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.
LITERATURE REVIEW ON THE BEHAVIOUR OF
Introduction
Tunnels are crucial for transportation and utility systems in urban and national infrastructures, particularly as their construction rates increase to meet the demands of expanding densely populated areas Due to their significant scale and construction costs, tunnels play a vital role in modern society However, even minor seismic loading can result in temporary shutdowns and considerable direct and indirect damages Therefore, it is essential to meticulously evaluate the impact of seismic loading throughout the analysis, design, construction, operation, and risk assessment of tunnels.
Tunnels exhibit greater resistance to earthquakes compared to surface structures due to their interaction with the surrounding soil and rock Historical earthquakes, such as those in Kobe, Japan (1995), Chi Chi, Taiwan (1999), Bolu, Turkey (1999), Baladeh, Iran (2004), and Sichuan, China (2008), have demonstrated damage to underground constructions, highlighting a lack of systematic investigation into the interaction effects between the ground and tunnels In 1993, Wang emphasized the importance of relative flexibility in understanding seismic-induced distortions of underground structures These incidents underscore that poorly designed underground structures are vulnerable to wave propagation effects.
Detailed reviews of the seismic performance of tunnels and underground structures are available in various publications These large-scale incidents serve as standard cases to analyze the interaction between structures and the ground, aiming to validate design methodologies Current analyses and validations of material models and interface conditions are essential to accurately capture observed responses However, there remains a significant need to enhance the design quality and computational approaches for underground structures in earthquake-prone regions.
Figure 1.1 Summary of observed bored/mined tunnel damage due to ground shakings [131]
Vietnam is uniquely situated on the Earth's tectonic map, resulting in a complex and diverse network of high-risk earthquakes Extensive research has been conducted on various aspects of earthquakes in Vietnam, including statistical analysis, localization, forecasting, risk assessment, and design considerations.
Figure 1.2 Typical failure modes of mountain tunnels reported during the 1999
Chi-Chi earthquake in Taiwan [160].
Seismic response mechanisms
Earthquake effects on underground structures can be categorized into two main types: ground shaking and ground failure, or expanded to four categories including land sliding and soil liquefaction Ground shaking is characterized by vibrations caused by seismic waves traveling through the earth's crust Understanding these effects is crucial for assessing the impact of earthquakes on subterranean infrastructure.
Body waves travel within the earth’s material They can contain either P waves (also known as primary or compressional or longitudinal waves) and
S waves (also known as secondary or shear or transverse waves) and they are able to travel in any ground direction
Surface waves travel along the earth’s surface They are Rayleigh waves or Love waves The velocity and frequency of these waves are slower than body waves
Figure 1.3 Ground response to seismic waves [159]
Underground structures experience deformation in response to ground movement caused by traveling seismic waves According to Owen and Scholl, the behavior of these structures during seismic events can be likened to that of an elastic beam reacting to surrounding ground deformations The response of underground structures to seismic motions is characterized by three distinct types of deformations.
Axial deformation along tunnel Bending (curvature) deformation along tunnel
Ovaling deformation of a circular tunnel Racking deformation of a rectangular tunnel
Figure 1.4 Type of tunnel deformations during a seismic event [123]
Axial tunnel deformations occur due to seismic wave components that create motions along the tunnel's axis, resulting in alternating compressive and tensile forces Additionally, bending strains are generated by these seismic wave components, further impacting the structural integrity of the tunnel.
Shear wave front Shear wave front
Positive curvature particle motions that are perpendicular to the tunnel's longitudinal axis were not considered in this study, as they typically align with the tunnel axis However, ovaling or racking deformations can occur in circular and rectangular tunnel linings when shear waves propagate almost perpendicular to the tunnel axis Research by Penzien and Hashash et al highlights that the most significant factor affecting tunnel lining behavior during seismic events is the ovaling or racking deformations induced by seismic shear wave (S-wave) propagation Therefore, these deformation modes are critical for understanding tunnel stability under seismic loading.
Earthquakes can lead to significant ground failures, primarily due to liquefaction, fault movements, or slope failures, which may result in substantial permanent deformations in tunnels.
Figure 1.5 Examples of the effects of seismically-induced ground failures on tunnels [155]
Research methods
Expression of underground structures under seismic loading was often studied using different methods, including analytical methods, experience, numerical methods
Underground structure deformation is frequently modeled using 2-dimensional plane strain problems under equivalent static loads, often neglecting the effects of inertial forces To address this, several analytical methods have been developed to calculate the internal forces in supporting structures, particularly for rectangular and circular tunnels However, these analytical approaches are typically constrained by certain assumptions, which can limit their applicability.
- Homogeneous isotropic soil masses, underground structures material with linear elastic behavior and mass lost;
- Circular tunnels usually are lining continuous structures with constant lining thicknesses;
- Construction procedures could not be considered
To address the limitations of analytical methods, researchers have employed experimental models to gain a deeper understanding of the physical processes involved, with a particular focus on soil-structure interaction during seismic loading conditions.
Various authors have conducted physical models to explore the functioning of underground structures and validate existing design and analysis methods These models primarily aimed to gather measurement data and assess design accuracy However, the complexity and high costs associated with physical modeling have resulted in a limited number of obtainable results.
Recently, the trend is to use a two-dimensional numerical model [70],[124], [125],[135] or a three-dimensional model [91],[137]
Numerical methods for simulating seismic loadings typically employ equivalent static loads, but these approaches share limitations with traditional analytical methods A significant drawback is that they fail to account for the time-dependent performance of tunneling under seismic conditions Additionally, using equivalent static loads often leads to underestimating internal forces compared to those calculated using actual seismic loadings.
The time history analysis method utilizing real earthquakes is the most complex among various techniques, yet it delivers the most precise outcomes Despite its accuracy, this method typically demands extensive computation time, which can restrict the availability of its results.
Most research on underground structures has focused on circular or rectangular tunnels, yet sub-rectangular tunnels also require attention Understanding the internal forces and deformations in these structures under seismic loading, along with the parameters affecting tunnel lining and surrounding soil, is crucial for both practical and scientific applications In earthquake-prone areas, the trend is shifting towards using highly flexible structures like segmental tunnel linings Initial findings highlight the critical role of joints in tunnel structures, emphasizing the need for comprehensive studies to ensure reliable tunnel lining designs.
The ovalling deformations are commonly simulated with a two-dimensional, plane strain configuration and are usually further simplified as a quasi-static case without taking into account the seismic interaction [70],[69]
Analytical solutions have led to the development of various elastic closed-form solutions for assessing internal forces on circular tunnel linings subjected to seismic loading Research by Hashash et al highlighted discrepancies among these methods and utilized numerical analyses to clarify the differences Their findings revealed that Wang’s solution accurately estimates normal forces on tunnel linings under no-slip conditions, while Peinzen’s solution is not recommended for such scenarios Additionally, other studies have shown strong agreement between their solutions and earlier findings, reinforcing the validity of established analytical approaches.
Generally, the analytical solutions are limited to the following assumptions [135]:
• The soil mass is assumed homogenous and the tunnel linings behavior have to be linearly elastic and mass-less materials;
• Tunnels are usually of circular shapes with an uniform thickness without joints;
• The effect of the construction sequence is not studied
A circular tunnel with radius R, situated underground, experiences seismic loading through shear waves, resulting in a shear-type stress state in the surrounding soil This condition is characterized by compressive and tensile principal stresses aligned at 45 degrees, indicating a pure shear direction.
Figure 1.7 Seismic shear loading and equivalent static loading (redrawn) [126]
The shear stresses can be estimated using the free-field shear strain γmax
[70],[69],[128]: τ = ( ) (1.1) where the shear strain 𝛾 can be determined as follows: γ = (1.2)
Table 1.1 Ratios of ground motion at depth to motion at ground surface (after
No Tunnel depth (m) Ratio of ground motion at tunnel depth to motion at ground surface
Table 1.2 Ratios of peak ground velocity to peak ground acceleration at surface in rock and soil (adapted from Sadigh and Egan [134])
Ratio of peak ground velocity (cm/sec) to peak ground acceleration (g) Source-to-Site Distance (Km)
The table categorizes sediment types based on low-strain shear wave velocity (Cm), with rock exhibiting velocities of 750 m/sec or higher, stiff soil ranging from 200 m/sec to 750 m/sec, and soft soil measuring below 200 m/sec Additionally, the correlation between peak ground velocity and peak ground acceleration in soft soils remains inadequately defined.
The peak shear wave velocity (Vmax), ground shear wave velocity (Vs), soil Young’s modulus (Es), and soil Poisson’s ratio (νs) are critical parameters in geotechnical engineering As illustrated in Figure 1.6, the maximum ovaling of circular tunnel linings occurs at the major and minor axes, specifically at an angle of θ = 45° relative to the spring line.
Vmax can be estimated using Tables 1.1 and 1.2 Table 1.1 helps establish the relationship between ground motion at depth and at the surface, while Table 1.2 relates known peak ground acceleration to estimates of peak ground velocity when site-specific data is unavailable.
1.3.1.1 Analytical solutions due to a seismic loading considering a circular tunnel
Wang [159] is recognized as the pioneer in proposing a closed-form solution for the forces acting on structural tunnel linings during seismic loading conditions In cases of full-slip soil-lining interaction, the maximum normal forces (Nmax) and maximum bending moment (Mmax) can be defined by specific expressions.
For the no-slip condition at the soil-lining interface, the formulation of Wang
[159] for the maximum normal forces (Nmax) can be expressed as follow:
K1 = full-slip lining response coefficient;
K2 = no-slip lining response coefficient;
E = tunnel lining Young’s modulus; ν = tunnel lining Poisson’s ratio;
R = tunnel radius; t = tunnel lining thickness;
I = inertia moment of tunnel lining per unit length of the tunnel (per unit width);
Es = soil Young’s modulus; νs = soil Poisson’s ratio; γmax = maximum free-field shear strain; θ = angle measured counter-clockwise from spring line on the right
Wang [159] did not provide a solution for calculating bending moments under no-slip conditions; however, it is proposed that the solution for full slip conditions can be applied in these scenarios The conservative estimates associated with full slip conditions are thought to compensate for any potential underestimations arising from the quasi-static representation of seismic issues [70],[159].
Recently, Kouretzis et al [90] proposed an expression of the maximum bending moment under the no-slip condition to improve the method proposed by Wang [159]:
Mmax = ±(1 − 𝐾 − 2𝐾 )𝜏 (1.10) where 𝜏 is the maximum free field seismic shear stress:
With ρmax is the density of the surrounding ground, Gmax is the maximum ground shear modulus, and Vmax is the peak seismic velocity due to shear wave propagation
1.3.1.2 Analytical solutions due to seismic loading for rectangular tunnel
The analysis of rectangular tunnels for imposed racking deformations can be conducted similarly to that of circular tunnels, taking into account the effects of propagating shear waves Additionally, it is essential to evaluate the walls and roof of the tunnel cross-section to assess seismic earth pressures.
Figure 1.8 Definition of terms used in racking analysis of a rectangular tunnel
Wang [159] introduced a simplified procedure for analyzing the racking deformation of rectangular tunnels, incorporating soil-structure interaction This method is based on extensive seismic finite element analyses that consider various seismic properties of both the soil and tunnel structure The analysis covers multiple conditions to ensure comprehensive evaluation.
Stiffness = τ/γ τ = P/w, S = P/∆st, τ = ∆st.S/w, γ = ∆st/h where ∆st = Structure displacement without Soil
The ratio of the depth to the center of the structure, H, to the structure height, h, ranged from 1.1 to 2.0 (Figure 1.8);
Soil shear modulus surrounding the structure between 11 to 72 MPa, corresponding to shear wave velocities of 75 to 200 m/sec;
The vertical distance between the bottom of the tunnel structure and the top of underlying stiff soils/rock was equal to or greater than the tunnel structure height;
Rigid body rotation was excluded;
Tunnel structures widths, w, ranged between 4.6 to 27.5 m, and tunnel structure heights, h, ranged from 4.6 to 8 m;
Time histories of artificial earthquake ground motion, illustrating western and northeastern U.S earthquakes, were used
Sub-rectangular tunnels
Modern tunneling often employs circular Tunnel Boring Machines (TBMs) for excavation; however, these machines may not always meet the specific underground space utilization requirements This has led to the emergence of non-circular TBMs, which are viewed as an ideal solution for tunneling with specialized cross-sections Recent studies have explored the effectiveness of these unique cross-section tunnels, highlighting their real and reduction ratios in various applications.
Figure 1.14 (a) Overlap cutter heads; (b) a copy cutter head [78]
Jianbin Li highlighted the technical features and current development of non-circular tunnel boring machines (TBMs), focusing on rectangular cross-section and horseshoe-shaped designs His work offers practical solutions for the design, manufacturing, and construction processes of non-circular TBMs Engineering practices indicate that these customized non-circular TBMs provide significant benefits, including improved construction schedules, enhanced settlement control, and better space utilization.
Figure 1.15 A photo showing the testing setup after fabrication [72]
Huang et al conducted a pioneering full-scale loading test to investigate the behavior of segmental lining in a sub-rectangular shield tunnel, marking the first experimental exploration of large cross-section tunnel linings under self-weight They addressed challenges in fabricating the testing setup and segmental linings, providing effective solutions The experimental findings were compared with numerical simulations, revealing that the influence of self-weight is crucial in structural loading tests, particularly for shallow-buried tunnels Future research is needed to further examine the mechanical behavior of this rectangular lining structure under various influential factors post-assembly.
Recent studies have focused on enhancing the performance of the HRM method for squared and sub-rectangular tunnels, with particular attention to the influence of tunnel shape and wall radii The numerical HRM model was validated against a finite element method (FEM), leading to a parametric study that examined how the earth pressure coefficient and soil Young’s modulus affect structural forces and deformations during tunnel excavation Additionally, research by Du et al optimized sub-rectangular tunnels by analyzing various parameters, including lateral earth pressure, soil Young’s modulus, tunnel depth, and surface loads on internal forces and tunnel shape Furthermore, Zhang et al investigated the impact of rotational stiffness on joint behavior and conducted an optimization study on longitudinal joints in sub-rectangular shield tunnels.
However, the above studies only study sub-rectangular cross-sections considering static loads but do not mention seismic ones, this issue should be further studied in the future.
Conclusions
Significant advancements have been made in understanding and predicting the seismic behavior of tunnels, primarily for circular and rectangular shapes However, the seismic responses of tunnels with different geometries to earthquake-induced ground failure remain underexplored Most research has concentrated on the transverse seismic response caused by S waves, typically under plane strain conditions, which effectively predicts seismic lining forces during transverse loadings Additionally, only a limited number of studies have examined the effects of surface waves on seismic tunnel responses, often dismissing their influence as negligible.
The behavior of tunnels under seismic loading can be analyzed through various methods, including analytical techniques, physical model tests, and numerical modeling While analytical solutions are straightforward and quick, they are constrained by oversimplified assumptions Physical model tests and numerical analyses address these limitations, but physical tests are often costly and yield limited results Recently, numerical models have gained popularity due to their reliability in full seismic analyses, effectively investigating factors such as spatially-variable ground motion, variations in layer boundaries, changes in structural properties, and near-fault effects along long tunnels However, the high computational costs associated with 3D numerical simulations mean that this approach is primarily utilized within academic and research settings.
Recent research has explored sub-rectangular tunnels in both field and laboratory settings, highlighting their potential to address the limitations of traditional circular and rectangular tunnels However, existing studies primarily concentrate on the performance of sub-rectangular tunnels under static loads, with little attention given to their behavior during seismic events This thesis aims to investigate the performance of sub-rectangular tunnels under seismic loading conditions.
NUMERICAL STUDY ON THE BEHAVIOR OF SUB-
Numerical simulation of the circular tunnel under seismic loading
2.1.1 Reference case study- Shanghai metro tunnel
Figure 2.1 Sub-rectangular express tunnel in Shanghai [48], distances in millimeters
This study references the parameters of a sub-rectangular express tunnel in Shanghai, China, which measures 9.7 meters in width and 7.2 meters in height, supported by a 0.5-meter segmental concrete lining To simplify the analysis, a continuous lining is assumed, excluding the effects of joints For comparative analysis, a circular tunnel with an external diameter of 4.89 meters, providing an equivalent utilization space area, is also considered.
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 created using the finite difference program FLAC 3D to analyze the performance of circular tunnel linings under quasi-static loading, allowing for a comparison with results derived from an analytical solution.
A 2D plane strain model was created (Figure 2.3 and Figure 2.4) The soil mass is discretized into hexahedral zones The tunnel lining is modeled using embedded
The circular tunnel utilizes R9450 liner elements attached to the zone faces along the tunnel boundary through interfaces Two interface conditions are analyzed: no slip and full slip between the soil and lining The liner–zone interface stiffness, including normal stiffness (kn) and tangential stiffness (ks), is determined based on established guidelines For the no-slip condition, kn and ks are set to 100 times the equivalent stiffness of the stiffest neighboring zone In contrast, for the full-slip condition, ks is assigned a value of zero The apparent stiffness of a zone, measured in stress-per-distance units, is calculated using a specific formula.
(2-1) where: K and G are the bulk and shear modulus, respectively; ∆zmin is the smallest dimension in the normal direction of zones that contact with the liner elements
Figure 2.3 The plane strain model under consideration
The mesh features a single layer of zones oriented in the y-direction, with element dimensions increasing as distance from the tunnel increases, ultimately reaching a maximum size of 2.6 x 1 m at the model boundary The numerical model's boundary conditions extend 120 m in the x-direction and 40 m in the z-direction.
(Es, νs) t direction It consists of approximately 4800 zones and 9802 nodes The bottom of the model was blocked in all directions, and the vertical sides were fixed in the horizontal one
Figure 2.4 Geometry and quasi-static loading conditions for the circular tunnel model
In this chapter, similar to the research work of Sederat et al [135], Do et al
In the study by Naggar and Hinchberger, seismic loading leads to ovaling deformations represented as inverted triangular displacements along the model's lateral boundaries Uniform horizontal displacements are applied at the top boundary, with the magnitude of these displacements determined by the maximum shear strain (γmax), which is estimated from the maximum horizontal acceleration (aH) Additionally, the model's bottom is restrained in all directions to ensure stability during the analysis.
Before applying ovaling deformation from seismic loading, a steady state of the excavated tunnel under static conditions was established In a 2D plane strain model, ground displacements around the tunnel boundary before lining installation are simulated using the convergence confinement method with a relaxation factor, λd, set to 0.3 The numerical simulation of tunnel ovaling is then conducted through a series of defined steps.
Step 1: In situ state of stresses before tunnel construction
Pr es cr ib ed d is pl ac em en t γ max
Step 2: Excavation of the tunnel and use of the convergence confinement process with a relaxation factor, λd, of 0.3 The concrete lining is then activated on the tunnel’s periphery
Step 3: Assigning ovaling deformations caused by the seismic loading on the model boundaries using the prescribed displacements previously mentioned
This study exclusively focuses on incremental internal forces, which are calculated by subtracting the total lining forces obtained at the conclusion of the static loading phase from those measured at the end of the ovaling phase.
2.1.3 Comparison of the numerical and analytical model for the circular tunnel case study
To validate the numerical model under quasi-static loading, the widely recognized analytical solution by Wang, later enhanced by Kouretzis et al., was employed for comparison with the numerical results This solution has proven effective for seismic design in circular tunnels, highlighting its significance in the field.
A recent study proposed a formula for calculating the maximum incremental bending moment under no-slip conditions, a topic not addressed by Wang Utilizing Wang's formulation, this research adapted it to determine the incremental internal forces along the tunnel lining perimeter, as referenced in FHWA guidelines The parameters outlined in Table 2.1 were used as the reference case, assuming the soil and tunnel lining materials to be linearly elastic and massless These assumptions were also consistent with the analytical solution, while an anisotropic stress field was implemented in the numerical model, featuring a lateral earth pressure coefficient at rest (K0) of 0.5.
Table 2.1 Input parameters for the reference case of seismic loading
Peak horizontal acceleration at ground surface aH g 0.5
Distance of site source Km 10
The study presents a deformed circular tunnel model and displacement vectors under prescribed boundary displacements for both no-slip and full slip conditions, as depicted in Figures 2.5 and 2.6 The ovaling deformation of the tunnel lining due to seismic loading is clearly illustrated, with Figure 2.7 showcasing the distribution of incremental internal forces within the tunnel lining The interaction conditions between the lining and soil were analyzed using the Wang solution and Finite Difference Method (FDM) for both no-slip and full slip scenarios The parameters for the soil and tunnel lining, detailed in Table 2.1, indicate that the results from the numerical and analytical models align closely.
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 In contrast, 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)
Figure 2.5 Deformed model and displacement contours in circular tunnel model for no-slip condition
Figure 2.6 Deformed model and displacement contours in circular tunnel model for full-slip condition
Wang solution: a) Incremental Bending Moments b) Incremental Normal Forces
Numerical solution (FDM): c) Incremental Bending Moments d) Incremental Normal Forces Figure 2.7 Distribution of the incremental internal forces in the circular tunnel by
Validation of circular tunnel under seismic loading
This study examines the behavior of circular tunnel linings under quasi-static loads, focusing on the influence of Young’s modulus (Es), horizontal seismic acceleration (aH), and the thickness of the tunnel lining.
No-slip case: Mmax = 0.738 MNm/m
Full slip case: Mmax = 0.845 MNm/m
No-slip case: Nmax = 0.894 MN/m Full slip case: Nmax = 0.173 MN/m
No-slip case: Mmax = 0.741 MNm/m
Full slip case: Mmax = 0.834 MNm/m
45 °No-slip case: Nmax = 0.903 MN/mFull slip case: Nmax = 0.169 MN/m t variations Parameters of the soil and tunnel lining presented in Table 2.1 are adopted for the reference case study
The article discusses the concept of extreme incremental bending moments, which includes both maximum and minimum values Additionally, it addresses extreme incremental normal forces, highlighting the maximum and minimum forces that affect the tunnel lining.
2.2.1 Effect of the peak horizontal seismic acceleration (aH)
A parametric study was performed to assess the effects of seismic loading magnitude, indicated by maximum horizontal acceleration (aH), which varied from 0.05 to 0.75 g, correlating to maximum shear strains (γmax) of 0.038% to 0.58% The reference parameters outlined in Table 2.1 were maintained throughout the study The findings illustrated in Figure 2.8 reveal significant impacts on extreme incremental bending moments and normal forces within the circular tunnel lining due to variations in aH.
E xt re m e In cr em en ta l B en di n g M om en t M (M N m /m ) a H (g)
Mmax_FDM_ns Mmax_FDM_fs Mmax_Wang_ns Mmax_Wang_fs Mmin_FDM_ns Mmin_FDM_fs Mmin_Wang_ns Mmin_Wang_fs
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m ) a H (g)
Nmax_FDM_ns Nmax_FDM_fsNmax_Wang_ns Nmax_Wang_fsNmin_FDM_ns Nmin_FDM_fsNmin_Wang_ns Nmin_Wang_fs
Numerical results for both no-slip and full-slip conditions demonstrate excellent agreement with the analytical solution, showing only a 1% discrepancy in extreme incremental bending moments and normal forces.
As the horizontal acceleration (aH) increases from 0.05 g to 0.75 g, the absolute values of extreme incremental bending moments and normal forces exhibit a gradual rise The incremental bending moments are significantly influenced by the aH value under both no-slip and full slip conditions In contrast, while the incremental normal forces in the tunnel lining under the no-slip condition are greatly affected by changes in aH, only minor variations in incremental normal forces are observed for the full slip condition.
2.2.2 Effect of the soil Young’s modulus, Es
The soil Young’s modulus is estimated to range between 10 and 350 MPa, utilizing additional parameters from Table 2.1 based on a reference case study as input data Numerical results were obtained through Flac 3D and compared with Wang’s analytical method under both full slip and no-slip conditions, as illustrated in Figure 2.9 Key insights can be drawn from these findings.
Figure 2.9 illustrates a strong correlation between the incremental bending moments and normal forces acting on the tunnel lining during seismic loading, as derived from both the numerical model and analytical solution This agreement holds true for both no-slip and full slip conditions while accounting for variations in Es, with the maximum discrepancy being less than 2%.
The extreme incremental bending moments are significantly influenced by the Es value, as illustrated in Figure 2.9a Notably, the maximum absolute values of these moments occur at Es values near 50 MPa Additionally, a sharp decline in the absolute extreme incremental bending moments is observed when the Es value decreases.
When the tunnel lining is stiffer than the surrounding ground, it effectively resists ground displacements, resulting in a pressure range of 25 to 10 MPa.
Es values are larger than 50 MPa, the tunnel structure is more flexible than the ground
The tunnel lining significantly amplifies distortions compared to soil shear distortions in the free field An increase in the elastic modulus (Es) leads to a reduction in the absolute extreme incremental bending moments This relationship is evident under both full slip and no-slip conditions Notably, for the same Es value, the absolute extreme incremental bending moments in the tunnel lining under no-slip conditions are consistently 10% to 15% smaller than those observed in full slip conditions.
In the analysis of tunnel lining forces, it is observed that extreme incremental normal forces under full slip conditions show minimal dependence on the Es value However, an increase in Es leads to a significant rise in both maximum and minimum incremental normal forces in the tunnel lining under no-slip conditions As anticipated, the incremental normal forces in no-slip conditions consistently exceed those in full slip conditions, highlighting the critical impact of Es on tunnel lining performance.
2.2.3 Effect of the lining thickness, t
The tunnel lining thickness is estimated to range from 0.2 to 0.8 meters, aligning with the typical lining thickness to tunnel dimension ratio of 3% to 8.5% Other parameters are derived from the reference case established in the study.
E xt re m e In cr em en ta l B en d in g M om en t M (M N m /m )
Mmax_FDM_ns Mmax_FDM_fs Mmax_Wang_ns Mmax_Wang_fs Mmin_FDM_ns Mmin_FDM_fs Mmin_Wang_ns Mmin_Wang_fs
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m )
Nmax_FDM_ns Nmax_FDM_fsNmax_Wang_ns Nmax_Wang_fsNmin_FDM_ns Nmin_FDM_fsNmin_Wang_ns Nmin_Wang_fs
Table 2.1 illustrates that, akin to the analysis of Young’s soil modulus (Es) and horizontal seismic acceleration (aH), the findings in Figure 2.10 demonstrate a strong correlation between analytical and numerical models under both no-slip and full slip conditions, with discrepancies remaining below 1% for incremental bending moments and normal forces.
As the lining thickness increases from 0.2 to 0.8 meters, both the absolute extreme incremental bending moments and normal forces rise, applicable to both full slip and no-slip conditions Notably, the incremental bending moments under no-slip conditions are consistently lower than those under full slip, with the largest difference of 14% observed at a lining thickness of 0.8 meters Additionally, the variations in incremental normal forces due to changes in lining thickness are less pronounced compared to the variations in incremental bending moments.
The comparison between the analytical solution and the numerical model reveals a strong correlation when analyzing key factors such as Young's modulus (Es), horizontal seismic acceleration (aH), and tunnel lining thickness (t) This alignment underscores the reliability of both approaches in assessing tunnel performance under seismic conditions.
E xt re m e In cr em en ta l B en d in g M om en t M (M N m /m )
Mmax_FDM_ns Mmax_FDM_fs Mmax_Wang_ns Mmax_Wang_fs Mmin_FDM_ns Mmin_FDM_fs Mmin_Wang_ns Mmin_Wang_fs
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m )
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
A numerical model was developed for sub-rectangular tunnels using consistent soil parameters, lining materials, and modeling processes to assess both static and seismic loads The model modifies the tunnel shape to sub-rectangular geometry while incorporating the effects of gravity The mesh features a single layer of zones in the y-direction, with element dimensions increasing away from the tunnel The geometry parameters for the sub-rectangular tunnels are illustrated in Figure 2.1, and additional parameters can be found in Table 2.1.
The model measures 120 meters in width along the x-direction and 40 meters in height along the z-direction, comprising approximately 5,816 elements and 11,870 nodes The bottom of the model is constrained in all directions, while the vertical sides are fixed horizontally.
Parametric study of sub-rectangular tunnels in quasi-static conditions
The prescribed displacement γ max for tunnel linings under seismic loading is analyzed, taking into account both no-slip and full-slip conditions The parameters from the reference case outlined in Table 2.1 are utilized for this evaluation.
Figure 2.12 Deformed model and displacement contours in Sub-rectangular tunnel model for no-slip condition
The analysis of the sub-rectangular tunnel model under full-slip conditions reveals significant insights into structural behavior Figure 2.13 illustrates the deformed model along with displacement contours, highlighting the incremental bending moments and normal forces Additionally, Figure 2.14 presents the distribution of these incremental forces, emphasizing their impact on the tunnel's stability and performance.
Figures 2.14 and 2.7 illustrate the behavior of circular and sub-rectangular tunnel linings under seismic loading, highlighting the locations of extreme incremental internal forces at the tunnel periphery Notably, Figure 2.14 indicates that the maximum bending moments and normal forces in the sub-rectangular tunnel occur at the corners with smaller lining radii A subsequent numerical investigation compares the performance of sub-rectangular and circular tunnels, both designed with the same utilization space area and subjected to seismic loadings This analysis considers various parameters, including horizontal seismic acceleration, soil deformation modulus, and lining thickness, while also examining the effects of soil-lining interface conditions.
2.4.1 Effect of the peak horizontal seismic acceleration (aH)
This study adopted shear strain values corresponding to maximum horizontal accelerations ranging from 0.05g to 0.75g Typically, higher seismic loadings are indicated by increased horizontal acceleration (aH).
No-slip case: Mmax = 0.900 MNm/m
Full slip case: Mmax = 0.807 MNm/m
The analysis of tunnel linings reveals that under no-slip conditions, the maximum shear force (Nmax) reaches 0.791 MN/m, while in full slip conditions, it drops to 0.159 MN/m This significant difference in shear strain values (γmax) leads to heightened absolute extreme incremental bending moments and normal forces, demonstrating a linear relationship as illustrated in Figure 2.15 The figure highlights the impact of the aH value on the internal forces experienced by both circular and sub-rectangular tunnel linings.
The analysis in Figure 2.15a indicates that under no-slip conditions, the absolute extreme incremental bending moments in sub-rectangular linings are 20% greater than those in circular linings Conversely, under full slip conditions, circular linings experience approximately 4% higher absolute extreme incremental bending moments compared to sub-rectangular linings Furthermore, for sub-rectangular linings, the absolute extreme incremental bending moments under full slip conditions are consistently about 10% lower than those observed under no-slip conditions, contrasting with the behavior seen in circular-shaped tunnels.
Figure 2.15b illustrates that the absolute extreme incremental normal forces for both tunnel shapes under no-slip conditions are roughly 80% greater than those under full slip conditions Additionally, the sub-rectangular lining experiences absolute extreme incremental normal forces that are about 9% lower than those of the circular lining, regardless of the slip condition, when varying horizontal acceleration.
E xt re m e In cr em en ta l B en d in g M om en t M (M N m /m ) a H (g)
Mmax_SR_ns Mmax_SR_fs
Mmax_Circular_ns Mmax_Circular_fs
Mmin_SR_ns Mmin_SR_fs
Mmin_Circular_ns Mmin_Circular_fs
E xt rm e I n cr em en ta l N or m al F or ce s N (M N /m ) a H (g)
Nmax_SR_ns Nmax_SR_fsNmax_Circular_ns Nmax_Circular_fsNmin_SR_ns Nmin_SR_fsNmin_Circular_ns Nmin_Circular_fs
2.4.2 Effect of the soil’s Young’s modulus (Es)
Soil Young's modulus values range from 10 to 350 MPa, with K0 set at 0.5 and aH at 0.5g, while other parameters are based on the reference case (Table 2.1) Figure 2.16 illustrates the impact of varying Es values on internal forces, highlighting a) incremental bending moments and b) incremental normal forces for both circular and sub-rectangular tunnel linings.
For low elastic modulus (Es) values below 50 MPa, an increase in Es leads to higher absolute extreme incremental bending moments Conversely, when Es exceeds 50 MPa, further increases result in a reduction of these bending moments This behavior highlights the critical relationship between Es and the bending moments in sub-rectangular tunnels.
The Es value is minimal when compared to circular-shaped tunnels Notably, the extreme incremental bending moments for circular tunnels under no-slip conditions are lower than those under full slip conditions In contrast, sub-rectangular tunnels exhibit higher extreme incremental bending moments under no-slip conditions compared to full slip conditions, indicating a distinct behavior for sub-rectangular tunnels.
E xt re m e I nc re m en ta l B en d in g M om en t M (M N m /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
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m )
Nmax_SR_ns Nmax_SR_fs
Nmax_Circular_ns Nmax_Circular_fs
Nmin_SR_ns Nmin_SR_fs
Nmin_Circular_ns Nmin_Circular_fs circular-shaped tunnels The same conclusion was also obtained when considering the horizontal seismic acceleration aH
Figure 2.16a illustrates that sub-rectangular tunnels experience greater absolute extreme incremental bending moments under no-slip conditions compared to circular tunnels with equivalent utilization space areas Conversely, in full slip conditions, circular tunnels exhibit higher absolute extreme incremental bending moments than sub-rectangular tunnels when the elastic modulus (Es) is below approximately 150 MPa However, when Es exceeds 150 MPa, the bending moments in circular tunnels become less than those in sub-rectangular tunnels.
An increase in Es value leads to a notable rise in absolute extreme normal forces in both sub-rectangular and circular tunnels under no-slip conditions However, this increase has little effect on absolute extreme incremental normal forces in full slip conditions Furthermore, the absolute extreme incremental normal forces in sub-rectangular tunnels are typically 9% lower than those in circular tunnels.
2.4.3 Effect of the lining thickness (t)
The lining thickness (t) varies between 0.2 to 0.8 m, maintaining a K0 value of 0.5, aH value of 0.5g, and an Es value of 100 MPa, along with other parameters from Table 2.1 Results shown in Figure 2.17 demonstrate that lining thickness significantly impacts the incremental internal forces in both sub-rectangular and circular tunnels, under no-slip and full slip conditions Furthermore, the relationship between lining thickness and incremental internal forces for the analyzed cases is notably linear.
In the no-slip condition, the absolute extreme incremental bending moments of sub-rectangular linings exceed those of circular linings, with the difference decreasing from 124% to 6% as lining thickness increases from 0.2 to 0.8 m Under full slip conditions, for lining thicknesses less than approximately 0.5 m, sub-rectangular linings again show larger bending moments compared to circular linings However, when the lining thickness exceeds 0.5 m, the trend reverses, indicating that circular tunnels experience larger absolute extreme incremental bending moments.
Figure 2.17b illustrates that the incremental normal forces in the no-slip condition are consistently greater than those in the full slip condition When comparing the incremental normal forces of circular linings to sub-rectangular linings, the latter exhibit reductions of approximately 9% under no-slip conditions and 25% under full slip conditions This highlights the impact of lining thickness on the incremental internal forces in both circular and sub-rectangular tunnel linings.
Conclusion
A 2D numerical study examined the performance of sub-rectangular tunnel linings subjected to seismic loads This research analyzed the effects of various parameters, including soil deformation, maximum horizontal acceleration, lining thickness, and soil-lining interface conditions, on the behavior of both circular and sub-rectangular tunnels during seismic events The findings revealed significant differences in performance based on these influencing factors.
E xt re m e In cr em en ta l B en d in g M om en t M (M N m /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
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m )
Nmax_SR_ns Nmax_SR_fs
Nmax_Circular_ns Mmax_Circular_fs
Nmin_SR_ns Nmin_SR_fs
Nmin_Circular_ns Mmin_Circular_fs response of these tunnels were observed Based on the research results, conclusions can be deducted as follows:
Horizontal acceleration (aH), the soil's Young modulus (Es), and lining thickness (t) significantly influence the incremental internal forces in both sub-rectangular and circular tunnels, regardless of whether there are no-slip or full slip conditions.
Higher seismic loading, resulting from increased horizontal acceleration (aH), leads to greater incremental bending moments and normal forces in both circular and sub-rectangular tunnels, demonstrating a linear relationship between the two.
The study highlights that soil-lining interface conditions significantly affect the behavior of sub-rectangular tunnels, contrasting sharply with circular tunnels Specifically, the maximum incremental bending moments for circular tunnels are smaller under no-slip conditions compared to full slip conditions In contrast, sub-rectangular tunnels exhibit greater maximum incremental bending moments under no-slip conditions than under full slip conditions, showcasing a distinct trend from that observed in circular tunnel linings.
In all examined case studies, the absolute incremental normal forces under no-slip conditions consistently exceed those under full slip conditions for both circular and sub-rectangular tunnels Notably, the absolute extreme incremental normal forces in sub-rectangular tunnels are roughly 9% smaller than those observed in circular tunnels.
The influence of the soil's Young modulus (Es) on the absolute extreme incremental bending moments in sub-rectangular tunnels is minimal when compared to circular tunnels A critical threshold for the soil's Young modulus is identified at 50 MPa; above this value,
An increase in the soil's Young modulus (Es) leads to a notable rise in the absolute extreme incremental normal forces for both sub-rectangular and circular tunnels under no-slip conditions, while only minimal changes are seen under full slip conditions.
The findings of this study provide valuable insights for the initial design of circular and sub-rectangular tunnel linings subjected to seismic loads These results will be referenced in Chapter 3, where the impact of joint distribution on the behavior of segmental linings will be explored in future research.
A NEW QUASI-STATIC LOADING SCHEME FOR THE
Fundamental of HRM method applied to sub-rectangular tunnel under static
The calculation scheme for support structures using the HRM method under static conditions is illustrated in Figure 3.1, where σv represents vertical loads and σh denotes horizontal loads Key parameters include kn for normal stiffness of springs and ks for shear stiffness, along with EI and EA indicating the bending and normal stiffness of the support The global Cartesian coordinates are defined by X and Y.
The HRM method, rooted in the Finite Element Method (FEM), effectively analyzes internal forces and displacements in tunnel linings Initially developed for segmental and continuous tunnel linings under static loads, this method has recently been applied by Do et al to investigate the behavior of square and sub-rectangular tunnels under similar conditions The authors introduced a calculation scheme for supporting structures, demonstrating the application of the HRM method for static scenarios, as illustrated in Figure 3.1.
Figure 3.2 depicts a beam-type element capable of generating internal forces The structure's interaction with the soil is facilitated by normal and tangential springs distributed across the nodes, allowing for the calculation of stresses on each element once the nodes' displacements are determined These unknown displacements can be identified by establishing the global stiffness matrix for all structural elements and their connections to the surrounding soil.
Figure 3.2 A finite element under the local Cartesian coordinates: i: initial node; i+1: final node; θ: rotation; x and y: local Cartesian coordinates; ν: transversal displacement; u: axial displacement; Li: element length [120]
The global stiffness matrix K is assembled by the local stiffness matrix ki (i=1,2
…, n) of the i th element in the global Cartesian reference system, n is the total number of elements The global stiffness matrix K is given as follows:
The local stiffness matrix \( k_i \) of the \( i \)th element in the global Cartesian coordinate system can be derived from the 3x3 sub-matrices \( k((\cdot)) \), \( k((\cdot)) \), \( k(\cdot)(\cdot) \), and \( k(\cdot) \), representing specific components of the local stiffness matrix \( k \) for the \( n \)th element.
[𝑘] = 𝜆 ⋅ [𝑘] ⋅ 𝜆 (3.2) where [𝑘] is the local stiffness matrix under the local Cartesian reference system and λi is the transformation matrix respectively:
Once the global stiffness matrix K is established, the unknown nodal displacement vector S and the nodal force vector F within the global Cartesian reference system can be determined using the appropriate relations.
Where F = [F1, F2, …, Fn] T is the vector of the nodal forces applied to the lining;
S = [S1, S2, …, Sn] T is the vector of nodal displacements Note that S1, S2, …, Sn are the sub-vectors composed of three displacements of each node, respectively; F1, F2,
…, Fn are the sub-vectors composed of three external forces applied to each node, respectively Once the vector S is calculated, a conversion of nodal displacements at
[𝑘] the local reference system of the element is easily calculated The characteristic of nodal stresses can immediately be determined at the nodal through the local stiffness matrix
The interaction between the ground and tunnel support occurs through normal springs (kn) and tangential springs (ks) connected to the structure's nodes, along with applied active loads The values of kn and ks can be derived from the normal and tangential ground stiffness (ηn and ηs, respectively) Unlike traditional methods that assume a constant ground stiffness, Oreste introduced a non-linear (hyperbolic) relationship between reaction pressure (p) and support displacement (δ).
Figure 3.3 Nonlinear relationship between the reaction pressure p and the support normal displacement δ: 𝜂 : initial spring stiffness; plim: maximum reaction pressure [121]
The maximum reaction pressure that the ground can provide, denoted as plim, and the initial stiffness of the ground, represented by 𝜂, form a fundamental relationship that effectively describes ground behavior when both parameters are known with confidence Conducting a plate load test reveals a load-displacement curve that closely resembles a hyperbolic shape, indicating the ground's response characteristics.
The apparent stiffness η* of the ground is given by the p/δ ratio that can be calculated at each node of the support structure:
The ground reaction is primarily influenced by the elasticity parameters of the ground and the radius of the tunnel This study utilizes a specific formula to calculate the initial normal stiffness of the ground.
In this context, νs represents the Poisson’s ratio, while Es denotes Young’s modulus of the soil The radius of each section of the tunnel boundary is indicated by Ri, with i corresponding to the crow, shoulder, and side wall of the tunnel (i=1, 2, and 3) Additionally, β is defined as a dimensionless factor.
This study examines the role of tangential springs, highlighting the challenges in estimating the frictional or shear stiffness at the ground support interface A straightforward relationship between normal stiffness (𝜂) and tangential stiffness (𝜂) is proposed for better understanding.
For non-cohesion soil, the maximum reaction pressure plim can be calculated based on the friction angle φ, Poisson’s ratio νs, and active loads [46],[48],[121]
Where Δ𝜎 conf is the confining pressure that acts on the tunnel perimeter It can be defined by the following equation: Δ𝜎 conf = ⋅ (3.11)
The value of the maximum shear reaction pressure ps,lim can be preliminarily estimated as followed:
𝑝 , = ⋅ tg𝜑 (3.12) where 𝜎 𝑎𝑛𝑑 𝜎 are the vertical and horizontal loads, respectively
The normal and shear stiffness of each spring can then be given by the formula
The internal forces in the tunnel lining are calculated using Eqs 3.1 to 3.14, which involve nodal displacements and the local stiffness matrices of each element It is important to note that normal springs will be absent in areas where the support structure shifts toward the tunnel, allowing only compressive loads in the normal direction as the tunnel support moves downward For further information on the HRM method, please refer to the relevant literature.
HRM method applied to sub-rectangular tunnel under seismic conditions
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
Applying shear stress to the far-field boundary leads to sub-ovaling deformation of the tunnel lining under seismic loading, as illustrated in Figure 3.4c This finding is derived from a finite-difference model (FDM), with incremental internal forces displayed in Figure 3.5, referencing Figure 2.14 for the no-slip condition.
In the HRM model, external loads are applied directly along the tunnel lining to account for seismic loading, rather than relying on shear stresses at the far-field boundary This study aims to introduce a loading scheme for the tunnel lining, illustrated in Figure 3.4c, to achieve sub-ovaling deformation, which is critical for sub-rectangular tunnels under seismic conditions Key points of interest include Point A, where the sidewall meets the shoulder; Point B, situated at the center of the shoulder; and Point C, where the crown intersects with the shoulder, as depicted in Figure 3.6.
Induced internal forces and deformations in tunnel linings are primarily affected by external loads and soil-lining interactions, taking into account the initial stiffness of the springs In the HRM method, normal springs function only under compression when the tunnel lining shifts towards the surrounding ground Compressive external loads applied in one direction generate tensile loads in the perpendicular direction Therefore, parameters (a) and (b) are essential for adjusting the external loads on sub-rectangular tunnel linings during seismic events, as illustrated in Figure 3.6.
The HRM method's equivalent static loading scheme is established by adjusting the loads on 360 nodes of the sub-rectangular tunnel lining until the incremental bending moment and normal forces align with those derived from the FDM model To implement and calibrate the HRM method effectively, a simplified chart format of the loading scheme is selected It is important to note that the sub-rectangular tunnel lining in this study is segmented into 360 elements, each with a maximum length of 0.17m, focusing on the parameters of incremental bending moment and 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 HRM, the interaction between the ground and tunnel support is facilitated through normal and tangential springs linked to the lining structure's nodes, denoted as kn and ks, and evaluated using the ground's initial stiffness, η0 Additionally, in sub-rectangular tunnels, the radius of the lining components varies along the tunnel's periphery.
Mmax = 0.900 (MNm/m) Nmax = 0.791 (MN/m) the initial stiffness of the ground η0 will then change depending on the radius as in
In static analyses, the dimensionless factor (β) impacting spring stiffness is typically assigned a value of 1 or 2 However, recent research by Sun et al introduced a method to estimate β based on the properties of soil and tunnel lining for circular tunnels under seismic loading This study also employs a varying dimensionless factor (β) to more accurately depict the interaction between soil and tunnel.
This study performs a numerical parametric analysis to identify the values of three dimensionless parameters: a, b, and β The findings derived from the HRM method are then validated against those produced by the FDM model, taking into account a wide variety of soil properties, lining characteristics, and tunnel geometries.
Numerical implementation
In this section, we utilize the FDM numerical model developed in Chapter 2 within FLAC 3D to calibrate the three dimensionless parameters (a, b, and β) essential for the HRM method Additionally, we present the numerical procedure for implementing the HRM in the context of sub-rectangular tunnels subjected to seismic loading, as detailed in Table 3.3 and illustrated in Figure 3.8.
Chapter 2 presents a 2D plane strain model using FLAC 3D, illustrating the geometry parameters of sub-rectangular tunnels in Figure 2.9 The soil and lining parameters utilized are detailed in Table 3.1 and Table 3.2, with relevant results found in section 2.4 and Figure 3.7 Additionally, it is important to note that the calibration process aims to determine the 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
Peak horizontal acceleration at ground surface aH g 0.5
Distance of site source Km 10
Table 3.2 Geometrical parameters of tunnel shape cases [48]
Figure 3.7 Shapes of tunnel cases (unit: m) [48]
3.3.2 Numerical procedure in HRM method
To effectively apply the HRM method for sub-rectangular tunnels under seismic loading, it is essential to establish the formulas for the three dimensionless parameters (a, b, and β) that characterize the external loads on the tunnel lining Comprehensive comparisons between the seismic-induced incremental internal forces predicted by the HRM method and those calculated using the FDM numerical model are performed to validate the approach.
During the initial calibration phase, parameters outlined in Table 3.1 were utilized, allowing for adjustments to tunnel dimensions to encompass various scenarios The tunnel width (w) was modified from 9.7 m to 15.52 m to create wider tunnels that maintain a uniform shape, as depicted in Figure 2.1, resulting in a t/w ratio ranging from 0.032 to 0.052 Additionally, various sub-rectangular shapes (SR1 to SR4) were implemented, with their dimensional characteristics detailed in Table 3.2 and Figure 3.7 The maximum horizontal acceleration was set at aH = 0.5g, leading to shear strains (γc) of 0.38% The calibration of the three parameters (a, b, and β) was then conducted, with the calibration process outlined in Table 3.3 and Figure 3.8.
Table 3.3 Overview of the calibration process
1 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
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
3 Determination of the relative error of incremental normal forces and bending moments obtained by two methods
4 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
5 Steps 2 to 4 repetitions until all cases scenarios of defined parameter ranges listed in Table 3.1 and Table 3.2 are considered
6 Determination of the formulas describing a, b, and β as functions of ti, hi, wi, Esi parameters by using regression analysis
Figure 3.8 Calibration flowchart of the three parameters
Upon completing the calibration process, equations can be formulated to represent the impact of three key parameters on soil, lining properties, and tunnel dimensions, based on optimal fitting as illustrated in Figures 3.9 and 3.10 The parameters β, a, and b are defined accordingly.
Generating soil and lining parameters { , } for all cases
Initial and computation using HRM
{ , } and computation using numerical solution
The coefficient \( a \) is determined solely by the soil’s Young’s modulus (Es), while coefficients \( \beta \) and \( b \) depend on the lining thickness (t), tunnel height (h), tunnel width (w), and Es As illustrated in Figure 3.9a, \( \beta \) increases significantly as Es rises from 10 to 150 MPa, but shows minimal growth beyond 150 MPa Conversely, parameters \( a \) and \( b_1 \) in Figures 3.10a and 3.10b exhibit a sharp decline when Es varies between 10 and 100 MPa, stabilizing at higher Es values Additionally, parameters \( \beta, \beta, \beta, b_2, b_3, \) and \( b_4 \) are heavily influenced by the ratios of t/h, t/w, and h/w, as depicted in Figures 3.9b, c, d and 3.10c, d, e It is important to note that the units for the parameters in Equations 3.15 to 3.25 align with those in Tables 3.1 and 3.2.
For a tunnel with an elastic modulus (Es) of 75 MPa and a thickness-to-height (t/h) ratio of 0.07, the values of b1 and b2 are zero, indicating no additional seismic loading on the tunnel lining This scenario occurs when the tunnel structure is more flexible than the surrounding ground, leading to amplified distortions in the tunnel lining compared to the shear distortions in the free field In contrast, when the elastic modulus is less than 75 MPa and the t/h ratio exceeds 0.07, the tunnel lining becomes stiffer than the ground, effectively resisting ground displacements.
Figure 3.9 Obtained numerical results and fitting curves adopted for the parameters β1, β2, β3 and β4 that created the parameter β
Figure 3.10 Coefficients fitting curves for the formulas of the parameters a and b1, b2, b3 and b4 that created the parameter b
Figure 3.11 illustrates a comparative analysis of incremental bending moments and normal forces in a sub-rectangular tunnel lining under seismic loading, with parameters set at Es = 100MPa and t = 0.5m, as referenced in Table 3.1 The results indicate minimal differences between the extreme incremental internal forces derived from the HRM method and the FDM model, with discrepancies of only 1.2% for bending moments and 0.6% for normal forces The variations in incremental bending moments and normal forces at the tunnel lining's top and bottom can be attributed to the simplified equivalent external loading depicted in Figure 3.6.
Validation of the HRM method
Extensive validations were conducted to confirm the effectiveness of the developed HRM method The initial validation focused on assessing the accuracy of the HRM method by analyzing various peak horizontal seismic acceleration (aH) values Subsequent validations involved altering Young’s modulus of the soil and the lining thickness Additionally, uniform tunnels with diverse cross-sections were examined to further validate the method.
In the study, the maximum values of Nmax for the Finite Difference Method (FDM) and the Hybrid Response Method (HRM) are reported as 0.791 MNm/m and 0.786 MNm/m, respectively Various sub-rectangular shapes based on tunnel geometrical parameters from Table 3.2 are utilized in validation 5, while validation 6 examines the impact of tunnel burial depth on lining behavior Validation 7 incorporates soil parameters from Hashash et al.'s research Throughout these validations, the seismic-induced incremental internal forces calculated by the HRM method are systematically compared with numerical solutions from the FDM and Finite Element Method (FEM).
3.4.1 Validation 1 a) Extreme incremental bending moments b) Extreme incremental normal forces Figure 3.12 Horizontal accelerations aH impact on extreme incremental internal forces of the sub-rectangular tunnel lining
The formulas for parameters a, b, and β, outlined in section 4.4, are based on a maximum horizontal acceleration (aH) of 0.5g This validation assumes that aH can range from 0.05 to 0.5g, which corresponds to shear strains (γmax) of 0.038% to 0.38% Additionally, the reference case parameters from Table 3.1 are utilized A comparison is made between the developed HRM method and FDM calculations, focusing on extreme incremental bending moments and normal forces.
E xt re m e In cr em en ta l B en di n g M om en t M ( M N m /m ) a H (g)
Mmax_FDM Mmin_FDM Mmax_HRM Mmin_HRM
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m ) a H (g)
Nmax_FDMNmin_FDMNmax_HRMNmin_HRM
The developed HRM method demonstrates strong agreement with numerical FDM results, as illustrated in Figure 3.12 Both methods effectively capture the trend that absolute extreme incremental bending moments and normal forces increase with rising maximum horizontal acceleration (aH) The discrepancies between the two methods are minimal, at under 2.2% for bending moments and 2% for normal forces This indicates that the HRM method is applicable across a broader range of horizontal accelerations.
Validation 2 is conducted for the soil Young’s modulus variation in a range from 10 to 350 MPa The lining thickness equals 0.5m and the other soil parameters, based on the reference case study and listed in Table 3.1 were used a) Extreme incremental bending moments b) Extreme incremental normal Forces Figure 3.13 Effect of Es on the extreme incremental internal forces of the sub- rectangular tunnel lining
Figure 3.13 illustrates a strong correlation between the extreme incremental bending moments and normal forces in tunnel lining, as calculated by the HRM method and the numerical FDM, particularly when accounting for changes in Es values The analysis indicates that the extreme incremental bending moments are significantly influenced by variations in Es.
E xt re m e In cr em en ta l B en d in g M om en t M ( M N m /m )
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m )
In Figure 3.13a, it is observed that when the elastic modulus (Es) is below 50 MPa, a decrease in Es leads to a reduction in the absolute extreme incremental bending moments Conversely, for Es values exceeding 50 MPa, an increase in Es results in lower absolute extreme incremental bending moments The HRM method demonstrates high accuracy in predicting these moments, with discrepancies from numerical FDM results generally under 3.5% Notably, for an Es value of 25 MPa, the differences rise to 6.8% and 7.2% for maximum and minimum incremental bending moments, respectively, suggesting potential issues with the accuracy of the fitting curves for parameters a, b, and β, as illustrated in Figures 3.9 and 3.10.
The tunnel lining thickness is assumed to vary between 0.3 to 0.8 m while the other parameters are based on the reference case described in Table 3.1 and Es is of
Figure 3.14 illustrates the comparison between the HRM method and numerical FDM techniques, highlighting the extreme incremental bending moments and normal forces The data emphasizes the impact of lining thickness on the extreme incremental internal forces within sub-rectangular tunnel linings, showcasing both bending moments and normal forces.
E xt re m e In cr em en ta l B en di ng M om en t M ( M N m /m )
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m )
The extreme incremental internal forces calculated using the HRM method show a consistent trend with numerical computations, as both absolute extreme normal forces and bending moments increase linearly with lining thickness However, variations in incremental normal forces due to lining thickness changes are less pronounced compared to those of bending moments Additionally, the HRM method's results align closely with numerical FDM computations, with discrepancies remaining under 2% for both extreme incremental bending moments and normal forces.
The validation process reveals that the uniform tunnel cross-section sizes vary from 1.0 to 1.6 times the original dimensions, resulting in a cross-section width (w) that ranges from 9.7 to 15.52 meters, as depicted in Figure 2.1 Other parameters, detailed in Table 3.1, maintain Es at 100 MPa Figure 3.15 illustrates the extreme incremental internal forces obtained via HRM in comparison to numerical FDM computations, highlighting the impact of cross-section dimensions on extreme incremental bending moments and normal forces in the sub-rectangular tunnel lining.
E xt re m e In cr em en ta l B en d in g M om en t M (M N m /m )
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m )
Tunnel width (m)Nmax_FDM Nmin_FDMNmax_HRM Nmin_HRM
The results in Figure 3.15 demonstrate a strong correlation between the HRM method and numerical modeling using FDM for extreme incremental internal forces Specifically, Figure 3.15a shows that the extreme incremental bending moments remain nearly constant as the tunnel width increases from 9.7 to 15.52 m, with a discrepancy of less than 1.7% between the HRM method and numerical computations In contrast, the absolute extreme incremental normal forces exhibit a linear increase with tunnel width, with a difference of under 1.4% between the HRM and FDM calculations, highlighting the HRM method's efficiency in varying tunnel widths.
The efficiency of the developed HRM method was validated against various tunnel geometries listed in Table 3.2, with other parameters specified in Table 3.1 (Es = 100 MPa, t = 0.5m) Figure 3.16 illustrates the extreme incremental internal forces obtained from HRM, compared to those derived from the numerical FDM method In the figure, different tunnel widths are utilized to differentiate between the various tunnel geometries.
Figure 3.16 illustrates the impact of tunnel shapes on extreme incremental normal forces and bending moments The extreme incremental bending moments experience a slight increase of 6% when transitioning from tunnel shape SR1 to SR4, while the extreme incremental normal forces remain nearly unchanged The extreme incremental normal forces calculated using the HRM method closely align with numerical results from the FDM, showing a discrepancy of less than 3% across all cases In comparison, the HRM method provides precise predictions for extreme incremental bending moments, with differences across all cases remaining under 2%.
Validation 6 is conducted considering the variation of burial depth of tunnel in a range from 5m to 20 m The other parameters of the tunnel lining and soil based on the reference case study listed in Table 3.1 (Es = 100 MPa, t = 0.5m) were used The results of the extreme incremental internal forces obtained by HRM compared with the numerical FDM ones are shown in Figure 3.17
The analysis presented in Figure 3.17 reveals a significant reduction in extreme incremental bending moments, decreasing by 24% as the tunnel burial depth increases from 5 to 20 meters Similarly, extreme incremental normal forces also show a gradual decline, with an 18% reduction observed at a burial depth of 5 meters compared to 20 meters Additionally, the extreme incremental internal forces calculated using the proposed HRM method closely align with numerical FDM computations, exhibiting discrepancies of less than 6% for normal forces and 4% for bending moments.
E xt re m e In cr em en ta l B en d in g M om en t M (M N m /m )
E xt re m e In cr em en ta l N or m al F or ce s N (M N /m )
Conclusions
This study introduces a novel numerical procedure utilizing the HRM method to effectively analyze the behavior of sub-rectangular tunnel linings under seismic loading The HRM method is comprehensively detailed, taking into account a wide array of soil parameters, lining properties, and tunnel geometries Additionally, the computation process is enhanced through a parametric analysis, complemented by a quasi-static loading scheme specifically applied to the sub-rectangular tunnel lining.
The developed HRM method underwent extensive validation through a series of numerical computations, comparing its results with a quasi-static numerical FDM model The comparative analyses demonstrated that the HRM method is effective for the preliminary seismic design of sub-rectangular shaped tunnels.
The current research indicates that when a tunnel structure is more flexible than the surrounding soil, the tunnel lining undergoes greater distortions than the shear distortions experienced by the soil in the free field Conversely, if the tunnel lining is stiffer than the adjacent soil, it is more effective at resisting ground displacements.
The proposed HRM method offers an innovative and efficient approach for the seismic design of sub-rectangular tunnels, leveraging advancements in numerical models and computing capabilities It is important to note that this method operates under the assumption of an elastic soil-tunnel configuration influenced by ground shaking, while it does not account for the nonlinearities of the soil.
The present thesis allowed achieving several conclusions that represent innovative contributions to the knowledge of the sub-rectangular tunnels considering seismic loadings
Chapter 1 reviewed the behavior of tunnels under seismic loadings, highlighting significant advancements in understanding and predicting the seismic response of both circular and rectangular tunnels However, the effects of earthquake-induced ground failures on tunnel performance remain inadequately explored.
Most research on seismic tunnel response primarily examines the effects of S waves on circular and rectangular tunnels under plane strain conditions This methodology effectively predicts seismic lining forces during transverse seismic loading Overall, the behavior of tunnels under seismic conditions can be analyzed through various approaches, including analytical methods, physical model tests, and numerical modeling.
Analytical solutions offer simplicity and speed but are constrained by oversimplified assumptions In contrast, physical model tests and numerical analyses address these limitations, although the complexity and expense of physical tests often yield limited results Recently, numerical models have gained popularity due to their effectiveness and reliability, particularly in full seismic analyses These models are valuable for studying spatially-variable ground motion in long tunnels, variations in layer boundaries among different geomaterials, changes in structural properties along the tunnel, including station boxes, and near-fault effects However, the high computational costs associated with numerical simulations typically restrict their use to academic and research settings.
Recent studies have focused on sub-rectangular tunnels through onsite and laboratory tests, highlighting their advantages over traditional circular and rectangular tunnel cross-sections in terms of space utilization However, it is important to note that these evaluations have thus far only addressed static loading conditions.
A 2D numerical study examined the behavior of sub-rectangular tunnel linings under seismic loads, focusing on key parameters such as soil deformations, maximum horizontal accelerations, lining thicknesses, and soil-lining interface conditions The investigation revealed significant differences in the response of circular and sub-rectangular tunnels during seismic events The findings highlight the importance of these parameters in understanding tunnel behavior under seismic loading.
The horizontal acceleration (aH), the soil's Young modulus (Es), and the lining thickness (t) significantly influence the incremental internal forces experienced in both sub-rectangular and circular tunnels, regardless of whether the conditions are no-slip or full slip.
Higher seismic loading, characterized by increased horizontal acceleration (aH), leads to greater incremental bending moments and normal forces in both circular and sub-rectangular tunnels, demonstrating a linear relationship between these factors.
The study revealed that the interface conditions between soil and lining significantly affect the behavior of sub-rectangular tunnels, contrasting sharply with circular tunnels Specifically, while circular tunnels exhibit smaller absolute extreme incremental bending moments under no-slip conditions compared to full slip conditions, sub-rectangular tunnels show the opposite trend, with greater bending moments under no-slip conditions than under full slip conditions This highlights a fundamental difference in the structural response of these two tunnel shapes.
In all examined case studies, the absolute incremental normal forces under no-slip conditions consistently exceed those under full slip conditions for both circular and sub-rectangular tunnels Additionally, the absolute extreme incremental bending moments experienced by sub-rectangular tunnels are influenced by the soil's Young modulus.
The Young's modulus of soil (Es) has a minimal impact when compared to circular tunnels However, an increase in Es leads to a notable rise in the absolute extreme incremental normal forces for both sub-rectangular and circular tunnels under no-slip conditions Conversely, under full slip conditions, the changes in absolute extreme incremental normal forces remain insignificant.
The numerical results obtained in the present study are useful for the preliminary design of circular and sub-rectangular shaped tunnel linings under seismic loadings
Chapter 3 proposed a new numerical procedure to calculate the seismic design in a sub-rectangular tunnel lining due to seismic loading, using the Hyperstatic Reaction Method (HRM) The mathematical formulas of the HRM method are introduced Then, the basic assumptions and algorithms for implementing the HRM method under seismic conditions were presented in details