Potential Roof Collapse Analysis of Tunnel Considering the Orthotropic Weak Interlayer on the Detaching Surface

e evaluation of the roof collapse in tunnels or cavities remains one of the most complex issues in geotechnical engineering. Taking the detaching surface of the tunnel roof collapse as an orthotropic weak interlayer, an analytical approach for determining the limit collapse range considering the arch eect of the tunnel is presented in this paper by the variation calculus. A discontinuity criterion moving from the anisotropic criterion proposed by the present authors is applied to the orthotropic interlayer. e phenomenon of sharp points in collapse blocks is further analyzed. Based on the proposed approach, illustrated examples are analyzed to investigate the eect of the strength parameters and the consideration of the collapse cusp, which show dierent inuence laws on the range of collapse blocks. ose interesting conclusions can provide guidance for the prediction of the collapse mechanism of the tunnel.


Introduction
e stability problems of tunnels have always been of overriding signi cance in geotechnical engineering. e possible collapse of the tunnel remains one of the most challenging problems. Due to the natural uncertainties of the properties of the rock mass in situ, such as mechanical parameters and the random variability of cracks or fractures [1][2][3][4][5][6][7][8], the collapse mechanism of a cavity roof has yet to be thoroughly grasped [9]. Because the limit analysis method requires no elastic characterization and only refers to the limit behavior, this approach can obtain more rigorous results with fewer assumptions [10]. As a result, the limit analysis method is very suitable for analyzing the collapse mechanism of tunnel roofs and has been rapidly developed in recent years.
Lippmann [11] rstly applied the limit analysis method to the roof stability problems of tunnels considering the Mohr-Coulomb (M-C) criterion. For many years, the roof stability of tunnels is analyzed in this framework [10]. Guarracino and Guarracino [12] made encouraging progress with the help of plasticity theory and calculus of variations, and a closed-form solution of the collapsed outline was obtained with the Hoek-Brown (H-B) criterion considered instead of the M-C rule. Since then, many researchers furthered their work by considering various cases of cavities such as di erent excavation pro les [13], layered rock masses or soils [9,[14][15][16][17], the presence of the karst cave [18], the solutions for shallow tunnels [19][20][21][22] or progressive collapse [23][24][25][26], consideration of the supporting pressure [27][28][29], and the case considering the groundwater [30][31][32][33].
ese extending works moving from the approaches of Guarracino and Guarracino [12] only focused on the H-B rule expressed in the M-C form (nonlinear). In fact, a weak interlayer may appear between the detaching surface when roof collapse occurs [34]. e rock mass at the detaching surface of the collapse zone can be taken as a weak interlayer with thin thickness, which is related to the failure mechanism of the surrounding rock [35,36]. Under the in uence of the dislocation of the rock masses, the weak interlayer exhibits di erent strength characteristics in the orthogonal direction. For this reason, analysis considering the orthotropic characteristics of the weak interlayer between the detaching surface can better describe the roof collapse problems of tunnels or cavities. is consideration requires a special criterion that can describe the failure behavior of the orthotropic weak interlayer on the detaching surface.
In addition, we notice that most researchers obtained a smooth collapse curve, which can be derivable at the axis of symmetry [24]. In fact, a collapse cusp (not derivable at the axis of symmetry) is usually observed in model tests or numerical analysis [34,37,38], which means that the condition at the axis of symmetry should be treated with caution ( Figure 1). To further explain this phenomenon, the sharp point of the collapse curve is discussed in our study. Once the assumption of a smooth curve (at the axis of symmetry) is not applied to the analysis, it becomes more difficult to get the collapse curve. As a result, we need to find a reasonable restriction as an alternative to the smooth assumption when considering the collapse cusp.
Based on the above considerations, a discontinuity yield criterion for an orthotropic interlayer that moves from a pressure-dependent, anisotropic criterion is applied in this research. en, the theoretical formulas for the cases with and without considering the collapse cusp are deduced to figure up the collapse block. Finally, some examples are analyzed, and the discrepancy between different cases is further discussed in this paper. e results can help constitute guidance for the prediction of the collapse range of tunnels or cavities.

Orthotropic Criterion at the Velocity Discontinuity.
e orthotropic yield criterion can move from the anisotropic criterion. Given the pressure-dependent of the rock material, Caddell et al. [39] proposed an anisotropic yield criterion in the following form: where the parameters A yz , A zx , A xy , B yz , B zx , B xy , K 1 , K 2, and K 3 characterize the properties of anisotropy. e subscript x, y, and z denote the reference axes of anisotropy. In consideration of the orthotropic materials, these parameters satisfy the following relations: Because the detaching surface is consistent with the weak layer, we take the normal direction of the detaching surface as the z axis ( Figure 2). As a result, the failure on the detaching surface only depends on σ z , τ zy, and τ zx , which leads to an orthotropic yield criterion in the degenerative form: For the plane strain problems, the equation (3) can be further simplified as where σ n denotes the stress of normal direction (the compressive stress is taken as positive in this paper), and the parameters B and K can be determined according to the shear (τ 0 ) and tensile (σ T ) strengths of the weak interlayer on the detaching surface.
On the basis of the above consideration, the discontinuity yield criterion at the detaching surface of velocity can be obtained as

Collapse Mechanism of the Tunnel Roof.
e key point about the roof stability of tunnels or cavities is to determine the shape and range of the potential collapsing blocks ( Figure 3). As it is usual, this paper considers the problem in a plane and only makes reference to the cross section of a long tunnel or cavity. e rock material is assumed to be ideally plastic, and the plastic strain rate follows the associated flow rule. Besides, strain within the collapsing body is regarded as insignificant when the roof collapse occurs (rigid-plastic behavior). Based on the above conditions, the shape of the potential collapsing region can be given by using the calculus of variations [12,40].
In order to investigate the roof collapse on account of the gravity field and refer to the upper bound principle [41], a kinematically admissible field of vertical velocity, which fulfills the compatibility with the strain rates, must be assumed at first [42]. As shown in Figure 3, the collapse velocity _ u v is in the negative direction of the y-axis, and the symmetrical collapse curve is expressed as f(x). Moreover, as shown in Figure 4, the value of the vertical velocity is considered a variable that decreases from _ u (x � 0) to zero (x � R) linearly. As a result, the field of the variable vertical velocity can be expressed as According to the geometric conditions, the plastic strain rate (the tensile strain rate is taken as negative) components in the tangential ( _ c) and normal (_ ε n ) directions can be obtained as Free boundary Sym Figure 3: Possible collapsing area of the tunnel roof.

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Coincident with the failure criterion mentioned in Section 2.1 (obeying to the associated flow rule), the plastic potential function ξ can be expressed as Further, the plastic strain rate can also be written in the form: e association of equations (7) and (9) leads to the following results: Finally, by substituting equations (10) into (6), we can obtain According to the equations (11) and (12), the tangential and normal stress components are expressed by using the derivative of the collapse function. Because a cusp ( Figure 1) can occur in roof collapse [34,37,38], the stress at the axis of symmetry should be treated with caution (no derivative). In particular, the shear stresses around the collapse cusp can be described in Figure 5. Based on symmetry, the magnitude of the shear stresses in the symmetrical tilt directions at the cusp point must be equal. As a result, the inner horizontal shear stress at the axis of symmetry naturally satisfies the condition of being equal to zero, so long as the collapse curves on both sides are symmetrical to each other.

Analysis without Considering Collapse Cusp
Because most researchers assumed a smooth collapse curve in their studies [23,24], the collapse curve is derivable at the axis of symmetry, which must lead to zero of derivative function f ′ (x). For comparison, we analyze the collapse curve without considering collapse cusp in this section. Meanwhile, a different criterion (i.e., the orthotropic yield criterion proposed in section 2.1) is applied at the velocity discontinuity.
Associating the equations (7), (11), and (12), the dissipated power density of the internal stresses at the discon- Besides, the power density of the applied loads is where c denotes the gravity per unit volume of the rock mass.
Here, we consider the right half of the symmetrical block (with respect to the y-axis). e total dissipated power of the collapse system is further deduced as where Because the effective collapse curve can be obtained when the total dissipation power makes a minimum [13], the problem can be solved by using the calculus of variations. In order to obtain an extremum of the total dissipated power _ U over the interval of 0-L, the functional F must satisfy Euler's equation:   .
. Figure 4: e field of the variable vertical velocity. 4 Advances in Civil Engineering From equation (14), we can deduce that By substituting equations (16) into (15), it is Integrating the equation (17), we can obtain the first derivative of f(x) as follows: Here, C 1 is an unknown parameter which needs to be further determined. Similar to existing studies, f ′ (x � 0) should be equal to zero because a smooth symmetrical collapse curve is assumed in this section, which results in C 1 � 0. en, the collapse curve f(x) can be deduced by integrating the equation (18): where C 2 is a pending parameter. Considering an implicit constraint f(x � L) � 0, we can obtain that en, L can be further determined by equating the total dissipated power to zero. Substituting equations (15) and (20)-(23) into (14), we can obtain the equation which L yields. It is Note that equation (23) can be easily solved by using the numerical method. After L is obtained, the collapse curve is finally written as

Analysis Considering Collapse Cusp
When a possible collapse cusp ( Figure 5) is considered in the collapse analysis, the condition at the axis of symmetry should be handled with care due to no derivative. As a result, C 1 in equation (19) cannot be simply determined by equating f ′ (x � 0) with zero. So, f(x) coming from equation (18) should be written as en, the pending parameter C 2 can be deduced by considering f(x � L) � 0. It results Substituting equations (17), (24), (27), and (28) into (14) and equating the total dissipated power to zero, C 1 and L yield In addition, the fracturing azimuth is related to the friction angle (φ) of the surrounding rock. Taking a stress element at the collapse cusp of the tunnel and considering the friction angle ( Figure 6), we can get the angle between directions of the fracture and maximum principal stress (the horizontal and vertical shear stresses, i.e., τ horiz and τ verti , are zero at the axis of symmetry) as (π/4 − φ/2). en, the onesided derivative f + ′ (x) can be expressed as

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e friction angle corresponding to the M-C criterion at the collapse cusp can be described in Figure 7. Considering equation (30) and the approximate geometric conditions of σ n and τ, the friction angle φ yields Finally, by considering equations (29), (30), and (31) together, we can get C 1 and L numerically. en, the collapse curve considering a collapse cusp at the axis of symmetry (x � 0) can be written as

e Discrepancy between Different Cases.
In the preceding sections, two different analytical results are obtained by considering and not considering the collapse cusp, respectively. e two results obtained under different conditions are compared through an example. e parameters involved in the example are τ 0 � 20 kPa, σ T � 22 kPa, c � 25 kN/m 3 , and R � 3 m. As a result, the comparison between these two analytical results of the collapse curve is shown in Figure 8.
e results indicate that the collapse curve obtained by considering the collapse cusp is higher than that obtained without considering the collapse cusp. According to the analytical results obtained in this paper, the height of the collapse block can increase by 0.78 m when considering the collapse cusp in the analysis. e span of the collapse curve does not change whether the cusp is considered or not. Due to the increase in the collapse height, the weight of the collapse block will also increase. e gravity of the collapse block can be calculated by using the following equation:   Advances in Civil Engineering Substituting equations (24) and (30) into (31), respectively, we can easily calculate the gravity corresponding to the collapse block for both cases. e results show that the gravity obtained by considering the collapse cusp is increased by 23% compared to the gravity obtained without considering the collapse cusp. erefore, taking the collapse cusp at the axis of symmetry into account when predicting the collapse block can help ensure the safety of the tunnel roof.

Comparison with Numerical Analysis.
A numerical analysis has been performed to further verify the above analyses. e numerical parameters are consistent with those mentioned in Section 5.1. e M-C friction angle can be calculated from equation (29). e results for the example in terms of vertical velocities have been obtained by FLAC3D. e collapse block described the sudden change of vertical velocities as shown in Figure 9.
By comparing the collapse block shapes as shown in Figure 9, it is worth highlighting the similarity of the collapse block shape from numerical analysis to that obtained by the proposed analytical method. Both analytical results can    Advances in Civil Engineering describe the collapse block shape well. As described in Section 5.1, the analytical result will lead to a wider range of collapse blocks when considering the phenomenon of sharp points in roof collapse behavior.

Influence of Strength Parameters of the Weak Interlayer.
In the process of predicting the potential collapse of the tunnel roof, the shear and tensile strengths of the weak interlayer on the detaching surface are involved in our analysis. In order to investigate the influence of these two important parameters, we first discuss the different cases without considering the collapse cusp. Figures 10(a) and 10(b) show the different results considering different values of the shear and tensile strengths, respectively. Figure 10(a) shows that the collapse curves obtained from our proposed analytical result are significantly affected by the shear strength of the weak interlayer on the detaching surface. e width and the height of the collapse block increase as the shear strength increases. Obviously, with the increase in shear strength, a greater gravity of surrounding rock can be maintained in the short term, but it also means that once the collapse occurs, there will be a wider range of primary failures. Figure 10(b) shows that the collapse curves obtained from our proposed analytical result are also significantly affected by the tensile strength of the weak interlayer on the detaching surface. e height of the collapse block increases as the tensile strength increases. However, the width of the collapse block decreases as the tensile strength increases, which is different from the effect of the shear strength. Similar to the influence of shear strength, although the increase in tensile strength may maintain a greater gravity of surrounding rocks, there will be a wider range of primary failure once the collapse occurs. Furthermore, in order to compare the different effects of shear and tensile strengths on the collapse block, the changes in height, width, and gravity of the collapse block are shown in Figures 11(a), 11(b), and 11(c), respectively. e example described in Section 5.1 is taken as the original case. By comparing the change rates of each index with different strength parameters, we can find that the height of the collapse block is more sensitive to the change in tensile strength. However, the width and the gravity of the collapse block are more sensitive to the change in shear strength.
As described in Section 5.1, the height and weight of the collapse block increase when considering the collapse cusp compared to those without considering the collapse cusp, while the span of the collapse curve does not change whether the cusp is considered or not. As a result, when considering the collapse cusp, only the changes in the collapse block height (ΔH) and gravity (ΔP) relative to the results without considering the collapse cusp are discussed. Figure 12 shows the comparison of the influences of shear and tensile strengths on the discrepancies of the collapse block height and gravity.
According to Figure 12, the discrepancies in the collapse block height and gravity between the two cases generally decrease with the increase in the tensile strength. On the contrary, the discrepancies in the collapse block height and gravity between the two cases increase with the increase in the shear strength. It is worth noting that these changes are more sensitive to the change of shear strength than the change of tensile strength, which is of directive functions when considering the effect of the consideration of the collapse cusp.

Conclusions
By using the orthotropic yield criterion which moves from the anisotropic criterion proposed by Caddell et al. [39] for the rock material, an exact solution to tunnel roof collapse has been obtained with the help of the traditional plasticity theory and the calculus of variations. In order to further illustrate the impact of collapse cusps which have been observed in previous studies [34,37,38], two different cases according to whether the collapse cusp is considered are analyzed in this paper. Our new theoretical results lead to the following conclusions: (1) Taking the detaching surface of the tunnel roof collapse as an orthotropic weak interlayer, the theoretical formulas figuring up the collapse block are obtained with and without considering the collapse cusp, respectively. A case analysis shows that considering the collapse cusp can lead to a higher range of collapse blocks. (2) e strength parameters of the weak interlayer have a significant impact on the range of collapse blocks. e shear and tensile strength have similar effects on the height of the collapse block, but their effects on the width have the opposite trend. Moreover, because the increase in shear and tensile strengths may maintain a greater gravity of surrounding rocks, there will be a wider range of primary failures once the collapse occurs. By sensitivity analysis, we can find that the height of the collapse block is more sensitive to the change in tensile strength, but the width and the gravity of the collapse block are more sensitive to the change of shear strength.
(3) e discrepancies between the two cases according to whether the collapse cusp considered are related to the strength parameters. e discrepancies between the two cases generally decrease with the increase in the tensile strength but increase with the increase in the shear strength. ese changes are more sensitive to the change of shear strength than the change of tensile strength, which is of directive functions when considering the effect of the consideration of the collapse cusp.
Our theoretical results can provide guidance on the collapse mechanism in tunnels or natural cavities, especially they can explain the phenomenon of sharp points in collapse blocks. Moreover, based on our proposed approach, many extensions including various cases such as layered rock masses and the presence of the karst cave can be further studied in future research.

Data Availability
All data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.