#### 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.

#### 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.

Figure 1 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 2 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*, and* D* 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.

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., . The angle of the midspan cross-section of rock beam is 0, i.e., . 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.

The origin of the coordinate in Figure 4 is located beneath the supercharged load peak of rock beam. The distance from the point to the coal wall is , and its distances to the hardened and softened zones of coal seam are and , respectively. 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 and , respectively. The length of support is , and the support resistances are and . 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 and . The following expressions can be used:to simulate the evenly distributed load and uplift-distributed load on rock beam ahead of and after supercharged load peak in Figure 4, where

In (3), and are the peak values of and , respectively, whereas and are the scale parameters. The value of supercharged load peak can be adjusted by changing and , and, in the process of , the speed of and can be adjusted by varying and .

The relationship between , , and even loads on the beam , is defined as follows:Equation (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 is and the average load on beam is . If , roof thickness and volumetric weight according to literature [26], . Equation (4) shows that . If in Figure 4, the hanging roof distance of hard roof is . 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 and . 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.

The load intensity 100 m ahead in Figure 5 is denoted by . The supercharged load peak is located at . The load curve is smooth and continuous, and . The midspan load intensity of rock beam is (~, i.e., 3.78 times of the roof weight), which is far less than the mean load on beam . 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 ( 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.

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

For the coordinate in Figure 4, the support pressure in the softened zone of coal seam can be expressed as follows:In (5), when , the maximum support pressure of coal seam is located at and is the scale parameter for . Equation (5) shows thatIn (5), if 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 can be expressed as follows:where is the scale parameter for . In (7), tends to the mean load when , where

By taking the derivative twice with respect to in (7) and letting it be 0, the relationship among , , and at the inflection point on support pressure curve of coal seam can be expressed as follows:

In (7), when the maximum support pressure curve of coal seam is , where is the maximum . Because and , let . The support pressure curve of coal seam is then smoothly connected at , and thus

For and 2 m, the inflection point location of support pressure curve of coal seam m can be obtained using (9). Suppose , with respect to and 8.6, 8.3, and 8 m and according to (5), (7), and (10), the support pressure curve (x is the m section) for the hardened zone of coal seam when 8 m and 2 m and the support pressure curve (x is the m section) for the softened zone of coal seam when and =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 on the support pressure curves. is the residual strength of coal seam at coal wall. can be varied in the range of by adjusting . 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, in Figure 7 is manually drawn the schematic diagram, whereas the value of 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.

#### 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 and 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 is directly proportional to the displacement of rock beam , i.e., , where is the elastic foundation constant and the minus represents the opposite direction of reaction force and displacement (the settlement amount of rock beam). A distributed force was applied above the rock beam in zone . The deflection of rock beam in this zone was denoted as . Following Timoshenko [27], the differential equation for the deflection of any infinitesimal segment in zone of a rock beam can be derived as follows:where is the elastic modulus of roof under the condition of plane strain and is the moment of inertia for the roof with unit width. If , (11) can be expressed as follows:

For the zone shown in Figure 4, the distributed load applied above on the elastic foundation beam is . The deflection of the rock beam in zone was denoted as . Similar to (12), the differential equation for the deflection of any infinitesimal segment in zone can be derived as follows:

Because 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 is the same as reported in the literature [20], and it is denoted as follows:whereas that in zone is also the same as that reported in the literature [20] and can be expressed as follows:The values of integral constants – in (14) and (15) are different from the values of – 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 load was applied above the rock beam in zone shown in Figure 4, and the support pressure of hardened zone of coal seam was applied below the rock beam. The deflection of the rock beam in zone was denoted as , and similar to (13), the differential equations for the deflection of the rock beam above the hardened zone of coal seam can be written as follows:

Figure 8(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 cross-section with respect to cross-section in Figure 8(a), the equation of the bending moment of the rock beam in zone can be obtained as follows:

**(a)**

**(b)**

The sum of vertical forces on the isolating body in Figure 8(a) was calculated, and the shearing force of the left-end cross-section of the rock beam in zone can be obtained from :

By calculating the moments of the forces of the isolating body in Figure 8(a) with respect to the left-end 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 :In (18) and (19), and 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 can be expressed as follows:where and are integral constants determined from the continuity conditions.

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

Figure 8(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 was applied above the rock beam in zone , and the support pressure of the softened zone of coal seam was applied beneath that. The deflection of rock beam in the softened zone of coal seam was denoted as , and the form of is the same as that reported in the literature [20]:

In (21), and 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 can be obtained from :

By calculating the moments of the forces with respect to the left-end cross-section shown in Figure 8(b), the equation for the bending moments of the rock beam in this zone can be obtained from :In (22) and (23), and are the shear forces and bending moments for the left end of the isolating body of rock beam shown in Figure 8(b) and 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 and are shown in Figure 4. , , , and are the same as those reported in the literature [20]:

##### 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 Figure 4 are listed as follows:

The minus symbols ahead of , , and 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 , , and 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, and , were added, and the continuity condition (26c) was added in (26a), (26b), (26c), (26d), (26e), (26f), and (26g). Therefore, the integral constants –, , , and – 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: The relationship between the equations of shear force and bending moment and deflection can be expressed as follows:

#### 4. Determination of Integral Constants in Deflection Equations

##### 4.1. Relationship among Integral Constants –, , , and

Because the rock beam model shown in Figure 4 is statically indeterminate, a supplementary equation should be established through the geometric condition of in (26g), defined as follows:to determine the relationship between of in (25) and right-end bending moment shown in Figure 8(b), where is an unknown variable to be determined. In (29),

According to the condition in (26g) and , the right-hand term, i.e., the shear force in (29), can be obtained as follows:

Using the first derivative of (24) and (25) and the condition in (26f), the following expression can be obtained:

Regarding (24) and (25), according to the condition in (26f), the following expression can be obtained:

By substituting (21) and (24) into the conditions and in (26e), the followings equations can be obtained:where

Equation (23) can be written in the following form:whereThe following expression can be obtained from (35) using (29) and (32).where

By substituting (20) and (21) into the conditions , in (26d), the following equations can be obtained:where

Using (40), (19) can be written in the following form:whereFrom (42), (45), and (50), the following equation can be obtained:where

##### 4.2. Relationship among Integral Constants – , , and

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 – can be derived. The following can be obtained by solving the equation as follows:The following has been shown in (54)–(57):

Using the 1st–3rd derivatives of (15) and (20), according to the first four conditions in (26c), four algebraic equations for –, , , , and for can be derived as follows: