1 Introduction

There has been a substantial interest among researchers in seismic risk, which is essential in selecting and designing the most appropriate short- and long-term earthquake mitigation and after-emergency management. The corresponding accuracy depends on the many input parameters defining the risk model. Evaluating fragility curves, which represent the probability of exceedance of a predefined performance level for a given intensity measure (IM), is one of the challenging tasks of seismic risk assessment.

In a large-scale application, to reduce the computational effort, the risk analysis is performed by applying existing fragility functions, like the ones created in the framework of the Global Earthquake Model (GEM) initiative (Yepes-Estrada et al. 2016). The analytical-derived fragility curves for large-scale analyses may have been estimated using records (generally a sizeable dataset) that do not correctly account for the variation in frequency and amplitude contents; those are imposed by local geotechnical and topographic conditions. Moreover, in large-scale applications, the complexity related to the characterization of the soil-foundation system, along with the common belief in the beneficial effects of soil-structure interaction (SSI), led to the development and application over the years of fragility functions following the fixed-base restrain hypothesis (Riga et al. 2017, 2021; Smerzini and Pitilakis 2018; De Risi et al. 2019, to mention few). Notwithstanding this, a series of research attempts recognized the modification of the fragility of structures founded on soft soil compared to the typical fixed-base assumption.

Incorporating such effects in the analytical computation of fragility functions has been accomplished by employing the simplified approaches, based on the uncoupled "substructure method", refined complete SSI models based on the "direct method", or "hybrid methods" based on domain reduction methods (NIST G 2012). Different results are obtained depending on the modeling approach adopted.

While site amplification (SAmp) is inherently taken into account when adopting the direct approach, the substructure method may be suitable for understanding the relevant contribution of SSI and SAmp to the modification of the dynamic response and, thus, the fragility compared to the fixed-base assumption. Site amplification is generally shown to have a negative impact, while SSI may play a beneficial role in the resulting fragility curves enhanced by soil hysteresis (Brunelli et al. 2022).

Some literature efforts (Karapetrou et al. 2015; Tomeo et al. 2018) compared fragility curves obtained by employing the decoupled approach for fixed-base structures subjected to free-field motions and the results from the direct method. For the high rise building designed with low seismic code provisions, Karapetrou et al. (2015) found that SSI may lead to an increase in the overall fragility with respect to the fixed-base model subjected to site amplification only when soil nonlinearity is considered. At the same time, no essential differences are observed when the soil profile is assumed linear elastic.

The assumption on the soil and, most notably, the soil-foundation behavior is found in all the studies to play a fundamental role in the resulting fragilities making their employment in a generic risk framework even more cumbersome. When considering soil nonlinearity, Saez et al. (2011) observed an increase in the overall dissipation capacity with respect to the pure linear SSI due to the hysteretic dissipation during earthquake loading. On the other hand, ignoring the nonlinear foundation effects (such as gapping, sliding, and uplift) may lead to an unconservative prediction of the structural demand and, consequently, a less fragile response (Figini et al. 2012; Rajeev and Tesfamariam 2012). It is also worth mentioning that some of these studies (e.g., Karapetrou et al. 2015; Karafagka et al. 2021) investigated the role of combined effects (e.g., aging or liquefaction), making the results strongly case-dependent. Even though the results of such studies provided the scientific community with valuable knowledge for site-specific vulnerability assessment, only recently some research efforts (Cavalieri et al. 2020; Petridis and Pitilakis 20202021; Pitilakis and Petridis 2022) have paved the way toward an integrated large-scale procedure providing, for example, secondary factors to shift the existing fragility functions to include nonlinear soil and SSI effects (Petridis and Pitilakis 20202021).

Despite this last effort and the previous investigations, not all possible superstructure-foundation-soil scenarios have been considered so far, making questionable the collection of a database and its implementation in a uniform approach as required, for example, from a city-scale or regional-level application.

Also, it is essential to have robust and reliable tools with a time-efficient process for the seismic risk assessment at urban level. All previous studies (Petridis and Pitilakis 2020; 2021; Pitilakis and Petridis 2022) rely upon detailed structural models, which require significant computational time and modeling effort (geometry, reinforcement, etc.). Such an approach is unfeasible for large-scale risk analyses (Pitilakis et al. 2022).

To this aim, the present study intends to start filling the existing gap by developing and proposing a systematic methodology for estimating fragility curves of different classes of buildings considering SSI and local SAmp effects. The proposed method applies to many soil-foundation systems encountered in an urban environment and is designed explicitly for urban-scale risk assessment. To demonstrate its potential, we use the proposed approach to calculate the failure rates of the buildings in the city of Thessaloniki in northern Greece.

2 Proposed method

This section proposes and quantifies an analytical methodology to assess the fragility functions of different building classes founded on shallow or embedded foundations considering SSI and local site amplification effects. All the analyses are conceived to be implemented in the open-source OpenSees software (Mazzoni et al. 2006). However, the proposed fragility assessment scheme is easily applicable using different software. Figure 1 summarizes the whole methodological framework, whereas each step is described below:

Fig. 1
figure 1

Flow chart of the proposed methodology for the fragility assessment of structures considering SSI and SAmp at an urban-scale

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2.1 Input:

  • Our procedure starts with the definition of the input parameters. An extensive set of input ground motions is selected to formally consider the randomness of ground motions (Jalayer et al. 2017). We choose input motions recorded on outcropping bedrock or very firm soil (i.e., with VS,30 greater than 700 m/s) since site amplification is directly simulated in the following step.

  • The soil profile can be simulated in any software able to perform 1D (or 2D/3D) site response analyses. We use OpenSees to run the 1D analyses, using appropriate material models to simulate sandy or clayey soil profiles.

  • To reduce the computational effort in large-scale applications, the structure is modeled as an equivalent single degree of freedom (ESDoF) system (Fajfar 2000; D’Ayala et al. 2014).

  • This ESDoF is placed on nonlinear horizontal and vertical springs and dashpots, simulating the compliance of the foundation subsoil using the Beam-on-Nonlinear-Winkler-Foundation (BNWF) concept (Harden et al. 2005).

2.2 Analysis:

  • This study presents effective probabilistic procedures for consideration of site amplification effects directly in the fragility curves rather than in the hazard (de Risi et al. 2019; Formisano and Chieffo 2022; Chieffo et al. 2022, just to mention a few). This approach provides more precise (i.e., with a minor degree of uncertainty) estimates of local site effects than when using amplification factors and/or functions for generic soil conditions. Since the uncertainties in the soil parameters play a secondary role compared to record-to-record variability (Bazzurro and Cornell 2004; Rajeev and Tesfamariam 2012), site amplification is quantified by running 1D wave propagation analyses by applying the selected earthquake records at the base of deterministically defined soil columns. The calculated free-field motion (FFM) inherently includes nonlinear site amplification effects.

  • The free-field motions are selected as input at the base of the flexible-base ESDoF models to run all the dynamic cloud analyses (CA) (Jalayer et al. 2017). The FFM is directly applied as input for the dynamic calculations for structures resting on surface foundations. In contrast, for embedded foundations, the free-field motion is further modified to consider the kinematic interaction before being applied at the base of the superstructure.

2.3 Output:

  • An improved fragility model, notably the Modified Cloud analysis (MCA) (Jalayer et al. 2017), is adopted herein to compute fragility functions that formally consider the collapse-inducing records.

  • The statistical treatment of the results allows calculating the probability of exceeding four different limit states (slight, moderate, extensive, and complete damage state, Martins and Silva 2020). This framework provides the user with fragility functions accounting for SAmp and SSI.

One of the novelties of the proposed approach is to provide fragility functions classified according to the averaged shear-wave velocity in the top 30 m (VS,30) and a foundation feature such as the slenderness ratio, H/B pertinent for SSI approximation. VS,30, and H/B are conceived in this work as proxies for the SAmp and SSI effects. As shown in the following, the advantage of the proposed method is the possibility of linking the so-modified taxonomy with actual soil conditions and foundation geometrical parameters, leading to a risk assessment framework in which SSI and SAmp are directly considered in the fragility analysis. In detail, such an improved taxonomy makes the proposed fragility functions accounting for SSI and SAmp implementable in the OpenQuake software (Amendola and Pitilakis 2022b).

It is worth mentioning herein that, in this specific application, site amplification includes only stratigraphic amplification. In a large-scale assessment, if relevant, topographic amplification can be considered in the hazard by employing the well-known curvature (second derivative of the topographic surface) from DEM approach with literature amplification factor values from numerical analyses (e.g., Torgoev et al. 2013) or those traditionally suggested by seismic codes.

2.4 General criteria for the selection of the input data

This section provides guidelines for selecting input parameters from globally available databases. These criteria make the proposed framework easily applicable for risk assessment in different cities.

2.4.1 Earthquake record

Following the MCA approach, we suggest selecting a large set of unscaled actual ground motions mostly recorded on outcrop rock to perform all the dynamic analyses (Jalayer et al. 2017). This selection criterion is twofold; it allows considering a unique set of records for all the spatially distributed building portfolios. At the same time, this approach reduces uncertainties when investigating the site amplification effects compared to the more simplified soil class-compatible records selection.

2.4.2 Soil profile

To consider local site amplification effects and determine the soil’s prominent features in the SSI modeling, it is necessary to define representative soil profiles. Following the city-scale approach, these profiles should be based on detailed microzonation studies (if available). As an alternative, to fully cover the spatial variability of site amplification, a sufficient number of simplified soil profiles can be defined, varying the mean shear wave velocity to 30 m depth, VS,30 to cover all the soil classes according to the specific country- regulation (e.g., CEN 2005 for Europe). On such a basis, the results of the analyses and, thus, the fragility functions will depend on VS,30. The latter is critical in implementing the proposed methodology, making the resulting fragility functions site-consistent. The updated taxonomy makes the proposed fragility functions accounting for SSI and SAmp implementable in the OpenQuake software (Pagani et al. 2014).

Among various site amplification proxies (Derras et al. 2017), we select VS,30 to represent soil conditions. Different approaches are currently available in the literature to compute VS,30 maps from globally available data. The VS,30-slope correlations proposed by (Wald and Allen 2007) are well-established and already used for large-scale (and not local or site-specific) studies. These correlations concern using the topographic slope from digital elevation models (DEM) obtained from remote sensing (satellite imaging). The VS 30 map estimates are also freely available on the USGS website (USGS) worldwide. Several studies have also attempted to correlate VS,30 with geological units (e.g., Forte et al. 2017), which are again available via geological maps at various scales for the globe via the One Geology portal (ONEGEO).

2.4.3 Foundation

Here, we account for SSI effects using the Beam-on-Nonlinear-Winkler-Foundation (BNWF) concept (Harden et al. 2005; Raychowdhury and Hutchinson 2008). The BNWF element is modeled in Opensees by investigating the non-dimensional parameters influencing the seismic response of the soil-structure system to earthquake ground motion. The interaction parameters can be presented and determined by basic as follows (Veletsos and Meek 1974):

  1. 1.

    The slenderness ratio of the building, H/B, where B is the characteristic foundation half-length and H is the structural or effective height. This parameter can be retrieved from the building footprint area as available in the OpenStreetMap. The whole foundation system is reduced to an equivalent rectangular surface foundation defined as a function of the footprint area. The structural height H can be retrieved from the same open data source or the number of stories known from the building asset.

  2. 2.

    The structure-to-soil mass ratio index, δ = m/(Hρπr2), where m is the effective structural mass and ρ expresses the soil mass per unit volume. The practical range of structure-to-soil mass ratio does not vary enormously for conventional building types. Stewart et al. (1999) suggest delta values between 0.1 and 0.2, while Veletsos and Meek (1974) recommended a delta equal to 0.15. The definition of the latter parameter is of significant importance in the proposed approach since it allows for the description of the structural mass.

  3. 3.

    The soil-to-structure relative stiffness ratio, σ = VS/(ƒH), where ƒ is the fundamental frequency of the fixed-base structure, which can be computed as a function of the structural height, H. Despite being the most effective index influencing the response of SSI systems, its definition is no longer necessary having already parametrized the SSI system as a function of δ, H/B, and VS,30. Classification as a function of σ could automatically follow.

2.4.4 Structure

To make the proposed approach computationally efficient, buildings are classified by combining attributes such as lateral force resisting mechanism, height, and seismic design code level. This classification is also known as taxonomy (e.g., GEM taxonomy Brzev et al. (2013)). To further reduce the computational effort (as implicitly demanded from seismic fragility assessment of building portfolios), each building class is modeled following the equivalent single degree of freedom, ESDoF systems approximation for the superstructure (Fajfar 2000; D’Ayala et al. 2014). Among the benefits, such an approach requires a limited number of parameters determined from globally available data. Capacity curves for the characterization of the ESDoF’s hysteretic low are available in the literature for a comprehensive set of building classes. For Europe, they are available in the GitHub repository by Martins and Silva (2020).

Once its parameters are defined, the ESDoF system can be easily implementable in OpenSees through a "zero-length" or the "twoNodeLink" element object. In SSI applications, while the former needs to be implemented with a rigid beam, the latter element can be preferred given its twofold nature (it can have zero or non-zero length). The "twoNodeLink" with a non-zero length set equal to the effective structural height, H, can be employed to consider the rocking-induced structural displacement. The element’s implementation is then finalized by assigning different hysteretic relationships selected from the OpenSees material library depending on the structural type at hand. Given the availability of the OpenSees library, it is possible to consider the potential presence of strength and/or cyclic stiffness degradation in the hysteresis most relevant to the building class under consideration. Despite its simplicity, this modeling approach has been successfully adopted by different authors and thus provided a pretty good representation of the dynamic response of actual structures subjected to seismic loading (Suzuki and Iervolino 2019).

3 Application

The objective of this section is twofold. First, the novel framework for the fragility assessment of structures considering the influence of SSI and local site amplification presented so far is applied to the site of Thessaloniki, in northern Greece, which belongs to one of the most seismo-tectonically active zones in Europe. The decision also stems from the fact that there is an available plethora of data regarding the local geology and the exposed building portfolio for Thessaloniki.

As a second objective, we will show how especially in soft soil formations, the conventional way of calculating fragility curves for large-scale applications, i.e., fixed-base structures subjected to free-field motion, may lead to a different evaluation of the seismic risk discussed in the following in terms of failure rates.

3.1 Input

3.1.1 Earthquake record selection

Within this application, records were selected from the ground motion database of the Pacific Earthquake Engineering Research center (NGAWEST) and the European (ISESD), and the Italian ITACA record (ITACA) databases. In detail, record selection was carried out following the general recommendation suggested by Jalayer et al. (2017). Ground motion selection should cover a vast range of intensity measure values (Fig. 2). Different intensity measures (IM) are considered in this study for the fragility computation. The IM we chose is the pseudo-spectral acceleration at periods close to the fundamental period of the structure, Sa(T), and the average spectral acceleration, AvgSa. The latter is computed from the geometrical mean of the spectral ordinates in a given interval of 10 discrete number periods evenly spaced in the range (0.2–1.5 T s). The average spectral acceleration is particularly interesting in SSI studies since it allows for comparing fragility functions developed for different compliant systems and the reference curves considering the fixed-base assumption.

Fig. 2
figure 2

Histogram of the selected set of records recorded on rock/firm soil plotted in terms of (a) their peak ground acceleration, PGA (b) pseudo-spectral acceleration at the fundamental period of T = 0.3 s and (c) T = 0.6 s. Each class interval is determined according to Sturges’ method

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Following the same state-of-the-art approach, records should be selected respecting the focal mechanisms and soil type. The latter was the key recommendation guiding the whole record selection. Only records recorded on rock should be considered since site amplification is numerically implemented within the proposed method. No explicit consideration of the type of faulting was carried out because of the limited number of records available for stations on rock/stiff soil. Also, the selected records could not lead the structure to pass the onset of the near-collapse capacity threshold for some structural typologies. Some records were scaled up to a factor of 2 maximum, to avoid the unrealistic, undesired modifications of scaled signals. Nevertheless, the proposed methodology could benefit from considering different databases to mitigate the need for scaling. The selected unscaled records are reported in Appendix A1 and their distribution shown in Fig. 2.

All the chosen IMs are computed for the set of input records (i.e., recorded on rock/very stiff soil) referred to in the following as PGAR, Sa(T)R, and AvgSaR respectively.

3.1.2 Soil profile modeling

For the site response analysis and to define the soil input parameters for the BNWF model, seven different representative clayey soil profiles were modeled in OpenSees. The same approach can be easily applied to a sandy soil profile; nevertheless, previous works (Pitilakis and Petridis 2022) on the topic recognized no critical differences in the final fragility curves. The selected soil profiles were conceived considering different average propagation velocities of the shear waves within the first 30 m of soil depth, VS,30 (i.e., ranging from 150 to 450 m/s) thus of soil types B, C and D according to the EC8 soil classification (CEN 2005).

As reported in Fig. 3, the adopted shear stiffness profile varies continuously with depth. A simplified distribution is considered in this study to describe the distribution of the soil shear stiffness, VS (z), with depth as follows:

$$V_{s} (z) = V_{S,z = 30} \left( {\left( {\frac{{V_{s,z = 0} }}{{V_{s,z = 30} }}} \right)^{\frac{1}{a}} + \frac{z}{30}\left( {1 - \left( {\frac{{V_{s,z = 0} }}{{V_{s,z = 30} }}} \right)^{\frac{1}{a}} } \right)} \right)^{a}$$
(1)

where, z stands for the depth measured from the soil surface, while VS,z=0 and VS,z=30 are the soil shear modulus at the ground surface and the depth of 30 m, respectively. In Eq. (1) the VS,z=0 and VS,z=30 are selected to ensure a VS,30 equal to the values reported in Table 1. The a coefficient was set equal to 0.25.

Fig. 3
figure 3

(a) Schematic view of the uncoupled SSI model comprising the structure foundation system and 1D soil column characterized by (b) the shear wave profiles of the seven different virtual soil profiles considered for the analysis ranging from very soft soil (light grey curves) to medium-stiff soil (dark grey curves)

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Table 1 Main soil parameters selected to characterize the soil profile
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As shown in Fig. 3a, the numerical model for the site response analysis is performed for a single column of soil deposit 30 m deep and modeled by "Quad" elements in OpenSees. The site response analysis is implemented in OpenSees using total stress analysis. The nonlinear behavior of the soil is modeled by assigning a nonlinear material to quad elements "nDMaterial PressureIndependMultiYield" (nonlinear constitutive law based on Von Mises criterion for clays). Table 1 summarizes the physical and mechanical properties assigned to the soil layers.

All the values reported in Table 1 have been selected from literature studies assessing the soil classification (e.g., Tropeano et al. 2018; Pitilakis et al. 2019). Such studies are based on a comprehensive analysis of a worldwide database of strong ground motions recorded on deeply characterized sites up to the bedrock. Finally, the energy dissipation in the soil is introduced through the Rayleigh damping formulation.

3.1.3 Foundation modeling

Nonlinear soil-foundation interaction through the BNWF model is automatically implemented in the OpenSees software through the "shallow foundation generator" command (Raychowdhury and Hutchinson 2008). It consists of elastic beam-column elements that capture the structural footing behavior with independent "zero-length" soil elements that model the soil-footing behavior. The stiffness used to calibrate the zero-length elements should consider the effect of soil inhomogeneity (Vratsikidis and Pitilakis 2019; Amendola et al. 2021).

Literature studies (Stewart et al. 2003; NIST G 2012) suggest to approximate soil inhomogeneity with an equivalent halfspace with representative values of VS averaged over depths equal to 0.75 rθx and 0.75 rx, where rθx and rx are the rocking and swaying equivalent radii. Accordingly, a unique value is selected as a function of a VS mobilized considering an interaction volume equal to the total foundation length (grey shadowed zone in Fig. 3a). An equivalent radiation damping is calculated from the impedance functions following the procedure described in Amendola and Pitilakis (2022a). Such a simplified approach allows considering the frequency dependence of the dissipation capacity of the foundation subjected to ground shaking, compared to the simplified assumption of fixed radiation damping.

To cover a wider variety of structures, various structural details/masses and aspect ratios are selected for each structural type or building class. The foundation parameters are defined from data gathered through a literature review of previous studies on SSI available for the case study of Thessaloniki, northern Greece (Karapetrou et al. 2015; Petridis and Pitilakis 2020, 2021; Karafagka et al. 2021). These studies mainly refer to moment resisting frame (LFM), or frame plus shear wall (LDUAL) reinforced concrete (RC) buildings.

Table 2 summarizes all the retrieved information, mainly the slenderness ratio, H/B and the structure-to-soil relative inertia, δ for different concrete building classes, i.e., for a different number of stories and resisting systems. The δ values reported in Table 2 are retrieved, considering a soil mass density ρ equal to 1.8 Mg/m3; whereas all the structural masses refer to the pure bare frame, the values reported in Table 2 are considered a lower boundary for the final δ selection. According to Table 2, the H/B and δ ratios are investigated in the range (1,2,3,4) and (0.05,0.1,0.2) respectively.

Table 2 Literature studies on SSI and site amplification available for Thessaloniki, Greece and main interaction parameters taken as reference
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3.1.4 Structure modeling

As shown in Fig. 4, for this application, three building classes of low, mid, and high rise regularly infilled structures designed with low-code prescriptions are selected from the Thessaloniki exposure. These are namely the CR-LFINF-DUL-H2, -H4, and -H6, following the GEM taxonomy Brzev et al. (2013). This selection arises out of a need to represent more than 40 percent of the total exposure of Thessaloniki (Crowley et al. 2020; 2021) to gain further insights into the urban structural fragility that constitute a prerequisite in the risk calculation. Figure 4 reports the spatial distribution of the chosen structural typologies and their concentration in the urban environment along with the VS,30 map from USGS computed following the slope-based approach proposed by Wald and Allen (2007) showing how the same building class can rest on extremely different soil profiles even within the same municipality. The ESDoF system representing the selected building classes is characterized by the corresponding elastic perfectly plastic (EPP) backbone curves and by a uniaxial bilinear hysteretic material object with pinching of force and deformation, damage due to ductility and energy, and degraded unloading stiffness based on ductility, as available in OpenSees.

Fig. 4
figure 4

Distribution of the building classes in Thessaloniki selected for the analysis along with the VS,30 map for the Thessaloniki municipality computed following the slope-based approach proposed by Wald and Allen (2007)

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3.2 Analysis

Fragility curves are calculated in this study, estimating damage distributions through nonlinear dynamic cloud analyses of the structural models discussed in Sect. 3.1.4 endowed with the BNWF (Sect. 3.1.3) subjected to seismic loading. The seismic action is estimated by propagating the selected input motions (Sect. 3.1.1) recorded on outcropping rock upward the free surface through one dimensional (1D) numerical simulations of seismic site response analysis for all the seven profiles listed in Table 1 and characterized in Sect. 3.1.2.

To provide an overview of the modification of the input motion due to local geotechnical conditions, Fig. 5 compares the bedrock and the free-field motions resulting from the performed analyses in terms of acceleration displacement response spectra (ADRS). Only two different soil deposits are shown for brevity, corresponding to soil classes C and D, respectively. Overall, site amplification effects are clearly recognizable from the difference between the mean values (thick red and black lines) for all the reference profiles considered in this study. In detail, the maximum amplification for the soft soil profile (i.e., class C in Fig. 5b) occurs for periods lower than 0.6 s (also evident in other studies on the topic, e.g., de Silva 2020). In comparison, the trend is inverted for very soft profiles (i.e., class D and Fig. 5a) where the seismic demand is maximized for larger periods and thus, is expected to influence more high rise buildings. The maximum amplification occurs when the soil attains its fundamental frequency, which can be assumed accordingly to the homogeneous soil approximation, around T = 0.4 s and T = 0.8 s for the ground classified as Class C and D, respectively, in the provided example.

Fig. 5
figure 5

Pseudo-acceleration-displacement response spectra for all the selected input records (thin black lines) and the corresponding free-field motion as resulting from the site response analysis (thin red lines) along with mean values (thick lines) for two soft soil profiles corresponding to (a) VS,30 = 150 and (b) VS,30 = 300 m/s

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By processing the results of the cloud analysis, it is possible to correlate the structural response variable and the seismic intensity measurement for all the limit states (LS) of interest. The engineering demand parameter (EDP) is assumed to be the critical demand-to-capacity ratio, DCRLS (Jalayer et al. 2007), i.e., the ratio between the structural response measure and the structural capacity for the performance of interest, which is equal to unity at the onset of failure (also known as the limit state). Following previous works (Martins and Silva 20202020), the slight limit state (SD) is assumed to be a fraction of the yield displacement (0.75Sdy). Moderate (MD) and extensive damage (ED) is assumed within a range defined from yield and ultimate displacement, equal to 0.5Sdy + 0.33Sdu and to 0.25Sdy + 0.67Sdu, respectively. Finally, complete damage (CD) is considered to be reached at the onset of the ultimate displacement capacity of the structure.

Figure 6 compares the performance of the fixed-base building resting on rock (grey data) with different SSI interacting systems (blue data) for two other soil profiles, corresponding to soil class D (Fig. 6a) and B (Fig. 6b), respectively. To improve the visual comparison, together with the cloud regressions is also reported the vertical line DCRLS = 1, corresponding to the attainment of the slight damage limit state (SD) and the so-called collapse cases (red data), i.e., the cases leading to structural collapse and/or reaching dynamic instability due to large deformations (Shome and Cornell 2000). The comparison is made considering the same intensity measure, i.e., spectral acceleration of the outcrop bedrock input motions. Local SAmp and SSI effects appear evident after a visual comparison among the data trends reported in the plot of Fig. 6a where the SSI + SAmp system generally exhibits a more significant displacement for the same level of intensity measure. As expected, the data points associated with the soil profile representative of soil type B almost overlap with the reference fixed-base data, thus confirming the desired negligible effects of very stiff soil on the seismic response of structures.

Fig. 6
figure 6

Cloud analysis results for the slight damage state (SD) for different building–foundation-soil combinations, i.e. for a profile with (a) VS,30 = 150 m/s and (b) VS,30 = 300 m/s; the vertical dashed red line indicates the attainment of the performance level (limit state function DCRLS = 1)

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Moreover, in all cases where SSI becomes essential, i.e., generally for very soft deposits, large slenderness, and inertia ratios (as in the case of Fig. 6a), the collapse cases tend to increase when nonlinear foundation behavior is allowed (red rhombus data) compared to the fixed-base assumption (red circles).

Thus, although the flexural displacement, responsible for the damage to the structures, generally tends to decrease when considering only SSI effects, the foundation translation and rocking due to soil compliance lead to an increase in the overall displacement (Karatzetzou and Pitilakis 2018). Consequently, even only SSI effects may lead to increased structural fragility for excessive displacement demand leading to structural instability.

The Modified Cloud analysis (MCA, Jalayer et al. (2017)) is used herein to compute fragility functions which formally considers the collapse-inducing records. Collapse cases are defined in literature as the input motions causing structural collapse and/or reaching dynamic instability due to large deformations. In this revised model, the collapse and non-collapse parts are mixed using the "Total Probability Theorem" as follows:

$$\begin{aligned} P(DCR_{LS} > 1|IM) \,\,=\,\, & P(DCR_{LS} > 1|IM,NoC) \cdot (1 - P(C|IM)) +\,\, P(DCR_{LS} > 1|IM,C) \cdot P(C|IM) \\ \end{aligned}$$
(2)

In detail, \(P(C|IM)\) is the probability of having collapse. From Eq. (2) it is relatively straightforward that as the number of records leading to collapse increases, the resulting structural fragility also increases. Although inherently different, the final regression is expected to diverge from the classical linear one when collapse cases occur, i.e., generally for larger IM values. This method is particularly effective in characterizing the fragility of compliant base structures, where collapse cases are likely to happen when SSI effects are significant. As the intensity measures reach higher levels, loss of numerical convergence occurs. When the compliance of the foundation soil is also taken into account, dynamic instability in the analysis may occur due to excessive settlements or rocking of the basement without significant damage to the superstructure.

From Fig. 6, it can be seen that for the exact value of IM, some records induce collapse cases when considering the interaction with the foundation subsoil. In contrast, they do not collapse when following the fixed-base assumption. It is well known in the literature (Gajan et al. 2010) that significant rotation of the foundation is more likely to occur for structures with high aspect ratios. As the soil becomes softer, this kinematic mechanism tends to enhance, being the impedance functions and the foundation stiffness directly proportional to the soil shear modulus. Therefore, this failure mechanism, and consequently collapse cases, are more expected in the case of high values of H/B and low values of VS,30.

In Fig. 7, the cases of collapse recorded during the dynamic analyses of the low, medium, and high rise reinforced concrete structures considering the foundation systems discussed in Sect. 3.1.3 characterized by different H/B and δ ratios are shown. For comparison, the collapse cases for all seven virtual profiles listed in Table 1 are reported, from very soft (red data) to stiff (blue points). As expected, for all the building classes under consideration, as the interaction factors H/B and δ increase and theVS,30 decreases, the cases of collapse increase. In particular, the cases of collapse for the mid rise system resting on soil class D (blue data point) identified by a foundation system H/B = 1 and δ = 0.05 are null. In contrast, for H/B = 3 and δ = 0.2, the cases go up to 14, confirming what stayed before.

Fig. 7
figure 7

Collapse cases recorded during the dynamic analyses of the low, medium, and high rise building class (CR-LFINF-DUL) identified by different soil-foundation systems, H/B = 1, 2, 3, 4 and δ = 0.05, 0.1, 0.2

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To conclude, even though it is common in design and assessment practice to neglect SSI because of the expected beneficial play on structural safety (mainly due to the predicted demand reduction coming with the inertial effects), in some cases, mostly when SSI effects are essential (high H/B and δ and low VS,30), the explicit consideration of nonlinear foundation compliance and consequent collapse cases, leads to an increase of the overall structural fragility.

Finally, the uncertainty of the fragility parameters is estimated through the standard deviation, βTOT, which is modeled by the combination of three different uncertainty sources. These are assumed statistically independent and include the variability related to the definition of the damage states and the building-to-building variability within the same building class. Furthermore, only one set of structural models is considered here, thus without explicitly capturing the epistemic (modeling) uncertainties. Besides, the third aleatory uncertainty source due to the random nature of the ground motion is, considered by assuming a set of different ground motions.

3.3 Output

The results of our methodology are fragility functions considering different SSI and SAmp scenarios.

Figure 8 reports the comparison of fragility functions developed for a low, mid, and high rise regularly infilled structure designed with low-code prescriptions (namely CR-LFINF-DUL-H2, -H4, and -H6) by changing the H/B and δ ratio and the VS,30 for all the predefined limit states. The fragility functions shown in Fig. 8 were estimated as a function of spectral acceleration at the conditioning period varying for each configuration (i.e., Sa(T = 0.3 s) for CR-LFINF-DUL-H2 and Sa(T = 0.6 s) for CR-LFINF-DUL-H4, -H6). All in all (see, for example, Fig. 8a), the result of the analyses for the flexible foundations, considering SSI and SAmp effects (dashed lines), produce a decrease of the median values (DCRLS,50th = 1) of the fragility curves compared to the fixed-base case (continuous lines), implying an increase in the structural fragility.

Fig. 8
figure 8

Comparison between fragility functions in terms of Sa(T)R developed for the selected building classes, i.e. CR-LFINF-DUL low (a, b), mid (c, d) and high rise (e, f) considering the structure fixed at its base (continuous lines) and SSI and SAmp for one BNWF system characterized by H/B = 1, δ = 0.05 and (a, c, e) VS,30 = 180 m/s and (b, d, f) VS,30 = 360 m/s (dashed lines)

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The fragility shift is more pronounced for very soft soil profiles. See for example Fig. 8c developed for the virtual soil profile corresponding to VS,30 180 m/s compared to Fig. 8d for VS,30 360 m/s. Indeed, for mid and high rise structures, i.e., for flexible systems characterized by a long fundamental period of T > 0.6 s, the spectral acceleration is generally expected to reduce for medium-soft profiles (see Fig. 5b) leading to an increase just in the case of Class D soil profiles (see Fig. 5a).

On the other hand, SSI may play a significant role in increasing the seismic base shear force for the low rise structure. Generally, the fixed-base period of low rise buildings being minimal may lie within the initial sharply-increasing zone of the response spectrum (red curve in Fig. 5). Hence, in such cases, site amplification effects may play an important role, but also, period elongation associated with SSI may cause an increase in the spectral acceleration ordinate. This results in an overall increase in the structural fragility, as shown in Fig. 8c,d.

When comparing the fragility functions developed for the selected building classes resting on the same soft soil profiles still accounting for different hypotheses on the BNWF systems by varying the H/B and δ ratio, it is possible to appreciate the variability associated with the SSI phenomenon in the fragility computation (see Fig. 9). This holds especially when SSI is expected to be significant (i.e., again for large values of H/B and δ ratios).

Fig. 9
figure 9

Fragility functions in terms of AvgSaR developed for the selected building classes, i.e. CR-LFINF-DUL low (a, b), mid (c, d) and high rise (e, f) considering SSI and SAmp accounting for different hypotheses on the BNWF system, i.e. by low H/B and low δ = 0.05 ratios (continuous lines), high H/B and low δ = 0.05 ratios (dashed lines), low H/B and high δ = 0.2 ratios (dotted lines) and high H/B and high δ = 0.2 ratios (dashed-dot lines) for (a, c, e) very soft soil profile characterized by VS,30 = 180 m/s and (b, d, f) soft profile characterized by VS,30 = 360 m/s

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The scatter between the results is likely to be more pronounced for high damage states due to the nonlinear soil-foundation phenomenon occurring under strong events, i.e., large IM values. The uncertainty in the definition of the soil-foundation configuration cannot be neglected for the complete damage state (red curves in Fig. 9) even for the medium soft profile characterized by VS,30 = 360 m/s.

Considering the number of records, intensity levels, and a possible parameterization of the structural characteristics, several thousands of time-history analyses are typically carried out in such studies.

For the seismic fragility curves including SAmp and SSI effects shown up to now, the given intensity measure refers to the analysis input records (i.e., as recorded on rock/stiff soil). These fragility curves can be used to compute the seismic risk by convolving it with the site-specific hazard curve at the bedrock level. Also, they can be adopted to gain insights into the differences concerning the typical assessment practice, which considers fixed-base structures and neglects the modification of the input motion due to the deformability of the soil profile (as in the case of Fig. 8a).

On the other hand, the fragility curves as a function of intensity measures defined from the free-field motions can also be used in the framework of a risk assessment where the hazard includes site effects adopting either code- or research-based amplification factors or, moreover where the hazard scenario comes directly from physics-based numerical simulations.

3.4 Risk model

The final goal of this work is to provide the reader with proof on the applicability of the proposed approach in the framework of an urban-scale risk assessment. The fragility functions accounting for different hypotheses on the soil-foundation configuration are integrated with the seismic hazard of Thessaloniki. The main results lead to understanding the role of SSI and SAmp in the final risk calculation, rather than being limited to the structural fragility assessment.

Following the performance-based earthquake engineering (PBEE) framework, (Cornell and Krawinkler 2001) the fragility curves shown in Sect. 3.3 are adopted to estimate the nominal probability of failure of the different building classes outlined in Sect. 3.1.4 supposed to be located in sites with other geotechnical conditions. The likelihood of failure can be quantified as the expected number in one year of earthquakes capable of causing the exceedance of a predefined performance level, also known as failure rate, λF as follows:

$$\lambda_{F} = \int\limits_{0}^{ + \infty } {P\left( {DCR > 1|IM = x} \right) \cdot \left| {d\lambda (x)} \right|}$$
(3)

where the first part of the integral represents the structural fragility and dλx the differential of the hazard curve. The intensity-based closed-form solution proposed by Jalayer et al. (2020) is adopted for the calculation of Eq. 3.

The hazard is computed using the OpenQuake platform (Pagani et al. 2014) for a reduced number probability of exceedance (POE) and then interpolated to reduce the computational burden. The open-access uniform European seismic risk model, ESRM20 (Crowley et al. 2021) recently developed in the Horizon 2020 EU SERA project framework (SERA) is adopted. Consistent with the proposed approach, the hazard is computed at the outcropping bedrock and is estimated in terms of spectral acceleration, Sa, at two oscillator periods, T = 0.3 s and T = 0.6 s for a reference site within the Thessaloniki municipality.

The so-calculated seismic hazard is integrated with the probabilistic representation of seismic fragility for the selected building classes to evaluate the failure rate. Figure 10 presents the computed rates with different colors and markers corresponding to different soil-foundation typologies and configurations (i.e., different VS,30, H/B, and δ ratio), with respect to the complete damage performance level.

Fig. 10
figure 10

Comparisons of failure rates for the low, mid, and high rise building class site computed with fragilities from different SSI scenarios

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From Fig. 10 appears that all the considerations made on SSI and SAmp effects on fragility functions are similarly reflected in λF. Failure rates are generally more significant when considering SSI and SAmp effects with respect to the reference fixed-base case neglecting both effects (black horizontal line in Fig. 10). Such an increase is again more pronounced for the very soft soil profiles (data corresponding to VS,30 = 150 or 180 m/s) and for large values of δ and H/B ratio. The blue, yellow, and grey data, which correspond to δ equal to 0.2 and H/B equal to 2, 3, and 4, respectively, always belong to the uppermost part of the graph. When the soil becomes stiffer, all the soil-foundation configurations show similar results, with a consequent overlap of the data points shown in Fig. 10.

From a visual comparison of Fig. 10, for the fixed-base reference case (black horizontal line), the failure rates increase as the structure’s height increases, i.e., from low to high rise. On the contrary, when considering the compliance of the foundation soil, this tendency is almost inverted. For the low rise case where SSI has been shown in the previous sections to play a crucial role, failure rates, especially in the soft-medium soil profile range, are higher than in mid and high rise buildings. For the latter case, the flexibility of the base may also lead to a decrease in the risk compared to the fixed-base case. It is worth remembering that the hazard is convoluted in the range of intensity measures where the fragility calculated considering SSI and site amplification effects, due to the larger uncertainties, becomes lower than the fixed-base case.

The failure rates of the analyzed building classes for all the soil classes are also summarized in Table 3. Even from Table 3 can be pointed out that although the estimated fragility parameters show a substantial dependency on the soil-foundation system, it can be observed that the resulting failure rates, thanks to the filtering effect of low exceedance rates of the most significant IM values (Suzuki and Iervolino 2019), are relatively close between each other except for the very soft soil profile corresponding to VS,30 = 150 m/s, where again the variability was already observed in the fragility curves.

Table 3 Failure rates corresponding to CD state calculated convoluting the hazard for Thessaloniki and the fragilities of the low, mid and high rise selected building class site (i.e. CR-LFINF-DUL) accounting for multiple SSI + SAmp scenarios
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To conclude, the difference in failure rates indicates that the standard way of fragility calculation, especially for a structure interacting with very soft soil profiles, may lead to different results in the final risk calculation.

4 Conclusions

Despite the great research effort in developing fragility functions accounting for soil-structure interaction (SSI) and site amplification (SAmp) effects, a thorough literature review revealed that they are still unavailable for many scenarios, rendering them inapplicable for large-scale applications. Typically, accounting for SAmp and SSI effects in fragility curves requires very detailed structural modeling, which requires extensive computational time and modeling effort. Such a procedure cannot be closely intertwined with the urban-scale risk assessment. For this reason, our work proposes a comprehensive novel framework for the fragility assessment of structures considering the influence of SSI and SAmp effects, suitable for urban-scale risk analyses.

We applied the proposed framework to investigate the impact of SAmp and SSI on the fragility curves derived by nonlinear dynamic analyses on simplified structural models of the most commonly met buildings of Thessaloniki’s exposure. Our results are also discussed in terms of failure rates.

The main novelties and conclusions of the present study are:

  • A novel computational and time-efficient methodology for the seismic risk assessment, including SSI and SAmp effects, suitable for city-scale application. The proposed method is intended to be used by stakeholders and local and/or national authorities interested in risk mitigation policies and post-seismic event emergency management. Because of the necessary simplified structural modeling of the structures, engineering judgment is required to accurately interpret the results at a city scale.

  • The notion of the enhanced taxonomy that includes proxies for the soil stiffness (VS,30) and SSI (H/B) in the fragility functions. This novel concept allows us to account for site amplification and SSI effects in an integrated manner within the urban-scale risk assessment.

  • One of the major impediments when accounting for SSI effects, especially in the assessment of a complex urban environment, is the definition of the main features of the whole interactive system. In this regard, we use freely available data to enhance our taxonomy with respect to soil stiffness and SSI. The proposed method is coded in Python and implemented in the OpenSees software, while the enhanced taxonomy can be included in OpenQuake analyses making it applicable for risk assessment in different cities.

  • The extensive examination of fragility curves and failure rates demonstrates that SSI and local SAmp effects are generally more pronounced in the case of soft soil formations and low rise structures, causing considerable modification to the resulting fragility functions compared to the fixed-base assumption.

  • In addition to site amplification, contrary to the common belief on beneficial SSI effects, the interaction with the foundation soil system may lead to higher seismic demand compared to the fixed base period, depending on the structure's dynamic characteristics and the shape of the free-field response spectra.

  • The sensitivity of the structural fragility to various configurations of the soil-foundation systems investigated allowed us to understand that the uncertainties associated with the definition SSI system can further affect the results.

  • Neglecting all these effects may lead to underestimating the seismic risk. Therefore, this study encourages the adoption of SSI and SAmp models in the fragility computation, contrary to more simplified approaches for large-scale applications, to promote a more accurate quantification of the potential fragility or failure estimates. Future applications of the proposed methodology may involve detailed local information on the foundation building inventory and site conditions from microzonation studies. This will promote a seismic risk assessment of cities associated with a lower degree of uncertainty.