LITERATURE REVIEW ON THE BEHAVIOUR OF
Introduction
Tunnels are vital for urban and national transportation and utility systems, with their construction increasing to meet the demands of densely populated areas Given their significant scale, construction costs, and essential functions, these infrastructures are crucial to modern society Even minor seismic loading can cause temporary shutdowns and considerable damages, making it essential to thoroughly evaluate the impact of seismic activity on 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 surrounding soil and rock Historical earthquake events, such as those in Kobe (1995), Chi Chi (1999), Bolu (1999), Baladeh (2004), and Sichuan (2008), have documented damage to underground constructions, highlighting a lack of systematic investigation into the interactions between ground motion and tunnels Wang (1993) emphasized the importance of relative flexibility in understanding the seismic-induced distortions of underground structures These incidents underscore that inadequately designed underground structures are vulnerable to the effects of wave propagation.
Comprehensive reviews of the seismic performance of tunnels and underground structures are available in various publications These studies utilize significant incidents as standard cases to analyze the interaction between structures and the ground, aiming to validate design methods Current analyses and validations of material models and interface conditions are essential for accurately capturing observed responses However, there remains a critical 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 situated in a unique location on the Earth's tectonic map, characterized by a complex and diverse network of high-risk earthquakes Various studies focus on earthquake statistics, localization, forecasting, risk assessment, and design to better understand and mitigate these seismic threats.
Figure 1.2 Typical failure modes of mountain tunnels reported during the 1999
Chi-Chi earthquake in Taiwan [160].
Seismic response mechanisms
The effects of earthquakes on underground structures can be classified into two main categories: ground shaking and ground failure, or expanded into four categories including land sliding and soil liquefaction Ground shaking is characterized by the vibrations caused by seismic waves traveling through the Earth's crust Understanding these effects is crucial for assessing the impact of seismic activity 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 due 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 responding to external 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 parallel to the tunnel's axis, resulting in alternating compressive and tensile forces Additionally, bending strains arise from 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 Ovaling or racking deformations, which can occur in circular and rectangular tunnel linings, arise when shear waves propagate nearly perpendicular to the tunnel axis Research by Penzien and Hashash et al has shown that these ovaling or racking deformations, induced by seismic shear wave (S-wave) propagation, significantly impact tunnel lining behavior under seismic loading, making them the most critical deformation modes for tunnels.
Earthquake-induced ground failures can result from liquefaction, fault movements, or slope failures, leading to significant 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 typically modeled using 2-dimensional plane strain problems with equivalent static loads, often neglecting inertial force effects To simplify this analysis, various analytical methods have been created to calculate internal forces in supporting structures, particularly for rectangular and circular tunnels However, these analytical approaches are frequently constrained by underlying assumptions.
- 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.
Numerous authors have conducted physical models to explore the operational mechanisms of underground structures and to assess existing design and analysis methodologies These models primarily aim to gather measurement data and validate design frameworks However, the complexity and expense associated with this approach have led to a limited number of 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 they share significant drawbacks with traditional analytical approaches A key limitation is that these methods overlook the time-dependent changes in tunneling performance under seismic conditions Additionally, calculations based on equivalent static loads often yield lower internal force estimates compared to those derived from actual seismic loadings.
The time history analysis method, which utilizes real earthquake data, is the most complex among various analytical techniques, yet it delivers the most precise results 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, overlooking the significance of sub-rectangular tunnel cross-sections Understanding the internal forces and deformations in these structures under seismic loads, as well as the influencing factors such as tunnel lining parameters and soil conditions, is crucial for both practical applications and scientific advancement In earthquake-prone areas, there is a growing trend to utilize highly flexible structures like segmental tunnel linings Initial findings indicate that the joints in tunnel structures play a vital role, highlighting the need for comprehensive studies to ensure the reliability of 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 been developed to assess internal forces on circular tunnel linings subjected to seismic loading, highlighting discrepancies between various methods Research by Hashash et al utilized numerical analyses to clarify these differences, revealing that Wang’s solution accurately estimates tunnel lining normal forces under no-slip conditions Conversely, Peinzen’s solution is not recommended for such scenarios Furthermore, studies have shown strong agreement between their proposed solutions and existing models, reinforcing the reliability of these 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 a radius R, situated beneath the ground surface, experiences seismic loading through shear waves, leading to a shear-type stress state in the surrounding soil This condition results in compressive and tensile principal stresses oriented at 45 degrees, aligning with the 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 sediment types in the table indicate specific low-strain shear wave velocity ranges: rock has a velocity of 750 m/sec or greater, stiff soil ranges from 200 m/sec to 750 m/sec, and soft soil has a velocity of less than 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 a circular tunnel lining occurs at its major and minor axes 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 illustrates the relationship between ground motion at depth and at the surface, while Table 1.2 connects 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 calculating structural tunnel lining forces subjected to seismic loading conditions Under full-slip conditions of soil-lining interaction, the maximum normal forces (Nmax) and maximum bending moment (Mmax) can be determined using 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 a no-slip condition However, it is proposed that the solution for a full slip condition could be applicable to a no-slip scenario The conservative estimates derived from the full slip condition 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
In addition to analyzing ovaling deformations in circular tunnels, it is essential to assess rectangular tunnels for imposed racking deformations caused by propagating shear waves Furthermore, the walls and roof of the tunnel's cross-section must be evaluated for seismic earth pressures to ensure structural integrity during seismic events.
Figure 1.8 Definition of terms used in racking analysis of a rectangular tunnel
Wang [159] introduced a simplified method for analyzing the racking deformation of rectangular tunnels, incorporating soil-structure interaction This approach was based on a series of seismic finite element analyses that considered various seismic properties of both the soil and the tunnel structure The analysis covered 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 Stiffness = S.h/w
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 excavating tunnels with circular cross-sections However, these circular designs may not always meet the specific requirements for underground space utilization As a solution, non-circular TBMs have emerged as an innovative option, offering enhanced adaptability for various cross-section shapes Recent studies have focused on special cross-section tunnels, analyzing their effective ratios and reduction capabilities to optimize space utilization in tunneling projects.
Figure 1.14 (a) Overlap cutter heads; (b) a copy cutter head [78]
Jianbin Li highlighted the technical features and current advancements of non-circular tunnel boring machines (TBMs), including special-shaped segment lining and soil conditioning in loess strata, specifically for pipe-jacking machines with rectangular cross-sections and horseshoe-shaped TBMs The study offered practical solutions for the design, manufacturing, and construction of non-circular TBMs Engineering practices demonstrated that these customized non-circular TBMs significantly enhance construction schedules, settlement control, and space utilization.
Figure 1.15 A photo showing the testing setup after fabrication [72]
Huang et al [72] 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 their own weight They addressed the challenges involved in fabricating both the testing setup and the segmental linings The experimental findings were compared with numerical simulations, revealing the necessity of considering self-weight effects in structural loading tests, particularly for shallow-buried tunnels Future research should focus on the mechanical behavior of this rectangular lining structure under various influential factors post-assembly (Figure 1.15).
Recent research by Do et al focuses on enhancing the performance of the HRM method for squared and sub-rectangular tunnels, emphasizing the impact of tunnel shape through varying wall radii Their numerical HRM model was validated against finite element method (FEM) comparisons, followed by a parametric study examining how the earth pressure coefficient and soil Young's modulus affect structural forces and deformations during tunnel excavation Additionally, Du et al utilized the HRM method to optimize sub-rectangular tunnels concerning lining forces, investigating the influence of parameters like lateral earth pressure coefficient, soil Young's modulus, tunnel depth, and surface loads on internal forces and tunnel shape Furthermore, Zhang et al explored the effect 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 achieved in understanding and predicting the seismic behavior of tunnels, primarily for circular and rectangular shapes However, the seismic responses of tunnels with different geometries during earthquake-induced ground failure remain underexplored Most research has concentrated on the transverse seismic response of tunnels to S waves, often under the assumption of plane strain conditions, which effectively predicts seismic lining forces under such loads Additionally, only a limited number of studies have examined the impact of surface waves on seismic tunnel responses, typically considering their effects to be minimal.
Tunnels' behavior under seismic loads can be examined through various methods, including analytical approaches, physical model tests, and numerical modeling While analytical solutions are quick and straightforward, they rely on oversimplified assumptions Physical model tests and numerical analyses address these limitations, but the high costs and complexities of physical tests often yield restricted results Currently, numerical modeling is the most favored method, providing reliable outcomes for comprehensive seismic analyses These models are particularly effective in studying spatially-variable ground motion in long tunnels, variations in geomaterial boundaries, structural property changes along the tunnel, and near-fault effects However, the high computational demands of 3D numerical simulations typically confine their use to academic and research settings.
Recent research has explored the use of sub-rectangular tunnels in both on-site and laboratory tests, highlighting their potential to address the limitations of traditional circular and rectangular tunnel designs However, existing literature primarily examines the performance of sub-rectangular tunnels under static loads, with a notable lack of studies addressing their behavior under seismic loading This thesis aims to investigate this gap in knowledge.
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 continuous segmental concrete lining of 0.5 meters 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 tunnel linings under quasi-static loading, with results compared to those 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 design incorporates R9450 liner elements that are affixed to the zone faces along the tunnel boundary using specific interfaces Two interface conditions are analyzed: one with no slip between the soil and the lining, and another with full slip The liner-zone interface stiffness, including normal stiffness (kn) and tangential stiffness (ks), is calculated based on established guidelines, where kn and ks are set to 100 times the equivalent stiffness of the stiffest adjacent zone for the no-slip scenario In contrast, for the full-slip condition, ks is designated as zero The apparent stiffness of a zone, measured in stress-per-distance units, follows a defined formula in the direction normal to the surface.
(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 expanding away from the tunnel, 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 seismic loading scenarios, ovaling deformations are represented by inverted triangular displacements along the lateral boundaries of the model, as noted by Naggar and Hinchberger 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 bottom of the model is constrained in all directions to ensure stability during analysis.
Before applying ovaling deformation from seismic loading, a steady state of the excavated tunnel was established under static conditions In a 2D plane strain model, ground displacements around the tunnel boundary before lining installation were simulated using the convergence confinement method with a relaxation factor, λd, set to 0.3 The numerical simulation of tunnel ovaling was 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 focuses exclusively on incremental internal forces, which are calculated by subtracting the total lining forces obtained after static loading (step 2) from those measured at the conclusion of the ovaling step.
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 analytical solution developed by Wang and enhanced by Kouretzis et al was utilized for comparison This solution has proven effective in the seismic design of circular tunnels, highlighting its relevance in structural analysis.
A new expression for the maximum incremental bending moment under no-slip conditions was proposed, which was not previously addressed by Wang This study utilized Wang's formulation to calculate 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, with the soil and tunnel lining materials assumed to be linearly elastic and massless These assumptions were consistent with the analytical solution, and 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 illustrated in Figures 2.5 and 2.6 Notably, the ovaling deformation of the tunnel lining due to seismic loading is evident Figure 2.7 depicts the distribution of incremental internal forces acting on the tunnel lining under seismic conditions The analysis considers lining and soil interaction 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 used in the model are detailed in Table 2.1, demonstrating strong agreement between numerical and analytical results.
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 investigates the response of circular tunnel linings under quasi-static loads, focusing on the influences 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 encompass both the maximum and minimum bending moments experienced Additionally, it highlights extreme incremental normal forces, representing the maximum and minimum normal forces exerted on the tunnel lining.
2.2.1 Effect of the peak horizontal seismic acceleration (aH)
A parametric study was performed to examine the effects of seismic loading magnitude, specifically focusing on maximum horizontal acceleration (aH), which varies from 0.05 to 0.75 g This range corresponds to maximum shear strains (γmax) between 0.038% and 0.58% Using the reference parameters outlined in Table 2.1, significant conclusions can be drawn from Figure 2.8, highlighting the impact of aH on extreme incremental bending moments and normal forces in circular tunnel linings.
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, revealing a discrepancy of about 1% in the 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 show a gradual rise The incremental bending moments for both no-slip and full slip conditions are significantly influenced by the aH value, as illustrated in Figure 2.8a In contrast, while the incremental normal forces in the tunnel lining under no-slip conditions are greatly affected by changes in aH, variations in normal forces due to aH are minimal for the full slip condition, as depicted in Figure 2.8b.
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, with additional parameters derived from a reference case study detailed in Table 2.1 Numerical results, obtained through Flac 3D and compared with the analytical Wang’s method under both full slip and no-slip conditions, are illustrated in Figure 2.9.
Figure 2.9 demonstrates a strong correlation between the incremental bending moments and normal forces experienced by 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 difference observed being less than 2%.
The extreme incremental bending moments are significantly influenced by the Es value, as illustrated in Figure 2.9a Notably, the highest absolute values of these moments occur at Es values near 50 MPa 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 pressure values ranging from 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 distortion compared to soil shear distortions in the free field An increase in the modulus of elasticity (Es) leads to a reduction in the absolute extreme incremental bending moments This relationship holds true 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 under full slip conditions.
In the context of tunnel lining mechanics, it is observed that while extreme incremental normal forces under full slip conditions show minimal dependence on the Es value, an increase in Es leads to a significant rise in both maximum and minimum incremental normal forces for no-slip conditions Notably, incremental normal forces in no-slip scenarios consistently exceed those in full slip conditions, highlighting the critical impact of Es on the internal forces within circular tunnel linings, as illustrated in Figure 2.9.
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
The results in Figure 2.10 demonstrate a strong correlation between analytical and numerical models regarding Young’s soil modulus Es and horizontal seismic acceleration aH effects, with discrepancies of less than 1% for both incremental bending moments and normal forces under no-slip and full slip conditions.
The analysis reveals that as the lining thickness increases from 0.2 to 0.8 m, both the absolute extreme incremental bending moments and normal force values rise This trend is observed under both full slip and no-slip conditions, with the no-slip condition consistently showing lower incremental bending moments compared to the full slip scenario Notably, a maximum difference of 14% was recorded at a lining thickness of 0.8 m Additionally, the variations in incremental normal forces due to changes in lining thickness are less pronounced than those of the incremental bending moments.
The comparison of the analytical solution and 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 demonstrates the effectiveness of both approaches in understanding the structural behavior 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 static and seismic loadings The model incorporates a modified sub-rectangular geometry while accounting for gravity effects The mesh features a single layer of zones in the y-direction, with element dimensions increasing away from the tunnel Geometry parameters for the sub-rectangular tunnels are illustrated in Figure 2.1, and additional parameters are detailed in Table 2.1.
The model measures 120 meters in width (x-direction) and 40 meters in height (z-direction), comprising approximately 5,816 elements and 11,870 nodes The base of the model is constrained in all directions, while the vertical sides are secured in the horizontal plane.
Parametric study of sub-rectangular tunnels in quasi-static conditions
The prescribed displacement γ max for tunnel linings under seismic loadings 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 assessment.
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, as illustrated in Figure 2.13, which presents the deformed model alongside displacement contours The figure highlights the incremental bending moments and normal forces acting on the structure Additionally, Figure 2.14 details the distribution of these incremental bending moments and normal forces, providing a comprehensive understanding of the tunnel's response to applied loads.
Figures 2.14 and 2.7 illustrate the behavior of circular and sub-rectangular tunnel linings under seismic loads, highlighting the locations of extreme incremental internal forces at the tunnel periphery Notably, Figure 2.14 reveals that the highest bending moments and normal forces in the sub-rectangular tunnel occur at the corners with smaller lining radii A subsequent numerical investigation compares the seismic performance of sub-rectangular and circular tunnels, both designed with the same utilization space area and subjected to seismic loads This study 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 Generally, higher horizontal accelerations (aH) indicate increased seismic loadings.
No-slip case: Mmax = 0.900 MNm/m
Full slip case: Mmax = 0.807 MNm/m
The analysis of tunnel linings reveals that the no-slip case exhibits a maximum shear force (Nmax) of 0.791 MN/m, while the full slip case shows a significantly lower maximum shear force of 0.159 MN/m These varying shear strain values (γmax) lead to substantial extreme incremental bending moments and normal forces, indicating a linear relationship between them, as illustrated in Figure 2.15 This figure demonstrates the impact of the aH value on the internal forces experienced by both circular and sub-rectangular tunnel linings.
The findings in Figure 2.15a indicate that under no-slip conditions, the absolute extreme incremental bending moments in sub-rectangular linings are 20% higher than those in circular linings Conversely, under full slip conditions, circular linings exhibit approximately 4% greater absolute extreme incremental bending moments compared to sub-rectangular linings Additionally, for sub-rectangular linings, the absolute extreme incremental bending moments under full slip conditions are consistently about 10% lower than those under no-slip conditions, contrasting with the behavior observed in circular tunnels.
Figure 2.15b illustrates that the absolute extreme incremental normal forces for both tunnel shapes under no-slip conditions are about 80% greater than those under full slip conditions Additionally, the sub-rectangular lining experiences absolute extreme incremental normal forces that are approximately 9% lower than the circular lining forces, regardless of the slip conditions and variations in 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 are estimated to 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 (see 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 values (Es) 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 decrease in these bending moments This behavior highlights the significant relationship between Es and the bending moments in sub-rectangular tunnels.
The value of Es 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 behavioral pattern 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 higher 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 greater absolute extreme incremental bending moments than their sub-rectangular counterparts when Es values are below approximately 150 MPa However, once Es values exceed 150 MPa, the absolute extreme incremental 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 change is minimal for absolute extreme incremental normal forces in full slip conditions Additionally, absolute extreme incremental normal forces in sub-rectangular tunnels are typically 9% smaller than those in circular tunnels.
2.4.3 Effect of the lining thickness (t)
The lining thickness (t) is varied between 0.2 to 0.8 m, with a K0 value of 0.5, aH value of 0.5g, and an Es value of 100 MPa, as outlined in Table 2.1 Results shown in Figure 2.17 reveal that lining thickness significantly influences incremental internal forces in both sub-rectangular and circular tunnels, under no-slip and full slip conditions Additionally, the relationship between lining thickness and incremental internal forces for the analyzed cases is notably linear.
In conditions without slip, the absolute extreme incremental bending moments of sub-rectangular linings consistently exceed those of circular linings, with the difference decreasing from 124% to 6% as lining thickness increases from 0.2 to 0.8 meters Under full slip conditions, for lining thicknesses below approximately 0.5 meters, sub-rectangular linings again show larger bending moments compared to circular ones However, when the lining thickness exceeds 0.5 meters, the trend reverses, revealing that circular tunnels exhibit greater absolute extreme incremental bending moments.
Figure 2.17b illustrates that incremental normal forces under no-slip conditions are consistently higher than those under full slip conditions When comparing the incremental normal forces of circular linings to sub-rectangular linings, it is observed that the latter experiences a reduction of approximately 9% in no-slip conditions and 25% in full slip conditions This highlights the impact of lining thickness on the incremental internal forces of both circular and sub-rectangular tunnel linings.
Conclusion
A 2D numerical study examined the performance of sub-rectangular tunnel linings subjected to seismic loads Key parameters analyzed included soil deformation, maximum horizontal acceleration, lining thickness, and soil-lining interface conditions The research revealed significant differences in the behavior of circular and sub-rectangular tunnels when exposed to seismic activity.
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 generated in both sub-rectangular and circular tunnels, under both no-slip and full slip conditions.
Higher seismic loading, characterized by increased horizontal acceleration (aH), results in greater incremental bending moments and normal forces in both circular and sub-rectangular tunnels This relationship is notably linear.
The study highlights that the soil-lining interface conditions 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 demonstrate the opposite trend, with greater bending moments in no-slip scenarios This finding underscores the unique response of different tunnel shapes to interface conditions.
In all case studies examined, 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 compared to their circular counterparts.
The relationship between the absolute extreme incremental bending moments in sub-rectangular tunnels and the soil's Young modulus (Es) is minimal when compared to circular tunnels A critical threshold for the soil's Young modulus is identified at 50 MPa for both tunnel shapes When the Young modulus exceeds this value, the absolute extreme incremental bending moments decrease Conversely, when the Young modulus falls below 50 MPa, an increase in Es results in higher absolute extreme incremental bending moments.
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 Conversely, under full slip conditions, there is only a minimal change in the absolute extreme incremental normal forces.
The findings of this study are essential for the preliminary design of circular and sub-rectangular tunnel linings subjected to seismic loads These results will also inform the upcoming discussions in Chapter 3 Future research will explore the impact of joint distribution on the behavior of segmental linings within tunnels.
A NEW QUASI-STATIC LOADING SCHEME FOR THE
Fundamental of HRM method applied to sub-rectangular tunnel under static
The HRM method is utilized to calculate support structures under static conditions, as illustrated in Figure 3.1 This method takes into account various forces, including vertical loads (σv) and horizontal loads (σh) Key parameters include the normal stiffness of springs (kn), shear stiffness of springs (ks), and the bending and normal stiffness of the support (EI and EA) The analysis is conducted within a global Cartesian coordinate system, represented by coordinates X and Y.
The HRM method, derived from the Finite Element Method (FEM), effectively analyzes internal forces and displacements in tunnel linings Originally developed for segmental and continuous tunnel linings under static loads, recent research by Do et al has applied the HRM method to investigate the behavior of square and sub-rectangular tunnels under similar conditions Their proposed calculation scheme for supporting structures highlights the application of HRM in static scenarios, as illustrated in Figure 3.1.
Figure 3.2 depicts a beam-type element capable of generating internal forces The structure interacts with the soil through normal and tangential springs at the nodes, allowing for the calculation of stresses on each element once the node displacements are determined To ascertain these unknown displacements, it is essential to establish the global stiffness matrix that encompasses 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 \( k_i \) for the \( i \)th element in the global Cartesian coordinate system can be derived using the 3x3 sub-matrices \( k(()) \), \( k(()) \), \( k(()) \), and \( k(()) \) of the local stiffness matrix \( k \) of 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 displacements S and nodal forces F within the global Cartesian reference system can be calculated using the appropriate relationships.
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 is facilitated by normal springs (kn) and tangential springs (ks) linked to the structural nodes, along with applied active loads The values of kn and ks can be derived from the normal ground stiffness (ηn) and tangential ground stiffness (ηs) Unlike traditional methods that treat ground stiffness as a constant, 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 of the ground, denoted as plim, along with the initial stiffness, represented by 𝜂, defines the simplest model for understanding ground behavior when these parameters are accurately known Conducting a plate load test reveals a load-displacement curve that closely resembles a hyperbolic shape.
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.
The Poisson's ratio (νs) and Young's modulus (Es) of soil play crucial roles in tunnel boundary analysis, where Ri represents the radius of different tunnel sections—specifically the crown, shoulder, and side wall Additionally, β serves as a dimensionless factor in this context.
This study addresses the role of tangential springs, highlighting the challenge of estimating the frictional or shear stiffness at the ground support interface A straightforward relationship between normal stiffness (𝜂) and tangential stiffness (𝜂) can be utilized for this purpose.
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 incorporate nodal displacements and the local stiffness matrices of each element It's important to note that normal springs will be absent in areas where the support structure moves toward the tunnel, allowing only compressive loads in the direction of the tunnel support's movement toward the ground For further information on the HRM method, 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 of a sub-rectangular tunnel during seismic loading leads to sub-ovaling deformation of the tunnel lining, as illustrated in Figure 3.4c This finding is derived from a finite-difference model (FDM), with incremental internal forces depicted in Figure 3.5, referencing the no-slip condition shown in Figure 2.14.
In the HRM model, external loads are applied directly along the tunnel lining to account for seismic loading, rather than using shear stresses at the far-field boundary A key objective of this study is to introduce an external loading scheme for the tunnel lining, as illustrated in Figure 3.4c, aimed at achieving sub-ovaling deformation typical of sub-rectangular tunnels under seismic conditions The critical points of interest include Point A, the intersection of the sidewall and shoulder; Point B, located at the center of the shoulder; and Point C, where the crown meets the shoulder, as depicted in Figure 3.6.
Induced internal forces and deformations of 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 in compression when the tunnel lining moves towards the surrounding ground Compressive external loads in one direction generate tensile loads in the perpendicular direction Therefore, parameters (a) and (b) are essential to adjust the external loads applied to sub-rectangular tunnel linings under seismic conditions.
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 moments and normal forces align with those derived from the FDM model To effectively implement and calibrate the HRM method, a simplified chart format of the loading scheme is selected It is important to note that the sub-rectangular tunnel lining is divided 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 occurs via normal and tangential springs linked to the lining structure's nodes, represented by kn and ks and estimated using the ground's initial stiffness, η0 Additionally, in sub-rectangular tunnels, the radius of the lining components varies around 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), which influences spring stiffness, is typically assigned values of 1 or 2 However, recent research by Sun et al introduced a method for estimating β based on the properties of soil and tunnel lining in circular tunnels under seismic loading This study also incorporates a variation of the dimensionless factor (β) to more accurately depict the interaction between soil and tunnel.
This study performs a numerical parametric analysis to evaluate three dimensionless parameters: a, b, and β Subsequently, the findings from the HRM method are validated against the FDM model results, taking into account a wide variety of soil properties, lining characteristics, and tunnel geometries.
Numerical implementation
This section discusses the development of the FDM numerical model in FLAC 3D, as outlined in Chapter 2, which is utilized to calibrate the three dimensionless parameters (a, b, and β) essential for the HRM method Additionally, it presents the numerical procedure for implementing the HRM in the context of sub-rectangular tunnels subjected to seismic loads, as illustrated in Table 3.3 and Figure 3.8.
Chapter 2 introduces a 2D plane strain model utilizing FLAC 3D, detailing the geometry parameters of sub-rectangular tunnels as shown in Figure 2.9 Key soil and lining parameters are outlined in Table 3.1 and Table 3.2, with relevant results discussed in section 2.4 of Chapter 2 and depicted in Figure 3.7 Additionally, the calibration process aims to establish the dimensionless parameters a, b, and β within 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 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 are made between the seismic-induced incremental internal forces predicted by the HRM method and those calculated using the FDM numerical model.
During the initial calibration phase, parameters outlined in Table 3.1 were utilized, allowing for modifications in tunnel dimensions to encompass a variety of scenarios The tunnel width (w) was adjusted from 9.7 m to 15.52 m to create wider, uniformly shaped tunnels, as depicted in Figure 2.1, resulting in a t/w ratio ranging from 0.032 to 0.052 Various sub-rectangular shapes, labeled SR1 to SR4, were also adopted, with their dimensional characteristics detailed in Table 3.2 and Figure 3.7 The maximum horizontal acceleration was set at aH = 0.5g, corresponding to shear strains (γc) of 0.38% The calibration of three parameters (a, b, and β) was subsequently executed, 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 derived to illustrate the impact of three key parameters on soil characteristics, lining properties, and tunnel dimensions These formulas are developed based on the best fit, as demonstrated in Figures 3.9 and 3.10 The parameters β, a, and b are defined as follows:
Generating soil and lining parameters { , } for all cases
Initial and computation using HRM
{ , } and computation using numerical solution
The coefficient a is solely dependent on the soil's Young's modulus (Es), while coefficients 𝛽 and b are influenced by the lining thickness (t), tunnel height (h), tunnel width (w), and Es As illustrated in Figure 3.9a, the value of 𝛽 significantly increases with Es, particularly from 10 to 150 MPa, but shows only a slight rise beyond 150 MPa Conversely, parameters a and b1, depicted in Figures 3.10a and 3.10b, exhibit a sharp decline when Es ranges from 10 to 100 MPa, stabilizing at higher Es values Additionally, parameters 𝛽, 𝛽, 𝛽, b2, b3, and b4 are substantially influenced by the ratios of t/h, t/w, and h/w, as shown in Figures 3.9b, c, d and 3.10c, d, e It is important to note that the units for 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 When the tunnel structure is more flexible than the surrounding ground, it amplifies distortions compared to soil shear distortions in the free field This scenario occurs for Es values greater than 75 MPa and t/h ratios less than 0.07 Conversely, when the tunnel lining is stiffer than the ground, with Es values below 75 MPa and t/h ratios exceeding 0.07, the lining effectively resists 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, based on the reference case in Table 3.1 The results show minimal differences between the extreme incremental internal forces calculated by the HRM method and the FDM model, with discrepancies of only 1.2% for bending moments and 0.6% for normal forces These variations, particularly at the top and bottom of the tunnel lining, can be attributed to the simplified equivalent external loading depicted in Figure 3.6.
Validation of the HRM method
Extensive validations were conducted to assess the effectiveness of the newly developed HRM method The initial validation focused on estimating its accuracy by analyzing various peak horizontal seismic acceleration (aH) values Subsequent validations examined the impact of varying Young's modulus of soil and lining thickness Additionally, uniform tunnels with different cross-sections were included in the validation process to ensure comprehensive applicability of the HRM method.
FDM: Mmax = 0.900 MNm/m HRM: Mmax = 0.911 MNm/m
In the validation process, the FDM method achieves a maximum value of Nmax = 0.791 MNm/m, while the HRM method reaches Nmax = 0.786 MNm/m Validation 4 utilizes various sub-rectangular shapes based on the geometrical parameters from Table 3.2 [48] The impact of tunnel burial depth on the behavior of tunnel lining is assessed in validation 6 Furthermore, validation 7 employs soil parameters from the research conducted by Hashash et al [70] and [145] Throughout each validation, the seismic-induced incremental internal forces derived from the HRM method are compared with the numerical solutions obtained from FDM and FEM approaches.
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 aH varies from 0.05 to 0.5g, which correlates to shear strains (γmax) ranging from 0.038% to 0.38% The reference case parameters, as detailed in Table 3.1, are utilized in this analysis 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 are minimal, at under 2.2% for bending moments and 2% for normal forces, confirming the HRM method's efficacy 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 demonstrates a strong correlation between the extreme incremental bending moments and normal forces in tunnel lining, as calculated using the HRM method and the numerical FDM, particularly when accounting for variations in Es values The analysis reveals that these extreme bending moments are significantly influenced by changes 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 )
When the elastic modulus (Es) is below 50 MPa, a decrease in Es leads to lower absolute extreme incremental bending moments Conversely, for Es values above 50 MPa, an increase in Es results in decreased extreme incremental bending moments The HRM method demonstrates high accuracy in predicting these moments, with discrepancies from numerical FDM results typically 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 This variation may be linked to the precision of the fitting curves for the parameters a, b, and β.
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 impact of lining thickness on extreme incremental bending moments and normal forces in sub-rectangular tunnel linings The analysis focuses on two key aspects: a) extreme incremental bending moments and b) extreme incremental 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 HRM method reveals that extreme incremental internal forces exhibit a linear increase in both absolute extreme normal forces and bending moments with rising lining thickness However, the variations in incremental normal forces due to lining thickness changes are less pronounced compared to those of the bending moments Notably, the extreme incremental internal forces obtained through the HRM method align closely with numerical FDM computations, showing a discrepancy of less than 2% for both extreme incremental bending moments and normal forces.
During the validation process, the uniform tunnel cross-section sizes vary from 1.0 to 1.6 times their original dimensions, with the cross-section width (w) ranging from 9.7 to 15.52 m, as shown in Figure 2.1 Other parameters are detailed in Table 3.1, with Es set at 100 MPa Figure 3.15 illustrates the extreme incremental internal forces derived from the HRM method in comparison to numerical FDM computations, highlighting a) extreme incremental bending moments and b) extreme incremental normal forces, thereby demonstrating the impact of cross-section dimensions on the internal forces of 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 analysis of extreme incremental internal forces demonstrates a strong correlation between the HRM method and numerical modeling via FDM, as illustrated in Figure 3.15 Specifically, Figure 3.15a reveals that extreme incremental bending moments remain nearly constant with an increase in tunnel width from 9.7 to 15.52 m, with discrepancies between the HRM method and numerical computations being less than 1.7% In contrast, absolute extreme incremental normal forces exhibit a linear increase corresponding to the tunnel width, with the difference between the HRM and FDM calculations remaining under 1.4% (Figure 3.15b) Overall, the HRM method proves to be highly efficient when assessing variations in tunnel width.
The efficiency of the developed HRM method was validated using various tunnel geometries outlined in Table 3.2, while maintaining other parameters as indicated in Table 3.1 (Es = 100 MPa, t = 0.5m) Figure 3.16 illustrates the comparison of extreme incremental internal forces obtained from the HRM method with those derived from the numerical FDM approach In this figure, 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 exhibit a slight increase of 6% when transitioning from SR1 to SR4, while the extreme incremental normal forces remain nearly constant The normal forces calculated using the HRM method align closely with numerical computations from the FDM, showing discrepancies of less than 3% across all cases Additionally, the HRM method provides precise predictions for extreme incremental bending moments, with differences remaining under 2% compared to numerical results.
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
Figure 3.17 shows 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 decline gradually with increased burial depth, with an 18% decrease observed when comparing depths of 5 meters and 20 meters Additionally, the extreme incremental internal forces calculated using the proposed HRM method align closely 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 for efficiently analyzing the behavior of sub-rectangular tunnel linings under seismic loading using the HRM method It provides a detailed examination of the HRM method in seismic conditions, taking into account various soil parameters, lining properties, and tunnel geometries Additionally, the computation procedure is enhanced by a parametric analysis and a quasi-static loading scheme specifically applied to sub-rectangular tunnel linings.
The newly developed HRM method underwent extensive validation through a series of numerical computations, demonstrating its effectiveness Validation involved comparisons with a quasi-static numerical Finite Difference Method (FDM) model The comparative analyses confirmed that the HRM method is a reliable tool for the preliminary seismic design of sub-rectangular shaped tunnels.
The study reveals 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 soil, it effectively resists ground displacements.
The HRM method introduced in this study offers an innovative and efficient approach for the seismic design of sub-rectangular tunnels, complementing the 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 and does not account for soil nonlinearities during ground shaking.
The present thesis allowed achieving several conclusions that represent innovative contributions to the knowledge of the sub-rectangular tunnels considering seismic loadings
Chapter 1 examined the seismic behavior of tunnels, highlighting significant advancements in understanding and predicting the responses of both circular and rectangular tunnels to seismic loads However, the effects of earthquake-induced ground failures on tunnel performance remain inadequately explored.
Research primarily examines the transverse seismic response of circular and rectangular tunnels subjected to S waves under plane strain conditions, effectively predicting seismic lining forces Various methodologies, including analytical techniques, physical model testing, and numerical modeling, can be employed to analyze tunnel behavior during seismic events.
Analytical solutions offer simplicity and speed but are constrained by oversimplified assumptions While physical model tests and numerical analyses address these limitations, the complexity and expense of physical tests result in limited outcomes Recently, numerical models have gained popularity, yielding reliable results, particularly for comprehensive seismic analyses These models are adept at investigating spatially-variable ground motion in long tunnels, variations in layer boundaries among different geomaterials, changes in structural properties along the tunnel, and near-fault effects However, the high computational costs associated with numerical simulations mean that their application primarily resides within academic and research settings.
Recent studies have focused on sub-rectangular tunnels through both onsite and laboratory tests, addressing the limitations of traditional circular and rectangular tunnel cross-sections in terms of space utilization However, it is important to note that these investigations have primarily examined static loadings for this innovative tunnel design.
A 2D numerical study examined the behavior of sub-rectangular tunnel linings under seismic loads, highlighting the effects of soil deformations, maximum horizontal accelerations, lining thicknesses, and soil-lining interface conditions The research revealed significant differences in the response of circular and sub-rectangular shaped tunnels when subjected to seismic loading Key conclusions were drawn from the findings of this investigation.
The horizontal acceleration (aH), the soil's Young modulus (Es), and the thickness of the lining (t) significantly influence the incremental internal forces generated in both sub-rectangular and circular tunnels, under conditions of no-slip and 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.
The study highlights that soil-lining interface conditions 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 display greater absolute extreme incremental bending moments in no-slip conditions than in full slip conditions This finding underscores a distinct difference in the response of these two tunnel shapes to varying interface conditions.
In all examined case studies, the absolute incremental normal forces under no-slip conditions consistently exceed those observed in full slip scenarios 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) plays a minor role compared to circular tunnels, as 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 However, under full slip conditions, changes in the absolute extreme incremental normal forces remain negligible.
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