Computers and Geotechnics 49 (2013) 338–351 Contents lists available at SciVerse ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo Effect of stress disturbance induced by construction on the seismic response of shallow bored tunnels Rui Carrilho Gomes ⇑ Civil Engineering and Architectural Dept., Technical Univ of Lisbon, IST, Av Rovisco Pais 1, 1049-001 Lisbon, Portugal a r t i c l e i n f o Article history: Received 28 July 2011 Received in revised form 26 June 2012 Accepted 13 September 2012 Available online 13 October 2012 Keywords: Tunnels Earthquake Stress disturbance Lining forces a b s t r a c t This paper examines the effect of the stress disturbance induced by tunnel construction on the completed tunnel’s seismic response The convergence-confinement method is used to simulate the tunnel construction prior to the dynamic analysis The analysis is performed using the finite element method and drained soil behaviour is simulated with an advanced multi-mechanism elastoplastic model, utilising parameters derived from laboratory testing of Toyoura sand The response of the soil and of the lining during dynamic loading is studied It is shown that stress disturbance due to tunnel construction may significantly increase lining forces induced by earthquake loading, and Wang’s elastic solution appears to underestimate the increase Ó 2012 Elsevier Ltd All rights reserved Introduction Historically, the general conviction has been that the effect of earthquakes on tunnels was not very important Nevertheless, some underground structures have experienced significant damage in recent earthquakes, including the 1995 Kobe Japan earthquake [1], the 1999 Chi-Chi Taiwan earthquake [2] and the 1999 Kocaeli Turkey earthquake [3] In recent decades, the number of large tunnels and underground spaces constructed has grown significantly In addition, the high cost of real estate has increased the demand for tunnels in large urban centres This work studies large-diameter tunnels at relatively shallow depth, commonly used in urban areas for metro structures, highway tunnels and large water and sewage transportation ducts In urban areas, tunnel excavation by boring maybe preferable to cut-and-cover excavation due to the existence of overlying structures Thus, excavation by boring is considered in this work Tunnels in earthquake prone areas are subjected to both static and seismic loading The most important static loads acting on underground structures are ground pressures and water pressure; in general, live loads can be safely neglected It is well known that tunnel excavation and the application of support measures induces three-dimensional (3D) deformation and stress redistribution during tunnel face advance The convergence-confinement method [4] is one of the most common assumptions used for considering the ⇑ Tel.: +351 218 418 420; fax: +351 218 418 427 E-mail address: ruigomes@civil.ist.utl.pt 0266-352X/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compgeo.2012.09.007 3D face effect in two-dimensional (2D) plane-strain analysis, as it approximates the stress relaxation of the ground due to the delayed installation of the lining [5,6] Before earthquake loading is applied to the tunnel, the initial stress field has already been disturbed by tunnel construction There are studies that simulate the tunnel construction prior to the seismic analysis of underground structures (e.g [7,8]), while other studies not simulate tunnel construction (e.g [9]) The main focus of this paper is to assess the influence of stress disturbance induced by construction on the seismic response of shallow tunnels The approaches used to quantify the seismic effect on an underground structure are summarized by Hashash et al [10] and include (i) closed-form elastic solutions to compute deformations and forces in tunnels for a given free-field deformation [11], (ii) numerical analysis to estimate the free-field shear deformations using one-dimensional (1D) wave propagation analysis (e.g Shake [12]), and (iii) 2D or 3D finite element or finite difference codes to simulate soil-structure response (e.g Flac [13], Flush [14], Gefdyn [15], CESAR-LCPC [16]) In this work, the finite element method is used, because an advanced constitutive model is required to evaluate accurately the effect of stress disturbance due to tunnel construction on the tunnel seismic response The advanced multi-mechanism elastoplastic model developed at École Centrale de Paris, ECP [17,18] is used, since this model can take into account important factors that affect soil behaviour, such as the strain level and the stress conditions, while the influence of other factors which control the stiffness degradation, such as the plasticity index and the initial 339 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 state (OCR, void ratio, stress state, etc.) are considered via the model parameters The ECP model has been widely used to simulate soil behaviour under static and cyclic loading (e.g [19,20]) and is implemented in the general purpose finite element code Gefdyn [15] This code is particularly suitable for modeling the cyclic behaviour of soils and soil-structure interaction, and it has been successfully used in the past to study the behaviour of geotechnical structures [21,22] In this work, the behaviour of Toyoura sand is simulated A 2D plane-strain finite element simulation of bored tunnel construction was performed using the convergence-confinement method [4] It was assumed that the stress disturbance induced by tunnel construction is controlled by a single parameter, namely the decompression level Three values of decompression level within the range of values usually adopted in practice are considered, to assess their effect on lining seismic response Subsequent to the simulation of tunnel construction, a set of eight dynamic analyses were performed in the time domain Ovaling deformation of the cross-section of circular tunnel due to ground shaking is studied, as it refers to the deformation of the ground produced by seismic wave propagation through the Earth’s crust – – – – – – rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 Àr0 2 ii jj qk ¼ þ ðr0ij Þ2 is the radius of the Mohr circle in the plane of the generic deviatoric mechanism of normal ~ ek Here i, j, k e {1, 2, 3}; i = + mod(k, 3), and j = + mod(k + 1, 3), with mod(k,0 i)0 representing the residue of the division of k by i; r þr p0k ¼ ii jj is the centre of the Mohr circle in the plane of the deviatoric mechanism of normal ~ ek ; p0c is the critical pressure that is linked to the volumetric plastic strain epv by the relation p0c ¼ p0co expðbepv Þ, where p0co represents the initial critical pressure, which is the critical mean effective stress that corresponds to the initial state defined by the initial void ratio, and b the plasticity compression modulus of the material in the isotropic plane (ln p0 , epv ); /0pp is the friction angle at the critical state; b is a numerical parameter which controls the shape of the yield surface in the ðp0k ; qk Þ plane (Fig 1) and varies from b = 0–1, passing from a Coulomb surface to Cam-Clay surface type; rk is an internal variable that defines the degree of mobilised friction of the mechanism k and introduces the effect of shear hardening of the soil The last variable is linked to the plastic deviatoric strain, according to the following hyperbolic function: epd;k , R r elk depd;k dt R a þ depd;k dt Constitutive model for soil rk ¼ 2.1 Model description where a is a parameter which regulates the deviatoric hardening of the material It varies between a1 and a2, such that: The soil behaviour is simulated over a large range of strains with ECP’s elastoplastic multi-mechanism model developed by Aubry et al [17] and extended to cyclic behaviour by Hujeux [18] The model is written in the framework of the incremental plasticity and is characterised by both isotropic and kinematic hardening It decomposes the total strain increment into elastic and plastic parts Whilst the elastic response is assumed to be isotropic, the plastic behaviour is considered to be anisotropic by superposing the response of three plane-strain deviatoric mechanisms (k = 1, 2, 3) and one purely isotropic (k = 4) With these assumptions, the total plastic strain increment deP is written as: deP ¼ depd þ depv ¼ p d X X k¼1 k¼1 epd;k þ epv ;k ð1Þ p where de and dev represent, respectively, the total deviatoric and volumetric plastic strain increments The former is given by the contributions, depd;k , of the three deviatoric mechanisms, the latter by the contributions of all four mechanisms The elastic response is assumed to be isotropic and non-linear with the bulk, K, and shear moduli, G, functions of the mean effective stress according to the relations: K ¼ K ref p0 p0ref !n e ; G ¼ Gref p0 p0ref þ ð4Þ a ¼ a1 þ ða2 À a1 Þak ðrk Þ ð5Þ where the intermediate of the parameter ak(rk) (Fig 2), integrates the decomposition of the behaviour domain into pseudo-elastic, hysteretic and mobilised domains, where: ak ðrk Þ ¼ ak ðrk Þ ¼  Àr hys rk r mob Àr hys ak ðrk Þ ¼ m if rel < r k < r hys if rhys < r k < rmob if r mob ð6Þ < rk < el r defines the extent of the elastic domain and is the minimum value that rk can take, while rhys and rmob designate the extent of the domain where hysteresis degradation occurs The evolution of the volumetric plastic strains is controlled by a flow rule based on Roscoe-type dilatancy rule:   q @ epv ;k ¼ @ epd;k ak ðr k Þ Á sin w À k0 Á aw pk ð7Þ with aw a constant parameter and w representing the characteristic angle [23] defining the limit between dilating ð@ epv ;k < 0Þ and contracting ð@ epv ;k > 0Þ of the soil (Fig 3) The primary yield function defined in Eq (3) can also be written in the form: !n e ð2Þ b=0 where Kref and Gref are respectively the bulk and shear modulus at the reference mean effective stress p0ref The degree of the non-linearity is controlled by the exponent ne During monotonic loading, the primary yield function associated to the generic mechanism k has the following expression:    À Á p Á rk fk qk ; p0k ; epv ; rk ¼ qk À p0k Á sin /0pp Á À b Á ln k0 pc where the following variables have been introduced: b=0.25 qk b=0.50 b=0.75 b=1 ð3Þ p'k Fig Influence of parameter b on the yield surface shape 340 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 Whenever a stress reversal occurs, the primary yield function (3) and (8) is abandoned and the cyclic surface becomes active The latter is defined by the function: À Á  n À h Á cyc h  p h h  fkc p0k ; sk ; r cyc À rcyc k ; ev ; dk ; nk ¼ sk À dk À r k nk k ð11Þ The surfaces associated to deviatoric yield functions are the circles of radius r cyc interior to circles of primary loading, both k tangents at the point dhk of exterior normal nhk , where the point dhk corresponds to the last load reversal h of the mechanism k (Fig 5): dhk ¼ Fig Evolution of ak(rk) qhk  p0  ; k p0h k sin /pp À ln p0 c rcyc k p'k Fig Critical state and characteristic state lines fk ðqk ; p0k ; epv ; r k Þ ¼ jsnk j À r k ð8Þ where the following variables have been introduced: – sk is the deviator stress vector in the k-plane of components r0 Àr0 sk1 ¼ ii jj and sk2 ¼ r0ij and norm jsk2 j ¼ qk The yield surfaces of each mechanism k can be interpreted in the normalised deviatoric plane ðsnk1 ; snk2 Þ: sk1 gk ; snk2 ¼ sk2 ð9Þ gk Á gk p0k ; epv ¼ p0k sin /0pp  ð13Þ in which ep;h d;k is the plastic deviatoric strain of the mechanism k at the last load reversal h The variable a(rk) obeys to the same relation as in monotonic loading (5) Finally, the constitutive equation set is completed with the equation describing the isotropic yield function which defines the last mechanism (k = 4) of the model The isotropic yield function is assumed to be: À Á f p0 ; epv ¼ p0 À p0c Á d Á r ð14Þ p0c where p and are respectively the current and the critical state mean effective pressure The parameter d represents the distance between the isotropic consolidation line and the critical state line in the (ln p0 , e) plane The internal variable r4 depends on model parameter c that controls the volumetric hardening of the soil as: r4 ¼   p À ln k0 pc ð10Þ In this plane, the deviatoric yield surfaces are circles of radius rk (Fig 4) (a) R   p p;h   ded;k dt À ed;k  R  ¼ r elk þ   a þ  depd;k dt À ep;h d;k  R with the normalisation factor gk given by: À ð12Þ h The vector ðdhk À rcyc k nk Þ corresponds to the vector going from the origin of the normalised deviatoric plane to the centre of the cyclic circle The vectors dhk and nhk are discontinuous parameters introducing kinematic hardening to the model [18] The hardening variable r cyc k can be expressed in terms of the position of the current stress state with respect to the position of the last load reversal with initial value equal to r elk : qk snk1 ¼ snh nhk ¼  knh  sk r el4 þ depd;4 dt R c Á pc =pref þ depd;4 dt The model parameter c is equal to c1 during monotonic loading, and equal to c2 during cyclic loading All the mechanisms are coupled through the total volumetric plastic strain given by: (b) τk ð15Þ s n k2 σ'ij s n k2 s sk n k σ'jj s p'k σ'ii σk n s k1 n k1 Fig Stress state representations for deviatoric mechanism k: (a) Mohr’s representation (normal stress, rk, vs shear stress, sk) in the i–j plane; (b) stress state in the normalised deviatoric plane ðsnk1 ; snk2 Þ 341 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 (a) s (b) n k2 s n k2 h nk h dk el rk cyc rk s el rk n k1 s n k1 m rk Fig Evolution of deviatoric yield surface in the normalised deviatoric plane of the mechanism k for: (a) monotonic loading; (b) cyclic loading depv ¼ depv ;1 þ depv ;2 þ depv ;3 þ depv ;4 ð16Þ Further analytical details concerning the model can be found in specific Refs [17,18] 2.2 Model’s parameters and laboratory tests simulations 2.2.1 Introduction Toyoura sand is a clean, uniform, fine sand with zero fines content, commercially available washed and sieved, which has been widely used for liquefaction and other studies, in Japan and worldwide [24,25] The behaviour of Toyoura sand has been characterised under a number of different stress paths, the results of which have been used to provide constitutive parameters for this study The soil is assumed to be homogeneous in each analysis performed The model’s parameters were evaluated for the sand with relative density equal to 40% (initial void ratio e0 = 0.833) The model’s parameters can be divided in those than can be directly measured: – Elasticity: Kref, Gref, ne and p0ref – Critical state and plasticity: /0pp , b, d and p0co – Yield function and hardening: w and those that are non-directly measured: – Critical state and plasticity: b – Yield function and hardening: a1, a2, c1, c2, aw and m – Threshold domains: rel, rhys, rmob and r el4 ð2:17 À e0 Þ2 0:4 ðp Þ ½kPaŠ þ e0 bffi þ e0 ¼ 21:5 k ð19Þ 2.2.3 Determination of non-directly measurable parameters As the paper is devoted to seismic applications, in the numerical simulations made for identification of the non-directly measurable parameters, simple shear loading was assumed The strategy for model parameter identification detailed in Santos et al [19] and Gomes [28] has already been explored and verified The strategy to derive model parameters related to shear hardening relies directly on experimental data represented by a strain-dependent stiffness degradation curve The calibration of the parameters is derived from the reference ‘‘threshold’’ shear strain, c0.7, that defines the shear strain for a stiffness degradation factor of G/G0 = 0.7 In effect, the reference threshold shear strain defines the beginning of significant stiffness degradation The ‘‘standard’’ shape for the ak = f(rk) curve proposed by Santos et al [19] is shown in Fig This curve can be determined by means of the point (r0.7; a0.7) obtained from the experimental strain-dependent stiffness curve, according to the following equations: r0:7 ¼ 2.2.2 Determination of directly measurable parameters The initial shear modulus, G0, is estimated according to the following equation from Iwasaki et al [26] for Toyoura sand: G0 ¼ 14100 For e0 = 0.833, the critical state line, CSL, presented by Ishihara [24] has abscissa p0co ¼ 1200 kPa and slope, k, equal to 0.0852 The parameter d (horizontal distance between the isotropic consolidation line and the critical state line) is equal to 5.8 Hajal [27] proposed the following relationship to calculate the plastic compressibility modulus, b: 0:7G0 c0:7  ; p0 p0k sin /0pp À b ln pk0 c 0:3c0:7 aðr 0:7 Þ ¼ 1 þ lnð1 À r Ã Þ Ã 1Àr rà with r à ¼ r 0:7 À r elas ð17Þ For e0 = 0.833 and p0ref ¼ MPa, Gref defined by Eq (2) becomes equal to 218 MPa and ne = 0.4 The initial bulk modulus, k0, was derived from the following relationship, valid for homogeneous isotropic linear elastic materials, assuming the Poisson’s ratio equal to m = 0.2: K0 ¼ 2G0 ð1 þ mÞ ¼ 1:333 Á G0 3ð1 À 2mÞ ð18Þ According to Eq (2), Kref becomes equal to 291 MPa Based on drained and undrained monotonic triaxial tests, Ishihara [24] determined that /0pp ¼ w ¼ 31 Fig Standard shape for the relationship rk À ak ð20Þ 342 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 Table Model parameters for Toyoura sand (Dr = 40%) Layer 0–5 m el 5–10 m À3 r a1 a2 10–15 m À3 2.18  10 1.50  10À5 2.06  10À3 15–20 m À3 1.13  10 2.50  10À5 3.13  10À3 1.01  10 2.50  10À5 2.60  10À3 20–25 m À4 25–30 m À4 8.18  10 2.50  10À5 3.13  10À3 7.61  10 2.50  10À5 4.44  10À3 6.81  10À4 2.50  10À5 5.33  10À3 À3 Gref = 218 MPa; Kref = 291 MPa; p0c0 ¼ 1200 kPa; p0ref ¼ MPa, m = 1; ne = 0.4; /0pp ¼ 31 ; w = 31°; b = 21.5; aw = 1; rmob = 0.8; b = 0.2; d = 5.8; c1 = 0.06; c2 = 0.03; r el ; ¼ 10 Coefficient of earth pressure at rest: k0 = 0.5 Volumetric unit mass: q = 1900 kg/m3 The parameters rel and a2 can be determined using the following relationship: r el ¼ p0k G0 cet  ; p0 À b ln pk0 sin /0pp a2 ¼ a0:7 ðr mob À rel Þ ðr 0:7 À rel Þ ð21Þ c where cet is the elastic threshold shear strain For Toyoura sand, cet ffi 10À6 For sands, b is small (b = 0.2) The parameter a1 assumes small values from matching simulated and experimental straindependent shear modulus curves The parameter rmob is usually taken equal to 0.8, and rel = rhys In addition, it was noticed that varying the parameters rel, a1 and a2 with mean effective stress improve the match between experimental and simulated results At last, the parameters c1, c2, rel4 and aw were determined to obtain best fitting of the experimental undrained triaxial tests [24] According to the proposed strategy and the available experimental data [24,26,29] the parameters summarized in Table were evaluated for the sand with relative density equal to 40% (initial void ratio e0 = 0.833) Fig shows experimental data from resonant column (RC) and cyclic torsional shear (CTS) tests [26,29] and the model response for a single element under stress controlled cyclic simple shear loading The model response is in good agreement with the experimental stiffness degradation curve, while damping tends to be underpredicted by the model Simulation of construction 3.1 Model The finite element mesh (Fig 8) simulates a soil mass 30 m thick in plane-strain conditions with 1254 isoparametric 4-node rectangular elements, overlying an impervious isotropic linear elastic half-space (G = 500 MPa, q = 2000 kg/m3) 0.5 0.8 0.4 increase p' increase p' 0.6 0.3 0.4 0.2 0.2 0.1 Damping ratio, Shear modulus ratio, G/G0 1.0 During the different stages of the static analysis that simulated tunnel construction, the nodes along lower boundary of the mesh were fixed in both the horizontal and vertical direction, the nodes along the lateral boundaries were fixed in the horizontal direction, while all other nodes were free in both the horizontal and vertical direction During the dynamic stage, the nodes along the lower boundary were freed in the horizontal direction in order to apply the seismic input motion and to activate the absorbing elements These are linear 2-node elements developed to simulate radiation conditions at the base of the finite element model by eliminating the elastic waves that would otherwise be reflected back into the interior of the finite domain by the artificial boundaries of the model [30] This objective is achieved by imposing additional actions reproducing the dynamic impedance at the nodes of the model boundary which characterises the interface between the finite and infinite domain [31] The latter is regarded as a 1-phase elastic medium Kuhlemeyer and Lysmer [32] suggested that the maximum element size, hmax, in the direction of wave propagation should be less than approximately one-tenth to one-eighth of the lowest wavelength of interest in the simulation The latter may be evaluated by the ratio between the minimum wave velocity, Vmin, and the highest frequency of the input wave, fmax In the model studied, Vmin is equal to 100 m sÀ1 near the surface, and fmax can be taken as equal to 10 Hz, thus, hmax should not exceed 1.25 m in the direction of wave propagation In this work, the vertical element size adopted is 1.0 m The lining is modelled as continuous and impervious circular ring with linear elastic behaviour using 40 beam elements No relative movement (no-slip) was allowed on the tunnel lining-soil interface Interface elements were not employed as there was no basis to determine their properties and they could potentially dominate the model response The potential separation of the two materials was examined by monitoring the normal stresses acting at the interface In all computations this potential separation never occurred The maximum mobilized strength ratio (ratio between mobilized strength to the available strength) was examined along the perimeter of the tunnel The maximum mobilized strength ratio reaches a value of only for the two strongest input motions This occurs in limited regions of the soil along the perimeter of the tunnel 100 m 0.0 1E-06 1E-05 1E-04 1E-03 1E-02 0.0 1E-01 Sand (30 m) Shear strain, γ Experimental 15 m Simulation Fig Toyoura sand (Dr = 40%, p0 = 25, 50, 100 and 200 kPa): strain-dependent shear modulus and damping curves from RC and CTS tests and model response for single element under stress controlled drained cyclic simple shear loading 10 m Elastic half-space Fig Finite element mesh R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 3.2 Lining properties A concrete lining subjected to bending and axial load often cracks and behaves in a non-linear fashion, and/or may have joints According to Wang [11], the ratio between lining effective stiffness and full-section stiffness is within the range 0.30–0.95 Lining material non-linearity effects were taken into account in an approximate manner by adopting an effective thickness of 0.35 m in the numerical simulations, which corresponds to fullsection thickness of 0.45 m So, a ratio between effective and fullstiffness of 0.48 was adopted for bending stiffness and 0.78 for axial stiffness, reflecting that the effect of cracking is greater for the bending stiffness than the axial stiffness The mechanical parameters are taken as the typical properties of a reinforced concrete lining: Young modulus, Ec = 32 GPa, Poisson ratio m = 0.2, volumetric unit mass q = 2550 kg/m3, compressive strength of concrete, fc = 38 MPa, and yield strength of the rebar, fsy = 500 MPa 343 as consequence of the tunnel construction The higher the k-parameter, the lower the vertical effective stress around the tunnel, as higher decompression of the soil is allowed The stress field at the end of the construction phase is the initial stress field for the subsequent seismic analysis To evaluate if the maximum strength of the concrete lining is exceeded, bending moment-axial load interaction diagrams for the lining were computed According to Eurocode [33], the maximum reinforcement area, As,max, should be 0.04Ac, where Ac is the cross sectional area of concrete Fig 11 plots the evolution of lining forces during the construction stages against the maximum strength of the concrete lining for As,max = 0.04Ac It is verified that maximum lining forces are about 5% of the maximum strength of the concrete The maximum compressive stress in the concrete lining from all analyses performed is about 4.8 MPa According to Eurocode [33], as the maximum compressive strength does not exceed 0.4fcm = 15.2 MPa, the hypothesis of linear elastic behaviour is valid 3.3 Simulation of construction Tunnel construction has been simulated using a procedure based on the convergence-confinement method [4] The first calculation phase of the procedure involves switching off both lining and ground elements inside the tunnel and applying an initial pressure p0 inside the tunnel to balance the initial geostatic stress field Afterward, the pressure p0 are reduced to (1 À k) Á p0, where k is the proportion of unloading before the lining is installed and is called the decompression level In the second calculation phase, the lining elements are activated, and the pressure applied inside the tunnel reduced to zero For stiff linings, the remaining ground stresses will largely go into the lining A large k-parameter corresponds to large unsupported lengths and/or late installation of the tunnel lining In this case, ground deformations will be relatively large, whilst structural forces in the lining will be relatively low Conversely, a smaller k-parameter leads to reduced ground deformations and larger structural forces in the lining In 2D numerical analyses of open face tunnels, a decompression level k of around 50% is commonly used Closed shield tunnelling, however, is typically represented by a reduced decompression level of around 20–30% [6] To cover a large range of tunnelling methods and to evaluate the influence of the k-parameter on the seismic response, three values were used in the simulations presented in this work: 5%, 20% and 50% 3.4 Tunnel construction results The convergence curve at the tunnel crown due to tunnel construction is shown in Fig 9a The dashed line represents the vertical stress evolution at the tunnel crown without lining placement, while the continuous line represents the vertical stress after lining placement Lining axial load, N, and bending moment, M, at the end of tunnel construction are shown in Fig 9b The curves in Fig 9b show that the less the decompression level is prior to lining installation, the higher the forces induced in the lining The sections with higher forces are in the crown (h % 0), floor (h % 180°) and sidewalls (h % 90° and 270°) The deformed shapes of the lining are shown in Fig 9c As expected, higher lining deformation of the soil occurs for larger values of k-parameter The profile of effective vertical stress, r0v , at the end of construction phase in the free-field and in the tunnel centre (Fig 10), are compared with the initial effective stress, r0v In the free-field r0v is equal to the initial vertical effective stress, r0v , as it is not affected by tunnel construction In the profile crossing the tunnel centre, r0v , diverges from r0v for depths greater than about m, Seismic response 4.1 Input earthquake motions The near source seismic scenario established by the Portuguese National Annex to Eurocode [34] was used as reference to select input earthquake motions from the European Strong-Motion Database [35] The following criteria were adopted: local magnitude between and 7, source-to-site distance from 15 to 35 km Eight seismic records from measurement sites on rock were available and the horizontal component with higher peak ground acceleration was used The properties of the selected records are presented in Table 2, namely the surface wave magnitude, Ms, the epicentral distance, R, the horizontal peak ground acceleration, PGA, the Arias Intensity, AI, and the mean period, Tm Fig 12 shows the pseudo-spectral acceleration (PSA) of all the time histories 4.2 Seismic analysis As tunnels are completely embedded in the ground and the inertial force induced by seismic wave propagation on the surrounding soil is large relative to the inertia of the structure, the model must be able to simulate the free-field deformation of the ground and its interaction with the structure In the analysis, where only vertically incident shear waves are introduced into the domain and the lateral limits of the problem are considered to be sufficient far not to influence the predicted response, the ground response is assumed to be the free-field response Thus, the width of the model plays an important role in ensuring the development of free-field deformation far away from the tunnel A sensitivity study based on similar conditions and using modal analysis, found that the soil-tunnel interaction region can extend up to three diameters from the tunnel centre [36] In this work, the lateral boundaries of the mesh were placed five diameters from the tunnel centre and the equivalent node condition was imposed at the nodes of the lateral boundaries, i.e the displacements of nodes at the same depth on the lateral boundaries are equal in all directions The incident waves defined at the outcropping bedrock (elastic half-space of the soil profile defined in Fig 8) are introduced into the base of the model after deconvolution performed in the linear range Thus, the obtained movement at the top of the elastic halfspace is composed of the incident waves and the reflected signal 344 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 (b) 300 N: 50% 50% M: 20% 20% 5% 5% 250 60 40 200 λ=5% 150 -400 λ=20% λ=50% 100 20 N (kN/m) Vertical stress (kPa) Sand - t=0.45m -800 -20 -1200 50 After lining placement Without lining 0.000 -40 -1600 0.005 0.010 0.015 0.020 0.025 -60 Crown Vertical displacement (m) (c) M (kNm/m) (a) 90 180 Floor 270 360 Crown θ (º) 6.0 Distance from tunnel centre (m) Displacement amplification factor = 100 cm 4.0 2.0 θ 0.0 -2.0 -4.0 -6.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 Distance from tunnel centre (m) Undeformed shape 20% 50% 5% Fig Effect of the decompression level on (a) the convergence curve at the tunnel crown, (b) lining forces and (c) lining deformed shape at the end of construction σ'v 200 σ'v 400 600 0.0 200 400 600 0.0 5.0 5.0 10.0 10.0 Depth (m) Depth (m) Free-field 15.0 20.0 15.0 20.0 Tunnel 25.0 25.0 σ'v0 σ'v0 30.0 30.0 50% 20% 5% 50% 20% 5% Fig 10 Effect of the decompression level on the vertical effective stress at the end of construction: profile in the free-field (lateral boundary), and profile in the centre of the tunnel 345 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 Bending moment (kNm/m) Maximum strength 1000 Evolution of construction stage 500 Start End -500 20 -2 Spectral acceleration (ms ) 1500 15 10 -1000 0.0 0.5 Maximum strength -1500 -2500 -2000 -1500 -1000 -500 1.0 1.5 2.0 2.5 Period (s) Fig 12 Pseudo-acceleration response spectra of the selected time-histories Axial force (kN/m) 50% 20% 5% Fig 11 Interaction diagrams: evolution of lining forces during the construction stages and the maximum strength of the concrete lining -2 Acc (ms ) In all the analyses, a time step of Dt = 0.005 s was used and an implicit Newmark numerical integration scheme with c = 0.625 and b = 0.375 is used in the dynamic analysis [37] -2 -4 4.3 Seismic response of the soil -6 4.3.1 Single earthquake time-history This section highlights the modifying effects of the stress disturbance induced by tunnel construction on the seismic response of the soil First, the details of the seismic response computed from a single time-history are presented, then the results from all eight records are collated and discussed The selected time-history is the Avej earthquake (number from Table 2, and Fig 13) This time-history has the highest Arias Intensity and the second highest PGA, thus it induces large deformations on the tunnel and in the ground Fig 14 presents the p0k À qk stress path of the activated deviatoric mechanism of two integration points at the depth of the tunnel centre (depth = 20 m): one point at the free-field (lateral boundary of the mesh, 50 m from the tunnel centre) and the other point is near the tunnel (1 m from the tunnel sidewall) In this figure the peak yield surface (Eq (3) with rk = 1), the mobilized yield surface, the critical state line, CSL, and the initial stress line in the free-field computed with coefficient of earth pressure at rest, k0, equal to 0.5 are presented The stress path starts from point The integration points in the free-field coincide with the initial stress line The integration points near the tunnel are affected by the stress disturbance induced by tunnel construction The position of point is consistent with the deformation mechanism shown in Fig 9c For low k-value, the tunnel deforms and pushes into soil at sidewalls, and thus the normal stress increases leading to reduced qk and increased p0k 10 t (s) Fig 13 Acceleration time-history number (Table 2): Avej earthquake The maximum extent of the mobilized yield surface is identified as Point 2, in some cases this coincides with the peak yield surface Point is reached simultaneously at t = 4.510 s in all the analyses From Point onwards the behaviour is highly non-linear near the tunnel, as it produces significant hysteresis loops The dynamic analysis ends at Point In the free-field, the integration points exhibit relatively narrow stress paths, with a single large loop preceded and followed by several small loops Near the tunnel, the integration points have more large loops and bigger variation of mean effective pressure The acceleration time histories at surface above the tunnel (1 m from the sidewall) and at the free-field are shown in Fig 15 The accelerations in the free-field and near the tunnel are similar The effect of the decompression level on the acceleration is relatively small The time that the stress path reaches Point (t = 4.51 s) is marked in Fig 15 This instant coincides with a negative peak acceleration that appears after a set of cycles of high peak acceleration So, the maximum extent of the mobilized yield surface (Point 2) is consequence of these cycles of high peak acceleration Table Properties of the selected records No Earthquake Country/year Station Ms R (km) PGA (m sÀ2) AI (m sÀ1) Tm (s) Friuli Montenegro Campano Lucano Campano Lucano Kozani Umbria Marche South Iceland Avej Italy, 1976 Serbia and Montenegro, 1979 Italy, 1980 Italy, 1980 Greece, 1995 Italy, 1997 Iceland, 2000 Iran, 2002 Tolmezzo-Diga Ambiesta Ulcinj-Hotel Albatros Bagnoli-Irpino Sturno Kozani-Prefecture Assisi-Stallone Thjorsarbru Avaj Bakhshdari 6.5 7.0 6.9 6.9 6.5 5.9 6.6 6.4 23 21 23 32 17 21 15 28 3.50 2.20 1.78 3.17 2.04 1.83 5.08 4.37 0.80 0.74 0.45 1.51 0.23 0.27 1.35 1.74 0.39 0.72 0.94 0.82 0.33 0.28 0.36 0.28 346 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 300 300 Initial stress line at the free-field Peak yield surface Mobilized yield surface CSL Initial stress line at the free-field Peak yield surface Mobilized yield surface CSL 200 Free-field 5% 100 qk (kPa) qk (kPa) 200 Tunnel 5% 100 0 100 200 300 400 500 100 200 p'k (kPa) 300 qk (kPa) qk (kPa) Free-field 20% 500 Initial stress line at the free-field Peak yield surface Mobilized yield surface CSL 200 100 400 300 Initial stress line at the free-field Peak yield surface Mobilized yield surface CSL 200 300 p'k (kPa) Tunnel 100 20% 1 0 100 200 300 400 500 100 200 300 300 Initial stress line at the free-field Peak yield surface Mobilized yield surface CSL 200 500 400 500 qk (kPa) qk (kPa) Free-Field 50% 400 Initial stress line at the free-field Peak yield surface Mobilized yield surface CSL 200 100 300 p'k (kPa) p'k (kPa) Tunnel 50% 100 3 0 100 200 300 400 500 100 200 p'k (kPa) 300 p'k (kPa) Fig 14 p0 –q stress paths of integration points at depth of the tunnel centre (z = 20 m) for various decompression levels (time-history 8) 8 Free-field (surface) Point (t = 4.51 s) Point (t = 4.51 s) -2 Acc (ms ) -2 Acc (ms ) Near the tunnel (surface) -2 -4 -8 -2 -4 50% 20% 5% -6 50% 20% 5% -6 -8 10 t (s) 10 t (s) Fig 15 Acceleration time histories for various decompression levels (time-history 8): free-field vs near the tunnel Fig 16 shows the acceleration response spectra in the free-field (lateral boundary) and above the tunnel at ground surface The response spectra in the free-field are similar for the three values of decompression level Above the tunnel, the response spectra are also similar in shape, but the effect of stress disturbance can be observed in the ordinates axe In general, a lower decompression level leads to slightly higher spectral acceleration, compared to the cases with larger decompression level 347 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 40 Free-field (surface) 35 30 Spectral acceleration (ms-2) Spectral acceleration (ms-2) 40 50% 20% 5% Input signal (earthquake 8) 25 20 15 10 0.0 Tunnel (surface) 35 30 50% 20% 5% Input signal (earthquake 8) 25 20 15 10 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 Period (s) 1.5 2.0 2.5 Period (s) Fig 16 Pseudo-response spectrum at ground surface for various decompression levels (time-history 8): free-field vs tunnel 4.3.2 All time-histories The influence of site effects on the seismic response of the ground is analysed in this section The PGA of the input motion is compared with the PGA at soil surface in the free-field and above the tunnel for all the selected time histories and for various decompression levels (Fig 17) In the free-field, the peak acceleration is nearly independent of the decompression level, no clear trend can be observed Above the tunnel, the influence of the decompression level can be seen, particularly at higher input PGA and, in general, the lower decompression level lead to slightly higher PGA at the soil surface 4.4 Seismic response of the lining 4.4.1 Single earthquake time-history This section highlights the modifying effects of stress disturbance induced by tunnel construction on the seismic response of the lining Again, the results based on time-history number are presented in detail Fig 18 shows the variation of the lining forces in the sidewall of the tunnel (h = 90°, depth = 20 m) during the seismic loading The axial force increases during the seismic action, because progressive plastification of the soil increases the vertical load transmitted to the lining This effect is greatest for the higher value of k The bending moment varies significantly during the seismic event, due to the ovalization of the tunnel lining The post-event bending moment increases in absolute value, approximately doubling the values prior to seismic loading The larger variation of the lining forces occurs between and s approximately, which correspond to the most intense part of the earthquake For t > s, the lining forces remain relatively constant Fig 19 plots the lining forces along the lining section (angle h) at the end of construction, at the end of the seismic analysis and the maximum forces envelop, for the analysis with decompression level equal to 50% and time-history number Just occasionally the envelop curves coincide with the end of construction or the end of the seismic analysis curves, which indicates that in general the maximum forces occur during earthquake loading The absolute maximum increment in lining forces along the lining section developed during the seismic loading are presented in Fig 20 The absolute maximum increment in bending moment, |DM|, and axial force, |DN|, due to seismic loading has peaks near the 45° diagonals (h % 45°, 135°, 225° and 315°) This is consistent with the ovalisation of a ring due to shear loading (e.g [10]) The maximum increment in axial force occurs near the floor (h % 135° and 225°), while the maximum increment in bending moment occurs near the crown (h % 45° and 315°) Higher k-parameter leads to higher increment in axial force for all lining sections, while for increment in bending moment no clear trend is noticeable 4.4.2 All time-histories The lining forces computed with all selected time-histories (see Section 4.1) are analysed in this section To assess the importance of taking into account the seismic loading in the lining design, Fig 21 shows the ratio between total maximum lining forces (construction + seismic loading) and the maximum lining forces induced by construction simulation The lining forces ratio grows proportionally to the PGA of the input motion and to the decompression level The bending moment ratio varies from 48% to 766%, while the axial force ratio varies from 7% to 126% When the decompression level changes from 1.0 1.0 0.8 50% Free-field (surface) PGA - soil surface (g) PGA - soil surface (g) 50% 20% 5% 0.6 1:1 0.4 0.2 Above the tunnel (surface) 20% 5% 0.8 0.6 1:1 0.4 0.2 0.0 0.0 0.0 0.1 0.2 0.3 0.4 PGA - input motion (g) 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 PGA - input motion (g) Fig 17 Peak ground acceleration (PGA) at the soil surface for various decompression levels and with all time-histories: free-field vs above the tunnel 348 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 50% 20% 100 5% M (kNm/m) N (kN/m) -500 -1000 -1500 50 -50 50% 20% 5% -100 -2000 10 10 t (s) t (s) Fig 18 Lining forces during seismic loading (sidewall h = 90°, z = 20 m) for various decompression levels (time-history 8) 200 λ = 50% λ = 50% -500 N (kN/m) M (kNm/m) 100 -100 -1500 End construction End seismic analysis Envelop -200 90 180 -1000 End construction End seismic analysis Envelop -2000 270 360 90 180 θ (º) Crown 270 360 θ (º) Floor Crown Crown Floor Crown Fig 19 Lining forces at end of construction phase, end of the seismic analysis and envelop for k = 50% and time-history 1000 200 20% 50% 5% 150 | N| (kN/m) | M| (kNm/m) 50% 100 20% 5% 750 500 250 50 0 90 180 270 360 90 Crown 270 180 θ (º) 360 θ (º) Floor Crown Crown Floor Crown Fig 20 Maximum increment in lining forces during earthquake loading for various decompression levels (time-history 8) 150% 50% 800% 20% 0.97 5% 600% 0.97 400% 0.98 200% R2 |Ntotal|/|Nconstruction|-1 |Mtotal|/|Mconstruction|-1 1000% 50% 125% 0.79 20% 100% 5% 75% 0.74 50% 0.76 25% R2 0% 0% 0.0 0.1 0.2 0.3 0.4 PGA input motion (g) 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 PGA input motion (g) Fig 21 Ratio between maximum total force and maximum forces during construction phase for various decompression levels and for all time-histories 349 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 5% to 20%, the axial force ratio increases in average 47% and the bending moment ratio increases 31% When the decompression level changes from 5% to 50%, the axial force ratio increases in average 184% and the bending moment ratio increases 126% Thus, larger construction induced stress disturbance lead to greater lining forces ratio The R-square coefficients are high (above 0.97 for bending moment ratio, and around 0.75 for axial force ratio), indicating strong correlation between the lining forces ratio and the PGA of the input motion Fig 22 plots the overall maximum lining forces computed for all time-histories, and compares them with the maximum strength of the concrete lining The maximum lining forces not exceed about 35% of the maximum strength of the concrete, indicating that a concrete lining can be designed to withstand the combined static and seismic loading The maximum compressive stress in the concrete lining in all analyses performed is about 12.6 MPa According to Eurocode [33], as the maximum compressive strength does not exceed 0.4fcm = 15.2 MPa, the hypothesis of linear elastic behaviour is valid – the ground is an infinite, elastic, homogeneous and isotropic medium; – the tunnel and the lining are circular and the lining thickness is small in comparison to the tunnel diameter Seismic actions are considered as external static forces acting on the tunnel lining, induced by the ground distortion related to a vertically propagating shear wave The resulting ovalisation of the tunnel lining is assumed to occur under plane strain conditions The detailed solution for the no-slip condition is summarised in Appendix A The maximum free-field shear strain at the tunnel depth, cff, introduced in the analytical solution (Table 3) is the average of the maximum shear strain on the lateral boundary of the numerical model, at the depth of the tunnel Fig 23 shows a strong rela- 2.E-03 R2 = 0.91 4.5 Analytical solutions 1.E-03 The closed-form solution to predict the transverse seismic response of the tunnel, summarised in [11], was adopted for comparison with the numerical analyses This solution takes into account explicitly the soil-structure interaction effect under both no-slip and full-slip conditions This method is based on the following assumptions: 0.E+00 0.0 0.1 Maximum strength 0.4 0.5 0.6 Fig 23 Relation between maximum free-field shear strain at the tunnel depth, cff with the peak acceleration of all time-histories 1000 500 Mobilized strength ratio Bending moment (kNm/m) 0.3 PGA input motion (g) 1500 -500 -1000 Maximum strength -1500 -2500 0.2 1.00 0.75 0.50 0.25 t=4.510 s 0.00 -2000 -1500 -1000 -500 0 180 90 Axial force (kN/m) 50% 20% 270 360 θ (º) 5% 50% Fig 22 Interaction diagrams for various decompression levels: overall maximum lining forces and the maximum strength of the concrete lining 20% 5% Fig 24 Mobilized strength ratio along the lining at t = 4.510 s for various decompression levels (time-history 8) Table Maximum increment in lining forces due to seismic loading computed by the numerical model and Wang solution (no-slip) for all time-histories TH PGA (m sÀ2) 3.50 2.20 1.78 3.17 2.04 1.83 5.08 4.37 cff 9.5  10À4 6.6  10À4 5.1  10À4 9.0  10À4 5.9  10À4 2.9  10À4 1.4  10À3 1.0  10À3 Gm (kPa) 56.7 67.3 74.8 58.4 70.3 91.2 46.0 54.6 |DMmax| (kN m/m) |DNmax| (kN/m) Wang 50% 20% 5% Wang 50% 20% 5% 83 58 45 78 53 26 119 89 120 66 43 112 64 27 189 152 138 72 45 116 65 27 206 178 134 71 45 121 71 31 233 188 363 296 252 352 278 172 432 377 837 764 473 781 500 258 1062 872 736 742 428 752 435 221 905 827 710 580 407 636 395 210 875 649 350 R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 tion between the PGA of the input motion and the maximum freefield shear strain at the tunnel depth The shear modulus of the ground, Gm, (Table 3) was estimated using the stiffness degradation curve (G/G0 À c, Fig 7) The G0 at the depth of the centre of the tunnel (%125 MPa) was multiplied by the shear modulus ratio corresponding to the value of cff to compute Gm According to Wang [11], the flexibility ratio F is the most important parameter to quantify the ability of the lining to resist the distortion imposed by the ground For the cases considered, F is between 17 and 34 with an average value of 24 Thus, according to this parameter, the lining deflects more than the soil being excavated Within this range of values of F, no relevant slippage between the soil and the tunnel is expected In fact, this feature becomes crucial only for F < 1, as, for example, in the case of the tunnel built in very soft ground Fig 24 presents the mobilized shear strength ratio, defined as the ratio between the mobilized strength to the available strength, distribution in the soil along the perimeter of the tunnel for t = 4.510 s, the instant where the maximum extent of mobilized yield surface occurs (Fig 14) Although the maximum mobilized strength ratio at t = 4.510 s reaches a value of 1.0, it occurs in confined regions of the soil No-slip assumption remains adequate, because slip in the interface soil-tunnel may occur only in these limited zones Since the effects of tunnel construction are not taken into account in analytical solutions, the comparison gives an indication of the significance of modelling the tunnel construction for practical applications Table summarises the increments in the axial force and bending moment in the tunnel lining, computed for no-slip conditions using Wang’s method and those obtained with the numerical model for various decompression levels Fig 25 shows the deviation between these two approaches, defined as: DeviationjDM max j ¼ jDM max jNumerical model À jDM max jWang jDMmax jWang are small, because the degree of non-linearity at tunnel depth is relatively small (e.g for cff %  10À4 the G/G0 % 0.7) The regression lines clearly diverge and the values of the deviation are larger for higher intensity motion, indicating that the decompression level and a larger degree of non-linearity (for cff % 10À3 the G/ G0 % 0.5) have an increasing influence The regression lines of the deviation of axial force are nearly parallel for the different decompression levels, indicating that the effect of decompression level has an important role for all input motions The progressive plastification of the soil above the tunnel induced by the seismic loading increases the axial force in the lining The deviation of axial force grows for higher input motion because this effect is not caught by the analytical solution Conclusions The effect of stress disturbance induced by tunnel construction on the seismic response of shallow bored tunnels was evaluated using numerical simulations The presence of tunnel and associated stress disturbance does not significantly affect the seismic response at the ground surface Some reduction in the peak acceleration occurs with the increasing k-value During seismic loading, stress paths in the soil close to the tunnel exhibit wider perturbations in terms of both qk and p0k than in the free-field The stress path perturbation is also wider for increasing k-value Seismic loading causes significant fluctuation in tunnel lining forces during the event, and higher permanent lining forces in the post-event state This is attributed to the progressive plastification of the soil that increases the vertical load transmitted to the lining and that increases the distortion of the tunnel Comparison of numerical predictions with an analytical solution highlights that the founding assumptions in the latter may result in the underestimate of tunnel lining forces resulting from seismic loading, particularly for higher intensity motions ð22Þ Appendix A In general, Wang’s solution underestimates the lining seismic forces in comparison with the numerical model The deviation varies up to 110% for increment in bending moment, and from 22% to 158% for increment in axial force The higher deviation in the incremental axial force occurs for k = 50%, while for the increment in bending moment it occurs for k = 5% The deviation grows with the intensity of the input motion and, thus, with the maximum free-field shear strain at the tunnel depth For low intensity motions, the regression lines of the deviation of the bending moment are close and the values of the deviation For no-slip conditions the maximum increments in the axial force and bending moment in the transverse direction of the tunnel are given by: DNmax ¼ Æ1:15 DMmax ¼ Æ 200% ðA1Þ K l Em R2 cff þ mm ðA2Þ 200% 50% 20% 150% 0.90 0.85 0.87 5% 100% 50% R 0% -50% 0.0 0.1 0.2 0.3 0.4 PGA input motion (g) 0.5 0.6 Deviation | N max | 50% Deviation | Mmax | K Em R Á cff þ mm 150% 0.40 20% 0.25 5% 0.40 100% 50% R2 0% -50% 0.0 0.1 0.2 0.3 0.4 0.5 0.6 PGA input motion (g) Fig 25 Deviation between the maximum increment in lining forces due to seismic loading computed with Wang solution and the numerical model for all time-histories and for various decompression levels R.C Gomes / Computers and Geotechnics 49 (2013) 338–351 where Em is the mobilised soil Young’s modulus (evaluated with reference to the previously calculated shear modulus Gm), mm indicates the corresponding Poisson’s ratio (here assumed equal to 0.3), R is the tunnel radius and cff the maximum free-field shear strain at the tunnel depth The lining response coefficients are given by the following expression: 12ð1 À mm Þ Kl ¼ 2F þ À 6mm K2 ¼ þ ðA3Þ F ½ð1 À 2mm Þ À ð1 À 2mm ÞC Š À 0:5ð1 À 2mm Þ2 þ F ½ð3 À 2mm Þ þ ð1 À 2mm ÞC Š þ Cð2:5 À 8mm þ 6m2m Þ þ À 8mm [13] [14] [15] [16] [17] [18] ðA4Þ where F is the flexibility ratio: F¼ t ÞR Em ð1 À m 6Et Ið1 þ mm Þ [19] ðA5Þ [20] with I corresponding to the moment of inertia of the tunnel lining in the transverse direction, Et is the lining Young’s modulus, mt the lining Poisson’s ratio and C the compressibility ratio: C¼ Em ð1 À m2t ÞR 2Gm ð1 À m2t ÞR ¼ Et tð1 þ mm Þð1 À 2mm Þ Et tð1 À 2mm Þ [21] [22] ðA6Þ [23] with t corresponding to the thickness of the tunnel lining in the transverse direction [24] [25] References [1] Iida H, Hiroto T, Yoshida N, Iwafuji M Damage to Dakai subway station Spec Issue Soils Found 1996:283–300 [2] Wang WL, Wang TT, Su JJ, Lin CH, Seng CR, Huang TH Assessment of damage in mountain tunnels due to the Taiwan Chi-Chi earthquake Tunnell Undergr Space Technol 2001;16(3):133–50 [3] O’Rourke TD, Goh SH, Menkiti CO, Mair RJ Highway tunnel performance during the 1999 Duzce earthquake In: Proceedings of the fifteenth international conference on soil mechanics and geotechnical engineering, Istanbul, Turkey; 2001 [4] Panet M Le calcul des tunnels par la méthode convergence confinement Paris: Ed Press de l’École National des Ponts et Chaussées; 1995 [5] Karakus M Appraising the methods accounting for 3D tunnelling effects in 2D plane-strain FE analysis Tunnell Undergr Space Technol 2007;22:47–56 [6] Moller SC, Vermeer PA On numerical simulation of tunnel installation Tunnell Undergr Space Technol 2008;23:461–75 [7] Kontoe S, Zdravkovic L, Potts DM, Menkiti CO Case study on seismic tunnel response Can Geotech J 2008;45:1743–64 [8] Corigliano M, Scandella L, Lai CG, Paolucci P Seismic analysis of deep tunnels in near fault conditions: a case study in Southern Italy 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F, Takagi T Shear moduli of sands under cyclic torsional shear loading Soils Found 1978;18(1):39–56 Hajal T Modélisation élastoplastique des sols par une loi multimécanismes PhD thesis, École Centrale de Paris France; 1984 Gomes RC Numerical modelling of the seismic response of the ground and circular tunnels PhD thesis, Instituto Superior Técnico, Technical University of Lisbon; 2009 [in Portuguese] Tatsuoka F, Iwasaki T, Takagi Y Hysteretic damping of sands under cyclic loading and its relation to shear modulus Soils Found 1978;18(2):25–40 Modaressi H, Benzenati I Paraxial approximation for poroelastic media Soil Dynam Earthq Eng 1994;13(2):117–29 Engquist B, Majada A Absorbing boundary conditions for the numerical simulation of waves Math Comput 1977;31(139):629–51 Kuhlemeyer RL, Lysmer J Finite element method accuracy for wave propagation problems J Soil Mech Found Div ASCE 1973;99(SM5):421–7 Eurocode Design of concrete structures General rules and rules for buildings EN 1992-1, CEN; 2004 Portuguese Annex to Eurocode Design of structures for earthquake resistance Part 1: general rules, seismic actions and rules for buildings NP EN 1998-1, CEN; 2010 Ambraseys N, Smit P, Berardi R, Rinaldis D, Cotton F, Berge C Dissemination of European strong-motion data CD-ROM collection European Council, Environment Programme; 2000 Gomes RC Seismic behaviour of tunnels under seismic loading MSc thesis, Instituto Superior Técnico, Technical University of Lisbon; 2000 [in Portuguese] Katona MG, Zienkiewicz OC A unified set of single step algorithms Part 3: the Beta-m method, a generalization of the Newmark scheme Int J Numer Methods Eng 1985;21(7):1345–59 [...]... effect of decompression level has an important role for all input motions The progressive plastification of the soil above the tunnel induced by the seismic loading increases the axial force in the lining The deviation of axial force grows for higher input motion because this effect is not caught by the analytical solution 5 Conclusions The effect of stress disturbance induced by tunnel construction on the. .. value of 1.0, it occurs in confined regions of the soil No-slip assumption remains adequate, because slip in the interface soil-tunnel may occur only in these limited zones Since the effects of tunnel construction are not taken into account in analytical solutions, the comparison gives an indication of the significance of modelling the tunnel construction for practical applications Table 3 summarises the. .. the seismic response of shallow bored tunnels was evaluated using numerical simulations The presence of tunnel and associated stress disturbance does not significantly affect the seismic response at the ground surface Some reduction in the peak acceleration occurs with the increasing k-value During seismic loading, stress paths in the soil close to the tunnel exhibit wider perturbations in terms of both... 338–351 tion between the PGA of the input motion and the maximum freefield shear strain at the tunnel depth The shear modulus of the ground, Gm, (Table 3) was estimated using the stiffness degradation curve (G/G0 À c, Fig 7) The G0 at the depth of the centre of the tunnel (%125 MPa) was multiplied by the shear modulus ratio corresponding to the value of cff to compute Gm According to Wang [11], the flexibility... strain at the tunnel depth, cff, introduced in the analytical solution (Table 3) is the average of the maximum shear strain on the lateral boundary of the numerical model, at the depth of the tunnel Fig 23 shows a strong rela- 2.E-03 R2 = 0.91 4.5 Analytical solutions 1.E-03 The closed-form solution to predict the transverse seismic response of the tunnel, summarised in [11], was adopted for comparison with... force The higher deviation in the incremental axial force occurs for k = 50%, while for the increment in bending moment it occurs for k = 5% The deviation grows with the intensity of the input motion and, thus, with the maximum free-field shear strain at the tunnel depth For low intensity motions, the regression lines of the deviation of the bending moment are close and the values of the deviation For... 10À4 the G/G0 % 0.7) The regression lines clearly diverge and the values of the deviation are larger for higher intensity motion, indicating that the decompression level and a larger degree of non-linearity (for cff % 10À3 the G/ G0 % 0.5) have an increasing influence The regression lines of the deviation of axial force are nearly parallel for the different decompression levels, indicating that the effect. .. is small in comparison to the tunnel diameter Seismic actions are considered as external static forces acting on the tunnel lining, induced by the ground distortion related to a vertically propagating shear wave The resulting ovalisation of the tunnel lining is assumed to occur under plane strain conditions The detailed solution for the no-slip condition is summarised in Appendix A The maximum free-field... indicating strong correlation between the lining forces ratio and the PGA of the input motion Fig 22 plots the overall maximum lining forces computed for all time-histories, and compares them with the maximum strength of the concrete lining The maximum lining forces do not exceed about 35% of the maximum strength of the concrete, indicating that a concrete lining can be designed to withstand the combined... becomes crucial only for F < 1, as, for example, in the case of the tunnel built in very soft ground Fig 24 presents the mobilized shear strength ratio, defined as the ratio between the mobilized strength to the available strength, distribution in the soil along the perimeter of the tunnel for t = 4.510 s, the instant where the maximum extent of mobilized yield surface occurs (Fig 14) Although the maximum ... 4.3 Seismic response of the soil -6 4.3.1 Single earthquake time-history This section highlights the modifying effects of the stress disturbance induced by tunnel construction on the seismic response. .. deviation of axial force grows for higher input motion because this effect is not caught by the analytical solution Conclusions The effect of stress disturbance induced by tunnel construction on the. .. of stress disturbance induced by tunnel construction on the seismic response of the lining Again, the results based on time-history number are presented in detail Fig 18 shows the variation of