TÓM TẮT NHỮNG KẾT LUẬN MỚI CỦA LUẬN ÁN Luận án đã phát triển mô hình số trong phần mềm sai phân hữu hạn (FDM) cho trường hợp đường hầm tiết diện chữ nhật cong chịu tải trọng động đất. Mô hình đã được sử dụng để khảo sát ảnh hưởng của các thông số như mô đun đàn hồi của đất, gia tốc ngang lớn nhất và chiều dày kết cấu chống đến ứng xử của đường hầm tiết diện chữ nhật cong chịu tải trọng động đất, đặc biệt chú ý đến điều kiện liên kết giữa kết cấu chống và khối đất (đá) xung quanh. Luận án đã xây dựng được một sơ đồ tải trọng tĩnh tương đương mới, tác dụng lên kết cấu chống trong đường hầm tiết diện chữ nhật cong chịu tải trọng động đất khi sử dụng phương pháp lực kháng đàn hồi HRM. Phương pháp có ưu điểm là thời gian tính toán ngắn và độ chính xác đã được kiểm chứng bằng cách so sánh với mô hình số FDM trong nhiều điều kiện đầu vào khác nhau. Kết quả nghiên cứu của luận án góp phần đa dạng hóa các phương pháp tiếp cận và nghiên cứu tính toán, thiết kế sơ bộ các đường hầm khi chịu tải trọng động đất.
MINISTRY OF EDUCATION AND TRAINING HA NOI UNIVERSITY OF MINING AND GEOLOGY PHAM VAN VI BEHAVIOR OF SUB-RECTANGULAR TUNNELS UNDER SEISMIC LOADING Major: Underground construction engineering Code: 9580204 THESIS SUMMARY HANOI – 2022 The thesis is completed at: Underground and Mining Construction Department, Faculty of Civil Engineering, Hanoi University of Mining and Geology, Vietnam Scientific supervisors: Asso.Prof., Dr Do Ngoc Anh Prof., Dr Daniel Dias Reviewer 1: Prof., Dr Do Nhu Trang University of Transport Technology Reviewer 2: Dr Ngo Ngoc Thuy Military Technical Academy Reviewer 3: Asso.Prof., Dr Nguyen Xuan Man Hanoi University of Mining and Geology The thesis will be defended before the Academic Review Board at the University level at Hanoi University of Mining and Geology at … , … of date … month … year ……… The thesis is available at the: National Library of Vietnam or the Library of Hanoi University of Mining and Geology GENERAL INTRODUCTION The necessity of the study Tunnels are an important component of the transportation and utility systems of cities They are being constructed at an increasing rate to facilitate the need for space expansion in densely populated urban areas and mega-cities Due to the interaction with the surrounding soil and rock, underground structures are more resistant to earthquakes than structures at the ground surface Despite this, the failure of underground structures was recorded for some earthquakes which occurred around the world and damages reports were reported Considering the substantial scale and construction cost, and their critical role, this kind of infrastructures play in modern society an important role Even slight seismic loading impacts can lead to short-time shutdowns and to substantial direct and indirect damages Therefore, it is very important to carefully consider the seismic loading effect on the design, construction, operation, and risk assessment of tunnels The behavior of underground structures under seismic loading was often studied by different methods including analytical methods, empirical methods, and numerical methods It should be noted that most of the research results were conducted considering circular or rectangular tunnels There are many other types of tunnel cross-sections, among them sub-rectangular tunnels were recently developed and are the object of this thesis The purpose of thesis This research aims to develop numerical methods used to calculate incremental internal forces arising in sub-rectangular tunnel lining under the seismic condition as well as investigation of the parameters (tunnel lining, soil mass, etc.) influencing the behavior of sub-rectangular tunnel subjected to seismic loading The main objectives of this thesis include the following: - Highlighting the behavior of sub-rectangular tunnels subjected to seismic loadings A special attention is paid to the soil-lining interface conditions - Investigating the influence of parameters, like soil Young’s modulus, maximum horizontal accelerations, and lining thickness on the sub-rectangular tunnel behavior under seismic loadings - Providing a new quasi-static loading scheme applied in the Hyperstatic Reaction Method (HRM) for sub-rectangular tunnels under seismic loading Scope of this study - Object of this study: Sub-rectangular tunnels supported by continuous lining The soil and tunnel lining material properties are assumed to be linearly elastic - Scope of this study: Calculate incremental internal forces arising in sub-rectangular tunnel lining under the seismic loading as well as investigation of the parameters (tunnel lining, soil mass, etc.) influencing the behavior of sub-rectangular tunnel structure subjected to seismic loadings Methodology of this study - Acquiring, inheriting: Synthesize, analyze and evaluate existing literature to absorb and inherit previous research results related to calculating internal forces in tunnel structures under seismic loading - Numerical method: Simulation of tunnel structures using FLAC3D (Itasca, 2012) (FDM), Plaxis V8.6, and Matlab software to calculate incremental internal forces in the sub-rectangular tunnel under seismic loading Scientific and practical meaning of this thesis - Scientific Meaning: The results of this thesis applied to sub-rectangular tunnels under seismic loading can be a useful reference for scientists, contributing to diversifying approaches to calculation, design for tunnels is subjected to seismic loading - Practical meaning: The thesis research results can be effectively used for the preliminary seismic design of sub-rectangular tunnels The new highlights of this thesis - Studying the behavior of sub-rectangular tunnels under seismic loading Special attention is paid to the soil-lining interface conditions - Providing a new quasi-static loading scheme applied in the HRM method for subrectangular tunnels under seismic loadings Primary argument of this thesis - Argument 1: A significant difference in the behavior of the sub-rectangular tunnel compared with the circular tunnel one when subjected to seismic loadings Special attention is paid to the soil-lining interface, i.e., full slip and no-slip conditions - Argument 2: Providing a new quasi-static loading scheme applied in the HRM method used for sub-rectangular tunnels under seismic loading The proposed equations are validated through numerical analyses Thesis outline This thesis consists of general introduction, chapters, general conclusions, perspectives, published manuscripts and references The whole content of this thesis is illustrated in 110 pages of A4 size, including tables and 47 figures CHAPTER 1: LITERATURE REVIEW ON THE BEHAVIOUR OF UNDERGROUND STRUCTURES UNDER SEISMIC LOADINGS 1.1 Introduction Tunnels are an important component of the transportation and utility systems in both urban and national systems They are being constructed at an increasing rate to facilitate the need for space expansion in densely populated urban areas Vietnam's territory is in a rather special position on the Earth's crust tectonic map and it exists a complex, diverse, and high-risk network of earthquakes There are studies of earthquakes such as statistics, localization, forecasting, assessment of the risk of earthquakes, and design (Nguyen et al., 2009; Bui, 2010; Mai Duc Minh, 2011; TCVN, 2012; Nguyen et al., 2012; Nguyen et al., 2014; Nguyen et al., 2015; Nguyen Đinh Xuyen, 2015; Le et al., 2015; Le Bao Quoc, 2015; Do Ngoc Anh, 2016) As tunnels are interacting with the surrounding soil and/or rock environment, they are more resistant to earthquakes than structures at the ground surface Despite this, the destruction of underground construction has been recorded at many earthquakes taking place around the world Other detailed reviews of the seismic performance of tunnels and underground structures can be found in relevant publications (Hashash et al., 2001; Gazetas et al., 2005; Lanzano et al., 2008; Roy and Sarkar, 2016; Yu et al., 2016; Jaramillo, 2017) Therefore, it is important to consider the influence of seismic loading on the analysis, design, construction, operation, and risk assessment of tunnel structures 1.2 Seismic response mechanisms Earthquake effects on underground structures can be grouped into two categories: ground shaking and ground failure (Wang, 1993) or four categories: ground shaking, ground failure, land sliding and soil liquefaction (FHWA, 2004) The ground response due to the various types of seismic waves: - Body waves travel within the earth’s material; - Surface waves travel along the earth’s surface Three types of deformations express the response of underground structures to seismic motions: - Axial compression/extension; - Longitudinal bending; - Ovalling/racking On the other hand, the ground failures induced by earthquakes: - Failures may be caused by liquefaction; - Fault motions; - Slope failure 1.3 Research methods Expression of underground structures under seismic loading is often studied using different methods: - Analytical methods - Experience - Numerical methods: quasi-static and Numerical full seismic analysis 1.4 Sub-rectangular Circular tunnels have a low cross-section space-utilization ratio while rectangular tunnels have low stability Recently, to overcome these limitations, sub-rectangular tunnels have been studied and applied The sub-rectangular tunnels have the following advantages: - Having great advantages in terms of underground space use; - Reduce the volume of earthwork excavation; - Avoid stress concentration at four corners compared with rectangular tunnels Sub-rectangular tunnels have been applied and studied with real ratio or reduction ratio (Liu et al., 2018; Zhang et al., 2017; Konstantin et al., 2017; Zhu et al., 2017; Zhang et al., 2019), numerical analyses (Do et al., 2020) However, the above studies only study the sub-rectangular cross-section works with static loads but not mention the works under seismic loading 1.5 Conclusions Many research works in the world on underground structures when subjected to seismic loading allow understanding and predicting behavior of underground structures However, these researches mainly focus on underground works with circular and rectangular sections, no research has been done for underground works with the subrectangular tunnel when subjected to seismic loading This is the main research object of this thesis CHAPTER 2: NUMERICAL STUDY ON THE BEHAVIOR OF SUBRECTANGULAR TUNNEL UNDER SEISMIC LOADING In this chapter, a 2D finite-difference numerical model of a sub-rectangular tunnel under seismic loading is proposed It is developed based on the modeling of a circular tunnel which is validated by comparing the results obtained using well-known analytical solutions (Wang, 1993; Hashash et al., 2005; Kouretzis et al., 2013) Such parameters as soil Young’s modulus, maximum horizontal acceleration, and lining thickness on the tunnel behavior under seismic loadings were carefully examined and their influence was evaluated In the study, particular attention was drawn to analyzing the soil-lining interface Different behaviors of sub-rectangular and circular shaped tunnels under seismic loadings were compared based on numerical modeling 2.1 Numerical simulation of the circular tunnel under seismic loading 2.1.1 Reference sub-rectangular tunnel case study- Shanghai metro tunnel Parameters of a sub-rectangular express tunnel in Shanghai, China are used as the reference case in this study (Do et al., 2020) The sub-rectangular tunnel dimensions are 9.7m in width, 7.2m in height and 60m2 in cross-section area (Figure 2.1) The tunnel is supported by a segmental concrete lining of 0.5m For simplification purposes, a continuous lining was adopted without considering the effect of joints Based on this reference sub-rectangular tunnel, a circular tunnel with an external diameter of 4.89m and 75m2 in cross-section area which has an equivalent utilization space area, is considered for comparison purposes (Figure 2.2) o1 (0, 6350) 500 R4850 o o (500, 0) (-3400, -1930) 6200 (-500, 0) 38 R4 utilization space area Circular tunnel (3400, -1930) o 500 6200 o (3400, 1930) 88 o (-3400, 1930) o 500 500 R4850 R500 R4 R500 500 R9450 R9450 500 8700 500 500 8700 500 o2 (0, -6350) Figure 2.1 Sub-rectangular express tunnel in Shanghai (Do et al., 2020), distances in millimeters Figure 2.2 Circular tunnel with the same utilization space area, distances in millimeters 2.1.2 Numerical model for the circular tunnel A numerical model for circular tunnels was developed using a finite difference program (FLAC3D) (Itasca, 2012) The purpose was to investigate the behavior of circular tunnel linings under quasi-static loading and make a comparison with those obtained by an analytical solution Similar to the research work of Sederat et al (2009), Naggar and Hinchberger (2012), and Do et al (2015), ovaling deformations due to the seismic loading are imposed as inverted triangular displacements, along with the model lateral boundaries Uniform horizontal displacements are applied along the top boundary (Figure 2.4) The magnitude of the prescribed displacements assigned at the top of the model is dependent on the maximum shear strain max, estimated based on the maximum horizontal acceleration aH The bottom of the model is restraint in all directions Figure 2.4 Geometry and quasi-static loading conditions for the circular tunnel model (Do et al., 2015) Table 2.1 Input parameters for the reference case of seismic loading Parameter Symbol Unit Soil properties Unit weight γ MN/m3 Young’s modulus Es MPa Poisson’s ratio νs Internal friction angle φ degrees Cohesion c MPa Lateral earth pressure coefficient K0 Depth of tunnel H m Peak horizontal acceleration at ground aH g surface Moment magitude Mw Distance of site source Km Tunnel lining properties Young’s modulus E MPa Poisson’s ratio ν Lining thickness t m External diameter D m Value 0.018 100 0.34 33 0.5 20 0.5 7.5 10 35000 0.15 0.5 9.76 2.2 Validation of circular tunnel under seismic loading For validation purposes of the numerical model subjected to quasi-static loading, the well-known analytical solution proposed by Wang, (1993) and thereafter improved by Kouretzis, (2013) was used for comparison with the results obtained from the numerical model The soil and tunnel lining material properties in numerical models are assumed to be linearly elastic Figure 2.7 illustrates the distribution of the incremental internal forces induced in the tunnel lining under seismic loading Conditions of lining and soil interaction, when using the Wang solution and FDM were considered for both cases of no-slip and full slip The soil and tunnel lining parameters fed into the model are presented in Table 2.1 - It can be seen that results obtained by numerical and analytical models are in very good agreement, The maximum difference is smaller than % - Figures 2.7a and 2.7c show that the maximum incremental bending moment in the full-slip case is 14% larger than the one obtained in the no-slip case - The maximum incremental normal forces in the full-slip case are smaller than that of the no-slip case (Figure 2.7b and Figure 2.7d) Wang solution: 45° 45° No-slip case: Mmax = 0.738 MNm/m No-slip case: Nmax = 0.894 MN/m Full slip case: Mmax = 0.845 MNm/m Full slip case: Nmax = 0.173 MN/m b) Incremental Normal Forces a) Incremental Bending Moments Numerical solution (FDM): 45° 45° No-slip case: Mmax = 0.741 MNm/m No-slip case: Nmax = 0.903 MN/m Full slip case: Mmax = 0.834 MNm/m Full slip case: Nmax = 0.169 MN/m c) Incremental Bending Moments d) Incremental Normal Forces Figure 2.7 Distribution of the incremental internal forces in the circular tunnel by Flac3D and Wang solution In the section, a parametric study is conducted to highlight the behavior of circular tunnel lining subjected to quasi-static loadings considering the effect of Young’s modulus Es, maximum horizontal seismic acceleration aH, and tunnel lining thickness t variations For both the no-slip and full-slip conditions, numerical results show a very good agreement with the analytical solution The difference under 2% for both the extreme incremental bending moments and normal forces is obtained 2.3 Numerical simulation of the sub-rectangular tunnel under seismic loading Figure 2.11 Geometry and quasi-static loading conditions in the numerical model of a sub-rectangular tunnel In this section, a numerical model was developed for the sub-rectangular tunnels cased using similar soil parameters, lining material, and modeling processes to consider the static and seismic loadings introduced above Only the tunnel shape is modified into a subrectangular geometry and the gravity effect is taken into consideration The mesh consists of a single layer of zones in the y-direction, and the dimension of the elements increases as one moves away from the tunnel (Figure 2.11) The geometry parameters of subrectangular tunnels are presented in Figure 2.1 and other parameters presented in Table 2.1 are adopted 2.4 Parametric study of sub-rectangular tunnels in quasi-static conditions Figure 2.14 introduces the incremental bending moments and normal forces induced in the sub-rectangular tunnel linings subjected to seismic loadings and considering both no-slip and full slip conditions Parameters of the reference case presented in Table 2.1 are adopted - Extreme incremental bending moments and normal forces observed in the subrectangular tunnel appear at the tunnel lining corners where the smaller lining radii are located - Absolute extreme incremental bending moments for the no-slip condition are always more than the full slip ones This relationship is opposite to the one observed in the cases of the circular-shaped tunnel It is clear that the behavior of sub-rectangular and circular tunnels are completely different under seismic loading 1.5 1 Extrme Incremental Normal Forces N (MN/m) Extreme Incremental Bending Moment M (MNm/m) 1.5 0.5 0.5 -0.5 -0.5 -1 -1.5 Mmax_SR_ns Mmax_Circular_ns Mmin_SR_ns Mmin_Circular_ns -2 -1 -1.5 Mmax_SR_fs Mmax_Circular_fs Mmin_SR_fs Mmin_Circular_fs -2.5 Nmax_SR_ns Nmax_Circular_ns Nmin_SR_ns Nmin_Circular_ns -2 Nmax_SR_fs Nmax_Circular_fs Nmin_SR_fs Nmin_Circular_fs -2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 aH (g) 0.3 0.4 0.5 0.6 0.7 0.8 aH (g) a) Incremental bending moments b) Incremental normal forces Figure 2.15 Effect of the aH value on the internal forces of circular and sub-rectangular tunnel linings 2.4.2 Effect of the soil’s Young’s modulus (Es) Soil Young’s modulus values are assumed to vary in the range from 10 to 350 MPa The other parameters based on the reference case are assumed (Table 2.1) It can be seen from Figure 2.16 that: 1.5 Extreme Incremental Normal Forces N (MN/m) Extreme Incremental Bending Moment M (MNm/m) 1.25 0.75 0.5 0.25 Mmax_SR_ns Mmax_Circular_ns Mmin_SR_ns Mmin_Circular_ns -0.25 Mmax_SR_fs Mmax_Circular_fs Mmin_SR_fs Mmin_Circular_fs -0.5 -0.75 0.5 -0.5 -1 -1.5 Nmax_SR_ns Nmax_Circular_ns Nmin_SR_ns Nmin_Circular_ns -2 -2.5 -1 -1.25 Nmax_SR_fs Nmax_Circular_fs Nmin_SR_fs Nmin_Circular_fs -3 50 100 150 200 250 300 350 50 100 150 200 250 300 350 Young's Modulus, Es (MPa) Young's Modulus, Es (MPa) a) Incremental bending moments b) Incremental normal forces Figure 2.16 Effect of the Es value on the internal forces for the circular and subrectangular tunnel linings - For the no-slip condition: Figure 2.16a also shows greater absolute extreme incremental bending moments induced in sub-rectangular tunnels compared with circular tunnels having the same utilization space area - For the full slip condition: + Absolute extreme incremental bending moments in the circular tunnel are greater than the sub-rectangular ones for Es values smaller than approximately 150 MPa + When Es values are larger than 150 MPa, absolute extreme incremental bending moments developed in circular tunnels are smaller than in sub-rectangular tunnels 10 - Figure 2.16b indicates that an increase of Es value causes a significant corresponding increase of the absolute extreme normal forces in both sub-rectangular and circular tunnels for the no-slip condition But it induces an insignificant change in absolute extreme incremental normal forces for the full slip condition - Absolute extreme incremental normal forces in the sub-rectangular tunnels are generally 9% smaller than for the circular ones 2.4.3 Effect of the lining thickness (t) The lining thickness t is assumed to vary in the range between 0.2 to 0.8 m, while other parameters introduced in Table 2.1 were adopted The results presented in Figure 2.17 indicate that the lining thickness has a great effect on the incremental internal forces for both sub-rectangular and circular tunnels under seismic loadings 1.5 Extreme Incremental Normal Forces N (MN/m) Extreme Incremental Bending Moment M (MNm/m) Mmax_SR_ns Mmax_SR_fs Mmax_Circular_ns Mmax_Circular_fs Mmin_SR_ns Mmin_SR_fs Mmin_Circular_ns Mmin_Circular_fs 0.5 -0.5 -1 -1.5 -2 0.5 -0.5 -1 Nmax_SR_ns Nmax_Circular_ns Nmin_SR_ns Nmin_Circular_ns -1.5 Nmax_SR_fs Mmax_Circular_fs Nmin_SR_fs Mmin_Circular_fs -2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Lining thickness (m) Lining thickness (m) a) Incremental bending moments b) Incremental normal forces Figure 2.17 Effect of the lining thickness on the incremental internal forces of the circular and sub-rectangular tunnel linings - For the no-slip condition, absolute extreme incremental bending moments of the sub-rectangular linings are always larger than the circular ones (Figure 2.17a) The discrepancy declined gradually from 124% to 6%, corresponding to the lining thickness increase from 0.2 to 0.8 m - In the full slip conditions: + When t < 0.5 m, the absolute extreme incremental bending moments of the subrectangular linings are still larger than the circular ones; + When t > 0.5 m, Figure 2.17a proves an opposite result Thus, larger absolute extreme incremental bending moments on the circular tunnels are observed - It can be seen in Figure 2.17b that the incremental normal forces in the no-slip condition are always larger than for the full slip ones - In comparison with the incremental normal forces of the circular lining, incremental normal forces in the sub-rectangular lining are lower by about 9% and 25% for no-slip and full slip conditions, respectively (Figure 2.17b) 11 The results indicated that an increase in lining thickness is not good solution for design of tunnel linings under seismic loadings 2.5 Conclusion Based on the research results, conclusions can be deducted as follows: - The horizontal acceleration aH, soil’s Young’s modulus Es, and lining thickness t have a great effect on the incremental internal forces induced in both sub-rectangular and circular tunnels for both no-slip and full slip conditions; - The results proved that the soil-lining interface conditions have a great influence on the behavior of sub-rectangular tunnels This is completely different when comparing the behavior circular-shaped tunnels Indeed, while the absolute extreme incremental bending moments of a circular tunnel for the no-slip condition are smaller than the corresponding full slip ones, the absolute extreme incremental bending moments of sub-rectangular tunnels for the no-slip condition are greater than the corresponding full slip ones That is opposite to the trend observed in circular tunnel linings; - Absolute extreme incremental normal forces in sub-rectangular tunnels are approximately 9% smaller than the circular ones; - The no-slip condition is the most unfavorable internal force case for the subrectangular tunnel subjected to seismic loading - The results indicated that an increase in lining thickness is not good solution for design of tunnel linings under seismic loadings CHAPTER 3: A NEW QUASI-STATIC LOADING SCHEME FOR THE HYPERSTATIC REACTION METHOD – CASE OF SUB-RECTANGULAR TUNNELS UNDER SEISMIC CONDITION The HRM method was successfully applied to estimate the seismic-induced structural forces in a circular tunnel lining considering pseudo-static condition (Do et al., 2015; Sun et al., 2020) Based on the in-plane shear stresses applied on the tunnel lining proposed by Peinzen and Wu (1998) and Naggar et al (2008), Do et al (2015) applied a set of dimensionless parameters to change the external loading magnitude of the seismic-induced shear stresses After Do et al (2015), Sun et al (2021) additionally considered the groundtunnel interaction effect on the applied external loadings by using a dimensionless factor to realistically describe the ground-tunnel interaction This chapter aims to introduce a new pseudo-static loading scheme acting on the tunnel lining, when using the HRM method for estimating seismic-induced structural forces in sub-rectangular tunnels in homogeneous isotropic grounds - Firstly, the mathematical formulation of this method is presented 12 - A new external loading scheme applied on sub-rectangular tunnels considering a pseudo-static condition is proposed Three dimensionless parameters are introduced to include the applied active loads and the ground-tunnel interactions - Then, the constitutive equations employing these parameters are proposed by calibrating the HRM solution applying numerical computations - Finally, an extensive validation of the developed HRM method is conducted considering several case scenarios including different seismic magnitudes, ground properties, lining thickness, tunnel geometry, tunnel dimensions and tunnel burial depth 3.1 Fundamental of HRM method applied to sub-rectangular tunnel under static loading The HRM method is based on the Finite Element Method (FEM) which can be used for analyzing the internal forces and displacements induced in the tunnel lining The method was developed for the analysis of segmental and continuous tunnel linings under static loads (Oreste, 2007; Du et al., 2018) Recently, Do et al (2020) have used the HRM method to study the behavior of square and sub-rectangular tunnels under static loading (Figure 3.1) O EI, EA Figure 3.1 Calculation scheme of support structures with the HRM method under static conditions With σv: the vertical loads; σh: the horizontal loads; kn: normal stiffness of springs; ks: shear stiffness of spring; EI and EA: bending and normal stiffness of the support; X and Y are the global Cartesian coordinates (Do et al., 2020) 3.2 HRM method applied to sub-rectangular tunnel under seismic conditions When using HRM for seismic tunnel design, it is necessary to define the external loads that act on the tunnel lining It is assumed that the ovaling deformation of the crosssection is the most critical one for circular tunnels subjected to seismic loadings (Hashash, 2001; Lu et al., 2017; Sun et al., 2019), as illustrated in Figure 3.4a Therefore, the seismicinduced stresses and deformations can easily be determined when the external shear 13 stresses are applied at the far-field boundary, as illustrated in Figure 3.4b The acting shear stress, τ, can be estimated using the free-field shear strain γmax (Penzien & Wu, 1998; Hashash et al, 2001): Figure 3.4 Transversal response in 2D plane strain conditions of the circular tunnel (a) ovaling deformation; (b) corresponding seismic shear loading; (c) sub-ovaling deformation; (d) corresponding seismic shear loading Similarly, when applying shear stress to the far-field boundary, the critical state of the sub-rectangular tunnel subjected to seismic loading causes a sub-ovaling deformation of the tunnel lining as seen in Figure 3.4c This result is obtained by using a finite-difference model (FDM) and incremental internal forces are presented in Figure 3.5 (from Figure 2.10 for no-slip condition) Base on the incremental internal forces in the sub-rectangular tunnel lining obtained using the FDM model (Figure 3.5) The equivalent static loading scheme for the HRM method in Figure 3.6 is determined which contains a couple of dimensionless parameters (a) and (b) 14 Nmax = 0.791 (MN/m) Mmax = 0.900 (MNm/m) a) Incremental Bending Moment b) Incremental Normal Forces Figure 3.5 Incremental bending moments and normal forces of sub-rectangular tunnel obtained using FDM model Figure 3.6 Equivalent static loading with the HRM method for sub-rectangular tunnel In the HRM, the ground interacts with the tunnel support through normal and tangential springs connected to the nodes of the lining structure (Figure 3.1) which are respectively represented by kn and ks, and estimated by the ground initial stiffness η0 In sub-rectangular tunnels, the lining parts radius vary along the tunnel periphery, the initial stiffness of the ground η0 will then change depending on the radius (Do et al., 2020): 𝜂 , =𝛽 (3.8) ⋅ Where νs and Es are respectively the soil Poisson’s ratio and Young’s modulus; Ri is the radius of part i (i=1, and corresponding to the crown, shoulder and sidewall of the tunnel boundary); β is a dimensionless factor 15 In static analyses, the value of dimensionless factor (β) which affects the spring stiffness was usually set to (Molins et al., 2011; Mashimo et al., 2005) or (Do et al., 2015) Recently, Sun et al (2020, 2021) estimated the β value, depending on properties of the soil and tunnel lining for the case of circular tunnels subjected to seismic loading In the present work, a variation of dimensionless factor (β) is also utilized to realistically represent the soil-tunnel interaction 3.3 Numerical implementation In this section, using the FDM numerical model in FLAC3D (Itasca, 2012) has been developed in chapter which was adopted to calibrate the three dimensionless parameters (a, b and β) used in the HRM method Then, the numerical procedure to implement the HRM in the case of sub-rectangular tunnels subjected to seismic loadings is presented (Table 3.3 and Figure 3.8) 3.3.1 FDM numerical simulations Using the FLAC3D, a 2D plane strain model is presented in Chapter The geometry parameters of sub-rectangular tunnels are presented in Figure 2.9 Other soil and lining parameters listed in Table 3.1 (the results in section 2.4 in Chapter 2) and Table 3.2 (Figure 3.7) are adopted It should be mentioned that for the calibration purpose to determine dimensionless parameters a, b and β in the HRM method Table 3.1 Input parameters for the reference case for developing the HRM method Parameter Symbol Unit Value or Range Soil properties Unit weight γ MN/m3 0.018 Young’s modulus Es MPa 10-350 Poisson’s ratio νs 0.34 Internal friction angle φ degrees 33 Cohesion c MPa Lateral earth pressure coefficient K0 0.5 Depth of tunnel H m 20 Peak horizontal acceleration at ground aH g 0.5 surface Moment magitude M 7.5 Distance of site source Km 10 Tunnel lining properties Young’s modulus E MPa 35000 Poisson’s ratio ν 0.15 Lining thickness t m 0.3-0.8 Tunnel height h m 7.2 Tunnel width w m 9.7 16 Table 3.2 Geometrical parameters of tunnel shape cases (Do et al., 2020) Case Tunnel Tunnel h/w R1 (m) R2 (m) R3 (m) width (w) height (h) ratio (m) (m) SR1 8.76 8.15 0.930 8.36 1.02 4.99 SR2 9.13 7.89 0.864 7.09 1.23 4.81 SR3 9.39 7.53 0.802 8.5 0.96 5.07 SR4 (reference 9.70 7.20 0.742 9.95 1.00 5.35 case) Figure 3.7 Shapes of tunnel cases (unit: m) (Do et al., 2020) 3.3.2 Numerical procedure in HRM method To implement the HRM method in the case of sub-rectangular tunnels subjected to seismic loading, it is necessary to determine the formulas of the three dimensionless parameters (a, b and β) which define the external loadings applied on the tunnel lining The main procedure to calibrate the three parameters is illustrated in Table 3.3 and Figure 3.8 After the calibration process is completed, the equations representing the influence of the three parameters on the soil, lining properties and tunnel dimensions can be established The formulas are proposed based on the best fit (Figure 3.9 and 3.10) The parameters β, a, and b can be given as follows: 𝛽 = 𝛽 +𝛽 +𝛽 +𝛽 (3.15) 𝛽 = −1.65 𝐸 + 1.477 (3.16) 𝛽 = 4.8002 (3.17) − 0.3333 17 𝛽 = 30644 − 4452.3 𝛽 = 7.9746 − 5.9192 (3.18) + 207.88( ) − 3.0828 (3.19) 𝑎 = 12.155 + 6.45 𝐸 𝑏 = 𝑏 +𝑏 +𝑏 +𝑏 𝑏 = 10 𝐸 − 0.305 (3.20) (3.21) (3.22) 𝑏 = 23.04 (3.23) − 1.6 𝑏 = −20461 𝑏 = 5.3148 + 3736 (3.24) − 192.79 ( ) + 2.8135 (3.25) − 3.945 Generating soil and lining parameters { , Initial and computation using HRM } for all cases Selection of parameters set { , } and computation using numerical solution Potential error computation Update a, b and β No = = } } Yes Output a, b and β No All cases are computed? i=i+1 Yes a = ƒ( , b = f( , β = f( , ) ) ) Figure 3.8 Calibration flowchart of the three parameters 18 Table 3.3 Overview of the calibration process Step Description Generating the input parameters of soil, lining and tunnel dimensions {ti, hi, wi, Esi} using defined parameter ranges listed in Table 3.1 and Table 3.2 Seismic-induced incremental normal forces and bending moments calculation {NFDM, MFDM} using FDM model, and computation of the initial values of {NHRM, MHRM} using the HRM method based on a=b=β=1 Determination of the relative error of incremental normal forces and bending moments obtained by two methods If eN ≤ 0.02 and eM ≤ 0.02, export a, b and β Otherwise, update these three parameters (i.e., a, b, β) until the target precision is reached Steps to repetitions until all cases scenarios of defined parameter ranges listed in Table and Table are considered Determination of the formulas describing a, b, and β as functions of ti, hi, wi, Esi parameters by using regression analysis 1.25 0.3 0.2 β2 0.1 β1 Numerical calculations 0.75 -0.1 Fitting curve Numerical calculations -0.2 0.5 50 100 150 200 250 300 Fitting curve -0.3 0.02 350 0.04 0.06 Young's Modulus, Es (MPa) 0.08 0.1 0.12 0.9 0.95 t/h a) b) 0.15 0.1 1.5 0.05 Fitting curve β4 β3 Numerical calculations 0.5 Numerical calculations -0.05 -0.1 0.03 Fitting curve -0.5 0.035 0.04 0.045 0.05 0.055 0.7 t/w 0.75 0.8 0.85 h/w c) d) Figure 3.9 Obtained numerical results and fitting curves adopted for the parameters β1, β2, β3 and β4 that created the parameter β 19 13 Parameter a 12.8 Numerical calculations Fitting curve 12.6 12.4 12.2 12 50 100 150 200 250 300 350 Young's Modulus, Es (MPa) a) Numerical calculations Numerical calculations 1.5 0.8 0.3 b2 0.5 b1 Fitting curve Fitting curve -0.2 -0.5 -0.7 -1 -1.5 50 100 150 200 250 300 -1.2 0.02 350 0.04 0.06 0.1 0.12 0.9 0.95 t/h Young's Modulus, Es (MPa) b) c) 0.1 1.1 Numerical calculations 0.08 Numerical calculations 0.9 Fitting curve Fitting curve -0.1 b4 b3 0.7 0.5 0.3 -0.2 0.1 -0.3 0.03 -0.1 0.035 0.04 0.045 0.05 0.055 0.7 t/w 0.75 0.8 0.85 h/w d) e) Figure 3.10 Coefficients fitting curves for the formulas of the parameters a and b1, b2, b3 and b4 that created the parameter b While the coefficient a is expressed as a function of soil’s Young’s modulus alone (Es), coefficient 𝛽 and b are the functions of the lining thickness (t), tunnel height (h), tunnel width (w), and soil’s Young’s modulus (Es), as shown in Figures 3.9 and 3.10 To have a clear understanding of the results obtained by HRM and the numerical FDM model, Figure 3.11 introduces a comparative example of the incremental bending moments 20 and normal forces distribution in the sub-rectangular tunnel lining subjected to a seismic loading when Es = 100MPa and t = 0.5m Other soil parameters of the reference case presented in Table 3.1 are adopted Figure 3.11 reveals insignificant differences between the extreme incremental internal forces obtained by the HRM method and the FDM model The differences are 1.2% and 0.6% corresponding to the extreme incremental bending moments and the normal forces FDM: Mmax = 0.900 MNm/m FDM: Nmax = 0.791 MNm/m HRM: Mmax = 0.911 MNm/m HRM: Nmax = 0.786 MNm/m a) Incremental Bending Moments b) Incremental Normal Forces Figure 3.11 Comparison of the incremental bending moments and normal forces calculated by the developed HRM method and numerical FDM calculation 3.4 Validation of the HRM method The extensive validations were carried out to demonstrate the applicability of the developed HRM method The first validation aims at estimating the accuracy of the developed HRM method, using a range of peak horizontal seismic acceleration (aH) Then, varying Young’s modulus of soil and lining thickness are used for validations and While the uniform tunnels with different cross-sections are considered in validation 4, different sub-rectangular shapes with geometrical parameters of tunnel shape cases from Table (Do et al., 2020) are used in validation The effect of the burial depth of the tunnel on behavior of tunnel lining is considered in validation Finally, validation is performed using soil parameters adopted in research by Hashash et al (2005) and Sun et al (2021) In each validation, the seismic-induced incremental internal forces obtained by the HRM method are compared with the numerical FDM solution The validation results are shown that the developed HRM method can be effectively used to estimate the incremental internal forces in sub-rectangular tunnel lining under seismic loading 3.5 Conclusions The novelty and the scientific contribution of this study lie in proposing a new numerical procedure to efficiently calculate the behavior of sub-rectangular tunnel linings subjected to seismic loading using the Hyperstatic Reaction Method 21 To verify the application capability of the developed HRM method, an extensive validation was performed considering series of numerical computations The developed HRM method was validated based on comparison with a quasi-static numerical FDM model The proposed HRM method in this study provides a new and alternative free method of very efficient seismic design of sub-rectangular tunnels GENERAL CONCLUSIONS AND PERSPECTIVES General conclusion Although there have been many studies evaluating the effects of seismic loading on circular and rectangular tunnels, no research has been conducted on the sub-rectangular tunnels The thesis has used FDM numerical method to investigate the behavior of subrectangular tunnels when subjected to seismic loading, compare and clarify the difference between the behavior of circular and sub-rectangular tunnels The thesis has also proposed a new equivalent static loading scheme acting on sub-rectangular tunnel lining subjected to seismic loading for the HRM method The novelty and the scientific contribution of this thesis in the behavior of subrectangular tunnel linings subjected to seismic loading: The horizontal acceleration aH, soil’s Young’s modulus Es, and lining thickness t have a great effect on the incremental internal forces induced in both sub-rectangular and circular tunnels for both no-slip and full slip conditions; The results proved that the soil-lining interface conditions have a great influence on the behavior of sub-rectangular tunnels This is completely different when comparing the behavior circular-shaped tunnels Indeed, while the absolute extreme incremental bending moments of a circular tunnel for the no-slip condition are smaller than the corresponding full slip ones, the absolute extreme incremental bending moments of subrectangular tunnels for the no-slip condition are greater than the corresponding full slip ones That is opposite to the trend observed in circular tunnel linings; Proposing a new numerical procedure to efficiently calculate the behavior of subrectangular tunnel linings subjected to seismic loading using the Hyperstatic Reaction Method; The present study also shows that in the case when a tunnel structure is more flexible than the soil mass, the tunnel lining will experience amplified distortions in comparison to the soil shear distortions in the free field By contrast, when a tunnel lining is stiffer than the surrounding soil, it tends to resist the ground displacements The proposed HRM method in this study provides a new and alternative free method of very efficient seismic design of sub-rectangular tunnels 22 It is important to note that all of the numerical models developed in this research were performed using drained conditions and with tunnels located at shallow depth Additionally, because no appropriate data exists in literature, all numerical results were not yet compared and validated with experimental data Perspectives The research works listed below are proposed for short-term: Validate all numerical models using real or laboratory data; Develop segment tunnel lining simulation considering the existence of joints; Improve the HRM method for the segmental lining with sub-rectangular tunnels; Develop and perform 3D numerical analyses for full-seismic with sub-rectangular tunnels; For the long-term, the perspectives will be the following ones: Develop and perform 3D numerical analyses considering the water effect on the tunnel lining behavior on undrained analyses; Study the surface structure effect on the tunnel response PUBLISHED AND SUBMITTED MANUSCRIPTS ISI papers: Do Ngoc Anh., Daniel Dias., Zhang ZX., Huang X., Nguyen Tai Tien., Pham Van Vi., Nait-Rabah O (2020) Study on the behaviour of squared and sub-rectangular tunnels using the Hyperstatic Reaction Method, Transp Geotech, 22, 10021 doi: 10.1016/j.trgeo.2020.100321 (ISSN: 2214-3912) Pham Van Vi, Do Ngoc Anh, Daniel Dias (2021) Sub-rectangular tunnels behavior under seismic loading, Appl Sci, 11, 9909 doi.org/10.3390/app11219909 (ISSN: 2076-3417) Pham Van Vi, Do Ngoc Anh, Dias Daniel, Nguyen Chi Thanh, Dang Van Kien (2022) Sub-rectangular tunnels behavior under static loading Transp Infrastruct Geotechnol doi.org/10.1007/s40515-022-00230-w (ISSN: 2196-7202) Do Ngoc Anh, Pham Van Vi, Dias Daniel A New Quasi-Static Loading Scheme for the Hyperstatic Reaction Method - Case of Sub-Rectangular Tunnels under Seismic Conditions, Comput Methods Appl Mech Eng (ISSN: 0045-7825) (under review) 23 Other papers: Nguyễn Chí Thành, Đỗ Ngọc Anh, Phạm Văn Vĩ (2021), Nghiên cứu tính tốn ảnh hưởng động đất đến kết cấu chống đường tàu điện ngầm Hà Nội, Tạp chí Khoa học Kỹ thuật Mỏ - Địa chất, Tập 62, Kỳ 2: 35 - 46 (in Vietnamese) Đặng Văn Kiên, Đỗ Ngọc Anh, Nguyễn Tài Tiến, Nguyễn Huỳnh Anh Duy, Phạm Văn Vĩ (2021), Nghiên cứu tổng quan vỏ hầm metro tiết diện ngang hình chữ nhật cong, Tạp chí Khoa học Kỹ thuật Mỏ - Địa chất, Tập 62, Kỳ 4: 68 - 78 (in Vietnamese) Pham Van Vi, Do Ngoc Anh, Vo Trong Hung, Daniel Dias, Nguyen Chi Thanh, Do Xuan Hoi (2021), Efect of Soil’s Young’s modulus on Sub-rectangular tunnels behavior under quasi-static loadings, International Conference on Underground and Mining Construction UMC 2021 (accepted) 24 ... Mmin_Circular_ns -0 .25 Mmax_SR_fs Mmax_Circular_fs Mmin_SR_fs Mmin_Circular_fs -0 .5 -0 .75 0.5 -0 .5 -1 -1 .5 Nmax_SR_ns Nmax_Circular_ns Nmin_SR_ns Nmin_Circular_ns -2 -2 .5 -1 -1 .25 Nmax_SR_fs... Moment M (MNm/m) 1.5 0.5 0.5 -0 .5 -0 .5 -1 -1 .5 Mmax_SR_ns Mmax_Circular_ns Mmin_SR_ns Mmin_Circular_ns -2 -1 -1 .5 Mmax_SR_fs Mmax_Circular_fs Mmin_SR_fs Mmin_Circular_fs -2 .5 Nmax_SR_ns Nmax_Circular_ns... Mmin_Circular_ns Mmin_Circular_fs 0.5 -0 .5 -1 -1 .5 -2 0.5 -0 .5 -1 Nmax_SR_ns Nmax_Circular_ns Nmin_SR_ns Nmin_Circular_ns -1 .5 Nmax_SR_fs Mmax_Circular_fs Nmin_SR_fs Mmin_Circular_fs -2 0.2 0.3 0.4 0.5 0.6