1 Introduction

Tunnels give continuity to road networks placed in mountainous lands and to urban transportation networks. In rural areas, tunnels are an alternative to large, sinuous, and complex geometrical routes that can compromise drivers’ safety, the traffic of vehicles, and the roads’ level of service. In urban areas, tunnels serve to subway networks and underground freeways that pass through the cities. Tunnels are classified into drilled/bored, cut and cover, and immersed. Drilled/bored tunnels are built by directly boring the soil, either bare rock or lined with concrete or other materials. Cut-and-cover tunnels are structures generally made of reinforced concrete, with a rectangular section bored in the ground and then roofed with structural elements. Immersed tunnels are those excavated in soil strata below water bodies (Argyroudis et al. 2019).

A tunnel’s total or partial lack of serviceability increases the operational costs of the transport system. This lack of serviceability can be due to human causes (e.g. traffic accidents, fires, or failures in the maintenance systems) or natural ones (e.g. landslides, seismic structural failures, underground water leakage, or deformations caused by geological faults). Structural or functional causes can also impair serviceability. The former refers to the loss of structural integrity by physical damage to the structure, preventing safe traffic. The latter occurs when the damage level does not affect the structure’s stability and integrity but prevents or limits vehicle traffic (Werner et al. 2006). The seismic activity is the main cause for tunnels losing their serviceability since it is a precursor of ground failures, liquefaction, or deformation that can affect the overall structure, the ventilation systems, and lining pavements or access portals (Wang and Zhang 2013; Roy and Sarkar 2017). The damage level and the probability of tunnel traffic interruption because of earthquakes depend on the interaction between the tunnels’ structural properties, the soil properties, and the magnitude and propagation of the seismic event (Fabozzi et al. 2018). Therefore, the tunnel management must consider procedures to evaluate traffic interruption scenarios induced by seismic events and adopt cost-efficient mitigation measures that guarantee the road network’s operational continuity.

Risk assessment offers a relevant tool for road managers to identify vulnerable segments in the road networks and estimate the probability and consequences of traffic interruptions induced by natural events (Argyroudis et al. 2019, 2020). The risk is estimated in terms of the occurrence probability of a natural event, the asset vulnerability, and the transport system’s losses due to the interruption of one or more network links (D’Andrea et al. 2005). The system’s losses are estimated in terms of the infrastructure recovery costs and travel time increase due to traffic re-routing (Deco et al. 2013; Selva et al. 2013; Argyroudis et al. 2019; Akiyama et al. 2020).

The tunnel’s seismic vulnerability is calculated from fragility curves, which estimate the probability that a tunnel reaches a specific damage state, given the earthquake’s intensity. The damage states group together the individual damages experienced by each component of the tunnel. Damages are obtained from damage reports or numerical simulations. In the first case, damages are grouped according to their magnitude and severity. In the second case, damages are grouped according to damage indexes. Argyroudis and Pitilakis (2012) proposed a damage index as the ratio between the actual bending moment of the tunnel cross section and the bending capacity to overcome the lack of indexes to characterize tunnels’ damage states. Andreotti (2019) proposed a similar damage index based on the cumulated damage measured through the relative difference between actual rotation and the ultimate plastic rotation.

Literature provided fragility curves of tunnels. Most are calibrated using data from tunnels affected by earthquakes (ALA 2001; Corigliano et al. 2007) and a combination of seismic records and numerical modelling (Huang et al. 2017, 2020; Qiu et al. 2018).

Strong earthquakes are uncommon events, and if the number of tunnels within the road networks is small, the existence of enough empirical evidence concerning the damage gains relevance. In countries with few tunnels exposed to seismic hazards, there are not always enough data of damages, geological and soil characteristics to perform an empirical or analytical calibration of fragility curves; consequently, it is not easy to estimate the vulnerability of tunnels. Likewise, inventory, repairing cost, and traffic data are not always available to estimate repairing and traffic re-routing costs.

The paper presents a simplified procedure to assess the seismic risk of highway tunnels. The method allows identifying and ranking the critical tunnels based on the risk, using existent shake maps and fragility curves to estimate vulnerability. Existing shake maps, such as those provided by the U.S. Geological Survey, can be used to estimate the earthquake intensity. The fragility curves were selected using the Rossetto et al. (2014) procedure for existing fragility curves. Considering the shake map and the fragility curves selected, the traffic interruption probability is estimated. The procedure was applied in Chile to identify the critical tunnels affected by the Maule earthquake.

2 Background on seismic risk assessment of tunnels

The risk is estimated by multiplying the hazard occurrence probability by the system’s physical and operational consequences. According to the above definition, the seismic risk in tunnels corresponds to a seismic event’s occurrence probability, multiplied by its vulnerability and the consequences on the road network (Nazari and Bargi 2012). Seismic events are described by recurrence models, attenuation models, and shake maps (Vanuvamalai et al. 2018). The vulnerability is obtained from fragility curves, in terms of the tunnels’ damage after an earthquake grouped into damage states, given the seismic event’s intensity (Kennedy et al. 1980; Choun and Elnashai 2010). The road network consequences are estimated in terms of repair costs, duration of repairs, and additional travel time costs due to re-routing traffic. The possibility of re-routing depends on the level of damage experienced by the tunnel and the existence of alternative links.

2.1 Earthquake-induced damages in tunnels

The damage induced by earthquakes in tunnels allows defining the damage states and, at the same time, damage states are essential to calibrate fragility curves and to estimate vulnerability. Tunnels’ seismic damages are caused by ground failure, which depends on the site’s geological and geotechnical characteristics, the displacement of active faults, and the ground shaking or vibrations because of the earthquake (Dowding and Rozen 1978). Its magnitude depends on the earthquake’s intensity and localization; the tunnel characteristics (geometry, design, construction, and condition); and the characteristics of the site (adverse geology, depth, presence of faults, susceptible to slope sliding) (D’Andrea et al. 2005; Wang and Zhang 2013). Table 1 summarizes a damage classification based on Wang et al. (2001), Asakura et al. (2007), Chen et al. (2012), Roy and Sarkar (2017), and Zhang et al. (2018).

Table 1 Damage classification in tunnels affected by earthquakes
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2.2 Damage states

A damage state consists of a grouping of different damages (each of a determined magnitude) experienced by various tunnel components. It makes it possible to characterize the tunnel’s overall condition, provide information to calibrate fragility curves, and assess the tunnel’s residual serviceability.

Dowding and Rozen (1978) proposed the damage levels “no damage”, “minor”, and “severe” using damage and PGA data from 71 tunnels affected by earthquakes with Richter magnitudes between 5.8 and 8.3, occurring between 1906 and 1971 in California, Alaska, and Japan. The authors concluded that nondamaged tunnels experienced PGA values lower than 0.2 g and that tunnels with a “severe” damage state experienced PGA values greater than 0.5 g. Sharma and Judd (1991) proposed the damage levels “severe”, “moderate”, “slight”, and “no damage” from 192 damage reports from 92 earthquakes. To group the damage states at each level, the authors used the criteria: tunnel depth, type of rock, internal support, geographic location, Richter magnitude, and epicentral distance of each earthquake.

Wang et al. (2001) proposed four damage states following the criteria of crack length (L) and width (W): “no damage” (no damage is detected by visual inspection; regular traffic is allowed); “slight” (slight damage is seen by visual inspection L < 5 m and W < 3 mm; regular traffic is permitted); “moderate” (L > 5 m and W > 3 mm; differential displacements due to deep cracks and exposed reinforcements, water leakage, and joint gaps; regular traffic with restrictions is allowed); and “severe” (liquefaction and slope sliding, structural lining collapse, floods, damage in the ventilation and lighting systems; traffic is interrupted).

Werner et al. (2006) proposed four damage states and a qualitative description as follows: “slight/minor” (minor cracking in the lining, slight settlement of ground in portals and small rockfalls in portals; full serviceability is attained after four days), “moderate” (moderate lining cracking and rockfalls in the portal; complete serviceability is attained after eleven days), and “major” (major lining cracking, and significant settlements in portals; full serviceability is attained after 30 days; before this time traffic should be re-routed).

Xiaoquing et al. (2008) prepared a damage state econometric model with data from 34 tunnels. They considered the same damage descriptors as Sharma and Judd (1991) and added tunnel length, geological faults, age of the tunnel, fortification level, and portals stability.

FEMA (2011) added to the Werner et al. (2006) damage states a fourth damage state and updated the traffic opening as follows: “slight/minor” (minor cracking in the lining, slight settlement of ground in portals, and small rockfalls in portals; full serviceability is attained after three days), “moderate” (moderate lining cracking and rockfalls in the portal; full serviceability is attained after seven days), “extensive” (major lining cracking, and significant settlements in portals; full serviceability is achieved after 90 days), and “complete” (major lining cracking with possible collapse; restoration period is higher than one year; traffic should be re-routed).

Wang and Zhang (2013) proposed tunnel damage states based on earthquake observations in Taiwan, Japan, and China. They extended the classification of Wang et al. (2001) to five damage states (“no damage”, slight, “moderate”, “severe”, and “collapse”). They proposed additional damage thresholds for “moderate” (3 mm < W < 30 mm and 5 m < L < 10 m; traffic is allowed) and “severe” (W > 30 mm, L < 10 m, lining displacements over 20 cm and pavement blowup over 20 cm; traffic is not allowed) damage states.

2.3 Fragility curves of tunnels

Fragility curves estimate the probability of exceeding a damage state, given a certain intensity of a natural event. The fragility curves of Fig. 1 allow determining the likelihood that a “Ds” damage on the tunnel exceeds a specific “ds” damage state, given a particular PGA value.

Fig. 1
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Example of fragility curves for bored tunnels (FEMA 2011)

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Fragility curves are calibrated using analytical, empirical, expert opinion, or hybrid approaches (Choun and Elnashai 2010). In the analytical approach, damages are estimated by structural and geotechnical modelling combined with real structural tunnel responses under accelerations obtained from real seismic responses (Argyroudis and Pitilakis 2012; Argyroudis et al. 2014a, b; Andreotti and Lai 2015; Qiu et al. 2018). The empirical approach uses historical tunnel damage records from different earthquakes and on-site identification of damage states using visual inspection protocols (Corigliano et al. 2007; Le and Huh, 2014). The expert opinion approach estimates subjective probabilities obtained by an expert panel and damage data for building fragility curves (FEMA 2011). It is used when the empirical information is scarce or analytical modelling is intricate (Rossetto et al. 2014). The hybrid approach compensates the lack of data and expert opinions’ subjectivity combining analytical and empirical methods.

Equation 1 shows the general model of a fragility curve. It expresses the probability that a damage state “Ds” exceeds the pre-established damage state ds due to an earthquake of intensity “A”, measured in terms of acceleration.

$$Pf\left( {Ds \ge ds|A} \right) = \Phi \left[ {\frac{{\ln \left( A \right) - \ln \left( {A_{mi} } \right)}}{{\beta_{{{\text{tot}}}} }}} \right] ; \beta_{{{\text{tot}}}} = \sqrt {\mathop \sum \limits_{j} \beta_{j}^{2} }.$$
(1)

It assumes that the distribution of the mean ground acceleration “A”, where the tunnel reaches the damage state threshold dsi, is a log-normal distribution. The damage threshold (Ami) is the mean value of the peak ground acceleration, where the tunnel reaches a specific damage state. The probability is calculated by a standard cumulative distribution function (Φ) normalized by the overall standard deviation (βtot). The overall standard deviation is estimated under the assumption that their components (βj) are statistically independent. In analytical and hybrid calibration approaches, the components of βtot included the uncertainty in the seismic demand (seismic records dispersion, attenuation laws, and the selection of the seismic demand variable), in the structural capacity (mechanics and geometric parameters of tunnels, structural characteristics) in the damage state thresholds and in the type of method used to calibrate the fragility curve (Rossetto et al. 2014; Argyroudis et al. 2019). Shinozuka et al. (2000) suggest estimating the confidence interval of the fragility curves’ parameters to obtain the calibration’s uncertainty.

Table 2 shows the fragility curves available in the literature, calibrated for different tunnel configurations, intensity measures, soil conditions, and construction qualities.

Table 2 Fragility curves available in the literature
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The following characteristics of input data are identified in the models of Table 2:

  1. (a)

    Type of tunnel: fragility curves are specific to each type of tunnel. There are different fragility curves for bored/drilled, cut and cover, and reinforced concrete box tunnels.

  2. (b)

    Construction quality: only models #1 and #8 consider construction quality. Other models assume good construct quality.

  3. (c)

    Hazard intensity measure: most models use PGA and PGV as measures of hazard intensity. Model #11 uses Arias’ intensity, and model #13 uses PSA.

  4. (d)

    Tunnel depth: models #1 and #6 consider tunnels depth between 20 and 330 m. Model #17 uses tunnels depths of 10, 15, 20, and 25 m. Model #16 uses tunnels depths of 80, 200, 300, and 400 m. Other models considered depths between 4 and 44 m. Tunnel cross section: most models used circular cross sections with diameters of 6, 9, and 10 m. Only models #11 and #13 propose fragility curves dependent on diameter.

  5. (e)

    Tunnel age: the tunnel age is considered only in models #4 and #14 regarding resistance degradation because of reinforcement corrosion. These fragility curves include tunnels age of 0, 50, 75, and 100 years.

  6. (f)

    Damage states: most models use three or four damage states. The criteria for defining damage states depend on the calibration method. In the empirical fragility curves, damage states are determined using damage reports from past earthquakes. In analytical or semi-analytical fragility curves, the damage states are defined by damage indices calculated by numerical simulation as the ratio between the actual and capacity bending moment of the tunnel cross-section. Models #2 and #16 use a damage index based on the cumulative plastic rotation. In model #5, the damage index combines lining strains with bending moment. The rotation, strains, and moments are obtained by structural modelling.

  7. (g)

    Soil conditions: it is considered through a single homogeneous soil layer (for instance, models #9, #10, and #13); a single fragility curve for different soil layers (for instance, model #15); and fragility curves for different soil types (for instance models #3, #4, #14 and #18). Model #16 considers the rock quality according to its geotechnical characteristics.

A critical aspect of calibrating fragility curves is the data availability, in both quantity and quality, concerning seismic events and damages experienced by tunnels obtained through systematic protocols. It is crucial in countries with a few tunnels in their road network or does not rely on maintenance management systems that systematically collect data to calibrate fragility curves (Selva et al. 2013). If there are no locally calibrated fragility curves, an alternative is to select the most suitable ones from a set of existing curves. However, inter-model variability is a practical difficulty that needs to be addressed. To this end, Shinozuka et al. (2000), Selva et al. (2013), and Rossetto et al. (2014) discuss procedures for selecting a fragility curve from existing curves.

Shinozuka et al. (2000) proposed a statistical procedure to combine bridge fragility curves. They establish that a combined fragility curve is the weighted sum of class “i” bridge fragility curves. The weights represent the proportion of class “i” bridges in the total population of bridges in the road network. They also warn that the combined curve’s total variance increases concerning the sum of the variances of each bridge class’s fragility curves. Selva et al. (2013) used a Bayesian approach to show the propagation of inter-model variability of fragility curves in loss functions.

Rossetto et al. (2014) propose a systematic procedure for selecting a suitable fragility curve, which discriminates between analytical and empirical curves and then classifies them according to their relevance and quality. The relevance criterion depends on each fragility curve’s ability to represent an asset’s damage states for a given range of hazard intensities. The quality criterion considers the quality of the input data, the rationality of the model (in terms of uncertainty and modelling approach), and the quality of the technical documentation used to calibrate the curves.

3 Procedure for tunnel risk assessment

The risk estimation procedure is organized into three steps: (1) hazard analysis, (2) vulnerability assessment, and (3) risk estimation (Fig. 2). The seismic scenario is configured in the first step, considering the earthquake’s location and magnitude, thereby obtaining seismic exposure maps to which the tunnels’ geolocation is superimposed. These maps allow determining seismic intensities values for each tunnel, such as peak ground displacements, velocities, or accelerations, among others. The second step determines suitable fragility curves for each tunnel, based on their characteristics (tunnel type and construction quality). From fragility curves, the vulnerability of tunnels is obtained. In the third step, the risk is estimated considering the seismic recurrence, the tunnel vulnerability, and the cost. Costs include repair costs and travel time cost increasing due to the re-routing of traffic during repairs.

Fig. 2
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Proposed method for risk assessment of highway tunnels

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3.1 Tunnel seismic exposure maps

This map requires the location and moment magnitude of the earthquake and the geolocation of highway tunnels. Existing exposure maps, such as those provided by the U.S.G.S. (Shedlock and Tanner 1999), are needed. The seismic intensity values in a specific location can also be estimated with acceleration attenuation models locally calibrated, but this type of model is intensive in input data. Afterwards, the exposure maps and georeferenced tunnels are superimposed, thus obtaining the seismic intensity value for each tunnel.

3.2 Estimate of the tunnel vulnerability

This stage infers the tunnel vulnerability based on the fragility curves. Three input data are required: (a) the seismic hazard map of the tunnels; (b) the inventory of georeferenced tunnels; and (c) tunnel fragility curves. The tunnels’ seismic hazard map is obtained from step 1 (Fig. 2). The inventory of tunnels is obtained directly on-site or from existing inventories and includes the following:

  • Location of each tunnel by georeferencing their portals.

  • Type of tunnel: bored in hard rock (bored/drilled) or covered trench (cut and cover).

  • General tunnel construction characteristics: portal, liner, and construction age.

  • Construction quality, according to ALA (2001):

  • Tunnel bored with poor-to-average construction quality: tunnels with poor-to-average quality rock, without lining, with treated wood or masonry lining, with nonreinforced or low-strength concrete.

  • Tunnel bored with good construction quality: tunnels in hard rock, designed for specific geological conditions (special supports or reinforced lining in weak areas), with nonreinforced lining and good strength concrete. It also includes a reinforced lining.

  • Cut-and-cover tunnel with poor-to-average construction quality: tunnels with a lining of treated wood, masonry, or nonreinforced concrete. Includes tunnels that were not designed to withstand excessive strains.

  • Cut-and-cover tunnel with good construction quality: tunnels designed for seismic load, excessive strain, with reinforced concrete lining.

The fragility curves are selected in three steps: first, to collect existent fragility curves based on the same seismic intensity measure; second, to choose fragility curves calibrated for similar conditions of the place in which will be applied; and third, to use the evaluation criteria set out by Rossetto et al. (2014) to select the proper fragility curve. The criteria of Rossetto et al. (2014) are shown in Fig. 3.

Fig. 3
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Adapted from Rossetto et al. (2014)

Criteria for selecting fragility functions.

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Each criterion is assessed based on three scores: “high” (score 3), “medium” (score 2), and “low” (score 1). The final score is the sum of these qualifications for each sub-criterion. The curve with the highest final score is selected. When there is more than one fragility curve with the highest score, they are combined through Eq. 2, where “n” is the number of fragility curves, Pj() is each fragility curve, and E[P()] is the combination of fragility curves.

$$E\left[ {P\left( {DS \ge dsi|A} \right)} \right] = \mathop \sum \limits_{j = 1}^{n} w_{j} Pj\left( {DS \ge dsi|A} \right).$$
(2)

The weighted factor wj represents the probability that the fragility curve is the true one among the “n” curves to be combined. It can be obtained by assigning the same weight to each curve, which is equivalent to the inverse of the total curves to be combined, weighting by the score corresponding to the overall quality criterion of the curves, considering the representativeness and reliability of each curve or through the opinion of experts (Scherbaum et al. 2005; Scherbaum and Kuehn 2011; Rossetto et al. 2014). From the selected fragility curve, vulnerability is obtained using the discrete probabilities between damage states.

3.3 Risk estimation

The risk is estimated by multiplying the earthquake’s recurrence period with the traffic interruption probability and the total cost. The earthquake’s recurrence period is calculated by models that correlate the earthquake magnitude and the mean annual exceeding events. The traffic interruption probability depends on the damage experienced by the tunnel. Traffic closure rules provide a proxy of this probability for each damage state. According to Wang and Zhang (2013), the following closure rules were proposed: in the damage states, “none” traffic is allowed, and the travel time increase is zero. In the damage state “slight”, the tunnel is closed for two hours during the inspection, cleaning, and systems verification. All the traffic is re-routed for two hours. For the damage state “moderate”, traffic is re-routed during minor repairs (two or three days). In other damage states, traffic is not permitted and is re-routed. Reconstruction time is approximately one year. The total cost is the cost of repair, plus the additional cost of travel time due to re-routing by assuming that there is no long-term modification of the travel matrix involving additional nontrip costs.

4 Case study: risk assessment of Chilean tunnels

4.1 Description of the case study

Chile is in the convergent border between the Nazca and the South American plates; therefore, most earthquakes are due to subduction. In the last 400 years, Chile has registered approximately 20 earthquakes with magnitudes above 7.5. The Valdivia earthquake of 1960 has the highest magnitude ever recorded and reached between 9.6 and 9.8 of the Richter scale. This case study uses the 2010 Maule earthquake data, which impacted Chile from Valparaiso’s region to La Araucanía, covering 30% of the Chilean territory. It had a moment magnitude of 8.8 and a rupture length of 500 km offshore, 25 km from the coastline. Chile has 31 bored highway tunnels, with a total of 34.2 km. The lengths vary between 142 and 4,520 m, and the commissioning dates oscillate from 1910 to 2017. Among these tunnels, nine have reinforced concrete lining, 19 have shotcrete, and 3 have no lining; 6 have one lane in both directions, 11 have one lane per direction, 1 has two lanes per direction, and 13 have two lanes in the same direction.

4.2 Drawing up of the seismic hazard map of tunnels

The seismic hazard map was based on the Maule earthquake PGA map, drawn up by the US Geological Survey. The tunnel locations were obtained from the tunnel inventory of the Ministry of Public Works of Chile. Both maps were superimposed to identify the 31 highway tunnels affected by the Maule earthquake. The coherence between each tunnel’s construction date and the date of occurrence of the earthquake was checked. Figure 4 shows the PGA map and the 20 tunnels affected by the Maule earthquake (red circles), and Table 3 shows the primary data for each tunnel. All the tunnels are bored/drilled. Tunnels #3 to #8 are in the coastal zone and the others in Chile’s central valley. The rupture distance was higher than 40 km for the nearby tunnels. According to Poulos et al. (2019), Maule’s earthquake’s recurrence period is 242 years.

Fig. 4
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PGA seismic exposure of tunnels during the Maule earthquake. a Shakemap, b location of affected tunnels

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Table 3 Characteristics of Chilean tunnels affected by Maule earthquake
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4.3 Determination of the expected damage level

Figure 5 consolidates the frequency at which the 20 tunnels of Table 3 experienced different acceleration ranges during the Maule earthquake. In two cases, 20% of the time, a tunnel experienced an acceleration below 0.2 g; 80% of the time (16 cases), a tunnel experienced an acceleration between 0.2 and 0.4 g; no tunnel experienced an acceleration equal or higher than 0.4 g. It means that 20% of the tunnels should not have suffered any damage and that the remaining 80% suffered moderate damage and, therefore, none of them should have lost their serviceability. The lack of post-earthquake damage data did not allow comparing the reliability of the Dowding and Rozen (1978) thresholds, which helped to elaborate Fig. 5.

Fig. 5
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Expected damage level in Chilean tunnels

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Reports concerning the transport infrastructure damage after the Maule earthquake were focused on bridges, roads, buildings, and historical buildings (Elnashai et al. 2010; Yen et al. 2011; Medina et al. 2010), so no specific reports are dealing with the tunnels’ behaviour that allows accurately determining their damage states. Because Elnashai et al. (2010), Yen et al. (2011), and Medina et al. (2010) did not report tunnel damaged, it was assumed that the damage ranges proposed by Dowding and Rozen (1978) were assimilated to “slight” and “moderate” damage states. The absence of data also prevented using, for example, the central damage index proposed by Argyroudis and Pitilakis (2012) to identify the damage states.

4.4 Selection of fragility curves

The first step was to identify the available fragility curves of tunnels from the literature. Table 2 shows the fragility curves preliminary identified and their general properties. We eliminate the fragility curves with hazard intensity units different to PGA, those calibrated for box tunnels, those calibrated with design response spectrums, and those based on Eurocode, according to the criteria “asset class” and “hazard intensity” of Fig. 3. The subset of fragility curves were: #1, #5, #8, #12, and #16. The qualification procedure of Fig. 3 was applied to this subset of fragility curves, obtaining the following ranking: #1, #8, #5, #16, and #12. Model #1 scored highest on the relevance criterion, while all other models scored slightly lower. In the general quality criterion, model #1 obtained a higher score than the rest of the models mainly because it considers a substantially higher number of seismic events than the rest of the models. Therefore, according to the criteria in Fig. 3, model #1 was selected to estimate the tunnel’s vulnerability.

Figure 6 summarizes the fragility curves recommended for Chile, according to the following damage states and construction qualities: (a) tunnels bored in rocks, poor-to-average construction quality for slight damage (1L), and moderate damage (1 M); (b) tunnels bored in rock, good construction quality for slight (2L) and moderate damage (2 M). These fragility curves were used to estimate the probability that the Maule earthquake’s tunnels exceeded the “slight” and “moderate” damage states.

Fig. 6
figure 6

Tunnel fragility curves proposed for Chile

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Table 4 shows that PGA ranges that tunnels experienced during the Maule earthquake fluctuated between 0.12 and 0.36 g. For this PGA range, the exceedance probability varies between 0.02 and 0.50 for the “slight” damage state and between 0.00 and 0.13 for the “moderate” damage state (poor-to-average constructive quality); between 0.02 and 0.08 for the “slight” damage state and between 0.00 and 0.02 for the “moderate” damage state (good constructive quality). Table 4 also shows the discrete probabilities associated with the damage states “slight”, “moderate”, and “no damage”.

Table 4 Exceedance and traffic interruption probabilities of Chilean tunnels
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4.5 Risk estimation

The risk was estimated for the 20 tunnels affected by the Maule earthquake. The Poulos et al. (2019) model was used to estimate the Maule earthquake’s recurrence period. The tunnels’ vulnerability was calculated from the fragility curves proposed for Chile, according to Fig. 6, and the PGA experienced by tunnels during the Maule earthquake according to Table 4 for each damage state. The tunnel’s repairing cost was estimated using cost data from MOP (2016). The operating cost was estimated using the unit time travel cost obtained from the Chilean Ministry of Social Development and Family (MSDF 2020). The travel time increase was calculated considering the best available route to re-route whether each tunnel is closed, the closure time, and the average travel speed along that route. The tunnel’s traffic was obtained using the National Traffic Survey data elaborated yearly by the Ministry of Public Works of Chile (MOP 2018, 2019).

The risk values obtained for each tunnel are summarized in Table 5. The risk in tunnels #1, #14, #15, and #16 is substantially higher than in other tunnels because Tunnel #1 has no alternatives and requires a detour with alternating traffic. In tunnels #14 and #15, the alternative route’s standard is lower than the standard of the tunnels’ route. The alternative route is bi-directional, sinuous, and with steep slopes, reducing traffic speed and increasing travel time. In tunnel #16, the alternative route is of a similar standard but passes through urban areas with high traffic, so the traffic re-routed increases travel times.

Table 5 Risk values calculated for each tunnel
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5 Conclusions

The work proposes a simplified procedure to assess highway tunnels’ seismic hazards based on damage studies and fragility curves obtained from the international experience. The outcome of the procedure is the traffic interruption probability. The procedure proposed aids road agencies in assessing a group of tunnels’ seismic risk and making decisions such as reinforcement or repairing, considering the existence or not of alternative routes to prevent the traffic interruption. From the study conducted, the following was concluded:

After a seismic event, the characterization of damage is a critical task to have high-quality data with which to define damage states. Therefore, it is necessary to have protocols that allow estimating the magnitude and intensity of the damage. However, road agencies do not always have such protocols. Due to the need to enable the tunnels to traffic, the damages survey is not a priority. Consequently, damage data were not always available, reducing the options of calibrating fragility curves and estimating tunnels’ seismic vulnerability.

The damage states are determined using the damage magnitudes of the tunnels’ various structural components obtained after an earthquake. For example, several authors propose damage states in terms of the width and length of cracks in the lining, exposure of reinforcements, collapse of portals, among others. For this reason, it is essential to have damage survey protocols. When these do not exist, it is not convenient to arbitrarily define damage states, considering that each damage state determines the possibility of restricting or enabling traffic through the tunnels and the restoration period. Without periodic damage reports, it is impossible to define damage states and calibrate fragility curves.

Due to the variability of input data and the diversity of fragility curve calibration considerations, it is not appropriate to arbitrarily use one or more fragility curves to estimate tunnel seismic vulnerability when locally calibrated fragility curves are not available. On the contrary, it is necessary to select fragility curves considering the characteristics of the tunnels of a road network, the type of earthquake, and the geological and soil conditions at the tunnel sites to ensure representativeness for the case study. The selection procedure adopted in the case study provides a systematic tool for a proper selection of fragility curves.

The case study results show that the 20 tunnels within the Maule earthquake area were exposed to PGA between 0.12 and 0.36 g. The vulnerability to “slight” damage is lower than 0.41 and lower than 0.15 for “moderate” damage. The estimation shows that the major losses are associated with the tunnels that combine the highest vulnerabilities and poor alternative routes. These theoretical results are consistent with the evidence obtained on-site by the Chilean Road Department after the Maule earthquake, which revealed that tunnels had not suffered any damage that compromises the serviceability.

One of the case study’s limitations was the lack of accurate inventory data and damage reports after the Maule earthquake, so it was difficult to estimate more precisely the actual damage states, the detailed geometry of the tunnels, and their construction quality. Thus, the quality of the available information is a relevant criterion in selecting the fragility curves.

The proposed procedure allows the risk evaluation of groups of road tunnels. Therefore, one of the main challenges is integrating into the risk assessment other road assets, such as bridges and road platforms, and other natural hazards such as floods, landslides, and lahar flows.

Additional research is required to improve the proposed procedure. The main issues are: to calibrate models of seismic event recurrence for assessing several seismic scenarios instead of a specific earthquake; to estimate the transportation costs considering the effects of earthquakes on the origin–destination travel matrixes; and to estimate more accurately the traffic on the road network.