Underground facilities are an integral part of the infrastructure of modern society and are used for a wide range of applications, including subways and railways, highways, material storage, and sewage and water transport.
ThThe 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, ChinaAN EARLY-STAGE DESIGN PROCEDURE FOR CIRCULAR TUNNEL LINING UNDER SEISMIC ACTIONS 1112E. Santucci de Magistris and F. Silvestri112DIGA, University of Napoli Federico II, Naples, SAVA, Structural and Geotechnical Dynamics Laboratory, University of Molise, Campobasso, Italy Email:ABSTRACT: The increments of the internal forces induced by an earthquake in the transverse section of a tunnel lining can be ascribed to the ovalisation of the section, induced by soil shear straining in the vertical plane. They can be assessed with several procedures at different levels of complexity. In this paper, two kind of analysis were per-formed on idealised geometry and soil conditions, considered representative of soil classes specified by Euro-code 8: pseudo-static analysis, where the seismic input was reduced to an equivalent peak strain amplitude, computed through a free-field pseudo-static analysis of the ground and then considered acting on the tunnel lining in static conditions; and full dynamic analysis, where the soil and tunnel responses were mechanically coupled and modelled by using FEM. Both were performed considering the soil as an equivalent linear medium.
On the basis of the comparison of the results of both approaches, modification factors of the usual pseudo-static formulae are proposed, which take into account the kinematic interaction between the tunnel and the ground during shaking. The method, based on the use of simple charts, can be easily adopted for early-stage design.KEYWORDS:tunnels, lining, seismic actions, pseudo-static, dynamicthThe 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China1. INTRODUCTION Civil infrastructures and lifelines in seismic areas need to be designed to support the extra loading produced by earthquakes. Some indications for such design can be found in Owen & Scholl (1981), JSCE (1992), AFPS/AFTES Guidelines (2001), ISO TC 98 (2003). Design rules for tunnels are not introduced in Eurocode 8 (EN 1998-5, 2003). This maybe because earthquake effects on underground structures were deemed to be negligible, in spite of the different evidences from several case-histories (see for instance Lanzano et al., 2008).
Research activities are in progress in Italy to refine the design methods for tunnels under seismic actions (e.g. Bilotta et al., 2007). The shear waves propagating during an earthquake perpendicularly to the tunnel axis, result in a distortion of the cross-section of the structure: in this paper a procedure to calculate the forces induced by ground shaking in the tunnel lining in simple subsoil conditions is illustrated. Such simplified procedure incorporates the results of finite elements dynamic analyses, which consider the kinematic interaction between the tunnel lining and the ground, in the framework of the pseudo-static approach commonly adopted for early-stage design. Three idealized ground conditions (Fig.
1) were considered: a 30 m thick layer of soft clay, medium dense sand or gravel, overlying a compliant rock bedrock (Vr= 800 m/s, γ=22 kN/m3, D0=0.5%). The tunnel has the following characteristics: circular shape with reinforced concrete lining (variable thickness from 0.1 to 1.3 m, diameter D=6 m); axis depth z0=15 m; VS (m/s) 0512 m 10z(m)t = 0.3 m 6margilla clay (D)(D) sabbia sand (C)(C) ghiaia (B) (B) gravel1520(riv. Cls Rbk concrete lining450 Rckkg/cmq) =45 MPa12 m 2530bedrock rock Bedrocksoft roccia teneraVS = 800 m/s3γ = 22 kN/mD0 = 0.5%Figure 1 - Ground conditionsThe values of small strain soil parameters have been chosen according to literature empirical relationships linking the shear modulus (G0) and the damping ratio (D0) to the lithostatic stress, the void ratio and intrinsic soil properties, such as particle size and plasticity index IP (Santucci de Magistris, 2005; d’Onofrio & Silvestri, 2001). The profiles of VS with depth adopted for each soil type are shown in Fig.
1, where the dashed lines represent the value of the so called ‘equivalent velocity’ VS,30 (EN 1998-1, 2003).
The complete enclosure in soil or rock makes their seismic behavior different than that of aboveground structures or superstructures. Underground structure seismic response is constrained by seismic response of the surrounding soil and cannot experience free vibrations as is the case for aboveground structures. This section focuses on the seismic analysis and design of large linear underground structures commonly used for metro structures, highway tunnels, and large water/sewage transportation ducts in urban areas and can be grouped into three broad categories: (1) bored or mined tunnels, (2) cut-and-cover tunnels, and (3) immersed tunnels (Fig. 1Typical cross sections of underground structuresThe section starts with a selected review of performance of underground structures during seismic events. This is followed by presentation of a performance-based framework for design and analysis of underground structures considering both permanent and transient deformations.
A number of seismic design issues are then discussed including vertical ground shaking and response, interaction of temporary and permanent structures, impact of superstructure and adjacent structures, transitions and tunnel joints, seismic retrofit of existing facilities, design considerations for structural support members, and precast tunnel lining. Performance of Underground Facilities During Seismic Events. Based on several studies that documented earthquake damage to underground facilities in the past and the behavior of underground facilities in recent large magnitude earthquakes (e.g., Tohoku, Japan, 2011; Maule, Chile, 2010), underground structures suffer appreciably less damage than surface structures. However, damage or failure of a limited section of an underground structure can be disruptive to post-earthquake recovery or operation in densely populated urban areas as such damage can interrupt the function of an entire system whether it is part of a mass transit or vehicular transportation network or large water or sewage transportation tunnels.Damage is related to a number of parameters including ground motion intensity, ground conditions, and structural support system.
Shallow tunnels tend to be more vulnerable to earthquake shaking than deep tunnels, and those constructed in soil can undergo more deformation than those constructed in competent rock. Circular bored tunnels are less susceptible to earthquake damage than cut-and-cover tunnels. Shaking damage can be reduced by stabilizing the ground around the tunnel and by improving the contact between the lining and the surrounding soil using grouting. Stiffening the lining without stabilizing the surrounding poor ground may only result in excessive seismic forces in the lining.
Damage at tunnel portals may be caused by slope instability (Hashash et al. Underground Structures in the United StatesThe Bay Area Rapid Transit system (BART) in San Francisco, California, sustained the 1989 Loma Prieta earthquake without damage and was operational after the earthquake.
It consists of underground stations and tunnels embedded in soft bay mud deposits connected to Oakland via the transbay-immersed-tube tunnel. It was one of the first underground facilities designed with seismic considerations: special seismic joints were designed to accommodate differential movements. Limited displacements were measured at that joint.During the same earthquake, the Alameda Tubes, a pair of immersed-tube tunnels that connect Alameda Island to Oakland in California, experienced structural cracking on the ventilation buildings and limited water leakage due to liquefaction of loose deposits above the tubes.The 1994 Northridge earthquake caused no damage to concrete lining of bored tunnels of the metro system in Los Angeles, California (Hashash et al. Underground Structures in Japan. The Dakai subway collapse (Fig. ) during the 1995 Hyogoken-Nambu earthquake in Kobe, Japan, was the first collapse of an urban underground structure due to earthquakes shaking.
The collapse experienced by the center concrete columns (Fig. ) was due to lack of shear reinforcement, leading to a collapse of the ceiling slab and settlement of the soil cover. The 1962 station design did not include specific seismic provisions. However, in the 2011 Tohoku earthquake, the underground subways in Sendai experienced strong shaking with no reports of damage.
Only a water distribution plant in Kashima City and a wastewater treatment plant in Itako City were observed to have been damaged by liquefaction. The damages include uplift of buried tanks, offsets in underground tunnels, damage to support utilities, and damage to major trunk lines on and off the site (Ashford et al. 5Chi-Shue tunnel before and after Chi-Chi earthquake (Wang et al. ) Underground Structures in TurkeyThe August 17, 1999, Kocaeli earthquake had minimal impact on the Bolu twin tunnels, a 1.5-billion-dollar project under construction at that time. It had an excavated section of 15 m tall by 16 m wide and crossed several minor faults parallel to the North Anatolian Fault. After the earthquake, continuous monitoring showed no movement due to the earthquake.
The November 12, 1999, earthquake caused collapse of both tunnels 300 m from its eastern portal, in a clay gauge material in the unfinished section (Hashash et al. Underground Structures in Chile.
Cut-and-cover highway box structures with three lanes of traffic in one direction and approximately 1 km long constructed in relatively stiff gravels, the Santiago Metro tunnels and underground stations, and highway tunnels on the southbound of Route 5 appeared to be undamaged by the 2010 Maule earthquake in Chile, as shown in Fig. In the northbound of Route 5, “La Calavera” tunnel near Calera has a rock block dislodged, but the tunnel was old and had problems before the earthquake (Elnashai et al. 6Left: Highway box structures in Santiago, Right: Highway tunnel in Route 5 SouthIn summary, well-engineered underground structures performed well even under strong shaking in recent earthquakes in different parts of the world. However, underground structures are vulnerable to permanent ground displacements such as liquefaction, slope stability, and fault displacement. There is also vulnerability to transient ground motions when insufficient structural detailing is provided, the underground structure is constructed in loose ground, masonry lining is used, or near field effects are present. Performance-Based Seismic Evaluation Framework.
![Tunnel Tunnel](/uploads/1/2/5/3/125383107/724236519.gif)
Underground structures under earthquake effects can undergo permanent deformations and/or transient deformations. Factors influencing these effects include shape, dimensions, and depth of the structure, properties of the surrounding soil or rock, properties of the structure, and severity of ground shaking (Hashash et al. Table summarizes a performance-based framework for seismic design and analysis of underground structures. The framework consists of three main steps: definition of seismic environment, evaluation of ground response to shaking, and assessment of structure response due to seismic shaking. Step 1: Definition of Seismic EnvironmentSeismic analysis of underground structures starts with site-specific definition of its seismic environment. A detailed field and laboratory investigation program is necessary; the field investigation program should include definition of the site stratigraphy and direct measurements of shear wave velocity profiles and cone penetration resistance of soft soils as well as assessment of potential geo-hazards including slope instability, fault displacement, and lateral spreading. Appropriate static and cyclic laboratory tests for major soil units are also required.Site-specific probabilistic and/or deterministic seismic hazard analyses, as well as hazard analyses using conditional (mean) spectra, are needed to define seismic hazard levels for permanent condition of the structure (operational and maximum levels).
Increasingly seismic hazard, using a shorter return period, is being considered for temporary conditions during construction. Often a two-level criterion is adopted: operating design earthquake (ODE) and maximum design earthquake (MDE). Those are defined using response spectra developed in the seismic hazard analysis. A suite of three component motions is needed for each of the design earthquake levels for site response analysis and soil-structure interaction modeling. It is preferred to use recorded motions instead of synthetic motions to spectrally match the target spectra. Ground motion spatial incoherence must be taken into account for long structures including (1) wave passage, (2) extended source effects, (3) ray-path effects, and (4) local site effects. One-dimensional equivalent linear and nonlinear site response analyses are conducted to assess how the ground motion is affected by the soil column.
One-dimensional site response analyses are used in the analysis of underground structures to. Step 2: Evaluation of Ground Response to ShakingEvaluation of ground response to shaking can be divided into permanent deformations or ground failure and transient deformations or ground shaking.Permanent deformations or ground failure includes liquefaction, slope instability, and fault displacement. Liquefaction, prevalent in loose sand and fill deposits, can result in generation of sand boils, loss of shear strength, lateral spreading, and slope failure. Tunnels in liquefiable deposits can experience increased lateral pressures, loss of lateral passive resistance, flotation or sinking, lateral displacements if lateral spreading happens, permanent settlement, and compression/tension failure after soil consolidation. A landslide intercepting a tunnel can result in concentrated shearing displacements and collapse of a cross section.
The potential for these failures is greatest when a pre-existing landslide intersects the tunnel, in shallower parts of tunnel, and at tunnel portals. An underground structure may have to pass across an active fault zone; in these situations the tunnel must tolerate the expected displacements. The design for permanent deformations is discussed in section “.”Transient deformations can be quite complex due to interaction of seismic waves with surficial deposits.
Underground structures undergo three primary modes of deformation during seismic shaking: compression-extension, longitudinal bending, and ovaling/racking. The design for transient deformations is discussed in section “.” Step 3: Assessment of Structure Behavior Due to Seismic ShakingThe evaluation of structure behavior will be primarily a deformation controlled soil-structure interaction problem. If fault displacements are small and/or distributed over a relatively wide zone, providing articulation of the tunnel lining through ductile joints is a possible solution. The closer the joint spacing, the better the tunnel performance will be; this is more viable in soft soils where displacements can be effectively redistributed over the tunnel lining.
The tunnel can then deform in an S-shape through the fault zone without rupture. It is always necessary to keep the tunnel watertight when using joints. If large displacements are concentrated in a narrow zone, retrofit will consist of enlarging the tunnel section across and beyond the displacement zone. The length over which the enlargement is made is a function of fault displacement and permissible curvature of the road or track; the longer the enlarged tunnel, the smaller the post-earthquake curvature (Power et al.
This solution has been implemented in the San Francisco BART system and Los Angeles Metro rapid transit tunnel system. Concrete-encased steel ribs provide sufficient ductility to accommodate distortions with little strength degradation.
Under axial displacements, even though compression is more damaging to the tunnel lining than extension, both will result in unacceptable water inflow. A solution for water tightness is flexible couplings (Wang ), used for the Southwest Ocean Outfall in San Francisco. Cellular concrete may also be used within the enlarged tunnel, because it has a low yield strength that can minimize the loads on the tunnel liner while also providing adequate resistance for normal soil pressures and other seismic loads. Estimating fault displacement is a key issue to design tunnels crossing active faults. One option to estimate fault displacement is using empirical relationships that express expected displacements in terms of some source parameter. Deterministic and probabilistic fault displacement hazard analyses can be used to assess fault displacement hazard where a displacement attenuation function is used in a probabilistic seismic hazard analysis (Coppersmith and Youngs; Youngs et al. Flotation in Liquefiable Deposits.
Liquefaction evaluation is discussed elsewhere. If liquefaction is limited to soil layers above the underground structure, then it is unlikely to influence the racking deformations of the structure. However, if the structure is partially or entirely embedded in liquefiable soil, additional evaluations are required. Underground structures may experience flotation in liquefiable deposits. As shown in Fig., when the tunnel experiences uplift due to flotation, the liquefied soil moves underneath the displaced tunnel and lifts it further up (Schmidt and Hashash ). Uplift can be prevented through isolation using cutoff walls, such as sheet pile walls; stone columns (Fig.
); or jet grout columns (Fig. Sheet piles with drainage capability can also reduce excess pore water pressure. The rise in excess pore water pressure is prevented at the bottom of the tunnel and in the soil underneath with these barrier walls. With longer barrier walls and a wider structure, uplift is more difficult. After the liquefaction potential is mitigated, flexible joints can be used to allow for differential displacements at tunnel connection joints.
The focus of underground structure seismic design is on free-field deformations of the ground and their interaction with the structure, since the inertia of the surrounding soil is large relative to the inertia of the structure. Figure shows response of underground structures to seismic motions: axial compression and extension, longitudinal bending, and ovaling/racking. Axial deformations are due to seismic waves producing motions parallel to the tunnel axis, bending is due to seismic waves producing particle motion perpendicular to the longitudinal axis, and ovaling/racking is due to shear waves propagating normally to the tunnel axis. Design considerations for axial and bending deformations are generally in the direction along the tunnel axis and in the transverse direction for ovaling/racking.
ApproachesAdvantagesDisadvantagesApplicabilityFree-field deformation methods1. Comparatively easy to formulate, many 1D wave propagation programs available1.
Nonconservative for tunnel structure more flexible than ground2. Conservative for tunnel structure stiffer than ground3. Overly conservative for tunnel structures significantly stiffer than ground4. Less precision with highly variable ground conditionsFor tunnel structures with equal stiffness to groundPseudo-static soil-structure interaction methods1. Good approximation of soil-structure interaction2. Comparatively easy to formulate3.
Reasonable accuracy in determining structure response4. Computationally efficient5. Sensitivity analysis can be easily performed1. Ignores inertial effects2.
Less precision with highly variable ground3. Shear displacement not transmitted uniformly to shallow box structuresMost conditions except for variable soil profile, shallow structuresDynamic soil-structure interaction finite-element analysis1. Best representation of soil-structure system2. Best accuracy in determining structure response3. Capable of solving problems with complicated tunnel geometry and ground conditions (significant variations in soil stiffness)1.
Computationally demanding2. Uncertainty of design seismic input parameters may be several times the uncertainty of the analysisAll conditionsDynamic earth pressure methods1. Serve as additional safety measures against seismic loading1. Lack of rigorous theoretical basis2. Resulting in excessive deformations for tunnels with significant burial3.
Use limited to certain types of ground propertiesNoneFree-field deformation methods assume that the underground structure deformations are identical to those of the surrounding ground. They do not take into account soil-underground structure interaction and are most appropriate when the structure (racking) stiffness is equivalent to that of the surrounding ground.Pseudo-static soil-structure interaction models account for the kinematic interaction between the soil and the underground structure neglecting inertial interaction. They are often used for practical design purposes when the structure is not too complex (NCHRP 611, Anderson et al.
( )).Nowadays, the ease of access to high performance computers makes it possible to perform dynamic soil-structure interaction analyses within a reasonable amount of time. These types of analyses allow problems with complicated tunnel geometry and ground conditions to be solved efficiently. However, the selection of parameters for a complex problem requires expertise; therefore, it is important to always verify the computer model solution with simpler pseudo-static or closed-form solutions.The presence of a rectangular frame structure in the ground will induce dynamic earth pressures acting upon the structure. Complex shear and normal stress distributions along the exterior surfaces of the structures are expected, but quantifying those distributions require rigorous dynamic soil-structure interaction, since they heavily depend on how the interface is modeled.In the past, the Mononobe-Okabe method was used to calculate the seismic-induced dynamic earth pressures on underground structures. The method assumes the earthquake load is caused by inertial forces of the surrounding soil and calculates the load using soil properties and a determined seismic coefficient. This method is not applicable in the case of underground structures, since they will move with the ground and will not form an active wedge.When designing underground structures for transient deformations, sufficient ductility is needed to absorb imposed deformations without losing the capacity to carry static loads.
Care should be exercised in not increasing the stiffness of the structure as this tends to attract additional loads thus increasing the demand on the structure. Free-Field Deformation ApproachFree-field deformations are the deformations caused by seismic waves on a given soil profile in the absence of structures or excavations. The interaction between the soil and the underground structure is neglected, but provides a first-order estimate of the underground structure deformation. Imposing the free-field deformations directly on the underground structure can underestimate or overestimate the structure deformations.
Closed-Form Elastic Solutions. 10Seismic waves causing longitudinal axial and bending strains (Power et al. )The strain bending component is relatively small compared to axial strains, but if the tunnel radius increases, the curvature contribution increases.
Tunnel cracks may open and then close in the lining due to the cyclic nature of the axial strains. As long as the cracks are small, are uniformly distributed, and do not affect the performance of the tunnel, even unreinforced concrete linings are considered adequate.
It is important to emphasize that the p- and s-wave velocities used are those of the deep rock. The range for s-wave is between 2 and 4 km/s and p-wave between 4 and 8 km/s (Power et al. Ovaling and Racking Deformation. Ovaling deformations, developed by waves acting perpendicular to the circular tunnel lining, are caused predominately by vertically propagating shear waves (Wang ). Ground shear distortions can be defined assuming a non-perforated ground or a perforated ground.
As shown in Fig., both cases ignore the tunnel lining (soil-structure interaction), where the maximum diametric strain is in terms of the maximum free-field shear strain ( γ max) and the Poisson ratio ( v m). The first can be used to approximate the behavior of a tunnel lining whose stiffness is equal to the medium it replaces. The second can be used to approximate the behavior of a tunnel lining whose stiffness can be neglected in comparison with the stiffness of the medium. 12Typical free-field racking deformation imposed on a rectangular frame (Wang ) Numerical AnalysisMany computer programs are available to estimate free-field shear distortions: SHAKE (Schnabel et al. ), FLUSH (Lysmer et al.
), D-MOD (Matasovic ), and DEEPSOIL (Hashash et al.; Hashash and Park ), among others. One-dimensional site response analyses can be used to characterize the change in the propagating ground motions on variable soil profiles, but these analyses only take into consideration vertically propagating shear waves. However, these are the waves that carry most of the seismic energy.
The analyses can be performed in equivalent linear frequency domain or nonlinear time domain. The resulting free-field shear distortion can be expressed in the form of shear strain or shear deformation profile with depth. Applicability of Free-Field Deformation ApproachThe free-field deformation approach is a simple and effective design tool when earthquake-induced ground motions are small. However, in structures located within soft soil profiles, the method gives overly conservative designs, because free-field ground distortions in these soils are large. It also neglects the difference in stiffness between the lining and the surrounding soil. The presence of an underground structure modifies the free-field deformations; methods to model this interaction will be described in the following sections.
Pseudo-static Soil-Structure InteractionIn pseudo-static soil-structure interaction analyses, the soil and structure inertia due to seismic shaking is neglected. The problem is simplified to that of a structure in a soil medium subjected to simple shear on horizontal and vertical planes.The beam-on-elastic foundation approach is used to model soil-structure interaction effects.
Both the lining and the medium are assumed to be linear elastic. Wang ( ) presents a summary of closed-form elastic solutions for axial force and moment developed for circular tunnels due to seismic waves propagating along and perpendicular to the tunnel axis. Adding stiffness and strength to the structure may attract more forces, so a better solution would be to add ductility.
These solutions are dependent on the estimates of appropriate spring coefficients compatible with anticipated displacements and wave lengths. They are often limited to idealized seismic wave forms.Most pseudo-static SSI analyses focus on the interaction of vertically propagating shear waves with the transverse section of a tunnel. These analysis approaches are described next.
Transverse: Ovaling Deformations of Circular TunnelsPeck et al. ( ) proposed closed-form solutions in terms of thrust, bending moments, and displacement under external loading.
The lining response was a function of structure compressibility and flexibility ratios, in situ overburden pressure, and at-rest earth coefficient. To adapt to seismic loading, the free-field shear stress replaces the in situ overburden pressure and earth coefficient. The stiffness of the tunnel relative to the ground is measured by the compressibility ( C) and flexibility ( F) ratios. Those are the extensional stiffness and flexural stiffness of the medium relative to the lining. Under this framework, Wang ( ) presented solutions for the diametric strain, the maximum thrust, and the bending moment under full-slip conditions, meaning normal force but no tangential shear force are present between the lining and the medium. For most cases the interface condition is between full slip and no slip.
Slip interface can only happen in tunnels in soft soils or cases of severe seismic loading and full-slip assumption may lead to underestimation of the maximum thrust. As shown in Wang ( ) and NCHRP 611 (Anderson et al. ), for ground Poisson’s ratio less than 0.5, thrusts decrease with decreasing compressibility ratio, but for Poisson’s ratio of 0.5, the thrust response is independent of compressibility.
The normalized lining distortion can be a plotted as function of flexibility ratio, as shown in Fig. When F 1.0, the lining is expected to deform more than the free field with an upper limit equal to the perforated ground case as described in Table. Penzien ( ) provides an analytical procedure to evaluate racking deformations of rectangular and circular tunnels. His solutions for ovaling deformations in terms of thrust and moment are very close to those of Wang ( ) for full-slip condition. However, the value of thrust for no-slip condition is much smaller in Penzien ( ) than in Wang ( ), differing in one order of magnitude. Flexibility ratio FMeaningF → 0The structure is rigid, so it will not rack regardless of the distortion of the groundF 1The structure racking stiffness is smaller than that of the soil.
Usually stiff soil, and racking deformations are smallF → ∞The structure has no stiffness, so it will undergo deformations identical to the perforated groundHashash et al. ( ) compared the two analytical solutions to finite-element method numerical analyses to validate which of the solutions provide the correct solution to this problem. The results from the numerical analyses agree with Wang ( ) solutions, highlighting the limitations of the other analytical solution. Sedarat et al. ( ) show that interface condition between the tunnel lining and the surrounding soil has an important impact on the computed thrust in the lining but limited impact on computed lining deformation. Transverse: Racking Deformations of Rectangular Tunnels. Box-shaped cut-and-cover tunnels, common for transportation tunnels, have seismic characteristics different from circular tunnels because the walls and slabs of the box-shaped tunnels are stiffer.
They are also often placed at shallower depths compared to circular tunnels. Therefore, it is important to carefully consider the soil-structure interaction due to increased stiffness and the increased seismic ground deformations at shallow depths (Hashash et al. Numerical analyses are often employed to compute the response of the tunnel structure to deformation of the surrounding soil. Wang ( ) and Anderson et al. ( ) employed such techniques to develop relationships between racking ratio and flexibility (Fig. 152D pseudo-static numerical analysis (Hashash et al. )To perform the 2D numerical analysis, the second step (Fig.
) is to define the elastic properties of a uniform soil medium as the average strain-compatible elastic properties of the selected soil layers (Anderson et al. As recommended in Hashash et al. ( ), layers 1–3 m above and below the structure should be included. Shear modulus values can be selected using the strain-compatible shear wave velocities from site response analysis in the selected layers, from which the average shear modulus over the selected layers can be calculated. In this step, the structural member properties are needed: E (stiffness), I (moment of inertia), and A (cross-sectional area).Finally, in a 2D numerical analysis, the lateral displacement (d im) time histories obtained from 1D site response analysis are applied at the left, right and top boundaries of the model to impose the free-field racking calculated in the first step on the model, as shown in Fig. With the numerical analysis, the soil medium will transmit shearing deformations to the box structure and the box racking deformation time history can be obtained.
This is used to obtain the racking ratio (R).Two-dimensional pseudo-static numerical analyses can be a very useful tool, but they have some limitations. The ground surface shear displacements for shallow box structures cannot be transmitted uniformly. The model can be artificially extended to address this problem. Racking deformations are assumed to vary uniformly over the height of the box structure. The response of individual layers is not represented. This becomes a problem when layers with very different stiffness are part of the soil profile.
Dynamic Soil-Structure InteractionThe complex soil-structure interaction of underground structures during seismic loading can be simulated using numerical analysis tools which include lumped mass/stiffness methods and finite-element/difference methods.Lumped mass/stiffness methods are useful to analyze the 3D behavior of a tunnel lining in a simplified manner. Many parameters for the springs that represent the structure stiffness and the soil stiffness must be defined to have a realistic model.In finite-element/difference models, the tunnel structure is discretized and the soil surrounding the tunnel can be either discretized or represented by springs. 2D and 3D models can also be used to analyze the inelastic sections of the tunnel cross section.
Hashash et al. ( ) performed a series of pseudo-static and dynamic soil-structure interaction analyses of single- and double-box structures in stiff and soft soil profiles using equivalent linear and nonlinear site response analysis with 14 ground motion time histories. The results are shown in Fig. The results of the study found that for F.
Pseudo-static longitudinal 3D models can be used to analyze axial and bending deformations in immersed-tube tunnels. In a lumped mass analysis approach, the tunnel lining is divided into individual segments with different masses and stiffnesses. The masses are then connected by springs that represent the axial, shear, and bending stiffness of the tunnel as shown in Fig. (Hashash et al. Free-field displacement time histories that consider effects of wave passage/phase shift and incoherence are calculated at selected locations along the tunnel’s length.
The computed free-field displacement time histories are then applied at the ends of springs to represent soil-tunnel interaction in a quasi-static analysis. If a dynamic analysis is needed, appropriate damping factors need to be incorporated into the structure as well as springs to represent the soil. 183D model for global response of immersed-tube tunnel (Hashash et al. )Recent work from Anastasopoulos et al.
( ) focuses on nonlinear response to strong seismic shaking of deep immersed tunnels (≈70 m). The free-field acceleration time histories are computed at the base of the tunnel through 1D wave propagation analysis using equivalent linear and nonlinear analyses. The computed free-field acceleration time histories are then imposed on the supports of the tunnel in the form of excitation. The tunnel is modeled as a multi-segment beam connected to the ground through calibrated springs, dashpots, and sliders. Wave passage effects are taken into account using Eurocode 8 (EC8 ); however, the geometric incoherence was not considered because it did not make a difference when added to wave passage. The soil is assumed to be uniform along the tunnel.A finite-element analysis is used to perform a nonlinear dynamic transient analysis of the tunnel.
Tunnel segments are modeled using beam elements that take into account shear rigidity. The joints are modeled with nonlinear hyperelastic elements. The bored tunnels at the end of both segments are incorporated in the analysis as beams on viscoelastic foundation. Influence of segment length and joint properties was then investigated parametrically.The results from Anastasopoulos et al. ( ) show that seismic response of immersed tunnel correlates better with PGV than PGA consistent with prior studies. There are some key conclusions applicable to immersed tunneling projects.
First, a properly designed immersed tunnel can resist near-fault soil-amplified excitation with a PGA as large as 0.6 g and PGV as large as 80 m/s and containing long period pulses. Also, the net tension and excessive compression between segments can be avoided by suitable design of joint gaskets and relatively small segment lengths. However, it is important to note that this study did not examine the heterogeneous nature of the soil conditions and ground motion incoherency and did not investigate the time-dependent stress relaxation on the rubber gasket or the effect of tectonic displacements from fault rupturing. Additional Seismic Performance Issues. Significant vertical ground motions often due to near-fault effects can impose vertical loads on the roof of box structures. Figure shows a schematic of the vertical loading on relatively shallow box structure (Hashash et al.
Two types of analyses can be performed to assess the vertical loads. The vertical acceleration near the ground surface can be estimated as part of the seismic hazard analysis and then be used to computer pseudo-static inertial load of the soil on the tunnel roof. Alternatively, 2D dynamic soil-structure interaction analysis representing both the underground structure and the soil can be performed to compute the vertical inertial loading on the roof. There remain significant uncertainties in selecting appropriate soil properties for propagating vertical ground motion in such a model.