# Bending Moment Characteristics of Hard Roof before First Breaking of Roof Beam Considering Coal Seam Hardening

Authors: Shitan Gu; Bangyou Jiang; Yue Pan; Zheng Liu
Journal: Shock and Vibration
 (2018)

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## Abstract

The mechanical properties of a coal seam affect the distribution of support pressure. Considering the strain hardening effect of coal seam, the support pressure relationship of three zones—softened, hardened, and elastic—of a coal seam with regard to a hard roof is proposed, and methods to determine an approximate expression for the support pressure of hardened zone of coal seam, the range of hardened zone, and the corresponding peak values of support pressure are provided. The deflection equations of a hard roof under three different support pressure relationships of coal seam before the first breaking were theoretically derived, and all the relevant integration constants were determined. Numerical examples of two cases are provided for calculating the bending moment of a hard roof and the support pressure of a coal seam. The analysis shows that as the working face advances, the maximum support pressure increases, the residual strength of coal seam at coal wall decreases, the overall deflection of roof gradually increases, the maximum bending moment of roof in the front of coal wall increases, and the advanced distance of roof bending moment peak gradually increases. As the depth of softened zone of coal seam increases, the similar conclusion is obtained, and the advanced distance of the roof bending moment peak increases at a relatively fast speed. Because the bending moment peak of hard roof is located near the support pressure peak in the softened zone of coal seam, the depth of softened zone of coal seam significantly affects the advanced distance of the bending moment peak of a roof. The actual advanced fracture distances of hard roof are distributed in a relatively broad range. The results indicate that there is a “large and long” type advanced fracture distance occurring in the actual stope. With the same overlying load and stope parameter conditions, the maximum support pressures of support roof in softened, hardened, and elastic zones considering a hardened coal seam are smaller than those in softened and elastic zones without hardening. However, the plumpness in front of the peak of the former support pressure curve is superior to that of the latter, and both the bending moment peak value and the advanced distance of bending moment peak of the former are higher than those of the latter.


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## Body

## 1. Introduction

In the early stage of working face mining, a large area of hard roof overhangs in the goaf. When the critical hanging roof distance is reached, the hard roof starts to rupture in front of the coal wall. For a hard roof with advanced fracture, when the working face advances to below the roof fracture line, if the supporting resistance of frame and friction between seams are insufficient, the roof easily subsides in a stepped style, causing collapse accidents under pressure and even rock burst disasters. Many studies have been conducted on the deformation, law of motion, and fracture of a hard roof under the overlying load by field observation, similar material simulation, numerical simulation, and theoretical analysis [1–14]. Qian et al. [15, 16], Miao et al. [17], and Li et al. [18] analyzed a hard roof by assuming the support pressure relationship between coal seam and immediate roof as an elastic foundation and obtained some basic results such as the deflection solution for a hard roof and the maximum bending moment in the front of a coal wall and for an advanced roof fracture.Studies on support pressure showed that the coal seam near the coal wall is softened before applying pressure; the coal seam is in a yielding and softened state. Therefore, the assumption that all the support pressure relationship of a coal seam on a hard roof is elastic foundation is far from the reality. Peng [19] reported that a coal seam is in a softened state from the coal wall to the support pressure peak, and the coal seam is in an elastic state ahead of the support pressure peak. Based on this understanding, Pan et al. [20] analyzed the bending moments of a supporting hard roof in the softened and elastic zones; they found that when the residual strength of a coal seam at the coal wall is sufficiently low, the maximum bending moment and advanced distance of bending moment peak of a support roof in the softened and elastic zones under the same overlying load are exponentially larger than the corresponding values of a completely elastic foundation support roof.Figure1 shows the distribution diagram of measured support pressures obtained by Shao et al. [21] for the third mining area of the west wing of Xieqiao Coal Mine in Huainan city. The curve section at the first measuring point ahead of support pressure peak is convex, while that ahead of the second measuring point is concave. The intermediate point between the convex and concave sections is the inflection point of curve. The support pressure ahead of the inflection point is relatively small, and the coal seam is in an elastic state. For the convex section between the inflection point and pressure peak, the coal seam has the hardening state (unlike elastic characteristics). The coal seam between the pressure peak and coal wall has the yield softening state. Because of the difference in the mechanical properties of coal seam, the coal seam in Figure 1 shows three different support pressure relationships for the roof, i.e., softened zone, hardened zone, and elastic zone, bounded by the support pressure peak and inflection point of curve, consistent with the division of four zones—flow zone, softened zone, hardened zone, and elastic zone—for the support pressure relationship of the immediate roof of coal seam on the main roof proposed by He et al. [22].Figure 1
Distribution picture of support pressure of each measured points [21].Figure2 shows the schematic diagram of support pressure of coal seam when the working face advances in a phase before the breaking of hard roof shown by Shao et al. [23].P,S, andD in Figure 2 are advanced support pressure, advanced distance, and load of powered support of working face, respectively. The distance of overhanging roof and support pressure peak increases with the advance of working face. In fact, with the increase in hanging roof distance, the total load caused by the overlying strata on the coal seam increases, the fracture state of coal seam at coal wall becomes severe, the residual strength decreases, and the depth of softened zone increases. A change in support pressure changes the bending moment and advanced distance of bending moment peak for a hard roof.Figure 2
Support pressure of coal seam when working face advancing [23].The mechanical properties of a coal seam in different mines are different; such a difference causes difference in the distribution pattern of support pressure of coal seam. The softened, hardened, and elastic three-zone support pressure relationship of hardening effect of coal seam can be better understood from the softened and hardened two-zone support pressure relationship of coal seam. The consideration of depth variation of hardened and softened zones of coal seam significantly affects the maximum bending moment and advanced distance of bending moment peak. Based on previous study [20], the relationship between a coal seam and roof was regarded as softened, hardened, and elastic three-zone support in this study. Focusing on a phase in which the working face advances and the hanging roof distance increases before the first breaking and the case in which the depth of softened zone of coal seam varies, the bending moment of hard roof was theoretically analyzed. The numerical example was provided by Matlab software, and the relevant values under the same overlying load conditions were compared with those only considering the softened and elastic support of coal seam for a hard roof. The results can lead to a better understanding of the variations in bending moment, advanced distance of bending moment peak, and deflection of hard roof when the hanging roof distance and depth of softened zone of coal seam increase.


## 2. Analytical Model

### 2.1. Analytical Model and Overlying Load Expression of Roof

Generally, the working face of a stope is relatively long. Therefore, the softened zone of a coal seam in the middle part of working face has the maximum depth and the largest bending moment of roof, and the fracture comes first. Usually, the bending moment characteristics of a roof in the middle part of a stope were studied using the strata structure of unit width—the system of a rock beam [10, 15–18] along the advancing direction in the middle part of a stope. Qian et al. [15] found it similar to the support pressure of coal seam: the hard roof in front of the coal wall bears a supercharged load of uplift distribution in addition to the average load, except that the uplift part is relatively flat and the supercharged load peak is located ahead of coal seam support pressure peak. Thus, the strata structure of unit width before the first breaking of hard roof can be determined by considering three support pressure relationships of coal seam, i.e., elastic, hardened, and softened, as shown in Figure 3. To simplify analysis, both sides of rock beam were assumed to have supports and both the structure and the loads in the picture were assumed to be symmetric with respect to the midspan of goaf. According to structural mechanics, the midspan cross-sectional shear force of rock beam in a symmetric system is 0, i.e., Q(l+L)=0. The angle of the midspan cross-section of rock beam is 0, i.e., y1′(L+l)=0. Hence, the semistructure, i.e., the model in Figure 4 whose right end is a directed support, can be used to analyze the bending moment characteristics of the middle part of hard roof before the first breaking.Figure 3
Surrounding rock constitution and load state of hard roof before first breaking.Figure 4
Semiconstitution model of hard roof.The origin of the coordinateo in Figure 4 is located beneath the supercharged load peak of rock beam. The distance from the point o to the coal wall is l, and its distances to the hardened and softened zones of coal seam are l4 and l3, respectively. l4 is the location coordinate of the inflection point on the support pressure curve of coal seam. The support pressures of the softened and hardened zones of coal seam on rock beam are denoted by f3(x) and F4(x), respectively. The length of support is Lk, and the support resistances are po and pk. The maximum supercharged load above the rock beam is higher than the evenly distributed load from the distant position, and the rock beam load far from load peak tends towards evenly distributed forces q1 and q2. The following expressions can be used:(1)F1x=q1+f1x=q1+k1x+xc1e-x+xc1/xc1,0≤x≤l+L(2)F2x=q2+f2x=q2+k2xc2-xex-xc2/xc2,x≤0to simulate the evenly distributed load and uplift-distributed load on rock beam ahead of and after supercharged load peak in Figure 4, where(3)k1=fc1exc1,k2=fc2exc2In (3), fc1 and fc2 are the peak values of f1(x) and f2(x), respectively, whereas xc1 and xc2 are the scale parameters. The value of supercharged load peak can be adjusted by changing fc1 and fc2, and, in the process of x→±∞, the speed of f1(x)→0 and f2(x)→0 can be adjusted by varying xc1 and xc2.The relationship betweenfc1, fc2, and even loads on the beam q1, q2 is defined as follows:(4)q1+fc1=q2+fc2,orfc1-fc2=q2-q1Equation (4) ensures smooth and continuous supercharged load peaks in Figures 2 and 4.In the case where the buried depth is ~350 m, the volumetric weight of overlying rock and soil mass isγ=20000N/m3 and the average load on beam is q2=γH=7MN/m. If fc2=0.22q2, roof thickness h=6m and volumetric weight γ′=25000N/m3 according to literature [26], q1=γ′h=0.15MN/m. Equation (4) shows that fc1=q2+fc2-q1=8.39MN/m. If l=18m in Figure 4, the hanging roof distance of hard roof is L=20m. The roof load slowly decreases ahead of supercharged load peak but becomes fast after the supercharged load peak in Figure 4. Therefore, the scale parameters in (3) are assigned as xc2=13m and xc1=8m. Based on the above values and according to (1)–(3), the load distribution on rock beam from −100 m to 38 m was plotted by Matlab and shown in Figure 5.Figure 5
Load on roof in analysis and calculation.The load intensity 100 m ahead in Figure5 is denoted by F1(100)→q2=7MN/m. The supercharged load peak is located at x=0. The load curve is smooth and continuous, and F1max/q2=1.22. The midspan load intensity of rock beam is F1min=0.58MN/m (~3.78q1, i.e., 3.78 times of the roof weight), which is far less than the mean load on beam q2=7MN/m. Figure 6 shows the results of finite element analysis conducted by Qian et al. [15] for the distribution pattern of overlying load of the key overlying strata (∑h3=0m is equivalent to 150 m). Compared with Figure 6, the load curve shown in Figure 5 can better simulate the roof load curve with a buried depth of 350 m.Figure 6
Relation of loads on the roof and buried depth [15].


### 2.2. Expressions of Support Pressures in Softened and Hardened Zones of Coal Seam

For the coordinate in Figure4, the support pressure in the softened zone of coal seam can be expressed as follows:(5)f3x=fc3exc3l3+xc3-xex-l3-xc3/xc3=k3l3+xc3-xex-l3-xc3/xc3,l3≤x≤lIn (5), when x=l3, the maximum support pressure of coal seam is located at f3(l3)=fc3 and xc3 is the scale parameter for f3(x). Equation (5) shows that(6)k3=fc3exc3In (5), if fc3 is large, the support pressure intensity in the softened zone of coal seam is also large.The support pressure in the hardened zone of coal seam[l4,l3] can be expressed as follows:(7)F4x=q2+f4x=q2+k4l3+xc4-xex-l3-xc4/xc4,0<l4≤x≤l3where xc4 is the scale parameter for f4(x). In (7), F4(x) tends to the mean load q2 when x→-∞, where(8)k4=fc4exc4By taking the derivative twice with respect tof4(x) in (7) and letting it be 0, the relationship among l4, l3, and xc4 at the inflection point on support pressure curve of coal seam can be expressed as follows:(9)l4=l3-xc4In (7), F4(l3)=q2+fc4 when the maximum support pressure curve of coal seam is x=l3, where fc4 is the maximum f4(x). Because f4′(l3)=0 and F4′(l3)=0, let F4(l3)=fc3. The support pressure curve of coal seam is then smoothly connected at x=l3, and thus(10)fc3=fc4+q2Forl3=10m and xc4=2 m, the inflection point location of support pressure curve of coal seam l4=8 m can be obtained using (9). Suppose fc3=11MN/m, with respect to q2=7MN/m and xc3=8.6, 8.3, and 8 m q2=7MN/m and according to (5), (7), and (10), the support pressure curve F4(x) (x is the [8,10] m section) for the hardened zone of coal seam when l4=8 m and xc4=2 m and the support pressure curve f3(x) (x is the [10,18] m section) for the softened zone of coal seam when l3=10m and xc3=8.6, 8.3, and 8 m were plotted by Matlab, as shown by curves 1, 2, and 3 in Figure 7. A horizontal tangent line is located at l3=10m on the support pressure curves. f3(l) is the residual strength of coal seam at coal wall. f3(l) can be varied in the range of [0~0.8]fc3 by adjusting xc3. The curves shown in Figure 7 can better simulate the support pressures of coal seam from the inflection point to the coal wall in Figures 1 and 4. In particular, fc3=11MN/m in Figure 7 is manually drawn the schematic diagram, whereas the value of fc3 in the numerical example was determined according to the method introduced in Section 5 by the varying overlying load corresponding to the increase in the hanging roof distance or the depth of softened zone of coal seam in Figure 4.Figure 7
Schematic drawings of support pressure to roof in softened zone and hardened zone of coal seam.


## 2.1. Analytical Model and Overlying Load Expression of Roof

Generally, the working face of a stope is relatively long. Therefore, the softened zone of a coal seam in the middle part of working face has the maximum depth and the largest bending moment of roof, and the fracture comes first. Usually, the bending moment characteristics of a roof in the middle part of a stope were studied using the strata structure of unit width—the system of a rock beam [10, 15–18] along the advancing direction in the middle part of a stope. Qian et al. [15] found it similar to the support pressure of coal seam: the hard roof in front of the coal wall bears a supercharged load of uplift distribution in addition to the average load, except that the uplift part is relatively flat and the supercharged load peak is located ahead of coal seam support pressure peak. Thus, the strata structure of unit width before the first breaking of hard roof can be determined by considering three support pressure relationships of coal seam, i.e., elastic, hardened, and softened, as shown in Figure 3. To simplify analysis, both sides of rock beam were assumed to have supports and both the structure and the loads in the picture were assumed to be symmetric with respect to the midspan of goaf. According to structural mechanics, the midspan cross-sectional shear force of rock beam in a symmetric system is 0, i.e., Q(l+L)=0. The angle of the midspan cross-section of rock beam is 0, i.e., y1′(L+l)=0. Hence, the semistructure, i.e., the model in Figure 4 whose right end is a directed support, can be used to analyze the bending moment characteristics of the middle part of hard roof before the first breaking.Figure 3
Surrounding rock constitution and load state of hard roof before first breaking.Figure 4
Semiconstitution model of hard roof.The origin of the coordinateo in Figure 4 is located beneath the supercharged load peak of rock beam. The distance from the point o to the coal wall is l, and its distances to the hardened and softened zones of coal seam are l4 and l3, respectively. l4 is the location coordinate of the inflection point on the support pressure curve of coal seam. The support pressures of the softened and hardened zones of coal seam on rock beam are denoted by f3(x) and F4(x), respectively. The length of support is Lk, and the support resistances are po and pk. The maximum supercharged load above the rock beam is higher than the evenly distributed load from the distant position, and the rock beam load far from load peak tends towards evenly distributed forces q1 and q2. The following expressions can be used:(1)F1x=q1+f1x=q1+k1x+xc1e-x+xc1/xc1,0≤x≤l+L(2)F2x=q2+f2x=q2+k2xc2-xex-xc2/xc2,x≤0to simulate the evenly distributed load and uplift-distributed load on rock beam ahead of and after supercharged load peak in Figure 4, where(3)k1=fc1exc1,k2=fc2exc2In (3), fc1 and fc2 are the peak values of f1(x) and f2(x), respectively, whereas xc1 and xc2 are the scale parameters. The value of supercharged load peak can be adjusted by changing fc1 and fc2, and, in the process of x→±∞, the speed of f1(x)→0 and f2(x)→0 can be adjusted by varying xc1 and xc2.The relationship betweenfc1, fc2, and even loads on the beam q1, q2 is defined as follows:(4)q1+fc1=q2+fc2,orfc1-fc2=q2-q1Equation (4) ensures smooth and continuous supercharged load peaks in Figures 2 and 4.In the case where the buried depth is ~350 m, the volumetric weight of overlying rock and soil mass isγ=20000N/m3 and the average load on beam is q2=γH=7MN/m. If fc2=0.22q2, roof thickness h=6m and volumetric weight γ′=25000N/m3 according to literature [26], q1=γ′h=0.15MN/m. Equation (4) shows that fc1=q2+fc2-q1=8.39MN/m. If l=18m in Figure 4, the hanging roof distance of hard roof is L=20m. The roof load slowly decreases ahead of supercharged load peak but becomes fast after the supercharged load peak in Figure 4. Therefore, the scale parameters in (3) are assigned as xc2=13m and xc1=8m. Based on the above values and according to (1)–(3), the load distribution on rock beam from −100 m to 38 m was plotted by Matlab and shown in Figure 5.Figure 5
Load on roof in analysis and calculation.The load intensity 100 m ahead in Figure5 is denoted by F1(100)→q2=7MN/m. The supercharged load peak is located at x=0. The load curve is smooth and continuous, and F1max/q2=1.22. The midspan load intensity of rock beam is F1min=0.58MN/m (~3.78q1, i.e., 3.78 times of the roof weight), which is far less than the mean load on beam q2=7MN/m. Figure 6 shows the results of finite element analysis conducted by Qian et al. [15] for the distribution pattern of overlying load of the key overlying strata (∑h3=0m is equivalent to 150 m). Compared with Figure 6, the load curve shown in Figure 5 can better simulate the roof load curve with a buried depth of 350 m.Figure 6
Relation of loads on the roof and buried depth [15].


## 2.2. Expressions of Support Pressures in Softened and Hardened Zones of Coal Seam

For the coordinate in Figure4, the support pressure in the softened zone of coal seam can be expressed as follows:(5)f3x=fc3exc3l3+xc3-xex-l3-xc3/xc3=k3l3+xc3-xex-l3-xc3/xc3,l3≤x≤lIn (5), when x=l3, the maximum support pressure of coal seam is located at f3(l3)=fc3 and xc3 is the scale parameter for f3(x). Equation (5) shows that(6)k3=fc3exc3In (5), if fc3 is large, the support pressure intensity in the softened zone of coal seam is also large.The support pressure in the hardened zone of coal seam[l4,l3] can be expressed as follows:(7)F4x=q2+f4x=q2+k4l3+xc4-xex-l3-xc4/xc4,0<l4≤x≤l3where xc4 is the scale parameter for f4(x). In (7), F4(x) tends to the mean load q2 when x→-∞, where(8)k4=fc4exc4By taking the derivative twice with respect tof4(x) in (7) and letting it be 0, the relationship among l4, l3, and xc4 at the inflection point on support pressure curve of coal seam can be expressed as follows:(9)l4=l3-xc4In (7), F4(l3)=q2+fc4 when the maximum support pressure curve of coal seam is x=l3, where fc4 is the maximum f4(x). Because f4′(l3)=0 and F4′(l3)=0, let F4(l3)=fc3. The support pressure curve of coal seam is then smoothly connected at x=l3, and thus(10)fc3=fc4+q2Forl3=10m and xc4=2 m, the inflection point location of support pressure curve of coal seam l4=8 m can be obtained using (9). Suppose fc3=11MN/m, with respect to q2=7MN/m and xc3=8.6, 8.3, and 8 m q2=7MN/m and according to (5), (7), and (10), the support pressure curve F4(x) (x is the [8,10] m section) for the hardened zone of coal seam when l4=8 m and xc4=2 m and the support pressure curve f3(x) (x is the [10,18] m section) for the softened zone of coal seam when l3=10m and xc3=8.6, 8.3, and 8 m were plotted by Matlab, as shown by curves 1, 2, and 3 in Figure 7. A horizontal tangent line is located at l3=10m on the support pressure curves. f3(l) is the residual strength of coal seam at coal wall. f3(l) can be varied in the range of [0~0.8]fc3 by adjusting xc3. The curves shown in Figure 7 can better simulate the support pressures of coal seam from the inflection point to the coal wall in Figures 1 and 4. In particular, fc3=11MN/m in Figure 7 is manually drawn the schematic diagram, whereas the value of fc3 in the numerical example was determined according to the method introduced in Section 5 by the varying overlying load corresponding to the increase in the hanging roof distance or the depth of softened zone of coal seam in Figure 4.Figure 7
Schematic drawings of support pressure to roof in softened zone and hardened zone of coal seam.


## 3. Deflection Equations and Continuity Conditions for Various Zones of Rock Beam

### 3.1. Deflection Equation of Rock Beam for Elastic Foundation Zone

For the zones(-∞,0] and (0,l4] in Figure 4, the coal seam is in the elastic state; therefore, the supporting relationship of coal seam on roof in both the zones was treated as elastic foundation. The reaction force of elastic foundation qc is directly proportional to the displacement of rock beam y, i.e., qc=-Cy, where C is the elastic foundation constant and the minus represents the opposite direction of reaction force and displacement y (the settlement amount of rock beam). A distributed force F2(x)=q2+f2(x) was applied above the rock beam in zone (-∞,0]. The deflection of rock beam in this zone was denoted as y2(x). Following Timoshenko [27], the differential equation for the deflection of any infinitesimal segment dx in zone (-∞,0] of a rock beam can be derived as follows:(11)EIy24x=q2+k2xc2-xex-xc2/xc2-Cy2x,x≤0where E is the elastic modulus of roof under the condition of plane strain and I is the moment of inertia for the roof with unit width. If β=[C/(4EI)]1/4, (11) can be expressed as follows:(12)y24x+4β4y2x=q2EI+k2EIxc2-xex-xc2/xc2,x≤0For the zone[0,l4] shown in Figure 4, the distributed load applied above on the elastic foundation beam is F1(x)=q1+f1(x). The deflection of the rock beam in zone [0,l4] was denoted as y21(x). Similar to (12), the differential equation for the deflection of any infinitesimal segment dx in zone [0,l4] can be derived as follows:(13)y214x+4β4y21x=q1EI+k1EIx+xc1e-x+xc1/xc1,0≤x≤l4Because the differential equations for deflections (12) and (13) are identical to that of the corresponding zones reported in literature [20], the solution form of the differential equation for the deflection of semi-infinite elastic foundation beam in zone (-∞,0] is the same as reported in the literature [20], and it is denoted as follows:(14)y2x=eβxd1cos⁡βx+d2sin⁡βx+q24β4EI+k2EI·xc251+4β4xc24xc2-xxc2+41+4β4xc24ex-xc2/xc2,x≤0whereas that in zone [0,l4] is also the same as that reported in the literature [20] and can be expressed as follows:(15)y21x=d3sin⁡βxsinh⁡βx+d4sin⁡βxcosh⁡βx+d5cos⁡βxsinh⁡βx+d6cos⁡βxcosh⁡βx+q14β4EI+d5cos⁡βxsinh⁡βx+d6cos⁡βxcosh⁡βx+q14β4EI+k1EIxc151+4β4xc14x+xc1xc1+41+4β4xc14e-x+xc1/xc1,0≤x≤l4The values of integral constants d1–d6 in (14) and (15) are different from the values of d1–d6 reported in the literature [20]; therefore, these values should be redetermined according to the parameters of the hardened zone of coal seam.


### 3.2. Deflection Equation of Rock Beam above the Hardened Zone of Coal Seam

The distributed loadF1(x) was applied above the rock beam in zone [l4,l3] shown in Figure 4, and the support pressure of hardened zone of coal seam F4(x) was applied below the rock beam. The deflection of the rock beam in zone [l4,l3] was denoted as y22(x), and similar to (13), the differential equations for the deflection of the rock beam above the hardened zone of coal seam [l4,l3] can be written as follows:(16)y224x=q1EI+k1EIx+xc1e-x+xc1/xc1-q2EI-k4EIl3+xc4-xex-l3-xc4/xc4,l4≤x≤l3Figure8(a) shows the chart of the isolating body of the rock beam above the hardened zone of coal seam in Figure 4. By calculating the moments of the forces from the left of x cross-section with respect to x cross-section in Figure 8(a), the equation of the bending moment of the rock beam in zone [l4,l3] can be obtained as follows:(17)M22x=Ml4-Ql4x-l4+q1-q22x-l42+k1xc13×x+xc1xc1+2e-x+xc1/xc1-l4+xc1xc12+2l4+xc1xc1+2-l4+xc1xc1+1x+xc1xc1e-l4+xc1/xc1-k4xc43×2-x-l3-xc4xc4ex-l3-xc4/xc4-l4-l3-xc4xc42-2l4-l3-xc4xc4+2-l4-l3-xc4xc4-1x-l3-xc4xc4×el4-l3-xc4/xc4,l4≤x≤l3Figure 8
Charts of isolating body of the rock beam before first breaking.
(a) (b)The sum of vertical forces on the isolating body in Figure8(a) was calculated, and the shearing force of the left-end cross-section of the rock beam in zone [l4,l3] can be obtained from ∑Fy=0:(18)Ql4=Ql3+q1-q2l3-l4+k1xc12l4+xc1xc1+1e-l4+xc1/xc1-l3+xc1xc1+1e-l3+xc1/xc1-k4xc422e-1+l4-l3-xc4xc4-1el4-l3-xc4/xc4By calculating the moments of the forces of the isolating body in Figure8(a) with respect to the left-end x=l4 cross-section of the rock beam, i.e., the bending moment of the left-end cross-section of the rock beam in this zone, can be obtained from ∑Ml4=0:(19)Ml4=Ml3+Ql3l3-l4+q1-q22l3-l42+k1xc13l4+xc1xc1+2×e-l4+xc1/xc1-l3+xc1xc12+2l3+xc1xc1+2-l3+xc1xc1+1×l4+xc1xc1e-l3+xc1/xc1+k4xc43e-15+2l4-l3-xc4xc4-2-l4-l3-xc4xc4el4-l3-xc4/xc4In (18) and (19), Ql3 and Ml3 are the shear forces and bending moments of the left end of the isolating body shown in Figure 8(a) or the right end of the isolating body shown in Figure 8(b).By computing twice the integral of (17), the equation of the deflection of the rock beam above the hardened zone of coal seam [l4,l3] can be expressed as follows:(20)EIy22x=Ml42x-l42-Ql46x-l43+q1-q224x-l44+k1xc15x+xc1xc1+4e-x+xc1/xc1-l4+xc1xc12+2l4+xc1xc1+2×12x+xc1xc12-16l4+xc1xc1+1x+xc1xc13e-l4+xc1/xc1-k4xc454-x-l3-xc4xc4ex-l3-xc4/xc4-12l4-l3-xc4xc42-2l4-l3-xc4xc4+2×x-l3-xc4xc42-16l4-l3-xc4xc4-1x-l3-xc4xc43el4-l3-xc4/xc4+j1x-l4+j2,l4≤x≤l3where j1 and j2 are integral constants determined from the continuity conditions.


### 3.3. Deflection Equation of Rock Beam above the Softened Zone of Coal Seam

Figure8(b) shows the chart of the isolating body of the rock beam above the softened zone of coal seam shown in Figure 4. The distributed load F1(x) was applied above the rock beam in zone [l3,l], and the support pressure f3(x) of the softened zone of coal seam was applied beneath that. The deflection of rock beam in the softened zone of coal seam [l3,l] was denoted as y23(x), and the form of y23(x) is the same as that reported in the literature [20]:(21)EIy23x=Ml32x-l32-Ql36x-l33+q124x-l34+k1xc15x+xc1xc1+4e-x+xc1/xc1-l3+xc1xc12+2l3+xc1xc1+2×12x+xc1xc12-16l3+xc1xc1+1x+xc1xc13e-l3+xc1/xc1-k3xc35l3+xc3-xxc3+4ex-l3-xc3/xc3+e-13l3+xc3-xxc33-52e-1l3+xc3-xxc32+g1x-l3+g2,l3≤x≤lIn (21), Ql3 and Ml3 are the shear forces and bending moments of the left-end cross-section of the isolating body shown in Figure 8(b). The sum of the vertical forces shown in Figure 8(b) was computed, and the shearing force of the left-end cross-section of the rock beam in zone [l3,l] can be obtained from ∑Fy=0:(22)Ql3=Ql+q1l-l3+k1xc12l3+xc1xc1+1e-l3+xc1/xc1-l+xc1xc1+1e-l+xc1/xc1-k3xc32l3+xc3-lxc3+1el-l3-xc3/xc3-2e-1By calculating the moments of the forces with respect to the left-endx=l3 cross-section shown in Figure 8(b), the equation for the bending moments of the rock beam in this zone can be obtained from ∑Ml3=0:(23)Ml3=Ml+Qll-l3+q12l-l32+k1xc13l3+xc1xc1+2×e-l3+xc1/xc1-l+xc1xc12+2l+xc1xc1+2-l+xc1xc1+1×l3+xc1xc1e-l+xc1/xc1-k3xc333e-1-l-l3-xc3xc32-l-l3-xc3xc3+1el-l3-xc3/xc3In (22) and (23), Ql and Ml are the shear forces and bending moments for the left end of the isolating body of rock beam shown in Figure 8(b) and x=l cross-section of the rock beam in the goaf shown in Figure 4.


### 3.4. Deflection Equation of Rock Beam in Goaf

The deflection equations of the rock beam in zones[l,l+Lk] and [l+Lk,l+L] are shown in Figure 4.  y11(x), y12(x), y11(x), and y12(x) are the same as those reported in the literature [20]:(24)EIy11x=Ml2x-l2-Ql6x-l3+q124x-l4+k1xc15×x+xc1xc1+4e-x+xc1/xc1-l+xc1xc12+2l+xc1xc1+2×12x+xc1xc12-16l+xc1xc1+1x+xc1xc13e-l+xc1/xc1-pox-l424-pk-po120Lkx-l5+c1x+c2,l≤x≤l+Lk(25)EIy12x=Ml2x-l2-Ql6x-l3+q124x-l4+k1xc15×x+xc1xc1+4e-x+xc1/xc1-l+xc1xc12+2l+xc1xc1+2×12x+xc1xc12-16l+xc1xc1+1x+xc1xc13e-l+xc1/xc1-poLkx-l36-Lk4x-l2-pk-poLk2x-l36-Lkx-l23+c3x+c4,l+Lk≤x≤l+L


### 3.5. Boundary and Continuity Conditions for Deflection of Rock Beam in All Zones

The boundary and continuity conditions for the rock beam model shown in Figure4 are listed as follows:(26a)y2-∞=q2C,y2′-∞=0(26b)y20=y210,y2′0=y21′0,y2′′0=y21′′0,y2′′′0=y21′′′0(26c)y21l4=y22l4,y21′l4=y22′l4,y21′′l4=Ml4EI,y21′′′l4=-Ql4EI,EIy214l4=EIy224l4(26d)y22l3=y23l3,y22′l3=y23′l3,y22′′l3=Ml3EI,y22′′′l3=-Ql3EI(26e)y23l=y11l,y23′l=y11′l,y23′′l=MlEI,y23′′′l=-QlEI(26f)y11l+Lk=y12l+Lk,y11′l+Lk=y12′l+Lk(26g)Q12l+L=0,y12′l+L=0The minus symbols ahead ofQl4, Ql3, and Ql in (26c), (26d), and (26e) are because of the following: the bending moment of fiber tension in the upper side of rock beam is positive, and the corresponding shear force is positive if anticlockwise. The shear forces Ql4, Ql3, and Ql shown in Figures 8(a) and 8(b) are clockwise; therefore, they are labeled with minus signs.By satisfying (26a), (26b), (26c), (26d), (26e), (26f), and (26g) for boundary and continuity conditions, 14 integral constants in (14), (15), (20), (21), (24), and (25) can be determined. Owing to the introduction of the hardened zone of coal seam, one more equation was added to the deflection equations of rock beam compared with literature [20], i.e., in (20), two more integral constants, j1 and j2, were added, and the continuity condition (26c) was added in (26a), (26b), (26c), (26d), (26e), (26f), and (26g). Therefore, the integral constants d1–d6, g1, g2, and c1–c4 in (14), (15), (21), (24), and (25) are totally different from those in the corresponding equations reported in the literature [20].The relationship between the equations of bending moment and deflection of rock beam in each zone can be expressed as follows:(27)Mx=EIy′′xThe relationship between the equations of shear force and bending moment and deflection can be expressed as follows: (28)Qx=dMxdx=EIy′′′x


## 3.1. Deflection Equation of Rock Beam for Elastic Foundation Zone

For the zones(-∞,0] and (0,l4] in Figure 4, the coal seam is in the elastic state; therefore, the supporting relationship of coal seam on roof in both the zones was treated as elastic foundation. The reaction force of elastic foundation qc is directly proportional to the displacement of rock beam y, i.e., qc=-Cy, where C is the elastic foundation constant and the minus represents the opposite direction of reaction force and displacement y (the settlement amount of rock beam). A distributed force F2(x)=q2+f2(x) was applied above the rock beam in zone (-∞,0]. The deflection of rock beam in this zone was denoted as y2(x). Following Timoshenko [27], the differential equation for the deflection of any infinitesimal segment dx in zone (-∞,0] of a rock beam can be derived as follows:(11)EIy24x=q2+k2xc2-xex-xc2/xc2-Cy2x,x≤0where E is the elastic modulus of roof under the condition of plane strain and I is the moment of inertia for the roof with unit width. If β=[C/(4EI)]1/4, (11) can be expressed as follows:(12)y24x+4β4y2x=q2EI+k2EIxc2-xex-xc2/xc2,x≤0For the zone[0,l4] shown in Figure 4, the distributed load applied above on the elastic foundation beam is F1(x)=q1+f1(x). The deflection of the rock beam in zone [0,l4] was denoted as y21(x). Similar to (12), the differential equation for the deflection of any infinitesimal segment dx in zone [0,l4] can be derived as follows:(13)y214x+4β4y21x=q1EI+k1EIx+xc1e-x+xc1/xc1,0≤x≤l4Because the differential equations for deflections (12) and (13) are identical to that of the corresponding zones reported in literature [20], the solution form of the differential equation for the deflection of semi-infinite elastic foundation beam in zone (-∞,0] is the same as reported in the literature [20], and it is denoted as follows:(14)y2x=eβxd1cos⁡βx+d2sin⁡βx+q24β4EI+k2EI·xc251+4β4xc24xc2-xxc2+41+4β4xc24ex-xc2/xc2,x≤0whereas that in zone [0,l4] is also the same as that reported in the literature [20] and can be expressed as follows:(15)y21x=d3sin⁡βxsinh⁡βx+d4sin⁡βxcosh⁡βx+d5cos⁡βxsinh⁡βx+d6cos⁡βxcosh⁡βx+q14β4EI+d5cos⁡βxsinh⁡βx+d6cos⁡βxcosh⁡βx+q14β4EI+k1EIxc151+4β4xc14x+xc1xc1+41+4β4xc14e-x+xc1/xc1,0≤x≤l4The values of integral constants d1–d6 in (14) and (15) are different from the values of d1–d6 reported in the literature [20]; therefore, these values should be redetermined according to the parameters of the hardened zone of coal seam.


## 3.2. Deflection Equation of Rock Beam above the Hardened Zone of Coal Seam

The distributed loadF1(x) was applied above the rock beam in zone [l4,l3] shown in Figure 4, and the support pressure of hardened zone of coal seam F4(x) was applied below the rock beam. The deflection of the rock beam in zone [l4,l3] was denoted as y22(x), and similar to (13), the differential equations for the deflection of the rock beam above the hardened zone of coal seam [l4,l3] can be written as follows:(16)y224x=q1EI+k1EIx+xc1e-x+xc1/xc1-q2EI-k4EIl3+xc4-xex-l3-xc4/xc4,l4≤x≤l3Figure8(a) shows the chart of the isolating body of the rock beam above the hardened zone of coal seam in Figure 4. By calculating the moments of the forces from the left of x cross-section with respect to x cross-section in Figure 8(a), the equation of the bending moment of the rock beam in zone [l4,l3] can be obtained as follows:(17)M22x=Ml4-Ql4x-l4+q1-q22x-l42+k1xc13×x+xc1xc1+2e-x+xc1/xc1-l4+xc1xc12+2l4+xc1xc1+2-l4+xc1xc1+1x+xc1xc1e-l4+xc1/xc1-k4xc43×2-x-l3-xc4xc4ex-l3-xc4/xc4-l4-l3-xc4xc42-2l4-l3-xc4xc4+2-l4-l3-xc4xc4-1x-l3-xc4xc4×el4-l3-xc4/xc4,l4≤x≤l3Figure 8
Charts of isolating body of the rock beam before first breaking.
(a) (b)The sum of vertical forces on the isolating body in Figure8(a) was calculated, and the shearing force of the left-end cross-section of the rock beam in zone [l4,l3] can be obtained from ∑Fy=0:(18)Ql4=Ql3+q1-q2l3-l4+k1xc12l4+xc1xc1+1e-l4+xc1/xc1-l3+xc1xc1+1e-l3+xc1/xc1-k4xc422e-1+l4-l3-xc4xc4-1el4-l3-xc4/xc4By calculating the moments of the forces of the isolating body in Figure8(a) with respect to the left-end x=l4 cross-section of the rock beam, i.e., the bending moment of the left-end cross-section of the rock beam in this zone, can be obtained from ∑Ml4=0:(19)Ml4=Ml3+Ql3l3-l4+q1-q22l3-l42+k1xc13l4+xc1xc1+2×e-l4+xc1/xc1-l3+xc1xc12+2l3+xc1xc1+2-l3+xc1xc1+1×l4+xc1xc1e-l3+xc1/xc1+k4xc43e-15+2l4-l3-xc4xc4-2-l4-l3-xc4xc4el4-l3-xc4/xc4In (18) and (19), Ql3 and Ml3 are the shear forces and bending moments of the left end of the isolating body shown in Figure 8(a) or the right end of the isolating body shown in Figure 8(b).By computing twice the integral of (17), the equation of the deflection of the rock beam above the hardened zone of coal seam [l4,l3] can be expressed as follows:(20)EIy22x=Ml42x-l42-Ql46x-l43+q1-q224x-l44+k1xc15x+xc1xc1+4e-x+xc1/xc1-l4+xc1xc12+2l4+xc1xc1+2×12x+xc1xc12-16l4+xc1xc1+1x+xc1xc13e-l4+xc1/xc1-k4xc454-x-l3-xc4xc4ex-l3-xc4/xc4-12l4-l3-xc4xc42-2l4-l3-xc4xc4+2×x-l3-xc4xc42-16l4-l3-xc4xc4-1x-l3-xc4xc43el4-l3-xc4/xc4+j1x-l4+j2,l4≤x≤l3where j1 and j2 are integral constants determined from the continuity conditions.


## 3.3. Deflection Equation of Rock Beam above the Softened Zone of Coal Seam

Figure8(b) shows the chart of the isolating body of the rock beam above the softened zone of coal seam shown in Figure 4. The distributed load F1(x) was applied above the rock beam in zone [l3,l], and the support pressure f3(x) of the softened zone of coal seam was applied beneath that. The deflection of rock beam in the softened zone of coal seam [l3,l] was denoted as y23(x), and the form of y23(x) is the same as that reported in the literature [20]:(21)EIy23x=Ml32x-l32-Ql36x-l33+q124x-l34+k1xc15x+xc1xc1+4e-x+xc1/xc1-l3+xc1xc12+2l3+xc1xc1+2×12x+xc1xc12-16l3+xc1xc1+1x+xc1xc13e-l3+xc1/xc1-k3xc35l3+xc3-xxc3+4ex-l3-xc3/xc3+e-13l3+xc3-xxc33-52e-1l3+xc3-xxc32+g1x-l3+g2,l3≤x≤lIn (21), Ql3 and Ml3 are the shear forces and bending moments of the left-end cross-section of the isolating body shown in Figure 8(b). The sum of the vertical forces shown in Figure 8(b) was computed, and the shearing force of the left-end cross-section of the rock beam in zone [l3,l] can be obtained from ∑Fy=0:(22)Ql3=Ql+q1l-l3+k1xc12l3+xc1xc1+1e-l3+xc1/xc1-l+xc1xc1+1e-l+xc1/xc1-k3xc32l3+xc3-lxc3+1el-l3-xc3/xc3-2e-1By calculating the moments of the forces with respect to the left-endx=l3 cross-section shown in Figure 8(b), the equation for the bending moments of the rock beam in this zone can be obtained from ∑Ml3=0:(23)Ml3=Ml+Qll-l3+q12l-l32+k1xc13l3+xc1xc1+2×e-l3+xc1/xc1-l+xc1xc12+2l+xc1xc1+2-l+xc1xc1+1×l3+xc1xc1e-l+xc1/xc1-k3xc333e-1-l-l3-xc3xc32-l-l3-xc3xc3+1el-l3-xc3/xc3In (22) and (23), Ql and Ml are the shear forces and bending moments for the left end of the isolating body of rock beam shown in Figure 8(b) and x=l cross-section of the rock beam in the goaf shown in Figure 4.


## 3.4. Deflection Equation of Rock Beam in Goaf

The deflection equations of the rock beam in zones[l,l+Lk] and [l+Lk,l+L] are shown in Figure 4.  y11(x), y12(x), y11(x), and y12(x) are the same as those reported in the literature [20]:(24)EIy11x=Ml2x-l2-Ql6x-l3+q124x-l4+k1xc15×x+xc1xc1+4e-x+xc1/xc1-l+xc1xc12+2l+xc1xc1+2×12x+xc1xc12-16l+xc1xc1+1x+xc1xc13e-l+xc1/xc1-pox-l424-pk-po120Lkx-l5+c1x+c2,l≤x≤l+Lk(25)EIy12x=Ml2x-l2-Ql6x-l3+q124x-l4+k1xc15×x+xc1xc1+4e-x+xc1/xc1-l+xc1xc12+2l+xc1xc1+2×12x+xc1xc12-16l+xc1xc1+1x+xc1xc13e-l+xc1/xc1-poLkx-l36-Lk4x-l2-pk-poLk2x-l36-Lkx-l23+c3x+c4,l+Lk≤x≤l+L


## 3.5. Boundary and Continuity Conditions for Deflection of Rock Beam in All Zones

The boundary and continuity conditions for the rock beam model shown in Figure4 are listed as follows:(26a)y2-∞=q2C,y2′-∞=0(26b)y20=y210,y2′0=y21′0,y2′′0=y21′′0,y2′′′0=y21′′′0(26c)y21l4=y22l4,y21′l4=y22′l4,y21′′l4=Ml4EI,y21′′′l4=-Ql4EI,EIy214l4=EIy224l4(26d)y22l3=y23l3,y22′l3=y23′l3,y22′′l3=Ml3EI,y22′′′l3=-Ql3EI(26e)y23l=y11l,y23′l=y11′l,y23′′l=MlEI,y23′′′l=-QlEI(26f)y11l+Lk=y12l+Lk,y11′l+Lk=y12′l+Lk(26g)Q12l+L=0,y12′l+L=0The minus symbols ahead ofQl4, Ql3, and Ql in (26c), (26d), and (26e) are because of the following: the bending moment of fiber tension in the upper side of rock beam is positive, and the corresponding shear force is positive if anticlockwise. The shear forces Ql4, Ql3, and Ql shown in Figures 8(a) and 8(b) are clockwise; therefore, they are labeled with minus signs.By satisfying (26a), (26b), (26c), (26d), (26e), (26f), and (26g) for boundary and continuity conditions, 14 integral constants in (14), (15), (20), (21), (24), and (25) can be determined. Owing to the introduction of the hardened zone of coal seam, one more equation was added to the deflection equations of rock beam compared with literature [20], i.e., in (20), two more integral constants, j1 and j2, were added, and the continuity condition (26c) was added in (26a), (26b), (26c), (26d), (26e), (26f), and (26g). Therefore, the integral constants d1–d6, g1, g2, and c1–c4 in (14), (15), (21), (24), and (25) are totally different from those in the corresponding equations reported in the literature [20].The relationship between the equations of bending moment and deflection of rock beam in each zone can be expressed as follows:(27)Mx=EIy′′xThe relationship between the equations of shear force and bending moment and deflection can be expressed as follows: (28)Qx=dMxdx=EIy′′′x


## 4. Determination of Integral Constants in Deflection Equations

### 4.1. Relationship among Integral Constantsc1–c4, Ml, g1, and g2

Because the rock beam model shown in Figure4 is statically indeterminate, a supplementary equation should be established through the geometric condition of y11′(l+L)=0 in (26g), defined as follows:(29)c3=QlL22-MlL-D1to determine the relationship between c3 of y12(x) in (25) and right-end bending moment Ml shown in Figure 8(b), where Ml is an unknown variable to be determined. In (29),(30)D1=q1L36+k1xc1412l+xc1xc1+1L+l+xc1xc12-L+l+xc1xc1l+xc1xc12+2l+xc1xc1+2e-l+xc1/xc1-L+l+xc1xc1+3e-L+l+xc1/xc1-poLkL22-LkL2-pk-poLk2L22-2LkL3According to the conditionQ12(l+L)/(EI)=0 in (26g) and ∑Fy=0, the right-hand term, i.e., the shear force in (29), can be obtained as follows:(31)Ql=q1L+k1xc12l+xc1xc1+1e-l+xc1/xc1-L+l+xc1xc1+1×e-L+l+xc1/xc1-poLk-pk-poLk2Using the first derivative of (24) and (25) and the condition y11′(l+Lk)=y12′(l+Lk) in (26f), the following expression can be obtained:(32)c1=c3+poLk36+pk-poLk38Regarding (24) and (25), according to the condition y11(l+Lk)=y12(l+Lk) in (26f), the following expression can be obtained:(33)c4=c1-c3l+Lk+c2-poLk48-11pk-poLk4120By substituting (21) and (24) into the conditions y23(l)=y11(l) and y23′(l)=y11′(l) in (26e), the followings equations can be obtained:(34)Ml32l-l32+D2+g1l-l3+g2=c1l+c2+D3(35)Ml3l-l3+D4+g1=c1+D5where(36)D2=-Ql36l-l33+q124l-l34+k1xc15×16l+xc1xc13l3+xc1xc1+1-12l+xc1xc12l3+xc1xc12+2l3+xc1xc1+2e-l3+xc1/xc1-k3xc35l3+xc3-lxc3+4el-l3-xc3/xc3+e-13l3+xc3-lxc33-52e-1l3+xc3-lxc32(37)D3=-k1xc1513l+xc1xc14+56l+xc1xc13+l+xc1xc12e-l+xc1/xc1(38)D4=-Ql32l-l32+q16l-l33+k1xc14×12l+xc1xc12l3+xc1xc1+1-l+xc1xc1l3+xc1xc12+2l3+xc1xc1+2e-l3+xc1/xc1-k3xc34l3+xc1-lxc1+3×el-l3-xc3/xc3-e-1l3+xc3-lxc32+5e-1l3+xc3-lxc3(39)D5=-k1xc1412l+xc1xc13+32l+xc1xc12+2l+xc1xc1e-l+xc1/xc1Equation (23) can be written in the following form:(40)Ml3=Ml+D6where(41)D6=Qll-l3+q12l-l32+k1xc13l3+xc1xc1+2e-l3+xc1/xc1-l+xc1xc12+2l+xc1xc1+2-l+xc1xc1+1×l3+xc1xc1e-l+xc1/xc1-k3xc333e-1-l-l3-xc3xc32-l-l3-xc3xc3+1el-l3-xc3/xc3The following expression can be obtained from (35) using (29) and (32).(42)g1=-Mll-l3+L+D7where(43)D7=QlL22-D1+poLk36+pk-poLk38-D4+D5-D6l-l3By substituting (20) and (21) into the conditions y22(l3)=y23(l3), y22′(l3)=y23′(l3) in (26d), the following equations can be obtained:(44)Ml42l3-l42+D8+j1l3-l4+j2=g2+D9(45)Ml4l3-l4+D10+j1=g1+D11where(46)D8=-Ql46l3-l43+q1-q224l3-l44-k1xc15×12l4+xc1xc12+2l4+xc1xc1+2l3+xc1xc12-16l4+xc1xc1+1l3+xc1xc13e-l4+xc1/xc1-k4xc45×5e-1-12l4-l3-xc4xc42-53l4-l3-xc4xc4+53×el4-l3-xc4/xc4(47)D9=-k1xc1513l3+xc1xc14+56l3+xc1xc13+l3+xc1xc12×e-l3+xc1/xc1-176k3xc35e-1(48)D10=-Ql42l3-l42+q1-q26l3-l43-k1xc14×l4+xc1xc12+2l4+xc1xc1+2l3+xc1xc1-12l4+xc1xc1+1×l3+xc1xc12e-l4+xc1/xc1-k4xc444e-1+l4-l3-xc4xc42-32l4-l3-xc4xc4+32el4-l3-xc4/xc4(49)D11=-k1xc1412l3+xc1xc13+32l3+xc1xc12+2l3+xc1xc1×e-l3+xc1/xc1-8k3xc34e-1Using (40), (19) can be written in the following form:(50)Ml4=Ml3+D12=Ml+D6+D12where(51)D12=Ql3l3-l4+q1-q22l3-l42+k1xc13l4+xc1xc1+2×e-l4+xc1/xc1-l3+xc1xc12+2l3+xc1xc1+2-l3+xc1xc1+1×l4+xc1xc1e-l3+xc1/xc1+k4xc43e-15+2l4-l3-xc4xc4-2-l4-l3-xc4xc4el4-l3-xc4/xc4From (42), (45), and (50), the following equation can be obtained:(52)j1=-Mll-l4+L+D13where(53)D13=D7-D10+D11-l3-l4D6+D12


### 4.2. Relationship among Integral Constantsd1– d6, j1, and j2

Using (14), (15), and their 1st–3rd derivatives, according to the deflection, dip angle, bending moment, and shear force continuity conditions in (26b), four algebraic equations for d1–d6 can be derived. The following can be obtained by solving the equation as follows:(54)d1=d6+q1-q24β4EI+A1xc13B1+1-A2xc23B2+1(55)d2=d3+A1xc1B1-1-A2xc2B2-12β2(56)d4=d3+D14(57)d5=d6+D15The following has been shown in (54)–(57):(58)B1=41+4β4xc14,B2=41+4β4xc24,A1=k1EIxc12e-11+4β4xc14,A2=k2EIxc22e-11+4β4xc24(59)D14=A1xc12B1+A2xc22B22β+A1xc1B1-1-A2xc2B2-12β2+A1B1-2+A2B2-24β3(60)D15=q1-q24β4EI+A1xc13B1+1-A2xc23B2+1+A1xc12B1+A2xc22B22β-A1B1-2+A2B2-24β3Using the 1st–3rd derivatives of (15) and (20), according to the first four conditions in (26c), four algebraic equations for d3–d6, j1, j2, Ml4=Ml+D6+D12, and Ql4 for x=l4 can be derived as follows:(61)d3sin⁡βl4sinh⁡βl4+d4sin⁡βl4cosh⁡βl4+d5cos⁡βl4sinh⁡βl4+d6cos⁡βl4cosh⁡βl4+D16=j2EI+D17(62)βcos⁡βl4sinh⁡βl4+sin⁡βl4cosh⁡βl4d3+cos⁡βl4cosh⁡βl4+sin⁡βl4sinh⁡βl4d4+cos⁡βl4cosh⁡βl4-sin⁡βl4sinh⁡βl4d5+cos⁡βl4sinh⁡βl4-sin⁡βl4cosh⁡βl4d6-D18=j1EI+D19(63)2β2d3cos⁡βl4cosh⁡βl4+d4cos⁡βl4sinh⁡βl4-d5sin⁡βl4cosh⁡βl4-d6sin⁡βl4sinh⁡βl4+D20=MlEI+D6EI+D12EI(64)2β3cos⁡βl4sinh⁡βl4-sin⁡βl4cosh⁡βl4d3+cos⁡βl4cosh⁡βl4-sin⁡βl4sinh⁡βl4d4-cos⁡βl4cosh⁡βl4+sin⁡βl4sinh⁡βl4d5-cos⁡βl4sinh⁡βl4+sin⁡βl4cosh⁡βl4d6-D21=-Ql4EIThe following has been shown in (61)–(64):(65)D16=q14β4EI+A1l4xc13B1+l4xc1+1(66)D17=k1xc15EI-13l4+xc1xc14-56l4+xc1xc13-l4+xc1xc12+l4+xc1xc1+4e-l3+xc1/xc1-k4xc45EI-13l4-l3-xc4xc44+56l4-l3-xc4xc43-l4-l3-xc4xc42-l4-l3-xc4xc4+4×el4-l3-xc4/xc4(67)D18=A1l4xc12B1+l4xc1(68)D19=-k1xc14EI12l4+xc1xc13+32l4+xc1xc12+3l4+xc1xc1+3e-l4+xc1/xc1-k4xc44EI-12l4-l3-xc4xc43+32l4-l3-xc4xc42-3l4-l3-xc4xc4+3el4-l3-xc4/xc4(69)D20=A1l4xc1B1+l4xc1-1(70)D21=A1l4B1+l4xc1-2(71)A1l4=k1EIxc121+4β4xc14e-l4+xc1/xc1


### 4.3. Determination ofc1–c4, d1–d6, g1, g2, j1, j2, and Ml

By substitutingj1=-Ml(l-l4+L)+D13 in (52) into (62) and adding to (63) to eliminate Ml, the following equation can be obtained:(72)cos⁡βl4sinh⁡βl4+sin⁡βl4cosh⁡βl4+2βl-l4+Lcos⁡βl4cosh⁡βl4d3+cos⁡βl4cosh⁡βl4+sin⁡βl4sinh⁡βl4+2βl-l4+Lcos⁡βl4sinh⁡βl4d4+cos⁡βl4cosh⁡βl4-sin⁡βl4sinh⁡βl4-2βl-l4+Lsin⁡βl4cosh⁡βl4d5+cos⁡βl4sinh⁡βl4-sin⁡βl4cosh⁡βl4-2βl-l4+Lsin⁡βl4sinh⁡βl4d6=D22where(73)D22=l-l4+LD6+D12+D13βEI+D18+D19β-l-l4+LβD20Q l 4 and D22 are known, and d4 and d5 can be expressed as d3 and d6 according to (56) and (57); therefore, (72) and (64) form a linear equation of two unknowns d3 and d6, and d3 and d6 can be solved as follows:(74)d3=b1a22-b2a12a11a22-a12a21,d6=a11b2-a21b1a11a22-a12a21where(75)a11=eβl4cos⁡βl4+sin⁡βl4+2βl-l4+Lcos⁡βl4(76)a12=eβl4cos⁡βl4-sin⁡βl4-2βl-l4+Lsin⁡βl4(77)a21=eβl4cos⁡βl4-sin⁡βl4(78)a22=-eβl4cos⁡βl4+sin⁡βl4(79)b1=D22-cos⁡βl4cosh⁡βl4+sin⁡βl4sinh⁡βl4+2βl-l4+Lcos⁡βl4sinh⁡βl4D14+sin⁡βl4sinh⁡βl4-cos⁡βl4cosh⁡βl4+2βl-l4+Lsin⁡βl4cosh⁡βl4D15(80)b2=D21EI-Ql42β3EI+sin⁡βl4sinh⁡βl4-cos⁡βl4cosh⁡βl4D14+sin⁡βl4sinh⁡βl4+cos⁡βl4cosh⁡βl4D15The right-hand terms of (75)–(80) are known; therefore, d1–d6 can be determined from (74) and (54)–(57). By substituting d3–d6 into (61)–(63), j1, j2, and Ml can be determined; Ml4, j1, j2, and Ml3 can be further determined from (50), (44), (45), and (40). Furthermore, using (29) and (32), c3 and c1 can be determined. Then, using (34) and (33), c2 and c4 can be determined. All these finally determine the deflection (14) and (15) of an elastic foundation beam, deflection (20) and (21) of a rock beam in the hardened and softened zones of coal seam, and deflection (20) and (21) of a rock beam in goaf.


## 4.1. Relationship among Integral Constantsc1–c4, Ml, g1, and g2

Because the rock beam model shown in Figure4 is statically indeterminate, a supplementary equation should be established through the geometric condition of y11′(l+L)=0 in (26g), defined as follows:(29)c3=QlL22-MlL-D1to determine the relationship between c3 of y12(x) in (25) and right-end bending moment Ml shown in Figure 8(b), where Ml is an unknown variable to be determined. In (29),(30)D1=q1L36+k1xc1412l+xc1xc1+1L+l+xc1xc12-L+l+xc1xc1l+xc1xc12+2l+xc1xc1+2e-l+xc1/xc1-L+l+xc1xc1+3e-L+l+xc1/xc1-poLkL22-LkL2-pk-poLk2L22-2LkL3According to the conditionQ12(l+L)/(EI)=0 in (26g) and ∑Fy=0, the right-hand term, i.e., the shear force in (29), can be obtained as follows:(31)Ql=q1L+k1xc12l+xc1xc1+1e-l+xc1/xc1-L+l+xc1xc1+1×e-L+l+xc1/xc1-poLk-pk-poLk2Using the first derivative of (24) and (25) and the condition y11′(l+Lk)=y12′(l+Lk) in (26f), the following expression can be obtained:(32)c1=c3+poLk36+pk-poLk38Regarding (24) and (25), according to the condition y11(l+Lk)=y12(l+Lk) in (26f), the following expression can be obtained:(33)c4=c1-c3l+Lk+c2-poLk48-11pk-poLk4120By substituting (21) and (24) into the conditions y23(l)=y11(l) and y23′(l)=y11′(l) in (26e), the followings equations can be obtained:(34)Ml32l-l32+D2+g1l-l3+g2=c1l+c2+D3(35)Ml3l-l3+D4+g1=c1+D5where(36)D2=-Ql36l-l33+q124l-l34+k1xc15×16l+xc1xc13l3+xc1xc1+1-12l+xc1xc12l3+xc1xc12+2l3+xc1xc1+2e-l3+xc1/xc1-k3xc35l3+xc3-lxc3+4el-l3-xc3/xc3+e-13l3+xc3-lxc33-52e-1l3+xc3-lxc32(37)D3=-k1xc1513l+xc1xc14+56l+xc1xc13+l+xc1xc12e-l+xc1/xc1(38)D4=-Ql32l-l32+q16l-l33+k1xc14×12l+xc1xc12l3+xc1xc1+1-l+xc1xc1l3+xc1xc12+2l3+xc1xc1+2e-l3+xc1/xc1-k3xc34l3+xc1-lxc1+3×el-l3-xc3/xc3-e-1l3+xc3-lxc32+5e-1l3+xc3-lxc3(39)D5=-k1xc1412l+xc1xc13+32l+xc1xc12+2l+xc1xc1e-l+xc1/xc1Equation (23) can be written in the following form:(40)Ml3=Ml+D6where(41)D6=Qll-l3+q12l-l32+k1xc13l3+xc1xc1+2e-l3+xc1/xc1-l+xc1xc12+2l+xc1xc1+2-l+xc1xc1+1×l3+xc1xc1e-l+xc1/xc1-k3xc333e-1-l-l3-xc3xc32-l-l3-xc3xc3+1el-l3-xc3/xc3The following expression can be obtained from (35) using (29) and (32).(42)g1=-Mll-l3+L+D7where(43)D7=QlL22-D1+poLk36+pk-poLk38-D4+D5-D6l-l3By substituting (20) and (21) into the conditions y22(l3)=y23(l3), y22′(l3)=y23′(l3) in (26d), the following equations can be obtained:(44)Ml42l3-l42+D8+j1l3-l4+j2=g2+D9(45)Ml4l3-l4+D10+j1=g1+D11where(46)D8=-Ql46l3-l43+q1-q224l3-l44-k1xc15×12l4+xc1xc12+2l4+xc1xc1+2l3+xc1xc12-16l4+xc1xc1+1l3+xc1xc13e-l4+xc1/xc1-k4xc45×5e-1-12l4-l3-xc4xc42-53l4-l3-xc4xc4+53×el4-l3-xc4/xc4(47)D9=-k1xc1513l3+xc1xc14+56l3+xc1xc13+l3+xc1xc12×e-l3+xc1/xc1-176k3xc35e-1(48)D10=-Ql42l3-l42+q1-q26l3-l43-k1xc14×l4+xc1xc12+2l4+xc1xc1+2l3+xc1xc1-12l4+xc1xc1+1×l3+xc1xc12e-l4+xc1/xc1-k4xc444e-1+l4-l3-xc4xc42-32l4-l3-xc4xc4+32el4-l3-xc4/xc4(49)D11=-k1xc1412l3+xc1xc13+32l3+xc1xc12+2l3+xc1xc1×e-l3+xc1/xc1-8k3xc34e-1Using (40), (19) can be written in the following form:(50)Ml4=Ml3+D12=Ml+D6+D12where(51)D12=Ql3l3-l4+q1-q22l3-l42+k1xc13l4+xc1xc1+2×e-l4+xc1/xc1-l3+xc1xc12+2l3+xc1xc1+2-l3+xc1xc1+1×l4+xc1xc1e-l3+xc1/xc1+k4xc43e-15+2l4-l3-xc4xc4-2-l4-l3-xc4xc4el4-l3-xc4/xc4From (42), (45), and (50), the following equation can be obtained:(52)j1=-Mll-l4+L+D13where(53)D13=D7-D10+D11-l3-l4D6+D12


## 4.2. Relationship among Integral Constantsd1– d6, j1, and j2

Using (14), (15), and their 1st–3rd derivatives, according to the deflection, dip angle, bending moment, and shear force continuity conditions in (26b), four algebraic equations for d1–d6 can be derived. The following can be obtained by solving the equation as follows:(54)d1=d6+q1-q24β4EI+A1xc13B1+1-A2xc23B2+1(55)d2=d3+A1xc1B1-1-A2xc2B2-12β2(56)d4=d3+D14(57)d5=d6+D15The following has been shown in (54)–(57):(58)B1=41+4β4xc14,B2=41+4β4xc24,A1=k1EIxc12e-11+4β4xc14,A2=k2EIxc22e-11+4β4xc24(59)D14=A1xc12B1+A2xc22B22β+A1xc1B1-1-A2xc2B2-12β2+A1B1-2+A2B2-24β3(60)D15=q1-q24β4EI+A1xc13B1+1-A2xc23B2+1+A1xc12B1+A2xc22B22β-A1B1-2+A2B2-24β3Using the 1st–3rd derivatives of (15) and (20), according to the first four conditions in (26c), four algebraic equations for d3–d6, j1, j2, Ml4=Ml+D6+D12, and Ql4 for x=l4 can be derived as follows:(61)d3sin⁡βl4sinh⁡βl4+d4sin⁡βl4cosh⁡βl4+d5cos⁡βl4sinh⁡βl4+d6cos⁡βl4cosh⁡βl4+D16=j2EI+D17(62)βcos⁡βl4sinh⁡βl4+sin⁡βl4cosh⁡βl4d3+cos⁡βl4cosh⁡βl4+sin⁡βl4sinh⁡βl4d4+cos⁡βl4cosh⁡βl4-sin⁡βl4sinh⁡βl4d5+cos⁡βl4sinh⁡βl4-sin⁡βl4cosh⁡βl4d6-D18=j1EI+D19(63)2β2d3cos⁡βl4cosh⁡βl4+d4cos⁡βl4sinh⁡βl4-d5sin⁡βl4cosh⁡βl4-d6sin⁡βl4sinh⁡βl4+D20=MlEI+D6EI+D12EI(64)2β3cos⁡βl4sinh⁡βl4-sin⁡βl4cosh⁡βl4d3+cos⁡βl4cosh⁡βl4-sin⁡βl4sinh⁡βl4d4-cos⁡βl4cosh⁡βl4+sin⁡βl4sinh⁡βl4d5-cos⁡βl4sinh⁡βl4+sin⁡βl4cosh⁡βl4d6-D21=-Ql4EIThe following has been shown in (61)–(64):(65)D16=q14β4EI+A1l4xc13B1+l4xc1+1(66)D17=k1xc15EI-13l4+xc1xc14-56l4+xc1xc13-l4+xc1xc12+l4+xc1xc1+4e-l3+xc1/xc1-k4xc45EI-13l4-l3-xc4xc44+56l4-l3-xc4xc43-l4-l3-xc4xc42-l4-l3-xc4xc4+4×el4-l3-xc4/xc4(67)D18=A1l4xc12B1+l4xc1(68)D19=-k1xc14EI12l4+xc1xc13+32l4+xc1xc12+3l4+xc1xc1+3e-l4+xc1/xc1-k4xc44EI-12l4-l3-xc4xc43+32l4-l3-xc4xc42-3l4-l3-xc4xc4+3el4-l3-xc4/xc4(69)D20=A1l4xc1B1+l4xc1-1(70)D21=A1l4B1+l4xc1-2(71)A1l4=k1EIxc121+4β4xc14e-l4+xc1/xc1


## 4.3. Determination ofc1–c4, d1–d6, g1, g2, j1, j2, and Ml

By substitutingj1=-Ml(l-l4+L)+D13 in (52) into (62) and adding to (63) to eliminate Ml, the following equation can be obtained:(72)cos⁡βl4sinh⁡βl4+sin⁡βl4cosh⁡βl4+2βl-l4+Lcos⁡βl4cosh⁡βl4d3+cos⁡βl4cosh⁡βl4+sin⁡βl4sinh⁡βl4+2βl-l4+Lcos⁡βl4sinh⁡βl4d4+cos⁡βl4cosh⁡βl4-sin⁡βl4sinh⁡βl4-2βl-l4+Lsin⁡βl4cosh⁡βl4d5+cos⁡βl4sinh⁡βl4-sin⁡βl4cosh⁡βl4-2βl-l4+Lsin⁡βl4sinh⁡βl4d6=D22where(73)D22=l-l4+LD6+D12+D13βEI+D18+D19β-l-l4+LβD20Q l 4 and D22 are known, and d4 and d5 can be expressed as d3 and d6 according to (56) and (57); therefore, (72) and (64) form a linear equation of two unknowns d3 and d6, and d3 and d6 can be solved as follows:(74)d3=b1a22-b2a12a11a22-a12a21,d6=a11b2-a21b1a11a22-a12a21where(75)a11=eβl4cos⁡βl4+sin⁡βl4+2βl-l4+Lcos⁡βl4(76)a12=eβl4cos⁡βl4-sin⁡βl4-2βl-l4+Lsin⁡βl4(77)a21=eβl4cos⁡βl4-sin⁡βl4(78)a22=-eβl4cos⁡βl4+sin⁡βl4(79)b1=D22-cos⁡βl4cosh⁡βl4+sin⁡βl4sinh⁡βl4+2βl-l4+Lcos⁡βl4sinh⁡βl4D14+sin⁡βl4sinh⁡βl4-cos⁡βl4cosh⁡βl4+2βl-l4+Lsin⁡βl4cosh⁡βl4D15(80)b2=D21EI-Ql42β3EI+sin⁡βl4sinh⁡βl4-cos⁡βl4cosh⁡βl4D14+sin⁡βl4sinh⁡βl4+cos⁡βl4cosh⁡βl4D15The right-hand terms of (75)–(80) are known; therefore, d1–d6 can be determined from (74) and (54)–(57). By substituting d3–d6 into (61)–(63), j1, j2, and Ml can be determined; Ml4, j1, j2, and Ml3 can be further determined from (50), (44), (45), and (40). Furthermore, using (29) and (32), c3 and c1 can be determined. Then, using (34) and (33), c2 and c4 can be determined. All these finally determine the deflection (14) and (15) of an elastic foundation beam, deflection (20) and (21) of a rock beam in the hardened and softened zones of coal seam, and deflection (20) and (21) of a rock beam in goaf.


## 5. Method to Determine the Maximum Support Pressurefc3

The values offc3 and fc4 are critical parameters after introducing the softened and hardened zones of coal seam. Eq. (10) fc4=fc3-q2 provides the relationship between both these parameters. In a specific calculation, the value of fc3 was empirically “determined”, and the criterion of such “determination” was to ensure the continuity of the counterforce of rock beam by coal seam at the junction between the elastic and hardened zones. The accurate method to determine fc3 is given as follows.The last condition in (26c) is defined by(81)A:EIy214l4=EIy224l4By comparing (13) and (16), (81) indicates the continuity of the sum of all the vertical distributed forces, including the counterforce of coal seam, on the rock beam at the junction between the elastic and hardened zones of coal seam (x=l4) shown in Figure 4. Notice that β=[C/(4EI)]1/4 in (12) and k3=fc3e/xc3 in (6). By substituting (13) and (16) into (81), the following relationship can be obtained:(82)Cy21l4=F4l4=q2+f4l4The left-hand term of (82) is the counterforce of coal seam in the elastic zone on the left side of x=l4 that is applied on the rock beam, while the right-hand term is the counterforce of coal seam in the hardened zone on the right side of x=l4 that is applied on the rock beam. Equation (82) clarifies the issue in the calculation that the continuity of the counterforce of the coal seam at the junction between the elastic and hardened zones on the rock beam after the introduction of the hardened zone of coal seam. By substituting (7) and (10) into (82) and using (9), the right-hand term of (82) can be expressed as the maximum support pressure fc3 as follows:(83)Cy21l4=F4l4=q2+2e-1fc3-q2The figure ofEIy(4)(x) is named as onbeam vertical distributed force curve. Equation (83) shows that the continuity condition of the counterforce of coal seam at the junction between the elastic and hardened zones (x=l4) can be expressed as fc3. Thus, method A used for determining the value of fc3 was obtained using (81): the value of fc3 to ensure the continuity of curve EIy(4)(x) at x=l4 is the accurate value of fc3 for the continuity of support pressure (or counterforce) of coal seam at the junction between the elastic and hardened zones (x=l4) on rock beam.The following example can be used to illustrate the application of method A to determine the value offc3. If q2=7MN/m, xc1=8m, xc2=13m, xc3=8.6 m, and xc4=2m and the foundation constant of elastic zone of coal seam C=0.4GPa and elastic modulus of roof E=25GPa, together with the equations and parameters described in Section 2, by substituting (15) and (20) into EIy(4)(x) in (13) and (16), the distribution curve chart of EIy(4)(x) can be plotted by Matlab for the elastic zone [0,l4] and the hardened zone [l4,l3] of coal seam, as shown in Figure 9.Figure 9
TheEIy(4)(x) curves at with different fc3 values.Figure9 shows that when the maximum support pressure fc3 in the softened zone of coal seam is 7×1.7 or 7×1 MN/m, EIy(4)(x) distribution curves a and b are discontinuous at the junction between the elastic and hardened zones of coal seam at x=8 m, while for fc3=7×1.435272=10.046906MN/m, EIy(4)(x) distribution curve 1 is continuous at the junction at x=8 m.


## 6. Numerical Examples

An increase in the advanced distance of working face or hanging roof area of goaf changes the parameters of the softened zone of coal seam. For instance, as the hanging roof distance increases, the overall load above the coal seam increases, and the maximum support pressure increases (Figure2), but the residual strength of coal seam at coal wall f3(l) gradually decreases. The depth of the softened zone of coal seam increases (i.e., l-l3 increased).As mentioned above, let the mean load in Figure4  q2=7MN/m, xc1=8m, and xc2=13m; concerning the rock beam with a hanging roof distance of L=20m, assuming that as the working face advances each 2 m, xc3 in (5) decreases from 8.6 m by 0.3 m. Using the relevant equations described in Sections 2 and 3 and the integral constants described in Section 4, concerning the rock beam in the goaf with a hanging roof distance of L=20, 22, 24 m, l=18 m, l3=10 m (i.e., the depth of softened zone remains constant, l-l3=8 m), xc3=8.6, 8.3, 8 m, xc4=2 m, the elastic modulus of roof is E=25 GPa, the elastic foundation constant C=0.4 GPa, po=0.9 MN/m, and pk=1.2po. According to (13), (15), and (16) and the method introduced in Section 5, after a couple of trial calculations, the maximum support pressure of coal seam was found to be accurate to the sixth decimal place:(84)fc3=10.046906,10.541588,11.076129MN/mSubstituting this set of fc3 values (5), the support pressure of coal seam at the coal wall can be obtained as follows:(85)f3l≈1.78,1.00,0MN/mEquations (84) and (85) show that as the hanging roof distance of goaf L increases, the maximum support pressure of coal seam fc3 increases, and the residual strength of coal seam at coal wall f3(l) decreases. When L=24 m, the residual strength of coal seam f3(l)≈0.Based on thefc3 values listed in (84) and the condition of constant depth of the softened zone of coal seam (l-l3)=8 m, numerical examples for the support roof model of elastic, hardened, and softened zones are shown in Figure 4. The results were compared with the roof bending moments without considering the hardening of coal seam [20] (coal seam can be divided into two zones: elastic and softened zones).


### 6.1. Support Pressure of Coal Seam as Hanging Roof DistanceL Increases

Curves 1, 2, and 3 in Figure10 show the support pressure curves of softened, hardened, and elastic zones of coal seam when the working face advances or the hanging roof distance of goaf increases. They were plotted according to fc3 values in (84) and (5), (7), (14), and (15). To compare with the support pressures in Figures 1 and 2, the positive direction of support pressure of coal seam is shown upward in the chart. Figure 10 shows that the curves 1, 2, and 3 are smooth and continuous at the junction between the hardened and softened zones of coal seam (location of support pressure peak) and the junction between two elastic zones (o1, o2, and o3). In Figure 10, as the hanging roof distance L increases (the total load above the corresponding coal seam increases), the maximum support pressure of coal seam fc3 gradually increases, while the residual strength of coal seam at the coal wall f3(l) decreases (among which f3(l) of curve 3 tends to be close to 0), similar to the variation trend of support curve shown in Figure 2.Figure 10
Support pressure curves of coal seam with the increasing of the distance of overhang roof.The curves 1, 2, and 3 in Figure11 are magnified curves corresponding to the range of support pressure peaks in Figure 10. The support pressure curves I, II, and III are also shown in the chart without considering coal seam hardening according to the literature [20] under the same conditions of overlying load and hanging roof distance (curves are not smooth at the junction between the softened and elastic zones). The peak values of curves I, II, and III in Figure 11 are defined as follows:(86)fc3=10.294614,10.838460,11.076129MN/mThey were obtained based on the continuity conditions of counterforces of coal seam at the junction between the elastic and softened zones according to literature [20]. Figure 11 shows that the peak values fc3 of curves I, II, and III are higher than those of curves 1, 2, and 3; however, their plumpness is inferior to that of curves 1, 2, and 3.Figure 11
Enlarged drawings at peak part of support pressure with the increasing of the distance of overhang roof.Figure11 also shows that owing to the parameter of hardened zone of coal seam xc4=2 m, the hardened zones of support pressure curves 1, 2, and 3 for three zones are all 2 m (see blue line segments in Figure 11; red and green line segments in Figure 11 denote the ranges of the elastic and softened zones, respectively).


### 6.2. Deflection of Rock Beam as Hanging Roof DistanceL Increases

Figure12 shows the deflection curves of rock beam with the increase in hanging roof distance corresponding to the support pressure curves of coal seam in Figure 10. Curves 1, 2, and 3 ahead of x=−25m tend to be 19 mm, close to the condition of y(-∞)=q2/C=17.5 mm in (26a). A horizontal tangential line exists on the deflection curve of rock beam at the midspan position of the goaf, where the location is (l+L)=38, 40, 42 m, satisfying the condition of y12′(l+L)=0 in (26g). Besides, y12(l+L)=116, 139, 165 mm, i.e., each increase of 2 m of the hanging roof distance at this advancing phase results in an increase of (23–26) mm of the deflection of rock beam at goaf midspan.Figure 12
Deflection curves of rock beam with the increasing of the distance of overhang roof.Additionally, the support pressure parts in(-∞,0] and (0,l4] of both sides of the three origins o1, o2, and o3 in Figure 10 are shown by C·y2(x) and C·y21(x). These are the deflections of rock beam y2(x) and y21(x) of the corresponding segments in Figure 12 by multiplying foundation constant C=0.4GPa, respectively.Figure13 shows the magnified chart of the deflections of rock beam at the junction between the elastic and hardened zones (red and blue lines) of coal seam in Figure 12, where the horizontal axis of intersection between three vertical dashed lines whose intervals between the neighboring lines are 2 m and curves 1, 2, and 3 are 23.10, 24.01 and 25.00 mm. By multiplying these three values by the elastic foundation constant C=0.4GPa, the counterforces or the support pressures of the coal seam on the right end of the elastic zone can be obtained: C·y21(l4)≈9.240, 9.604, 10.00 MN/m. By substituting q2 and fc3 in Figures 12 and 13 into the right term of (83), the support pressure on the left end of the hardened zone of coal seam can be computed: F4(l4)≈9.242, 9.606, 9.999 MN/m, almost equal to the support pressures on the right end of the elastic zone of coal seam, indicating the validity of (82) for the continuity of counterforce of coal seam on rock beam at the junction between the elastic and hardened zones of coal seam described in Section 5 and the sufficient accuracy of the maximum support pressure fc3 given in (86).Figure 13
Enlarged drawings of rock beam deflection curves at the junction between elastic area and hardening area of coal seam.


### 6.3. Bending Moment of Rock Beam as Hanging Roof DistanceL Increases

Figure14 shows the bending moment curves of rock beam corresponding to the deflection curves shown in Figure 11. The curves tend to 0 ahead of x=−25 m, and a horizontal tangential line M12′(l+L)=0 exists at the midspan of goaf on each of the curves. Based on (28) for bending moment and shear force, it shows that the shearing force of rock beam on the midspan cross-section of goaf satisfies Q12(l+L)=0, conforming to the boundary condition of (26g). The peak values of bending moment of curves 1, 2, and 3 ahead of coal wall are Mmax(x^)=1.62×108, 1.83×108, 2.06×108 N·m, and the values of bending moment at the midspan of goaf are M12(L+l)= −1.61×108, −1.72×108, and −1.81×108 N·m; i.e., the maximum bending moment of rock beam ahead of coal wall remains higher than the absolute value of bending moment at the midspan of goaf.Figure 14
Bending moment curves of rock beam with the increasing of the distance of overhang roof.Curves 1, 2, and 3 in Figure15 show the magnified chart for peak parts of the bending moment curves of rock beam shown in Figure 14. Curves 1, 2, and 3 achieved their peak values of bending moment Mmax(x^) at x^=10.59, 10.55, 10.51 m. The position at the junction between the hardened and softened zones of coal seam is l3=10 m, the abovementioned three x^ values are all higher than 10 m. Thus, the peaks of the three bending moments of rock beam are located in the softened segment of coal seam (green line segments in Figure 14; the blue and red line segments in Figure 15 are the hardened and elastic ranges of coal seam, respectively). The advanced distance of bending moment peak of rock beam was denoted as l^, and each x^ value at the coal wall position l=18 m was subtracted. The advanced distance of bending moment peak of rock beam l^=l-x≈7.41, 7.45, 7.49 m can be obtained, indicating that the advanced distance of bending moment peak of rock beam increases as the hanging roof distance increases.Figure 15
Enlarged drawings at peak part of rock beam bending moment with the increasing of the distance of overhang roof.Curves I, II, and III in Figure15 show the magnified chart of the bending moment peaks of rock beam corresponding to the two-zone support pressure curves shown in Figure 11. Curves I, II, and III achieved their bending moment peaks at x^=10.86, 10.84, 10.81 m. By subtracting each x^ value at the coal wall position l=18 m, the advanced distance of bending moment peak of rock beam Mmax(x^) can be obtained as l^= 7.14, 7.16, 7.19 m.By comparing thel^ values among curves 1, 2, and 3 and curves I, II, III, it was found that the advanced distance of bending moment peak of the support rock beam with elastic-hardened-softened three-zone coal seam with a hardened zone parameter length of xc4=2 m exceeds those of the support rock beam with elastic-softened two-zone coal seam by about 27, 29 and 30 cm. Figure 15 also shows that, under identical conditions of overlying load and hanging roof distance, the peak values of bending moments for the support roof of three zones considering coal seam hardening are slightly higher than those of two zones without considering coal seam hardening.


## 6.1. Support Pressure of Coal Seam as Hanging Roof DistanceL Increases

Curves 1, 2, and 3 in Figure10 show the support pressure curves of softened, hardened, and elastic zones of coal seam when the working face advances or the hanging roof distance of goaf increases. They were plotted according to fc3 values in (84) and (5), (7), (14), and (15). To compare with the support pressures in Figures 1 and 2, the positive direction of support pressure of coal seam is shown upward in the chart. Figure 10 shows that the curves 1, 2, and 3 are smooth and continuous at the junction between the hardened and softened zones of coal seam (location of support pressure peak) and the junction between two elastic zones (o1, o2, and o3). In Figure 10, as the hanging roof distance L increases (the total load above the corresponding coal seam increases), the maximum support pressure of coal seam fc3 gradually increases, while the residual strength of coal seam at the coal wall f3(l) decreases (among which f3(l) of curve 3 tends to be close to 0), similar to the variation trend of support curve shown in Figure 2.Figure 10
Support pressure curves of coal seam with the increasing of the distance of overhang roof.The curves 1, 2, and 3 in Figure11 are magnified curves corresponding to the range of support pressure peaks in Figure 10. The support pressure curves I, II, and III are also shown in the chart without considering coal seam hardening according to the literature [20] under the same conditions of overlying load and hanging roof distance (curves are not smooth at the junction between the softened and elastic zones). The peak values of curves I, II, and III in Figure 11 are defined as follows:(86)fc3=10.294614,10.838460,11.076129MN/mThey were obtained based on the continuity conditions of counterforces of coal seam at the junction between the elastic and softened zones according to literature [20]. Figure 11 shows that the peak values fc3 of curves I, II, and III are higher than those of curves 1, 2, and 3; however, their plumpness is inferior to that of curves 1, 2, and 3.Figure 11
Enlarged drawings at peak part of support pressure with the increasing of the distance of overhang roof.Figure11 also shows that owing to the parameter of hardened zone of coal seam xc4=2 m, the hardened zones of support pressure curves 1, 2, and 3 for three zones are all 2 m (see blue line segments in Figure 11; red and green line segments in Figure 11 denote the ranges of the elastic and softened zones, respectively).


## 6.2. Deflection of Rock Beam as Hanging Roof DistanceL Increases

Figure12 shows the deflection curves of rock beam with the increase in hanging roof distance corresponding to the support pressure curves of coal seam in Figure 10. Curves 1, 2, and 3 ahead of x=−25m tend to be 19 mm, close to the condition of y(-∞)=q2/C=17.5 mm in (26a). A horizontal tangential line exists on the deflection curve of rock beam at the midspan position of the goaf, where the location is (l+L)=38, 40, 42 m, satisfying the condition of y12′(l+L)=0 in (26g). Besides, y12(l+L)=116, 139, 165 mm, i.e., each increase of 2 m of the hanging roof distance at this advancing phase results in an increase of (23–26) mm of the deflection of rock beam at goaf midspan.Figure 12
Deflection curves of rock beam with the increasing of the distance of overhang roof.Additionally, the support pressure parts in(-∞,0] and (0,l4] of both sides of the three origins o1, o2, and o3 in Figure 10 are shown by C·y2(x) and C·y21(x). These are the deflections of rock beam y2(x) and y21(x) of the corresponding segments in Figure 12 by multiplying foundation constant C=0.4GPa, respectively.Figure13 shows the magnified chart of the deflections of rock beam at the junction between the elastic and hardened zones (red and blue lines) of coal seam in Figure 12, where the horizontal axis of intersection between three vertical dashed lines whose intervals between the neighboring lines are 2 m and curves 1, 2, and 3 are 23.10, 24.01 and 25.00 mm. By multiplying these three values by the elastic foundation constant C=0.4GPa, the counterforces or the support pressures of the coal seam on the right end of the elastic zone can be obtained: C·y21(l4)≈9.240, 9.604, 10.00 MN/m. By substituting q2 and fc3 in Figures 12 and 13 into the right term of (83), the support pressure on the left end of the hardened zone of coal seam can be computed: F4(l4)≈9.242, 9.606, 9.999 MN/m, almost equal to the support pressures on the right end of the elastic zone of coal seam, indicating the validity of (82) for the continuity of counterforce of coal seam on rock beam at the junction between the elastic and hardened zones of coal seam described in Section 5 and the sufficient accuracy of the maximum support pressure fc3 given in (86).Figure 13
Enlarged drawings of rock beam deflection curves at the junction between elastic area and hardening area of coal seam.


## 6.3. Bending Moment of Rock Beam as Hanging Roof DistanceL Increases

Figure14 shows the bending moment curves of rock beam corresponding to the deflection curves shown in Figure 11. The curves tend to 0 ahead of x=−25 m, and a horizontal tangential line M12′(l+L)=0 exists at the midspan of goaf on each of the curves. Based on (28) for bending moment and shear force, it shows that the shearing force of rock beam on the midspan cross-section of goaf satisfies Q12(l+L)=0, conforming to the boundary condition of (26g). The peak values of bending moment of curves 1, 2, and 3 ahead of coal wall are Mmax(x^)=1.62×108, 1.83×108, 2.06×108 N·m, and the values of bending moment at the midspan of goaf are M12(L+l)= −1.61×108, −1.72×108, and −1.81×108 N·m; i.e., the maximum bending moment of rock beam ahead of coal wall remains higher than the absolute value of bending moment at the midspan of goaf.Figure 14
Bending moment curves of rock beam with the increasing of the distance of overhang roof.Curves 1, 2, and 3 in Figure15 show the magnified chart for peak parts of the bending moment curves of rock beam shown in Figure 14. Curves 1, 2, and 3 achieved their peak values of bending moment Mmax(x^) at x^=10.59, 10.55, 10.51 m. The position at the junction between the hardened and softened zones of coal seam is l3=10 m, the abovementioned three x^ values are all higher than 10 m. Thus, the peaks of the three bending moments of rock beam are located in the softened segment of coal seam (green line segments in Figure 14; the blue and red line segments in Figure 15 are the hardened and elastic ranges of coal seam, respectively). The advanced distance of bending moment peak of rock beam was denoted as l^, and each x^ value at the coal wall position l=18 m was subtracted. The advanced distance of bending moment peak of rock beam l^=l-x≈7.41, 7.45, 7.49 m can be obtained, indicating that the advanced distance of bending moment peak of rock beam increases as the hanging roof distance increases.Figure 15
Enlarged drawings at peak part of rock beam bending moment with the increasing of the distance of overhang roof.Curves I, II, and III in Figure15 show the magnified chart of the bending moment peaks of rock beam corresponding to the two-zone support pressure curves shown in Figure 11. Curves I, II, and III achieved their bending moment peaks at x^=10.86, 10.84, 10.81 m. By subtracting each x^ value at the coal wall position l=18 m, the advanced distance of bending moment peak of rock beam Mmax(x^) can be obtained as l^= 7.14, 7.16, 7.19 m.By comparing thel^ values among curves 1, 2, and 3 and curves I, II, III, it was found that the advanced distance of bending moment peak of the support rock beam with elastic-hardened-softened three-zone coal seam with a hardened zone parameter length of xc4=2 m exceeds those of the support rock beam with elastic-softened two-zone coal seam by about 27, 29 and 30 cm. Figure 15 also shows that, under identical conditions of overlying load and hanging roof distance, the peak values of bending moments for the support roof of three zones considering coal seam hardening are slightly higher than those of two zones without considering coal seam hardening.


## 7. Numerical Examples of Bending Moment of Roof as Depth of Softened Zone Varies

Concerning the roof, the mean load isq2=7MN/m, xc1=8m, and xc4=13m; the length of rock beam is L=22m, l=18m, l3=10.5, 10, 9.5 m. When the scale parameters of the softened and hardened zones of coal seam are xc3=8.5 m and xc4=2 m, based on the methods and equations described in Sections 1–5, the peak values of the support pressure of coal seam can be obtained for the depth of softened zone of coal seam (l-l3)=7.5, 8, 8.5 m:(87)fc3=10.062068,10.387880,10.814946MN/mCurves 1, 2, and 3 in Figure16 show the support pressure curves of the softened, hardened, and elastic zones when the depth of the softened zone of coal seam is (l-l3)= 7.5, 8, 8.5 m, based on the fc3 values of (87) and (5), (7), (14), and (15). Figure 16 shows that as the depth of the softened zone (l-l3) (the range denoted by green lines) increases, the peak values fc3 of the curves increase, while the residual strengths of coal f3(l) at the coal wall gradually decrease, among which the residual strength of curve 3 is f3(l)→0.Figure 16
Support pressure curves of coal seam for different softening area depths.Curves 1, 2, and 3 in Figure17 show the magnified chart of the peak parts of support pressures in Figure 16. This chart also shows the support pressure curves of coal seams I, II, and III for the two-zone relationship of softened and elastic zones with reference to the literature [16] under identical conditions of overlying load and depth of the softened zone of a coal seam. The peak values of curves I, II, and III in Figure 17(88)fc3=10.346857,10.674489,11.112275MN/mwere obtained based on the continuity condition of the counterforce at the junction between the elastic and softened zones according to the literature [20]. Figure 17 shows that the peak values fc3 of curves I, II, and III are higher than those of curves 1, 2, and 3, but the plumpness is inferior.Figure 17
Enlarged drawings at peak part of coal seam support pressure for different softening area depths.As space is limited, Figure18 only shows the magnified chart for the peak parts of the bending moment curves of rock beam corresponding to the support pressure curves in Figures 16 and 17. In Figure 18, the peak values of the bending moments for curves 1, 2, and 3 are slightly higher than those of curves I, II, and III. Curves 1, 2, and 3 in Figure 18 reach the peak values of bending moment at x^≈11.08, 10.64, 10.14 m, and each of x^ is located in the corresponding softened segment of coal seam (each range is denoted by green lines). By subtracting each x^ value at the coal wall position l=18 m, the advanced distances of the bending moment peak of rock beam can be obtained as l^≈6.92, 7.36, 7.86 m, i.e., each 0.5 m increase of the depth of the softened zone of coal seam leads to an increase in the advanced distance of bending moment peak by l^2-l^1≈44cm and l^3-l^2≈50cm. Curves I, II, and III in Figure 18 reach the peak values of bending moment at x^≈ 11.38, 10.93, 10.42 m corresponding to the advanced distances of bending moment peak of rock beam l^≈6.62, 7.07, 7.58 m. A comparison shows that, under the same depth condition of the softened zone of a coal seam, the advanced distances of bending moment peak of support rock beam with elastic-hardened-softened three zones are higher than those with elastic-softened two zones by about 30, 29, and 28 cm.Figure 18
Enlarged drawings at peak part of rock beam bending moment for different softening area depths.


## 8. Discussion

### 8.1. Relationship between Advanced Distance of Bending Moment Peak of Rock Beam and Depth of Softened Zone of Coal Seam

The advanced distance of bending moment peak of a hard roof is directly associated with the advanced fracturing distance of the hard roof. The bending moment peaks shown in Figure15 in Section 6 and Figure 18 in Section 7 are located slightly behind the softened zone of coal seam or the support pressure peak, consistent with the breaking position of main roof located slightly behind the support pressure peak reported by He et al. [22] (Figure 19).Figure 19
The model of double peaks of support pressure [22].Table1 shows nine engineering examples of the advanced fracturing distance of a hard roof predicted by the bounce information of a stope roof and proved by exposing the coal rock seam. Table 1 shows that the actual advanced fracturing distance of hard roof has a relatively broad range, where the advanced fracturing distance of East 708 working face roof of Zhang Xiao Lou Coal Mine reaches 11.56 m. In the statistical data provided by Shao [23], the advanced fracturing distance of a hard roof is 3–8 times the working height. The reason for such a relatively broad range of advanced fracturing distance of a hard roof is not further addressed either in the literature [23] or in other literatures related to roof fracture.Table 1
Engineering example of hard roof fracturing ahead of working face.
Hard roof of working face Advanced fracturing distance Old roof of coal mine in western of Xuzhou About 4m [1] Roof of 741003 working face of Muchengjian mine 5-8 m [24] Roof of coal seam 2 of Beijing Mining Bureau About 6m [24] Old roof of coal seam of Xinqiu mine 6.3m, 8m [23] Roof of 8320 working face of Dafeng mine 9.98m [25] Old roof of Sanhejian mine in Xuzhou About 10m [26] Roof of east 708 working face of Zhangxiaolou mine 11.56m [25]The advanced distance of the bending moment peak in Figure18 described in Section 7 is l^≈6.92, 7.36, 7.86 m, comparable to (6.3–8) m in the middle of Table 1. Figure 18 shows that the depth of the softened zone of a coal seam significantly affects the advanced distance of bending moment peak. The measured depth of the softened zone of coal seam in Figure 1 is 15 m, while those on the bending moment curve in Figure 18 are only 7.5, 8, and 8.5 m. If the data of the softened zone of coal seam with a higher depth are used, according to the methods and equations described in Sections 1–5, the calculation results and charts for the advanced distance of bending moment peak of more than 10 m can be acquired.


### 8.2. Quantitative Relationship between Overlying Load of Rock Beam and Support Pressure of Underlying Coal Seam

Figure20 shows the shearing force curve of rock beam corresponding to the support pressure curve shown in Figure 16 and the bending moment curve shown in Figure 18.Figure 20
Abutment forces curves of coal seam with the increasing of the distance of overhang roof.Figure20 shows that the shearing force of rock beam on the midspan surface of goaf is 0, conforming to the condition Q12(l+L)=0 in (26g). The abscissae for the curves reaching 0 at 11.08, 10.64 and 10.14 m correspond to those of the peaks of the bending moment curves shown in Figure 18; a peak value of curve appears ahead of coal wall, and the maximum Ql is obtained at the coal wall (x=18m). The shear force curves are not smoothly connected at both the ends of roof control area (x=18m, x=22m) owing to a sudden change in the distributed force at both the ends of roof control area beneath the rock beam shown in Figure 4, and the consequent bending angle appears at the sudden change place of distributed force in the shear force chart according to the mechanics of materials. The shear force charts of the three curves from the coal wall to the midspan segment [18,40] m of goaf are identical, because the overlying rock beam loads q1, f1(x), and support forces are identical in this segment and the directed support at the right end of the mechanical model (x=40 m) in Figure 4 can move upwards or downwards without a vertical counterforce.Figure20 shows that the shear force curves tend to be zero in the front segment of x<-40m, indicating that the overlying uplift load, hanging roof, and support resistances slightly affect the shear force of rock beam ahead of x=-40m. Thus, x=-40m cross-section is treated as the boundary in Figure 4, and to maintain the balance of force in y direction within the rock seam system of front or rear of the cross-section, the shear force of the rock beam Q(-40)≈0 should be transferred to the other side of the cross-section. Therefore, regarding the isolating body of the entire rock seam system in the rear of x=-40m cross-section in Figure 4, the relationship between the vertical loads along the y direction can be expressed as follows:(89)∑Fy≈0,or∑Fyabove≈∑Fybelowwhere(90)∑Fyabove=∫-400f2xdx+q2×40+∫0l+Lf1xdx+q1×l+L(91)∑Fybelow=∫-400Cy2xdx+∫0l4Cy21xdx+∫l4l3F4xdx+∫l3lf3xdx+po×Lk+pk-poLk2By substituting the onbeam mean loadq2=7MPa, fc2=0.22q2, q1=0.15MPa, fc1=q2+fc2-q1, po=0.9MN/m, pk=1.2po, Lk=4m and l3=10.5, 10, 9.5 m, l=18m, L=22 m, xc1=8 m, xc2=13 m, xc3=8.5 m, xc4=2 m, l4=l3-xc4, and y2(x) from (14), y21(x) from (15), f3(x) from (5), F4(x) from (7), and fc3 values from (87) into (90) and (91), the following expressions can be obtained using Matlab:(92)∑Fyabove=3.5354+28+13.107+0.6×107=45.2424×07N(93)∑Fybelowl3=10.5=30.435+6.475+1.9489+5.9945+0.396×107N=45.2494×107N(94)∑Fybelowl3=10=30.333+6.2102+2.0074+6.3023+0.396×107N=45.2489×107N(95)∑Fybelowl3=9.5=30.207+5.9585+2.0839+6.603+0.396×107N=45.2484×107NThe relative error between each∑Fybelow in (93)–(95) and ∑Fyabove in (92) is less than 2/1000 0. ∑Fyabove is the area enveloped by the curves of overlying loads F2(x)=q2+f2(x) and F1(x)=q1+f1(x) in [-40,l+L] m segment of rock beam and the horizontal axis in Figure 4, and it is the sum of the distributed loads of rock beam in this segment. ∑Fybelow in (93)–(95) is the sum of the support pressures of coal seam in [-40,l] m segment and the support resistance in [l,l+Lk] segment in Figure 4. The overlying loads of rock beam in [-40,l+L] m segment in Figure 4 are the same as the distributed load. Although the depths of the softened zone of coal seam (l-l3) in (93)–(95) are different, the sum of the support pressures of coal seam (or counterforce of coal seam) and the support resistance under the same rock beam load are very close.In addition, the values of∑Fyabove and ∑Fybelow are close to each other. Thus, the high accuracy of selecting the peak values of the support pressure of coal seam fc3=10.062068, 10.387880, 10.814946 in (87) together with l3=10.5, 10, 9.5 m was again validated.Concerning an elastic-softened two-zone support rock beam, the expression of overlying load is the same as that in (90), and that of the sum of loads is the same as that in (92). Because of the absence of a hardening zone in coal seam, the sum of the support pressures and resistances of the coal seam beneath the rock beam can be expressed as follows:(96)∑Fybelow=∫-400Cy2xdx+∫0l3Cy21xdx+∫l3lf3xdx+po×Lk+pk-poLk2By substituting fc3=10.346857, 10.674489, 11.112275 MN/m into (88) and the above data into (92), the sum of the support pressures and resistances of the coal seam beneath the rock beam when the depth of the softened zone of coal seam is (l-l3)=7.5, 8, 8.5 m can be obtained as follows:(97)∑Fybelowl3=10.5=30.490+8.2003+6.1641+0.396×107N=45.2504×107N(98)∑Fybelowl3=10=30.388+7.9898+6.4762+0.396×107N=45.25×107N(99)∑Fybelowl3=9.5=30.261+7.8071+6.7845+.396×107N=45.2486×107NThe relative error between∑Fybelow in (97)–(99) and ∑Fyabove in (92) is less than 2/1000 0. Specifically, the relative errors in (97) and (93), (98) and (94), and (99) and (95) are below 3/1000 00, i.e., for the same depth of softened zone, the relative error between the sum of the support pressures (counterpressures) of the coal seam with three-zone relationship and that with two-zone relationship is less than 3/100000. This is because although the peak values of the support pressure fc3 of the elastic-softened two-zone relationship are slightly higher, the parts ahead of peaks on the support pressure curves of elastic-hardened-softened three-zone relationship are plumper, resulting in an identical area enveloped by support pressure curves of [-40,l] m segment.


## 8.1. Relationship between Advanced Distance of Bending Moment Peak of Rock Beam and Depth of Softened Zone of Coal Seam

The advanced distance of bending moment peak of a hard roof is directly associated with the advanced fracturing distance of the hard roof. The bending moment peaks shown in Figure15 in Section 6 and Figure 18 in Section 7 are located slightly behind the softened zone of coal seam or the support pressure peak, consistent with the breaking position of main roof located slightly behind the support pressure peak reported by He et al. [22] (Figure 19).Figure 19
The model of double peaks of support pressure [22].Table1 shows nine engineering examples of the advanced fracturing distance of a hard roof predicted by the bounce information of a stope roof and proved by exposing the coal rock seam. Table 1 shows that the actual advanced fracturing distance of hard roof has a relatively broad range, where the advanced fracturing distance of East 708 working face roof of Zhang Xiao Lou Coal Mine reaches 11.56 m. In the statistical data provided by Shao [23], the advanced fracturing distance of a hard roof is 3–8 times the working height. The reason for such a relatively broad range of advanced fracturing distance of a hard roof is not further addressed either in the literature [23] or in other literatures related to roof fracture.Table 1
Engineering example of hard roof fracturing ahead of working face.
Hard roof of working face Advanced fracturing distance Old roof of coal mine in western of Xuzhou About 4m [1] Roof of 741003 working face of Muchengjian mine 5-8 m [24] Roof of coal seam 2 of Beijing Mining Bureau About 6m [24] Old roof of coal seam of Xinqiu mine 6.3m, 8m [23] Roof of 8320 working face of Dafeng mine 9.98m [25] Old roof of Sanhejian mine in Xuzhou About 10m [26] Roof of east 708 working face of Zhangxiaolou mine 11.56m [25]The advanced distance of the bending moment peak in Figure18 described in Section 7 is l^≈6.92, 7.36, 7.86 m, comparable to (6.3–8) m in the middle of Table 1. Figure 18 shows that the depth of the softened zone of a coal seam significantly affects the advanced distance of bending moment peak. The measured depth of the softened zone of coal seam in Figure 1 is 15 m, while those on the bending moment curve in Figure 18 are only 7.5, 8, and 8.5 m. If the data of the softened zone of coal seam with a higher depth are used, according to the methods and equations described in Sections 1–5, the calculation results and charts for the advanced distance of bending moment peak of more than 10 m can be acquired.


## 8.2. Quantitative Relationship between Overlying Load of Rock Beam and Support Pressure of Underlying Coal Seam

Figure20 shows the shearing force curve of rock beam corresponding to the support pressure curve shown in Figure 16 and the bending moment curve shown in Figure 18.Figure 20
Abutment forces curves of coal seam with the increasing of the distance of overhang roof.Figure20 shows that the shearing force of rock beam on the midspan surface of goaf is 0, conforming to the condition Q12(l+L)=0 in (26g). The abscissae for the curves reaching 0 at 11.08, 10.64 and 10.14 m correspond to those of the peaks of the bending moment curves shown in Figure 18; a peak value of curve appears ahead of coal wall, and the maximum Ql is obtained at the coal wall (x=18m). The shear force curves are not smoothly connected at both the ends of roof control area (x=18m, x=22m) owing to a sudden change in the distributed force at both the ends of roof control area beneath the rock beam shown in Figure 4, and the consequent bending angle appears at the sudden change place of distributed force in the shear force chart according to the mechanics of materials. The shear force charts of the three curves from the coal wall to the midspan segment [18,40] m of goaf are identical, because the overlying rock beam loads q1, f1(x), and support forces are identical in this segment and the directed support at the right end of the mechanical model (x=40 m) in Figure 4 can move upwards or downwards without a vertical counterforce.Figure20 shows that the shear force curves tend to be zero in the front segment of x<-40m, indicating that the overlying uplift load, hanging roof, and support resistances slightly affect the shear force of rock beam ahead of x=-40m. Thus, x=-40m cross-section is treated as the boundary in Figure 4, and to maintain the balance of force in y direction within the rock seam system of front or rear of the cross-section, the shear force of the rock beam Q(-40)≈0 should be transferred to the other side of the cross-section. Therefore, regarding the isolating body of the entire rock seam system in the rear of x=-40m cross-section in Figure 4, the relationship between the vertical loads along the y direction can be expressed as follows:(89)∑Fy≈0,or∑Fyabove≈∑Fybelowwhere(90)∑Fyabove=∫-400f2xdx+q2×40+∫0l+Lf1xdx+q1×l+L(91)∑Fybelow=∫-400Cy2xdx+∫0l4Cy21xdx+∫l4l3F4xdx+∫l3lf3xdx+po×Lk+pk-poLk2By substituting the onbeam mean loadq2=7MPa, fc2=0.22q2, q1=0.15MPa, fc1=q2+fc2-q1, po=0.9MN/m, pk=1.2po, Lk=4m and l3=10.5, 10, 9.5 m, l=18m, L=22 m, xc1=8 m, xc2=13 m, xc3=8.5 m, xc4=2 m, l4=l3-xc4, and y2(x) from (14), y21(x) from (15), f3(x) from (5), F4(x) from (7), and fc3 values from (87) into (90) and (91), the following expressions can be obtained using Matlab:(92)∑Fyabove=3.5354+28+13.107+0.6×107=45.2424×07N(93)∑Fybelowl3=10.5=30.435+6.475+1.9489+5.9945+0.396×107N=45.2494×107N(94)∑Fybelowl3=10=30.333+6.2102+2.0074+6.3023+0.396×107N=45.2489×107N(95)∑Fybelowl3=9.5=30.207+5.9585+2.0839+6.603+0.396×107N=45.2484×107NThe relative error between each∑Fybelow in (93)–(95) and ∑Fyabove in (92) is less than 2/1000 0. ∑Fyabove is the area enveloped by the curves of overlying loads F2(x)=q2+f2(x) and F1(x)=q1+f1(x) in [-40,l+L] m segment of rock beam and the horizontal axis in Figure 4, and it is the sum of the distributed loads of rock beam in this segment. ∑Fybelow in (93)–(95) is the sum of the support pressures of coal seam in [-40,l] m segment and the support resistance in [l,l+Lk] segment in Figure 4. The overlying loads of rock beam in [-40,l+L] m segment in Figure 4 are the same as the distributed load. Although the depths of the softened zone of coal seam (l-l3) in (93)–(95) are different, the sum of the support pressures of coal seam (or counterforce of coal seam) and the support resistance under the same rock beam load are very close.In addition, the values of∑Fyabove and ∑Fybelow are close to each other. Thus, the high accuracy of selecting the peak values of the support pressure of coal seam fc3=10.062068, 10.387880, 10.814946 in (87) together with l3=10.5, 10, 9.5 m was again validated.Concerning an elastic-softened two-zone support rock beam, the expression of overlying load is the same as that in (90), and that of the sum of loads is the same as that in (92). Because of the absence of a hardening zone in coal seam, the sum of the support pressures and resistances of the coal seam beneath the rock beam can be expressed as follows:(96)∑Fybelow=∫-400Cy2xdx+∫0l3Cy21xdx+∫l3lf3xdx+po×Lk+pk-poLk2By substituting fc3=10.346857, 10.674489, 11.112275 MN/m into (88) and the above data into (92), the sum of the support pressures and resistances of the coal seam beneath the rock beam when the depth of the softened zone of coal seam is (l-l3)=7.5, 8, 8.5 m can be obtained as follows:(97)∑Fybelowl3=10.5=30.490+8.2003+6.1641+0.396×107N=45.2504×107N(98)∑Fybelowl3=10=30.388+7.9898+6.4762+0.396×107N=45.25×107N(99)∑Fybelowl3=9.5=30.261+7.8071+6.7845+.396×107N=45.2486×107NThe relative error between∑Fybelow in (97)–(99) and ∑Fyabove in (92) is less than 2/1000 0. Specifically, the relative errors in (97) and (93), (98) and (94), and (99) and (95) are below 3/1000 00, i.e., for the same depth of softened zone, the relative error between the sum of the support pressures (counterpressures) of the coal seam with three-zone relationship and that with two-zone relationship is less than 3/100000. This is because although the peak values of the support pressure fc3 of the elastic-softened two-zone relationship are slightly higher, the parts ahead of peaks on the support pressure curves of elastic-hardened-softened three-zone relationship are plumper, resulting in an identical area enveloped by support pressure curves of [-40,l] m segment.


## 9. Conclusions

( 1 ) Based on the distribution pattern of measured support pressure and considering the hardening effect of coal seam, the support pressure relationship of three zones—softened, hardened, and elastic—for a coal seam and hard roof is proposed, and the methods for determining the approximate expressions for the support pressure of the hardened zone of a coal seam, the range of hardened zone, and the corresponding maximum support pressure are presented.( 2 ) By measuring the deflection curves of rock beam, all the selected three peak values of support pressure fc3 in (84) were shown to satisfy the requirement of continuity of counterforce (support pressure) on rock beam at the junction between the elastic and hardened zones of coal seam.( 3 ) As the working face advances and the hanging roof distance L increases (the total load above the corresponding coal seam increases), the peak values of the support pressure fc3 of a coal seam gradually increase and the residual strengths f3(l) of a coal seam at the coal wall decrease, similar to the trend of moving support pressure shown in Figure 2. As the hanging roof distance increases, the overall deflection of rock beam slowly increases, the peak values of the bending moment of rock beam ahead of the coal wall increase, and the advanced distance of bending moment peak slowly increases, and the peak values of the bending moment of rock beam ahead of the coal wall are always higher than the absolute value of the bending moment of rock beam at the midspan of goaf in this phase.( 4 ) With the increase in the depth of a softened zone (l-l3) of coal seam, the peak values of support pressure fc3 of coal seam increase, the residual strengths of coal seam at the coal wall gradually decrease, the peak values of the bending moment of rock beam ahead of the coal wall increase, and the advanced distance of the bending moment peak of rock beam increases at a relatively fast rate. The bending moment peaks of roof are located near the support pressure peaks in the softened zone coal seam shown in Figures 15 and 18. The depth of the softened zone of coal seam significantly affects the advanced distance of the bending moment peak of hard roof.( 5 ) The advanced distance of bending moment peak is directly associated with the advanced fracturing distance of a hard roof. The advanced fracturing distances of hard roof that occur in a stope are distributed in a relatively broad range. The results in (3) indicate a “large and long” type advanced fracturing distance occurring in a stope.( 6 ) With identical stope parameter conditions, the peak values of the support pressure of a softened-hardened-elastic three-zone support roof are smaller than those of a softened-elastic two-zone support roof. However, the plumpness ahead of the support pressure curve peak of the former is better than that of the latter, and the peak values of bending moment and the advanced distances of bending moment peak of the former are higher than the corresponding peak values of the latter. The degree of superiority is related to the parameter xc4 of the hardened zone of a coal seam.



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*Source: 7082951-2018-07-10.xml*