J. Cent. South Univ. (2018) 25: 949-960

DOI: https://doi.org/10.1007/s11771-018-3796-6

Catastrophe stability analysis for shallow tunnels considering settlement

HUANG Xiao-lin(黄晓林)¹, ZHANG Rui(张睿)²

1. School of Traffic and Transportation Engineering, Changsha University of Science and Technology,Changsha 410114, China;

2. Department of Civil and Structural Engineering, University of Sheffield, UK

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: Limit analysis of the stability of geomechanical projects is one of the most difficult problems. This work investigates the influences of different parameters in NL failure strength on possible collapsing block shapes of single and twin shallow tunnels with considering the effects of surface settlement. Upper bound solutions derived by functional catastrophe theory are used for describing the distinct characteristics of falling blocks of different parts in twin tunnels. Furthermore the analytical solutions of minimum supporting pressures in shallow tunnels are obtained by the help of the variational principle. Lastly, the comparisons are made both in collapsed mechanism and stability factor with different methods. According to the numerical results in this work, the influences of different parameters on the size of collapsing block are presented in the tables and the limit supporting loads are illustrated in the form graphs that account for the surface settlement.

Key words: collapse mechanism; twin shallow tunnel; failure criterion; surface settlement; functional catastrophe theory

Cite this article as: HUANG Xiao-lin, ZHANG Rui. Catastrophe stability analysis for shallow tunnels considering settlement [J]. Journal of Central South University, 2018, 25(4): 949–960. DOI: https://doi.org/10.1007/s11771-018- 3796-6.

1 Introduction

To estimate the catastrophe collapse of shallow tunnel is one of the most difficult problems in the process of tunnel construction. The shallow twin tunnels are often chosen in the urban subway projects because of the small underground space. Due to the limitations of the limit equilibrium method and other methods, some scholars adopted the limit analysis approach to predict the stability and failure modes of the face and crown of tunnels, which shows an extreme simplicity and great effectiveness [1]. For instance, the rigorous bound of supporting pressure was obtained by SLOAN and ASSADI [2] with the help of limit analysis theory.

In 1970’s, the upper bound theorem was proposed by CHEN [3]. Then this theorem had great importance in the field of geotechnical engineering because of its great validity in dealing with the stability problems in underground structures. With the extensive use of it, it was improved by many scholars. By introducing a linear multi-block collapse mechanism in 1980, DAVIS et al [4] obtained the upper bound solutions of the stability coefficient. In 1994, the accuracy of supporting pressure derived under the three-dimensional collapse mechanism with the centrifugal model test was studied by CHAMBON et al [5]. YANG et al [6] studied the influences of dilatancy angle on the slope stability by limit analysis method, and obtained the critical stability number. The upper bound theorem will be illustrated at length in the following.

With the development of the limit analysis method, the evaluation of the stability problems with linear criterion [7–9] has been replaced by the nonlinear criterion in terms of the nonlinear mechanical characteristics of geotechnical material in tunnel project. The different forms of Drucker- Prager [10] failure criterion were recommended for stability analysis because of the more realistic results compared with linear strength criterion. The Modified-Lade failure criterion was developed by EWY [11]. By considering that the Hoek-Brown failure criterion [12] has been widely used in hard rocks, this work uses a new nonlinear strength function proposed by BAKER[13] to analyze the stability of tunnel roof. The NL failure criterion will be presented at length in the following.

According to the upper bound theory and variational approach, the curved failure mechanism of deep tunnel was constructed by FRALDI and GUARRACINO [14–16]. Owing to a big difference in the tunnel roof collapse mechanism between shallow tunnels and deep tunnels, a new curved failure mechanism should be proposed to reflect the identities more appropriately. YANG et al [17] put forward a new failure mechanism of shallow tunnels by considering the surface settlement. On the basis of previous work which a kinematic plastic solution with ground movements is derived by OSMAN et al [18], a compatible displacement field to calculate the stability problem of shallow twin tunnels excavated in the soft layer is constructed by OSMAN [19].

The catastrophe theory has been proved to be effective in analyzing the stability problems in geology and geomechanics. Many scholars have applied this theory in prediction of the stability in the engineering. A fold catastrophe model of a tunnel rock burst was established by PAN et al [20] to predict the occurrences of a rock burst. ZHANG et al [21] used the functional catastrophe theory to predict the collapse mechanisms for deep tunnels based on the Hoek-Brown failure criterion.

According to the introduction of the previous works, this paper establishes a functional catastrophe collapse mechanism of shallow tunnels with regard to surface settlement. Referring to the NL failure criterion, upper bound solutions are used for describing the distinct characteristics of falling blocks of middle and side parts in twin tunnels. Moreover, the limit supporting loads in shallow tunnels are obtained by the help of the variational principle. Lastly, the comparisons are made both in collapsed mechanism and stability factor with different methods.

2 Nonlinear limit analysis with variational method

2.1 Upper bound theorem

The theory of the upper bound has been widely used to the predictions of the stability of the tunnels. The upper bound theorem of limit analysis can be depicted as: when the velocity boundary conditions and consistency conditions for strain and velocity are satisfied by the maneuvering-allowable velocity field which is built, the actual loads should be no more than the values of the calculated loads which is derived from the equation constituted by equating the external rate of work and the rates of the internal energy dissipation.

According to CHEN [3], the upper bound theorem with seepage forces effect can be written as follows:

                        (1)

where σ_ij is the stress tensor, is the strain rate in the kinematically admissible velocity field, respectively. T_i is the limit load exerted on the boundary surface. S is the length of velocity discontinuity, X_i is the body strength which is caused by weight, V is the volume of the plastic zone, v_i is the velocity along the velocity discontinuity surface.

2.2 NL strength law

Many scholars utilized a simple power law relation of the form to describe the nonlinear properties of the strength of the soil. Based on the work of BAKER [13], the present work employs a slight generalization of this nonlinear relation expressed in the form:

                         (2)

where P_a stands for atmospheric pressure; {A, n, T} are nondimensional strength parameters; and strength function is defined by Eq. (2). A is a scale parameter controlling the magnitude of shear strength, T is a shift parameter controlling the location of the envelope on the σ axis; is the tensile strength, which T representing a nondimensional tensile strength, n controls the curvature of the envelope.

By assuming the plastic potential, ξ, to be coincident with the Mohr envelope and considering without any loss of generality, τ_n is positive, it is

                      (3)

So, that the plastic strain rate can be written as follows:

              (4)

where λ is a scalar parameter. According to ZHANG et al [22], the plastic strain rate components can be written in the form of

                 (5)

where w is the thickness of the plastic detaching zone. A dot denotes differentiation with respect to time and a prime with respect to x, i.e.,

In order to enforce compatibility, from Eqs.(3) and (4), it follows

                  (6)

And the normal component of stress can be written as:

                  (7)

So that, by virtue of the Greenberg minimum principle, the effective collapse mechanism can be found by minimizing the total dissipation, the dissipation density of the internal forces on the detaching surface, results in

                      (8)

where is normal plastic strain;is shear plastic strain rates, w is the thickness of the plastic detaching zone, andis the velocity of the collapsing block.

3 Upper bound analysis of collapse for single shallow tunnel

3.1 Assumptions for analysis

On the basis of this theory, some assumptions should be made: 1) The behavior of the soil is ideally plastic and rigid; 2) The yield surface in the stress space is convex and the rates of plastic deformation are obtained from the yield function. 3) The collapsing block can be seen as a rigid block without considering the arch effect of shallow circle tunnels. The width of collapsing block at the ground level, 5i, is only determined by the depth from the center of tunnel to ground surface. Moreover, the relationship of the i and H can be described as where i stands for the distance between the tunnel centerline and the point of the trough inflexion, H is the depth of the tunnel, k₀ is a coefficient. Since the shape of collapse blocks should be symmetrical with respect to the y-axis, for simplicity, only the half cross-section is considered.

Figure 1 Potential falling blocks with consideration of effects of surface settlement

3.2 Analytic solution for characterizing collapsing shape and supporting pressure

Due to the presence of velocity detaching line existed in the soil layer of tunnel roofs, the impending failure would slide in a limit state along with the velocity discontinuous surfaces. During the process of the impending collapse, the dissipation densities of the internal forces on the detaching surface is

    (9)

where L characterizes the collapsing width of block illustrated in Figure 1. y=g′(x) is the first derivative of y=g(x).

The work rate of failure block produced by weight can be calculated by integral process.

(10)

where γ is the weight per unit volume of the soil. The function c(x) is the function describing the circular tunnel profile. s(x) characterizes the shape of surface settlement.

                   (11)

According to the research of OSMAN [19], the surface settlement is defined by a Gaussian distribution curve. The area of surface settlement trough can be obtained by calculation.

                      (12)

Due to the fact that the tunnel is buried in the shallow strata, the supporting structure is unavoidable for the requirement of safety and stability. Therefore, the work rate of supporting pressure in the shallow circular tunnel is

                             (13)

where q is the supporting pressure exerting on the circumference of tunnel lining.

Meanwhile, because of the fact that the external force always exerts on the underground structure, the work rate of extra force which puts on the ground surface cannot be ignored. The expressions can be written as

                            (14)

where σ_s stands for the surcharge load put on the ground surface.

In order to describe the shape and extension of the failure collapsing block, it is of essential to obtain the explicit expression of g(x) by constructing an objective function consisting of the external rate of work and the rate of the internal energy dissipation,

                     (15)

So that, the effective collapse mechanism can be obtained by minimizing the objective function Λ according to the kinematic theorem of limit analysis.

Then substituting Eqs. (9), (10), (13) and (14) into Eq. (15), the expression of objective function is given:

            (16)

in which

                              (17)

In this technical note, the analytical solutions of collapse dimension could be obtained by seeking for the minimum value of objective function ψ with the method of variational principle in the realm of plasticity theory. As a consequence, the expression of ψ should be turned into Euler’s equation through the variational method. The expression of variational equation of ψ on stationary conditions can be written as:

                    (18)

By the variational calculation, the explicit form of the Euler’s equation for the Eq. (17) can be obtained as:

(19)

Obviously, Eq. (19) is nonlinear second-order homogeneous differential equation. By the integral calculation process, the expression of velocity discontinuity surface is

                     (20)

in which

                      (21)

where θ and h stand for the integration constants coefficients determined by mechanical and geometric boundary conditions, respectively.

From what mentioned above, given the detaching curves are supposed symmetric with respect to the y-axis. It can be seen from Figure 1 that the shear stress on the ground surface equals zero at the point where its x-height is 2.5i.

                     (22)

Furthermore, the explicit expressions of the function of velocity discontinuity surface should fulfill other boundary conditions.

                           (23)

                       (24)

Substituting Eqs. (22) and (23) into Eq. (20), the surface function turns into

                       (25)

On the basis of the expression of the profile of the circular tunnel, the piece of external work can be calculated by integrating c(x) over the interval [0, L]:

 (26)

Therefore, the objective function Λ can be calculated, which is

               (27)

Let Λ=0, the equation can be written as

                   (28)

For the purpose of getting the explicit form of detaching curve profile g(x), the values of L and q must be obtained by combining and solving Eqs. (24) and (28). Then it is not difficult to draw the shape of failure surface by Eq. (25). Meanwhile, influences of different factors on the upper bound solution q can be analyzed.

4 Analytic solution for characterizing collapsing shape and supporting pressure of twin shallow tunnel

By considering the fact that a large number of shallow-buried tunnels are constructed in close proximity to each other, this work investigates the failure mechanism of twin shallow tunnels considering the surface settlement. It is necessary to construct distinct failure mechanisms of different parts of the small-spacing twin shallow tunnels with considering the different mechanical properties of different parts of small-spacing shallow tunnels. In order to investigate the difference between the failure modes of middle part and the side part of the twin shallow tunnels. Two different curves, y=f₁(x) and y=f₂(x) in the symmetrical coordinate system, are used to describe the failure modes of two parts. L₁ is the width of collapsing block around the circumference of tunnel lining in side part, L₂ is the width of the collapse block of middle part, q is the supporting force on the tunnel, as shown in Figures 2 and 3.

On the basis of the proposed failure mechanism of single tunnel introduced above, some assumptions should be made in the analysis of collapse mechanism of twin tunnel: 1) The behavior of the soil in middle part and side part is ideally plastic. 2) The presence of an existing tunnel does not alter the expected profile of surface settlement into a new tunnel and the pattern of the ground movements of each tunnel is deemed independent of others. Only half structure is considered in calculation in terms of the assumption that the failure mechanism of twin shallow tunnels is bilaterally symmetrical and 2S is the distance between centers of tunnels, as shown in Figures 2 and 3.

Figure 2 Curved failure mechanism of twin shallow tunnels

Figure 3 Curved collapse mechanism of left hole

During the process of the impending collapse, the dissipation densities of the internal forces on the detaching surface, and , are

         (29)

       (30)

whereandare normal plastic strain rates respectively, and are shear plastic strain rates, respectively.

  (31)

The expression of left hole is considered in the failure mechanism of middle part and the expression of right hole is considered in the failure mechanism of side part in this work.

The work rate of failure block produced by weight can be calculated by integral process:

                        (32)

The work rate of supporting pressure in the shallow circular tunnel is

                        (33)

In the meantime, the work rate of surcharge load can be computed as

                        (34)

Then the expression of objective function is given

       (35)

in which

               (36)

             (37)

According to the assumption above, given the detaching curves are supposed symmetric with respect to the y-axis and the settlement profile of each tunnel is independent. It can be seen from Figure 2 that the shear stresses on the ground surface equal zero at the points where x-heights are S+2.5i and 2.5i–S, respectively.

                  (38)

                  (39)

Furthermore, the explicit expressions of the function of velocity discontinuity surface should fulfill other boundary conditions.

                       (40)

                       (41)

               (42)

               (43)

The expressions of the function of velocity discontinuity surface can be obtained:

                   (44)

                  (45)

in which

                     (46)

                   (47)

Let Λ=0, the equation can be written as

     (48)

For the purpose of getting the explicit forms of detaching curve profile consisting of f₁(x) and f₂(x), the values of L₁, L₂ and q must be obtained by combining and solving Eqs.(42) and (48). Then it is not difficult to draw the shape of failure surface by Eqs. (44) and (45). Meanwhile, the influences of different factors on the upper bound solution q can be analyzed.

5 Catastrophe analysis of stability of shallow tunnel

Due to the complexity and diversity of the constitutive relation of the engineering soils and the uncertainty of the boundaries of the plastic during the process of the tunnel excavation, it is difficult to describe this process with the method of classical mathematical method. As a branch of nonlinear theory, the catastrophe theory was put forward by THOM [23]. Many scholars have adopted the catastrophe theory to illustrate the failure mechanics of the geotechnical structures in nonlinear system. In order to explain the phenomenon of various mutations, THOM put forward seven different kinds of models, and these models are generally based on elementary catastrophe theory (ECT). In ECT, the potential function of the system is one of the seven elementary functions defined by the polynomial functions shown in Table 1.

However, the total potential of the system has a complex mathematical, such as a functional of the state function f(x), rather than the elementary functions. In order to simplify the tunnel stability problems, ZHANG et al [21] successfully solved the stability problem of the single tunnel in the FCT framework. In this study, the detaching zone of a collapsing block is the studied system. Referring to the NL strength law, the FCT is used to explore the mechanisms of collapse in shallow-buried tunnel with regard of surface settlement.

Table 1 Potential functions used in elementary catastrophe theory

If the potential function of the system is defined by a function [V=f(x₁, x₂)], as in Eq. (49), determining the non-Morse critical point f_c(x) of the potential function of the system becomes challenging. Consider

             (49)

in which the primes indicate the derivatives of the functions with respect to their subscript coordinates; namely

In order to get the non-Morse critical point f_c(x) of the potential function of the system, it is necessary to expand the increment of function to the forms of a two-variable Taylor series in small perturbations of y. This work makes facilitate the derivation:

                     (50)

The function J synchronously reaches the catastrophic point when F reaches the catastrophic point. According to the Thom splitting lemma [23], F has a non-Morse critical point if the following equation is satisfied:

                                (51)

                            (52)

where Df and det(Hf) denote the gradient and determinant of the Hesse matrix of potential function f(x₁, x₂), respectively.

According to Eqs. (51) and (52), the conditions of function J[y] are illustrated below:

   (53)

The form of catastrophic conditions of function J[y] is illustrated in the following:

  (54)

During the process of the catastrophe analysis of the stability of a shallow tunnel, the specific expressions of g(x), f₁(x) and f₂(x) can be obtained by substituting Eqs. (25), (44) and (45) into Eq. (54), the results can be written as follows:

 (55)

Given any values of x, Eq. (55) must be satisfied. Therefore, the value of n in Eq. (55) must be 0.5 synchronously.

The value of n in this work is consistent with the general requirements specified in nonlinear Mohr Envelopes-General considerations. According to BAKER [13], it is necessary to impose the restrictionswhich also satisfies the real experimental data [24–33].

6 Numerical results and discussions

6.1 Effects of different parameters on shape of roof collapse block

In order to explore the influence of different parameters such as T, A and γ on the shape of roof collapse, the difference of failure mechanism of falling blocks for single and twin shallow tunnels should be considered. Table 2 shows the failure mechanism of tunnel roof corresponding to n=0.5, P_a=100 kPa, σ_s=50 kPa, R=5 m, s_m=0.1 m, k₀=0.2, H=13 m. According to OSMAN et al [19], the width of roof collapse extended to the ground surface is merely associated with the depth H. It can be concluded that when H is fixed, the collapsing width stays a constant no matter how the other parameters vary. In terms of the twin tunnel, Table 3 shows the failure mechanism of tunnel roof corresponding to n=0.5, P_a=100 kPa, σ_s=50 kPa, R=5 m, S=6.5 m, s_m=0.1 m, H=10 m. According to Tables 2 and 3, it can be obviously seen that the width of collapsing block around the circumference of tunnel lining tends to decrease with the increase of parameters A and γ.

Table 2 Impending roof failure for single tunnel with regard to different parameters

Table 3 Impending roof failure for twin tunnel with regard to different parameters

6.2 Effects of different parameters on supporting pressure

To endure the surcharge load, soil weight and other external forces more efficiently, it is meaningful to obtain the minimal upper bound supporting pressure exerted on the tunnel lining within the framework of upper bound theorem with considering the diverse impacts which different parameters have. Parametric analysis is conducted, such as σ_s, s_m, T with respect to the depth H and surcharge load σ_s, which are illustrated in Figure 4.

According to the Figures 4(c) and (d), the burial depth H has a positive correlation with the supporting force in twin shallow tunnel consistently. When the burial depth H is fixed, different parameters have different influences on the supporting pressure corresponding to A₁=0.7, A₂=0.8, R=5 m, S=6.5 m, H=10 m, P_a=100 kPa, n=0.5, γ′=18 kN/m³. It can be concluded that both in the single tunnel and twin tunnel, the supporting force tends to decrease with the increase of the value of maximum surface settlement, while the influences of surcharge load and the parameter T in NL criterion on supporting pressure presents the opposite trend. The figure shows that the value of maximum surface settlement s_m makes a bigger influence on the supporting force. The supporting force of twin shallow tunnel will be discussed in forms of stability factor in comparison part. From the perspective of engineering, the shallow-buried circular tunnels with heavier ground load need larger supporting forces and supporting system of deeper tunnels should be intensified to keep the stability.

7 Comparisons in stability analysis

The stability number is a good index to describe the shallow tunnel stability in nonlinear soils under limit states. The stability number N_c is defined by BROMS et al [34], as shown below:

                    (56)

where s_u,T is the undrained shear strength at the tunnel axis level.

The undrained strength profile can vary linearly with depth according to

                             (57)

where s_uo is the undrained strength at the ground surface, and t=ds_u/dh is the rate of change of undrained strength with depth. So the stability number N_c can be changed to:

           (58)

Figure 4 Effects of different parameters on supporting force of shallow tunnels:

In many design situations, the quantities tR/s_uo, γR/s_uo and H/R are known, and the problem can be regarded as finding the lowest value of (σ_s–q)/s_uo.

In this work, comparisons of stability number for twin tunnels constructed in weightless soil are made to verify the new collapse mechanism as mentioned above.

Figure 5 Comparison of stability number N_c with varying tunneling spacing

8 Summary and conclusions

In this paper, new curved failure mechanisms of circular single and twin shallow tunnels considering the effect of surface settlement were proposed to estimate the stability of tunnel crown within the framework of the functional catastrophe theory. Furthermore, this work proved the validity of the NL strength function in predicting the stability of the tunnel roof by comparing with the numerical methods. With NL failure criterion and FCT theory the analytical solution for the minimum supporting forces in shallow tunnels are obtained with the help of the variational principle. Some conclusions are drawn from above:

1) With the increase of the shear strength controlling coefficient A and unit weight γ in NL failure criterion, the value of the width of collapsing block around the circumference of tunnel lining presents to decrease in both single shallow tunnel and twin tunnels.

2) The supporting forces tend to increase with the increase of the surcharge load exerted on the single tunnel and the surface settlement and the parameter T both have a positive correlation with the scope of supporting forces in twin tunnels. Furthermore, the size of the failure collapsing block and the supporting forces both increase as the tunnels bury deeper. Especially the upper width of collapsing block is proportional to the depth H.

3) The influence of the clear distance between neighborhood tunnels on their stability could not be ignored. When the clear distance ratio S/R<4, the stability number has a positive relation with the clear distance and embedded depth. However, when the clear distance is big enough, the clear distance could not affect the stability of the neighborhood tunnels obviously.

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(Edited by HE Yun-bin)

中文导读

基于沉降影响的浅埋双孔隧道突变稳定性分析

摘要：极限分析是预测岩土力学性质稳定性的一种实用且有意义的方法。本研究在极限分析上限理论的框架下，考虑到地表沉降的影响，研究了NL破坏准则中的不同参数对双孔隧道可能的崩塌块体形状的影响。另外，采用NL非线性破坏准则考察了不同因素对最小支护压力的影响。通过功能突变理论推导出两种不同形状曲线的解析解，再通过变分原理下的虚功方程，用两种形状曲线描述了双孔隧道中不同部位塌落岩体块的不同特征。最后，本文使用不同的方法对崩溃机制和稳定性因素进行比较。根据这项工作的数值结果，本文用表格的形式列出了不同参数对塌落岩体块尺寸的影响，并导出了抵抗倒塌所需荷载的上限，并以图的形式对埋藏深度等因素进行了探究。

关键词：塌陷机理；双孔浅埋隧道；破坏准则；表面沉降；功能突变理论

Foundation item: Project(2017zzts157) supported by the Innovation Foundation for Postgraduate of Central South University, China

Received date: 2016-01-08; Accepted date: 2016-05-04

Corresponding author: ZHANG Rui, PhD; Tel: +86–18538707962; E-mail: 345766359@qq.com