A New Closed-Form Solution of the Side Abutment Pressure Distribution of Roadway

Instability of coal wall is one of the hot-button and difficult issues in the study of coal mine ground control.(e shallow side coal of roadway in the coal measures is usually weak and consequently easy to bring about failure. Hence, the side abutment pressure redistributes and dramatically influences the roadway stability. Since the previous closed-form solutions of the side abutment pressure do not take into account all the necessary parameters which include the properties of the coal and the interface between coal and roof/floor, the roadway height, and the support strength, a mechanical model is established based on the equilibrium of the plastic zone, and a new closed-form solution is derived in this paper. Moreover, a numerical investigation is conducted to validate the accuracy of the closed-form solution. (e numerical results of the side abutment pressure distribution are in good agreement with the closed-form solution. Afterwards, a parametric analysis of the width of the plastic zone is carried out, and the results show that the width of the plastic zone is nearly negatively linearly correlated with the friction angle and the cohesion of the coal, the interfacial cohesion, and the support strength. By contrast, it is positively linearly correlated with the roadway height and negatively exponentially correlated with the interfacial friction angle. (e results obtained in the present study could be useful for the evaluation process of roadway stability.


Introduction
Underground coal is an integral part of the primary energy sources and is used for a wide range of applications.Physical extraction of coal is still an important method in the mining industry.Before the physical extraction, a group of roadways needs to be constructed for the transportation of the workers, equipment, tools, and raw coal.Along with the mining production, the excavation of roadway is continuing.erefore, to some extent, the stability of roadway has a large effect on the efficiency of underground coal mining.e roadway stability is dramatically influenced by the abutment pressure distribution [1][2][3].e stress distribution of surrounding rock of roadway could be calculated by using elasticity when it is assumed that the surrounding rock follows the assumptions of elasticity [4].However, roadway is excavated in a coal seam, and the shallow coal is usually weak and consequently easy to bring about failure [5,6].Neither the laws of soil mechanics nor the theories of elasticity explain completely the behavior of the relatively soft coal [7].In this condition, the side abutment pressure distribution can be obtained roughly based on the closedform solutions of (1)∼( 4) and (7) in Table 1 [8][9][10].e above closed-form solutions neglect the interface between coal and roof/floor.Actually the interface between coal and roof/floor is usually weak and has a large effect on the side abutment pressure distribution, which usually degrades roadway stability.Shou demonstrated that the lesser angle between the axial direction of the entry and the dip direction of the weak plane contributed larger deformation for the roof [11].Jeon et al. found that the shear deformation at the weak plane leads to a large displacement of the rock around the entry [12].Zhao et al. revealed that the shear slipping mainly occurred along the weak plane due to the failure of K s , shear stiffness between coal and rock Advances in Civil Engineering weak plane [13].Zhao et al. pointed out that the coal-rock combination with weak interface trended to isotropic rock when the confining pressure is raised to a specific value [14].Shen et al. indicated that the side abutment pressure could result in a relative displacement for entry roof with the weak plane [2,15].Taking into account the interfacial properties, the closed-form solutions ( 5), (6), and ( 8) in Table 1 [8][9][10] were derived.
Although the closed-form solutions in Table 1 revealed some characteristics of the side abutment pressure distribution, none of these closed-form solutions include all the necessary parameters which include the properties of the coal, the properties of the interface between coal and roof/floor, the roadway height, and the support strength.To overcome the deficiency of the closed-form solutions of the side abutment pressure distribution, a new closed-form solution was derived in the paper.Moreover, the accuracy of the closed-form solution was validated by modelling.

Mechanical Model.
e side abutment pressure redistributes after roadway excavation.e coal in the plastic zone subjects to limit equilibrium state and deforms toward to the roadway.erefore, the shear stress occurs along the interface.e mechanical model is shown in Figure 1.

Basic Assumptions.
To solve this problem, several assumptions are made as follows: (1) e roadway height is equal to the thickness of the coal seam.(2) e support strength distributes evenly at the wall surface.(3) e effect of the failure of roof/floor on the coal is neglected.(4) e dead weight of the coal is neglected.(5) e properties of the coal and the interface remain the same after yielding.(6) e shear stress at the horizontal center line of the plastic zone is zero.(7) e interface is perfectly elastic-plastic and follows Mohr-Coulomb failure criterion.
where τ xy is the shear stress.( 8) e constitutive model of the coal follows Mohr-Coulomb failure criterion.( 9) e maximum vertical stress is given by Substituting ( 1) into (4) yields e vertical stress could be assumed as follows: From ( 5), it can be assumed that Assuming that ( 7) is equal to a constant of W 2 , the following equation can be obtained:

e Equation of Limit Equilibrium Zone.
Based on the mechanical model in Figure 1, the resultant force in the limit equilibrium zone ABCD is zero.
where P x is the support strength.e interfacial friction angle and cohesion are much lower than that of coal.erefore, the vertical and horizontal stress in the plastic zone are assumed as the principle stresses, and the relationship is shown as follows based on Mohr-Coulomb failure criterion: e derivation of each side of ( 12) is as follows: Substituting ( 1) into ( 14) yields Solving ( 15) yields

e Side Abutment Pressure Distribution in Elastic Zone.
e side abutment pressure distribution in the elastic zone follows the negative exponent distribution [8].
Advances in Civil Engineering where σ ye is the side abutment pressure in the elastic zone; χ is the factor to describe the peak side abutment pressure; and α 0 is the factor to describe the change of the side abutment pressure.
To assure the stress continuum, substituting x � x s and f(x) � k 0 cH into (24) yields e side abutment pressure in the elastic zone is shown as 2.3.5.Solving α 0 .e vertical counterforce at the interface should be equal to the weight of the overlying strata.

Numerical Validation
Figure 2 shows the 2D numerical model used in the present study, using the finite difference program FLAC 3D.It is assumed that the behavior of the roof and floor is linear-elastic, and the coal follows Mohr-Coulomb failure criterion.Embedded interfaces are attached to the zone faces along the coal and the roof/floor.e interface stiffness (normal stiffness k n and tangential stiffness k s ) is estimated using a rule-of-thumb in which k n and k s are set to 10 times the equivalent stiffness of the stiffest neighboring zone [16], and the constitutive model is perfectly elastic-plastic and follows Mohr-Coulomb failure criterion.
e numerical model is 100 m wide in the x-direction and 50 m high in the z-direction and consists of approximately 50000 zones.e width and height of roadway are 4.8 m and 4 m, respectively.e bottom and sides of the model are fixed in displacement and the top is applied with a load of 10 MPa.
An anisotropic stress field has been assigned in the model with a lateral earth pressure factor of 1. e parameters of the surrounding rock and the interface are from a case found in a coal mine in China as shown in Table 2.
e numerical results of the roadway stability are compared with the above closed-form solution, which is shown in Figure 3.Note that k 0 in the closed-form solution is obtained from the modelling.e following can be seen from Figure 3: (1) e side abutment pressure can be divided into two areas: plastic zone and elastic zone.In the plastic zone, the side abutment pressure increases from the support strength at the wall surface to the peak pressure at the boundary of the plastic zone and elastic zone.In the elastic zone, the side abutment pressure decreases to the in situ stress gradually.(2) e width of the plastic zone decreases and the width of the elastic zone increases with the increase of the   properties of the coal and the interface, the support strength, and the decrease of the roadway height.(3) From the comparison between the side abutment pressure of the closed-form solution and FLAC 3D, the numerical results of the side abutment pressure are generally in good agreement with the one obtained using the closed-form solution when changing the properties of the coal and the interface, the roadway height, and the support strength.However, the error between the side abutment pressure of the closedform solution and FLAC 3D is relatively large when the interfacial cohesion and friction angle are large.

Parametric Analysis of the Width of Plastic Zone
Based on (23), the influence of the properties of the side coal and the interface, the roadway height and the support strength on the width of the plastic zone is conducted as shown in Figure 4. e initial parameters are the same of Table 2, and the side abutment pressure concentration factor is set as 1.6.It can be seen from Figure 4 that the width of the plastic zone is nearly negatively linearly correlated with the friction angle from 20 °to 35 °and cohesion from 0.5 MPa to 2.5 MPa of the coal and the interfacial cohesion from 0.02 MPa to 1 MPa and the support strength from 0 MPa to 1 MPa.By contrast, it is positively linearly correlated with the roadway height from 2 m to 4 m and negatively exponentially correlated with the interfacial friction angle from 10 °to 30 °.In roadway support, the support strength is usually less than 0.2 MPa, and therefore the support strength has little effect on the width of the plastic zone especially in roadway with hard coals.

Summary and Conclusions
e side abutment pressure distribution is fundamental to evaluate the roadway stability.e side abutment pressure can be divided into two areas: plastic zone and elastic zone.
e side abutment pressure increases in the plastic zone and decreases in the elastic zone.e width of the plastic zone decreases and the width of the elastic zone increases with the increase of the properties of the coal and the interface, the support strength, and the decrease of the roadway height.
is paper derived a new closed-form solution which takes into consideration the properties of the side coal and the interface, the roadway height, and the support strength.Moreover, a numerical investigation was conducted to validate the accuracy of the closed-form solution.e numerical results of the side abutment pressure are generally in good agreement with the one obtained using the closed-form solution when changing the properties of the coal and the interface, the roadway height, and the support strength.However, the error between the side abutment pressure of the closed-form solution and FLAC 3D is relatively large when the interfacial cohesion and friction angle are large.
e results also show that the effect of the properties of the coal and the interface, the roadway height, and the support strength on the width of the plastic zone in detail when the shallow side coal is subject to plastic state.e results obtained in the present study could be useful for the evaluation process of roadway stability.

Table 1 :
A summary of research status of the side abutment pressure distribution.

Table 2 :
Properties of the surrounding rock and the interface.