1 Introduction
The intersection is widely utilized in tunnel and roadway engineering. However, in the context of complex geological conditions, such as those involving soft rock, deep burial, and high stress, the stability control of the roof is particularly severely tested [1-4]. As a traffic planning line structure, the intersection has a large section, a long-span roof, and a surrounding rock subjected to superimposed stress deformation [5, 6]. The roof at this position often undergoes extensive deformation damage, especially in the deep soft rock environment [7, 8]. Therefore, to realize the stability control of the intersection roof in different environments, it is necessary to pay attention to the research of fundamental theories. It needs to establish a more reasonable mechanics theory model and explore the significant deformation mechanism of the intersection roof to clarify the critical areas of its roof stability control.
Many scholars have carried out extensive and profound theoretical and applied research on the destruction of the intersection roof under various stress and lithologic environments and have achieved many valuable research results. After a detailed review and summary, the scholars’ research results are briefly described in the following four aspects.
1) The theory, model, and case of roadway roof breaking. In the study of the mechanism of roof breaking, TIAN et al [9] established a fixed beam model at both ends to study the breaking of the massive thick roof along the empty roadway. YAN et al [10] applied continuous and discontinuous beam mechanics models to the thick roof. LI et al [11] introduced energy indicators and obtained the expression of the roof-breaking criterion. MA et al [12] considered the lateral pressure and stress concentration coefficients to establish a modified support arch model. In the numerical analysis of roof fracture, GAO et al [13, 14] successfully simulated the initial expansion of roadway roof fracture with the discrete element simulation method. KANG et al [15] used numerical methods to explain the through-failure form of large-scale roof collapse. HE et al [16] performed a discrete element simulation analysis on the situation where a strong mine earthquake occurred when the thick-hard roof broke.
2) The instability mechanism and control technology of the surrounding rock of the large-section roadway. Regarding the disaster-causing mechanism of the large-section attribute, LI et al [17] believed that the large section intensifies the failure of the weak surrounding rock with the development of multiple joints in the roadway. LIU et al [18] designed a similar simulation test to illustrate that the roof of the large-section roadway is more prone to failure under dynamic load. CHENG et al [19] analyzed the disaster-causing factors of the deformation of the surrounding rock at the intersection of the large section by numerical method. TAI et al [20] expressed the separation and leakage characteristics of the roof of the large-section roadway from the actual measurement. Regarding large-section roadway control technology, XIE et al [21] proposed a comprehensive treatment technology of full-section pressure regulation and grouting for a kilometer large-section chamber. ZHU et al [22] applied joint support technology, while WANG et al [23] applied a new constrained concrete composite arch type.
3) The failure model, mechanism, and control method of the surrounding rock of the deep roadway. On the deep roadway model, ZHU et al [24, 25] established an accurate triaxial model to study the deep roadway’s deformation. WANG et al [26] established a three-dimensional similarity model of the self-formed roadway in the deep roadway. On the instability mechanism of the surrounding rock of deep roadways, ZHAN et al [27] used the revised Drucker-Praeger criterion to study the disaster mechanism caused by the coupling of multiple factors. ZHANG et al [28] and WANG et al [29] studied the instability mechanism of the concrete support of deep roadways to reflect the deformation of the weak surrounding rock. However, the large deformation of deep roadways is related to the weak nature of surrounding rock, excavation path, and support conditions [30-32]. In the control of surrounding rock in deep roadways, strong anchor, and injection support [33], concrete-filled steel tube support [23, 28, 29], optimized combination support [26, 31], double arch cooperative support [30], and other means have been applied under various field case conditions.
4) The surrounding rock’s failure mechanism and control scheme at the intersection. Aiming at the failure mechanism of the surrounding rock at the intersection, CHENG et al [34] believed that the surrounding rock’s expansion characteristics and tectonic stress were the main disaster-causing factors. FAN et al [35] proposed that the super-large cross-section attribute and the energy accumulation at the intersection caused the large deformation of the surrounding rock at the intersection. LIU et al [36] studied the joint instability of the narrow rock column and the large-span roof at the intersection. WU et al [37] comprehensively integrated the structural properties of the intersection, the characteristics of the deep surrounding rock, and the centralized layout to analyze the instability of the intersection. WANG et al [38] applied concrete-filled steel tube composite support to control the surrounding rock at the intersection. XIE et al [39] applied grouting and high-elongation anchor cables to reinforce the roof.
The scholars mentioned above have made meaningful research results on the disaster mechanism of deep soft rock, large-span roofs, large-section, and deteriorated surrounding rock on the destruction of roadway surrounding rock and the control technology of specific cases. The intersection of the deep roadway has the attributes of the weak surrounding rock, large-span roof, the large section at the intersection and stress superposition, and the surrounding rock at the common intersection is mixed with the layer fracture zone. Under multiple factors, the roof at the branch roadway intersection is prone to large subsidence deformation, as shown in Figure 1.
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This paper will extend the maximum equivalent span beam (MESB) theory [40] widely accepted in roadway intersection research and derive the bending moment and limit deflection of the MESB under two boundary conditions. According to its ideas, the maximum equivalent triangular plate (METP) theory is innovatively proposed, and the formula for calculating geometric parameters and calculating deflection is derived. Then, a numerical model of the intersection considering different intersection angles, roadway types, and lithology is established. The METP is constructed from its plastic zone and roof displacement, and the applicability of the theory is verified by simulation. In addition, to verify the rationality of the application of the METP theory, the comprehensive index of the second variable of partial stress is introduced to investigate the mechanical mechanism of the large deformation area of the intersection roof. The research results can guide the field to accurately control the location, peak value, and range of the large deformation area of the roof at the intersection, to obtain the key control area for the roof reinforcement support at the intersection.
2 MESB theory for deformation analysis of the intersection roof
2.1 The MESB theory
The MESB theory [40] holds that the research on the breaking of the intersection roof needs to determine the maximum span of the roof at the intersection area of the branch roadway, as shown in Figure 2.
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In the MESB theory, the ultimate span Ld0 is the maximum span value without considering the plastic failure of the triangular rock column at the intersection, not the maximum span value. Due to the high stress concentration and superposition of plastic zones on the acute angle side of the triangular rock column, it is theoretically assumed that this side is the dominant deformation. Simultaneously, the rock at the tip of the triangular rock column is fragmented by the high pressure and loses its effective bearing capacity. Therefore, it is considered that the maximum equivalent span (Ldmax) is the maximum span value tangent to the boundary line of the plastic zone of the triangular rock column on the acute angle side, and the minimum equivalent span (Ldmin) is the span value perpendicular to the boundary line of the plastic zone of the triangular rock column on the obtuse angle side.
The calculation formula of the three roof span values at the intersection is:
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where a1 and a2 are the widths of the plastic zone at the side of the main roadway and the branch roadway; b1 and b2 are the driving widths of the main roadway and branch roadway; Δr1 and Δr2 are the width of the plastic zone of the midline of the acute triangular rock pillar and the obtuse triangular rock pillar.
2.2 Bending of the MESB model
According to the MESB theory of the intersection roof, the bending property of the MESB model is solved. For hard rock roadways, the equivalent beam of the roof at the intersection can be considered as the fixed support at both ends; for soft rock roadways, this MESB can be considered as the simple support at both ends [40]. Thus, the solution of the MESB deflection can be analyzed by the beam model under two types of boundary conditions, as shown in Figure 3.
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1) Fixed beam model
The expressions of bending moment M and deflection ω of the roof beam model under the condition of fixed ends are:
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where M is the bending moment, ω is the deflection, qd is the overburden load borne by the beam, E is the elastic modulus of the beam, I is the moment of inertia of the rectangular section beam. The maximum displacement ωmax in the middle of the beam is:
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2) Simply supported beam model
The expressions of bending moment M and deflection ω of the roof beam model under the condition of simple support at both ends are:
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The maximum displacement ωmax in the middle of the beam is:
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2.3 Key points of roof support under MESB theory
The maximum sinking amount of the roof is positively correlated with the length of the beam model. For the intersection roof, due to the failure of the triangular rock column, the distance Δr of the linear plastic zone is expanded. Currently, the length of the MESB of the roof evolves from Ld0 to the maximum equivalent span Ldmax, and its bending moment Mmax and maximum displacement ωmax increase. Once the structural stability limit is reached, the roof will collapse significantly.
Therefore, the equivalent span theory attributed the failure of the intersection roof to the instability of the large-span equivalent beam and pointed out the roof area where the end of the rock column needs to be supported for the intersection.
The maximum displacement position of the whole roof of the intersection is often not in the middle of the MESB. Using the beam model to calculate the maximum sinking amount is widely applicable to the roof of a single roadway. However, the intersection roof is irregular. Currently, there is a significant error in the calculation of the beam model, which is not fully applicable to the calculation of the sinking amount and the maximum sinking position of the overall roof, so it is necessary to establish a further mechanical model to match it.
3 METP theory for deformation analysis of the intersection roof
3.1 Establishment of the METP theory
Geometric analysis of the roof is carried out with the idea of integrity, and the METP structure of the intersection roof is established by using the thin plate structure in elastic mechanics combined with the plastic failure partition of the surrounding rock at the intersection in rock mechanics, as shown in Figure 4.
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In the METP structure of the intersection roof, the equivalent triangular ABC is established by the three boundaries of the plastic zone of the acute-angle triangular rock column, the obtuse-angle triangular rock column and the main roadway plastic zone. Among them, the intersection angle of the two roadways is α0, ∠ABC=α0. The side AC is the maximum equivalent span (Ldmax) in the equivalent span theory of the roof. The apex of the plastic zone of the acute-angle triangular rock column is connected with the apex of the rock column to form two included angles, and one of them is set to be β0. From this, the length and angle of the three sides of the METP can be calculated. The geometric calculation process is as follows.
From the similarity principle of triangles:
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Therefore, the three angles and one side of the triangle ABC are known. According to the sine theorem of triangles:
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The length calculation formula of the sides AB and BC can be obtained:
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According to the theorem, a triangle’s center of gravity is the intersection of three midlines, and there is only one intersection. Hence, the position of the center of gravity of the equivalent triangle ABC can be determined by geometric methods, which is the position of the peak sinking area of the intersection roof.
Particularly, when the two roadways are perpendicularly (α0=90°), both sides are right-angled triangular rock pillars and there is a maximum equivalent span Ldmax, i.e., there are two equivalent triangles. Therefore, the area of the actual equivalent triangle of the roof at this time should be one of them, and the center of gravity of the sinking roof at the right-angled intersection should be the midpoint of the center of gravity of the two equivalent triangles.
3.2 Bending of the METP model
Similarly, by referring to the method of solving the bending of the beam model with the METP theory [40], the bending properties of the mechanical model of the METP structure are solved. Consequently, the mechanical models of arbitrary triangular thin plates under two types of boundary conditions are established respectively, and the expression of the maximum sinking amount of the triangular plate is derived, as shown in Figure 5.
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1) Model of fixed boundary
When the boundary is fixed, the approximate expression of the first-order deflection of any trapezoidal plate [41, 42] is expressed as:
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where k1=tanφ1, k2=tanφ2, φ1 and φ2 are the angles between the two oblique sides of the trapezoid and the vertical coordinates, d is the length of the short base of the trapezoid, h is the height of the trapezoid, C is the undetermined parameter. The least squares collocation method is used to obtain its value, that is, k1=k2=0.5773, d=0, which is the bending of the triangular plate with three sides fixed.
With points on the line y=0.75h [41], the deflection of any point of the triangular thin plate is obtained:
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where q0 is a uniform load, D is the flexural stiffness. From this, the deflection of the center of gravity of the triangular plate fixed on three sides can be obtained:
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Therefore, the subsidence of the center of gravity of the METP of the intersection roof can be calculated under the condition of three-sided fixed support.
2) Model of simply supported boundary
The approximate expression of deflection ω(x, y) of arbitrary trapezoidal plate with simply supported boundary [41]:
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where k1=tanφ1, d1 is half of the short base of the trapezoid, α1 and β1 are the adjustment parameters. Let k1=0.5773, d1=0, α1=β1=1, it is a triangular plate simply supported by three sides.
The least squares collocation method is used to calculate the bending deflection on the y=0.7h line [41]:
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From this, the deflection of the center of gravity of the triangular plate simply supported by three sides can be obtained:
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Therefore, the subsidence of the center of gravity of the METP of the intersection roof can be calculated under simply support on three sides. The deflection calculation for the triangular plate, as presented in this paper, is based on the least squares collocation method, and its calculation can also use other methods [42].
3.3 Key points of roof support under METP theory
As a theory for analyzing the sinking law of the intersection roof, the METP theory clarifies that the high displacement area of the intersection roof is not at the center of the maximum equivalent span beam (i.e., the triangular rock column roof), but at the center of the equivalent triangular plate (i.e., the roof area at the intersection of two roadways). Hence, its guiding points for reinforcing the surrounding rock at the intersection are shown in Figure 6.
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Firstly, the center of gravity of the METP is in a high displacement area, which has a high risk of subsidence and needs to be reinforced. At the same time, to prevent the area of the METP from increasing, it is necessary to reduce the length expansion of the MESB, so the stability of the triangular rock column should be strongly controlled. In addition, consideration must also be given to reinforcing the surrounding rock of the roadway near the intersection to prevent the increase of the plastic zone caused by superimposed stress.
4 Numerical analysis and verification of METP theory
4.1 Establishment of numerical analysis model
The fundamental viewpoint of the METP theory is that the intersection roof roughly exhibits the sinking characteristics of the triangular plate structure, verified by numerical simulation. According to the relevant literature, angle, roadway type, and surrounding rock lithology are the main factors influencing the stability of the intersection [3, 36, 38]. Thus, three types of 18 intersection models are established with the angle of branch roadway intersection, roadway type, and surrounding rock lithology as the main influencing factors, as shown in Figure 7.
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The intersection is set in a single rock layer to avoid the difference in results caused by special surrounding rock conditions such as the layered roof. The model is a cube with a side length of 40 m, its bottom and surrounding are fixed, and the buried depth of the roadway is 600 m. The upper surface is applied with a uniform load of 15.08 MPa. The radius of the circular roadway is 2.0 m, the width of the arched and rectangular roadways is 4.0 m, and the sections of the branch roadway and the main roadway are equal. The model applies the Mohr-Coulomb constitutive, and the selected parameters and the parameters of the soft rock and hard rock settings are shown in Table 1.
| Object | Bulk modulus/GPa | Shear modulus/GPa | Cohesion/MPa | Friction angle/(°) | Tensile strength/MPa |
|---|---|---|---|---|---|
| Selected rock formation | 9.39 | 6.18 | 3.0 | 30 | 1.5 |
| Soft rock | 4.11 | 2.06 | 2.4 | 27 | 1.2 |
| Hard rock | 9.66 | 6.65 | 4.0 | 37 | 1.8 |
4.2 METP structure under different intersection angles
According to the simulation analysis results, three-dimensional cloud images of the plastic zone section and roof displacement at intersections with different angles are obtained. Observing the plastic zone cloud diagram of the intersection side, the parameters a1, a2, b1, b2, Δr and β0 can be measured. Among them, the most important measurement content is the depth of the plastic zone on the bisector of the corner of the triangular rock column at the intersection, and the specific values are shown in Figure 8.
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The smaller the intersection angle of the roadway, the more extensive the superposition range of the plastic zone of the triangular rock column at the intersection. During the process of increasing the intersection angle from 15° to 90°, the depth of the plastic zone at the end of the triangular rock column at the intersection is reduced by 86%. In addition, according to the displacement cloud diagram of the intersection roof (Figure 9), it can be obtained that the maximum displacement point of the intersection roof is not at the maximum equivalent span beam. However, the range of the high displacement area is at the roof of the branch lane intersection, and its value gradually decreases in the direction of the roof on the roadway side.
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The section of the roof displacement cloud image is semi-transparent, and the vertical projection is made to the plastic zone section at the intersection. Then, the triangular contour of the structure is calculated and drawn according to the theoretical formula of the METP at the intersection, and the stack display of the plastic zone-the vertical displacement-the METP structure is formed, as shown in Figure 10.
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By obtaining the center of gravity at the intersection of the midline of the triangle, the center of gravity of the METP structure in Figure 10 can be marked. It can be clearly obtained that the center of gravity of the METP on the intersection roof is precisely in the peak area of its positional shift cloud diagram at different intersection angles, which shows that the METP theory of the intersection roof can successfully calculate the high displacement area position of the roof under different intersection angles.
The size of the roof’s MESB and METP at intersections with different intersection angles are analyzed, as shown in Figure 11.
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Figure 11 illustrates that with the increase of the intersection angle of the roadway, the MESB length of the intersection roof increases, while the METP area gradually decreases. The intersection angle increases from 15° to 90°, and the MESB length increases by 31.85%, while its METP area decreases by 77.13%. According to the MESB theory and the rock beam model of the roof subsidence, the more significant the equivalent span beam value, the greater the partial subsidence. However, the increase of the intersection angle increases the equivalent span value of the roof while the displacement of the roof is relatively reduced. Therefore, the MESB theory cannot be reasonably explained, but the METP theory can be better solved.
4.3 METP structure under different roadway types
The three-dimensional cloud diagrams of the plastic zone section and the roof displacement of intersections under different roadway types are obtained. The relevant parameters of the plastic region of the intersection side are also measured according to as shown in Figure 12.
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It can be observed that the type of roadway has a significant impact on the plastic zone of the triangular rock column at the intersection. Taking the intersection of 45° and 90° as an example, from circle to arch to rectangle, the depth of the plastic zone at the end of the triangular rock column increases by 36.04%, 66.59%, 19.66% and 62.92%, respectively. In addition, according to the roof’s displacement cloud diagram at intersection (Figure 13), it is also obtained that the range of the high displacement area is all at the roof of the branch roadway intersection. Meanwhile, the roof’s peak displacement at the intersection of the rectangular roadway is significantly greater than the intersection of the circular roadway and the intersection of the arched roadway.
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Similarly, the METP of the roof under different roadway types is constructed according to the METP theory of the intersection so that the lamination of the plastic zone of the upper part-vertical displacement-the METP structure is displayed, as shown in Figure 14.
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Figure 14 illustrates that the equivalent triangle center of gravity of the intersection roof under different roadway types is in the peak area of its displacement cloud image. Hence, the METP theory of the intersection roof can be applied to analyzing the roof displacement at the intersection of different roadway types. When the crossing angles are 45° and 90°, the cross-section of the roadway changes from circular to arched to rectangular, and the area of the METP increases by 33.55%, 64.31%, 16.66%, and 46.20%, as shown in Figure 15.
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Based on the triangular plate mechanical model, the increase of the area increases the peak displacement of the roof, which better explains the reason for the extensive range of the displacement area of the rectangular roadway and the high peak value.
4.4 METP structure under different surrounding rock lithology
The three-dimensional cloud diagram of the plastic zone section and roof displacement of the roadway intersections under different surrounding rock lithology are obtained. The relevant parameters are also measured according to the plastic zone of the intersection, as shown in Figure 16.
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The plastic zone of the surrounding rock at the intersection of soft rock and hard rock is significantly different from the superimposed plastic zone in the triangular rock column. Taking the intersection of 30°, 60° and 90° as an example, the depth of the plastic zone at the end of the hard triangular rock column is only 0.427, 0.433 and 0.282 times that of soft rock. In addition, according to the displacement cloud diagram of the intersection roof (Figure 17), the high displacement area of the roof is also obtained with the peak value at the intersection of the branch roadway, and it diffuses to the side of the roadway. Simultaneously, with the increase of the intersection angle, the high displacement range and peak value of the roof decrease for the intersections of different lithologies.
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Similarly, the METP of the roof under different surrounding rock lithology is constructed according to the METP theory of the intersection so that the lamination of the plastic zone of the upper part-vertical displacement-the METP structure is displayed, as shown in Figure 18.
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For the soft and hard surrounding rock intersections, the center of gravity of the METP on its top plate is in the peak area of the displacement cloud image. Thus, the METP theory can be used to calculate the peak position of roof displacement at intersections under different surrounding rock lithology. Figure 19 illustrates that when the intersection angles are 30°, 60° and 90°, the METP area of the intersection roof under soft rock lithology is significantly more significant, and its values are 2.01, 1.89 and 2.06 times that of hard rock. The increase in the area increases the peak displacement of the roof, which explains why the weaker the lithology, the greater the subsidence of the intersection roof.
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4.5 Comparative advantages of the METP theory
To explain the analysis and application advantages of the METP theory of intersection roofs more clearly, the MESB theory is compared on the basic assumptions, mechanical models, main viewpoints, and matching degrees with numerical analysis, as shown in Figure 20.
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1) The METP theory establishes the mechanical model of the plate structure from the actual shape of the intersection roof, which is more holistic than the local beam model of the MESB theory; 2) The METP theory calculates the maximum subsidence of the plate structure under two boundary conditions. In contrast, the MESB theory is mainly based on the calculation of two types of beam models; 3) The METP theory holds that the center of the equivalent triangular plate is in the maximum sinking area of the intersection roof, and it is not in the middle of the maximum span beam in the MESB theory; 4) Through numerical analysis, the simulation results of the intersections under different intersection angles, roadway types, and surrounding rock lithology conditions are well matched with the METP theory, which can form an effective calculation process. In contrast, the MESB cannot explain most of the conclusions in the simulation.
5 Mechanical analysis of regional large deformation of intersection roof
5.1 Analysis index based on the second invariant of deviatoric stress
After the METP theory is used to determine the high displacement zone and peak subsidence of the intersection roof, further exploration of its deformation’s driving mechanism is necessary. Due to the well-matching degree between the theoretical and numerical analysis results, a comprehensive and appropriate stress index is selected to analyze it, as shown in Figure 21.
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Vertical stress is a standard index to analyze the stress concentration in the rock mass of the roadway side, but it is a pressure relief zone on the roof, which makes it difficult to compare and analyze. The maximum shear stress is an important index to describe the shear failure of the surrounding rock mass, but it ignores the influence of intermediate stress. The maximum deviatoric stress can better describe the potential area of rock mass failure, only lacking comprehensiveness. Therefore, the second invariant of deviatoric stress (J2), which has the characteristics of the above indexes and is more comprehensive, is selected as the deformation analysis index of the complex situation of the intersection roof.
5.2 Relationship between deformation, theoretical model, and mechanical index
According to the J2 expression, the interface for extracting the data in the finite element numerical software is written and imported into the data analysis software, which displays the J2 cloud diagram of the intersection roof at different intersection angles. Meanwhile, the intersection contour is superimposed at the bottom to observe the position of the J2 concentration area more clearly at the intersection, as shown in Figure 22.
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As the intersection angle increases, the peak value of J2 is reduced by 32.8% and the range of the peak area is reduced, indicating that the internal driving capacity for deformation is reduced. This explains the law that the angle increases and the roof subsidence of the intersection decreases. Simultaneously, in the METP theory, as the angle increases, the area of the METP decreases, and the deflection calculation value decreases, respectively.
Therefore, the impact of the angle on the roof is verified in terms of mechanical index, displacement index, and theoretical calculation. The internal relationship between the roof displacement, J2, and the METP structure is shown in Figure 23.
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The J2 peak area drives the local large deformation area of the intersection roof, and the J2 peak area at different intersection angles is not above the triangular rock pillar of the intersection, but in the roof area at the front of its end, which is consistent with the center of gravity of the METP. So, the center of gravity of the METP is the peak position of the roof displacement, and its internal drive is the peak area of J2, which makes the roof in this area have a strong deformation ability and prone to large deformation. Thus, the rationality and applicability of the METP theory to determine the large deformation area of the intersection roof from the mechanical mechanism is verified.
6 Conclusions
1) The METP theory based on the overall structural instability of the intersection roof is established. The large deformation area of the intersection roof is located at the center of gravity of the METP, that is the roof area in front of the end of the triangular rock pillar at the intersection of the two roadways.
2) As the intersection angle gradually increases, the MESB value of the intersection roof increases, while the actual high displacement area and peak value gradually decrease. Therefore, the beam model alone cannot be fully explained, while the METP theory expounds that its essence is that the area of the METP structure is gradually reduced, resulting in a decrease in the deflection value of the plate structure.
3) When only the roadway section is different, the shape and angle of the METP structure of the intersection roof are the same but the area difference is noticeable, the rectangular type > the arch type > the circular type. Hence, the roof-high displacement area of the rectangular section intersection is large, and the peak value is high. This is also applicable to explain the mechanism that the weaker the lithology is, the more significant the roof subsidence at the intersection is.
4) The J2 peak area of the intersection roof is in the roof area at the front position of the end of the triangular rock pillar, which makes the area have a strong deformation capability, which is consistent with the position of the center of gravity of the METP, this point is also the peak position of the analyzed roof displacement. So, the rationality of the METP theory to determine the large deformation area of the intersection roof is verified by the mechanical mechanism.
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武毅艺,高玉兵,马翔等.确定交岔点顶板大变形区域的新理论模型及验证分析[J].中南大学学报（英文版）,2025,32(2):656-677.