Calculation method for roof movement and deformation characteristics in paste-filled mining

Abstract

This invention relates to the field of constructing mechanical models for paste-filled mining, and particularly to a method for calculating the roof movement and deformation characteristics in paste-filled mining. The method includes establishing a mechanical model for upper-layer paste-filled mining, treating the immediate roof as a roof beam after upper-layer paste-filled mining, simplifying the upper layer as an elastic foundation, and establishing a mechanical model for the first layer of mining; solving the upper-layer mining mechanical model. A mechanical model for lower-layer paste-filled mining is then established, based on the elastic foundation theory, treating the immediate roof as a roof beam, and considering the two layers of filling, the solid coal support area, and the lower-layer filling hydraulic support roof control area as elastic foundations, establishing a segmented non-uniform combined elastic foundation beam model; solving the lower-layer paste-filled mining mechanical model. This invention studies the deflection and deformation law of the immediate roof after working face mining, especially after layered mining, based on the basic principles of elastic foundation beams, laying a theoretical foundation for selecting process parameters for paste-filled mining in working faces.

Description

Technical Field

[0001] This invention relates to the field of mechanical model construction for paste backfill mining, and in particular to a method for calculating the roof movement and deformation characteristics in paste backfill mining. Background Technology

[0002] For backfill mining of extra-thick coal seams, due to limitations in mining technology and equipment, layered backfill mining is currently the most common method, which is divided into upward backfill mining and downward backfill mining. Upward backfill mining uses the backfill body as a false floor, recovering coal resources from bottom to top. This method can eliminate the need for bottom coal and is widely used in the mining of coal seams prone to rock impact. Downward mining uses the backfill body as a false roof; the roof of the downward working face is entirely composed of upper-layered backfill, avoiding problems such as roof fracture and support difficulties. Therefore, it is used by the vast majority of mines.

[0003] The manifestation of mine pressure and strata control during coal mining have always been key focuses of green coal mining research. To analyze the impact of backfill mining on mine pressure and overburden movement and deformation, many scholars have conducted research on the backfill itself, including its creep and strain hardening characteristics under long-term loads, strength variation characteristics at different ages, and the influence of moisture content on the strength properties of the backfill. Based on this, creep models of backfill under long-term loads have been constructed, providing a theoretical basis for predicting the age-related deformation of backfill. However, existing mechanical models oversimplify actual working conditions, their computational accuracy needs improvement, and few mechanical models have been established for the layered mining of paste-like materials. Summary of the Invention

[0004] To study the impact of paste backfilling mining, especially layered paste backfilling mining, on the movement and deformation of the immediate roof, this invention is based on the fundamental principle of elastic foundation beams. The immediate roof strata are regarded as beam structures. The stress state of the immediate roof and the characteristics of the foundation structure under different support structures are analyzed. The deflection deformation law of the immediate roof after mining, especially after layered mining, is studied, laying a theoretical foundation for the selection of paste backfilling process parameters for the working face.

[0005] Specifically, the method for calculating the roof movement and deformation characteristics in paste-filled mining proposed in this invention includes the following steps:

[0006] S1: Calculation of roof movement and deformation characteristics during upper-layer paste-filled mining

[0007] S11: Establish a mechanical model for mining of upper-layer paste backfill.

[0008] When using paste-filled mining for upper-layer mining, the bottom plate of the working face is the lower-layer coal seam. The total length of the upper-layer working face is set as l0 + l3, where l0 is the length of the filling area, l3 is the length of the hydraulic support controlling the roof, and a uniformly distributed load q1 is applied to it. l1 represents the stress-increasing area of ​​the solid coal on both sides, with a stress concentration factor of k1; l2 represents the stress-reducing area of ​​the solid coal on both sides, with a stress concentration factor of k0, and the original rock stress value is q0.

[0009] After the upper layer of paste filling is mined, the immediate roof is regarded as the roof beam, and the upper layer is simplified as an elastic foundation. Let the elastic foundation coefficients of the filling body, the filling hydraulic support, and the solid coal before and after be k respectively. c k z and k m Based on this, a mechanical model for the first layer of mining was established;

[0010] S12: Solving the mechanical model of upper-layer mining

[0011] The direct top beam is divided into 6 segments, and a coordinate system is established to calculate the deflection differential equation of each segment of the top beam.

[0012] (a) -l2-l1-l0≤x<-l1-l0 segment

[0013]

[0014] In the formula: E—Elastic modulus of the top beam, N / m 3 I—Moment of inertia of the top beam section, m 4 ;

[0015] (b) -l1-l0≤x<-l0 segment

[0016]

[0017] (c) -l0≤x<0 segment

[0018]

[0019] In the formula:

[0020] (d) 0≤x<l3 segment

[0021]

[0022] In the formula:

[0023] (e) l3≤x<l1+l3 segment

[0024]

[0025] (f) segment l1+l3≤x<l1+l2+l3

[0026]

[0027] Based on the continuity between the segments of the top beam and the boundary conditions of the top beam, the integral constants A0~A4, B0~B4, C0~C4, and D0~D4 of each segment of the top beam are solved, and then the deflection equations of each segment of the top beam are obtained.

[0028] Preferably, it also includes S2: calculation of the roof movement and deformation characteristics during the mining of lower-layer paste filling.

[0029] S21: Establish a mechanical model for mining of lower-layer paste backfill.

[0030] For the upper layered working face filled with paste, its total length is l0+l3, where l0+l3 is the length of the filling area. For the lower layered working face filled with paste, its total length is l0+l3, where l0 is the length of the filling area, l3 is the length of the filling hydraulic support for roof control, and a uniformly distributed load q1 is distributed on it; l1 is the stress-increasing area of ​​the solid coal on both sides, with a stress concentration factor of k1; l2 is the stress-reducing area of ​​the solid coal on both sides, with a stress concentration factor of k0, and the original rock stress value is q0.

[0031] An elastic foundation beam model for the segmented combination of the direct roof after the lower layer paste filling mining is established. Based on the elastic foundation volume theory, the direct roof is regarded as the roof beam, and the two layers of filling body and solid coal support area and the lower layer filling hydraulic support roof control area are regarded as elastic foundation. A segmented non-uniform combination elastic foundation beam model is established.

[0032] S22: Solving the mechanical model for mining of lower-layer paste backfill

[0033] The elastic foundation coefficients for each stratigraphic combination are calculated as follows:

[0034]

[0035] In the formula, E m E c E z —Elastic modulus of coal seam, filling material, and filling support (N / m) 2 h1—thickness of the upper layer, m; h2—thickness of the lower layer, m; k2—elastic foundation coefficient of the solid coal assembly; k3—elastic foundation coefficient of the filling body assembly; k4—elastic foundation coefficient of the filling body and filling hydraulic support assembly;

[0036] The direct top beam is divided into 6 segments, and a coordinate system is established to calculate the deflection differential equation of each segment of the top beam.

[0037] (a) -l2-l1-l0≤x<-l1-l0 segment

[0038]

[0039] In the formula:

[0040] (b) -l1-l0≤x<-l0 segment

[0041]

[0042] (c) -l0≤x<0 segment

[0043]

[0044] In the formula:

[0045] (d) 0≤x<l3 segment

[0046]

[0047] In the formula:

[0048] (e) l3≤x<l1+l3 segment

[0049]

[0050] In the formula:

[0051] (f) segment l1+l3≤x<l1+l2+l3

[0052]

[0053] In the formula:

[0054] Based on the continuity between the segments of the top beam and the boundary conditions of the top beam, solve for the integral constants A5 to A5 of each segment of the top beam. 11 B5~B 11 C5~C 11 D5~D 11 This leads to the deflection equations for each segment of the top beam.

[0055] This invention has the following innovative points: Based on the basic principle of elastic foundation beams, this invention treats the direct roof rock strata as a beam structure, analyzes the stress state and foundation structure characteristics of the direct roof in different support structures, studies the deflection deformation law of the direct roof after mining, especially after layered mining, and lays a theoretical foundation for the selection of process parameters for paste filling in the working face. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the foundation structure state after the layered paste filling mining according to the present invention;

[0057] Figure 2 This is a schematic diagram of the mechanical model of the layered paste filling after mining according to the present invention;

[0058] Figure 3 This is a schematic diagram of the foundation structure state after the layered paste filling mining according to the present invention;

[0059] Figure 4 This is a schematic diagram of the mechanical model after the layered paste filling mining according to the present invention. Detailed Implementation

[0060] The method for preventing insufficient negative pressure in the pipeline conveying of paste according to the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0061] Example 1

[0062] The method for calculating the roof movement and deformation characteristics in paste-filled mining proposed in this invention, using downward paste-filled mining, includes the following steps:

[0063] S1: Calculation of roof movement and deformation characteristics during upper-layer paste-filled mining, including steps S11-S12.

[0064] S11: Establish a mechanical model for mining of upper-layer paste backfill.

[0065] During the mining of a layered paste backfilling working face, the main load-bearing structures directly below the top along the working face direction are, in sequence, the coal pillar behind the working face, the paste backfill, the backfill support, and the coal body in front of the working face.

[0066] like Figure 1 As shown, when the upper layer is mined with paste filling, the bottom plate of the working face is the lower layer coal seam. The total length of the upper layer working face is set to l0+l3, where l0 is the length of the filling area, l3 is the length of the filling hydraulic support for roof control, and a uniformly distributed load q1 is distributed on it; l1 is the stress-increasing area of ​​the solid coal on both sides, with a stress concentration factor of k1; l2 is the stress-reducing area of ​​the solid coal on both sides, with a stress concentration factor of k0, and the original rock stress value is q0.

[0067] After the upper layer of paste filling is mined, the immediate roof is considered as the roof beam. Ignoring the influence of factors such as the filling material, time, and friction between strata, the upper layer is simplified as an elastic foundation. Let the elastic foundation coefficients of the filling material, the filling hydraulic support, and the preceding and following solid coal be k, respectively. c k z and k m Based on this, a mechanical model for the first layer of mining was established;

[0068] S12: Solving the mechanical model of upper-layer mining

[0069] according to Figure 2The direct top beam is divided into 6 segments, and a coordinate system is established accordingly to derive the deflection differential equation of each segment of the top beam.

[0070] (a) For the segment -l2-l1-l0≤x<-l1-l0, the differential equation for the deflection of the top beam is:

[0071]

[0072] The load q2(x) above the top beam is:

[0073]

[0074] Substituting equation (12-2) into equation (12-1) and integrating, we obtain the deflection equation for this segment as follows:

[0075]

[0076] In the formula: E—Elastic modulus of the top beam, N / m 3 I—Moment of inertia of the top beam section, m 4 ;

[0077] Combination Figure 1 It can be seen that when x approaches -∞, the top beam can be considered as semi-infinite, and its settlement value approaches a certain fixed value. Therefore, in equation (12-3), C0 and D0 are 0, resulting in:

[0078]

[0079] (b) For the segment -l1-l0≤x<-l0, the differential equation for the deflection of the top beam is:

[0080]

[0081] The load q3(x) above the top beam is:

[0082]

[0083] Substituting equation (12-6) into equation (12-5) and integrating, we get...

[0084]

[0085] (c) For the segment -l0≤x<0, the differential equation for the deflection of the top beam is:

[0086]

[0087] Integrating equation (12-8) yields:

[0088]

[0089] In the formula:

[0090] (d) For the segment 0 ≤ x < l3, the differential equation for the deflection of the top beam is:

[0091]

[0092] Integrating equation (12-10) yields:

[0093]

[0094] In the formula:

[0095] (e) For the segment l3≤x<l1+l3, the differential equation for the deflection of the top beam is:

[0096]

[0097] The load q4(x) above the top beam is:

[0098]

[0099] Substituting equation (12-13) into equation (12-12) and integrating, we get:

[0100]

[0101] (f) For the segment l1+l3≤x<l1+l2+l3, the differential equation for the deflection of the top beam is:

[0102]

[0103] The load q5(x) above the top beam is:

[0104]

[0105] Integrating equation (12-15) yields:

[0106]

[0107] Combination Figure 1 It can be seen that when x approaches +∞, the top beam can be considered as semi-infinite, and its settlement value approaches a certain fixed value. Therefore, in equation (12-17), A5 and B5 are 0, resulting in:

[0108]

[0109] The relationships between the bending moment M(x), shear force Q(x), rotation angle θ(x), and bending moment w(x) at any section on the top beam are as follows:

[0110]

[0111] Based on the continuity between the segments of the top beam and the boundary conditions of the top beam, it can be concluded that:

[0112]

[0113] Based on equation (12-20) and combined with relevant geological parameters of specific engineering problems, the integral constants A0~A4, B0~B4, C0~C4, and D0~D4 of each segment of the top beam can be solved. Specifically, the deflection equations of each segment of the top beam can be obtained. The solution process is calculated using Maple software.

[0114] S2: Calculation of roof movement and deformation characteristics during lower-layer paste-filled mining, including steps S21-S22.

[0115] S21: Establish a mechanical model for mining of lower-layer paste backfill.

[0116] During the lower-layer paste filling mining, the upper-layer filling body serves as a false roof for the lower-layer mining. For the immediate roof, the supporting system below it is a combination of the paste filling bodies filled in the upper and lower goaf areas. At this time, the elastic foundation coefficient of each segment is a combination of the elastic foundation coefficients of multiple strata. To study the subsidence changes of the immediate roof after the lower-layer paste filling mining, an elastic foundation beam model of the segmented combination of the immediate roof after the lower-layer paste filling mining is established. The structural state of the foundation after the lower-layer mining is as follows: Figure 3 As shown;

[0117] For the upper layered working face filled with paste, its total length is l0+l3, where l0+l3 is the length of the filling area. For the lower layered working face filled with paste, its total length is l0+l3, where l0 is the length of the filling area, l3 is the length of the filling hydraulic support for roof control, and a uniformly distributed load q1 is distributed on it; l1 is the stress-increasing area of ​​the solid coal on both sides, with a stress concentration factor of k1; l2 is the stress-reducing area of ​​the solid coal on both sides, with a stress concentration factor of k0, and the original rock stress value is q0.

[0118] Based on the theory of elastic foundation, the immediate roof is considered as the roof beam, and the two-layer filling body, the solid coal support area, and the lower-layer filling hydraulic support roof control area are considered as elastic foundations. A segmented non-uniform composite elastic foundation beam model is established, and the simplified mining mechanics model is as follows: Figure 4 As shown;

[0119] S22: Solving the mechanical model for mining of lower-layer paste backfill

[0120] The reciprocal of the elastic foundation coefficient for a multi-stratum combination is equal to the sum of the reciprocals of the elastic foundation coefficients of each stratum. Based on the definition of the elastic foundation coefficient, we can conclude that:

[0121]

[0122] In the formula: E m Ec E z —Elastic modulus of coal seam, filling material, and filling support (N / m) 2 h—thickness of the corresponding stratum, in meters.

[0123]

[0124] Combining equations (2-1) and (2-2), the elastic foundation coefficients for each stratigraphic combination can be obtained as follows:

[0125]

[0126] In the formula, h1—thickness of the upper layer, m; h2—thickness of the lower layer, m; k2—elastic foundation coefficient of the solid coal assembly; k3—elastic foundation coefficient of the filling body assembly; k4—elastic foundation coefficient of the filling body and filling hydraulic support assembly;

[0127] The direct top beam is divided into 6 segments, and a coordinate system is established to calculate the deflection differential equation of each segment of the top beam.

[0128] (a) For the segment -l2-l1-l0≤x<-l1-l0, the deflection differential equation of the top beam is:

[0129]

[0130] The deflection equation for this segment can be obtained as follows:

[0131]

[0132] in:

[0133] As x approaches -∞, w2(x) approaches C, where C is a constant. Therefore, C6 and D6 are 0, and equation (2-5) can be simplified to:

[0134]

[0135] In the formula:

[0136] (b) For the segment -l1-l0≤x<-l0, the differential equation for the deflection of the top beam is:

[0137]

[0138] In the formula: The deflection equation is:

[0139]

[0140] (c) For the segment -l0≤x<0, the differential equation for the deflection of the top beam is:

[0141]

[0142] The deflection equation is:

[0143]

[0144] In the formula:

[0145] (d) For the segment 0 ≤ x < l3, the differential equation for the deflection of the top beam is:

[0146]

[0147] The deflection equation is:

[0148]

[0149] In the formula:

[0150] (e) For the segment l3≤x<l1+l3, the differential equation for the deflection of the top beam is:

[0151]

[0152] The deflection equation is:

[0153]

[0154] In the formula:

[0155] (f) For the segment l1+l3≤x<l1+l2+l3, the differential equation for the deflection of the top beam is:

[0156]

[0157] The deflection equation is:

[0158]

[0159] In the formula:

[0160] As x → +∞, w2(x) → C, where C is a constant, then A 11 B 11 When the value is 0, equation (2-16) can be simplified to:

[0161]

[0162] The relationships between the bending moment M(x), shear force Q(x), rotation angle θ(x), and bending moment w(x) at any section on the top beam are as follows:

[0163]

[0164] Based on the continuity between the segments of the top beam and the boundary conditions of the top beam, it can be concluded that:

[0165]

[0166] Based on equation (2-19) and combined with relevant geological parameters for specific engineering problems, the constant parameters A5 to A6 of each segment of the top beam can be obtained. 11 B5~B 11 C5~C 11 D5~D 11 Specifically, the deflection curve equations for each segment of the top beam can be obtained, and the solution process is calculated using Maple software.

[0167] It should be noted that the upper-layer mining mechanical model established in steps S11-S12 of Example 2 and its solution can be used as an independent technical solution, only for solving certain deformation characteristics of the roof during upper-layer paste filling mining.

[0168] Example 2

[0169] The method for calculating the roof movement and deformation characteristics in paste-filled mining proposed in this invention adopts a single-stage full-thickness paste-filled mining method and includes the following steps:

[0170] S1: Calculation of roof movement and deformation characteristics during paste-filled mining, including steps S11-S12

[0171] S11: Establishing a mechanical model for paste backfilling mining

[0172] During the longwall mining of the paste backfilling face, the main load-bearing structures directly below the top along the strike of the face are, in sequence, the coal pillar behind the face, the paste backfilling body, the backfilling support, and the coal body in front of the face.

[0173] When using paste backfill mining, the bottom plate of the working face is the bottom rock stratum. The total length of the working face is set as l0 + l3, where l0 is the length of the backfill area, l3 is the length of the backfill hydraulic support for roof control, and a uniformly distributed load q1 is distributed on it; l1 is the stress concentration factor of the solid coal on both sides; l2 is the stress concentration factor of the solid coal on both sides; l2 is the stress concentration factor of the solid coal on both sides; and the original rock stress value is q0.

[0174] After the paste backfilling mining, the immediate roof is considered as the roof beam. Ignoring the influence of factors such as the backfill, time, and friction between strata, the backfill, the hydraulic support, and the preceding and following solid coal seams are simplified as an elastic foundation. Let the elastic foundation coefficients of the backfill, the hydraulic support, and the preceding and following solid coal seams be k. c k z and k m Based on this, a mechanical model for full-thickness paste backfilling mining was established;

[0175] S12: Solving the mining mechanics model

[0176] refer to Figure 2 The direct top beam is divided into 6 segments, and a coordinate system is established accordingly to derive the deflection differential equation of each segment of the top beam.

[0177] (a) For the segment -l2-l1-l0≤x<-l1-l0, the differential equation for the deflection of the top beam is:

[0178]

[0179] The load q2(x) above the top beam is:

[0180]

[0181] Substituting equation (12-2) into equation (12-1) and integrating, we obtain the deflection equation for this segment as follows:

[0182]

[0183] In the formula: E—Elastic modulus of the top beam, N / m 3 I—Moment of inertia of the top beam section, m 4 ;

[0184] As x approaches -∞, the top beam can be considered semi-infinite, and its settlement value approaches a certain fixed value. Therefore, in equation (12-3), C0 and D0 are 0, resulting in:

[0185]

[0186] (b) For the segment -l1-l0≤x<-l0, the differential equation for the deflection of the top beam is:

[0187]

[0188] The load q3(x) above the top beam is:

[0189]

[0190] Substituting equation (12-6) into equation (12-5) and integrating, we get...

[0191]

[0192] (c) For the segment -l0≤x<0, the differential equation for the deflection of the top beam is:

[0193]

[0194] Integrating equation (12-8) yields:

[0195]

[0196] In the formula:

[0197] (d) For the segment 0 ≤ x < l3, the differential equation for the deflection of the top beam is:

[0198]

[0199] Integrating equation (12-10) yields:

[0200]

[0201] In the formula:

[0202] (e) For the segment l3≤x<l1+l3, the differential equation for the deflection of the top beam is:

[0203]

[0204] The load q4(x) above the top beam is:

[0205]

[0206] Substituting equation (12-13) into equation (12-12) and integrating, we get:

[0207]

[0208] (f) For the segment l1+l3≤x<l1+l2+l3, the differential equation for the deflection of the top beam is:

[0209]

[0210] The load q5(x) above the top beam is:

[0211]

[0212] Integrating equation (12-15) yields:

[0213]

[0214] As x approaches +∞, the top beam can be considered semi-infinite, and its settlement value approaches a certain fixed value. Therefore, in equation (12-17), A5 and B5 are 0, resulting in:

[0215]

[0216] The relationships between the bending moment M(x), shear force Q(x), rotation angle θ(x), and bending moment w(x) at any section on the top beam are as follows:

[0217]

[0218] Based on the continuity between the segments of the top beam and the boundary conditions of the top beam, it can be concluded that:

[0219]

[0220] Based on equation (12-20) and combined with relevant geological parameters for specific engineering problems, the integral constants A0~A4, B0~B4, C0~C4, and D0~D4 of each segment of the top beam can be solved. Specifically, the deflection equations of each segment of the top beam can be obtained. The solution process is calculated using Maple software.

Claims

1. A method for calculating the moving and deforming characteristics of the roof in paste backfill mining, characterized in that, Includes the following steps: S1: Calculation of roof movement and deformation characteristics during upper-layer paste-filled mining S11: Establish a mechanical model for mining of upper-layer paste backfill. When using paste-filled mining for upper-layer mining, the bottom plate of the working face is the lower-layer coal seam. The total length of the upper-layer working face is set as l0 + l3, where l0 is the length of the filling area, l3 is the length of the hydraulic support controlling the roof, and a uniformly distributed load q1 is applied to it. l1 represents the stress-increasing area of ​​the solid coal on both sides, with a stress concentration factor of k1; l2 represents the stress-reducing area of ​​the solid coal on both sides, with a stress concentration factor of k0, and the original rock stress value is q0. After the upper slice paste filling mining, the immediate roof is regarded as the roof beam, and the upper slice is simplified as the elastic foundation. The elastic foundation coefficients of the filling body, the filling hydraulic support and the front and rear solid coal are respectively k c , k z and k m . According to the above, the mechanical model of the first slice mining is established. S12: Solving the mechanical model of upper-layer mining The direct top beam is divided into 6 segments, and a coordinate system is established to calculate the deflection differential equation of each segment of the top beam. (a) -l2-l1-l0≤x<-l1-l0 segment wherein: E - top beam modulus of elasticity, N / m 3 ; I - top beam section moment of inertia, m 4 ; (b) -l1-l0≤x<-l0 segment (c) -l0≤x<0 segment In the formulae: (d) 0≤x<l3 segment In the formulae: (e) l3≤x<l1+l3 segment (f) segment l1+l3≤x<l1+l2+l3 Based on the continuity between the segments of the top beam and the boundary conditions of the top beam, the integral constants A0~A4, B0~B4, C0~C4, and D0~D4 of each segment of the top beam are solved, and then the deflection equations of each segment of the top beam are obtained.

2. The method of claim 1, wherein, Also includes S2: Calculation of roof movement and deformation characteristics during lower-layer paste-filled mining. S21: Establish a mechanical model for mining of lower-layer paste backfill. For the upper layered working face filled with paste, its total length is l0+l3, where l0+l3 is the length of the filling area. For the lower layered working face filled with paste, its total length is l0+l3, where l0 is the length of the filling area, l3 is the length of the filling hydraulic support for roof control, and a uniformly distributed load q1 is distributed on it; l1 is the stress-increasing area of ​​the solid coal on both sides, with a stress concentration factor of k1; l2 is the stress-reducing area of ​​the solid coal on both sides, with a stress concentration factor of k0, and the original rock stress value is q0. An elastic foundation beam model for the segmented combination of the direct roof after the lower layer paste filling mining is established. Based on the elastic foundation volume theory, the direct roof is regarded as the roof beam, and the two layers of filling body and solid coal support area and the lower layer filling hydraulic support roof control area are regarded as elastic foundation. A segmented non-uniform combination elastic foundation beam model is established. S22: Solving the mechanical model for mining of lower-layer paste backfill The elastic foundation coefficients for each stratigraphic combination are calculated as follows: wherein E m , E c , E z — elastic modulus of coal body, filling body, filling support, N / m 2 ; hi — thickness of upper layer, m; h2 — thickness of lower layer, m; k2 — elastic foundation coefficient of solid coal combination; k3 — elastic foundation coefficient of filling body combination; k4 — elastic foundation coefficient of filling body and filling hydraulic support combination; The direct top beam is divided into 6 segments, and a coordinate system is established to calculate the deflection differential equation of each segment of the top beam. (a) -l2-l1-l0≤x<-l1-l0 segment In the formulae: (b) -l1-l0≤x<-l0 segment (c) -l0≤x<0 segment In the formulae: (d) 0≤x<l3 segment In the formulae: (e) l3≤x<l1+l3 segment In the formulae: (f) segment l1+l3≤x<l1+l2+l3 In the formulae: Based on the continuity between each segment of the roof beam and the boundary conditions of the roof beam, the integral constants A5-A 11 , B5-B 11 , C5-C 11 , D5-D 11 of each segment of the roof beam are solved, and then the deflection equation of each segment of the roof beam is obtained.

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