Analytical solution of surrounding rock stress and plastic zone of rectangular roadway under non-uniform stress field

Abstract

The shape and extent of the plastic zone can effectively reflect the degree of deformation and failure of the surrounding rock. By applying complex variable functions and the theory of elasticity, a mechanical model for the surrounding rock of a rectangular roadway is established. The analytical solution for the stress in the surrounding rock of a rectangular roadway under a non-uniform stress field is derived, and the spatial stress distribution pattern in the surrounding rock of the rectangular roadway is obtained. Furthermore, using the Mohr-Coulomb criterion, the boundary equation of the plastic zone in the surrounding rock of a rectangular roadway is derived, and the morphological characteristics of the plastic zone are obtained. The results indicate that the plastic zones on both sides of the rectangular roadway are arc-shaped. The asymmetry of the stress environment has a significant impact on the plastic zones in the roof and floor of the rectangular roadway, causing the shape of the plastic zones to resemble a “petal” pattern. The lateral pressure coefficient and the height-to-width ratio have a significant impact on the stress distribution and the morphological characteristics of the plastic zone in the surrounding rock of the rectangular roadway. By validating the computational accuracy of the analytical solution through numerical simulations and conducting an engineering applicability analysis based on a coal mine roadway, it has been demonstrated that the analytical calculation method is both rational and possesses good engineering applicability. This provides reliable theoretical guidance for the design calculations and construction planning of similar roadway projects.

Introduction

With the rapid economic development across the world, the demand for energy is increasingly rising. After experiencing high-intensity and prolonged mining, coal resources gradually shift from shallow to deeper deposits¹. The stress distribution in the surrounding rock of the roadway becomes more complex, leading to potential issues such as roof falls and large deformations. This poses higher demands for the safe mining of coal^2,3. Deformation of the surrounding rock in a roadway under complex stress conditions can directly lead to roadway disasters, making the study of deformation and failure characteristics of the surrounding rock highly significant^4,5.

Currently, scholars both domestically and internationally have applied complex variable function theory to study the stress and deformation of coal mine roadways. Muskhelishvili et al.⁶ categorized the mechanical analysis of roadway stress and deformation as a problem of an infinite plane with an opening. Zhou et al.⁷ provided analytical solutions for the stress in rectangular roadways with different lateral pressure coefficients. Building on this, Kargar et al.⁸ employed the Cauchy method of complex variable functions to analyze the stress in non-circular roadways and derived an analytical solution for the stress, though this approach is relatively complex.

In the process of solving for the stress in the surrounding rock of a rectangular roadway, it is essential to first map the rectangular roadway onto the boundary of a unit circle using conformal mapping. Therefore, the determination of the mapping function is particularly important. Zhao et al.⁹ solved the mapping function for rectangular roadways using the polygonal method and found that retaining the first three terms ensures sufficient accuracy. Building on this, Zan et al.¹⁰ analyzed the stress in the surrounding rock of rectangular roadways. However, their analysis has mostly been limited to the stress concentration phenomena on the surface of the roadway surrounding rock, with less focus on the stress distribution in the deeper parts of the surrounding rock.

Research on the deformation and failure characteristics of roadway surrounding rock primarily focuses on the plastic zone, as it effectively represents the extent and mode of rock failure. The study of the shape of the plastic zone in circular roadways has been extensive. Kastner derived the radius of the circular plastic zone in a uniform stress field¹¹, and many scholars have subsequently revised this formula^{12,13,14,15,16}. For circular roadways in a non-uniform stress field, Abdel-Meguid et al.^17,18 conducted theoretical analyses and concluded that the plastic zone should be elliptical. Later, Zhao et al.¹⁹ examined the boundary equations for circular roadways and observed that, as the lateral pressure coefficient changed, the plastic zone shape transitioned sequentially from circular to elliptical to petal-like. Building on this, Guo et al.^20,21 conducted extensive research on the critical conditions for changes in the plastic zone shape and developed standards and mathematical expressions for the shape of the plastic zone.

The aforementioned studies primarily address the plastic zones in circular roadways. In contrast, rectangular roadways are more prevalent in practical engineering due to their advantages, such as faster excavation rates and higher tunnel formation efficiency^22,23. However, the non-uniform stress distribution on the roof of rectangular roadways often leads to significant stress concentration at the corners. Modeling rectangular roadways as circular can neglect the stress concentrations and associated damage at these corners, resulting in potential inaccuracies in the analysis.

Analytical solution for the stress in the surrounding rock of a rectangular roadway

The roadway model is simplified to a rectangular aperture problem within an infinite plane model. The inverse and semi-inverse methods in classical elastoplastic mechanics cannot effectively solve this problem²⁴. Therefore, complex variable functions are introduced to solve the stress in the surrounding rock of a rectangular roadway. The complex variable function expression for the stress around the aperture is given by^25,26:

$$\left\{ \begin{gathered} {\sigma _\theta }+{\sigma _\rho }=2\left[ {\Phi (\xi )+\overline {{\Phi \left( \xi \right)}} } \right]=4\operatorname{Re} [\Phi (\xi )] \hfill \\ {\sigma _\theta } - {\sigma _\rho }+2i{\tau _{\rho \theta }}=\frac{{2{\xi ^2}}}{{{\rho ^2}\overline {{\omega ^{\prime}\left( \xi \right)}} }}[\overline {{\omega \left( \xi \right)}} \Phi ^{\prime}(\xi )+\omega ^{\prime}\left( \xi \right)\Psi (\zeta )] \hfill \\ \end{gathered} \right.$$

(1)

where:

$$\left\{ \begin{gathered} \Phi (\xi )=\frac{{\varphi ^{\prime}(\xi )}}{{\omega ^{\prime}(\xi )}} \hfill \\ \Psi (\xi )=\frac{{\phi ^{\prime}(\xi )}}{{\omega ^{\prime}(\xi )}} \hfill \\ \end{gathered} \right.$$

(2)

To solve Eq. (1), it is necessary to simplify the boundary of the rectangular roadway. This is achieved by applying conformal mapping to map it onto the boundary of a unit circle in a complex plane. Subsequently, the stress in the surrounding rock of the rectangular roadway is determined^27,28.

Conformal transformation of rectangular roadway

The area in the surrounding rock of the rectangular roadway in the Z plane is transformed to the unit circle area of the$\xi$plane by the mapping function²⁹:

$$z=\omega (\xi )=R(\frac{1}{\xi }+\sum\limits_{{k=0}}^{n} {{c_k}{\xi ^k}} )$$

(3)

The Cartesian positions of any point in the Z plane are represented as $z=x+iy$, and the positions of any point in the$\xi$plane, after being transformed into polar coordinates, are expressed as $\xi =\rho (\cos \varphi +i\sin \varphi )=\rho {e^{i\varphi }}$.

For the sake of computational simplicity, and given that previous studies have shown that using n = 3 ensures the accuracy of theoretical calculations³⁰, so take n = 3 and substitute the Cartesian coordinates into Eq. (3) to obtain:

$$\left\{ \begin{gathered} x=R(\frac{1}{\rho }\cos \varphi +{c_1}\rho \cos \varphi +{c_3}{\rho ^3}\cos 3\varphi ) \hfill \\ y=R( - \frac{1}{\rho }\sin \varphi +{c_1}\rho \sin \varphi +{c_3}{\rho ^3}\sin 3\varphi ) \hfill \\ \end{gathered} \right.$$

(4)

The values of R, ${c_1}$, and ${c_3}$ are determined by assigning values to the mapping function, and their values are determined by:

$$\left\{ \begin{gathered} R=\frac{{\frac{{a - b}}{2}\cos \theta +\frac{{a+b}}{2}\cos 3\theta - a}}{{\cos 3\theta - \cos \theta }} \hfill \\ {c_1}=\frac{{a - b}}{{2R}} \hfill \\ {c_3}=\frac{{a+b}}{{2R}} - 1 \hfill \\ \end{gathered} \right.$$

(5)

Where: θ is the angle of a point on the plane corresponding to the plane $\xi$, and $- \frac{\pi }{2}<\theta <0$, which is determined by:

$$\frac{{\left( {\frac{{a - b}}{2}\cos \theta +\frac{{a+b}}{2}\cos 3\theta - a} \right)}}{{\cos 3\theta - \cos \theta }}=\frac{{\left( {\frac{{a - b}}{2}\sin \theta +\frac{{a+b}}{2}\sin 3\theta - b} \right)}}{{\sin 3\theta +\sin \theta }}$$

(6)

Determination of boundary conditions

The rectangular roadway is exposed to the far-field vertical stress P₁ and the horizontal stress P₃, with the lateral pressure factor is λ, then the following can be obtained:

$$\left\{ \begin{gathered} {\alpha _1}=\frac{{1+\lambda }}{4}{P_1} \hfill \\ {\alpha _2}=\frac{{1 - \lambda }}{2}{P_1} \hfill \\ \end{gathered} \right.$$

(7)

In the existing roadway engineering, the support resistance provided by the conventional anchor cable ranges from 0.05 to 0.8 MPa, which is far less than the ground stress field of the roadway³¹. In order to simplify the calculation, the support resistance is ignored in the calculation. Without considering the support resistance, ${f_x}={f_y}=0$, ${F_x}={F_y}=0$,then:

$${f_0}= - 2{\alpha _1}R(\frac{1}{\sigma }+{c_1}\sigma +{c_3}{\sigma ^3}) - {\alpha _1}^{\prime }R(\sigma +\frac{{{c_1}}}{\sigma }+\frac{{{c_3}}}{{{\sigma ^3}}})$$

(8)

$\varphi (\xi )$ and $\phi (\xi )$ are calculated, and the solution formula is:

$$\left\{ \begin{gathered} \varphi (\xi )={\alpha _1}R\xi +{\varphi _0}(\xi ) \hfill \\ \phi (\xi )={\alpha _2}R\xi +{\phi _0}(\xi ) \hfill \\ \end{gathered} \right.$$

(9)

$$\left\{ \begin{gathered} {\varphi _0}(\xi )=\frac{1}{{2\pi i}}\int {\frac{{{f_0}}}{{\sigma - \xi }}} {\text{d}}\sigma - \frac{1}{{2\pi i}}\int {\frac{{\omega (\sigma )\overline {{{\varphi _0}^{\prime }(\sigma )}} }}{{\overline {{\omega ^{\prime}(\sigma )}} (\sigma - \xi )}}} {\text{d}}\sigma \hfill \\ {\phi _0}(\xi )=\frac{1}{{2\pi i}}\int {\frac{{\overline {{{f_0}}} }}{{\sigma - \xi }}} {\text{d}}\sigma - \frac{1}{{2\pi i}}\int {\frac{{\overline {{\omega (\sigma )}} {\varphi _0}^{\prime }(\sigma )}}{{\omega ^{\prime}(\sigma )(\sigma - \xi )}}} {\text{d}}\sigma \hfill \\ \end{gathered} \right.$$

(10)

Expand ${\varphi _0}(\xi )$ and ${\phi _0}(\xi )$ into the Laurent series^32,33:

$$\left\{ \begin{gathered} {\varphi _0}(\xi )={a_0}+{a_1}\xi + \cdots +{a_{n - 1}}{\xi ^{n - 1}} \hfill \\ {\phi _0}(\xi )={b_0}+{b_1}\xi + \cdots +{b_{n - 1}}{\xi ^{n - 1}} \hfill \\ \end{gathered} \right.$$

(11)

Equation (8), Eq. (9) and Eq. (10), substituting Eq. (11), solve $\varphi (\xi )$ and $\phi (\xi )$:

$$\left\{ \begin{gathered} \varphi (\xi )={\alpha _1}R(\frac{1}{\xi }+{c_1}\xi +{c_3}{\xi ^3})+{a_1}\xi - 2{\alpha _1}R{c_3}{\xi ^3} \hfill \\ \phi (\xi )=\frac{{{\alpha _1}^{\prime }R - {a_1}{c_3}}}{\xi } - 2{\alpha _1}R\xi - \frac{{{\xi ^4}+{c_1}{\xi ^2}+{c_3}}}{{3{c_3}{\xi ^3}+{c_1}{\xi ^2} - 1}}(\frac{{{a_1}}}{\xi } - 6{\alpha _1}{c_3}R\xi ) \hfill \\ \end{gathered} \right.$$

(12)

Analytical formula of surrounding rock stress in rectangular roadway

Inserting Eq. (2) and Eq. (12) into Eq. (1), the calculated resolution of the surrounding rock stress along the rectangular roadway is as follows:

$$\left\{ \begin{gathered} {\sigma _\theta }+{\sigma _\rho }=4\operatorname{Re} \left( {{\alpha _1}+\frac{{\frac{{{a_1}}}{R} - 6{\alpha _1}{c_3}{\xi ^2}}}{{ - \frac{1}{{{\xi ^2}}}+{c_1}+3{c_3}{\xi ^2}}}} \right) \hfill \\ {\sigma _\theta } - {\sigma _\rho }+2i{\tau _{\rho \theta }}=\frac{{2({K_1}{\xi ^2}+{K_2}{\xi ^4}+{K_3}{\xi ^6}+{K_4}{\xi ^8}+{K_5}{\xi ^{10}})}}{{{R^2}{{\left( {{c_1}{\xi ^2}+3{c_3}{\xi ^4} - 1} \right)}^2}\left( {{c_1}{\rho ^4}{\xi ^2}+3{c_3}{\rho ^8} - {\xi ^4}} \right)}} \hfill \\ \end{gathered} \right.$$

(13)

where Re is the real part of the imaginary number, and the rest of the arguments can be obtained from Eq. (14):

$$\left\{ \begin{gathered} {K_1}= - {\alpha _2}R{\rho ^4} - 2{a_1}{c_3}{\rho ^{10}} \hfill \\ {K_2}=2{\alpha _2}{c_1}R{\rho ^4}+{\rho ^4}\left( {{a_1}{c_1}\left( {1+{c_3} - 2{\rho ^2}} \right)+2{\alpha _1}R\left( { - 1+3{c_3}^{2}\left( { - 1+4rh{o^6}} \right)} \right)} \right) \hfill \\ {K_3}= - {\alpha _2}{c_1}^{2}R{\rho ^4}+6{\alpha _2}{c_3}R{\rho ^4}+{a_1}\left( { - 2+{\rho ^4}\left( {4 - {c_1}+{c_1}^{2}\left( {1+{c_3}} \right)+{c_3}^{2}\left( {9 - 6{\rho ^6}} \right)} \right)} \right)+ \hfill \\ {\text{ }}2{\alpha _1}R{\rho ^4}\left( {3{c_3}^{2}+{c_1}\left( {2 - 3{c_3}\left( {4+{c_3} - 4rh{o^2}+2{c_3}{\rho ^6}} \right)} \right)} \right) \hfill \\ {K_4}= - 6{\alpha _2}{c_1}c3R{\rho ^4} - {a_1}{c_1}{\rho ^4}\left( {4 - 3{c_1} - 3{c_3}\left( {3+2{c_3} - 2{\rho ^2}} \right)} \right) - \hfill \\ {\text{ }}2{\alpha _1}R\left( {{c_1}^{2}{\rho ^4}+3{c_3}\left( { - 4+{\rho ^4}\left( {3+3{c_1}{c_3}+9{c_3}^{2}+2{c_1}^{2}\left( { - 2+{\rho ^2}} \right)} \right)} \right)} \right) \hfill \\ {K_5}= - 9{\alpha _2}{c_3}^{2}R{\rho ^4}+3{c_3}\left( { - 2\left( {{a_1}+2{\alpha _1}{c_1}R} \right)+\left( {{a_1}\left( { - 4+5{c_1}+3{c_3}^{2}} \right)+2{\alpha _1}\left( {{c_1}+12{c_1}{c_3} - 15{c_3}^{2}} \right)R} \right){\rho ^4}} \right) \hfill \\ {a_1}=\frac{{{\alpha _2}+2{\alpha _1}{c_1}}}{{{c_3} - 1}}R \hfill \\ \end{gathered} \right.$$

(14)

From Eq. (13), the ${\sigma _\theta }$, ${\sigma _\rho }$ and ${\tau _{\rho \theta }}$ in the surrounding rock of the roadway under the curvilinear coordinates can be obtained. The stress is then transformed into Cartesian coordinates by applying the appropriate coordinate transformation formulas.

From the stress coordinate transformation formula, it can be obtained that:

$$\left\{ \begin{gathered} {\sigma _x}=\frac{1}{2}\left[ {{\sigma _\theta }+{\sigma _\rho } - \operatorname{Re} \left( {\frac{{{\sigma _\theta } - {\sigma _\rho }+2i{\tau _{\rho \theta }}}}{{{e^{2i\lambda }}}}} \right)} \right] \hfill \\ {\sigma _y}=\frac{1}{2}\left[ {{\sigma _\theta }+{\sigma _\rho }+\operatorname{Re} \left( {\frac{{{\sigma _\theta } - {\sigma _\rho }+2i{\tau _{\rho \theta }}}}{{{e^{2i\lambda }}}}} \right)} \right] \hfill \\ {\tau _{xy}}=\frac{1}{2}\operatorname{Im} \left( {\frac{{{\sigma _\theta } - {\sigma _\rho }+2i{\tau _{\rho \theta }}}}{{{e^{2i\varphi }}}}} \right) \hfill \\ \end{gathered} \right.$$

(15)

In the formula, Im denotes the imaginary part of the imaginary number. As indicated by Eq. (15), once the burial depth and dimensions of the roadway are established, the stress dispersion in the surrounding rock of the rectangular roadway can be derived using the aforementioned formula.

Stress distribution pattern in the roadway

Upon specifying the subsurface depth and size of the roadway, the spatial stress dispersion state in the surrounding rock of the rectangular roadway can be ascertained using Eq. (15). For instance, considering a roadway buried at a depth of 400 m and a bulk density of 20kN/m³, then P₁ = 8 MPa, lateral pressure factor$\lambda =0.5,1$, the size of the roadway is taken as a×b = 4m×4m, the spatial dispersion of the stress in the surrounding rock of the roadway is displayed in Fig. 1:

Fig. 1

Spatial dispersion of stress in the surrounding rock of rectangular roadway. (a) λ=0.5 Horizontal Stress, (b) λ=0.5 Vertical Stress, (c) λ=0.5 Shear Stress, (d) λ=1 Horizontal Stress, (e) λ=1 Vertical Stress, (f) λ=1 Shear Stress.

As shown in Fig. 1, the analytical solution for the stress in the surrounding rock of a rectangular roadway, obtained using complex variable functions and elasticity methods, effectively reflects the stress concentration at the corners of the roadway. Horizontal stress concentration increases with the lateral pressure coefficient, while vertical stress concentration decreases as the lateral pressure coefficient increases. Shear stress remains largely unchanged and exhibits opposite values at adjacent corners. Therefore, further quantitative analysis of horizontal and vertical stresses is warranted.

The stress concentration coefficient of the surrounding rock in coal mine tunnels primarily reflects the stress concentration phenomenon that occurs in the rock mass due to tunnel excavation. Studying this coefficient can aid in predicting the stability of the surrounding rock and the design of support structures. Particularly under non-uniform stress fields and complex geological conditions, analyzing the stress concentration coefficient allows for effective assessment of the shape of the plastic zone and the areas of stress release around the tunnel, thereby providing a basis for safe mining operations. To analyze the influence of different factors on the stress distribution in the surrounding rock of the roadway. The ratio of the surrounding rock stress after tunnel excavation to the original rock stress prior to excavation is defined as the stress concentration coefficient. The horizontal stress concentration factor ${t_x}$ and the vertical stress concentration factor ${t_y}$ in the surrounding rock of the roadway are defined as shown in Eq. (16):

$$\left\{ \begin{gathered} {t_x}=\frac{{{\sigma _x}}}{{\lambda {P_1}}} \hfill \\ {t_y}=\frac{{{\sigma _y}}}{{{P_1}}} \hfill \\ \end{gathered} \right.$$

(16)

Owing to the symmetry of the stress dispersion in the roadway, the center of the roadway’s roof, the top angle, and the center of the right side were selected for analysis.

Four stress directions ${\sigma _1}$, ${\sigma _2}$, ${\sigma _3}$, and ${\sigma _4}$ are defined in Fig. 2, and the variation of the stress concentration factors of horizontal stress and vertical stress with depth-width ratio and lateral pressure factor in these directions is studied.

Fig. 2

Stress calculation model of surrounding rock of rectangular roadway.

Effect of lateral pressure factor

It is found that the variation range of lateral pressure coefficient in shallow rock mass is relatively large, and with the increase of buried depth of rock mass, the lateral pressure coefficient will continue to decrease³⁴. The depth-width ratio k = 1, and the lateral pressure factor λ = 0.4, 0.5, ., 1 are chosen to examine the impact of varying lateral pressure factors on the stress concentration in a rectangular roadway. The variations of ${t_x}$ and ${t_y}$ in the four directions are calculated and presented in Fig. 3.’

Fig. 3

Variation of ${t_x}$ and ${t_y}$ with λ in different directions. (a) Stress ${t_x}$ in the ${\sigma _1}$ direction, (b) Stress ${t_y}$ in the ${\sigma _1}$ direction, (c) Stress ${t_x}$ in the ${\sigma _2}$ direction, (d) Stress ${t_y}$ in the ${\sigma _2}$ direction, (e) Stress ${t_x}$ in the ${\sigma _3}$ direction, (f) Stress ${t_y}$ in the σ₃ direction, (g) Stress ${t_x}$ in the σ₄ direction, (h) Stress ${t_y}$ in the σ₄ direction

At the center of the roadway roof (${\sigma _1}$ direction), although both horizontal and vertical stress concentration coefficients increase with distance, they exhibit different growth trends. The horizontal stress concentration coefficient shows a gradual increase, following a logarithmic growth trend. In contrast, the vertical stress concentration coefficient increases according to a quadratic growth trend. As the lateral pressure coefficient decreases, the stress concentration coefficients decrease linearly.

In the ${\sigma _2}$ direction at the upper corner of the roadway, the horizontal stress concentration coefficient decreases with both increasing distance and decreasing lateral pressure coefficient. In contrast, the vertical stress concentration coefficient increases with decreasing lateral pressure coefficient within a distance of 0.6 m, but decreases beyond this distance. Both coefficients show a significant decrease with increasing distance up to 4 m, after which the decrease becomes more gradual.

In the ${\sigma _3}$ direction at the upper corner of the roadway, the horizontal stress concentration coefficient decreases with a reduction in the lateral pressure coefficient. However, within 3 m from the roadway, the effect of the lateral pressure coefficient on horizontal stress is more pronounced. When $0.6 \leqslant \lambda \leqslant 1$, the maximum horizontal stress concentration coefficient occurs at the roadway boundary, and as the lateral pressure coefficient decreases, the change in horizontal stress concentration coefficient with distance becomes less pronounced. Conversely, when the lateral pressure coefficient is below 0.5, the maximum horizontal stress concentration coefficient is observed approximately 2 m from the roadway boundary. Regarding vertical stress concentration, it initially increases with a decrease in the lateral pressure coefficient, then decreases after reaching a critical value around 1.8 m from the roadway. In this direction, the influence of the lateral pressure coefficient on horizontal stress is significantly greater than on vertical stress.

At the center of the roadway wall (${\sigma _4}$ direction), as the distance increases, both the horizontal and vertical stress concentration coefficients initially increase with a decrease in the lateral pressure coefficient and then decrease after reaching a critical value located approximately 2 m to 3 m from the roadway. Prior to this critical point, the horizontal stress concentration coefficient exhibits a quadratic growth trend, followed by a logarithmic growth trend beyond the critical point. The vertical stress concentration coefficient is more significantly influenced by the lateral pressure coefficient. Before the critical point, as the lateral pressure coefficient increases, the rate of growth of the vertical stress concentration coefficient also increases. After the critical point, for lower values of λ, the vertical stress concentration coefficient significantly decreases with increasing distance, indicating the formation of a stress release zone away from the roadway boundary³⁵. For higher values of λ, the vertical stress concentration coefficient tends to stabilize.

Effect of depth-width ratio

The lateral pressure factor λ = 0.7, and the depth-width ratio k = 0.4, 0.5, …, 1 are selected for analysis to examine the impact of varying height-span ratios on the stress dispersion in the surrounding rock of the roadway The outcomes of these calculations are displayed in Fig. 4.

Fig. 4

Variation of ${t_x}$ and ${t_y}$ with k in different directions. (a) Stress ${t_x}$ in the ${\sigma _1}$ direction, (b) Stress ${t_y}$ in the ${\sigma _1}$ direction, (c) Stress ${t_x}$ in the ${\sigma _2}$ direction, (d) Stress ${t_y}$ in the ${\sigma _2}$direction, (e) Stress ${t_x}$ in the ${\sigma _3}$ direction, (f) Stress ${t_y}$ in the ${\sigma _3}$ direction, (g) Stress ${t_x}$ in the ${\sigma _4}$ direction, (h) Stress ${t_y}$ in the ${\sigma _4}$direction.

At the midpoint of the roadway roof (${\sigma _1}$ direction), the variation trends of the horizontal and vertical stress concentration coefficients with respect to the height-to-width ratio are similar to those observed with the lateral pressure coefficient. The horizontal stress concentration coefficient increases with distance following a logarithmic growth trend, while the vertical stress concentration coefficient exhibits a quadratic growth trend with distance.

In the ${\sigma _2}$ direction at the upper corner of the roadway, When $0.8 \leqslant k \leqslant 1$, the horizontal stress concentration coefficient decreases with increasing distance. For $0.4 \leqslant k \leqslant 0.7$, the horizontal stress concentration coefficient initially increases and then decreases with distance. The maximum value shifts further away from the roadway boundary as the height-to-width ratio decreases. Within the surrounding rock up to 1 m from the roadway, the vertical stress concentration coefficient increases with decreasing height-to-width ratio until it reaches a peak at k = 0.7, after which it decreases. At locations further from the roadway, the vertical stress concentration coefficient is inversely proportional to the height-to-width ratio.

In the ${\sigma _3}$ direction at the upper corner of the roadway, as the distance increases, the horizontal stress concentration coefficient initially decreases with a decreasing height-to-span ratio, then increases with a decreasing lateral pressure coefficient, with a critical value occurring around 3 m from the roadway. When λ = 0.4, the horizontal stress concentration coefficient first increases with distance, reaching a maximum at approximately 2 m from the roadway, and then decreases as the distance further increases. Within 3 m of the roadway, the vertical stress concentration coefficient increases as the height-to-width ratio decreases, reaching a maximum when k = 0.8, and then decreases as the height-to-width ratio continues to decrease. Beyond 3 m, the vertical stress concentration coefficient increases with a decreasing height-to-width ratio.

At the center of the roadway wall (${\sigma _4}$ direction), the horizontal and vertical stress concentration coefficients increase with an increasing height-to-width ratio. As the distance increases, the horizontal stress concentration coefficient tends to stabilize beyond 4 m. The vertical stress concentration coefficient initially increases with distance, reaching its maximum between 1 m and 2 m, and then decreases with further distance. It can be observed that when the vertical stress concentration coefficient is less than 1, it indicates the presence of a stress release zone near the roadway sidewalls.

Morphological study on the plastic area of the surrounding rock

Determination of the plastic area range

Existing research has found that the Mohr-Coulomb criterion provides sufficient accuracy in determining the boundaries of the plastic zone^36,37,38. So after obtaining the stress in the surrounding rock of the rectangular roadway through Eq. (14), the M-C criterion is used to determine whether the stone aggregate transitions into the plastic state, and when the rock mass reaches the elastic limit, the stress is³⁹:

$$\tau =C+\sigma \tan \varphi$$

(17)

where: $\tau$ represents the tangential stress, C represents the cohesion, $\sigma$ represents the normal stress, and $\varphi$ is the angle of internal friction.

In an image, it can be formulated as:

The limit equilibrium condition of the Mohr-Coulomb strength theory is derived from the geometric relation:

$$\sigma _{1} - \sigma _{3} - (\sigma _{1} + \sigma _{3} )\sin \varphi - 2C\cos \varphi = 0$$

(18)

Inserting Eq. (15) into Eq. (18) yields the plastic area boundary equation:

$$\begin{gathered} \frac{1}{4}\left[ {\operatorname{Re} {{\left( {\frac{{{\sigma _\theta } - {\sigma _\rho }+2i{\tau _{\rho \theta }}}}{{{e^{2i\lambda }}}}} \right)}^2}+\operatorname{Im} {{\left( {\frac{{{\sigma _\theta } - {\sigma _\rho }+2i{\tau _{\rho \theta }}}}{{{e^{2i\lambda }}}}} \right)}^2}} \right] \hfill \\ - {(\frac{{{\sigma _\theta }+{\sigma _\rho }}}{2})^2}{\sin ^2}\varphi - ({\sigma _\theta }+{\sigma _\rho })C\sin \varphi \cos \varphi - {C^2}{\cos ^2}\varphi =0 \hfill \\ \end{gathered}$$

(19)

After determining the buried depth, size, cohesion of the surrounding rock and the internal friction angle of the roadway, the boundary position of the plastic area in the surrounding rock of the rectangular roadway can be calculated by the above formula.

Effect of lateral pressure factor

To study the impact of different lateral pressure coefficients on the shape characteristics of the plastic zone in a rectangular roadway, consider a roadway with a burial depth of 400 m as an example. Assuming the rock unit weight is 20 kN/m³, the overburden pressure is calculated as 8 MPa. By selecting height-to-width ratios k = 0.4, 0.7, and 1, and lateral pressure coefficients λ = 0.4, 0.5, …, 1, the plastic zone shapes in the surrounding rock are computed, as shown in Fig. 5.

Fig. 5

M-C theory.

As observed in Fig. 6, the influence of the lateral pressure coefficient on the plastic zone in the surrounding rock of a rectangular roadway is primarily evident at the roof corners. When vertical stress is constant, a decrease in the λ value corresponds to a reduction in horizontal stress in the model. As λ decreases, the asymmetry in the stress environment significantly impacts the shape of the plastic zone in the surrounding rock, resulting in a “petal” shape. The areas of the petal-shaped zones on both sides noticeably increase with the reduction of the lateral pressure coefficient. To quantify the plastic zone changes at the roof corners, the maximum depth of the plastic zone in the middle of the roadway roof is defined as R_ce, and the maximum depth at the roof corners as R_co. The ratio of these two values is used to describe the variation in the plastic zone at the roof corners. The relationship between this ratio and the lateral pressure coefficient is shown in Fig. 7.

Fig. 6

Morphological features of the plastic area of rectangular roadway. (a) The plastic area varies with lambda morphology, (b) plastic area boundaries.

Fig. 7

Morphological changes of the plastic area under different λ values. (a) The greatest depth of the plastic area of roadway roof and wall (b) R_ce/R_co

Observation Fig. 6b and Fig. 7a, it can be concluded that the maximum depths of the plastic zones at the center of the roadway roof and the sidewalls are approximately the same when λ = 1. However, as λ decreases, these depths change independently. The maximum depth of the plastic zone in the roof increases as λ decreases, while the maximum depth of the plastic zone in the sidewalls decreases with the reduction of λ. Both exhibit a clear linear relationship with the lateral pressure coefficient λ.

In Fig. 7b, the ratio R_ce/R_co decreases as the λ value decreases. When k = 1, 0.7, and 0.4, as λ decreases from 1 to 0.4, R_ce/R_co decreases by 55.3%, 40.4%, and 30.7%, respectively. When λ is large, the k value significantly influences R_ce/R_co. For instance, when λ = 1 and k = 0.7 or 0.4, R_ce/R_co decreases by 28.1% and 40.4%, respectively, compared to when λ = 1 and k = 1. As λ continues to decrease, the R_ce/R_co value approaches 1, indicating that the depths and areas of the three “petals” of the plastic zone in the surrounding rock at the top of the roadway become nearly equal.

Considering different values of λ and k, it is evident that R_ce/R_co exhibits a clear linear relationship with the lateral pressure coefficient λ. By selecting k = 1, 0.7, and 0.4 for representative analysis and performing regression fitting, the linear relationship between R_ce/R_co and λ can be expressed by Eq. (22). As the height-to-span ratio k decreases, the slope of the linear relationship significantly decreases, indicating that the influence of the lateral pressure coefficient on R_ce/R_co diminishes correspondingly.

$$\left\{ \begin{gathered} {R_{{\text{ce}}}}/{R_{{\text{co}}}}=2.973\lambda +0.1088{\text{ }}(k=1,{R^2}=0.9789) \hfill \\ {R_{{\text{ce}}}}/{R_{{\text{co}}}}=1.2934\lambda +0.5672{\text{ }}(k=0.7,{R^2}=0.9941) \hfill \\ {R_{{\text{ce}}}}/{R_{{\text{co}}}}=0.8092\lambda +0.7414{\text{ }}(k=0.4,{R^2}=0.9979) \hfill \\ \end{gathered} \right.$$

(22)

In scenarios with higher λ values, the roadway is generally safer, with less deformation at the corners. Conversely, when λ is lower, the R_ce/R_co ratio is relatively smaller, resulting in greater deformation at the corners. In such cases, decreasing the height-to-span ratio k further exacerbates the deformation at the corners and increases the risk of surrounding rock failure. Therefore, in practical engineering, it is crucial to consider the magnitude of the in-situ stress comprehensively to accurately assess the development of the plastic zone in the surrounding rock. This approach ensures that proper guidance is provided for engineering practice.

Effect of depth-width ratio

To analyze the evolution of the plastic zone morphology in the surrounding rock of the roadway under different height-to-span ratios, representative analyses were conducted with λ = 1, 0.7, and 0.4, and k = 1, 0.9, …, 0.4. The resulting plastic zone morphologies are shown in Fig. 7, which reveals significant differences in the plastic zone shapes for various k values. When the lateral pressure coefficient is constant, the area of the plastic zone in the roof increases as the k value decreases. This indicates that a too-small k value leads to stress concentration in the roadway roof, exacerbating deformation and failure of the surrounding rock, which is detrimental to the stability of the rock mass. Further analysis of the maximum depth of the plastic zone in the roof and sidewalls, as well as the variation of R_ce/R_co, is presented in Fig. 8.

Fig. 8

Morphological changes of the plastic region with λ.

From Fig. 9a, it can be observed that the maximum depth of the plastic zone at the center of the roadway roof and sidewalls both increase as λ decreases. Both of these depths exhibit an approximately linear relationship with k. In Fig. 9b, when λ = 1, the relationship between R_ce/R_co and k follows a quadratic function. As λ decreases, this quadratic relationship gradually transitions into a linear one, with the critical point occurring around λ = 0.7. As λ decreases, the influence of k on R_ce/R_co significantly diminishes, which is reflected in the decreasing slope of the curve. Representative analyses for λ = 1, 0.7, and 0.4 were conducted, with the regression fitting results shown in Eq. (23). When λ = 1, reducing the k value from 1 to 0.4 results in a 40.4% change in R_ce/R_co. When λ = 0.4, the same reduction in k causes only a 7.7% change in R_ce/R_co, with the slope of the curve becoming nearly flat.

$$\left\{ \begin{gathered} {R_{{\text{ce}}}}/{R_{{\text{co}}}}=2.2099{\lambda ^2} - 1.414\lambda +1.7915{\text{ }}(\lambda =1,{R^2}=0.996) \hfill \\ {R_{{\text{ce}}}}/{R_{{\text{co}}}}=0.6716\lambda +1.0039{\text{ }}(\lambda =0.7,{R^2}=0.9728) \hfill \\ {R_{{\text{ce}}}}/{R_{{\text{co}}}}=0.1437\lambda +1.0208{\text{ }}(\lambda =0.4,{R^2}=0.9781) \hfill \\ \end{gathered} \right.$$

(23)

Fig. 9

Morphological changes of the plastic area in diverse k values. (a) The greatest depth of the plastic area of roadway roof and wall (b) R_ce/R_co

When the lateral pressure coefficient λ is relatively large, R_ce/R_co exhibits high sensitivity to variations in the k value. Even minor changes in k can result in significant damage depth at the corners of the roadway. Therefore, during roadway dimension design, it is crucial to fully consider the coupling effects between the lateral pressure coefficient and the height-to-width ratio to optimize the design and enhance the stability of the roadway.

Comparison with the plastic area of the circular roadway

At present, numerous research focus on the plastic area of circular roadways, and its boundary equation can be expressed by Eq. (24)²⁰.

$$\left\{ \begin{gathered} f = B_{1} T^{4} + B_{2} T^{3} + B_{3} T^{2} + B_{4} T + B_{5} = 0 \hfill \\ B_{1} = 9(1 - \lambda )^{2} \hfill \\ B_{2} = - 12(1 - \lambda )^{2} - 6(1 - \lambda ^{2} )\cos 2\theta \hfill \\ B_{3} = 2(1 - \lambda )^{2} [\cos ^{2} 2\theta (4 + 2\cos 2\varphi ) - \sin ^{2} 2\theta ] + (1 + \lambda )^{2} + 4(1 - \lambda ^{2} )\cos 2\theta \hfill \\ B_{4} = - 4(1 - \lambda )^{2} \cos 4\theta - 2(1 - \lambda ^{2} )\cos 2\theta \cos 2\varphi + \frac{{4C}}{{P_{1} }}(1 - \lambda )\cos 2\theta \sin 2\varphi \hfill \\ B_{5} = (1 - \lambda )^{2} - \sin ^{2} \varphi (1 + \lambda + \frac{{2C}}{{P_{1} \tan \varphi }})^{2} \hfill \\ \end{gathered} \right.$$

(24)

where $T={(\frac{R}{r})^2}$,r is the depth of the plastic area at the angle θ, and R is the radius of the circular roadway.

In the context of the applicability of circular tunnel plastic zone calculation methods to rectangular tunnels, the “circumscribed circle” method is frequently used (Zhao et al. 2016). For circular tunnels under a non-uniform stress field, as the lateral pressure coefficient decreases (from 1 to 0.4), the shape of the plastic zone transitions from circular to elliptical and eventually to a “butterfly” shape. To compare the differences in plastic zone morphology between rectangular and circular tunnels, a non-uniform coefficient R_max/R_min is defined for circular tunnels, where R_max represents the maximum plastic zone radius and R_min represents the minimum plastic zone radius. The comparison of the plastic zone shapes between equivalent circular tunnels and rectangular tunnels is illustrated in Fig. 9.

Figure 10a reveals that the shape of the plastic zone in a rectangular tunnel changes uniformly, with R_ce/R_co increasing linearly as the lateral pressure coefficient λ increases. In contrast, the plastic zone shape of a circular tunnel exhibits a “butterfly” shape when λ = 0.4, and undergoes a sudden change at λ = 0.5, where the area and maximum depth of the “butterfly” rapidly decrease. Following this, the non-uniform coefficient decreases slowly with the increase of λ until the plastic zone returns to a circular shape at λ = 1.

Fig. 10

Morphological comparison of the plastic area. (a) Morphological comparison of plastic areas (b) The depth of plastic area in the center of roadway roof.

Comparing the two in Fig. 10b, it is evident that the plastic zone depth at the center of the roof in a rectangular tunnel decreases slowly with increasing λ, but the change is minimal and remains relatively stable. On the other hand, the maximum depth of the plastic zone at the roof center in a circular tunnel is zero when λ = 0.4, and undergoes a sudden change at λ = 0.5, followed by a slow increase with further increases in λ. This indicates that when using equivalent methods to calculate the plastic zone in rectangular tunnels, the estimation of the plastic zone extent at the roof center is limited and does not accurately describe the plastic zone’s morphology. This limitation can negatively impact construction guidance and the control of the plastic zone extent in practical engineering, underscoring the need for precise calculations using the boundary equation of the plastic zone in rectangular roadways.

Theoretical validation and Engineering Application

Numerical simulation verification

To more accurately and intuitively reflect the rationality of the analytical calculation method for the distribution range of the plastic zone presented in this paper, a comparative analysis was conducted between the analytical results and numerical simulation results using FLAC^3D finite element software. The differences between the analytical solution and numerical solution were examined. The “petal” shape of the plastic zone is most pronounced when λ = 0.4. Based on this, we selected λ = 0.4 and utilized the Mohr-Coulomb yield criterion for numerical modeling, with the boundary conditions illustrated in Fig. 11. The numerical model dimensions are 50 m × 50 m in length and height, with a length of 20 m along the roadway’s central axis (z-axis). The rectangular roadway dimensions are 5 m × 4 m, resulting in a total of 74,840 elements in the model.

Fig. 11

Numerical model boundary conditions.

Based on the numerical simulation calculations of the model shown in Fig. 11, a comparative analysis was conducted between the analytical calculation results of the plastic zone distribution range and the numerical simulation results, as illustrated in Fig. 12. In Fig. 12b, the scatter plot of the numerical simulation results represents the extracted boundary coordinate information of the plastic zone obtained using Origin software.

Fig. 12

Comparative verification of distribution range of plastic region. (a) Numerical simulation result, (b) Comparison of analytical solutions and numerical simulation result.

From Fig. 12, it can be seen that the analytical results obtained in this study align well with the numerical simulation results, both exhibiting a consistent trend in the form of a “petal” shape. This effectively validates the rationality of the analytical method for calculating the distribution range of the plastic zone in rectangular roadways, demonstrating its high computational precision.

Engineering Application

Building on the validation of the aforementioned rationality, we further examined the application effectiveness of the proposed method in actual engineering by relying on the return airway of a deep coal mine project. The lithology of the strata where the roadway is located is illustrated in Fig. 13.

Fig. 13

Column diagram of roadway roof and floor strata.

The roadway is approximately 400 m deep, with dimensions of 5 m × 4 m. Since the monitoring points are located relatively far from the main roadway, their influence is minimal, allowing us to simplify the analysis to a single rectangular roadway excavation process. The main support locations in coal mine roadways are the roof and sidewalls. Therefore, based on the characteristics of the analytical calculation method presented in this paper, we selected a specific monitoring section, where five boreholes were arranged in the roof and sidewalls. To ensure that each borehole effectively identifies the damaged areas and covers the plastic zone boundary, the depth of each borehole was set to 8 m. The arrangement of measurement points and monitoring is illustrated in Fig. 14.

Fig. 14

Measuring point arrangement and monitoring. (a) monitoring point arrangement (b) drilling detection.

Due to the considerable thickness of the coal seam in the roadway roof, the plastic zone exists only within the coal seam. The physical and mechanical parameters of the coal seam are shown in Table 1. Based on the results of in-situ stress testing, the lateral pressure coefficient λ was set to 0.8. Using the method presented in this paper, we calculated the theoretical distribution range of the plastic zone in the roadway surrounding rock under the specified geological conditions. A comparative analysis of the analytical calculation results and the field measurement results is illustrated in Fig. 15.

Table 1 Physical and mechanical parameters of coal.

From Fig. 15, it can be observed that the measured depth of the plastic zone at the locations of the boreholes around the rectangular roadway aligns well with the overall trend of the theoretical depth calculated using the method presented in this paper, with relatively small errors between the two. The average error between the theoretical and measured values of the plastic zone distribution range is 5%, indicating that the analytical calculation method effectively addresses the prediction of the plastic zone distribution range in rectangular roadways, demonstrating good engineering applicability. By understanding the distribution range of the plastic zone, valuable insights can be provided for support design.

Fig. 15

Comparison of the theoretical and measured values of the distribution range of the plastic region.

Conclusion

(1) Using complex variable functions and elasticity theory, a mechanical model of the surrounding rock for rectangular tunnels was established, and an analytical solution for the stress distribution in the surrounding rock of rectangular tunnels under non-uniform stress fields was derived. The spatial stress distribution patterns for rectangular tunnel surrounding rock were obtained. The stress is minimal at the center of the roof and side walls, with stress release zones appearing within the surrounding rock. Stress concentration is notably prominent at the corners, and the deep rock stress in the tunnel is significantly influenced by both the lateral pressure coefficient and the height-to-width ratio.

(2) By applying the M-C criterion, the boundary equations for the plastic zone of the surrounding rock in rectangular tunnels were derived, and the morphological characteristics of the plastic zone were obtained. It was found that the plastic zones on the two sides of the rectangular tunnel are arc-shaped. The asymmetry in the stress environment significantly impacts the plastic zones at the roof and floor of the rectangular tunnel, resulting in a “petal” shape. When using the boundary equations for circular tunnel plastic zones for equivalent calculations, the central “petal” is overlooked, leading to an inaccurate assessment of the plastic zone morphology.

(3) Based on the boundary equations for the plastic zone of the surrounding rock in rectangular tunnels, the effects of different lateral pressure coefficients and height-to-width ratios on the morphological characteristics of the plastic zone were investigated. The results indicate that the influence of lateral pressure coefficient and height-to-width ratio on the plastic zone is most pronounced at the corners. As the lateral pressure coefficient and height-to-width ratio decrease, the depth and area of the “petals” on the sides of the roof and floor plastic zones significantly increase, while the changes in the plastic zones at the sides remain minimal. Additionally, a smaller height-to-width ratio leads to a greater depth of damage to the roof.

(4) Utilizing FLAC^3D finite element software for numerical simulation analysis, the results indicate that the analytical method’s calculation results align well with the numerical simulation results, exhibiting a consistent trend in the form of a “petal.” This effectively validates the rationality of the analytical method for calculating the distribution range of the plastic zone in rectangular roadways, demonstrating high computational precision. Additionally, the engineering applicability of the analytical calculation method was validated based on the return airway of a coal mine project. The results show that the overall trend of the measured values aligns well with the theoretical values, with relatively small errors between them. This indicates that the analytical calculation method can effectively address the prediction of the plastic zone distribution range in rectangular roadways, demonstrating good engineering applicability and providing a theoretical foundation for subsequent support design.