Shear-Compression Failure Condition of Key Strata Under Elastic Support During Periodic Breakage

Abstract

The shear-compression failure of key strata leads to stair-step collapse and severe mine pressure, posing significant safety risks in coal mines. Existing theories fail to account for the boundary conditions and breaking sizes of key strata, making accurate description of shear-compression failure difficult. A periodic breakage mechanics model for key strata was developed using Timoshenko Beam and Winkler Foundation Theory, incorporating transverse shear deformation. The deflection, rotation angle, bending moment, and shear force were calculated, and a shear-compression failure criterion function f(x) was derived. The main conclusions include the following: (1) shear-compression failure is influenced by the thickness–span ratio, cohesion, internal friction angle, and elastic modulus of the key strata, but not by the elastic foundation coefficient and shear modulus; (2) shear-compression failure occurs when the thickness–span ratio reaches 0.4; (3) when the internal friction angle is 25°, 30°, 35°, or 40°, shear-compression failure does not occur if cohesion exceeds 8.0, 7.5, 7.0, or 6.5 MPa, respectively, with a larger internal friction angle corresponding to a smaller critical cohesion; (4) when cohesion is 6 MPa, 8 MPa, 10 MPa, or 12 MPa, shear-compression failure does not occur if the internal friction angle exceeds 44°, 32°, 19°, or 8°, respectively, with larger cohesion correlating to a smaller critical internal friction angle; and (5) once cohesion or internal friction angle surpasses a critical value, the failure criterion approaches a constant value, preventing failure; the elastic modulus has a greater effect on shear-compression failure than the shear modulus, with higher elastic modulus increasing the likelihood of failure.

1. Introduction

In recent years, with the gradual depletion of coal resources in eastern China, the focus of coal mining has shifted to western regions such as Inner Mongolia and Shaanxi Province [1]. These shallowly buried mining areas are characterized by shallow burial depth, thin bedrock, and thick loose layers [2]. High-intensity mining can easily cause discontinuous surface subsidence in these shallowly buried mining areas, forming stair step collapse, which may be accompanied with hydraulic support pushed down [3]. The key strata refer to the strata that control the movement of the overlying rock strata after mining, either locally or all the way to the surface [4]. Numerous studies have shown that surface step subsidence is caused by shear-compression failure of the key strata [5,6,7]. Therefore, it is essential to understand the conditions for shear-compression failure of the key strata and to analyze the influencing factors.

Research on the conditions for shear-compression failure of the key strata needs to consider three aspects: boundary conditions [8], transverse shear deformation (caused by breaking size) [9], and failure criteria [10]. Current research by scholars can be categorized into three types: (1) Fixed constraint for the key strata, ignoring transverse shear deformation, using a single tensile failure criterion: For example, Yin et al. [11] used the Euler Beam without considering transverse shear deformation, and the boundary was equivalent to a fixed constraint. They established initial and periodic breakage models of the key strata and provided the initial and periodic breakage spacing. Similarly, Wang et al. [12] treated the boundary conditions as fixed, simply supported, and free, and based on elastic thin plate theory, they established a mechanical model of key strata fracture. They analyzed the fracture characteristics of the key strata under tensile failure in a variable length working face, concluding that a large working face first forms an “O”-shaped fracture around the plate edge and then a planar “X”-shaped fracture in the middle, while a small working face does the opposite. Cao et al. [13], based on elastic thin plate theory, studied the impact of island face mining on impact mine pressure under thick hard rock layers, concluding that sub-key strata fractures form “vertical O”-shaped fractures, and giant thick primary key strata form large-scale “vertical O + X” layered fractures. Wang et al. [14], based on thick plate theory, established initial and periodic breakage mechanical models for the basic roof of steep coal seams, proposing initial fracture “V + Y” patterns and periodic breakage “quadrilateral” patterns for steep coal seams. (2) Elastic support of the key strata, ignoring transverse shear deformation, using a single tensile failure criterion: For example, Zhou et al. [15] adopted the Euler Beam elastic foundation beam theory to establish a mechanical model of fracture for thick hard igneous rock (key strata), analyzing the initial and periodic breakage size and location under tensile failure conditions. Similarly, Xie et al. [16], based on the finite difference method, established a mechanical model of tensile fracture for key strata thin plates supported by elastic coal rock bodies during filling mining. They analyzed the effects of parameters such as the elastic foundation coefficient of the filling body, the elastic boundary foundation coefficient, and the basic roof thickness on the main bending moment distribution and fracture characteristics of the basic roof, concluding that under dense filling, the basic roof undergoes incomplete fracture, i.e., the basic roof fractures only on the upper surface of the leading coal wall on the long side or the lower surface in the middle. Similarly, Xie et al. [17] used the finite difference theory to establish mechanical models for initial and periodic breakage of basic roof thin plates supported by elastic coal rock bodies, concluding that the key strata initially fracture in “O + X” patterns with three types of fracture sequences, and fracture periodically with two types of sequences. He et al. [18] used the finite difference principle to establish a mechanical model of initial fracture for key strata thin plates under elastic-plastic foundation boundaries, analyzing factors and weight relationships affecting fracture location, sequence, and regional fracture patterns, concluding that the basic roof initially fractures in “O + X” patterns. Chen et al. [19] used the finite difference method to analyze initial fracture characteristics of basic roof thin plates with elastic foundation boundaries on one side of the goaf (coal pillar), concluding that the fracture pattern is asymmetrical “O + X” or “U + X”. (3) Simplified boundary conditions, considering transverse shear deformation, and using multiple failure criteria: For example, Zuo et al. [20]. established a mechanical model of key strata fracture with four fixed constraints based on thick plate theory. They analyzed three failure modes: tensile fracture, tensile-shear fracture, and shear fracture, corresponding to three structural types of the key strata: masonry beam structure, layered fracture, and step rock beam. Similarly, Shi et al. [21] used a plane strain model to establish a mechanical model of key strata fracture with both edges fixed. They provided the criteria for three failure modes: shear-compression failure along the rock fracture surface, shear failure along a weak layer of the rock, and tensile failure at both ends. The internal force state within the key stratum is the main cause of its failure. However, there are few studies that combine the internal force derived from Timoshenko beam theory with the Mohr–Coulomb criterion to investigate shear-compression failure in key strata. This paper integrates Timoshenko beam theory with the Mohr–Coulomb failure criterion to identify the dominant factors controlling shear-compression failure in key strata.

None of the above studies simultaneously consider elastic support, transverse shear deformation, and shear-compression failure. In existing studies, no research combines elastic support, transverse shear deformation, and a Mohr–Coulomb-based shear-compression criterion to model the periodic failure of thick key strata. Moreover, studies integrating the Timoshenko beam model with key strata are rare, and the mechanism by which shear-compression failure in key strata leads to key stratum fracturing and consequent mining disasters remains insufficiently investigated. Therefore, this paper aims to use the Timoshenko Beam model on a Winkler foundation to establish a periodic breakage mechanical model of the key strata considering the impact of transverse shear deformation. The model will solve for deflection, rotation, bending moment, and shear force of the key strata periodic breakage. Then, the Mohr–Coulomb criterion will be used to construct a shear-compression failure criterion function f(x) for thick–hard key layers. The study will investigate the characteristics of shear-compression failure in key strata under different geometric and mechanical parameters and different mechanical parameters of supports, providing techniques to control shear-compression failure and prevent strong mine pressure disasters.

2. Periodic Breakages of a Key Stratum During Mining of the Working Face

2.1. Boundary Conditions During Periodic Breakages of a Key Stratum

After the initial fracture of the key strata, periodic breakages occur as the working face advances a certain distance, causing periodic strong mining pressure and potentially dynamic disasters throughout the mining process. Taking a typical working face in the Tashan Coal Mine as an example, there are three key strata above the working face: K1, K2, and K3. Field monitoring of the movement of the key strata and the resistance of the working face supports revealed the following: (1) As the range of the goaf area behind the working face increases, the K1 key strata experience an initial fracture, causing pressure on the working face, as shown in Figure 1a. After fracturing, the K1 key strata form a cantilever state, creating a low-level composite cantilever beam structure with the layers above and below it. During the advancement of the working face through this composite cantilever beam structure, the working face remains under pressure. (2) As shown in Figure 1b, with the continued mining of the working face, the K2 key strata undergo periodic breakages, forming a cantilever beam and masonry beam composite structure. Both the fracture spacing and strength of the rock layers increase. The instability of the mid-level cantilever beam structure affects the underlying low-level composite cantilever beam structure, causing rotational forces on the working face, leading to relatively strong periodic pressure on the working face. (3) As shown in Figure 1c, when the K3 key strata fracture, the large fracture spacing and significant space result in a high intensity of energy release and a wide impact range. This causes instability and rotation of the underlying masonry beam structure and composite cantilever beam structure, collectively affecting the working face and leading to very strong mining pressure on the working face. (4) As shown in Figure 1d, with the continued mining of the working face, the K3 key strata undergo periodic breakages and instability, affecting the mid- and low-level rock layers and causing dynamic disasters in the working face.

2.2. Failure Characteristics of a Key Stratum

When using beam theory to establish a fracture model of the overburden in a mining area, it is necessary to consider the size characteristics of the key strata. This involves determining the impact of transverse shear deformation on the accuracy of beam analysis and the failure criteria employed. Zuo et al. [20] compiled the fracture sizes of key strata in some mining areas, as detailed in

Table 1. Statistics of periodic compression failure modes of the key stratum.

Mine	Rock Property	Thickness/m	Breaking Step Distance/m	Rock Structure	Thickness-to-Span Ratio
Bulianta Coal Mine 22303 Workface	Coarse-grained sandstone	10.10	25.70	Masonry beam	0.40
Xinyuan Mine 3108 Workface	Fine-grained sandstone	2.00	8.00	Masonry beam	0.25
Haragou Coal Mine 12 Upper 101 Workface	Fine-grained sandstone	2.98	8.00	Masonry beam	0.37
Tashan Mine 8108 Workface	Medium-grained sandstone	11.00	52.75	Masonry beam	0.21
Daliuta Mine 20604 Workface	Sandstone	16.00	14.60	Stepped rock beam	1.10
Daliuta Mine C202 Workface	Sandstone	17.30	9.00	Stepped rock beam	1.92
Wulanmulun 12307 Workface	Siltstone	14.82	13.00	Stepped rock beam	1.14
Huojitu Mine 21305 Workface	Medium-grained sandstone	11.60	12.40	Stepped rock beam	0.94
Rongda Coal Mine No. 6 coal heading working face	Medium-grained sandstone	18.22	18.90	Stepped rock beam	0.96
Rongda Coal Mine No. 6 coal heading Workface	Fine-grained sandstone	7.88	6.88	Stepped rock beam	1.15
Ningtiaota Mine N1200 Workface	Coarse-grained sandstone	15.00	11.78	Stepped rock beam	1.25
Ningtiaota Mine N1208 Workface	Coarse-grained sandstone	13.87	15.80	Stepped rock beam	0.88
Ningtiaota Mine N1200-1 Workface	Medium-grained sandstone and Siltstone	18.00	18.00	Stepped rock beam	1.00
Ningtiaota Mine N1212 Workface	Coarse-grained sandstone and Siltstone	15.60	14.28	Stepped rock beam	1.09

.

The thickness-to-span ratio in the table refers to the ratio of the key strata thickness to the breaking step distance. When this ratio exceeds 0.2, the influence of transverse shear deformation on the key strata must be considered. Thus, most key strata’s periodic breakages should be analyzed using beams that account for transverse shear deformation. Moreover, when the thickness-to-span ratio exceeds 0.4, the key strata transition from tensile failure to shear-compression failure. Therefore, this paper adopts the Timoshenko beam theory, introducing a variable to describe the transverse shear behavior of the beam, enabling a more accurate description of the periodic breakage of the key strata. Additionally, the commonly used Mohr–Coulomb criterion is employed to establish the shear-compression failure conditions for the key strata.

3. Key Strata’s Periodic Breakage Model via Timoshenko Beam Theory

In this chapter, the Timoshenko beam on Winkler foundation is used to establish a mechanical model for the periodic breakage of the key strata. The deflection, rotation, bending moment, and shear force of the key strata during periodic breakages are provided, laying the foundation for calculating the shear-compression failure.

3.1. Periodic Breakage Mechanical Model of Thick–Hard Roof

During the periodic breakage of the key strata, one end is clamped by the overlying and underlying rock layers, while the other end is free, with the cantilever position subjected to the gravity of the overlying rock layers within a certain range. To accurately characterize the boundary conditions and stress state of the key strata, a Timoshenko beam mechanical model is developed, as depicted in Figure 2. In this model, AB represents the cantilever section, while segment BC corresponds to the clamped section. The outermost point A of the cantilever section is designated as the origin O of the coordinate system. The x-axis coincides with the central axis of the key strata, extending to the right, and the y-axis is oriented downward. The fundamental rock layer beneath the key strata is assumed to behave as an elastic body.

During the periodic breakage of the key strata, its thickness-to-span ratio ranges between 1/5 and 1/10, rendering shear deformation significant. Unlike the classical assumption that the cross-section remains perpendicular to the central axis after deformation, Timoshenko beam theory incorporates transverse shear deformation to capture the transverse shear behavior of the key strata. In this theory, in addition to deflection w, a variable ψ is introduced to represent the cross-sectional rotation angle, effectively describing the shear behavior of the beam.

During periodic breakage, one end of the key strata in the goaf area is free, while the other end is elastically supported by the underlying rock layers. The Winkler foundation beam theory models the elastic relationship in which the support force is directly related to the deflection. Consequently, this paper applies Timoshenko beam theory on a Winkler foundation. The micro-element of a Timoshenko beam on Winkler foundation is illustrated in Figure 3 below, with its corresponding equilibrium equations provided in Equation (1).

$$\left\{\begin{array}{l}-{\displaystyle \frac{d}{dx}}\left[\kappa GA\left({\displaystyle \frac{dw}{dx}}-\psi \right)\right]+kw=q\\ -{\displaystyle \frac{d}{dx}}EI\left({\displaystyle \frac{d\psi }{dx}}\right)-\kappa GA\left({\displaystyle \frac{dw}{dx}}-\psi \right)=m\end{array}\right.$$

(1)

In the formula, k = k₀b, where k₀ is the subgrade coefficient, and b is the width of the beam; E is the elastic modulus; I is the moment of inertia of cross-section; G is the shear modulus; A is the cross-sectional area; κ is the shear correction factor, which depends on the shape of the section (here taken as 5/6); q represents the vertical distributed load; and m is the distributed bending moment, where w represents the deflection of the beam and ψ denotes the rotation angle of the beam.

3.2. Timoshenko Beam on Winkler Foundation

The disturbance caused by coal mining is confined to a limited area, with the boundary far from the working face treated as a fixed boundary. When the elastic support section of a beam is infinitely long, one end of the beam can be considered fixed. At this fixed point, both the deflection w and the cross-sectional rotation angle ψ are assumed to be zero.

The boundary limitation and stress status of the model for the crucial strata during periodic breakage are complex, which makes it challenging to directly derive its deflection and rotation equations. As shown in Figure 4, the mechanical model for periodic breakage is divided into the cantilever section AB and the clamped section BC. The BC section is addressed using the superposition method to tackle this complexity.

The beam section AB is subjected to an evenly distributed load q along its length, with a concentrated force $Q_0$ and a moment $M_0$ acting at point B. The restrained section BC is treated as two parts: one being an infinitely long Timoshenko beam on an elastic foundation with a concentrated force $Q_0$ at one end and the other as an infinitely long Timoshenko beam on an elastic foundation with a moment $M_0$ at one end. In all three models, point A is chosen as the coordinate origin O, with the x-axis aligned along the central axis of the beam, pointing to the right, and the y-axis directed downward.

According to the generalized displacement and deformation equations and internal force equations of the Timoshenko beam theory, in the cantilever section AB of the key strata, the internal force solution is as follows:

$$\left\{\begin{array}{l}{\omega }_1={\displaystyle \frac{q{x}^4}{24D}}+a_3{x}^3+a_2{x}^2+a_{1}x+a_0\\ {\psi }_1={\displaystyle \frac{q{x}^3}{6D}}+3a_3{x}^2+2a_{2}x+{\displaystyle \frac{qx}{C}}+a_1+{\displaystyle \frac{6D{a}_3}{C}}\\ M_1=-{\displaystyle \frac{q{x}^2}{2}}-6a_{3}x-2a_{2}D-{\displaystyle \frac{qD}{C}}\\ Q_1=-qx-6D{a}_3\end{array}\right.$$

(2)

In the equation, $C=\kappa GA$, $D=EI$.

At the free end of the cantilever section of the key strata, $M_1{|}_{x=0}=0$, $Q_1{|}_{x=0}=0$. These are solved as follows:

$$a_2=-{\displaystyle \frac{q}{2C}},\hspace{1em}{a}_3=0$$

(3)

Reference [1] provides the four inherent mechanic equations for an infinitely distance elastic foundation Timoshenko beam subjected to a shear force $Q_0$ at one side:

$$\left\{\begin{array}{l}{\omega }_2={\displaystyle \frac{Q_0}{4D\beta {\left({\alpha }^2+{\beta }^2\right)}^2}}{e}^{-\alpha \left(x-L\right)}\left[2\alpha \beta \cos (\beta x-\beta L)-\left({\alpha }^2-{\beta }^2\right)\sin (\beta x-L)\right]\\ {\psi }_2={\displaystyle \frac{Q_0}{4D\beta \left({\alpha }^2+{\beta }^2\right)}}{e}^{-\alpha \left(x-L\right)}\left[\beta \cos (\beta x-\beta L)+\alpha \sin (\beta x-\beta L)\right]\\ Q_2=-{\displaystyle \frac{Q_0}{\beta }}{e}^{-\alpha \left(x-L\right)}\left[\beta \cos (\beta x-\beta L)+\alpha \sin (\beta x-\beta L)\right]\\ {\mathrm{M}}_2=-{\displaystyle \frac{Q_0}{\beta }}{e}^{-\alpha \left(x-L\right)}\sin (\beta x-\beta L)\end{array}\right.$$

(4)

In the equation, $\alpha =\sqrt{\sqrt{{\displaystyle \frac{k}{4D}}}+{\displaystyle \frac{k}{4C}}},\beta =\sqrt{\sqrt{{\displaystyle \frac{k}{4D}}}-{\displaystyle \frac{k}{4C}}}$.

Reference [22] also provides four inherent mechanic equations for an infinitely distance elastic foundation Timoshenko beam concentrated to a moment $M_0$ at one side:

$$\left\{\begin{array}{l}{\omega }_3={\displaystyle \frac{M_0}{D\beta \left({\alpha }^2+{\beta }^2\right)}}{e}^{-\alpha \left(x-L\right)}\left[-\beta \cos (\beta x-\beta L)+\alpha \sin (\beta x-\beta L)\right]\\ {\psi }_3={\displaystyle \frac{M_0}{D\beta \left({\alpha }^2+{\beta }^2\right)}}{e}^{-\alpha \left(x-L\right)}\left[2\alpha \beta \cos (\beta x-\beta L)+\left({\alpha }^2-{\beta }^2\right)\sin (\beta x-\beta L)\right]\\ Q_3=-{\displaystyle \frac{M_0\left({\alpha }^2+{\beta }^2\right)}{\beta }}{e}^{-\alpha \left(x-L\right)}\sin (\beta x-\beta L)\\ M_3=M_0{e}^{-\alpha \left(x-L\right)}\left[\cos (\beta x-\beta L)+{\displaystyle \frac{\alpha }{\beta }}\sin (\beta x-\beta L)\right]\end{array}\right.$$

(5)

Based on the force balance, we can obtain:

$$\left\{\begin{array}{l}{Q}_0=qL\\ M_0=-{\displaystyle \frac{q{L}^2}{2}}\end{array}\right.$$

(6)

According to Equation (2), at point $x=L$ in the cantilever section of the key strata, solutions are obtained as follows:

$$\left\{\begin{array}{l}{\omega }_1={\displaystyle \frac{q{L}^4}{24D}}-{\displaystyle \frac{q{L}^2}{2C}}+a_{1}L+a_0\\ {\psi }_1={\displaystyle \frac{q{L}^3}{6D}}+a_1\\ M_1=-{\displaystyle \frac{q{L}^2}{2}}\\ Q_1=-qL\end{array}\right.$$

(7)

According to the continuity conditions, at $x=L$, we have the following conditions:

$$\left\{\begin{array}{l}{\omega }_1={\omega }_2+{\omega }_3\\ {\psi }_1={\psi }_2+{\psi }_3\end{array}\right.$$

(8)

Hence, the solution leads to:

$$\begin{gathered} a_0={\displaystyle \frac{qL\left\{\left.q{L}^{3}C{({\alpha }^2+{\beta }^2)}^2+8q{L}^{2}C({\alpha }^2+{\beta }^2)+6({\alpha }^2+{\beta }^2)\left[C+\frac{2D({\alpha }^2+{\beta }^2)}{5}\right]qL+4\alpha C\right\}\right.}{8CD{({\alpha }^2+{\beta }^2)}^2}} \\ {a}_1=-{\displaystyle \frac{qL+4q{L}^2}{4D({\alpha }^2+{\beta }^2)}}-{\displaystyle \frac{q{L}^3}{6D}} \end{gathered}$$

(9)

The four mechanical parameters of the Timoshenko beam on elastic foundation, derived by integrating Equations (2), (4)–(6) and (9), are as follows:

$$\omega =\left\{\begin{array}{l}\left\{\begin{array}{l}{\displaystyle \frac{q{x}^4}{24D}}-{\displaystyle \frac{q{x}^2}{2C}}-\left({\displaystyle \frac{qL+4q{L}^2}{4D({\alpha }^2+{\beta }^2)}}+{\displaystyle \frac{q{L}^3}{6D}}\right)x+\\ {\displaystyle \frac{qL\left\{\left.q{L}^{3}C{({\alpha }^2+{\beta }^2)}^2+8q{L}^{2}C({\alpha }^2+{\beta }^2)+6({\alpha }^2+{\beta }^2)\left[C+\frac{2D({\alpha }^2+{\beta }^2)}{5}\right]qL+4\alpha C\right\}\right.}{8CD{({\alpha }^2+{\beta }^2)}^2}}\end{array}\right.,0\le x\le L\\ \left\{\begin{array}{l}{\displaystyle \frac{qL}{4D\beta {\left({\alpha }^2+{\beta }^2\right)}^2}}{e}^{-\alpha \left(x-L\right)}\left[2\alpha \beta \cos (\beta x-\beta L)-\left({\alpha }^2-{\beta }^2\right)\sin (\beta x-L)\right]+\\ {\displaystyle \frac{q{L}^2}{2D\beta \left({\alpha }^2+{\beta }^2\right)}}{e}^{-\alpha \left(x-L\right)}\left[\beta \cos (\beta x-\beta L)-\alpha \sin (\beta x-\beta L)\right]\end{array}\right.,x>L\end{array}\right.$$

(10)

$$\psi =\left\{\begin{array}{l}{\displaystyle \frac{q{x}^3}{6D}}-{\displaystyle \frac{qL+4q{L}^2}{4D({\alpha }^2+{\beta }^2)}}-{\displaystyle \frac{q{L}^3}{6D}},0\le x\le L\\ \left\{\begin{array}{l}{\displaystyle \frac{qL}{4D\beta \left({\alpha }^2+{\beta }^2\right)}}{e}^{-\alpha \left(x-L\right)}\left[\beta \cos (\beta x-\beta L)+\alpha \sin (\beta x-\beta L)\right]-\\ {\displaystyle \frac{q{L}^2}{2D\beta \left({\alpha }^2+{\beta }^2\right)}}{e}^{-\alpha \left(x-L\right)}\left[2\alpha \beta \cos (\beta x-\beta L)+\left({\alpha }^2-{\beta }^2\right)\sin (\beta x-\beta L)\right]\end{array}\right.\end{array}\right.,x>L$$

(11)

$$M=\left\{\begin{array}{l}-{\displaystyle \frac{q{x}^2}{2}},0\le x\le L\\ -{\displaystyle \frac{q{L}_1}{\beta }}{e}^{-\alpha \left(x-L\right)}\sin (\beta x-\beta L)+{\displaystyle \frac{q{L}^2\left({\alpha }^2+{\beta }^2\right)}{2\beta }}{e}^{-\alpha \left(x-L\right)}\sin (\beta x-\beta L),x>L\end{array}\right.$$

(12)

$$Q=\left\{\begin{array}{l}-qx,0\le x\le L\\ -{\displaystyle \frac{q{L}_1}{\beta }}{e}^{-\alpha \left(x-L\right)}\left[\beta \cos (\beta x-\beta L)+\alpha \sin (\beta x-\beta L)\right]+{\displaystyle \frac{q{L}^2\left({\alpha }^2+{\beta }^2\right)}{2\beta }}{e}^{-\alpha \left(x-L\right)}\sin (\beta x-\beta L),x>L\end{array}\right.$$

(13)

4. Shear-Compression Failure Condition of Key Strata

4.1. Stress of the Key Stratum

Shear failure in rock layers is a common form of structural failure occurring under shear stress or shear force [23]. Shear failure exhibits the following characteristics: (1) Failure morphology: Shear failure often presents as a shear direction shear fracture surface, where the structure exhibits a fracture plane in the direction of the shear force [24]. (2) Failure modes: Shear failure can be categorized into bending failure and shear failure [25]. Under large shear stress, structures typically undergo bending failure, characterized by excessive bending leading to failure [26]. Under favorable support conditions and high shear strength, shear failure tends to manifest as shear failure, where a portion of the structure breaks away.

Axial stresses at different positions of the key strata are as follows:

$${\sigma }_x=-Ey{\displaystyle \frac{d\psi }{dx}}$$

(14)

Substituting Equation (11) into Equation (14) yields:

$${\sigma }_x=\left\{\begin{array}{l}-Ez{\displaystyle \frac{q{x}^2}{2D}},0\le x\le L\\ -Ey\left\{-{\displaystyle \frac{qL}{\beta }}{e}^{-\alpha \left(x-L\right)}\sin (\beta x-\beta L)-{\displaystyle \frac{q{L}^2}{2}}{e}^{-\alpha \left(x-L\right)}\left[\cos (\beta x-\beta L)+{\displaystyle \frac{\alpha }{\beta }}\sin (\beta x-\beta L)\right]\right\},x>L\end{array}\right.$$

(15)

According to (14), for the beam model with elastic supports at both ends, the beam at y = h/2, x = $L+{\displaystyle \frac{\arctan \left(\frac{\beta }{{\alpha }^{2}L+{\beta }^{2}L+2\alpha }\right)}{\beta }}$ is under compression, with the maximum principal stress being:

$${\sigma }_{t\max }={\displaystyle \frac{qhEL{e}^{-\frac{\alpha }{\beta }\arctan \left(\frac{2\beta }{{\alpha }^{2}L+{\beta }^{2}L+2\alpha }\right)}\left[4+\left({\alpha }^2+{\beta }^2\right)L^2+4\alpha L\right]}{4\sqrt{\frac{\left({\alpha }^2+{\beta }^2\right)\left[4+\left({\alpha }^2+{\beta }^2\right)L^2+4\alpha L\right]}{{\left[\left({\alpha }^2+{\beta }^2\right)L+2\alpha \right]}^2}\left[\left({\alpha }^2+{\beta }^2\right)L+2\alpha \right]}D}}$$

(16)

4.2. Conditions for Shear-Compression Failure of a Key Stratum

According to the Coulomb strength criterion [27], the angle between the fracture plane of the rock and the direction of the maximum principal stress is $\theta =45°-\phi /2$ (as shown in Figure 3), and $\phi$ represents the internal friction angle. Therefore, the compressive stress and shear stress on the failure surface are, respectively:

$$\left\{\begin{array}{l}{\sigma }_{\theta }={\displaystyle \frac{1-\sin \phi }{2}}{\displaystyle \frac{qhEL{e}^{-\frac{\alpha }{\beta }\arctan \left(\frac{2\beta }{{\alpha }^{2}L+{\beta }^{2}L+2\alpha }\right)}\left[4+\left({\alpha }^2+{\beta }^2\right)L^2+4\alpha L\right]}{4\sqrt{\frac{\left({\alpha }^2+{\beta }^2\right)\left[4+\left({\alpha }^2+{\beta }^2\right)L^2+4\alpha L\right]}{{\left[\left({\alpha }^2+{\beta }^2\right)L+2\alpha \right]}^2}\left[\left({\alpha }^2+{\beta }^2\right)L+2\alpha \right]}D}}\\ {\tau }_{\theta }={\displaystyle \frac{\cos \phi }{2}}{\displaystyle \frac{qhEL{e}^{-\frac{\alpha }{\beta }\arctan \left(\frac{2\beta }{{\alpha }^{2}L+{\beta }^{2}L+2\alpha }\right)}\left[4+\left({\alpha }^2+{\beta }^2\right)L^2+4\alpha L\right]}{4\sqrt{\frac{\left({\alpha }^2+{\beta }^2\right)\left[4+\left({\alpha }^2+{\beta }^2\right)L^2+4\alpha L\right]}{{\left[\left({\alpha }^2+{\beta }^2\right)L+2\alpha \right]}^2}\left[\left({\alpha }^2+{\beta }^2\right)L+2\alpha \right]}D}}\end{array}\right.$$

(17)

According to the Mohr–Coulomb strength criterion [28], the mechanical criteria for rock mass failure under compressive-shear conditions:

$$f(x)={\displaystyle \frac{{\tau }_{\theta }}{{\sigma }_{\theta }\tan \phi +c}}-1>0$$

(18)

In the equation, c is the cohesive strength.

5. Analysis of Factors Influencing Key Strata Shear Failure

The shear failure of the key stratum will cause step-like subsidence of the working face, thereby inducing strong mining pressure and even dynamic disasters [5]. By analyzing the influencing factors of shear failure in the key stratum, we can understand the conditions under which shear failure occurs, thereby preventing shear failure from happening.

5.1. Research Plans

According to Equations (17) and (18) above, the key stratum transitions from tensile failure to shear failure mode, which correlates with the elastic modulus, shear modulus, elastic foundation stiffness, thickness, and cantilever length (strength) of the key stratum. These factors can be categorized into three types: the first type includes geometric parameters of the key stratum, specifically cantilever length and thickness, which can be controlled through methods such as hydraulic fracturing and blasting; the second type comprises mechanical parameters such as elastic modulus, shear modulus, cohesive strength, and internal friction angle; the third type involves mechanical parameters of the support body, reflecting the elastic foundation stiffness of the underlying rock strata. In coal mining, hard roofs are often composed of high-strength rock materials such as sandstone, limestone, and conglomerate. The elastic modulus E and shear modulus G of these materials can vary over a wide range, both of which have significant effects on their failure behavior.

To determine reasonable values for the thickness and cantilever length of the key stratum, as well as for parameters such as elastic modulus, shear modulus, cohesive strength, and internal friction angle, a statistical analysis of these parameter ranges was conducted [29]. Typically, the thickness of the key stratum ranges from 10 to 30 m, while the elastic foundation stiffness of coal and rock masses ranges from 0.2 to 1 GN/m. Additionally, the elastic modulus, shear modulus, ultimate cantilever length, cohesive strength, and internal friction angle were analyzed. To address geometric parameters, mechanical parameters, and support body mechanical parameters, a single-variable method was employed, resulting in the establishment of four research scenarios, as detailed in

Table 2. Research plans.

No.	Cantilever Length L and Thickness h	Elastic Modulus E and Shear Modulus G	Cohesive Strength c and Internal Friction Angle φ	Elastic Foundation Stiffness k/GN/m
Plan I	L = 0~150 m, h = 0~150 m	E = 4 GPa, G = 1.67 GPa	φ = 30°, c = 6 MPa	1.00
Plan II	L = 0~150 m, h = 0~150 m	E = 4 GPa, G = 1.67 GPa	φ = 30°, c = 6 MPa	0.25, 0.50, 0.75, 1.00
Plan III	L = 50 m, h = 20 m	E = 4 GPa, G = 1.67 GPa	φ = 0~90°, c = 0~10 MPa	0.75
Plan IV	L = 50 m, h = 20 m	E = 0~90 GPa, G = 0~45 GPa	φ = 30°, c = 6 MPa	0.75

.

5.2. Geometric Parameters

According to Plan I, with an elastic foundation stiffness of 1.0 GN/m, elastic modulus of 4.0 GPa, shear modulus of 1.67 GPa, uniform distributed load of 1 MN/m² on the overlying key strata, cohesion of 6.0 MPa, and friction angle of 30°, the values of the shear failure criterion function f(x) are provided for various cantilever lengths and thicknesses, as shown in Figure 5a,c. The ratio of key strata thickness to cantilever length is denoted as v, where v = h/L, and curves for v₀ = 0.40, v₁ = 0.30, v₂ = 0.35, v₃ = 0.47, v₄ = 0.57, and v₅ = 1.08 are marked on Figure 5.

Based on Figure 5: (1) When the v, ratio of key strata thickness to cantilever length, is constant for a key stratum, the value of f(x) remains constant, indicating that whether shear failure occurs in the key strata depends on v. (2) When v = 0.40, f(x) = 0, indicating that the key strata undergo shear failure. (3) Using v = 0.40 as a boundary, when v ≤ 0.40, f(x) decreases rapidly with increasing v; when v > 0.40, the increase in f(x) with decreasing v is slower. When v = 1.08, the value of f(x) no longer changes.

To further visually explore the influence of thickness and cantilever length on key strata shear failure, profiles were analyzed at thicknesses of 10 m, 20 m, 30 m, and 40 m for their effects on cantilever length in Figure 5b, where h = 10 m, 20 m, 30 m, and 40 m. Similarly, the effects of thickness on key strata shear failure were analyzed at lengths of 20 m, 30 m, 40 m, and 50 m in Figure 5c, along L = 20 m, 30 m, 40 m, and 50 m. The results are shown in Figure 5d.

From Figure 5b: (1) The value of criterion function f(x) for key strata shear failure increases with increasing cantilever length, reaching critical failure when the cantilever length reaches its maximum. (2) When thicknesses are 10 m, 20 m, 30 m, and 40 m, the critical cantilever lengths for shear failure in the key strata are 25 m, 50 m, 75 m, and 100 m, respectively, corresponding to the ratios v = 0.25, 0.40, 0.40, and 0.40. This indicates that as the key strata thickness increases, the ratio quickly stabilizes at 0.40, suggesting shear failure occurs when the thickness exceeds 10 m.

From Figure 5d: (1) The criterion value f(x) for key strata shear failure decreases with increasing thickness, indicating that shear failure occurs when thickness reaches its critical value. (2) For cantilever lengths of 20 m, 30 m, 40 m, and 50 m, with thicknesses of 10 m, 20 m, 30 m, and 40 m, respectively, and corresponding thickness-to-cantilever ratios v = 0.25, 0.40, 0.40, and 0.40, key strata shear failure occurs. This shows that as the key strata thickness increases, the ratio stabilizes at 0.40, indicating shear failure occurs when the thickness exceeds 10 m.

In conclusion, the ratio of the key strata, known as the thickness–span ratio, is the determining factor for shear-compression failure in the key strata. When selecting an elastic foundation stiffness of 1 GN/m, an elastic modulus of 4.0 GPa, and a shear modulus of 1.67 GPa, with the key strata subjected to a uniformly distributed load of 1 MN/m², shear-compression failure typically occurs when the ratio reaches 0.40.

5.3. Elastic Foundation Stiffness

To explore the shear failure patterns in the key strata under different foundation stiffnesses, the conditions of Plan II are studied, selecting elastic foundation stiffness values of 0.25, 0.50, 0.75, and 1.00 GN/m. The values of the shear failure criterion function f(x) are shown in Figure 6.

From Figure 6a, we can observe the following: For different elastic foundation stiffnesses, the value of the criterion function f(x) remains constant for the ratio v. Overall, the effect of elastic foundation stiffness on the shear failure criterion function f(x) is negligible.

Further investigation of the effect of elastic foundation stiffness on shear failure for a cantilever length of 50 m is shown in Figure 6b. (1) When the cantilever length is 50 m and the key strata thickness is 20 m, shear failure occurs with a ratio v = 0.40. The influence of elastic foundation stiffness on shear failure is not significant. (2) Although higher elastic foundation stiffness slightly increases the f(x) value, indicating a greater tendency for shear failure, the difference in f(x) values is minimal.

When the key strata thickness is consistently 20 m, the impact of elastic foundation stiffness on shear failure is further explored, as shown in Figure 6c. The results indicate that: (1) Across different elastic foundation stiffnesses, a key strata thickness of 20 m with a cantilever length of 50 m results in shear failure at a thickness-to-span ratio v = 0.40. This indicates that the influence of elastic foundation stiffness on shear failure is negligible. (2) As the elastic foundation stiffness increases, the thickness-to-span ratio for shear failure slightly decreases, but the extent of this decrease is minor.

In conclusion, the influence of elastic foundation stiffness on shear failure in the key strata is negligible.

5.4. Intrinsic Mechanical Parameters

Equations (17) and (18) show that the shear-compression failure of the key layer is also related to the mechanical parameters intrinsic to the rock layer. These parameters include the internal friction force, internal friction angle, elastic modulus, and shear modulus. In this study, we group internal friction force and internal friction angle together, while the elastic modulus and shear modulus form another group. According to Plan III, with a key layer cantilever length of 50 m, a thickness of 20 m, and an elastic foundation stiffness of 0.75 GN/m, E = 4 GPa, and G = 1.67 GPa, Equations (16) and (17) are used to calculate the variations in the shear-compression failure criterion function f(x) under different combinations of internal friction force and internal friction angle. The results are shown in Figure 7a,c, leading to the following conclusions: (1) Shear-compression failure of the key layer tends to occur more easily under lower internal friction angles and internal friction forces. (2) When the ratio of internal friction force to internal friction angle is constant, the value of the shear-compression failure criterion function f(x) remains essentially unchanged, indicating that whether shear-compression failure occurs is related to the ratio c/φ. (3) When the ratio of internal friction force to internal friction angle is of less than −5, f(x) > 0, making shear-compression failure likely. When it is greater than −5, f(x) < 0, and shear-compression failure does not occur.

To further analyze the influence of individual factors—internal friction force and internal friction angle—on the shear-compression failure of the key layer, we obtained the values of the shear-compression failure criterion function f(x) under different internal friction forces at angles of 25°, 30°, 35°, and 40°, as shown in Figure 7b. Additionally, we obtained the values of the shear-compression failure criterion function f(x) under internal friction forces of 6 MPa, 8 MPa, 10 MPa, and 12 MPa at different internal friction angles, as shown in Figure 7d.

From Figure 7b,d, we can deduce the following:

(1) When the internal friction angles are 25°, 30°, 35°, and 40°, the key layer’s shear-compression failure criterion function f(x) < 0 if the internal cohesion of the rock layer exceeds 8.0, 7.5, 7.0, and 6.5, respectively. This indicates that the key layer does not undergo shear-compression failure. Additionally, the larger the internal friction angle, the lower the corresponding critical internal cohesion. Similarly, when the internal cohesion values are 6 MPa, 8 MPa, 10 MPa, and 12 MPa, the key layer does not undergo shear-compression failure if the internal friction angles of the rock layer exceed 44°, 32°, 19°, and 8°, respectively. Moreover, the larger the internal cohesion, the lower the corresponding critical internal friction angle.

(2) When the internal cohesion or internal friction angle increases to a certain critical value, the shear-compression failure criterion function f(x) approaches a constant value, and shear-compression failure does not occur.

The shear modulus, elastic modulus, and Poisson’s ratio are related as follows:

$${\displaystyle \frac{E}{G}}=2\left(1+\mu \right)$$

(19)

In the equation, μ is Poisson’s ratio, which ranges from 0 to 0.5. Therefore, E/G falls between 2 and 3.

Similarly, according to Scenario 4, with a key layer cantilever length of 50 m, thickness of 20 m, elastic foundation stiffness of 0.75 GN/m, φ = 30°, and c = 6 MPa, the shear-compression failure criterion function f(x) is calculated for various combinations of elastic modulus and shear modulus (with elastic modulus ranging from 0 to 90 GPa and shear modulus from 0 to 45 GPa) based on Equations (16) and (17). The results are shown in Figure 8.

From Figure 8, we can make the following observations: (1) The elastic modulus has a more significant impact on the value of the shear-compression failure criterion function f(x) compared to the shear modulus. (2) The larger the elastic modulus of the key layer, the more prone it is to shear-compression failure. At higher elastic modulus values, the key layer will undergo shear-compression failure regardless of the shear modulus.

6. Discussion

Shear-compression failure in key strata is a primary cause of large-scale roof collapse, which poses a serious threat to the safety of working faces. By integrating Timoshenko beam theory with the Mohr–Coulomb criterion, this study investigates the shear-compression failure behavior of key strata, aiming to clarify the dominant controlling factors during coal mining. This understanding enables the development of targeted prevention and control strategies to ensure safe and continuous mining operations and protect lives and property. When the parameters of the key stratum are clearly identified, the potential range of shear-compression failure can be estimated. Accordingly, techniques such as surface hydraulic fracturing and composite blasting can be employed to weaken the structural integrity of the key stratum, thereby eliminating the mechanical and geometric conditions necessary for shear-compression failure. However, as the key strata are composed of rock materials, their precise failure modes still require further in-depth investigation.

(1) In practical predictions of shear-compression failure in coal mine working faces, key strata may contain joints, fissures, and other discontinuities. This study, however, models the rock layer as an intact body. Given the current technological limitations in accurately detecting such features in real mines, discrepancies may arise between theoretical predictions and actual outcomes.

(2) The cohesion values used in this study are based on data from previous research. As key strata are typically composed of low-cohesion rock masses, the cohesion values considered here range from 0 to 10 MPa. While this range is applicable to most coal mining regions in China, it may be insufficient for igneous rocks with significantly higher strength.

(3) Shi et al. [21] applied Euler beam theory to the study of key strata and proposed three failure modes: tensile failure, brittle failure, and shear-compression failure. Most previous studies have focused on the first two, with limited research on shear-compression failure, despite its potential to cause more severe hazards. This study contributes by exploring the mechanical mechanism and key controlling factors of shear-compression failure, which is crucial for preventing mining pressure disasters associated with key stratum instability.

(4) In coal-bearing formations, the stiffness of the elastic foundation typically ranges from 0.25 to 1 GN/m. Since the shear modulus of rock is related to its elastic modulus, and as shown in Equation (18), the shear modulus of key strata is generally less than half of the elastic modulus, and the variation range of the shear modulus is inherently linked to that of the elastic modulus. When analyzing the influence of foundation stiffness k and shear modulus, we find that their impact on the failure range of key strata is relatively minor under constant values of other parameters. It is speculated, however, that under conditions of higher loads and greater thickness, this failure range may increase. Nevertheless, it is difficult to provide precise numerical predictions, and only a qualitative assessment can currently be made.

7. Conclusions

(1) The Timoshenko Beam model on Winkler foundation was adopted to establish a periodic failure mechanics model for the key strata considering transverse shear deformation. The deflection, rotation, bending moment, and shear force of the key strata were solved, and the criterion for key strata compression-shear failure was provided.

(2) Whether the key strata undergo compression-shear failure is related to the thickness-span ratio, cohesion and cohesion ratio, and elastic modulus, while it is minimally affected by the elastic foundation coefficient and shear modulus.

(3) When the elastic foundation stiffness is 1 GN/m, elastic modulus is 4.0 GPa, the shear modulus is 1.67 GPa, and the key strata is subjected to a uniform load of 1 MN/m², compression-shear failure occurs when the thickness–span ratio of the critical hard layer reaches 0.4.

(4) At friction angles of 25°, 30°, 35°, and 40°, when the cohesion of the rock exceeds 8.0, 7.5, 7.0, and 6.5 MPa, respectively, the key strata do not undergo compression-shear failure. Moreover, as the friction angle increases, the corresponding critical cohesion decreases. Similarly, at cohesion values of 6 MPa, 8 MPa, 10 MPa, and 12 MPa, when the friction angle of the rock exceeds 44°, 32°, 19°, and 8°, respectively, the key strata do not undergo compression-shear failure, and the larger the cohesion, the smaller the corresponding critical friction angle.

(5) When cohesion or friction angle increases to a critical value, the compression-shear failure criterion function f(x) approaches a constant value, and compression-shear failure does not occur.

(6) Compared to the shear modulus, the elastic modulus has a greater influence on the value of the compression-shear failure criterion function f(x). A higher elastic modulus of the key strata increases the likelihood of compression-shear failure. Under a high elastic modulus, compression-shear failure occurs regardless of the shear modulus of the key strata.