrm{Z}2}\sin {\theta }_{\mathrm{Z}2}-{\displaystyle \frac{a}{2}}\right)-T_{\mathrm{A}1}\left(h_4-L_{\mathrm{Z}1}\sin {\theta }_{\mathrm{Z}1}-{\displaystyle \frac{a}{2}}\right)\\ M_{\mathrm{Z}2\text{-}2}=-Q_{\mathrm{Z}1}\left(L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}+h_4\cot {\beta }_{\mathrm{Z}1}\right)\\ M_{\mathrm{Z}2\text{-}3}={\displaystyle \frac{\left(P_{\mathrm{ZJ}1}+P_{\mathrm{Z}1}\right)\left(L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}+h_4\cot {\beta }_{\mathrm{Z}1}\right)+\left(P_{\mathrm{Z}4}-P_{\mathrm{Z}3}-P_{\mathrm{ZJ}4}\right)L_{\mathrm{Z}3}}{2}}\\ M_{\mathrm{Z}2\text{-}4}=-\left(P_{\mathrm{Z}2}+P_{\mathrm{ZJ}2}\right)\left(L_{\mathrm{Z}3}+{\displaystyle \frac{L_{\mathrm{Z}2}\cos {\theta }_{\mathrm{Z}2}+h_4\cot {\beta }_{\mathrm{Z}2}}{2}}\right)\\ M_{\mathrm{Z}2\text{-}5}=Q_{\mathrm{Z}2}\left(L_{\mathrm{Z}3}+L_{\mathrm{Z}2}\cos {\theta }_{\mathrm{Z}2}+h_4\cot {\beta }_{\mathrm{Z}2}\right)\end{array}\right.$$

(9)

According to Formula (1), ${\displaystyle \sum F_{\mathrm{Zx}}}=0$ and ${\displaystyle \sum F_{\mathrm{Zy}}}=0$, can be derived as follows:

$$T_{\mathrm{A}1}+T_{\mathrm{A}2}=0$$

(10)

$$Q_{\mathrm{Z}1}+Q_{\mathrm{Z}2}+P_{\mathrm{Z}4}=P_{\mathrm{Z}1}+P_{\mathrm{Z}2}+P_{\mathrm{Z}3}+P_{\mathrm{ZJ}1}+P_{\mathrm{ZJ}2}+P_{\mathrm{ZJ}3}$$

(11)

According to the breaking characteristics of rock strata, key block C has been fully broken and collapsed, and then:

$$P_{\mathrm{Z}4}=P_{\mathrm{Z}3}+P_{\mathrm{ZJ}3}$$

(12)

Incorporating it into Formula (7), which is the second formula, and Formula (9), which is the third formula, yields:

$$M_{\mathrm{Z}1\text{-}2}=\left(P_{\mathrm{Z}2}+P_{\mathrm{ZJ}2}\right)\left(L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}+h_4\cot {\beta }_{\mathrm{Z}1}+{\displaystyle \frac{L_{\mathrm{Z}3}}{2}}\right)$$

(13)

$$M_{\mathrm{Z}2\text{-}3}={\displaystyle \frac{\left(P_{\mathrm{ZJ}1}+P_{\mathrm{Z}1}\right)\left(L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}+h_4\cot {\beta }_{\mathrm{Z}1}\right)}{2}}$$

(14)

Substituting Formulas (6) and (12) into Formula (11) yields Q_Z1 as follows.

$$Q_{\mathrm{Z}1}=P_{\mathrm{Z}1}+P_{\mathrm{Z}2}+P_{\mathrm{ZJ}1}+P_{\mathrm{ZJ}2}-{\displaystyle \frac{M_{\mathrm{Z}1\text{-}1}+M_{\mathrm{Z}1\text{-}2}+M_{\mathrm{Z}1\text{-}3}}{L_{\mathrm{Z}1}\cos {\theta }_{\mathrm{Z}1}+h_4\cot {\beta }_{\mathrm{Z}1}+L_{\mathrm{Z}3}+L_{\mathrm{Z}2}\cos {\theta }_{\mathrm{Z}2}+h_4\cot {\beta }_{\mathrm{Z}2}}}$$

(15)

Substituting Formulas (9) and (10) into Formula (8) yields T_A1 and T_A2, where the positive and negative signs in the equation represent the direction of the force.

$$\left\{\begin{array}{l}{T}_{\mathrm{A}1}={\displaystyle \frac{M_{\mathrm{Z}2\text{-}2}+M_{\mathrm{Z}2\text{-}3}+M_{\mathrm{Z}2\text{-}4}+M_{\mathrm{Z}2\text{-}5}}{L_{\mathrm{Z}1}\sin {\theta }_{\mathrm{Z}1}-L_{\mathrm{Z}2}\sin {\theta }_{\mathrm{Z}2}}}\\ T_{\mathrm{A}2}=-{\displaystyle \frac{M_{\mathrm{Z}2\text{-}2}+M_{\mathrm{Z}2\text{-}3}+M_{\mathrm{Z}2\text{-}4}+M_{\mathrm{Z}2\text{-}5}}{L_{\mathrm{Z}1}\sin {\theta }_{\mathrm{Z}1}-L_{\mathrm{Z}2}\sin {\theta }_{\mathrm{Z}2}}}\end{array}\right.$$

(16)

According to the masonry arch theory and the S-R criterion, it is necessary to ensure the stability of the full-trapezoidal hinged arch structure. The arch foot must satisfy the following:

$$T_{\mathrm{A}1}\tan \phi \ge Q_{\mathrm{Z}1}$$

(17)

$$T_{\mathrm{A}2}\tan \phi \ge Q_{\mathrm{Z}2}$$

(18)

where $\tan \phi$ is the friction coefficient, take 0.5.

At this time, the arch height h_q of the fully trapezoidal hinged arch structure is:

$$h_{\mathrm{q}}=\max \left\{L_{\mathrm{Z}1}\sin {\theta }_{\mathrm{Z}1},L_{\mathrm{Z}2}\sin {\theta }_{\mathrm{Z}2}\right\}$$

(19)

4.3. Mechanical Analysis of Load-Bearing Structure of Trapezoid-like Hinged Arch

For the dual key strata load-bearing structure, they jointly bear the load of overlying rock and soil layers, controlling surface subsidence. Based on the results of physical similarity simulations and the mechanical characteristics of key strata fracture-bearing structures, the load-bearing structure of the sub-key strata is simplified into a trapezoidal-like hinged arch mechanical calculation model, as shown in Figure 8.

With the same applied force as the main key strata, the sub-key strata-fractured key blocks B₂, C₂, C₃, and B₃ form a unified force system at hinge points F₅, F₆, F₇, and F₈. Based on the stress equilibrium condition, it can be derived that:

$$\left\{\begin{array}{l}{\displaystyle \sum M_{\mathrm{F}5}=0}\\ {\displaystyle \sum M_{\mathrm{F}6}=0}\\ {\displaystyle \sum F_{\mathrm{Yy}}=0}\\ {\displaystyle \sum F_{\mathrm{Yx}}=0}\end{array}\right.$$

(20)

Based on the relationship between the uniform load and point load on the key blocks, the structural loads are simplified as point loads acting at the center of the key blocks, resulting in W_C1, W_C2, W_C3, and W_C4:

$$\left\{\begin{array}{l}{W}_{\mathrm{C}1}=\left(Q_{\mathrm{Z}1}+P_{\mathrm{C}1}\right)\cos {\theta }_{\mathrm{Y}1}\\ W_{\mathrm{C}2}=\left(Q_{\mathrm{Z}2}+P_{\mathrm{C}2}\right)\cos {\theta }_{\mathrm{Y}2}\\ W_{\mathrm{C}3}={\gamma }_3{h}_3{L}_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}\\ W_{\mathrm{C}4}={\gamma }_3{h}_3{L}_{\mathrm{Y}4}+P_{\mathrm{Z}3}+P_{\mathrm{ZJ}3}\end{array}\right.$$

(21)

where ${\gamma }_3$ is the interlayer rock bulk density (kN/m³); P_C1 is the load of interlayer rock to the key stratum fracture block B₂ (kN); and P_C2 is the load of the interlayer rock layer on the key layer fracture block B₃ (kN).

Based on the fracture angle of the sub-key strata rock layers and the fracture length of the key block, L_Y4 can be determined as follows.

$$L_{\mathrm{Y}4}=L-{\displaystyle \sum _{n=t}^3{h}_{\mathrm{t}}}\left(\cot {\beta }_{\mathrm{Y}1}+\cot {\beta }_{\mathrm{Y}2}\right)-\left(L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}+\left(L_{\mathrm{Y}2}-d\right)\cos {\theta }_{\mathrm{Y}2}+L_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}\right)$$

(22)

According to Formula (20), ${\displaystyle \sum M_{\mathrm{F}5}}=0$ can be derived as follows:

$$Q_{\mathrm{Y}2}={\displaystyle \frac{M_{\mathrm{Y}1\text{-}1}+M_{\mathrm{Y}1\text{-}2}+M_{\mathrm{Y}1\text{-}3}+M_{\mathrm{Y}1\text{-}4}}{L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}+h_2\cot {\beta }_{\mathrm{Y}1}+L_{\mathrm{Y}4}+L_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}+L_{\mathrm{Y}2}\cos {\theta }_{\mathrm{Y}2}-d\cot {\beta }_{\mathrm{Y}2}+L_{\mathrm{Y}2}\cot {\theta }_{\mathrm{Y}2}}}$$

(23)

wherein:

$$\left\{\begin{array}{l}{M}_{\mathrm{Y}1\text{-}1}=\left(P_{\mathrm{Y}1}+W_{\mathrm{C}1}\right)\left({\displaystyle \frac{L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}}{2}}+h_2\cot {\beta }_{\mathrm{Y}1}\right)\\ M_{\mathrm{Y}1\text{-}2}=\left(W_{\mathrm{C}3}+P_{\mathrm{Y}4}-P_{\mathrm{R}}\right)\left(L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}+h_2\cot {\beta }_{\mathrm{Y}1}+{\displaystyle \frac{L_{\mathrm{Y}4}}{2}}\right)\\ M_{\mathrm{Y}1\text{-}3}=-\left(W_{\mathrm{C}3}+P_{\mathrm{Y}3}+W_{\mathrm{C}2}+P_{\mathrm{Y}2}\right)\left(L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}+h_2\cot {\beta }_{\mathrm{Y}1}+L_{\mathrm{Y}4}+{\displaystyle \frac{L_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}}{2}}\right)\\ M_{\mathrm{Y}1\text{-}4}=\left(W_{\mathrm{C}2}+P_{\mathrm{Y}2}\right)\left({\displaystyle \frac{d\cot {\beta }_{\mathrm{Y}2}-L_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}-L_{\mathrm{Y}2}\cos {\theta }_{\mathrm{Y}2}}{2}}\right)\end{array}\right.$$

(24)

where P_R is the supporting load of the interlayer rock stratum after the key block C₂ is broken (kN); and d is the vertical height of key blocks B₃ and C₃ (m).

According to Formula (20), ${\displaystyle \sum M_{\mathrm{F}6}}=0$ can be derived as follows:

$$M_{\mathrm{Y}2\text{-}1}=M_{\mathrm{Y}2\text{-}2}+M_{\mathrm{Y}2\text{-}3}+M_{\mathrm{Y}2\text{-}4}+M_{\mathrm{Y}2\text{-}5}$$

(25)

wherein:

$$\left\{\begin{array}{l}{M}_{\mathrm{Y}2\text{-}1}=T_{\mathrm{B}2}\left(h_2-L_{\mathrm{Y}2}\sin {\theta }_{\mathrm{Y}2}-{\displaystyle \frac{a}{2}}\right)-T_{\mathrm{B}1}\left(h_2-L_{\mathrm{Y}1}\sin {\theta }_{\mathrm{Y}2}-{\displaystyle \frac{a}{2}}\right)\\ M_{\mathrm{Y}2\text{-}2}=-Q_{\mathrm{Y}1}\left(L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}+h_2\cot {\beta }_{\mathrm{Y}1}\right)\\ M_{\mathrm{Y}2\text{-}3}={\displaystyle \frac{\left(W_{\mathrm{C}1}+P_{\mathrm{Y}1}\right)\left(L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}+h_2\cot {\beta }_{\mathrm{Y}1}\right)+\left(P_{\mathrm{R}}-W_{\mathrm{C}4}-P_{\mathrm{Y}4}\right)L_{\mathrm{Y}4}}{2}}\\ M_{\mathrm{Y}2\text{-}4}=-\left(W_{\mathrm{C}3}+P_{\mathrm{Y}3}\right)\left(L_{\mathrm{Y}4}+{\displaystyle \frac{L_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}+h_2\cot {\beta }_{\mathrm{Y}2}}{2}}\right)\\ M_{\mathrm{Y}2\text{-}5}=-\left(W_{\mathrm{C}2}+P_{\mathrm{Y}2}\right)\left(L_{\mathrm{Y}4}+L_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}+h_2\cot {\beta }_{\mathrm{Y}2}+{\displaystyle \frac{L_{\mathrm{Y}2}\cos {\theta }_{\mathrm{Y}2}-d\cot {\beta }_{\mathrm{Y}2}}{2}}\right)\\ M_{\mathrm{Z}2\text{-}6}=Q_{\mathrm{Y}2}\left(L_{\mathrm{Y}4}+L_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}+h_2\cot {\beta }_{\mathrm{Y}2}+L_{\mathrm{Y}2}\cos {\theta }_{\mathrm{Y}2}-d\cot {\beta }_{\mathrm{Y}2}\right)\end{array}\right.$$

(26)

According to Formula (20), ${\displaystyle \sum F_{\mathrm{Yx}}}=0$ and ${\displaystyle \sum F_{\mathrm{Yy}}}=0$ can be derived as follows:

$$T_{\mathrm{B}1}+T_{\mathrm{B}2}=0$$

(27)

$$Q_{\mathrm{Y}1}+Q_{\mathrm{Y}2}+P_{\mathrm{R}1}=W_{\mathrm{C}1}+W_{\mathrm{C}2}+W_{\mathrm{C}3}+W_{\mathrm{C}4}+P_{\mathrm{Y}1}+P_{\mathrm{Y}2}+P_{\mathrm{Y}3}+P_{\mathrm{Y}4}$$

(28)

Based on the characteristics of rock fracture, it can be inferred that key block C₂ has completely fractured and collapsed.

$$P_{\mathrm{R}}=W_{\mathrm{C}4}+P_{\mathrm{Y}4}$$

(29)

Substituting Formula (29) into Formulas (24) and (26) yields:

$$M_{\mathrm{Y}1\text{-}2}=0$$

(30)

$$M_{\mathrm{Y}2\text{-}3}={\displaystyle \frac{\left(W_{\mathrm{C}1}+P_{\mathrm{Y}1}\right)\left(L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}+h_2\cot {\beta }_{\mathrm{Y}1}\right)}{2}}$$

(31)

By substituting Formulas (23) and (29) into Formula (28), Q_Y1 can be determined as follows:

$$Q_{\mathrm{Y}1}=W_{\mathrm{C}1}+W_{\mathrm{C}2}+W_{\mathrm{C}3}+P_{\mathrm{Y}1}+P_{\mathrm{Y}2}+P_{\mathrm{Y}3}-{\displaystyle \frac{M_{\mathrm{Y}1\text{-}1}+M_{\mathrm{Y}1\text{-}3}+M_{\mathrm{Y}1\text{-}4}}{L_{\mathrm{Y}1}\cos {\theta }_{\mathrm{Y}1}+h_2\cot {\beta }_{\mathrm{Y}1}+L_{\mathrm{Y}4}+L_{\mathrm{Y}3}\cos {\theta }_{\mathrm{Y}3}+L_{\mathrm{Y}2}\cos {\theta }_{\mathrm{Y}2}-d\cot {\beta }_{\mathrm{Y}2}+L_{\mathrm{Y}2}\cot {\theta }_{\mathrm{Y}2}}}$$

(32)

By substituting Formulas (26) and (27) into Formula (25), T_B1 and T_B2 are obtained as:

$$\left\{\begin{array}{l}{T}_{\mathrm{B}1}={\displaystyle \frac{M_{\mathrm{Y}2\text{-}2}+M_{\mathrm{Y}2\text{-}3}+M_{\mathrm{Y}2\text{-}4}+M_{\mathrm{Y}2\text{-}5}}{L_{\mathrm{Y}1}\sin {\theta }_{\mathrm{Y}1}-L_{\mathrm{Y}2}\sin {\theta }_{\mathrm{Y}2}}}\\ T_{\mathrm{B}2}=-{\displaystyle \frac{M_{\mathrm{Y}2\text{-}2}+M_{\mathrm{Y}2\text{-}3}+M_{\mathrm{Y}2\text{-}4}+M_{\mathrm{Y}2\text{-}5}}{L_{\mathrm{Y}1}\sin {\theta }_{\mathrm{Y}1}-L_{\mathrm{Y}2}\sin {\theta }_{\mathrm{Y}2}}}\end{array}\right.$$

(33)

According to the masonry beam theory, step rock beam structure, and S-R criteria, it can be inferred that to ensure the stability of the trapezoidal-like hinged arch structure in advance, the arch base must meet the following requirements:

$$T_{\mathrm{B}1}\tan \phi \ge Q_{\mathrm{Y}1}$$

(34)

$$T_{\mathrm{B}2}\tan \phi \ge Q_{\mathrm{Y}2}$$

(35)

Based on Formulas (16) and (33), it is evident that when the denominator of the equation approaches zero, the horizontal stress T tends toward infinity. Clearly, achieving this condition under practical circumstances is challenging. Therefore, the structure must still satisfy the criterion for lateral–torsional buckling:

$$T\le a\eta {\sigma }_{\mathrm{c}}$$

(36)

where $\eta {\sigma }_{\mathrm{c}}$ is the extrusion strength of the key block at the corner end of the key strata (MPa).

At this time, the arch height h_l of the trapezoidal-like hinged arch structure is determined as:

$$h_l=\max \left\{L_{\mathrm{Y}1}\sin {\theta }_{\mathrm{Y}1},L_{\mathrm{Y}2}\sin {\theta }_{\mathrm{Y}2}+L_{\mathrm{Y}3}\sin {\theta }_{\mathrm{Y}3}+d\right\}$$

(37)

A semi-trapezoidal hinged arch structure can be deduced using the trapezoid-like hinged arch structure, but it is not deduced in this paper.

5. Analysis of Rock Movement and Surface Subsidence Coupling Characteristics in Key Strata-Bearing Structures

5.1. Analysis of Settlement of Key Strata-Bearing Structure

The extraction of mines causes damage to the surface, which represents a prominent contradiction between coal resource extraction and the ecological environment. The fractured load-bearing structure of the overlying strata plays a key role in controlling the subsidence of the loose layer beneath the surface. The loose layer at the surface can mitigate the uneven subsidence caused by the fractured load-bearing structure of the key strata. It can be seen that the fractured load-bearing structure of the key strata determines the state of surface subsidence, and both factors reduce the overall subsidence space during mining operations. Based on the analysis of the fractured structure of the key strata in this paper, the trapezoidal hinged arch structure exhibits strong symmetry. To simplify the analysis process, we only conducted an analysis of the left side of the structure. In the research area of this study, the inclined length of the working face is 294 m, with an average fracture length of 15 m for the key blocks. Based on the mechanical model of masonry beams [38], a mechanical transmission structure comprising seven key blocks was established. The mechanical transfer model of the key block is shown in the mechanical model of the masonry beam structure in Figure 9.

In Figure 8, Q_ir is the self-weight load of the ith key strata acting on the middle of rock block r (kN); m_irQ_ir is the force acting on the middle of rock block r by the key strata of layer i (kN); R_ir is the key strata of the first layer, which acts on the support force of the rock block in the middle of the rock block r, which is approximately equal to m_irQ_ir (kN); and T is the horizontal thrust (kN).

By considering the equilibrium of the force system in the masonry beam mechanical model, it can be derived that:

$$\left[\begin{array}{ccccccc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{7}{2}}& {\displaystyle \frac{9}{2}}& {\displaystyle \frac{11}{2}}& {\displaystyle \frac{13}{2}}\\ 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{7}{2}}& {\displaystyle \frac{9}{2}}& {\displaystyle \frac{11}{2}}\\ 0& 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{7}{2}}& {\displaystyle \frac{9}{2}}\\ 0& 0& 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{7}{2}}\\ 0& 0& 0& 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{3}{2}}\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{2}}\end{array}\right]\cdot \left[\begin{array}{c}{Q}_{i1}\\ Q_{i2}\\ Q_{i3}\\ Q_{i4}\\ Q_{i5}\\ Q_{i6}\\ Q_{i7}\end{array}\right]\cdot L_{ir}+T\cdot \left[\begin{array}{ccccccc}1& 1& 1& 1& 1& 1& 0\\ 0& 1& 1& 1& 1& 1& 0\\ 0& 0& 1& 1& 1& 1& 0\\ 0& 0& 0& 1& 1& 1& 0\\ 0& 0& 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]\cdot \left[\begin{array}{c}{c}_{i1}-h_2\\ c_{i2}\\ c_{i3}\\ c_{i4}\\ c_{i5}\\ c_{i6}\\ c_{i7}\end{array}\right]=\left[\begin{array}{ccccccc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{7}{2}}& {\displaystyle \frac{9}{2}}& {\displaystyle \frac{11}{2}}& {\displaystyle \frac{13}{2}}\\ 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{7}{2}}& {\displaystyle \frac{9}{2}}& {\displaystyle \frac{11}{2}}\\ 0& 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{7}{2}}& {\displaystyle \frac{9}{2}}\\ 0& 0& 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{7}{2}}\\ 0& 0& 0& 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{3}{2}}& {\displaystyle \frac{5}{2}}\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{3}{2}}\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{2}}\end{array}\right]\cdot \left[\begin{array}{c}0\\ R_{i2}\\ R_{i3}\\ R_{i4}\\ R_{i5}\\ R_{i6}\\ R_{i7}\end{array}\right]\cdot L_{ir}$$

(38)

where c_ir is the relative subsidence at both ends of the key block of the r block in the key strata of the i layer (m); and L_ir is the length of the key block r of the key strata i (m).

From the relationship between stress–strain and load–displacement [39], it can be obtained that:

$$R_{ir}=k_i{\Delta }_{ir}^3$$

(39)

where k_i is the ratio coefficient of the key strata of the i layer; ${\Delta }_{ir}$ is the subsidence of the key block of the r block in the key strata of the i layer (m); and h_i is the thickness of the ith key strata (m).

The maximum subsidence ${\Delta }_{\max }$ of the key strata is:

$${\Delta }_{\max }=h-H_{\mathrm{K}}\left(K_{\mathrm{P}}-1\right)$$

(40)

where H_K is the height of the coal seam roof from the key strata (m); and K_P is the broken expansion coefficient of the lower part of the key strata.

For the overlying rock condition of double-key strata, the maximum subsidence ${\Delta }_{\mathrm{smax}}$ of the upper key strata is:

$${\Delta }_{\mathrm{smax}}=\left(h-{\Delta }_{\max }\right)-h_3\left(K_{\mathrm{Pc}}-1\right)$$

(41)

where K_Pc is the coefficient of interlayer rock fragmentation.

Since the first rock mass in the masonry beam mechanical model is in a suspended state with no interaction with the underlying rock layers, the formula for calculating the subsidence of the key block is as follows:

$$\left[\begin{array}{c}{\Delta }_{i2}\\ {\Delta }_{i3}\\ {\Delta }_{i4}\\ {\Delta }_{i5}\\ {\Delta }_{i6}\\ {\Delta }_{i7}\end{array}\right]=\left[\begin{array}{cccccc}1& 0.5& 0& 0& 0& 0\\ 1& 1& 0.5& 0& 0& 0\\ 1& 1& 1& 0.5& 0& 0\\ 1& 1& 1& 1& 0.5& 0\\ 1& 1& 1& 1& 1& 0.5\\ 1& 1& 1& 1& 1& 1\end{array}\right]\cdot \left[\begin{array}{c}{c}_{i2}\\ c_{i3}\\ c_{i4}\\ c_{i5}\\ c_{i6}\\ c_{i7}\end{array}\right]$$

(42)

Based on extensive field production and laboratory experiments, it was observed that the quantity of fractured blocks in the key stratum does not remain constant. Therefore, the formula for calculating the subsidence of the key stratum, derived from the masonry beam theory, needs to be adjusted to account for varying conditions of fractured blocks in different key strata. At this point, the formula for calculating the subsidence of the key block is as follows:

$$\left[\begin{array}{cccc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& \cdots & {\displaystyle \frac{2n-1}{2}}\\ 0& {\displaystyle \frac{1}{2}}& \cdots & {\displaystyle \frac{2n-3}{2}}\\ ⋮& ⋮& \cdots & ⋮\\ 0& 0& \cdots & {\displaystyle \frac{1}{2}}\end{array}\right]\cdot \left[\begin{array}{c}{Q}_1\\ Q_2\\ ⋮\\ Q_n\end{array}\right]\cdot L_{it}+T\cdot \left[\begin{array}{cccc}1& 1& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \cdots & ⋮\\ 0& 0& \cdots & 0\end{array}\right]\cdot \left[\begin{array}{c}{c}_{i1}-h\\ c_{i2}\\ c_{i3}\\ ⋮\\ c_{ir}\end{array}\right]=\left[\begin{array}{cccc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& \cdots & {\displaystyle \frac{2n-1}{2}}\\ 0& {\displaystyle \frac{1}{2}}& \cdots & {\displaystyle \frac{2n-3}{2}}\\ ⋮& ⋮& \cdots & ⋮\\ 0& 0& \cdots & {\displaystyle \frac{1}{2}}\end{array}\right]\cdot \left[\begin{array}{c}0\\ k{\Delta }_{i2}^3\\ ⋮\\ k{\Delta }_{in}^3\end{array}\right]\cdot L_{it}$$

(43)

where h_I is the key strata thickness (m).

The relative subsidence of the key strata fracture block can be obtained from Formulas (38)~(43). According to the rotary compaction characteristics of the rock strata of the masonry beam structure, the arch height h_g of the trapezoidal hinged arch structure is calculated as follows.

$$h_g=\max \left\{{\displaystyle \sum _{n=t}^i{L}_{\mathrm{t}}\sin {\theta }_{\mathrm{t}}},{\displaystyle \sum _{n=j}^i{L}_{\mathrm{j}}\sin {\theta }_{\mathrm{j}}}\right\}$$

(44)

$${\theta }_t=\arcsin {\displaystyle \frac{d_t}{L_t}}$$

(45)

$${\theta }_j=\arcsin {\displaystyle \frac{d_j}{L_j}}$$

(46)

where L_t and L_j are the breaking lengths of each key block of the masonry beam (m); and ${\theta }_{\mathrm{t}}$ and ${\theta }_{\mathrm{j}}$ are the fracture rotation angles of each key block of the masonry beam (°).

5.2. Analysis of Surface Subsidence Based on Key Strata-Bearing Structure Control

The settlement of the fractured bearing structure of the key strata can be obtained from Formulas (43) and (44). The subsidence curve of the key strata can be plotted based on each settlement point. It is known that the subsidence curve of the key strata and the overlying loose layer exhibit similarities; therefore, it is assumed that their subsidence curve is as depicted in Figure 10, as shown in the key strata and surface subsidence calculation model.

In Figure 9, ${\delta }_0$ is the boundary moving angle (°); ${\beta }_0$ is the angle between the key stratum subsidence curve ${\mathrm{D}}_1^{\prime }{\mathrm{D}}_{2\mathrm{k}}^{\prime }{\mathrm{D}}_5^{\prime }$ and the surface subsidence curve D₁D_2sD₅ moving boundary in the vertical direction (°); H is the buried depth of the coal seam (m); H_Z is the key strata buried depth (m); and H_d is the depth of the connection between the key stratum subsidence curve and the surface subsidence curve and the maximum subsidence point of the subsidence curve (m).

Based on the probability integration method, it can be determined that when a mining unit volume dv is extracted at a depth of H, surface subsidence will inevitably occur, and the subsidence probability $P\left(\mathrm{d}v\right)$ is:

$$P\left(\mathrm{d}v\right)=\varsigma \left(x,y,z\right)\mathrm{d}v$$

(47)

where $\varsigma \left(x,y,z\right)$ is the density function.

At this point, the subsidence curves for the key strata and the surface are assumed to be ${\mathrm{D}}_2^{\prime }{\mathrm{O}}_1{\mathrm{D}}_4^{\prime }$ and D₂O₂D₄, respectively. These subsidence curves are connected at the boundary of movement, with the intersection point denoted as O. For nearly horizontal coal seams, it can be assumed that the subsidence curve is symmetrical about its maximum subsidence point, so only half of it needs to be studied. Let us assume D₁D₃ = l₁, and D₁O = l₂. Based on trigonometric geometry and functional relationships, we can derive:

$$l_1=L/2+H\cot {\delta }_0$$

(48)

$$l_2=l_1\sec {\beta }_0$$

(49)

Through Formula (48), the calculation formula of H_d can be obtained as follows:

$$H_{\mathrm{d}}=\left(L/2+H\cot {\delta }_0\right)\tan {\beta }_0$$

(50)

When the mining depth is dv of the mining unit ore body at H, the distance between the moving boundary of the key strata and the surface subsidence curve and the maximum subsidence point D₂D₃ is set to x_d, and ${\mathrm{D}}_2^{\prime }{\mathrm{D}}_3^{\prime }$ is set to x_k. At this time, the subsidence value ${\mathrm{D}}_2{\mathrm{D}}_{2\mathrm{s}}=w_{\mathrm{d}}\left(x_{\mathrm{d}}\right)$ of any point x_d in the surface subsidence curve D₁D_2sD₅, and the subsidence value ${\mathrm{D}}_2^{\prime }{\mathrm{D}}_{2\mathrm{k}}^{\prime }=w_{\mathrm{k}}\left(x_{\mathrm{k}}\right)$ of any point x_k in the key strata subsidence curve ${\mathrm{D}}_1^{\prime }{\mathrm{D}}_{2\mathrm{k}}^{\prime }{\mathrm{D}}_5^{\prime }$. The relationship between the surface abscissa D₂ and the key strata abscissa ${\mathrm{D}}_2^{\prime }$ is:

$${\displaystyle \frac{x_{\mathrm{d}}}{x_{\mathrm{k}}}}={\displaystyle \frac{H_{\mathrm{d}}}{H_{\mathrm{d}}-H_{\mathrm{Z}}}}$$

(51)

Substituting Formula (50) into Formula (51) results in the following:

$$x_{\mathrm{d}}={\displaystyle \frac{\left(L/2+H\cot {\delta }_0\right)\tan {\beta }_0}{\left(L/2+H\cot {\delta }_0\right)\tan {\beta }_0-H_{\mathrm{Z}}}}{x}_{\mathrm{k}}$$

(52)

Due to the deformation and failure of the overlying rock and soil layers of the key strata, the change in the damaged volume within a certain range is referred to as volume deformation, denoted as $\zeta$. Therefore, the relationship between the area under the subsidence curve of the key strata and the undisturbed subsidence curve of the surface can be expressed as:

$${\displaystyle \underset{0}{\overset{\left|D_2{D}_3\right|}{\int }}{w}_{\mathrm{d}}\left(x_{\mathrm{d}}\right)d}{x}_{\mathrm{d}}={\displaystyle \underset{0}{\overset{\left|D_2^{\prime }{D}_3^{\prime }\right|}{\int }}\zeta \left(x_{\mathrm{k}}\right)w_{\mathrm{k}}\left(x_{\mathrm{k}}\right)d}{x}_{\mathrm{k}}$$

(53)

Substituting Formula (52) into Formula (53), we obtain:

$${\displaystyle \underset{0}{\overset{\left|D_2{D}_3\right|}{\int }}{w}_{\mathrm{d}}\left(x_{\mathrm{d}}\right)d}{x}_{\mathrm{d}}={\displaystyle \underset{0}{\overset{\left|D_2^{\prime }{D}_3^{\prime }\right|}{\int }}\zeta \left(x_{\mathrm{k}}\right)\frac{\left(L/2+H\cot {\delta }_0\right)\tan {\beta }_0}{\left(L/2+H\cot {\delta }_0\right)\tan {\beta }_0-H_{\mathrm{Z}}}{x}_{\mathrm{k}}d}{x}_{\mathrm{k}}$$

(54)

In order to simplify the complexity of calculations, the overlying rock strata above the key strata are considered homogeneous, isotropic, and volume-conserving ideal bodies. Using the integral calculation method, the subsidence model of the key strata and the surface can be expressed as follows:

$$w_{\mathrm{d}}\left(x_{\mathrm{d}}\right)={\displaystyle \frac{\left(L/2+H\cot {\delta }_0\right)\tan {\beta }_0}{\left(L/2+H\cot {\delta }_0\right)\tan {\beta }_0-H_{\mathrm{Z}}}}{w}_{\mathrm{k}}\left(x_{\mathrm{k}}\right)$$

(55)

The connection between the key stratum and the surface subsidence points is the subsidence curve; that is, the subsidence curve of the key stratum can be expressed as $w_{\mathrm{k}}\left(x_{\mathrm{k}}\right)$, and the surface subsidence curve is $w_{\mathrm{d}}\left(x_{\mathrm{d}}\right)$. Through calculation, the formula for the movement and deformation of any point on the surface under the bearing structure of the key strata can be obtained, respectively, but this is not investigated further in this paper.

5.3. Prediction of Key Strata and Surface Subsidence

The 1–2 coal seam in the Longhua coal mining research area is located at a depth of 71.08 m with a mining height of 2.95 m. The working face has a length of 294 m, and the main key stratum is situated 36.35 m above the coal seam, with a thickness of 16.63 m. The sub-key stratum is located 11.31 m above the coal seam, with a thickness of 10.77 m. The thickness of the loess layer is 18.1 m, and the interlayer thickness between rock strata is 14.27 m. The average length of fractured blocks in the key stratum is 15 m. The lower part of the sub-key stratum has an average fragmentation coefficient of 1.1, while the interlayer rock strata have an average fragmentation coefficient of 1.05. The average bulk density of the bedrock layer is 26 kN/m³, and for the loess layer, it is 23 kN/m³. The boundary movement angle is set at 63°, and the angle between the sinking curve ${\mathrm{D}}_1^{\prime }{\mathrm{D}}_{2\mathrm{k}}^{\prime }{\mathrm{D}}_5^{\prime }$ of the key stratum and the subsurface sinking curves D₁D_2sD₅ connecting the boundary movement is 52° in the vertical direction.

The horizontal thrust T and the overlying load Q can be obtained by substituting the above geological conditions and related parameters into calculation Formulas (6), (15), (16), (23), (32), and (33) of the trapezoidal hinged arch structure. The subsidence of each key block can be obtained from calculation Formulas (38)~(42) of the subsidence of the key strata, and the subsidence curve can be drawn. The surface subsidence curve can be obtained by calculation Formula (55) of the key strata subsidence curve and the surface subsidence curve. Due to the complexity and difficulty of the aforementioned calculations, this study employs Matlab and Origin 2022 software to perform the calculations and create the subsidence curve, as illustrated in Figure 11.

The theoretically predicted key strata and surface subsidence curve indicate that the working face between 97.5 and 196.5 m falls within the zone of excessive mining stability, where the maximum subsidence value has been reached. Due to the influence of fractured key blocks in the key strata, the primary subsidence zones are from 0 to 97.5 m and from 196.5 to 294 m. The subsidence curves of the main key strata and the sub-key strata exhibit strong similarities; however, the supporting effect of the sub-key strata significantly reduces the subsidence value of the main key strata, with the subsidence points of the curves aligning. The surface subsidence curve is flatter compared to that of the main key strata, indicating a reduction in subsidence and suggesting that the surface plays a role in mitigating and weakening subsidence. Field measurements show a maximum surface subsidence of 1.93 m, with a subsidence coefficient of 0.65. Since this study did not consider the volumetric deformation rate $\zeta$ of the loess layer, the measured maximum subsidence is greater than the theoretical predicted value. However, the overall predicted results are similar, validating the scientific and rational basis of this study on the interaction between the key strata’s bearing structure and surface subsidence.

To obtain more scientifically sound and reasonable research conclusions, the study of the interaction between key strata movement and surface subsidence should be based on the mining conditions of shallow coal seams. Research should focus on the coordinated movement of multiple key strata and coal seams, as well as the repeated surface disturbance and subsidence under the mining of coal seam groups. This will enable better solutions to practical issues encountered in the field.

6. Conclusions

(1) Through physical similarity simulations, it was determined that the key strata load-bearing structure exhibits a shape resembling a trapezoid but with stress transfer characteristics resembling an arch, referred to as the trapezoidal hinged arch structure. Based on the key strata fracture patterns, the trapezoidal hinged arch structure is categorized into three types: a full-trapezoidal hinged arch structure, a trapezoidal-like hinged arch structure, and a semi-trapezoidal hinged arch structure. It is observed from the key strata and surface subsidence curves that the key strata load-bearing rock layers effectively reduce subsidence among them, and the surface loess layer has a restraining and weakening effect on both subsidence shape and magnitude;

(2) Based on the fracture characteristics of the key strata, a mechanical calculation model of the trapezoidal hinged arch structure was established, and the structural mechanical formulas and hinged arch height calculation formula were derived;

(3) According to the characteristics of key strata and surface subsidence, the surface subsidence calculation model controlled by the bearing structure of key strata was obtained, and the rock movement–surface subsidence linkage evolution characteristics of the bearing structure of key strata were revealed;

(4) Through theoretical predictions and field measurements, the subsidence curves for the sub-key strata, main key strata, and surface were obtained. The predicted surface subsidence values closely match the maximum measured subsidence, with the measured maximum surface subsidence being 1.93 m and a subsidence coefficient of 0.65. The surface soil layer plays a role in suppressing and reducing the amount of subsidence, validating the scientific and rational basis of this study on the interaction between key strata-bearing structures and surface subsidence.

7. Shortcomings and Innovations of This Manuscript

(1) Based on the key strata fracture characteristics observed in physical similarity simulation experiments, a trapezoidal hinged arch support structure inclined toward the key strata of the working face is proposed;

(2) Following the stress transmission conditions of key block fractures within the key strata, a mechanical computational model for the failure-bearing structure inclined toward the key strata of the working face was constructed;

(3) Utilizing a masonry beam mechanical model, calculation formulas for the subsidence of each key block within the failure-bearing structure inclined toward the key strata of the working face were derived;

(4) Drawing from the characteristics of the key strata and surface subsidence, a surface subsidence calculation model controlled by the key strata-bearing structure was established, revealing the interactive evolutionary mechanism between the key strata-bearing structure and surface subsidence.