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Nghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đấtNghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đất

LITERATURE REVIEW ON THE BEHAVIOUR OF

Introduction

Tunnels are an important component of the transportation and utility systems in urban and national systems They were being constructed at an increasing rate to facilitate the space expansion need in densely populated urban areas Considering their substantial scale and cost of construction, and their critical role, this kind of infrastructures play an important role in modern societies Even slight seismic loading impacts can lead to short-time shutdowns and substantial direct and indirect damages Therefore, it is very important to carefully consider the effect of seismic loading on the analysis, design, construction, operation, and risk assessment of tunnels

As tunnels are interacting with the surrounding soil and/or rock environment, they are more resistant to earthquakes than structures at the ground surface The destruction of underground constructions has recorded for many earthquakes taking place around the world (e.g., Kobe, Japan 1995; Chi Chi, Taiwan 1999; Bolu, Turkey Period 1999; Baladeh, Iran 2004; and more recently Sichuan, China in 2008) Reports of damages [50],[158],[167] show that the effects interacting between the ground and tunnels were not conducted a systematic investigation In 1993, Wang highlighted the notion of relative flexibility as a key parameter for understanding the seismic-induced distortions of underground structures interacting with the surrounding ground Considering these incidents, it is clear that an improper designed underground structures are susceptible to wave propagation effects [14],[15],[16],[74],[75],[111], [166] (see Figure 1.1), Other detailed reviews of the seismic performance of tunnels and underground structures can be found in relevant publications [61],[69],[77],[94],[133],[168] (see Figure 1.2) These incidents were used as large- scale 'standard cases', to understand the interplay between structure and ground, with the ultimate goal of verifying the capabilities of design methods Existing analysis and validation of material models involved [81],[89] or interface conditions [74],[90],[135] needed to capture the observed response It was shown that there is still a need to improve the quality of the design and computation of underground structures in the areas that can suffer from earthquakes

Figure 1.1 Summary of observed bored/mined tunnel damage due to ground shakings [131]

Vietnam's territory is in a rather special position on the Earth's crust tectonic map and it exists a complex, diverse, and high-risk network of earthquakes There are studies of earthquakes such as statistics, localization, forecasting, assessment of the risk of earthquakes, and design [1],[2],[3],[4],[5],[6],[7],[8],[117],[118]

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 grouped into two categories: ground shaking and ground failure [159] or four categories: ground shaking, ground failure, land sliding and soil liquefaction [59] Ground shaking refers to the ground vibration induced by seismic waves that propagate through the earth’s crust Figure 1.3 shows the ground response due to the various types of seismic waves:

 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]

The underground structures will be deformed when the ground will deform due to the traveling waves Owen and Scholl [123] claimed that the behavior of an underground structure during a seismic event can be approximated to the one of an elastic beam subjected to deformations imposed by the surrounding ground Three types of deformations express the response of underground structures to seismic motions (see Figure 1.4):

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 are generated by seismic wave components that induce motions parallel to the tunnel's axis and cause alternatively compressive and tension forces Bending strains are caused by seismic wave components that induce

Shear wave front Shear wave front

Positive curvature particle motions perpendicular to the tunnel longitudinal axis They were not taken into account in this study, as they are generally oriented along the tunnel axis [159] Ovaling or racking deformations corresponding to circular and rectangular tunnel linings can be developed when shear waves propagate normally or nearly normally to the tunnel axis Penzien [129] and Hashash et al [70] indicated that the component that has the most significant influence on the tunnel lining behavior under seismic loading is the ovaling or racking deformations caused by the seismic shear wave or S-wave propagation In other words, they are the most crucial deformation modes for tunnels

On the other hand, the ground failures induced by earthquakes, may be caused by liquefaction, fault motions, or slope failure (Figure 1.5) They may induce large permanent ground deformations to 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

The deformation of underground structures is often simulated in 2-dimensional plane strain problems under equivalent static load, not paying attention to the inertial force effects [70] Due to their simplicity, various analytical methods were developed that allow the determination of the internal forces generated in the supporting structures such as for rectangular and circular tunnels [21],[24],[43],[90], [126],[128],[129];[159] In general, analytical methods are often limited by assumptions [135]:

- 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 overcome the analytical method drawbacks, experimental models were used to better understand the physical nature of the processes and have paid a special attention to the soil-structure interaction under seismic loading conditions

Physical models were performed by various authors to investigate the functioning mode of underground structures and verify the current design/analysis methods Most physical models were constructed to collect measurement data and to verify design model [19],[30],[39],[42],[96] However, due to the complexity and high cost of this method, the results obtained are currently very limited

Recently, the trend is to use a two-dimensional numerical model [70],[124], [125],[135] or a three-dimensional model [91],[137]

When using a numerical method, seismic loadings are usually simulated with equivalent static loads However, most numerical methods using equivalent static loads still have the drawbacks of the analytical methods presented above The main disadvantage of using equivalent static loads is that it does not pay attention to the tunneling performance change considering time under the seismic loadings Besides, these methods using the equivalent static load also often result in calculating internal forces lower than when applying real seismic loadings [132]

Compared to other methods, the time history analysis method using real earthquakes are the most complicated but therefore they are also able to provide the most accurate results However, these methods often require a long computation time and that is why their results are limited

It should be noted that most of the researches were conducted for underground structures with circular or rectangular tunnels In fact that, there are many other types of tunnel cross-sections works, including sub-rectangular tunnels The research and development of methods to calculate internal forces and deformations arising in sub- rectangular structures under seismic loadings as well as surveying the influencing parameters (parameters of the tunnel lining structure, of the soil mass environment, ) is a matter of practical and scientific significance Besides, the current trend when holding underground works for construction in earthquake areas is to use highly flexible structures such as segmental tunnel linings The initial research results mentioned above have shown the indispensable effect of the joints in the tunnel structures in this case and need to be fully studied to achieve reliable lining tunnel 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]

Due to the simplicity of analytical solutions, various elastic closed-form solutions were developed to determine internal forces induced on circular tunnel linings due to seismic loading [21],[43],[126],[128],[129],[159] The works conducted by Hashash et al [69],[70] indicated the discrepancies between the methods [128],[159], and used numerical analyses under the same assumptions to better understand the differences between the two solutions and their causes The comparisons demonstrated that the Wang’s solution provides a realistic estimation of the tunnel lining normal forces for a no-slip condition It has been recommended that Peinzen’s solution should not be used for a no-slip condition [70],[69] The works [17],[125],[126],[150] indicated a good agreement between their solution with the previous solutions [21],[159]

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

Considering a circular tunnel and its radius R located under the ground surface and subjected to a seismic loading using shear waves (Figure 1.6), the seismic- induced soil stress state can be treated as a shear-type stress This corresponds to compressive and tensile free-field principal stresses at 45 0 with a pure shear direction, as shown in Figure 1.7 [126]

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)

242 Note: In this table, the sediment types represent the following low-strain shear wave velocity (Cm) ranges: rock ≥ 750 m/sec; stiff soil 200 m/sec – 750 m/sec; and soft soil < 200 m/sec The relationship between peak ground velocity and peak ground acceleration in soft soils is poorly constrained

Where Vmax is the peak shear wave velocity, Vs is the ground shear wave velocity, Es is the soil Young’s modulus, and νs is the soil Poisson’s ratio The maximum circualr tunnel lining ovaling in Figure 1.6 will occur at its major and minor axes at θ = 45 0 with respect to the spring line [126]

Vmax can be estimated through Table 1.1 and Table 1.2 [59],[69] Table 1.1 can be used to determine the relationship between the ground motion at depth and at the ground surface Table 1.2 can be used to relate the known peak ground acceleration to estimates the peak ground velocity in the absence of site-specific data

1.3.1.1 Analytical solutions due to a seismic loading considering a circular tunnel

Wang [159] may be the first person who proposed a closed-form solution for the structural tunnel lining forces under seismic loading conditions For a full-slip condition of the soil-lining interaction, the maximum normal forces (Nmax) and maximum bending moment (Mmax) can be expressed as follows:

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

Note that no solution was developed for calculating bending moments under a no-slip condition by Wang [159] It is suggested that the solution for full slip condition may be used for no-slip condition The more conservative estimations of the full slip condition are considered to offset the potential underestimation due to quasi-static representation of the seismic problem [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

Likewise with the analysis of ovaling deformations for circular tunnels, rectangular tunnels could be analyzed for the imposed racking deformations assuming propagating shear waves Besides, the walls and roof of the tunnel cross- section should be analyzed for seismic earth pressures

Figure 1.8 Definition of terms used in racking analysis of a rectangular tunnel

Racking Deformation Analysis: Wang [159] developed a simplified procedure comprising the soil-structure interaction for the analysis of racking of rectangular tunnels The proceeding was developed based on a series of seismic finite element analyses with various soil and tunnel structure's seismic properties The cases analyzed included the following conditions:

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

Today, the tunnel with circular cross-sections are excavated by circular TBMs Sometimes they are not well adapted to the underground space exploitation requirements concerning the cross-section and space-utilization ratio Non-circular TBMs became a new development, and can be considered as the perfect cross-section tunneling equipment Recently, special cross-section tunnels were used and studied with real ratio or reduction ratio [72],[87],[99],[100],[170],[171],[175],[176],[177]

Figure 1.14 (a) Overlap cutter heads; (b) a copy cutter head [78]

Jianbin Li [78] gave out the technical features and development status of non- circular TBMs, special-shaped segment lining, and soil conditioning in loess strata for a pipe-jacking machine performed with a rectangular cross-section and a horseshoe-shaped TBM It provided a set of feasible solutions for the design, manufacture, and construction of non-circular TBMs Relevant engineering practice showed that non-circular TBMs with customized design and manufacture had great advantages in terms of the construction schedule, settlement control, and space utilization [78] (see Figure 1.14).

Figure 1.15 A photo showing the testing setup after fabrication [72]

Huang et al [72] developed a full-scale loading test to study the segmental lining of a sub-rectangular shield tunnel For the first time, the behavior of segmental tunnel lining with a large cross-section under self-weight was explored experimentally through full-scale loading tests The challenges of fabricating the testing setup and the segmental linings as well as their solutions are solved The experimental data were compared with the numerical simulation results which indicate that the self-weight influence should be considered in structural loading tests, especially for shallow buried tunnels The mechanical behavior of this rectangular lining structure subjected to various influential factors after assembly needs to be further studied in the future (Figure 1.15)

Recently, Do et al [48] focuses on improving the performance of the HRM method for the case of squared or sub-rectangular tunnels A special attention was paid to the tunnel shape influence by using various tunnel wall radii The numerical HRM model was validated based on a comparison with a finite element method (FEM) Then a parametric study was conducted to highlight the influences of the earth pressure coefficient and the soil Young’s modulus on the structural forces, and deformations induced by the tunnel excavation Du et al [52] used the HRM method for the optimization process of sub-rectangular tunnels in term of tunnel lining forces The effect of different parameters, such as the coefficient of lateral earth pressure, Young’s modulus of soil, tunnel depth, surface loads on the internal forces and shape of sub-rectangular tunnel was investigated Zhang et al [172],[173] studied the rotational stiffness influence on the joint behavior and 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

Considerable progresses were made in understanding and predicting the seismic behavior of tunnels but mostly in the case of circular and rectangular tunnels The responses of other shaped tunnels to earthquake-induced ground failure were not thoroughly investigated Most of the researches done focus on the transverse seismic tunnel response induced by S waves assuming plane strain conditions This approach seems to reasonably predict the seismic lining forces under transverse seismic loadings It is worth noting that only a few studies focused on the surface waves effect on the seismic tunnel responses obecause of their negligible influence

In general, the tunnels behavior under seismic loadings can be studied using a variety of approaches: analytical methods, physical model tests, and numerical modeling The analytical solutions have the advantage of being simple and fast but are limited by simple assumptions Physical model tests and numerical analysis have overcome the shortcomings of the analytical solutions However, due to the complexity and high cost of the tests, the results obtained from the physical tests are still quite limited The most popular recent trend is the use of numerical models They gave good results and their design seem to be reliable especially for full seismic analyses They can be potentially used for the investigation of spatially-variable ground motion along long tunnels, variations in layer boundaries between different geomaterials, variations in structural properties along the tunnel length, including the presence of station boxes, and near-fault effects However, due to the computational costs of the 3D numerical simulations, running this kind of simulations rests mostly in the academic and research community

Recently, sub-rectangular tunnels were considered in on site or laboratory tests Sub-rectangular tunnels allow shortcomings of the circular and rectangular tunnels to be solved Unfortunately, studies in the literature have just only focused on the applicability of sub-rectangular tunnels under static loadings but do not mention the works under seismic loading This issue will be further studied in this thesis.

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

Parameters of a sub-rectangular express tunnel in Shanghai, China are used as the reference case in this study [48] The sub-rectangular tunnel dimensions are 9.7m in width and 7.2m in height (Figure 2.1) The tunnel is supported by a segmental concrete lining of 0.5m For simplification purposes, a continuous lining was adopted without considering the effect of joints Based on this reference sub-rectangular tunnel, a circular tunnel with an external diameter of 4.89m, which has an equivalent utilization space area, is considered for comparison purposes (Figure 2.2)

Figure 2.2 Circular tunnel with the same utilization space area, distances in millimeters

2.1.2 Numerical model for the circular tunnel

Firstly, a numerical model for circular tunnels was developed using a finite difference program (FLAC 3D ) [76] The purpose was to investigate the behavior of circular tunnel linings under quasi-static loading and make a comparison with those obtained by an analytical solution

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

Circular tunnelR9450 liner elements, which are attached to the zone faces along the tunnel boundary with interfaces Two interface conditions were considered: no slip between the soil and lining and full slip The liner–zone interface stiffness (normal stiffness kn and tangential stiffness ks) is determined based on a given rule of thumb [76] kn and ks are set equal to 100 times the equivalent stiffness of the stiffest neighboring zone [76] for the no-slip condition case When considering the full-slip condition, ks is assigned to be equal to zero [76] The apparent stiffness (expressed in stress-per-distance units) of a zone in the direction normal to the surface has followed the 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 is composed of a single layer of zones in the y-direction, and the element’s dimension increases as one moves away from the tunnel It reaches the maximum size of 2.6 x 1 m at the model boundary (Figure 2.4) The boundary conditions of the numerical model are 120 m in the x-direction and 40 m in the z-

(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 [47], and Naggar and Hinchberger [113], ovaling deformations due to the seismic loading are imposed as inverted triangular displacements, along with the model lateral boundaries Uniform horizontal displacements are applied along the top boundary (Figure 2.4) The magnitude of the prescribed displacements assigned at the top of the model is dependent on the maximum shear strain max, estimated based on the maximum horizontal acceleration aH (see section 1.3.1 Chapter 1) The bottom of the model is restraint in all directions

Before applying the ovaling deformation due to seismic loading, a steady state of the excavated tunnel under static conditions was established When the tunneling process is performed in a 2D plane strain model, it is assumed that ground displacements surrounding the tunnel boundary prior to the lining installation are simulated by using the convergence confinement method with a relaxation factor, d, equal to 0.3 [47],[110] The numerical simulation of the tunnel ovaling was therefore performed through the following 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

It should be noted that only incremental internal forces are presented in this study They are determined by subtracting the total lining forces computed at the end of the static loading (step 2) from those determined at the end of the ovaling step (step 3)

2.1.3 Comparison of the numerical and analytical model for the circular tunnel case study

For validation purposes of the numerical model subjected to quasi-static loading, the well-known analytical solution proposed by Wang [159] and thereafter improved by Kouretzis et al [90] was used for comparison with the results obtained from the numerical model This solution was demonstrated to efficiently use in seismic design of circular tunnels [70],[47] It should be noted that Kouretzis et al [90] proposed an expression of the maximum incremental bending moment under the no-slip condition which was not mentioned by Wang [159] This study used Wang's formulation which has been adapted to calculate incremental internal forces distributed along the tunnel lining perimeter in FHWA [59] Parameters shown in Table 2.1 were adopted in this section as the reference case The soil and tunnel lining material properties in numerical models are assumed to be linearly elastic and massless These assumptions were also adopted in the analytical solution An anisotropic stress field was assigned in the numerical model with 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

Deformed circular tunnel model and displacement vectors after assigning prescribed displacements on the boundaries in cases of no-slip and full slip conditions are shown in Figures 2.5 and 2.6, respectively The ovaling deformation of the tunnel lining under seismic loading can be clearly seen Figure 2.7 illustrates the distribution of the incremental internal forces induced in the tunnel lining under seismic loading Conditions of lining and soil interaction, when using the Wang solution and FDM were considered for both cases of no-slip and full slip The soil and tunnel lining parameters fed into the model are presented in Table 2.1 It can be seen that results obtained by numerical and analytical models are in very good agreement Figures

2.7a and 2.7c show that the maximum incremental bending moment in the full-slip case is 14% larger than the one obtained in the no-slip case 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

In the section below, a parametric study is conducted to highlight the behavior of circular tunnel lining subjected to quasi-static loadings considering the effect of Young’s modulus Es, horizontal seismic acceleration aH, and tunnel lining thickness

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

Both maximum and minimum incremental bending moments are presented They are named ‘extreme incremental bending moment’ Similarly, extreme incremental normal forces representing both the maximum and minimum incremental normal forces induced in the tunnel lining

2.2.1 Effect of the peak horizontal seismic acceleration (aH)

A parametric study was conducted to investigate the seismic loading magnitude effects represented here by the maximum horizontal acceleration, aH The maximum horizontal acceleration is varied in the range between 0.05 and 0.75 g, corresponding to the respectively maximum shear strains, γmax, of 0.038 and 0.58% The other parameters of the reference case in Table 2.1 are used The following conclusions can be deduced from Figure 2.8: a) Extreme incremental bending moments b) Extreme incremental normal forces Figure 2.8 Effect of aH on the extreme incremental internal forces of the circular tunnel lining

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

- For both the no-slip and full-slip conditions, numerical results show a very good agreement with the analytical solution A discrepancy of approximately 1% for both the extreme incremental bending moments and normal forces is obtained,

- The absolute values of the extreme incremental bending moments and normal forces increased gradually with the aH increase from 0.05 g to 0.75 g Incremental bending moments for both no-slip and full slip conditions are strongly dependent on the aH value (Figure 2.8a) However, while incremental normal forces in the tunnel lining for the no-slip condition are strongly affected by the aH value, insignificant incremental normal forces variations due to aH for the full slip condition are observed (Figure 2.8b)

2.2.2 Effect of the soil Young’s modulus, Es

The soil Young’s modulus is assumed to fall in a range from 10 to 350 MPa The other parameters presented in Table 2.1, based on the reference case study were used as the input data The numerical results obtained by using the Flac 3D comparison with the analytical Wang’s method for the full slip and no-slip conditions are presented in Figure 2.9 The following comments can be deduced:

- Figure 2.9 shows a very good agreement of the incremental bending moments and normal forces induced in the tunnel lining under seismic loading, obtained by the numerical model and the analytical solution for both no-slip and full slip conditions when considering the Es variation The maximum difference is smaller than 2 %

- The extreme incremental bending moments are strongly dependent on the Es value as seen in Figure 2.9a The absolute values of the extreme incremental bending moments were reached for Es values close to 50 MPa There was a rapid decrease of the absolute extreme incremental bending moments when the Es value reduces from

25 to 10 MPa This can be explained by the fact that when the tunnel lining is stiffer than the ground, the tunnel lining tends to resist the ground displacements When the

Es values are larger than 50 MPa, the tunnel structure is more flexible than the ground

As a consequence, the tunnel lining will amplify the distortion compared with the soil shear distortions in the free field An increase of Es induces a decrease of the absolute extreme incremental bending moments This correlation of the extreme incremental bending moments is observed in both full slip and no-slip conditions It should be noted that for the same Es value, the absolute extreme incremental bending moments induced in the tunnel lining for the no-slip condition are always 10% to 15% smaller than the full slip ones

- While the extreme incremental normal forces in the full slip conditions are insignificantly dependent on the Es value (Figure 2.9b), an increase of Es can cause a rapid increase of the maximum and minimum incremental normal forces in the tunnel lining for the no-slip condition As predicted, incremental normal forces for the no- slip condition are always larger than the full slip ones a) Extreme incremental bending moments b) Extreme incremental normal Forces Figure 2.9 Effect of Es on the incremental internal forces of the circular tunnel lining

2.2.3 Effect of the lining thickness, t

The tunnel lining thickness was assumed to vary between 0.2 to 0.8 m, corresponding to the common lining thickness vs tunnel dimension ratio of 3% to 8.5% [60], while the other parameters are based on the reference case assumed in

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 Similar to what happens when considering Young’s soil modulus Es and horizontal seismic acceleration aH effects, the results presented in Figure 2.10 show a very good agreement between analytical and numerical models for both no-slip and full slip conditions The discrepancy is under 1% for both the incremental bending moments and normal forces

In general, the absolute extreme incremental bending moments and normal forces values gradually increase when the lining thickness t increases from 0.2 to 0.8 m This concerns both the full slip and no-slip conditions The incremental bending moments for the no-slip condition are always smaller than the full slip ones (Figure 2.10a) The biggest difference of 14% was obtained for a lining thickness of 0.8m It should be noted that the incremental normal forces variations caused by the lining thickness increase are less significant when compared to the incremental bending moment ones (Figure 2.10a and Figure 2.10b) a) Extreme incremental bending moments b) Extreme incremental normal forces Figure 2.10 Effect of the lining thickness on the incremental internal forces in the circular tunnel lining

Based on the above comparison between the analytical solution and numerical model when considering Young’s modulus Es, horizontal seismic acceleration aH, and tunnel lining thickness t, which show a very good agreement between the analytical

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

In this section, a numerical model was developed for the sub-rectangular tunnels cased using similar soil parameters, lining material, and modeling processes to consider the static and seismic loadings introduced above Only the tunnel shape is modified into a sub-rectangular geometry and the gravity effect is taken into consideration The mesh consists of a single layer of zones in the y-direction, and the dimension of the elements increases as one moves away from the tunnel (Figure 2.11) The geometry parameters of sub-rectangular tunnels are presented in Figure 2.1 and other parameters presented in Table 2.1 are adopted The numerical model is

120 m wide in the x-direction, and 40 m high in the z-direction and consists of approximately 5816 elements and 11870 nodes The bottom of the model was blocked in all directions and the vertical sides were fixed in the horizontal one.

Parametric study of sub-rectangular tunnels in quasi-static conditions

Pr es cr ib ed d is pl ac em en t γ max tunnel linings subjected to seismic loadings and considering both no-slip and full slip conditions Parameters of the reference case presented in Table 2.1 are adopted

Figure 2.12 Deformed model and displacement contours in Sub-rectangular tunnel model for no-slip condition

Figure 2.13 Deformed model and displacement contours in Sub-rectangular tunnel model for full-slip condition a) Incremental Bending Moment b) Incremental Normal Forces Figure 2.14 Distribution of the incremental bending moments and normal forces in the sub-rectangular tunnel

Figures 2.14 and 2.7 allow having a clear understanding of the behavior of circular and sub-rectangular tunnel linings under seismic loadings The positions at the tunnel periphery, where the extreme incremental internal forces are reached, are positioned It can be seen from Figure 2.14 that extreme incremental bending moments and normal forces observed in the sub-rectangular tunnel appear at the tunnel lining corners where the smaller lining radii are located In the following sections, a numerical investigation was conducted to highlight the behavior of a sub- rectangular tunnel compared with a circular shape These two tunnels have the same utilization space area and are twice subjected to seismic loadings while considering the effect of parameters, like the horizontal seismic acceleration, soil deformation modulus, and lining thickness Effects of the soil-lining interface condition are also investigated

2.4.1 Effect of the peak horizontal seismic acceleration (aH)

Shear strain values corresponding to a range of a maximum horizontal acceleration varying from 0.05g and 0.75g were adopted in this study In general, high seismic loadings are implied by a high horizontal acceleration aH, and therefore

No-slip case: Mmax = 0.900 MNm/m

Full slip case: Mmax = 0.807 MNm/m

No-slip case: Nmax = 0.791 MN/mFull slip case: Nmax = 0.159 MN/m shear strain values of γmax, result in high absolute extreme incremental bending moments and normal forces The relationship is quite linear (Figure 2.15) a) Incremental bending moments b) Incremental normal forces Figure 2.15 Effect of the aH value on the internal forces of circular and sub- rectangular tunnel linings

The results presented in Figure 2.15a show that, for the no-slip condition, absolute extreme incremental bending moments in the sub-rectangular lining are 20% larger than the circular ones Nevertheless, for the full slip condition, absolute extreme incremental bending moments in the circular lining are approximately 4% greater than the sub-rectangular ones In the case of sub-rectangular linings, absolute extreme incremental bending moments for the full slip condition are always lower by about 10% than the no-slip ones This relationship is opposite to the one observed in the cases of the circular-shaped tunnel (Figure 2.15a)

It can be seen in Figure 2.15b that for both shapes of tunnels, the absolute extreme incremental normal forces for the no-slip condition are approximately 80% larger than the full slip ones Unlike the incremental bending moments mentioned above, the absolute extreme incremental normal forces of the sub-rectangular lining are approximately 9% lower than the circular lining ones, for both the no-slip and full slip conditions and when changing the horizontal acceleration (Figure 2.15b)

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 assumed to vary in the range from 10 to 350 MPa while keeping K0 equal to 0.5 and aH of 0.5g The other parameters based on the reference case are assumed (Table 2.1) It can be seen from Figure 2.16 that: a) Incremental bending moments b) Incremental normal forces Figure 2.16 Effect of the Es value on the internal forces for the circular and sub- rectangular tunnel linings

- For low Es values smaller than 50 MPa, an increase of Es induces an increase of the absolute extreme incremental bending moments When the Es value is greater than 50 MPa, the increase of Es causes a decrease of the absolute extreme incremental bending moments (Figure 2.16a) It should be noted that the dependency of the absolute extreme incremental bending moments in the sub-rectangular tunnels on the

Es value is insignificant compared to the circular-shaped tunnels (Figure 2.16a) It is also worth highlighting that, while the absolute extreme incremental bending moments of the circular tunnel for the no-slip condition are smaller than the corresponding full slip ones [47],[159], the absolute extreme incremental bending moments of the sub-rectangular tunnel for the no-slip condition are greater than the corresponding full slip ones The behavior of sub-rectangular tunnels is different from

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 also shows greater absolute extreme incremental bending moments induced in sub-rectangular tunnels for the no-slip condition compared with circular tunnels having the same utilization space area However, in the full slip condition, absolute extreme incremental bending moments in the circular tunnel are greater than the sub-rectangular ones for Es values smaller than approximately 150 MPa When Es values are larger than 150 MPa, absolute extreme incremental bending moments developed in circular tunnels are smaller than in sub-rectangular tunnels

- Figure 2.16b indicates that an increase of Es value causes a significant corresponding increase of the absolute extreme normal forces in both sub-rectangular and circular tunnels for the no-slip condition But it induces an insignificant change in absolute extreme incremental normal forces for the full slip condition Absolute extreme incremental normal forces in the sub-rectangular tunnels are generally 9% smaller than for the circular ones

2.4.3 Effect of the lining thickness (t)

The lining thickness t is assumed to vary in the range between 0.2 to 0.8 m while keeping K0 value of 0.5, and aH value of 0.5g, and Es value of 100 MPa Other parameters introduced in Table 2.1 were adopted The results presented in Figure 2.17 indicate that the lining thickness has a great effect on the incremental internal forces for both sub-rectangular and circular tunnels and in both no-slip and full slip conditions The relationship between the lining thickness and the incremental internal forces for the considered cases is quite linear

For the no-slip condition, absolute extreme incremental bending moments of the sub-rectangular linings are always larger than the circular ones (Figure 2.17a) The discrepancy declined gradually from 124% to 6%, corresponding to the lining thickness increase from 0.2 to 0.8 m In the full slip conditions, considering a small lining thickness smaller than approximately 0.5m, the absolute extreme incremental bending moments of the sub-rectangular linings are still larger than the circular ones, just like for the no-slip condition presented earlier However, when the lining thickness is larger than 0 5m, Figure 2.17a proves an opposite result Thus, larger absolute extreme incremental bending moments on the circular tunnels are observed

It can be seen in Figure 2.17b that the incremental normal forces in the no-slip condition are always larger than for the full slip ones In comparison with the incremental normal forces of the circular lining, incremental normal forces in the sub- rectangular lining are lower by about 9% and 25% for no-slip and full slip conditions, respectively (Figure 2.17b) a) Incremental bending moments b) Incremental normal forces Figure 2.17 Effect of the lining thickness on the incremental internal forces of the circular and sub-rectangular tunnel linings

Conclusion

A 2D numerical study allowed investigating the behavior of sub-rectangular tunnel linings under seismic loadings The influences of parameters, like the soil deformation, the maximum horizontal acceleration, the lining thickness, and soil- lining interface conditions, on the circular and sub-rectangular shaped tunnel behavior under seismic loading, were investigated Considerable differences in the

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:

- The horizontal acceleration a H , soil’s Young modulus Es, and lining thickness t have a great effect on the incremental internal forces induced in both sub-rectangular and circular tunnels for both no-slip and full slip conditions;

- In general, a higher seismic loading induced by a higher horizontal acceleration aH, will induce higher incremental bending moments and normal forces in both circular and sub-rectangular tunnels The relationship is quite linear;

- The results proved that the soil-lining interface conditions have a great influence on the behavior of sub-rectangular tunnels This is completely different when comparing the behavior circular-shaped tunnels Indeed, while the absolute extreme incremental bending moments of a circular tunnel for the no-slip condition are smaller than the corresponding full slip ones, the absolute extreme incremental bending moments of sub-rectangular tunnels for the no-slip condition are greater than the corresponding full slip ones That is opposite to the trend observed in circular tunnel linings;

- For all investigated case studies, absolute incremental normal forces for the no-slip conditions are always larger than the full slip ones, for both circular and sub- rectangular tunnels cases Absolute extreme incremental normal forces in sub- rectangular tunnels are approximately 9% smaller than the circular ones;

- The dependency of the absolute extreme incremental bending moments induced on sub-rectangular tunnels on the soil’s Young modulus (Es) is insignificant compared with the circular ones Soil’s Young modulus of 50 MPa could be considered as a critical value for both tunnel shape cases Beyond this value, the (Es) increase induces a decrease of the absolute extreme incremental bending moments However, below this value, an increase in (Es) value induces an increase of the absolute extreme incremental bending moments;

- An increase of the soil’s Young modulus (Es) causes a significant corresponding increase of the absolute extreme incremental normal forces for both sub-rectangular and circular tunnels (no-slip condition) An insignificant change of the absolute extreme incremental normal forces is observed for the full slip conditions

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 These results also will be used in the next content in chapter 3 The joint distribution influence, in the segmental lining on the tunnel behavior, will be considered in a future research.

A NEW QUASI-STATIC LOADING SCHEME FOR THE

Fundamental of HRM method applied to sub-rectangular tunnel under static

Figure 3.1 Calculation scheme of support structures with the HRM method under static conditions With σv: the vertical loads; σh: the horizontal loads; kn: normal stiffness of springs; ks: shear stiffness of spring; EI and EA: bending and normal stiffness of the support; X and Y are the global Cartesian coordinates [48]

The HRM method is based on the Finite Element Method (FEM) which can be used for analyzing the internal forces and displacements induced in the tunnel lining The method was developed for the analysis of segmental and continuous tunnel linings under static loads [121],[51],[53] Recently, Do et al [48] have used the HRM method to study the behavior of square and sub-rectangular tunnels under static loading The author has proposed a calculation scheme of supporting structures and applied HRM for static conditions what is presented in Figure 3.1

Figure 3.2 illustrated the scheme of a beam-type element, which can develop internal forces Since the structure interacts with the soil by the normal and tangential springs distributed over the nodes, one can obtain the stresses on each element once the displacements of the nodes are known The unknown displacements can be achieved by defining the global stiffness matrix of the whole structure elements as well as the 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:

(3.1) where the terms 𝑘 ( ( ) )( ) , 𝑘 ( ( ) ) , 𝑘 ( ) ( ) , and 𝑘 ( ) represent the 3x3 sub- matrices of the local stiffness matrix 𝑘 of the (n) th element under the global Cartesian coordinates The local stiffness matrix ki of the i th element under the global Cartesian reference system can be obtained as follows:

[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:

After the global stiffness matrix K is obtained, and the vectors of the unknown nodal displacement S and the nodal forces F under the global Cartesian reference system can be determined by the following relation:

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 ground interacts with the tunnel support through the normal springs (kn) and tangential springs (ks) connected to the nodes of the structure (Figure 3.1) and through the applied active loads The values of kn and ks could be determined on the basis of the normal ground stiffness (ηn) and of the tangential ground stiffness (ηs) Unlike other methods, in which the ground stiffness is set as a constant value and is determined, Oreste [121] introduced a non-linear (hyperbolic) relationship between the reaction pressure p and the support displacement δ (Figure 3.3):

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]

Where: plim is the maximum reaction pressure that the ground can offer, and 𝜂 is the initial stiffness of the ground This relation represents the simplest way to describe the behavior of the ground when the initial stiffness and the limit pressure are known with a certain confidence By performing a plate load test in the ground, it is possible to note a curve load-displacement that is very similar to the hyperbolic one [11],[46]

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 generally depends on the ground elasticity parameters and tunnel radius [11],[121] In this study, the following formula is applied to determine the initial normal stiffness of the ground [48]

Where νs and Es are respectively the Poisson’s ratio and Young’s modulus of soil; Ri is the radius of part i (i=1, 2 and 3 which corresponds to the crow, shoulder and side wall of the tunnel boundary); β is a dimensionless factor

The presence of tangential springs has been taken into consideration in this study The frictional or shear stiffness at the ground support interface is very difficult to estimate The following simple relationship between the normal stiffness (𝜂 ) and tangential stiffness (𝜂 ) can be adopted [46],[108],[109]

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

Using Eqs 3.1 to 3.14, the internal forces in the tunnel lining are therefore calculated through the nodal displacements and the local stiffness matrices of each element It should be emphasized that the normal springs will disappear in zones in which the support structure moves towards the tunnel Therefore, only compressive loads are possible in the normal direction, in which the tunnel support moves towards the ground More details of the HRM method can be found in the literature [44],[46],[47],[51],[52],[53],[54],[98],[120],[121].

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

Similarly, when applying shear stress to the far-field boundary, the critical state of the sub-rectangular tunnel subjected to seismic loading causes a sub-ovaling deformation of the tunnel lining as seen in Figure 3.4c This result is obtained by using a finite-difference model (FDM) and incremental internal forces are presented in Figure 3.5 (from Figure 2.14 for no-slip condition)

Instead of the shear stresses applied at the far-field boundary, external loads acting directly along the tunnel lining are assumed in the HRM model to consider the seismic loading [47] One of the main goals of this study is to propose an external loading scheme applied to the tunnel lining as presented in Figure 3.4c This is to obtain a sub-ovaling deformation as observed at the critical state of sub-rectangular tunnels subjected to seismic loading Point A is the intersection between the sidewall and shoulder, point B is located at the center of shoulder parts, and point C is the intersection between the crown and shoulder as seen in Figure 3.6

As previously mentioned, induced internal forces and deformations of tunnel linings are mainly influenced by the external loads and soil-lining interactions considering the initial spring's stiffness It should be mentioned that the normal springs in the HRM method can only work in compression when the tunnel lining is moving toward the surrounding ground The compressive external loads applied to the tunnel in a certain direction will automatically produce tensile loads in the perpendicular direction [47],[145] That is why a couple of parameters (a) and (b) are necessary to be used to modify the external loads that are applied to the sub- rectangular tunnel lining under seismic conditions, as shown in Figure 3.6

The equivalent static loading scheme for the HRM method in Figure 3.6 is determined by varying the loads acting on 360 nodes on the sub-rectangular tunnel lining, until the incremental bending moment and normal forces in the sub- rectangular tunnel lining approach the incremental bending moment and normal forces in the sub-rectangular tunnel lining obtained using FDM model in Figure 3.5 Then choose the simplest chart format possible of loading scheme for the HRM method to implement and calibrate the HRM method in the following steps It should be noted that the sub-rectangular tunnel lining for the HRM method in this study is divided into 360 elements corresponding to 360 nodes with a maximum element length is 0.17m a) Incremental Bending Moment b) Incremental Normal Forces

Figure 3.5 Incremental bending moments and normal forces of sub-rectangular tunnel obtained using FDM model

Figure 3.6 Equivalent static loading with the HRM method for sub-rectangular tunnel

In the HRM, the ground interacts with the tunnel support through normal and tangential springs connected to the nodes of the lining structure (Figure 3.1) which are respectively represented by kn and ks, and estimated by the ground initial stiffness η0 In sub-rectangular tunnels, the lining parts radius vary along the tunnel periphery,

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 value of dimensionless factor (β) which affects the spring stiffness was usually set to 1 [108],[109] or 2 [47] Recently, Sun et al [145] estimated the β value, depending on properties of the soil and tunnel lining for the case of circular tunnels subjected to seismic loading In the present work, a variation of dimensionless factor (β) is also utilized to realistically represent the soil-tunnel interaction

A numerical parametric analysis is conducted in this study to determine the value of the three dimensionless parameters (i.e., a, b and β) After that, the results obtained from the HRM method are verified against the ones given by FDM model, considering a large range of soil properties, lining characteristics and tunnel geometry.

Numerical implementation

In this section, using the FDM numerical model in FLAC 3D [76] has been developed in chapter 2 which was adopted to calibrate the three dimensionless parameters (a, b and β) used in the HRM method Then, the numerical procedure to implement the HRM in the case of sub-rectangular tunnels subjected to seismic loadings is presented (Table 3.3 and Figure 3.8)

Using the FLAC 3D , a 2D plane strain model is presented in Chapter 2 The geometry parameters of sub-rectangular tunnels are presented in Figure 2.9 Other soil and lining parameters listed in Table 3.1 (the results in section 2.4 in Chapter 2) and Table 3.2 (Figure 3.7) are adopted It should be mentioned that for the calibration purpose to determine dimensionless parameters a, b and β in the HRM method

Table 3.1 Input parameters for the reference case for developing the HRM method

Parameter Symbol Unit Value or Range

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 implement the HRM method in the case of sub-rectangular tunnels subjected to seismic loading, it is necessary to determine the formulas of the three dimensionless parameters (a, b and β) which define the external loadings applied on the tunnel lining Extensive comparisons between the seismic-induced incremental internal force predicted by the HRM method and the ones calculated using the FDM numerical model are conducted

In the initial calibration phase, the parameters of Table 3.1 were adopted Based on that, parameters of tunnel dimensions are changed in a certain range to cover most of the cases that occurred The tunnel width (w) is assumed to be modified from 9.7 m to 15.52 m to make wider tunnels that have a uniform shape with the reference tunnel illustrated in Figure 2.1 The t/w ratio, therefore, varied from 0.032 to 0.052, respectively In addition, different sub-rectangular shapes, i.e., SR1 to SR4, with all dimension characteristics presented in Table 3.2 and Figure 3.7 are adopted The maximum horizontal acceleration, aH = 0.5g, corresponding to the shear strains, γc of 0.38% The calibration process of the three parameters (i.e., a, b and β) can then be performed The main procedure to calibrate the three parameters is illustrated 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

After the calibration process is completed, the equations representing the influence of the three parameters on the soil, lining properties and tunnel dimensions can be established The formulas are proposed based on the best fit (Figure 3.9 and 3.10) The parameters β, a, and b can be given as follows:

Generating soil and lining parameters { , } for all cases

Initial and computation using HRM

{ , } and computation using numerical solution

While the coefficient a is expressed as a function of soil’s Young’s modulus alone (Es), coefficient 𝛽 and b are the functions of the lining thickness (t), tunnel height (h), tunnel width (w), and soil’s Young’s modulus (Es), as shown in Figures 3.9 and 3.10 In Figure 3.9a, the value of the parameter 𝛽 increases rapidly when Es value increases from 10 to 150 MPa, and its value rises slightly when values of Es are larger than 150 MPa However, for parameters a and b1 in Figure 3.10a and Figure 3.10b, respectively, there was a rapid decrease of these parameters when Es values changed within a range of 10 and 100 MPa When Es values continue to increase, values of these two parameters stay constant Parameters 𝛽 , 𝛽 , 𝛽 , b2, b3, and b4 are greatly affected by t/h, t/w, and h/w ratios (Figure 3.9b, c, d and Figure 3.10c, d, e) It should be noted that units of parameters in Eq 3.15 to Eq 3.25 are the same as those used in Table 3.1 and Table 3.2

It is interesting to note that for Es value of 75 MPa and t/h ratio of 0.07 (Figure 3.10b, c), b1 and b2 values are equal to zero and therefore do not cause additional seismic loading acting on the tunnel lining through the b value in Eq 4.21 In other words, when the tunnel structure is more flexible than the ground, the tunnel lining will amplify the distortion compared with the soil shear distortions in the free field This case corresponds to Es values larger than 75MPa in Figure 3.10a, the t/h ratios are smaller than 0.07 in Figure 3.10c and Figure 3.10d In contrast, when a tunnel lining is stiffer than the ground, corresponding to Es values smaller than 75MPa in Figure 3.10b, the t/h ratio is larger than 0.07 in Figure 3.10c and Figure 3.10e, the lining tends to resist the ground displacements a) b) c) d)

Figure 3.9 Obtained numerical results and fitting curves adopted for the parameters β1, β2, β3 and β4 that created the parameter β

Figure 3.10 Coefficients fitting curves for the formulas of the parameters a and b1, b2, b3 and b4 that created the parameter b

To have a clear understanding of the results obtained by HRM and the numerical FDM model, Figure 3.11 introduces a comparative example of the incremental bending moments and normal forces distribution in the sub-rectangular tunnel lining subjected to a seismic loading when Es = 100MPa and t = 0.5m Other soil parameters of the reference case presented in Table 3.1 are adopted Figure 3.11 reveals insignificant differences between the extreme incremental internal forces obtained by the HRM method and the FDM model The differences are 1.2% and 0.6% corresponding to the extreme incremental bending moments and the normal forces The discrepancies of the incremental bending moments and normal forces observed at the top and bottom of the tunnel lining could be explained by the fact that the chosen equivalent external loading presented in Figure 3.6 is a simplified one a) Incremental Bending Moments b) Incremental Normal Forces Figure 3.11 Comparison of the incremental bending moments and normal forces calculated by the developed HRM method and numerical FDM calculation

Validation of the HRM method

The extensive validations were carried out to demonstrate the applicability of the developed HRM method The first validation aims at estimating the accuracy of the developed HRM method, using a range of peak horizontal seismic acceleration (aH) Then, varying Young’s modulus of soil and lining thickness are used for validations 2 and 3 While the uniform tunnels with different cross-sections are

FDM: Nmax = 0.791 MNm/mHRM: Nmax = 0.786 MNm/m considered in validation 4, different sub-rectangular shapes with geometrical parameters of tunnel shape cases from Table 3.2 [48] are used in validation 5 The effect of the burial depth of the tunnel on behavior of tunnel lining is considered in validation 6 Finally, Validation 7 is performed using soil parameters adopted in research by Hashash et al [70] and [145] In each validation, the seismic-induced incremental internal forces obtained by the HRM method are compared with the numerical FDM and FEM solutions

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 established for parameters a, b and β and presented in section 4.4 are proposed based on the case of aH of 0.5g In this particular validation, the maximum horizontal acceleration, aH, is assumed to vary in the range from 0.05 to 0.5g, corresponding to the shear strains, γmax, of 0.038% and 0.38% The other parameters of the reference case presented in Table 3.1 are adopted The comparison between the developed HRM method and the FDM calculations in terms of extreme incremental bending moments and normal forces is performed

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

As shown in Figure 3.12, both the extreme incremental normal forces and bending moments computed using the developed HRM method show a very good agreement with the numerical FDM results These two methods can capture the same trend, i.e., the absolute extreme incremental bending moments and normal forces increase with the applied maximum horizontal acceleration (aH) rise The discrepancy is under 2.2% and 2% corresponding to the extreme incremental bending moments and normal forces It proves that the developed HRM method can be effectively applied in a larger range of horizontal accelerations, aH

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 shows a very good agreement of the extreme incremental bending moments and normal forces induced in the tunnel lining calculated using HRM method and the numerical FDM when considering the Es changes The extreme incremental bending moments are strongly dependent on the Es values as seen in

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 )

Figure 3.13a With Es values smaller than 50 MPa, a decrease of Es induces a decrease of the absolute extreme incremental bending moments When the Es value is greater than 50 MPa, an increase of the Es values causes a decrease of the absolute extreme incremental bending moments Compared to the numerical FDM results, the HRM method precisely predicts the extreme incremental bending moments For almost all the Es values, the difference is smaller than 3.5% Except for the case of an Es value equal to 25 MPa, the discrepancies are equal to 6.8% and 7.2% for the maximum and minimum incremental bending moments, respectively The observation is that it could be related to the accuracy of the fitting curves of the parameters a, b and β (Figure 3.9 and Figure 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

100 MPa The comparisons between the HRM method and the numerical FDM ones are given in Figure 3.14, in terms of the extreme incremental bending moments and normal forces a) Extreme incremental bending moments b) Extreme incremental normal forces Figure 3.14 Effect of the lining thickness on the extreme incremental internal forces of the sub-rectangular tunnel lining

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 exhibit the same trend as those using numerical computations: both the absolute extreme normal forces and bending moments increase linearly with the lining thickness increase It should be noted that the incremental normal forces variations caused by the lining thickness increase are less significant when compared to the incremental bending moment ones (Figure 3.14a and Figure 3.14b) Figure 3.14 indicates that the extreme incremental internal forces determined by the developed HRM method agree well with the numerical FDM computations The discrepancy is under 2% for both the extreme incremental bending moments and normal forces

In this validation process, the sizes of the uniform tunnel cross-sections change from 1.0 to 1.6 times compared to the original one (Figure 2.1), corresponding to the tunnel cross-section width (w) changes from 9.7 to 15.52 m The other parameters are adopted as shown in Table 3.1, although Es equals 100 MPa The extreme incremental internal forces obtained by HRM compared with the numerical FDM computations are illustrated in Figure 3.15 a) Extreme incremental bending moments b) Extreme incremental normal forces Figure 3.15 The cross-section dimensions influence on the extreme incremental 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

In terms of the extreme incremental internal forces, the results presented in Figure 3.15 show a very good agreement between the HRM method and the numerical modeling using FDM Figure 3.15a also indicated that the extreme incremental bending moments are almost constant when the tunnel width increased from 9.7 to 15.52 m The difference between the extreme bending moments obtained using the developed HRM method and the numerical computations is lower than 1.7% In contrast to the extreme incremental bending moments, the absolute extreme incremental normal forces tend to increase linearly as the tunnel width increases The discrepancy between the extreme normal forces calculated by the HRM method and the numerical FDM ones is under 1.4% (Figure 3.15b) When the tunnel width changes, the HRM method shows very good efficiency too

Another step was to validate the efficiency of the developed HRM method with the different tunnel geometry presented in Table 3.2 [48], while the other parameters are as shown in Table 3.1 (Es = 100 MPa, t = 0.5m) The results of the extreme incremental internal forces obtained by HRM compared with the numerical FDM ones are shown in Figure 3.16 It should be mentioned that tunnel widths are used in Figure 3.16 to distinguish different tunnel geometries

Figure 3.16 indicates how the tunnel shapes influence the extreme incremental normal forces and bending moments The extreme incremental bending moments increase slightly by 6% when the tunnel shape changes from SR1 to SR4 By contrast, the extreme incremental normal forces are almost constant The extreme incremental normal forces in the tunnel lining obtained by the HRM method are matching well the numerical computations using FDM, the discrepancy is lower than 3% for all the cases Compared with the numerical results, the HRM method allows accurate prediction of the extreme incremental bending moments, the differences of all cases are under 2% a) Extreme incremental bending moments b) Extreme incremental normal forces Figure 3.16 Effect of the shape of cross-section on the extreme incremental internal forces of the sub-rectangular tunnel lining

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 indicates that the extreme incremental bending moments decrease significantly by 24% when the burial depth of the tunnel changes from 5 to 20 m Likewise, the extreme incremental normal forces are decreased gradually when the burial depth of the tunnel increases A decrease of 18% of extreme incremental normal forces obtained at a burial depth of 5m compared to the one of 20m is observed Figure 3.17 also indicates that the extreme incremental internal forces determined by the proposed HRM method are in well agreement with the numerical FDM computations The discrepancies are less than 6% and 4% corresponding to the extreme incremental normal forces and 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

The novelty and the scientific contribution of this study lie in proposing a new numerical procedure to efficiently calculate the behavior of sub-rectangular tunnel linings subjected to seismic loading using the HRM method The HRM method under seismic conditions is presented in details, considering a large range of soil parameters, lining properties and tunnel geometry The entire computation procedure is assisted by a parametric analysis and a quasi-static loading scheme which was applied on the sub-rectangular tunnel lining is also proposed

To verify the application capability of the developed HRM method, an extensive validation was performed considering series of numerical computations The developed HRM method was validated based on comparison with a quasi-static numerical FDM model Comparative analyses revealed that the developed HRM method can be efficiently used in preliminarily seismic design of sub-rectangular shaped tunnels

The present study also shows that in the case when a tunnel structure is more flexible than the soil mass, the tunnel lining will experience amplified distortions in comparison to the soil shear distortions in free field By contrast, when a tunnel lining is stiffer than the surrounding soil, it tends to resist to ground displacements

Although with the current development of numerical models and computing capacities of full dynamic or pseudo-static model of the soil-tunnel system, the proposed HRM method in this study provides a new and alternative free method of very efficient seismic design of sub-rectangular tunnels It is necessary to emphasize that the proposed method assumes an elastic soil-tunnel configuration subjected to the effect of ground shaking, nonlinearities of soil are not covered by the presented method

The present thesis allowed achieving several conclusions that represent innovative contributions to the knowledge of the sub-rectangular tunnels considering seismic loadings

Firstly, the behavior of tunnels subjected to seismic loadings were reviewed in Chapter 1 Currently, considerable progresses were made in understanding and predicting the seismic behavior of circular and rectangular tunnels In particular, the tunnels response to earthquake-induced ground failures are not thoroughly investigated

 Most of the researches in the literature focuses on the transverse seismic tunnel response induced by S waves under assumed plane strain conditions for circular and rectangular tunnels This approach seems to be able to reasonably predict the seismic lining forces under transverse seismic loadings In general, the tunnel behavior under seismic loadings can be studied using a variety of approaches: analytical methods, physical model tests, and numerical modeling;

 Analytical solutions have the advantage of being simple and quick but are limited by their simplified assumptions Physical model tests and numerical analysis overcome these shortcomings However, due to the complexity and high cost of the tests, the results obtained from the physical tests are still quite limited The most popular recent trend is the use of numerical models They gave good results and reliable results especially for full seismic analyses They can potentially be used for the investigation of spatially-variable ground motion along long tunnels, variations in layer boundaries between different geomaterials, variations in structural properties along the tunnel length, including the presence of station boxes, and near-fault effects Nevertheless, due to the computational costs of the numerical simulations, the ability to run such simulations mostly stays in the academic and research community;

 Recently, sub-rectangular tunnels were considered in onsite and laboratory tests Sub-rectangular tunnels solved the shortcomings of the circular and rectangular tunnels cross-section (space utilization ratio) However, only static loadings were considered for such tunnel shapes

A 2D numerical study allowed investigating the behavior of sub-rectangular tunnel linings under seismic loadings in chapter 2 The influences of parameters, like the soil deformations, the maximum horizontal accelerations, the lining thicknesses, and soil-lining interface conditions, on the circular and sub-rectangular shaped tunnel behavior under seismic loading, were investigated Considerable differences in the response of these tunnels were observed Based on the research results obtained, some conclusions can be drawn:

 The horizontal acceleration aH, soil’s Young modulus Es, and lining thickness t have a great effect on the incremental internal forces induced in both sub- rectangular and circular tunnels for both no-slip and full slip conditions;

 In general, a higher seismic loading induced by a higher horizontal acceleration aH, will induce higher incremental bending moments and normal forces in both circular and sub-rectangular tunnels The relationship is quite linear;

 The results proved that the soil-lining interface conditions have a great influence on the behavior of sub-rectangular tunnels This is completely different when comparing the behavior circular-shaped tunnels Indeed, while absolute extreme incremental bending moments of a circular tunnel for the no-slip condition are smaller than the corresponding full slip ones, the absolute extreme incremental bending moments of sub-rectangular tunnels for the no-slip condition are greater than the corresponding full slip ones That is opposite to the trend observed in circular tunnel linings;

 For all investigated case studies, absolute incremental normal forces for the no-slip conditions are always larger than the full slip ones, for both circular and sub- rectangular tunnels cases The dependency of the absolute extreme incremental bending moments induced on sub-rectangular tunnels on the soil’s Young modulus

(Es) is insignificant compared with the circular ones An increase of the soil’s Young modulus (Es) causes a significant corresponding increase of the absolute extreme incremental normal forces for both sub-rectangular and circular tunnels (no-slip condition) An insignificant change of the absolute extreme incremental normal forces is observed for the full slip conditions;

 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