J. Cent. South Univ. (2019) 26: 1706-1718

DOI: https://doi.org/10.1007/s11771-019-4127-2

Reliability analysis for seismic stability of tunnel faces in soft rock masses based on a 3D stochastic collapse model

ZHANG Jia-hua(张佳华)¹, ZHANG Biao(张标)²

1. Work Safety Key Lab on Prevention and Control of Gas and Roof Disasters for Southern Coal Mines, Hunan Provincial Key Laboratory of Safe Mining Techniques of Coal Mines, Hunan University of

Science and Technology, Xiangtan 411201, China;

2. School of Civil Engineering, Hunan University of Science and Technology, Xiangtan 411201, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: A new horn failure mechanism was constructed for tunnel faces in the soft rock mass by means of the logarithmic spiral curve. The seismic action was incorporated into the horn failure mechanism using the pseudo-static method. Considering the randomness of rock mass parameters and loads, a three-dimensional (3D) stochastic collapse model was established. Reliability analysis of seismic stability of tunnel faces was presented via the kinematical approach and the response surface method. The results show that, the reliability of tunnel faces is significantly affected by the supporting pressure, geological strength index, uniaxial compressive strength, rock bulk density and seismic forces. It is worth noting that, if the effect of seismic force was not considered, the stability of tunnel faces would be obviously overestimated. However, the correlation between horizontal and vertical seismic forces can be ignored under the condition of low calculation accuracy.

Key words: 3D stochastic collapse model; pseudo-static method; response surface method; reliability index; safety factor; support pressure

Cite this article as: ZHANG Jia-hua, ZHANG Biao. Reliability analysis for seismic stability of tunnel faces in soft rock masses based on a 3D stochastic collapse model [J]. Journal of Central South University, 2019, 26(7): 1706-1718. DOI: https://doi.org/10.1007/s11771-019-4127-2.

1 Introduction

At present, underground space exploration has achieved unprecedented development in China. Along with the massive construction, however, many problems have been exposed [1-4]. The most common and serious problem is the collapse accidents. According to statistics, the face collapse accounts for about 33% in the tunnel collapse accidents. It can be seen that the collapse accidents of tunnel faces are very common, and the consequences are very serious [5-7]. Therefore, the stability of tunnel faces is a key factor for the safe construction of tunnels, and it is also a difficult problem to be urgently solved at present [8-11].

For the problem of the stability of tunnel faces, a large number of studies have been performed and great achievements have been obtained. LECA et al [12] proposed a failure mechanism of tunnel faces and the collapse pressure was solved in the light of the upper bound theorem of limit analysis.

However, the failure mechanism is too simple and the accuracy of the results still needs to be improved. Subsequently, SOUBRA [13, 14] introduced a logarithmic spiral curve into the failure mechanism and obtained the upper bound solution of collapse pressure, which is closer to the results of centrifuge test. Realizing the advantage of logarithmic spiral curves, SUBRIN et al [15] constructed a failure mechanism only consisting of the logarithmic spiral curves. Nevertheless, some assumptions inevitably lead to significant errors. LIU et al [16] constructed a cone failure mechanism of tunnel faces and solved the collapse pressure using the upper bound method of limit analysis. KHEZRI et al [17] also discussed the influence of soil heterogeneity on the collapse pressure of tunnel faces based on the limit analysis and strength reduction methods. SONG et al [18] improved the cone failure mechanism by replacing the cylinder with a truncated elliptical column, and then the upper bound solution of collapse pressure was solved. Although the result was improved, the effect was not obvious. In addition, some scholars used the spatial discretization technique to construct a failure mechanism of tunnel faces, which is very similar to the horn in shape. However, due to the extremely complex construction method, it is not convenient to apply in the practical engineering [19-22].

The above literatures all use the limit analysis method to study the stability of tunnel faces. However, the limit analysis method belongs to the fixed value method, mainly relying on the engineering experience to consider the influence of uncertain factors. In other words, this method can only solve the collapse pressure when the tunnel face collapses, but it cannot obtain a reasonable safety factor and support pressure. Therefore, the upper bound theorem and the response surface method were combined to analyze the stability of tunnel faces in this work. The dynamic effect of seism was considered and a 3D stochastic collapse model of tunnel faces was established in soft rock masses. Not only the collapse pressure can be solved when the face is destroyed, but also the reasonable safety factor, the support pressure and the failure probability of tunnel faces can be obtained. The approach can provide a new idea for the support and aseismic design of tunnel faces in soft rock masses.

2 Hoek-Brown failure criterion

In 1980, HOEK and BROWN proposed a criterion for the plastic failure of rock masses. After several amendments, the Hoek-Brown failure criterion can be expressed as [23, 24]:

                   (1)

where σ₁ and σ₃ are the maximum and minimum principal stresses, respectively, σ_ci is the uniaxial compressive strength, m_b, s and a are dimensionless parameters, and they can be determined by the following expressions:

                    (2)

                        (3)

         (4)

where GSI, m_i and D are the geological strength index, rock mass constant and disturbance factor, which respectively characterize the degree of fracture and weathering, soft and hard, and disturbance of the rock mass.

The tangential technique is widely used to obtain the equivalent relation of parameters between Hoek-Brown failure criterion and Mohr- Coulomb failure criterion [25-27]. As shown in Figure 1, a tangent line passes any point M on the Hoek-Brown strength envelope line. The tangent expression is τ=c_t+σ_ntanφ_t, which denotes the Mohr- Coulomb strength envelope line. Obviously, the M point is the common point on the two envelope lines, meaning the common solution of the two equations. Then c_t and φ_t can be obtained through this common solution, and the corresponding expression is shown in Eq. (5) [28, 29], where c_t and φ_t represent respectively the intercept and inclination of the tangent, and they are also the equivalent cohesive force and internal friction angle of the Hoek-Brown failure criterion.

 (5)

Figure 1 Hoek-Brown failure criterion and its tangent

3 3D stochastic collapse model under seismic effect

3.1 Horn failure mechanism

A large number of studies show that the failure shape of tunnel faces is extremely similar to the logarithmic spiral curve [15, 19, 22]. Therefore, the 3D construction method proposed by MICHALOWSKI et al [30, 31] was employed in this work and a horn failure mechanism was established for the tunnel face in soft rock masses. As shown in Figure 2, the diameter of tunnel face AB is d and the buried depth is C. The collapsing block of horn AEB slides into tunnel at an angular velocity w around O point, where, AE, BE and O are all on the central plane, and AE and BE are two logarithmic spiral curves. The length of OA is r_a, the length of OB is r_b, and the angles between OB, OA, OE and vertical line are θ₁, θ₂ and θ₃, respectively. A line passing point O intersects the logarithmic spiral curves AE and BE at two points, and the corresponding distances from these two points to point O are r₁ and r₂, respectively. In addition, a circle is drawn by taking the two points as a diameter, which is the cross section of the horn. The radius of the cross section is R and the distance between the center of the cross section and point O is r_m.

The expressions of logarithmic spiral curves AE and BE are respectively:

                  (6)

                  (7)

The cross section 1-1 and 2-2 are both shown in Figure 2. Taking the circle 2-2 as an example, a rectangular coordinate system was set up with the direction from O point to the circle center as the positive direction of the y axis. In the coordinate system, the angle between any point on the circle and the positive direction of the y axis is α, the linear velocity of the point is v, and the microelement volume of the point is dV. In addition, the angle of the end point of the arc is α₀, and the corresponding coordinate of y axis is l. According to the geometric relations in Figure 2, they can be obtained as follows:

                         (8)

                         (9)

           (10)

                          (11)

                           (12)

                            (13)

                     (14)

                          (15)

                   (16)

Figure 2 Horn failure mechanism and cross sections

3.2 Seismic force

The quasi-static method is an approximate calculation method to solve the dynamic problem, and its essence is equivalent to the dynamic effect to the static force acting on a body. Owing to the clear mechanics concept and the simple calculation process, the method has been widely applied in the aseismatic design of geotechnical engineering [32, 33]. As shown in Figure 2, seismic forces k_uG and k_vG were introduced into tunnel faces by the quasi-static method, and the relation of two seismic forces is as follows [34, 35]:

                            (17)

where k_h and k_v are the horizontal and vertical seismic effect coefficients, respectively; ζ is the proportion coefficient; and G is the horn weight.

3.3 Assumptions

The assumptions were made using the upper bound theorem: 1) Soft rock masses obey the Hoek-Brown failure criterion and the associated flow rule; 2) The volume of AEB is constant in the sliding process, and the energy is dissipated along the discontinuous surface of velocity; 3) The collapse pressure σ_c is uniformly distributed on tunnel faces and solved according to the equation σ_T=σ_c, where σ_T is the support pressure at the ultimate failure state and 4) Statistical characteristics of parameters of soft rock masses were presented referring to Ref. [36].

3.4 Collapse pressure

As shown in Figure 2, during the sliding process of the horn, the external force power includes the power generated by the weight of the horn, the power generated by the seismic force and the power generated by the support pressure. And the internal energy dissipation rate occurs on the velocity discontinuity surface of the horn.

The power generated by the weight of the horn is:

    (18)

The total power generated by the horizontal and vertical seismic forces is:

  (19)

The power generated by the support pressure is:

  (20)

The internal energy dissipation rate is:

         (21)

According to the principle of virtual work, the collapse pressure under seismic effect can be determined by equating the external force power with the internal energy dissipation rate. The expression of the collapse pressure σ_c is as follows:

 (22)

                     (23)

In order to avoid the collapse accident of tunnel faces, the maximum collapse pressure σ_c must be solved, which is the minimum support pressure in the condition of ultimate failure state. It can be obtained by the sequential quadratic programming algorithm in the Matlab software.

3.5 Reliability model

The support pressure σ_T of the tunnel design can be obtained according to the collapse pressure σ_c by introducing the safety factor F_s:

 (24)

The ultimate state equation of tunnel faces can be obtained by the support pressure and the collapse pressure:

  (25)

To ensure the stability of tunnel faces, the function must conform to the following condition:

 (26)

The expressions of the 3D stochastic collapse model of tunnel faces in soft rock masses under seismic effect are as follows:

                (27)

                              (28)

                           (29)

where R_s is the reliability; P_f is the failure probability; β is the reliability index; Φ^-1 is the inverse function of the standard normal distribution; and X is the vector composed of random variables with the following expression:

            (30)

4 Comparison and verification

The statistical characteristics of parameters of soft rock masses and the support pressure are shown in Table 1, and the remaining parameters are as follows: the diameter of tunnel faces is d=10 m, and the disturbance factor is D=0.7. Based on the 3D stochastic collapse model of tunnel faces in soft rock masses, the Monte Carlo method (MCM) and the response surface method (RSM) were respectively applied to solving the reliability of tunnel faces. As shown in Figure 3, the curves obtained by using these two methods begin to completely coincide, and then the curve from the MCM fluctuates on both sides of the curve from the RSM. Obviously, the time and magnitude of the fluctuation are related to the sample size n. More specifically, the smaller n is, the earlier and the larger the curve fluctuates, meaning the lower calculation accuracy. Contrarily, the larger n is, the later and the less the curve fluctuates, meaning the higher calculation accuracy. It can be seen that the results obtained by these two methods are completely consistent when the accuracy is lower. So the correctness of the RSM is verified. In addition, when the calculation accuracy is higher (especially the failure probability P_f reaches to 1×10^-4), the results from the MCM have large errors through the comparison with the results of the RSM. Therefore, the RSM is better than the MCM, and it is used to solve the reliability of tunnel faces in the following sections.

Table 1 Statistical characteristics I of random variables

Figure 3 Comparison of results obtained by response surface method and Monte Carlo method:

5 Analysis of results

5.1 Influence of mean

The strength of rock masses determines the ability of rock masses to resist failure. The effect of mean of strength parameters of soft rock masses on the reliability of tunnel faces was discussed in this section. Table 2 shows the statistical characteristics of 3 groups of strength parameters of soft rock masses. The results are shown in Figure 4, the mean of strength parameters in group 1 is the smallest, meaning that the surrounding rock is the worst, but the collapse pressure 29.4 kPa is the largest. And then the mean of strength parameters in group 2 is the second largest, indicating the strength of surrounding rock ranks the second, and the collapse pressure 14.0 kPa also ranks the second. Finally, the mean of strength parameters in group 3 is the largest, showing the surrounding rock is the best, but the collapse pressure 7.6 kPa is the smallest. In addition, with the increasing of support pressure σ_T applied to tunnel faces, the failure probability P_f all decreases with the concave curve, but the reliability index β increases with the convex curve. Note that significant differences exist among the results of 3 groups. The specific values are shown in Table 3, under 3 security levels, although the safety factor F_s of 3 groups are relatively close, the difference of support pressure of 3 groups is very conspicuous, and the relative error is more than 70% and 260%, respectively. It can be seen that the mean of strength parameters of soft rock masses determines the magnitude of collapse pressure, and it has a significant impact on the failure probability, reliability index and support pressure of tunnel faces.

Table 2 Statistical characteristics Ⅱ of random variables

5.2 Influence of coefficient of variation

Three kinds of the discreteness are given in Table 4, and the coefficients of variation are ideal,general, and unideal, respectively. The results are shown in Figure 5, as the safety factor F_s increases, the failure probability P_f decreases but the reliability index β increases. Comparing the results from 3 kinds of discreteness, under the ideal condition (the smallest Cov), the failure probability P_f is the least and the reliability index β is the largest. On the contrary, under the unideal condition (the largest Cov), the failure probability P_f is the greatest and the reliable index β is the smallest. It can be seen that the reliability of tunnel faces is significantly affected by the discreteness of parameters of soft rock masses and loads. To be more specific, the larger Cov is, the smaller the reliability is. In addition, the safety factor and support pressure of tunnel faces with different security levels were given in Table 5 based on the 3 kinds of the discreteness.

Figure 4 Effect of mean of strength parameters of soft rock masses on reliability of tunnel faces:

Table 3 Collapse pressure, safety factor and support pressure of tunnel faces meeting 3 security levels under 3 groups of strength parameters of soft rock masses

Table 4 Statistical characteristics III of random variables

Figure 5 Effect of 3 kinds of discreteness on reliability of tunnel faces:

Table 5 Safety factor and support pressure of tunnel faces with different security levels based on 3 kinds of discreteness

In order to separately analyze the influence of discreteness of each random variable on the reliability of tunnel faces, 5 groups of Cov are presented in Table 6. As shown in Figure 6, in the corresponding discrete range, when Cov of each random variable increases, the failure probability P_f increases in different trends, but the reliability index β decreases with different trends. Note that the effect of support perssure σ_T is the most significant, unit weight of rock mass γ, geological strength index GSI and uniaxial compressive strength σ_ci take the second place, and the effect of rock mass constant m_i is not obvious.

5.3 Sensitivity analysis

The expression of the sensitivity coefficient α_Xi of the random variable X_i is given by

         (31)

whereis the partial derivative of function g_X to random variables X_i, and σ_Xi is the standard deviation of random variables X_i.

Table 7 shows the statistical characteristics of random variables, and the sensitivity coefficient of each random variable can be solved by Eq. (31). The results are shown in Table 8. To facilitate analysis, the data of Table 8 are plotted as shown in Figures 7 and 8. It can be seen from Figure 7 that,sensitivity coefficients α_σT, α_σci, α_GSI and α_mi are all negative, which means a positive effect on the reliability of tunnel faces. Contrarily, the sensitivity coefficient α_γ is positive and it has a negative effect on the reliability of tunnel faces. As shown in Figure 8, the impact of random variables on the reliability of tunnel faces according to the absolute value of the sensitivity coefficient is successively the support pressure σ_T, uniaxial compressive strength σ_ci, geological strength index GSI, unit weight of rock mass γ and rock mass constant m_i. Furthermore, with the increase of the support pressure, the absolute value of sensitivity coefficient |α_σT| increases, while |α_σci|, |α_GSI|, |α_γ| and |α_mi| decrease. Note that the effects of the uniaxial compressive strength σ_ci, support pressure σ_T, geological strength index GSI and unit weight of rock mass γ are obvious, while the effect of rock mass constant m_i is not obvious.

Table 6 Statistical characteristics IV of random variables

Figure 6 Effect of discreteness of each random variable on reliability of tunnel faces:

Table 7 Statistical characteristics V of random variables

Table 8 Sensitivity coefficients of random variables under 3 kinds of security levels

Figure 7 Comparison of sensitivity coefficients under 3 kinds of security levels

Figure 8 Change rule of absolute value of sensitivity coefficients with support pressure

5.4 Influence of seismic force

5.4.1 Influence of horizontal seismic force

Table 9 shows the statistical characteristics of random variables when only the horizontal seismic force is considered, and the results are shown in Figure 9. As the support pressure σ_T increases, the failure probability P_f decreases and the reliability index β increases. In addition, when the horizontal seismic effect coefficient k_h increases, the failure probability P_f increases and the reliability index β decreases. The required safety factor and support pressure of tunnel faces with 3 kinds of security levels under different horizontal seismic effect coefficients k_h were presented in Table 10. The safety factor F_s is relatively close, but the support pressure σ_T increases with k_h, and the increasing effect is significant. For example, comparing the results from k_h=0.5 and k_h=0, the relative errors of the support pressures with 3 kinds of security levels are 204.7%, 206.5% and 199.3%, respectively. It can be seen that the horizontal seismic force has a significant impact on the reliability of tunnel faces. If the impact of the horizontal seismic force was ignored (k_h=0), the reliability of tunnel faces will be significantly overestimated, resulting in a seriously insufficient support pressure in the aseismic design.

Table 9 Statistical characteristics VI of random variables

Figure 9 Influence of horizontal seismic effect coefficient k_h on reliability of tunnel faces:

5.4.2 Influence of vertical seismic force

Table 11 shows the statistical characteristics of random variables under the condition of considering the horizontal and vertical seismic forces, and the results are shown in Figure 10. As the proportion coefficient ζ increases, the failure probability P_f increases but the reliability index β decreases. The required safety factor and support pressure of tunnel faces with 3 kinds of security levels under different proportion coefficients ζ were presented in Table 12. The safety factor F_s is relatively close, but the support pressure σ_T increases with the proportion coefficient ζ, and the increasing effect is obvious. Comparing the results from ζ=1.0 and ζ=0, the relative errors of the support pressures with 3 kinds of security levels are 28.6%, 28.6% and 32.5%, respectively. It can be seen that the vertical seismic force also has a great influence on the reliability of tunnel faces. If the impact of the vertical seismic force was ignored (ζ=0), the results will have a great error.

Table 10 Required safety factor and support pressure of tunnel faces with three kinds of security level under different k_h

Table 11 Statistical characteristics VII of random variables

5.4.3 Influence of correlation of seismic effect coefficients

As for parameters of rock mass and loads, some are independent of each other, while the others are correlated. To simplify the calculation, they are generally assumed to be independent of each other, resulting in a certain error in results. Therefore, the influence of the correlation of seismic effect coefficients on the reliability of tunnel faces was discussed in this section.

Figure 10 Effect of proportion coefficient ζ on reliability of tunnel faces:

Table 12 Required safety factor and support pressure of tunnel faces with three kinds of security levels under different ζ

Table 13 shows the statistical characteristics of random variables considering the correlation between the horizontal and vertical seismic forces. The results are shown in Figure 11 and Table 14, the failure probability P_f increases but the reliability index β decreases linearly with the increase of the correlation coefficient ρ_kh,kv, but the change effect is not obvious. Comparing the result from and the maximum relative errors of the reliability index under 3 kinds of security levels are 0.9%, 0.8% and 0.7%, respectively. It can be seen that the correlation between the horizontal and vertical seismic force has little impact on the reliability index of tunnel faces. Specifically, the negative correlation slightly improves the reliability index, but the positive correlation slightly reduces the reliability index. If in the condition of lower accuracy, the horizontal and vertical seismic forces can be assumed to be independent of each other, and the influence of the correlation can be also ignored.

Table 13 Statistical characteristics VIII of random variables

6 Conclusions

1) Based on a 3D stochastic collapse model of tunnel faces in soft rock masses, the upper bound theorem of limit analysis and the response surface method were used to obtain the collapse pressure, failure probability, reliability index, reasonable safety factor and support pressure with three kinds of security levels. The approach provides a new idea for the aseismic design of tunnel faces in soft rock masses.

2) The results from the response surface method (RSM) and the Monte Carlo method (MCM) were compared. When the calculation accuracy is lower, the results obtained by the two methods are completely consistent, and the correctness of the response surface method (RSM) is verified. However, when the calculation accuracy is higher, the results from the Monte Carlo method have a large error, so the response surface method (RSM) is more superior.

Figure 11 Effect of correlation coefficienton reliability of tunnel faces:

Table 14 Reliability of tunnel faces under different correlation coefficients ρ_kh,kv

3) The mean of strength parameters of soft rock masses determines the collapse pressure of tunnel faces, and it has a significant impact on the failure probability, reliability index and required support pressure. The coefficient of variation characterizing the discreteness also has significant influence on the results, so that it cannot be neglected. According to the sensitivity analysis, the impact of these factors on the reliability of tunnel faces is successively the support pressure σ_T, geological strength index GSI, uniaxial compressive strength σ_ci, unit weight of rock mass γ and rock mass constant m_i.

4) Both the horizontal and vertical seismic forces have significant effects on the reliability of tunnel faces. If the horizontal seismic force was ignored (k_h=0), the reliability index will be obviously overestimated, resulting in a serious deficiency of support pressure. If the vertical seismic force was ignored (ζ＝0), the results will also have a large error. However, the correlation of them has little effect on the reliability of tunnel faces. Specifically, the negative correlation improves slightly the reliability index and the positive correlation reduces slightly the reliability index. If the desired accuracy was lower, the horizontal and vertical seismic forces can be assumed to be completely independent of each other, and the influence of their correlation can be also ignored.

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(Edited by HE Yun-bin)

中文导读

基于三维随机坍塌模型的软岩隧道掌子面抗震稳定性的可靠度分析

摘要：采用对数螺旋曲线构建了软岩隧道掌子面的牛角破坏模式，利用拟静力法将地震力引入到牛角破坏模式中，考虑岩体参数及荷载的随机性，建立地震效应下软岩隧道掌子面的三维随机坍塌模型。将极限分析上限法和响应面法有机结合，对软岩隧道掌子面的抗震性进行了可靠度分析。研究表明，支护力、地质强度指标、单轴抗压强度、岩体重度以及地震力对隧道掌子面的可靠度均有显著影响。尤其是水平地震力和竖直地震力的影响更不容忽视，否则会明显高估计算结果，但是在计算精度要求不高的条件下可以忽略其相关性的影响。

关键词：三维随机坍塌模型；拟静力法；响应面法；可靠指标；安全系数；支护力

Foundation item: Projects(51804113, 51434006, 51874130) supported by the National Natural Science Foundation of China; Project(E51768) supported by the Doctoral Initiation Foundation of Hunan University of Science and Technology, China; Project(E61610) supported by the Postdoctoral Research Foundation of Hunan University of Science and Technology, China; Project(E21734) supported by the Open Foundation of Work Safety Key Lab on Prevention and Control of Gas and Roof Disasters for Southern Coal Mines, China

Received date: 2019-01-03; Accepted date: 2019-01-24

Corresponding author: ZHANG Biao, PhD, Lecturer; Tel: +86-15211030424; E-mail: 1020176@hnust.edu.cn; ORCID: 0000-0001- 8195-8744