ORAL PAPER PROCEEDINGS

Transcription

1 ITA - AITES WORLD TUNNEL CONGRESS April 2018 Dubai International Convention & Exhibition Centre, UAE ORAL PAPER PROCEEDINGS

2 The reliability analysis of tunnel lining sections during environmental disturbance based on stochastic response surface method YONG Yuan1, HAN Wang2, HAIXI Jiang3 1 State Key Laboratory for Hazard Reduction in Civil Engineering, Tongji University, Shanghai, China yuany@tongji.edu.cn 2 Department of Geotechnical Engineering, Tongji University, Shanghai, China wanghan_2016@163.com 3 Shanghai SMI Highway (Group) Co., Ltd., 99 Hami Rd, Shanghai, China @qq.com ABSTRACT In the process of operation and management of shield tunnel project, the unloading and reloading caused by deep excavation and superstructure will lead to significant effect on the adjacent shield tunnel. In this study, the reliability variation of lining sections during environmental disturbance is analyzed by non-intrusive stochastic response surface method. Based on Latin Hypercube Sampling (LHS) and Hermite polynomial expansion, the implicit limit state function is represented by explicit one, and it is feasible to develop the limit state functions of lining section in multi-failure modes, and analyze the reliability index with the method of Monte Carlo Simulation (MCS) and First Order Reliability Method (FORM). A case of shield tunnel in metro line is investigated by the method mentioned above, and the validity of the method for structural response approximation and reliability analysis are verified. The reliability index and failure probability varying with loading are highlighted in this study. The research shows that the reliability of lining section is influenced by the environmental disturbance, and drastic change in loading will increase the operational risk and failure probability of tunnel lining section, so limiting the range of environmental disturbance is a necessary measure to guarantee the reliability level of shield tunnel in the life cycle. Keyword: Reliability variation; Stochastic response surface method; Environmental disturbance; Shield tunnel lining 1. INTRODUCTION Composed of precast concrete segments and joints, the reliability of tunnel lining structure is closely linked to the performance of joints and lining segments, when shield tunnel is surrounded by given engineering geology and load conditions. Considering the numerous uncertainty in shield tunnel projects, such as variability in geotechnical parameters, randomness of lining material and uncertainty in the process of construction, the reliability methods have been conducted to analyze the effect of uncertainty in shield tunnel projects. Moment method (Rackwitz et al.,1978), surrogate model method (MOLLON et al.,2009; LÜ et al.,2011) and Monte Carlo simulation method (Wang,2011) are the most commonly used in structural reliability analysis. In these reliability analysis of shield tunnel, the little failure probability and implicit limit state function limit the application and effectiveness of Moment method and Monte Carlo simulation method. However, surrogate models (Hosder et al., 2001; Isukapalli and Georgopoulo,1998; Papadrakakis et al.,1996; Cheng et al., 2008; Rocco and Moreno,2002; Hurtado and Alvarez,2003) build up the bridge between deterministic model and probability analysis where complicated implicit functions are replaced by highly approximate explicit ones, and promote the development of reliability analysis, Based on rigorous mathematical derivation and scientific sampling method (Hu et al., 2010), the stochastic response surface method (SRSM) avoids complicated iteration process which is commonly referred in traditional response surface method, and reduces the programming difficulty for engineers, so it becomes an easy and widely used method for mechanical engineering and geotechnical engineering. With the development of urbanization in China, shield driven method is now the first choice for construction of urban railways. There have been many cases that shield tunnel lining structure is damaged by environmental disturbance in the process of operation, for example the unloading and re-loading effect (Sharma et al.,2001; Huang et al.,2013) induced by deep excavation and superstructure, which even affect the normal function and safety level of tunnel lining. Considering the fact that the inadequate studies for reliability variation during environmental disturbance, it is necessary to analyze the reliability variation in the process of operation. In this study, stochastic response surface method is conducted to analyze the lining structural reliability of shield tunnel. The explicit limit state function developed by the stochastic response surface method, and the lining structural reliability in the process of design and operation is studied based on the limit state equation of concrete lining segment, which will provides evidence and basis for reasonable protection measures for service performance of tunnel projects in life cycle. The rest of this paper is organized as follows. Section 2 introduces stochastic response surface method and Latin Hypercube sampling method (LHS). Section 3 gives the limit state equation of concrete lining segment as well as structural reliability calculation procedure for shield tunnel lining. Section 4 gives a real case study for reliability analysis of tunnel lining. Section 5 represents the reliability variation of shield tunnel lining caused by environmental disturbance. Section 6 draws the final conclusion. 2. STOCHASTIC RESPONSE SURFACE METHOD (SRSM) Stochastic response surface method, an efficient surrogate model, is composed of polynomial chaos expansion and collocation method. Based on the analysis of chaos process in Brownian motion, the mathematician Wiener (1938) developed a new method to approximate the exact response in the random variable space by polynomial expansion. Isukapalli et al. (1999) and Huang et al. (2009) introduced the method into the field of uncertainty analysis and showed the uniform flowchart of the stochastic response surface method. The core parts of the stochastic response surface method are as following: 2.1 Polynomial Chaos Expansion (PCE) The precision of approximation in stochastic response surface method is clearly influenced by the form of polynomial chaos expansion, and Hermite polynomials expansion is an effective and widely adopted method for functional approximation. Wiener (1938) proved the conclusion that functions with Gaussian measures could be approximated by orthogonal Hermite polynomial expansions in Hilbert space, and the accuracy of approximation increases as the order of expansion increase. Supposed that F is the output random variable and X=[x1, x2,, xn] is the n-dimensional input random variable vector which can be expressed by independent standard normal variables U=[U 1, U 2,, U n ], and the relationship 1 2

3 between X and U will be written as: (1) X= TU ( ) Then the function between output variables and input ones will be expressed as: (2) F= G( X) = GTU ( ( )) = HU ( ) Using the Hermite polynomial expansion, F = H(U) will be written as: (3) is the n-dimensional p-order Hermite polynomial expressed by: (4) Compared with the traditional response surface method, SRMS avoids the shortcoming of poor computational accuracy and complexity in twofold. On the one hand, the response function, which is approximated by Hermite polynomial expansion in the global scope, decouple the deterministic model calculation and probability statistics in the analysis of structural reliability, so it avoids the complex iteration in the analysis of structural reliability. On the other hand, stochastic response surface method gives theoretically convergence properties in Hilbert space, and reasonable order of expansion will provide a guarantee for effective approximation and accurate reliability calculation(hu et al, 2010; Wang and Li,2017). It is an easy and efficient method of reliability analysis for civil engineer and mechanical engineer. 2.2 Selection of collocation points based on HLS method As we all know that the calculation accuracy increases with higher order of Hermite polynomial expansions, and reasonable collocation method in global scope is another key factor for function approximation. Probabilistic collocation method and Monte Carlo sampling method (Isukapalli,1999; Jiang et al., 2013) are widely used in collocation method for multiple random variables and multiple orders of stochastic response surface method. Based on the analysis of selection of collocation (Li et al., 2013), it is difficult to select linearly independent collocation points and ineffective for regression calculation in probabilistic collocation method. And Monte Carlo sampling method with the deficiency of cluster of collection points, which can not evenly distribute collocation point in global scope of random variable space. So a better method for collocation points should be considered and developed to enhance the effectiveness of stochastic response surface method. According to the principle of even distribution in global scope of random space, MaKay (1979) developed Latin Hypercube sampling method (HLS) for uncertainty analysis. With the function of sampling memory, it is easy for LHS method to evenly distribute the collocation points in the global scope. And some real cases (Jiang et al., 2013) have been studied and proved the fact that HLS method is an effective collocation method for stochastic response surface method. And the steps for HLS method in M-dimensional space with N level are also defined: (2) Randomly select one value from each interval, and sort N values randomly to form a vector for random variables; (3) Repeat steps 1 and 2, and pair M column vector with N values. The steps for LHS method provide collocation points in M dimensional space with N level, and the responses at the collocation point are calculated (e.g., by numerical approaches in the analysis of structural reliability). In this way the unknown coefficients are computed as following: T a = ( H H) H F (5) T 1 Where a is the unknown coefficient vector, H is the matrix of Hermite polynomial expansions and F is the vector of system responses at the collocation points. Considering the robustness of regression calculation, more collection points should be provided to ensure the accuracy of unknown coefficients in Eq (5). In this study, twice or more collocation points than the number of unknown coefficients will be calculated for the regression fitting. 3. LIMIT STATE FUNCTION AND STRUCTURAL RELIABILITY CALCULATION PROCEDURE 3.1 Limit state functions for tunnel lining structure In structural reliability analysis for tunnel lining structure in the life cycle, two failure modes will be referred : insufficient of bearing capacity and defects in the state of serviceability. In this study, the limit state equation and reliability index of ultimate bearing capacity will be developed and analyzed. According to the failure characteristics of bearing capacity in tunnel lining section, two models are studied and applied to real cases: eccentric compression failure and shear failure. Limit state function of eccentric compression failure can be presented as following (Zhao et al., 2009): (6) Where Z is the limit state function of bearing capacity, KR and Ks are uncertain coefficients of bearing capacity and moment calculating mode of tunnel lining segments, Mu is the bending bearing capacity of segment; M is bending effect of segment (GB ,2010; ITA, 2000). The bending bearing capacity of tunnel lining segment Mu is determined by the material and geometric sizes of lining structure as well as axial force of tunnel lining section, and Mu can be expressed as: (7) Where A, B, C are parameters related to the mode of eccentric compression, material and geometric sizes of lining segments; Nc is the axial force of lining section. As well, limit state function of shear failure mode can be presented as following: (8) Where Z is the limit state function of bearing capacity, KVR and KVs are uncertain (1) Divide the range of each variable into N nonoverlapping intervals on the coefficient of bearing capacity and segments shear calculating mode, Vu is shear principle of equal probability; bearing capacity of segment; V is shear effect of segment (GB ,2010; 3 4

4 ITA, 2000). More information will be referred in the Refs (Yuan and Zhao, 2009). 3.2 Structural reliability calculation procedure for tunnel lining Based on the stochastic response surface method with Latin Hypercube sampling method, structural response and reliability analysis of tunnel lining can be studied following the steps below: In the procedure for structural reliability of tunnel lining structure, finite element software ANSYS will be applied to deterministic model and math software MATALB will be applied to HLS method, regression calculation and probability calculation. And some attention should be paid to the calculation algorithm, for example, in the process of transformation from the U-space to the original space, Nataf transformation will be used to transform collocaction points into input random variables with statistical properties, and the selection of order of Hermite polynomial expansions should be determined by trial calculation and previous research (Phoon and Huang, 2007) Concrete grade of the segment is C55, elastic modulus of which is 35.5GPa, and the density of concrete is 2500kg/m3, as well the reinforcement bar is HRB 335 (II degree reinforcement bar). Preventing the steel from corroding, concrete cover is 45mm on each side. The neighboring segments are connected by three 5.8 M36 bending bolts in circumferential direction and four 5.8 M30 bending bolts in axial direction. Diameter of the bolts are 36mm and 30mm, while yield limit and strength of bolts are 400Mpa and 500Mpa, respectively. The cycle of staggerjointed segmental linings is two lining rings, and the angle of neighboring joints is Figure 2. Schematic diagram of tunnel lining In this study, the typical load condition with the deepest cover soil will be analysis, and depth of cover soil is 13.9m with 1.5m depth of ground water level. The statistical parameter for the basic random variables in the shield tunnel lining response model are listed in Table 1 and 2, and each random variable is independent from other ones. Table 1 Summary statistical of basic random variable of tunnel lining material Figure 1. The flow chart for structural reliability calculation procedure Monte Carlo simulation (MCS) and first order reliability method (FORM) is applied for probability calculation of limit state function represented by Hermite polynomial expansions of tunnel lining section, and makes it possible to study complex multidimension variables reliability analysis with limit number of deterministic model simulation. 4. AN CASE STUDY FOR RELIABILITY ANALYSIS OF SHIELD TUNNEL LINING SECTION Where fc and ft are the compressive and tensile strength of concrete, fy and fy are the compressive and tensile strength of reinforcement bar, fyv is the tensile strength of hooping, while Asi and Aso are the section area of inside and outside of longitudinal ribs, Asv is the area of hooping. Table 2 Summary statistical of basic random variable of tunnel lining response model 4.1 Background As illustrated in Figure 2, the tunnel lining segments represent a typical cross section of shield tunnel of Metro 16 in Shanghai. The lining structure is composed of 8 similar segments, and the central angle of segment is 45 degrees; the outer diameter and thickness of lining segment are 11360mm and 1500mm, while width of whicht is 1200mm. Where γ is the density of the cover soil, H is the depth cover soil (from ground surface to the top of tunnel lining), k0 is the coefficient of lateral earth pressure and K is the coefficient of lateral ground resistance. 5 6

5 4.2 Deterministic model of structural response for tunnel lining section Based on the traditional load-structure model for underground structure, a deterministic model is developed to simulate structural response of shield tunnel lining structure (ITA, 2000). The loads on the segmental ring are mainly earth pressures, water pressures and ground reactions, as shown in Figure Approximation of structural response for tunnel lining According to the flow chart for structural reliability analysis, the explicit function can be established by he Nataf transform and Hermite polynomial expansions with HLS method. To verify the accuracy of stochastic response surface method for structural response and determine the order of Hermite polynomial expansions, probability calculation, which are conducted by the PDS module in ANASY, are simulated with the deterministic model. And the simulation result of probability calculation for structural response is compared with the one of 2nd and 3rd Hermite polynomial expansions (HPE) with 70 collocation points, take the section of maximum negative bending moment as an example, the simulation results and this results fitted by stochastic response surface method are shown as Figure 4.. Figure 3. Loads on the shield tunnel segmental ring In this deterministic model, the magnitude of earth pe and water vertical load pw are the same as the downward earth and water pressure in the upper part. The lateral earth qe and water pressures qw are calculated by multiplying the vertical loads by the coefficient of lateral earth pressure k0. The ground reaction qr is the soil resistance caused by the lateral deformation of segmental ring, and its value corresponds to the lateral tunnel deformation d multiplied by a coefficient of ground reaction K, which is distributed at the middle of the ring, as shown in Figure 3. The surcharge P0 is loads from the road, buildings and other facilities on the ground, which is uniformly distributed on the top of the tunnel with a deterministic magnitude of 20 Kpa. The factor pg is the self-weight pressure of the tunnel uniformly distributed on the structure. More information will be referred in ITA report (ITA, 2000). In the beam-spring model built by ANSYS for tunnel lining structure (Zhu and Tao, 1998), the tunnel segmental lining is modelled as Beam4 element, and joints are modelled as Matrix27 spring elements. Meanwhile, the ground reaction qr is applied by springs between tunnel lining and soil. In this deterministic model, the structural response of tunnel lining is affected by the stiffness of joints, and the stiffness is given by the structural experiment of joints. In this study, the positive bending stiffness is 3 105kN m/rad and the negative one is 2 105kN m/rad, while axial and shear stiffness of which are infinity. The validity of the model developed is verified with means of random variables, and the extreme region of structural response, for example the lining sections with maximum (minimum) bending moment, axial force and shear, are paid more attention. These regions define the critical regions for deterministic design, and show the dangerous section for structural reliability analysis (Li et al., 2013). Figure 4. The cumulative function of structural response fitted by different methods According to Fig.4, the cumulative distribution functions (CDF) of tunnel lining structural response fitted by Hermite polynomial expansions are similar to the ones calculated by stochastic simulation in ANSYS, and the structural response can be effectively approximated by stochastic response surface method in the global scope of random variable space. In the analysis of structural reliability, 3rd and 4th order Hermite polynomial expansions are adequate and effective for calculation accuracy, and higher order will sharply increases the difficult of simulation and regression calculation with limited improvement in accuracy of structural response (Hu et al., 2010; Phoon and Huang, 2007). Fig.4. shows clearly that the curve fitted by 2nd order Hermite polynomial expansions is similar to 3rd order, and 3rd order Hermite polynomial expansions is adequate for structural response. To balance the difficulty of simulation and accuracy of the approximation for structural response, 3rd order of Hermite polynomial expansions will be conducted to the structural reliability analysis of tunnel lining. 4.4 Reliability analysis of tunnel lining section According to the flow chart shown by Fig.1, following the approximation of structural response conducted by stochastic response surface method, failure probability and reliability index are calculated by calculation method of reliability (MCS and FORM) in the failure modes of ultimate bearing capacity. The reliability index of tunnel lining in two failure modes are shown in Figure 5 and Figure

6 Figure.5 Reliability index of eccentric failure model Figure.6 Reliability index of shear failure model lining (Liu et al., 2014), the conditions of environment disturbance are simplified as following: (1) reloading condition: increasing the vertical earth and water pressure with constant lateral load; (2) unloading condition: decreasing the lateral earth and water pressure with constant vertical load. These simplified models for tunnel lining ensure the validity of structural response in beam-spring model, and balance the difficulty with the accuracy of structural simulation. According to the reliability analysis of tunnel lining section, the reliability index in two failure modes is dominated by the eccentric compression failure mode in the normal condition. To simplify the reliability analysis of tunnel lining section in the condition of reloading and unloading, reliability index in eccentric compression failure mode will be explored. Similar to the condition of design, 3rd order of Hermite polynomial expansions with 70 collocation points is applied to approximate the structural response of the tunnel lining, and the reliability index is calculated by first order reliability method (FORM). The reliability variation of tunnel lining section in surcharge condition are shown as Figure 7. As shown in Fig.5, the minimum reliability index of tunnel lining section is 2.87, and the maximum failure probability is , the region of which is the same as the most dangerous one with the maximum negative bending moment and the minimum axial force in deterministic model. And the maximum reliability index is 12.54, the region of which is the same as the safest one with minimum bending moment. By comparing the reliability index in probability analysis with structural response in deterministic model, it is clear that the reliability index of tunnel lining section is negatively correlated with the bending moment, which means that the larger bending moment is, the less reliability index is. The extra region of structural response is similar to maximum and minimum reliability index, and the difference may be caused by the limit state function of eccentric compression failure mode. As illustrated in Fig.6, the reliability indexes of shear failure mode changes a little in different section of tunnel lining with tiny failure probability, and the minimum value of which is Be similar to the reliability index of eccentric failure mode, and reliability index is negatively correlated with the shear in deterministic model, and the larger shear of tunnel lining section is, the less reliability index is. When analyzing the reliability indexes of tunnel lining section in two failure modes, tandem model can be applied to analyze the reliability index in multi-failure modes. Margin estimation method will be a practical method for structural reliability with two failure mode. Considering the fact that the failure probability is governed by the one of eccentric failure mode, in this study the reliability index of lining section is replaced by the one of eccentric failure mode. 5 RELIABILITY VARIATION DURING ENVIRONMENTAL DISTURBANCE The structural response of tunnel lining influenced by environment disturbance, such as deep excavation and superstructure construction, is studied in the form of deformation analysis (DoležalovÁ, 2001; Chang et al.,2001) and ultimate bearing capacity analysis (Liu et al.,2014). The reliability analysis of tunnel lining during environment disturbance is inadequate and important for the reliability evaluation in life cycle, and the reliability of tunnel lining section in two conditions of environment disturbance, which refers to reloading caused by superstructure and unloading resulted by deep excavation, is studied. Based on the analysis and experiments for structural response of tunnel Figure. 7 Reliability index of tunnel lining section in surcharge condition As shown in Fig. 7, the distribution of reliability indexes of different section in surcharge condition are similar to the ones in normal condition, and the regions with minimum reliability index are close to the angle region of 45, 135, 225 and 315, which are similar to the ones with maximum bending moment in deterministic model. The reliability indexes of tunnel lining section in surcharge condition are less than the ones in normal condition, and the minimum reliability indexes decrease with the increase of the value of reloading, which means that the risk level of local damage for tunnel lining section increases with the development of environment disturbance. Four typical and dangerous sections of tunnel lining with small reliability index are also expressed in Fig.7. It is clear that the minimum reliability index of tunnel lining section during environment disturbance locates in the top section with 0 degree, and the reliability indexes of four typical section are strongly negative linear correlation with the change value of surcharge condition, When the change value is 40KPa, the reliability index of top section is 1.54, and the failure probability is 6.18%, which is 25 times more than these (0.227%) in normal condition. 9 10

7 Figure. 8 Reliability index of tunnel lining section in unloading condition The reliability indexes of tunnel lining section in unloading condition are similar with the ones in surcharge condition, and more information is illustrated in Fig.8. When the change value of unloading condition is 40KPa, the minimum reliability index is 1.30, and the maximum probability of structure failure is 17.14%, which is 75.5 times more than these in normal condition. Considering the lasting time of environment disturbance, there is great risk for local damage of tunnel lining section in the process of operation with great load value changing. Based on the reliability analysis for tunnel lining section during environment disturbance, it is clear that the change condition of load will increase the failure probability of tunnel lining, and more attention should be applied to the structural response and reliability of tunnel lining during environment disturbance, and some measure must be taken to reduce the loading effect on tunnel lining during environment disturbance and strength the ultimate limit state of lining structure. 6. CONCLUSION Stochastic response surface method is conducted to analyze the reliability of tunnel lining section, and reliability variation in the condition of environment disturbance is studied, the results show that: (1) Stochastic response surface method (SRSM) makes it possible to decouple the deterministic model simulation and probability statistics, and the accuracy of SRMS is similar to the Monte Carlo simulation, which provides a new method for structural response and reliability analysis. (2) The accuracy of SRMS is determined by the order of Hermite polynomial expansions, and the 3rd order of Hermite polynomial expansion is adequate for the approximation accuracy of structural response and the reliability analysis of tunnel lining section. (3) Based on the reliability analysis of tunnel lining in two failure modes, the reliability index is negative correlation with the bending moment and shear in deterministic model, and the reliability indexes in multi-failure modes is governed by eccentric compression failure mode. In this study, more attention will be paid to the failure mode of eccentric compression. (4) The failure probability of tunnel lining section is less than 0.3% in normal condition, and the change of load will decrease the reliability of tunnel lining. According to the study of reliability variation during environment disturbance, the reliability index is negative correlation with the change value of load, and maximum failure probability in the condition with a change value of 40KPa is 25times more than that in normal condition. It is suggested that some reasonable measures should be taken to control the environment disturbance and strengthen the ultimate bearing capacity of tunnel lining. In practice, the operation and maintenance of shield tunnel lining are referred to different fields of engineering, and the results of the reliability variation lay the foundation to estimate the state of tunnel lining section, so further study will be proposed in the future. 7. CITATIONS AND REFERENCES [1] Rackwitz R, Flessler B. Structural reliability under combined random load sequences[j]. Computers & Structures, 1978, 9(5): [2] MOLLON G, DIAS D, SOUBRA A H. Probabilistic analysis of circular tunnels in homogeneous soil using response surface methodology[j]. Journal of Geotechnical and Geoenvironmental Engineering, 2009, 135(9): [3] LÜ Q, SUN H Y, LOW B K. Reliability analysis of ground-support interaction in circular tunnels using the response surface method[j]. International Journal of Rock Mechanics & Mining Sciences, 2011, 48: [4] WANG Hui, HUANG Hong-wei. Reliability evaluation method for segment joints of metro shield tunnels in soft soils[j]. Chinese Journal of Geotechnical Engineering, 2011, 33(S1): (in Chinese) [5] Hosder, S., Watson, L.T., Grossman, B., Mason, W.H., Kim, H., Haftka, R.T., Cox, S.E. Polynomial Response surface approximations for the multidisciplinary Design optimization of high speed civil transport. Optimization Engineering. 2001, 2(4): [6] Isukapalli S S, Roy A, Georgopoulos P G. Stochastic Response Surface Methods (SRSMs) for Uncertainty Propagation: Application to Environmental and Biological Systems[J]. Risk Analysis An Official Publication of the Society for Risk Analysis, 1998, 18(3): [7] Papadrakakis M, Papadopoulos V, Lagaros N D. Structural reliability analyis of elastic-plastic structures using neural networks and Monte Carlo simulation[j]. Computer Methods in Applied Mechanics & Engineering, 1996, 136(1 2): [8] Cheng J, Li Q S, Xiao R. A new artificial neural network-based response surface method for structural reliability analysis[j]. Probabilistic Engineering Mechanics, 2008, 23(1): [9] Rocco C M, Moreno J A. Fast Monte Carlo reliability evaluation using support vector machine[j]. Reliability Engineering & System Safety, 2002, 76(3): [10] Hurtado J E, Alvarez D A. Classification Approach for Reliability Analysis with Stochastic Finite-Element Modeling[J]. Journal of Structural Engineering, 2003, 129(8): [11] Ran, H. U., Dian-Qing, L. I., Zhou, C. B., & Chen, Y. F. (2010). Structural reliability analysis using stochastic response surface method. Engineering Mechanics, 27(9), [12] Sharma, J. S., Hefny, A. M., Zhao, J., & Chan, C. W. (2001). Effect of large excavation on deformation of adjacent mrt tunnels. Tunnelling & Underground Space Technology Incorporating Trenchless Technology Research, 16(2), [13] Huang, X., Schweiger, H. F., & Huang, H. (2013). Influence of deep excavations on nearby existing tunnels. International Journal of Geomechanics, 13(2),

8 [14] Wiener N. The Homogeneous Chaos[J]. American Journal of Mathematics, 1938, 60(4): [15] Isukapalli S S. Uncertainty Analysis of Transport-Transformation Models[J]. Dissertations & Theses - Gradworks, 1999, 57(1): [16] S.P. Huang, B. Liang, K.K. Phoon. Geotechnical probabilistic analysis by collocation-based stochastic response surface method: An Excel addin implementation[j]. Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2009, 3(2): [17] Wang, F., & Li, H. (2017). Stochastic response surface method for reliability problems involving correlated multivariates with non-gaussian dependence structure: analysis under incomplete probability information. Computers & Geotechnics, 89, [18] Jiang, S. H., Dian-Qing, L. I., & Zhou, C. B. (2013). Non-intrusive stochastic finite element method for slope reliability analysis based on latin hypercube sampling. Chinese Journal of Geotechnical Engineering, 35, (in Chinese) [19] Mckay, M. D., Beckman, R. J., & Conover, W. J. (2000). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. American Society for Quality Control and American Statistical Association. [20] Choi S K, Grandhi R V, Canfield R A, et al. Polynomial Chaos Expansion with Latin Hypercube Sampling for Estimating Response Variability[J]. Aiaa Journal, 2004, 42(6): [21] Zhao, Q., Yuan, Y., Gao, S., & Yu, G. (2009). Reliability of a Tunnel Segment Account for Section Bearing Capacity. International Conference on Transportation Engineering,2009,345, [22] Working Group No. 2, ITA, Guidelines for the design of shield tunnel lining,tunneling and Underground Space Technology, 2000, 15(3): [23] GB : Code for design of concrete structures. China Architecture & Building Press, Beijing, (in Chinese) [24] Yuan, Y., & Zhao, Q. L. (2009). Functions for the bearing capacity of lining crosssection of shield tunnel. Journal of Railway Engineering Society,2009,5: [25] Phoon, K. K., & Huang, S. P. (2007). Uncertainty Quantification Using Multi- Dimensional Hermite Polynomials. Geo-Denver,2007,170:1-10. [26] Zhu, H., & Tao, L. (1998). Study on two beam-spring models for the numerical analysis of segments in shield tunnel. Rock & Soil Mechanics (in Chinese). (in Chinese) [37] Xiao-Jun, L. I., Chen, X. Q., & Zhu, H. H. (2013). Reliability analysis of shield lining sections using spreadsheet method. Journal of Geotechnical Engineering, 35(9), (in Chinese). [28] DoležalovÁ, M. (2001). Tunnel complex unloaded by a deep excavation. Computers & Geotechnics, 28(6), [29] Chang, C. T., Sun, C. W., Duann, S. W., & Hwang, R. N. (2001). Response of a taipei rapid transit system (trts) tunnel to adjacent excavation. Tunnelling & Underground Space Technology Incorporating Trenchless Technology Research, 16(3), [30] Liu, X., Bai, Y., Yuan, Y., & Mang, H. A. (2014). Experimental investigation of the ultimate bearing capacity of continuously jointed segmental tunnel linings. China Civil Engineering Journal, 12(10), (in Chinses) 13 14

9 THANK YOU FOR VISITING ITA AITES