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May 13, 2014 - A numerical Round Robin on tunnels under seismic actions. Emilio Bilotta, Giovanni Lanzano,. S. P. Gopal Madabhushi & Francesco. Silvestri ...
A numerical Round Robin on tunnels under seismic actions

Emilio Bilotta, Giovanni Lanzano, S. P. Gopal Madabhushi & Francesco Silvestri Acta Geotechnica ISSN 1861-1125 Volume 9 Number 4 Acta Geotech. (2014) 9:563-579 DOI 10.1007/s11440-014-0330-3

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Author's personal copy Acta Geotechnica (2014) 9:563–579 DOI 10.1007/s11440-014-0330-3

RESEARCH PAPER

A numerical Round Robin on tunnels under seismic actions Emilio Bilotta • Giovanni Lanzano • S. P. Gopal Madabhushi • Francesco Silvestri

Received: 23 December 2013 / Accepted: 24 April 2014 / Published online: 13 May 2014 Ó Springer-Verlag Berlin Heidelberg 2014

Abstract Although the seismic behaviour of shallow circular tunnels in soft ground is generally safer than aboveground structures, some tunnels were recently damaged during earthquakes. In some cases, damage was associated with strong ground shaking and site amplification, which increased the stress level in the tunnel lining. Pseudo-static and simplified dynamic analyses enable to assess transient changes in internal forces during shaking. Nevertheless, experimental evidences of permanent changes in internal loads in the tunnel lining would suggest that a full dynamic analysis including plastic soil behaviour should be performed when modelling the dynamic interaction between the tunnel and the ground. While sophisticated numerical methods can be used to predict seismic internal forces on tunnel structures during earthquakes, the accuracy of their predictions should be validated against field measurements, but the latter are seldom available. A series of centrifuge tests were therefore carried out at the University of Cambridge (UK) on tunnel models in sand, in the framework of a research project funded by the Italian Civil Protection Department. A numerical Round Robin on Tunnel Tests was later promoted among some research

E. Bilotta (&)  F. Silvestri University of Napoli Federico II, Naples, Italy e-mail: [email protected] F. Silvestri e-mail: [email protected] G. Lanzano University of Molise, Termoli (CB), Italy e-mail: [email protected] S. P. G. Madabhushi University of Cambridge, Cambridge, UK e-mail: [email protected]

groups to predict the observed behaviour by means of numerical modelling. In this paper, the main results of five selected numerical predictions are summarized and compared with the experimental results. Keywords Centrifuge experiments  Earthquakes  Numerical analysis  Soil–tunnel interaction  Tunnels

1 Introduction The seismic response of shallow circular tunnels in soft ground generally does not create much concern in design, in comparison with that of the aboveground structures; it is generally considered that tunnels are fully embedded in the ground, and hence, they tend to move with it and do not experience strong inertial loading. Nevertheless, some tunnels suffered severe damage in past and recent earthquakes [9, 56, 57], in most cases due to ground failure or liquefaction; however, sometimes, damage has occurred also associated with strong ground shaking and possibly amplified by local site conditions. As a consequence, the tunnel linings locally cracked when the inertial loading caused an increase in the stress level under the static regime, which exceeded the concrete tensile strength [26, 40, 56]. Several procedures exist to evaluate the variation in internal forces induced in a tunnel lining during earthquakes [26, 44]. The level of complexity of calculation must be consistent with the level of detail of the investigations and definition of seismic actions. For preliminary design stages, pseudo-static and simplified dynamic analyses with uncoupled approaches are in fact suggested by various guidelines [4, 5, 30, 39]. Such analyses enable to assess transient changes in internal forces during shaking, provided that the mobilized shear strain around the tunnel

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Table 1 Main features of RRTT numerical analyses Group

Reference paper

Adopted constitutive law

Numerical code

AUTH (Greece)

Tsinidis et al. [52]

Visco-elasto-plastic model [7]

ABAQUS (FEM)

TUD (Germany)

Hleibieh et al. [28]

Hypoplastic [54]

TOCHNOG (FEM)

TVG (Italy)

Conti et al. [22]

M1—Bounding surface plasticity [8]

FLAC (FDM)

UTL (Portugal)

Gomes [25]

Elastoplastic multimechanism [29]

GEFDYN (FEM)

BaBo (Italy)

Amorosi et al. [6]

‘Small-strain’ elasto-plastic hardening soil model [45]

PLAXIS (FEM)

M2—Perfect plasticity with embedded hysteretic behaviour [31]

is predicted. On the other hand, experimental and numerical evidences of permanent changes in internal loads in the tunnel lining [18, 32, 33, 40] have indicated that a full dynamic analysis including soil–structure interaction and plastic soil behaviour should be performed in order to achieve the most reliable seismic design of such underground structures. Numerical analyses should use, in this case, sophisticated soil models, which are usually calibrated based on the results from laboratory element tests. While sophisticated numerical procedures exist, the accuracy with which they can predict the seismic performance of field tunnels needs to be demonstrated. Measurements of seismic internal forces on real-scale tunnel structures during earthquakes loadings are seldom available. Complication arises not only due to the random occurrence of earthquakes, but also because the routine monitoring instrumentation of the existing tunnels has too low sampling frequency. Hence, such instrumentation is not able to capture the transient nature of dynamic soil– tunnel kinematic interaction. Due to the substantial lack of well-documented full-scale case histories, it is difficult to validate any given design procedure or guidelines. Although centrifuge studies had been previously accomplished to assess the earthquake performance of tunnels in liquefiable sand [3], the extensive use of centrifuge modelling to investigate the seismic behaviour of tunnels in dry sand was first carried out at Cambridge University [19–21]. In this research, the seismic behaviour of square and circular tunnels was investigated, but the tunnels themselves were not instrumented to measure internal forces such as lining hoop forces or bending moments. To bridge this gap, centrifuge seismic tests on an instrumented model tunnel [36] were carried out in 2007 at the Schofield Centre of Cambridge University (UK) in the framework of an Italian research project (ReLUIS). A numerical Round Robin on Tunnel Tests (RRTT) was then promoted and officially launched at the TC28 (now Technical Committee 204 of the International Society of Soil Mechanics and Geotechnical Engineering) symposium on ‘Underground constructions in soft ground’ in Rome [11]. Several groups attended the RRTT, initially by

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performing ‘blind’ numerical modelling of two centrifuge tests. In this context, ‘blind’ means that soil characterization was based on laboratory element tests [53] and the results of the centrifuge tests were not known: only the geometry of the model, the boundary conditions and the signal of the input earthquake loading were delivered to the participants. The first results were presented during a Workshop held at the II International Conference on Performance Based Design in Earthquake Geotechnical Engineering at Taormina [2, 12, 13, 24, 27, 51]. Later, the teams were enabled to know the results of the centrifuge tests, and hence, they could perform back-analyses to improve the numerical predictions. In the latter case, the dynamic response of the sand layer was used for calibration, while the dynamic changes in bending moment and hoop force in the tunnel lining were the actual target of the computation. Such a numerical Round Robin on the prediction of centrifuge results follows the established methodology of similar projects such as VErification of Liquefaction Analysis by Centrifuge Studies (VELACS), which used the concept of numerical predictions of centrifuge test data [46, 47]. Also, a new project called Liquefaction Experiments and Analysis Projects for validation (LEAP) is currently being planned between research centres from USA, UK, Japan, Taiwan and China. In this paper, the main results of five selected numerical predictions are summarized (Table 1). They were all carried out in the framework of RRTT, and each one is detailed in a standalone paper [6, 22, 25, 28, 52]. A comparative discussion is proposed in the final section of the paper, which highlights major implications that the extensive back-analysis of the experimental centrifuge tests may have on applications.

2 Background Assuming that during earthquake a circular tunnel is subjected to vertical propagating shear waves, it is generally admitted that the cross section must be analysed for the ovalling deformation imposed by the ground [41].

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Simplified pseudo-static methods of analyses are generally used to predict the transient changes in bending moment and hoop force associated with the lining ovalization [43, 55], using an equivalent linear approach to model the dependency of stiffness and damping on strain [37]. To better take into account the nonlinear behaviour of soil, numerical analyses must be performed, which implement appropriate constitutive models. This may be helpful to capture some aspects of the dynamic behaviour of the tunnel observed in documented case histories, which a linear approach would be unable to predict. Centrifuge modelling is traditionally considered a valid alternative to well-documented case histories to highlight critical aspects of soil–structure interaction. Due to the poor availability of abovementioned full-scale data, the centrifuge test presents also an opportunity for a proper calibration of numerical models. Centrifuge tests are typically carried out on model n times smaller than the real problem; therefore, scaling laws apply. The basic scaling law for centrifuge modelling derives from the need to ensure the stress and strain similarity between the model and corresponding prototype. An acceleration n times larger than the Earth’s gravity, g, must be applied to obtain an overburden vertical stress, rvm, in the model equal to the vertical stress, rvp, in the prototype. According to the scaling law for length, displacements will also have a scale factor of 1:n, and consequently, strains have a scale factor of 1:1. Consistently, the stress– strain behaviour of soil does not scale in centrifuge models [49, 50]. Simulation of dynamic events such as earthquakes requires special consideration to define appropriate scaling laws [38]. Since linear dimensions and accelerations in the model have scale factors 1:n and n:1, respectively, the amplitude of seismic motion, Am, is equal to n-1Ap and its frequency at the model scale, fm, is equal to nfp (Ap and fp being the amplitude and frequency at the prototype scale). It follows that the velocity magnitude will then be the same in the model and the prototype, i.e. velocity has a scale factor 1:1. The time scaling factor for dynamic events is therefore 1:n. Similarly, bending moments and hoop forces measured in the lining must be scaled up to be expressed at real scale. Therefore, the bending moment per unit length (plane strain) at prototype scale, Mp, is equal to n2Mm, and the hoop force per unit length at prototype scale Np is equal to nNm. Since in this paper the results of numerical analyses are all shown and compared with the results of centrifuge tests at model scale, the above scaling factors must be always used to compare them with typical ranges of values at the real (or prototype) scale.

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Fig. 1 Instrumented model tube with locations of the strain gauges

3 Reference centrifuge tests 3.1 Materials All the models tested in this experimental campaign were made of dry Leighton Buzzard sand (grade E) reconstituted at two different relative densities Dr (about 50 and 80 %). A characterization of the sand used in tests was performed in laboratory by means of triaxial and resonant column– torsional shear (RC–TS) tests [53]. The RC–TS apparatus was an upgrading of a Stokoe-type fixed-free model [42], originally developed at the University of Napoli Federico II [23]. The tunnel lining was modelled using an aluminium alloy tube having an external diameter D = 75 mm and a thickness t = 0.5 mm. At n = 80 g, the model would represent a 6 m diameter prototype tunnel with an equivalent bending stiffness corresponding, e.g., to a shotcrete lining of about 6 cm. 3.2 Testing programme The tube has been instrumented to measure bending moments (BM) and hoop forces (HS) at four locations along two transverse sections (Fig. 1). The main instrumented section (A) was located at the mid-span of the tube and a second section (B) 50 mm aside. In total, 16 Wheatstone bridges (4 locations 9 2 sections 9 2 force measurements) were installed. The vertical displacement of the surface during centrifuge tests was measured by linear variable differential transformers (LVDTs). The tests selected for this benchmark, T3 and T4, are two models of deep tunnel in dense and loose sand, respectively; a schematic layout of both of them is drawn in Fig. 2. A laminar box [14, 15] was adopted as container for the model. Miniature piezoelectric accelerometers were used to measure horizontal and vertical acceleration in the soil and on the model container during earthquakes.

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Fig. 2 Models T3/T4: layout of the instrumentation

The device has a resonant frequency of about 50 kHz and maximum error of 5 %; the transducer weight is about 5 g. Both models were prepared by pluviation of about 50 kg of sand in the container, obtaining the desired average void ratio and relative density. The procedure of model making was carefully controlled, since it is well known that the mechanical behaviour of a reconstituted granular soil is strongly dependent on the deposition procedure both in the lab [48] and in the centrifuge [17]. Four earthquakes were fired at n = 80 and one at n = 40, with variable nominal peak acceleration amplitude and frequency. Only the artificial earthquakes under 80 g are considered in this paper. The main features of each earthquake are shown in Table 2, at the model and prototype (bracketed values) scales; the time histories of those fired under 80 g, as recorded by the reference accelerometer (Acc13 in T3 and Acc 12 in T4 in Fig. 2) are summarized in Fig. 3a, b.

4 Results The five papers presented in this Special Issue show the predictions of teams belonging to academic departments of several European countries. Each group adopted a different numerical code and a different constitutive model for the soil, as shown in Table 1. A selection of the results of their predictions is shown in this paper and compared each other and with the experimental results. In order to simplify the comparison, only one set of numerical results will be

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shown for TVG, since the Authors observed that the two constitutive models, M1 [8] and M2 [31], predicted almost the same dynamic behaviour of the soil [22]; in fact, the only significant difference was observed in terms of permanent internal hoop force in the lining, lower values being calculated by M2, which has been considered in the following comparisons. A significant issue for all the participants was the calibration of the constitutive model on the results of both laboratory and centrifuge tests in order to reproduce the decay of soil stiffness from small to large shear strain. In fact, the interpretation of laboratory tests to calibrate an advanced constitutive model commonly is based on the assumption that a laboratory test is equivalent to the application of a uniform and controlled stress (or strain) path to a soil element. This assumption is commonly accepted, provided that the stress paths followed in the laboratory tests fit in the range of those followed in the physical models. However, it may complicate the assessment of the constitutive relationship, especially far from soil failure, where some details affecting small-strain behaviour may be crucial (e.g. fabric or stress-induced anisotropy). On the other hand, the direct interpretation of the centrifuge tests, as a back-analysis of a nonlinear boundary value problem, incorporates in the calibration the detail of the actual history and stress paths, leading to a more reliable estimate of the geotechnical behaviour [37]. Hence, most groups performed a first calibration on the laboratory tests only (i.e. blind predictions of dynamic behaviour observed in the centrifuge models), and then, they tuned the calibration on the basis of the backanalysis of the centrifuge tests.

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Table 2 Earthquakes fired in test T3–T4 Earthquake #

n

Frequency (Hz)

Duration (s)

Nominal PGA (g)

Model

[Prototype]

Model

[Prototype]

Model

[Prototype]

1

80

30

[0.375]

0.4

[32]

4

[0.05]

2

80

40

[0.5]

0.4

[32]

8

[0.10]

3

80

50

[0.625]

0.4

[32]

9.6

[0.12]

4

80

60

[0.75]

0.4

[32]

12

[0.15]

5

40

50

[1.25]

0.4

[16]

6

[0.15]

Fig. 3 Shaking applied to model T3 (a) and T4 (b)

A comparison between laboratory data (RC–TS tests) and the numerical predictions in terms of dependency of the maximum shear modulus, G0, on the mean effective stress, p0 , is shown in Fig. 4. The only constitutive law apparently underestimating the small-strain stiffness is the visco-elasto-plastic model adopted by AUTH. Similar comparisons are shown in Fig. 5 in terms of G(c)/G0 and D(c). All the predictions lead to comparable trends, except those by TGV, showing a more pronounced

stiffness decay and increase in damping at smaller strain levels. Note that the D(c) curves by BaBo show a cut-off; the predictions by TUD were instead not delivered by this team. A comparison between the experimental time history of acceleration measured at the top of the reference array (black line, ACC15 in Fig. 2) and the corresponding calculated time histories (coloured lines) is shown in Figs. 6 and 7 for model T3 and model T4, respectively. Only the

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Fig. 4 Comparison between the curves G0(p0 ) measured in RC–TS tests and those predicted by the constitutive models

Fig. 5 Comparison between the curves G(c)/G0 and D(c) measured in RC–TS tests and those predicted by the constitutive models

time histories relevant to the event EQ1 and the event EQ4 have been presented in the figure, corresponding to the weakest and the strongest events. The predictions of four out of five teams are satisfactory, since the main features of the experimental record are caught, both in terms of amplitude and frequency content. Only one numerical prediction (UTL) shows an unexpected amplification of the signal at high frequencies after a few cycles, particularly in the weakest event (EQ1). This is also evident in Fig. 8, where the corresponding experimental and numerical response spectra are shown. It is worth noting that the initial nonnull (finite) damping D0, which has been measured in the laboratory tests on Leighton Buzzard sand [53], was not introduced in the UTL analyses; this might have led UTL to overestimate the amplification of the input signal at high frequencies. Also, BaBo decided not to model the initial damping D0: in this case, however, the input signal was preprocessed and the frequency content higher than 180 Hz was filtered out, preventing overamplification at high frequency. The three remaining teams (AUTH, TUD and TVG) all modelled the small-strain viscous damping by

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adopting a Rayleigh formulation, which, as well known, attenuates the energy content at high frequencies. Profiles of amax with depth are shown in the plots of Fig. 9, as measured and calculated along the ‘reference’ and ‘tunnel’ vertical arrays during EQ1 and EQ4 in both T3 and T4. The predictions of all groups, except UTL, are very close to measurements and each other along both arrays and in both examined events. Since BaBo modelled the problem by setting lateral boundaries far away from the tunnel, in the paper they showed the results along the ‘free-field’ array, instead of ‘reference’, and these are shown in the figure. The dispersion among the numerical predictions is lower for the weakest earthquake (EQ1) and the denser model (T3), being generally larger for the strongest event (EQ4). Consistently with the time histories of acceleration shown in Figs. 6 and 7 (ACC15 is close to ground surface), UTL overpredicts also the maximum accelerations in depth. The experimental time histories of the displacements, u(t), were obtained from double integration of the accelerograms [16]. Consequently, the experimental time histories of shear strain have been calculated by

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Fig. 6 Model T3: comparison between the experimental (ACC15) and the numerical time histories of acceleration during EQ1 (a) and EQ4 (b) (color figure online)

differentiating such displacements, u(t), with respect to depth, z, using a second-order approximation over three instruments positioned in the ‘reference’ array: h cð z i Þ ¼

i zi1 Þ ðziþ1 zi Þ ðuiþ1  ui Þ ððzziiþ1 zi Þ þ ðui  ui1 Þ ðzi zi1 Þ ðziþ1  zi1 Þ

ð1Þ

in which the index i is relative to the position of the central instrument and i - 1 and i ? 1 to the upper and lower accelerometer.

Profiles of the maximum experimental values of shear strain, cmax, with depth are shown in the plots of Fig. 10, for EQ1 and EQ4 along the ‘reference’ arrays of T3 and T4. They are compared with the corresponding numerical results. Similar to the acceleration plots (see Fig. 9), BaBo values along the ‘free-field’ array, rather than ‘reference’, are plotted in the figure. In some cases (TUD, UTL and TVG), the numerical analyses underpredicted the shear strain in the model, while the average experimental value was fairly well predicted by

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Fig. 7 Model T4: comparison between the experimental (ACC15) and the numerical time histories of acceleration during EQ1 (a) and EQ4 (b) (color figure online)

AUTH. BaBo was the only predicting, quite singularly, shear strains decreasing with depth. Their predictions for T4 EQ1 are not shown in the figure since they are out of the adopted scale. Measured and calculated time histories of surface settlement during the first 4 events (EQ1 to EQ4) are shown in Fig. 11. The total duration represented in the figure is fictitious since the time lag between subsequent events was shortened to make the plot more readable. The measured settlement (black line) accumulates during the events,

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indicating the densification of the sand layer due to the accumulation of plastic volumetric strain during shaking. The amount of predicted densification is extremely variable among the different numerical models, indicating their different ability to simulate the volumetric plastic straining induced by cyclic shear loads. All the numerical predictions underestimate the measured settlement. This is in part consistent with the underestimation of the shear strain, which may fall below the threshold of cyclic degradation, at least for the weaker earthquakes fired; in this

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Fig. 8 Response spectra at surface (EQ1 and EQ4): model T3 (a) and T4 (b)

case, shaking would not produce any plastic volumetric strain. However, this cannot be the only reason, as it is evident by comparing the results of TUD and TVG: although they are quite similar in terms of predicted shear strain, they look very different in terms of predicted settlement. The difference in the constitutive models which have been adopted by the two groups (Mohr–Coulomb elastic-perfectly plastic with hysteretic behaviour for TVG and hypo-plastic with inter-granular strain extension for TUD) appears the key factor in this aspect. In addition, local nonhomogeneities of the sand density may be responsible of larger permanent changes in measured volume strains compared with the computed ones. A representative selection of time histories of bending moment in section A (cf. Fig. 1) is plotted in Fig. 12 for T3

and Fig. 13 for T4. To enhance the comparison, only three significant windows are shown, corresponding to the initial, central and final part of the signal. The experimental data (black line) refer to the NW transducer, shown in Fig. 1; they show a trend to increase during shaking, which is generally larger for the stronger events. In addition, reversible change in bending moment can be observed, which is associated with the cyclic loading. Since all the numerical analyses were performed by assuming a positive acceleration at the base opposite to the test convention, in the following figures the calculation at the mirror point (symmetrical with respect to the vertical axis of the tunnel) was used for comparison. Some numerical models (AUTH and TUD namely) are able to predict with different approximation both the cyclic

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Fig. 9 Comparison between the experimental and numerical profiles of maximum acceleration along the reference and the tunnel arrays

Fig. 10 Comparison between the experimental and numerical profiles of maximum shear strain along the reference array

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Fig. 11 Comparison between the experimental (LVDT059) and numerical time history of surface ground settlement

Fig. 12 Model T3 (dense): comparison between the experimental (NW) and numerical time history of bending moments during EQ1 (a) and EQ4 (b)

reversible change and the permanent accumulation of bending moment. The latter ability seems to correlate well with the ability to predict sand densification. For the T3 test, the same model which best approximates the sand densification (TUD) results in an amplitude of permanent change in bending moment at the end of EQ4, which is closer to the experimental one. On the other hand, for the loose sand test (T4), the same model seems to excessively overestimate the experimental residual bending moment. The sign of such

permanent change is a minor issue since its actual value and sign are strongly affected by the exact position of the transducer: even a small difference between the position of the node of calculation and the experimental point of measurement may justify an opposite sign of the residual values. The experimental time histories of hoop forces in section A (cf. Fig. 1) at the location NE are plotted in Figs. 14 and 15 and compared with the numerical predictions. Reversible and permanent changes in hoop forces with

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Fig. 13 Model T4 (loose): comparison between the experimental (NW) and numerical time history of bending moments during EQ1 (a) and EQ4 (b)

Fig. 14 Model T3 (dense): comparison between the experimental (NE) and numerical time history of hoop forces during EQ1 (a) and EQ4 (b)

different magnitude can be observed in the experimental data of both T3 and T4 and in the relevant numerical predictions. Due to the scale of the plots, the change with time of the measured hoop forces and of some of the predictions (AUTH and BaBo) is less evident. The value of the hoop force is particularly influenced by the roughness of the interface between the aluminium lining and the sand, this contact being very smooth.

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The numerical analyses where a full contact between the sand and the lining was assumed, such as UTL and TUD, predicted larger values of hoop force changes compared to those modelling intermediate conditions of roughness, such as TVG, or very smooth interface, such as AUTH and BaBo. A better match with the centrifuge data is achieved in the latter cases. Figure 16 shows a comparison between the experimental and numerical distribution of reversible increments in

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Fig. 15 Model T4 (loose): comparison between the experimental (NE) and numerical time history of hoop forces during EQ1 (a) and EQ4 (b)

Fig. 16 Model T4: comparison between the experimental and numerical distribution of reversible increments in bending moments during EQ1 and EQ4

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Fig. 17 Model T4: comparison between the experimental and numerical distribution of reversible increments in hoop forces during EQ1 and EQ4

bending moments during EQ1 and EQ4. Such increments were calculated as half of the average values of the peakto-peak amplitudes of the stationary oscillations in the time histories of bending moment [34]. The experimental values along the lining are shown for all the measuring points along both transverse sections A and B of the model tunnel (cf. Fig. 1). The inequality of the increments measured at top (NE vs NW) and bottom (SE vs SW) locations in both sections is a measure of the experimental error. The differences between the values measured at the same locations appear appreciable only for the NW position, while in the others they do not appear affected by boundary effects. On the average, the numerical predictions underestimate the mean experimental measurements (dashed lines) for the denser model T3, being however quite different each other, due to the different way to model the contact between the soil and the tunnel. The dispersion is even larger for model T4. In this case, however, the closeness to the experimental data is better, particularly for TUD. In Fig. 17, the similar comparison between the experimental and numerical distribution of reversible increments in hoop forces (calculated according to the same procedure adopted for bending moment [34]) during EQ1 and EQ4 is shown, in a logarithmic scale, due to the large difference

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among the numerical predictions. The results of the two analyses modelling a very smooth interface (AUTH and BaBo) approximate better the experimental values. AUTH team has performed a parametric analysis indicating that assuming interface friction angles as low as 5.7° still produces increments in hoop forces higher than measured, while by using a full slip contact law [1] the experimental values are better caught. BaBo used interface elements with very low shear strength and stiffness [45] to mimic the full slip conditions. The way a frictionless contact is modelled by different numerical codes is recognized as a reason for discrepancy among the calculated values of hoop forces, particularly for very thin lining such as the one at hand, as observed by Bilotta et al. [10]. Nevertheless, as stated also in the paper TVG, the assumption of zero friction between the tunnel and the soil may be unrealistic in this case, suggesting instead that the experimental values might be unreliable.

5 Discussion The numerical analyses which have been compared in this paper to the experimental results of a set of centrifuge tests

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differ from each other in various aspects. The analyses were performed at model scale and at prototype scale, with different approaches for modelling the boundary conditions. The adopted constitutive laws were rather different (see Table 1), simulating in a different fashion the transition from linear/reversible to nonlinear/irreversible behaviour of sand: their formulations have different ability to model plastic accumulation of both shear and volumetric strain. The selection of the initial profile of small-strain shear modulus in the sand layer was not univocal. Finally, the assumptions on the contact between the sand and the tunnel lining were different. In spite of the differences among the analyses, a general comment on the whole set shows that in most cases, the dynamic response in terms of acceleration of the sand layer is captured fairly well in terms of both amplitude and frequency content (cf. Figs 6, 7, 8, 9). A deeper insight into the numerical analyses may highlight the influence on the results of the way the small-strain damping is accounted for. Since all the adopted constitutive models dissipate energy by reproducing the hysteretic behaviour, linear and reversible stress–strain cycles are associated with zero damping, unless viscous damping is numerically introduced. A small Rayleigh damping is needed to overcome such a limitation. TVG, for instance, perform calculations which show that the choice of viscous damping in a range between 0 and 4 % does not affect the numerical results in the range of frequency associated with the larger part of the energy content of the input signal. On the contrary, such additional damping is able to smooth the overamplified signal at higher frequency. This problem affects clearly the predictions by UTL, who did not use any viscous damping. BaBo on the other hand decided not to introduce any viscous damping, but preliminarily processed the input signals by applying a bandpass filter (15–180 Hz) to overcome unrealistic amplification at high frequency. A further observation on the Rayleigh formulation concerns the use of a single or double frequency approach. It is useful to compare (Fig. 8) the response spectra at surface by AUTH and TVG. The former calibrated the Rayleigh damping on two control frequencies (i.e. 80 and 480 Hz at the model scale) where small-strain damping was equal to 6 %; the latter instead calibrated the Rayleigh damping on a single frequency (the nominal frequency of the input signal, varying between 30 and 60 Hz) where damping was equal to 4 %. As a consequence of such different choices, TVG damped at high frequencies more than AUTH. In both models for the lower intensity signals (EQ1), both sets of analyses predicted well the amplitude of the peak corresponding to the main frequency of the signal. At higher frequency, the single frequency approach (TVG) seems to underestimate the experimental spectral amplitude, while the double frequency method (AUTH) matches better the experimental spectra and

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somewhere overestimates. For the higher intensity event (EQ4), such a difference is less evident, possibly due to the predominant effect of hysteretic damping. Looking at the distributions of shear strain (Fig. 10) and at the time histories of settlement (Fig. 11), one gets a perception of the capability of different models of capturing the dynamic stress–strain behaviour of sand. By reducing the initial shear stiffness of sand in the model layer compared to the element tests (Fig. 4), AUTH is able to match at the same time the average shear strain mobilized in the layer and the acceleration peak values along depth, both at the weakest and at the strongest events. BaBo overpredict shear strain, although they assumed a G0 profile very close to that from the laboratory tests (cf. Fig. 4). It must be, however, observed that the strain profiles ‘BaBo’ shown in the figure have been on purpose calculated according to the same procedure followed for the experimental data [16, 36, 58] for a consistent comparison. BaBo reported in the paper strain values calculated as cyclic reversible shear strain at an integration point at the tunnel axis depth in free-field conditions, which are around 10-2 %. These values are lower than the mean experimental values, ranging between 0.35 and 0.1 %. The difference between total and reversible part of the shear strain would indicate a progressive accumulation of plastic shear strain during the event. Such a fact could not be measured in the tests, but it has been numerically simulated by TVG using the model by Andrianopoulos et al. [8], which has not been included in the comparison shown in this paper (see the paper by TVG for further details [22]). All the other teams generally underestimate the shear strain, predicting values of about 20–30 % of the experimental amplitudes. It has been observed that the accumulation of surface settlement during shaking is underestimated by the numerical models. Coupling between volumetric and shear strain is crucial in this aspect, since densification can be induced by shaking only if the volumetric threshold shear strain is trespassed. Although the mobilized shear strain in the experiments was high enough to produce irreversible volumetric change, either the volumetric threshold predicted was too high by the constitutive models or the calculated shear strain was too low. On the other hand, possible local variation in sand density at the tunnel ankles cannot be excluded. Since a homogeneous distribution of void ratio in the sand layer was assumed in the analyses, local nonuniformity in tests could have induced higher densification than it could be predictable. The different prediction of shear strain in the different analyses should in part justify the different assessment of peak increments in bending moment shown in Fig. 16: the larger the average shear strain, the larger the peak increments in bending moment. This is, however, not always the case, indicating that larger ovalization of the tunnel lining

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can be predicted compared to what expected on the basis of the free-field calculated shear strain. The increment in hoop forces due to ovalization is highly affected by the assumption on the soil–lining interface, as observed before; therefore, no particular correlation with the predictions of shear strain could be evaluated.

6 Conclusions This paper summarized and discussed the results of several sets of numerical analyses, which were performed with different constitutive models and numerical codes to predict the behaviour of a set of centrifuge models of tunnels under seismic loading in sand. The results of the numerical analyses are discussed in details in the individual papers of this Special Issue; the reference experimental set of data was published elsewhere [35, 36]. The comparison among the numerical results has shown that the amplification of ground acceleration is relatively well matched by very different constitutive models, provided that the model is able to mobilize a shear stiffness and a damping ratio in the soil layer, which are close to those mobilized in the centrifuge test. On the opposite, the calculation of shear strain is pretty much affected by the constitutive behaviour, since a significant accumulation of plastic shear strain, which may have occurred in the physical model during shaking, could be underestimated by the numerical analysis. In addition, permanent volumetric changes were observed in the tests and partially reproduced by the numerical models. The coupling between plastic shear and volume strains is a key factor in this respect, since those models which are able to associate a reasonable amount of permanent volume changes to shear strain are more likely able to predict the observed densification. Plastic soil strains are responsible for irreversible load increments on the tunnel lining after shaking. Yielding near the tunnel may result in different stress redistribution, and hence, it affects the static performance of the tunnel lining. The analysis of the numerical results (cf. Sect. 4) would suggest that plastic volume change in the soil may induce a significant change in bending moments and hoop forces in the lining after earthquake, while the effect of plastic shear strains seems of minor importance. Although mechanical characterization of sand along laboratory element test paths was rather complete and all the adopted constitutive models were rather sophisticated, the numerical calculations were often not satisfactory in predicting permanent changes in internal forces in the tunnel lining. They were in fact rather dispersed, particularly when higher levels of yielding were involved due to higher intensity of shaking.

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Finally, as far as the reversible changes in internal forces are concerned, changes in hoop force showed a major dependence on the assumption about the contact between the lining and the sand, while changes in bending moment were predicted fairly well by most models. Considering the observed differences in the prediction of the average shear strain in the sand layer, the latter changes seem to depend prevalently on the cyclic increments in shear stresses around the tunnel lining.

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