An Analytical Elastic Solution for Right-Angle Trapezoidal Opening in Steeply Inclined Coal Seam

Abstract

In the process of underground mining, steeply inclined rocks or coal seams are often encountered, forming the openings of right-angle trapezoid. According to the geological conditions of a mining project in China, an analytical elastic solution of stress and displacement around right-angle trapezoidal opening in a homogeneous, isotropic, and linear elastic geomaterial is presented, which is based on the evaluation of the conformal mapping representation by an appropriate numerical calculation and the complex potential functions. The different results from other shaped openings are shown as follows. In a right-angle trapezoidal opening, the maximum displacements of roof falling occur on the low side, while the most horizontal displacements on the low side are around the roof and the most horizontal displacements on the high side are around the middle of the high side in this opening. These results are also compared with the numerical calculations in FLAC software, illustrating that the solution may be easily applied to rock mechanics or rock engineering for understanding the deformation of floor heave and roof falling down. The solution is also suitable for optimum design of bolt supporting in a right-angle trapezoidal opening, which is different from the traditional concept of symmetrical bolt supporting. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of an opening without symmetrical axis.

1. Introduction

Underground openings in rocks and coal seams are excavated in a wide range of geometries, such as circle [1,2,3,4], ellipse [5], ovaloid [6], rectangle [7], etc. The common feature of all these openings is that the confining stress on the boundary of the opening drops to zero and the rock or coal seam deforms elastically at the very least. An understanding of the elastic deformation around the opening is quite important for underground engineering problems, especially for mining engineering problems. The majority of elastic solutions are applied to underground openings with one or two axes of symmetry [8,9]. Although the visualizing solutions for openings without symmetrical axes can be obtained by the use of numerical modeling, the analytical solution for openings is helpful to understand the solution. On the other hand, the engineer can analyze the correctness of numerical calculations because of the simplification of numerical modeling; however, the analytical solution provides a basis for grasping the correctness.

During underground mining, steeply inclined rocks and coal seams are often encountered and may result in instability problems, which has become a significant research topic in recent years [10,11,12,13,14]. Because of the presence of steeply inclined dips, the openings in rocks or coal seams do not have symmetrical axes, such as right-angle trapezoidal shape, for decreasing the exposed area of the opening roof and withstanding great lateral pressure on both sides of the opening. It should be noticed that the right-angle trapezoidal shape is not the only option for opening without symmetrical axes, but most openings are designed as right-angle trapezoidal shapes in steeply inclined rocks or coal seams in China, and some numerical calculations are carried out as shown in references from [15,16]. However, there are few analytical results of stresses or displacements in the right-angle trapezoidal openings. The excavation-induced asymmetrical elastic deformation in surrounding rock or coal seam is a difficult problem for the stability of supporting the openings. Therefore, considering the excavation of openings in steeply inclined rocks or coal seams, the analytical solution for right-angle trapezoidal openings should be given to obtain the stress or displacement on the boundary of the opening.

Based on the geological conditions in the Chen-man-zhuang mining project in China, a theoretical solution of right-angle trapezoidal opening in elastic, isotropic, and homogeneous coal seam is proposed based on the powerful conformal mapping method. Subsequently, the deformation on the boundary of opening is obtained and analyzed, which may bring about a better insight to the stability of opening in underground rock (coal seam).

2. Chen-Man-Zhuang Mining Project

The Chen-man-zhuang underground mining project is situated in the East of Shanxian Coalfields in Heze City of Shandong Province, China. The center of the Chen-man-zhuang project is 16 km away from Shanxian and 90 km from Heze Economic Development Zone, which is shown in Figure 1. It is also interconnected by an excellent highway system with quick access to Jiangsu Province, Henan Province, and Anhui Province.

It is reported that the recoverable reserves of coal seam are 50.766 million tons, and the average thickness of coal seam is more than 3.50 m in the Chen-man-zhuang project. The total mining area of the Chen-man-zhuang project is 25.69 km². There is a coal seam named 3-top coal seam, which is 3.50 m thick and at a depth of 900 m. It belongs to steeply inclined seam with an average inclined dip angle of 40°. The immediate roof of 3-top coal seam is mudstone with 0.9–1.3 m thick and low strength, and the main roof is full medium sandstone with high strength. The floor of 3-top coal seam is mudstone with low strength, and contains small amounts of sandstone. The opening often needs to be excavated before mining the coal resource. In this project, the opening in 3-top coal seam is designed for mining the coal resource in 3402 working face. According to the characteristics of steeply inclined seam and the common design method in this mining project, the opening is designed to be a right-angle trapezoidal shape, which is shown in Figure 2. Based on the previous mining experience in this project, there is always great deformation and invalid bolt supporting because of the steeply inclined rock (seam) and low strength of floor and roof for the opening. Therefore, it is very important to predict the deformation and provide a bolt supporting scheme for the opening before mining, which can provide the basis for safety mining.

3. The Conformal Mapping Representation of Right-Angle Trapezoidal Opening

In the theory of plane elasticity [17], elastic stress/displacement around an opening can be approximated by that of a hole, having the same shape, in an infinitely large and elastic plate subjected to edge loads. As a matter of fact, classical solutions for the stresses/displacements around a circular, elliptic, oval, or rectangular hole have interesting applications in rock engineering. For the solution of plane elasticity problems with complicated shapes, the conformal mapping methodology with complex variables is a feasible technology, in which the problem geometry involving an awkwardly shaped region is transformed into one of a simple shape [18,19,20].

In this section, a plane strain elastic model is considered for the opening with a right-angle trapezoidal shape on a homogeneous infinitely large and elastic plate subjected to principal stresses σ_x∞ and σ_y∞ referred to a Cartesian coordinate system Oxy. That is, the opening axis is assumed to be aligned with the direction of the third out-of-plane principal stress σ_z∞. The direction of σ_x∞ is parallel to the Ox-axis. The methodology starts with the conformal mapping of the boundary of the right-angle trapezoidal opening and its exterior region S (Figure 3a) into the interior region Σ of the circle with unit radius (Figure 3b). The position of every point in the physical z-plane with z = x + iy = re^iα, where r, α denote polar coordinates and $i=\sqrt{-1}$ is the imaginary unit, is mapped into the unit circle in the ζ-plane with ζ = ξ + iη = ρe^iθ with complex function:

$$z=x+iy=\omega \left(\mathsf{\zeta }\right)=c_{-1}\frac{1}{\zeta }+{\displaystyle \sum }_{k=0}^n{c}_k{\zeta }^k,\left|\zeta \right|\le 1$$

(1)

where, the constant coefficient c₋₁ is a real number and the constant coefficients c_k are in general complex numbers with c_k = a_{k +} ib_k (k = 0, 1, 2, 3, …, n). These coefficients should be chosen in such a way that the curve is fully close to the right-angle trapezoidal shape in Figure 3a. Because of the exterior region of opening in z-plane being mapped into the interior region in ζ-plan, the point moves along the outline C in a counterclockwise direction in z-plane; however, this point moves along the clockwise direction in ζ-plane (Figure 3b). Meanwhile, the outline C of a right-angle trapezoidal shape is mapped into the outline γ of unit circle (for ζ = e^iθ along γ).

The parametric representation of the curves in the Oxy-plane transformed by Equation (1) are expressed as follows:

$$\{\begin{array}{c}x=c_{-1}\frac{cos\theta }{\rho }+{\displaystyle {\displaystyle \sum }_{k=0}^n}{\rho }^k\left(a_{k}cosk\theta -b_{k}sink\theta \right)\\ y=-c_{-1}\frac{sin\theta }{\rho }+{\displaystyle {\displaystyle \sum }_{k=0}^n}{\rho }^k\left(a_{k}sink\theta +b_{k}cosk\theta \right)\end{array}$$

(2)

By setting ρ = 1 in Equation (2), the parametric equations defining the boundary of the opening are obtained. Then, it is possible to define some tunnels or cavern shapes widely used in mining and civil engineering if a certain number of terms and their values are assigned to the coefficients of Equation (2).

The computation of constant coefficients c₋₁, a_k, b_k (k = 0, 1, 2, 3, …, n) are shown as follows:

(i) First, the outline C of the opening is divided into m + 1 points (m ≥ 1). It should be noticed that the first point and the last point must be the same point. For approximate calculations, it is supposed that these discrete points match with θ = 2π/m equidistant points along the outline γ of unit circle;

(ii) Next, the constant coefficients of Equation (2) are obtained by solving the 2 m linear equations with 2n + 3 unknown numbers with least square method:

$${\mathrm{A}}_{pq}{\mathrm{X}}_q=R_p\iff {\mathrm{X}}_q={\left(A_{pq}^T{\mathrm{A}}_{pq}\right)}^{-1}{A}_{pq}^T{R}_p$$

(3)

where m is the number of points and the coefficient matrix is shown as follows:

$$\begin{gathered} {\mathrm{A}}_{pq}=\left[\begin{array}{cccccccccc}cos{\theta }_1& 1& 0& cos{\theta }_1& -sin{\theta }_1& cos\left(2{\theta }_1\right)& -sin\left(2{\theta }_1\right)& \cdots & cos\left(n{\theta }_1\right)& -sin\left(n{\theta }_1\right)\\ cos{\theta }_2& 1& 0& cos{\theta }_2& -sin{\theta }_2& cos\left(2{\theta }_2\right)& -sin\left(2{\theta }_2\right)& \cdots & cos\left(n{\theta }_2\right)& -sin\left(n{\theta }_2\right)\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ cos{\theta }_m& 1& 0& cos{\theta }_m& -sin{\theta }_m& cos\left(2{\theta }_m\right)& -sin\left(2{\theta }_m\right)& \cdots & cos\left(n{\theta }_m\right)& -sin\left(n{\theta }_m\right)\\ -sin{\theta }_1& 0& 1& sin{\theta }_1& cos{\theta }_1& sin\left(2{\theta }_1\right)& cos\left(2{\theta }_1\right)& \cdots & sin\left(n{\theta }_1\right)& cos\left(n{\theta }_1\right)\\ -sin{\theta }_2& 0& 1& sin{\theta }_2& cos{\theta }_2& sin\left(2{\theta }_2\right)& cos\left(2{\theta }_2\right)& \cdots & sin\left(n{\theta }_2\right)& cos\left(n{\theta }_2\right)\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ -sin{\theta }_m& 0& 1& sin{\theta }_m& cos{\theta }_m& sin\left(2{\theta }_m\right)& cos\left(2{\theta }_m\right)& \cdots & sin\left(n{\theta }_m\right)& cos\left(n{\theta }_m\right)\end{array}\right] \\ {\mathrm{X}}_q={\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{ccc}{c}_{-1}& a_0& b_0\end{array}& a_1\end{array}& \begin{array}{cc}{b}_1& a_2\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}{b}_2& \cdots \end{array}& \begin{array}{cc}{a}_n& b_n\end{array}\end{array}\end{array}\right]}^T \\ {\mathrm{R}}_p={\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}{x}_1& x_2\end{array}& \begin{array}{cc}\cdots & x_m\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}{y}_1& y_2\end{array}& \begin{array}{cc}\cdots & y_m\end{array}\end{array}\end{array}\right]}^T \end{gathered}$$

(iii) After the first results of the coefficients c₋₁, a_k, b_k (k = 0, 1, 2, 3, …, n), the new points on the outline C of opening in the z-plane are obtained by the mapping function Equation (1). Subsequently, the x coordinate or y coordinate of every point obtained by Equation (1) is substituted into Rp and Equation (3). The new values are solved again for modified values of c₋₁, a_k, b_k (k = 0, 1, 2, 3, …, n);

(iv) This overlay process is carried out until an allowable error is achieved.

Figure 4 shows the predictions of the opening outline through the above calculation. It is shown that the prediction of opening outline is completely close to the real boundary when n = 10.

4. Elastic Solution for Right-Angle Trapezoidal Opening

Based on elastic mechanics, the boundary condition for unit circle can be taken as follows:

$${\phi }_0\left(\zeta \right)+\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{\omega \left(\mathsf{\sigma }\right)}{\overline{{\omega }^{\prime }\left(\mathsf{\sigma }\right)}}\frac{\overline{{{\phi }^{\prime }}_0\left(\sigma \right)}}{\sigma -\zeta }d\sigma =\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{f_{0}d\sigma }{\sigma -\zeta }$$

(4)

in which $\sigma =e^{i\theta }$ represents an arbitrary point along the outline γ, and:

$$f_0=i{{\displaystyle \int }}^{}\left(\overline{X}+i\overline{Y}\right)ds+\frac{X+iY}{2\pi }ln\sigma -\frac{1+\mu }{8\pi }\left(X-iY\right)\frac{\omega \left(\sigma \right)}{\overline{{\omega }^{\prime }\left(\sigma \right)}}\sigma -2B\omega \left(\sigma \right)-\left(B^{\prime }-i{C}^{\prime }\right)\overline{\omega \left(\sigma \right)}$$

(5)

where X and Y is the sum of surface traction on the contour C along Ox-axes and Oy-axes, respectively.

Additionally:

$$B=\frac{1}{4}\left({\sigma }_{x\infty }+{\sigma }_{y\infty }\right),B^{\prime }+i{C}^{\prime }=-\frac{1}{2}\left({\sigma }_{x\infty }-{\sigma }_{y\infty }\right)$$

(6)

where σ_x∞, σ_y∞ denotes to the primary stress at infinity along x-axis and y-axis, respectively (e.g., Figure 3a).

An expansion of φ₀(ζ) should be employed for solving the Cauchy integral Equation (4), which has been presented elegantly in the solution for the elliptical opening, rectangular opening, and notched circular opening with one or two axis of symmetry:

$${\phi }_0\left(\zeta \right)={\displaystyle \sum }_{k=0}^n{\alpha }_k{\zeta }^k$$

(7)

where α_k (k = 0, 1, 2, …, n) are unknown complex numbers and are obtained by the boundary conditions.

Proceeding formally, according to Equations (1) and (7), the left integral item of Equation (4) may be derived that we may write:

$$\begin{array}{c}\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{\omega \left(\mathsf{\sigma }\right)}{\overline{{\omega }^{\prime }\left(\mathsf{\sigma }\right)}}\frac{\overline{{{\phi }^{\prime }}_0\left(\sigma \right)}}{\sigma -\zeta }d\sigma =-\frac{1}{c_{-1}}{\displaystyle {\displaystyle \sum }_{k=1}^9}[k{\overline{\alpha }}_k\left(a_{k+1}+i{b}_{k+1}\right)]\hfill \\ +\frac{1}{-c_{-1}+{{\displaystyle \sum }}_{k=0}^{10}k\left(a_k-i{b}_k\right)\frac{1}{{\zeta }^{k+1}}}\{{\overline{\alpha }}_1{\displaystyle {\displaystyle \sum }_{k=1}^8}[{\zeta }^k\left(a_{k+2}+i{b}_{k+2}\right)]+2{\overline{\alpha }}_2{\displaystyle {\displaystyle \sum }_{k=1}^7}[{\zeta }^k\left(a_{k+3}+i{b}_{k+3}\right)]\hfill \\ +3{\overline{\alpha }}_3{\displaystyle {\displaystyle \sum }_{k=1}^6}[{\zeta }^k\left(a_{k+4}+i{b}_{k+4}\right)]+4{\overline{\alpha }}_4{\displaystyle {\displaystyle \sum }_{k=1}^5}[{\zeta }^k\left(a_{k+5}+i{b}_{k+5}\right)]+5{\overline{\alpha }}_5{\displaystyle {\displaystyle \sum }_{k=1}^4}[{\zeta }^k\left(a_{k+6}+i{b}_{k+6}\right)]\hfill \\ +6{\overline{\alpha }}_6{\displaystyle {\displaystyle \sum }_{k=1}^3}[{\zeta }^k\left(a_{k+7}+i{b}_{k+7}\right)]+7{\overline{\alpha }}_7{\displaystyle {\displaystyle \sum }_{k=1}^2}[{\zeta }^k\left(a_{k+8}+i{b}_{k+8}\right)]+8{\overline{\alpha }}_8\zeta \left(a_{10}+i{b}_{10}\right)\}\hfill \end{array}$$

(8)

The bars “¯” in Equation (8) denote complex conjugates and the primes “′” denote differentiation. Simplifying the right items of Equation (8) into polynomials it can be given as the following relation:

$$\begin{array}{c}\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{\omega \left(\mathsf{\sigma }\right)}{\overline{{\omega }^{\prime }\left(\mathsf{\sigma }\right)}}\frac{\overline{{{\phi }^{\prime }}_0\left(\sigma \right)}}{\sigma -\zeta }d\sigma =-\frac{1}{c_{-1}}{\displaystyle {\displaystyle \sum }_{k=1}^9}[k{\overline{\alpha }}_k\left(a_{k+1}+i{b}_{k+1}\right)]-{\overline{\alpha }}_1{\displaystyle {\displaystyle \sum }_{k=0}^8}{d}_{1k}{\zeta }^k-{\overline{\alpha }}_2{\displaystyle {\displaystyle \sum }_{k=0}^7}{d}_{2k}{\zeta }^k-{\overline{\alpha }}_3{\displaystyle {\displaystyle \sum }_{k=0}^6}{d}_{3k}{\zeta }^k\hfill \\ -{\overline{\alpha }}_4{\displaystyle {\displaystyle \sum }_{k=0}^5}{d}_{4k}{\zeta }^k-{\overline{\alpha }}_5{\displaystyle {\displaystyle \sum }_{k=0}^4}{d}_{5k}{\zeta }^k-{\overline{\alpha }}_6{\displaystyle {\displaystyle \sum }_{k=0}^3}{d}_{6k}{\zeta }^k-{\overline{\alpha }}_7{\displaystyle {\displaystyle \sum }_{k=0}^2}{d}_{7k}{\zeta }^k-{\overline{\alpha }}_8{\displaystyle {\displaystyle \sum }_{k=0}^1}{d}_{8k}{\zeta }^k+O(\frac{1}{\zeta })\hfill \end{array}$$

(9)

in which the constant complex coefficients d_1k (k = 0, 1, 2, …, 8), d_2k (k = 0, 1, 2, …, 7), d_3k (k = 0, 1, 2, …, 6), d_4k (k = 0, 1, 2, …, 5), d_5k (k = 0, 1, 2, …, 4), d_6k (k = 0, 1, 2, 3), d_7k(k = 0, 1, 2), and d_8k(k = 0, 1) are represented in closed form in terms of the constant conformal mapping coefficients c₋₁, a_k, b_k (k = 0, 1, 2, 3, …, 10) and O(·) represents high order of magnitude.

For solving the complex coefficients α_k, the right integral item of Equation (4) should be given according to the expression f₀ in Equation (5). In this paper, we mainly research the case that there is no surface traction on the contour C in Figure 3a. Therefore, the expression f₀ is given as the following relation:

$$f_0=-\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })[c_{-1}\frac{1}{\mathsf{\sigma }}+{\displaystyle \sum }_{k=0}^{10}(a_k+i{b}_k){\mathsf{\sigma }}^k]+\frac{1}{2}({\sigma }_{x\infty }-{\sigma }_{y\infty })[c_{-1}\mathsf{\sigma }+{\displaystyle \sum }_{k=0}^{10}(a_k-i{b}_k)\frac{1}{{\mathsf{\sigma }}^k}]$$

(10)

The right integral item of Equation (4) is written as follows.

$$\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{f_{0}d\sigma }{\sigma -\zeta }=-\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })[{\displaystyle \sum }_{k=0}^{10}(a_k+i{b}_k){\mathsf{\zeta }}^k]+\frac{1}{2}({\sigma }_{x\infty }-{\sigma }_{y\infty })c_{-1}\mathsf{\zeta }+\frac{1}{2}({\sigma }_{x\infty }-{\sigma }_{y\infty })(a_0-i{b}_0)$$

(11)

By substituting Equations (7), (9), and (11) into Equation (4), we obtain the relation for evaluating the unknown complex coefficients α_k in Equation (7):

$$\begin{array}{c}{\displaystyle {\displaystyle \sum }_{k=0}^n}{\alpha }_k{\zeta }^k-\frac{1}{c_{-1}}{\displaystyle {\displaystyle \sum }_{k=1}^9}[k{\overline{\alpha }}_k(a_{k+1}+i{b}_{k+1})]-{\overline{\alpha }}_1{\displaystyle {\displaystyle \sum }_{k=0}^8}{d}_{1k}{\zeta }^k-{\overline{\alpha }}_2{\displaystyle {\displaystyle \sum }_{k=0}^7}{d}_{2k}{\zeta }^k-{\overline{\alpha }}_3{\displaystyle {\displaystyle \sum }_{k=0}^6}{d}_{3k}{\zeta }^k-{\overline{\alpha }}_4{\displaystyle {\displaystyle \sum }_{k=0}^5}{d}_{4k}{\zeta }^k\hfill \\ -{\overline{\alpha }}_5{\displaystyle {\displaystyle \sum }_{k=0}^4}{d}_{5k}{\zeta }^k-{\overline{\alpha }}_6{\displaystyle {\displaystyle \sum }_{k=0}^3}{d}_{6k}{\zeta }^k-{\overline{\alpha }}_7{\displaystyle {\displaystyle \sum }_{k=0}^2}{d}_{7k}{\zeta }^k-{\overline{\alpha }}_8{\displaystyle {\displaystyle \sum }_{k=0}^1}{d}_{8k}{\zeta }^k\hfill \\ =-\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })[{\displaystyle {\displaystyle \sum }_{k=0}^{10}}(a_k+i{b}_k){\mathsf{\zeta }}^k]+\frac{1}{2}({\sigma }_{x\infty }-{\sigma }_{y\infty })c_{-1}\mathsf{\zeta }+\frac{1}{2}({\sigma }_{x\infty }-{\sigma }_{y\infty })(a_0-i{b}_0)\hfill \end{array}$$

(12)

By comparing the same powers of the variable ζ and separating the real values and imaginary values in Equation (12), the linear equations are gained for solving the coefficients α_k:

$$B_{\alpha q}{X}_{\alpha }=R_{\alpha q}\iff X_{\alpha }={(B_{\alpha q})}^{-1}{R}_{\alpha q}$$

(13)

$$\begin{gathered} B_{aq}=\left[\begin{array}{ccccccccc}1& 0& -\frac{a_2}{c_{-1}}-Re\left(d_{10}\right)& -\frac{b_2}{c_{-1}}-Im\left(d_{10}\right)& \cdots & -\frac{9a_{10}}{c_{-1}}& -\frac{9b_{10}}{c_{-1}}& 0& 0\\ 0& 1& -\frac{b_2}{c_{-1}}-Im\left(d_{10}\right)& \frac{a_2}{c_{-1}}+Re\left(d_{10}\right)& \cdots & -\frac{9b_{10}}{c_{-1}}& \frac{9a_{10}}{c_{-1}}& 0& 0\\ 0& 0& -Re\left(d_{11}\right)+1& -Im\left(d_{11}\right)& \cdots & 0& 0& 0& 0\\ 0& 0& -Im\left(d_{11}\right)& Re\left(d_{11}\right)+1& \cdots & 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right] \\ {X}_{\alpha }={[\begin{array}{cc}\begin{array}{c}\begin{array}{cc}\begin{array}{ccc}Re({\alpha }_0)& Im({\alpha }_0)& Re({\alpha }_1)\end{array}& Im({\alpha }_1)\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\cdots \end{array}& Im({\alpha }_{10})\end{array}\end{array}]}^T \\ {R}_{\alpha q}={[\begin{array}{ccc}-{\sigma }_{y\infty }{a}_0& -{\sigma }_{x\infty }{b}_0& \begin{array}{ccc}-\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })a_1+\frac{1}{2}({\sigma }_{x\infty }-{\sigma }_{y\infty })c_{-1}& -\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })b_1& \begin{array}{cc}\cdots & -\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })b_{10}\end{array}\end{array}\end{array}]}^T \end{gathered}$$

In the above relations, Re(·) and Im(·) denote the real and imaginary value of what it encloses, respectively. By setting the value of σ_x∞ and σ_y∞ is equal to −33.75 MPa and −22.5 MPa according to the in situ stress measurement in the Chen-man-zhuang mining project, respectively, and using the values of coefficients in

Table 1. The values of c₋₁, a_k, b_k (k = 0, 1, 2, 3, …, 10).

Coefficient	Value	Coefficient	Value
c−1	1.943007098870969
a0	0.123864538423751	b0	−0.343130047633050
a1	−0.762775765573139	b1	0.229952727807971
a2	−0.183253154614749	b2	−0.108592194784094
a3	−0.210508431449733	b3	−0.119129909218219
a4	0.059058447475317	b4	0.114679930116000
a5	0.041584641384929	b5	0.012230502784915
a6	0.009023747775452	b6	−0.041311547624255
a7	−0.017357739598054	b7	0.013639095406708
a8	−0.003902025809779	b8	−0.013828080675274
a9	−0.002407424676143	b9	0.004481421558265
a10	−0.009939070300865	b10	0.017387660145686

and

Table 2. The values of the complex coefficients d_1k, d_2k, d_3k, d_4k, d_5k, d_6k, d_7k, d_8k.

dik	Value	dik	Value
d10	−0.00613127527563420 − 0.0212816433307378i	d11	−0.117842059038110 − 0.0680388091648117i
d12	0.0293443760564685 + 0.0707224081150705i	d13	0.0274885674993676 + 0.00902292634928232i
d14	0.00303458256474199 − 0.0218746391688664i	d15	−0.00820945608106655 + 0.00400098771607415i
d16	0.00105898412971797 − 0.0100245445605539i	d17	−0.00123902001055050 + 0.00230643601913176i
d18	−0.00511530313329306 + 0.00894884025683173i
d20	−0.0190009864787558 − 0.0134533527904577i	d21	0.0586887521129370 + 0.141444816230141i
d22	0.0549771349987352 + 0.0180458526985646i	d23	0.00606916512948398 − 0.0437492783377329i
d24	−0.0164189121621331 + 0.00800197543214830i	d25	0.00211796825943594 − 0.0200490891211079i
d26	−0.00247804002110100 + 0.00461287203826352i	d27	−0.0102306062665861 + 0.0178976805136635i
d30	−0.00315302474218401 + 0.0351015869896333i	d31	0.0824657024981029 + 0.0270687790478470i
d32	0.00910374769422596 − 0.0656239175065993i	d33	−0.0246283682431996 + 0.0120029631482225i
d34	0.00317695238915391 − 0.0300736336816618i	d35	−0.00371706003165149 + 0.00691930805739527i
d36	−0.0153459093998792 + 0.0268465207704952i
d40	0.0243454394191507 + 0.0109132018455532i	d41	0.0121383302589680 − 0.0874985566754658i
d42	−0.0328378243242662 + 0.0160039508642966i	d43	0.00423593651887188 − 0.0400981782422158i
d44	−0.00495608004220199 + 0.00922574407652703i	d45	−0.0204612125331722 + 0.0357953610273269i
d50	−0.00804817520189555 − 0.00306491820535773i	d51	−0.0410472804053327 + 0.0200049385803708i
d52	0.00529492064858985 − 0.0501227228027697i	d53	−0.00619510005275249 + 0.0115321800956588i
d54	−0.0255765156664653 + 0.0447442012841587i
d60	0.00434391049468474 − 0.0181115587543296i	d61	0.00635390477830782 − 0.0601472673633236i
d62	−0.00743412006330299 + 0.0138386161147905i	d63	−0.0306918187997584 + 0.0536930415409904i
d70	0.0214705733520627 − 0.0203538957756759i	d71	−0.00867314007385349 + 0.0161450521339223i
d72	−0.0358071219330514 + 0.0626418817978221i
d80	0	d81	−0.0409224250663445 + 0.0715907220546539i

, the real values and imaginary values of the complex coefficients α_k are listed in

Table 3. The real values and imaginary values of the complex coefficients α_k.

αk	Real Value	Imaginary Value
α0	6,048,774.18703198	−10,298,069.9676466
α1	−31,352,104.1442811	9,158,635.93617443
α2	−5,619,295.03514821	−4,860,490.56892524
α3	−6,630,981.33006717	−3,869,348.92054447
α4	1,524,019.59546642	4,091,707.21872871
α5	1,662,297.65768788	494,696.756351372
α6	118,164.270916096	−1,132,658.59704894
α7	−457,719.446320244	172,337.556709757
α8	132,590.210627846	−622,630.541593787
α9	−67,708.8190165163	126,039.981326195
α10	−279,536.352211837	489,027.941597429

.

Finally, the second complex potential function is also supposed in a Laurent series expansion, which is written as follows:

$${\psi }_0(\zeta )={\displaystyle \sum }_{k=0}^n{\beta }_k {\zeta }^k$$

(14)

where β_k (k = 0, 1, 2, …, n) are unknown coefficients and can be obtained using the next relation:

$${\psi }_0(\zeta )+\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{\overline{\omega (\mathsf{\sigma })}}{\omega \prime (\mathsf{\sigma })}\frac{\phi {\prime }_0(\sigma )}{\sigma -\zeta }d\sigma =\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{\overline{f_0}d\sigma }{\sigma -\zeta }$$

(15)

Proceeding similarly, according to Equations (1) and (7), the left integral item of Equation (15) may be derived, which we may write as follows:

$$\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{\overline{\omega (\mathsf{\sigma })}}{\omega \prime (\mathsf{\sigma })}\frac{\phi {\prime }_0(\sigma )}{\sigma -\zeta }d\sigma =\frac{9c_{-1}{\alpha }_9}{10(a_{10}+i{b}_{10})}+\frac{10{\alpha }_{10}{c}_{-1}\zeta }{-c_{-1}{\zeta }^{-11}+{\sum }_{k=0}^{10}k(a_k+i{b}_k){\zeta }^{k-10}}+\frac{{\alpha }_{10}(a_0-i{b}_0)}{(a_{10}+i{b}_{10})}$$

(16)

Additionally, by simplifying the right items of Equation (16) into polynomials it can be given as the following relation:

$$\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{\overline{\omega (\mathsf{\sigma })}}{\omega \prime (\mathsf{\sigma })}\frac{\phi {\prime }_0(\sigma )}{\sigma -\zeta }d\sigma =\frac{{\alpha }_{10}(a_0-i{b}_0)}{(a_{10}+i{b}_{10})}+g_{11}\zeta +O(\frac{1}{\zeta })$$

(17)

in which, $g_{11}=\frac{{\alpha }_{10}{c}_{-1}}{(a_{10}+i{b}_{10})}$.

According to Equations (5) and (6), the right integral item of Equation (15) is written as follows:

$$\frac{1}{2\pi i}{{\displaystyle \int }}_{\sigma }^{}\frac{\overline{f_0}d\sigma }{\sigma -\zeta }=-\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })c_{-1}\mathsf{\zeta }-\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })(a_0-i{b}_0)+\frac{1}{2}({\sigma }_{x\infty }-{\sigma }_{y\infty }){\displaystyle \sum }_{k=0}^{10}(a_k+i{b}_k){\mathsf{\zeta }}^k$$

(18)

By substituting Equations (14), (17), and (18) into Equation (15), the value of the unknown complex coefficients β_k in Equation (7) can be obtained:

$$\begin{array}{l}{\displaystyle \sum }_{k=0}^n{\beta }_k {\zeta }^k+\frac{{\alpha }_{10}(a_0-i{b}_0)}{(a_{10}+i{b}_{10})}+g_{11}\zeta \\ =-\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })c_{-1}\mathsf{\zeta }-\frac{1}{2}({\sigma }_{x\infty }+{\sigma }_{y\infty })(a_0-i{b}_0)+\frac{1}{2}({\sigma }_{x\infty }-{\sigma }_{y\infty }){\displaystyle \sum }_{k=0}^{10}(a_k+i{b}_k){\mathsf{\zeta }}^k\end{array}$$

(19)

Next, by comparing the same powers of the variable ζ, the linear equations are also gained for solving the coefficients β_k. By using the values of σ_x∞ and σ_y∞ and the values of the constant conformal mapping coefficients a_k, b_k (k = 0, 1, 2, 3, …, 10) in Table 1, the real values and imaginary values of the complex coefficients β_k are listed in

Table 4. The real values and imaginary values of the complex coefficients β_k.

βk	Real Value	Imaginary Value
β0	−696,738.028633599	1,930,106.51793591
β1	4,290,613.68134891	−1,293,484.09391984
β2	1,030,798.99470796	610,831.095660530
β3	1,184,109.92690475	670,105.739352481
β4	−332,203.767048660	−645,074.606902502
β5	−233,913.607790227	−68,796.5781651473
β6	−50,758.5812369165	232,377.455386432
β7	97,637.2852390529	−76,719.9116627314
β8	21,948.8951800059	77,782.9537984135
β9	13,541.7638033033	−25,207.9962652390
β10	55,907.2704423675	−97,805.5883194858

.

Then, in polar coordinates (ρ,θ) the radial, tangential, and shear stresses are denoted as σ_r, σ_θ, and τ_ρθ, respectively; the radial and tangential incremental displacements u_ρ, u_θ due to stress relief at the breakout boundary, may be computed by virtue of the following formulae:

$$\{\begin{array}{c}{\mathsf{\sigma }}_{\rho }+{\mathsf{\sigma }}_{\theta }=4Re(B+\frac{{\phi }_0\prime (\zeta )}{\omega \prime (\zeta )})\\ {\mathsf{\sigma }}_{\theta }-{\mathsf{\sigma }}_{\rho }+2i{\tau }_{\rho \theta }=\frac{2{\zeta }^2}{{\rho }^2\overline{{\omega }^{\prime }(\zeta )}}\{\overline{\omega (\zeta )}\frac{{\phi }_0^{\prime \prime }(\zeta ){\omega }^{\prime }(\zeta )-{\phi }_0^{\prime }(\zeta ){\omega }^{″}(\zeta )}{{[{\omega }^{\prime }(\zeta )]}^2}+{\omega }^{\prime }(\zeta )[(B^{\prime }+i{C}^{\prime })+\frac{{\psi }_0\prime (\zeta )}{{\omega }^{\prime }(\zeta )}]\}\\ \frac{E}{1+\mu }(u_{\rho }+i{u}_{\theta })=\frac{\overline{\zeta }}{\rho }\frac{\overline{\omega \prime (\zeta )}}{\left|\omega \prime (\zeta )\right|}\{\kappa [B\omega (\zeta )+{\phi }_0(\zeta )]-\frac{\omega (\mathsf{\zeta })}{\overline{{\omega }^{\prime (\zeta )}}}\overline{B{\omega }^{\prime }(\zeta )+{\phi }_0^{\prime }(\zeta )}-[(B^{\prime }+i{C}^{\prime })\omega (\zeta )+{\psi }_0(\zeta )]\}\end{array}$$

(20)

where primes denote differentiation, |·| represents the absolute value for real number or the modulus for complex number, κ equals to 3−4μ for plane strain problem or κ equals to (3 − μ)/(1 + μ) for plain stress problem, E and μ is the Young’s modulus and Poisson’s ratio of rock (coal seam), respectively. Finally, the displacements are given in the Cartesian coordinate system as follows:

$$\frac{E}{1+\mu }(u_x+i{u}_y)=\kappa [B\omega (\zeta )+{\phi }_0(\zeta )]-\frac{\omega (\mathsf{\zeta })}{\overline{{\omega }^{\prime (\zeta )}}}\overline{B{\omega }^{\prime }(\zeta )+{\phi }_0^{\prime }(\zeta )}-[(B^{\prime }+i{C}^{\prime })\omega (\zeta )+{\psi }_0(\zeta )]$$

(21)

5. Displacements Analysis for Right-Angle Trapezoidal Opening

By setting the Young’s modulus E = 1.5 GPa and the Poisson’s ratio μ = 0.16 according to laboratory testing with the rock specimen from the Chen-man-zhuang mining project, the displacements on opening boundary can be obtained with the computational software (e.g., Matlab, Mathcad, Maple, etc.) according to the implementation of Equation (20) or (21). Geotechnical engineers have begun to take advantage of the results of elasticity theory in rock mechanics or rock engineering applications. The displacements on the boundary of right-angle trapezoidal openings are illustrated below for the applications in rock engineering or mining engineering. Meanwhile, the numerical model in FLAC software was developed for comparing the results of displacement according to the analytical elastic solution. The length of numerical model in horizontal and vertical direction is 100 m and 100 m, respectively; the overall mesh of numerical model is 500 × 500. The constraints on the left and right boundary of the numerical model are both that the horizontal displacement is equal to zero, while the constraint on the bottom of the numerical model is that the vertical displacement is equal to zero. Uniform loading is applied on the top of the numerical model to simulate the load of overlying strata. The value of the Young’s modulus E and Poisson’s ratio μ in numerical model is the same with the value in analytical computation, respectively.

Figure 5 and Figure 6 show the vertical displacement (y-displacement for short) on the floor and roof for right-angle trapezoidal openings as shown in Figure 3a, respectively. The positive value of y-displacement denotes that the vertical displacement is along Oy-axis; on the contrary, the negative value of y-displacement indicates that the vertical displacement is the opposite direction to Oy-axis. It can be seen that the most values of y-displacement on the floor are positive numbers, which is consistent with the phenomenon of floor heave for opening in mining engineering. The y-displacement on the floor around corner A or corner B, as shown on Figure 3a, are negative numbers because of stress concentration applied by principal stresses σ_x∞ and σ_y∞. The most values of y-displacement on the roof are negative numbers, which indicates that the roof is almost falling down for opening in mining engineering. The most displacements of falling down on the roof are around corner E, which is close to the left side of the opening (also called “low side” in right-angle trapezoidal openings). It can be concluded that the roof near the low side in right-angle trapezoidal openings need more bolt supporting to resist downward deformation, which is an important difference from the deformation of opening with other shapes.

Figure 7 and Figure 8 show the horizontal displacement (x-displacement for short) on the left side (also called “low side”) and right side (also called “high side”) for right-angle trapezoidal openings as shown in Figure 3a, respectively. In the same way, the positive value of x-displacement denotes that the horizontal displacement is along the Ox-axis, and the negative value indicates the opposite direction to the Ox-axis. It can be clearly obtained that the values of x-displacement on the left side are all positive numbers and these are almost negative numbers on the right side, which indicate that the left and right sides of opening are close to each other. The x-displacements on the left side around corner E are bigger than that around corner A and the x-displacements on the middle of the right side are bigger than that around corner B and corner F. This indicates that more bolt supporting is needed around corner A on the low side and the middle of the high side for right-angle trapezoidal openings, which is the other difference from the deformation of opening with other shapes.

In conclusion, the geotechnical engineers should change the traditional concept of symmetrical bolt supporting for right-angle trapezoidal openings because there is not a symmetrical axis in right-angle trapezoidal shapes and no symmetrical deformation on the boundary of right-angle trapezoidal openings.

6. Conclusions

An analytical elastic solution for stresses and displacements around right-angle trapezoidal openings is presented, which is usually applied to steeply inclined coal seam or rock. It can be shown that the conformal mapping and the complex potentials can be used to analyze the displacements or stresses in plane elasticity problems for the opening without a symmetrical axis. The different results from other shaped openings are shown as follows. The most displacements of falling down on the roof are around the low side in right-angle trapezoidal openings, but the most horizontal displacements on the low side are around the roof and the most horizontal displacements on the high side are around the middle of the high side in this opening. Meanwhile, the deformation results are consistent with those of the numerical calculation in FLAC software. The formulation employed here is a good method to gain insight into deformation and the optimum design of bolt supporting in right-angle trapezoidal openings. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of an opening without a symmetrical axis.