Dynamic soil-structure interaction models for fragility characterisation of buildings with shallow foundations

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx

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Soil Dynamics and Earthquake Engineering journal homepage: http://www.elsevier.com/locate/soildyn

Dynamic soil-structure interaction models for fragility characterisation of buildings with shallow foundations �nio A. Correia b, Helen Crowley c, Rui Pinho d, e Francesco Cavalieri a, *, Anto a

European Centre for Training and Research in Earthquake Engineering (EUCENTRE), Via Adolfo Ferrata 1, 27100, Pavia, Italy National Laboratory for Civil Engineering (LNEC), Av. do Brasil 101, 1700-066, Lisboa, Portugal c Seismic Risk Consultant, 27100, Pavia, Italy d University of Pavia, Civil Engineering Department, via Adolfo Ferrata 5, 27100, Pavia, Italy e Mosayk Ltd., via Fratelli Cuzio 42, 27100, Pavia, Italy b

A R T I C L E I N F O

A B S T R A C T

Keywords: Fragility curves Soil-structure interaction (SSI) Shallow foundations Impedance functions Lumped-parameter model Nonlinear macro-element Unreinforced masonry

Dynamic Soil–Structure Interaction (SSI), involving the coupling of structure, foundation and soil, is a crucial and challenging problem, especially when soil nonlinearity plays an important role. This paper shows the impact of adopting different SSI models on the assessment of seismic fragility functions. The linear substructure approach is initially adopted by implementing two different models, the first of which is one-dimensional and includes, between the foundation node and the ground, a translational elastic spring and a dashpot, whose stiffness and viscous damping are retrieved from the real and imaginary parts of the dynamic impedance at the first natural frequency of the structure. The second and more refined model is a Lumped-Parameter Model (LPM) accounting for frequency dependence of the impedance. In order to explore the sensitivity of fragility functions to the linearity assumption, an additional approach, including soil nonlinearities, is employed. A nonlinear footing macro-element is adopted to model the near-field behaviour by condensing the entire soil-foundation system into a single nonlinear element at the base of the superstructure. Energy dissipation through radiation damping is also accounted for. The superstructure response is simulated in all approaches as a simple nonlinear single-degree-offreedom (SDOF) system. The comparison between the adopted approaches is evaluated in terms of their effects on the characterisation of fragility functions for unreinforced masonry buildings (URM) on shallow foundations.

1. Introduction Recent earthquake occurrence in northern Netherlands has been attributed to gas production activity in the Groningen field, the largest of which to date has been the Huizinge event of August 2012 with a magnitude ML 3.59 (Mw 3.53: Dost et al., 2018) [1]. In response to this induced seismicity, the operators of the field, NAM - Nederlandse Aar­ dolie Maatschappij B.V., have been developing a comprehensive seismic hazard and risk model for the region, which comprises the entire gas field plus a 5 km buffer zone onshore (van Elk et al., 2019) [2]. A key component of the risk assessment involves the definition of fragility functions (which describe the probability of exceeding a given damage or collapse state, conditional on the intensity of input ground motion) for each building type that has been identified within the re­ gion, and included in the exposure model. At least one real

representative building from the region was found for each building type (the so-called index building) and structural drawings were used to develop a multi-degree-of-freedom (MDOF) numerical model of the structural system that included also the predominant non-structural el­ ements (such as partition and external façade walls). Nevertheless, running nonlinear dynamic analyses of many such numerical models, each subjected to tens or hundreds of records, was deemed to be a too large computational effort to allow fragility functions to be directly developed from these analyses. A simplified single-degree-of-freedom (SDOF) equivalent system approach, which considers dynamic SoilStructure Interaction (SSI), was thus used instead to analytically derive the fragility functions for the structural system of each building typology (Crowley et al., 2017 [3], 2019 [4]). Khosravikia et al. (2018) [5] investigated the effect of SSI on seismic risk, the latter interpreted as the probability distribution of seismic

* Corresponding author. E-mail addresses: [email protected] (F. Cavalieri), [email protected] (A.A. Correia), [email protected] (H. Crowley), [email protected], [email protected] (R. Pinho). https://doi.org/10.1016/j.soildyn.2019.106004 Received 31 July 2019; Received in revised form 15 November 2019; Accepted 8 December 2019 0267-7261/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Francesco Cavalieri, Soil Dynamics and Earthquake Engineering, https://doi.org/10.1016/j.soildyn.2019.106004

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Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx

monetary loss due to structural and nonstructural damage; the findings show that structures (with shallow foundations) on very soft soils are expected to experience reduced losses due to SSI, whereas the presence of moderately soft soils leads to considerable probability that SSI has detrimental effects and thus increases the seismic losses. It confirms the well-known fact that SSI can be beneficial, detrimental, or uninfluential on the seismic vulnerability, and thus risk, of structures, depending on the soil and structural characteristics. By modelling SSI through a finite element soil-block (i.e. using the direct approach) and performing In­ cremental Dynamic Analysis (IDA) to retrieve fragility curves of rein­ forced concrete (RC) buildings, Pitilakis et al. (2014) [6] found that the consideration of SSI effects may significantly affect the expected per­ formance of structures founded on soft soils producing an important shift to the left of fragility curves, i.e. towards more fragile response, in comparison to the fixed-base case. The crucial role of SSI under linear or nonlinear soil behaviour in altering the expected structural performance and fragility curves of high-rise fixed-base RC structures was also pointed out by Karapetrou et al. (2015) [7], who concluded that the hypothesis of fixed-base structure may lead to unconservative results. Nevertheless, it should be recognised that SSI will, in general, reduce the structural deformation demands at the cost of larger overall displacements. Lesgidis et al. (2017) [8] quantified the impact of the frequency dependence of the (linear) SSI on the fragility of RC bridges. By comparing the predicted vulnerability of a reference bridge using both a conventional, frequency-independent, Kelvin–Voigt model and the lumped parameter formulation by the same authors, it was found that the actual fragility curves of a bridge can be both underestimated or overestimated by the simplified, frequency-independent approach, and thus the latter may lead to a bridge behaviour significantly diverging from the actual one. An interesting study highlighting the importance of taking into ac­ count soil nonlinearities in SSI was presented by Bolisetti et al. (2018) [9], with reference to risk assessment of nuclear structures. Results from the nonlinear time-domain SSI analysis in LS-DYNA (LSTC, 2013) [10] for high intensity shaking were compared with those from an equivalent-linear analysis using a frequency-domain code, namely Sys­ tem for Analysis of Soil-Structure Interaction (SASSI) [11], finding that the equivalent-linear and nonlinear responses are significantly different: it was concluded that ignoring the nonlinear effects, including gapping, sliding and uplift, may lead to an unconservative prediction of the su­ perstructure response and of its seismic risk. The relevance of a nonlinear approach was also highlighted by Petridis and Pitilakis (2018) [12], who retrieved fragility curves for a set of RC moment resisting frames using the Beam on Nonlinear Winkler Foundation (BNWF) model and lumped individual elastic springs. Rajeev and Tesfamariam (2012) [13] using the BNWF model provided another evidence of the impact of (nonlinear) SSI on the seismic vulnerability of RC frames. The objective of the current work is to investigate the impact of adopting different SSI models of shallow foundations on the collapse fragility functions for the unreinforced masonry (URM) building typol­ ogies found in the Groningen exposure model (Crowley et al., 2019) [4], and, to this end, three different SSI modelling approaches were adopted. The first two, namely a one-dimensional frequency-independent model and a Lumped-Parameter Model (LPM) accounting for frequency dependence of the impedance, belong to the linear substructure approach, considering kinematic and inertial interaction effects by the principle of superposition. Instead, the third approach relies on the adoption of a nonlinear macro-element and belongs to the class of hybrid methods, combining the features of sub-domain decomposition and finite element modelling, including soil nonlinearities. Sections 2 and 3 of this manuscript summarise the properties of the investigated index buildings and of the soil in Groningen, respectively. The two linear SSI models are described in Section 4, dedicated to the substructure approach, while Section 5 presents the nonlinear SSI approach with the developed footing macro-element. Section 6

addresses the adopted methodology to retrieve fragility functions and the comparison between the obtained fragility curves for the investi­ gated index buildings, whilst final discussion and conclusions are given in Section 7. 2. Investigated index buildings Ten different index buildings (Arup, 2017) [14], all typically con­ structed with shallow foundations, have been considered herein, with the characteristics summarised in Table 1. These residential buildings are either detached or terraced (with units varying from 2 to 8) and, depending on their age, they are constructed with timber or concrete floors, and solid or cavity URM walls. Fig. 1 shows screenshots of the numerical models for the buildings. As mentioned above, in all SSI models considered in this work, the superstructure is represented in a simplified way as a SDOF system, whose behaviour is described in SeismoStruct (Seismosoft, 2019) [15] with the multi_lin model (Sivaselvan and Reinhorn, 1999) [16]. The latter, characterised by a polygonal hysteresis loop, can simulate the deteriorating behaviour of strength and stiffness. The sixteen parameters needed to fully define the response curve, for which an example is shown in Fig. 2, are related either to the backbone curve or to the hysteretic rules. In order to calibrate this hysteretic model, and with the exception of LNEC-BUILD3 (for which shake-table test results were employed), fixedbase MDOF models for each index building were produced in LS-DYNA [10] or ELS [17] and were subjected to nonlinear dynamic analyses using 11 training records (see Arup, 2017 [18] for further details). The maximum attic displacement of a given MDOF model under each training record was converted to the equivalent SDOF displacement (see Crowley et al., 2019 [4]) and then compared with the displacement obtained under the same records for the fixed-base SDOF model in SeismoStruct. These displacements were plotted against the average spectral acceleration (AvgSa) of each record, defined as the geometric mean of ten spectral acceleration ordinates from 0.01 to 1 s, and the linear regressions, in log-log space, of each model were compared (Fig. 3 shows such comparison for one of the structural models); the SDOF model was iteratively adapted until a reasonable match was obtained. The adopted properties for the SDOF systems for each index building are reported in Table 2. The symbol Heff denotes the effective height of the SDOF (see Crowley et al., 2019 [4]), representing the building centroid height. The sixteen parameters of the multi_lin hysteretic model are defined as follows: EI is the initial stiffness (kN/m), PCP and PCN are the positive and negative “cracking” force (kN), PYP and PYN are the positive and negative yield force (kN), UYP and UYN are the positive and negative yield displacement (m), UUP and UUN are the positive and negative ultimate displacement (m), EI3P and EI3N are the positive and negative post-yield stiffness as percent of elastic, HC is the stiffness degrading parameter, HBD is the ductility-based strength decay parameter, HBE is the hysteretic energy-based strength decay param­ eter, HS is the slip parameter, and IBILINEAR is a model parameter equal Table 1 Summary of the URM index buildings with shallow foundations.

2

Index Building Name

System type

Floor type

Wall type

Number of storeys

Mass (tonnes)

Zijlvest Julianalaan E45 Patrimoniumstraat Kwelder Badweg LNEC-BUILD3 Dijkstraat Solwerderstraat De Haver (house)

Terraced Terraced Terraced Terraced Detached Detached Detached Detached Detached Detached

Concrete Concrete Concrete Timber Concrete Timber Timber Timber Timber Timber

Cavity Cavity Cavity Cavity Cavity Cavity Solid Solid Solid Solid

2 þ attic 2 þ attic 2 þ attic 2 þ attic 1 þ attic 1 þ attic 1 þ attic 2 þ attic 2 þ attic 2þ mezzanine

219 252 315 148 96 44 44 138 106 159

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Fig. 1. Screenshots of LS-DYNA or ELS models of URM index buildings with shallow foundations (Arup, 2017) [18].

Fig. 3. Comparison of displacements from a MDOF (transformed to SDOF) LSDYNA model and a corresponding SDOF SeismoStruct model with calibrated multi_lin hysteretic model.

Fig. 2. Example response curve for the multi_lin hysteretic model.

to 0 for trilinear model, 1 for bilinear model, and 2 for vertex-oriented model. The typical foundations of both detached and terraced building types consist of a grid of continuous beams oriented in two orthogonal di­ rections, of either unreinforced masonry or concrete. Fig. 4a shows the schemes of foundations considered for bearing capacity calculation by Crux Engineering (2014) [18] for both unrein­ forced masonry and concrete foundations. The typical width is 600 mm, whereas the foundation level ranges between 0.2 and 1 m. A typical unreinforced masonry foundation is a strip foundation that is achieved by a widening of the load bearing walls; a representative foundation section is shown in Fig. 4b (Arup, 2015a) [19]. The inertia characteristics of the foundation were evaluated considering a rectan­ gular section 660 mm wide, characterised by the same moment of inertia of the section showed in Fig. 4b. The resulting height of the equivalent

square section is equal to 221 mm. A typical reinforced concrete foun­ dation is a strip foundation with width 600 mm and height 330 mm; only in some cases, for non-bearing walls of terraced buildings, a square cross section 330 mm wide was considered, as shown in Fig. 4c. For both unreinforced masonry and concrete, a foundation level at 600 mm depth was considered herein. 3. Soil characterisation in Groningen In order to account for SSI it is first required to define representative soil profiles that may be used for assessment of the input parameters of the different models used (one-dimensional frequency-independent, LPM, macro-element). The selection of representative soil profiles takes advantage of the detailed microzonation carried out in recent years for the Groningen region, resulting in maps of the site response 3

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Table 2 Adopted properties for the SDOF systems. Index Building Name

Mass (tonnes)

Period (s)

Heff (m)

Zijlvest Julianalaan E45 Patrimoniumstraat Kwelder Badweg LNEC-BUILD3 Dijkstraat Solwerderstraat De Haver

219 248 308 148 96 44 44 138 106 159

0.34 0.15 0.24 0.10 0.08 0.13 0.08 0.36 0.30 0.13

3.75 4.01 4.99 2.85 2.75 2.81 2.72 6.70 5.40 3.70

Index Building Name

multi_lin hysteretic model parameters (in base units of kN and m) [continued from table above] PCN

Zijlvest Julianalaan E45 Patrimoniumstraat Kwelder Badweg LNEC-BUILD3 Dijkstraat Solwerderstraat De Haver

150 800 467 400 300 150 138 400 150 400

PYN 320 1300 900 800 500 151 246 401 600 900

UYN 0.020 0.034 0.010 0.007 0.008 0.002 0.014 0.010 0.033 0.010

multi_lin hysteretic model parameters (in base units of kN and m) EI

PCP

PYP

UYP

UUP

EI3P

75000 448000 202900 571429 600000 100000 253906 41366 46875 400000

150 800 467 400 300 150 138 400 150 400

320 1300 900 800 500 151 246 401 600 900

0.020 0.034 0.010 0.007 0.008 0.002 0.014 0.010 0.033 0.010

0.109 0.050 0.098 0.017 0.118 0.020 0.052 0.154 0.123 0.061

0.012 1.00E-09 1.00E-09 0.018 1.00E-09 0.054 1.00E-09 1.00E-09 1.00E-09 1.00E-09

UUN 0.109 0.050 0.098 0.017 0.118 0.020 0.052 0.154 0.123 0.061

EI3N

HC

HBD

HBE

HS

IBILINEAR

0.012 1.00E-09 1.00E-09 0.018 1.00E-09 0.054 1.00E-09 1.00E-09 1.00E-09 1.00E-09

1 200 1 1 1 1 200 1 1 1

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0

Fig. 4. a) Schemes of foundations for bearing capacity calculation; b) Typical masonry and c) concrete foundations for both detached and terraced buildings.

Amplification Factor (AF) for several spectral ordinates (RodriguezMarek et al., 2017) [21]. The examination of AF distributions shows that in general the patterns of high and low AF are well reflected by the geological zonation model (Bommer et al., 2017) [22]. Therefore, the AF represents well the soil behaviour of the shallow deposits, and it can be considered a reliable parameter for the identification of representative soil profiles. The site response analysis study (carried out for ten levels of input motion) was performed for a grid of about 1400 000 points homoge­ neously distributed in the Groningen area. Fig. 5a shows the distribution of AFs for the highest input motion level spectral ordinates for a period of 0.5 s. Due to the non-negative values of the AF, it was assumed that AF

follows a lognormal distribution. A representative shear wave velocity (VS) profile was evaluated as the mean of VS profiles around the median AF (equal to 2.25) consid­ ering all sites with AFs in an interval of amplitude equal to 0.2. The AFs corresponding to the largest input motion level were considered. Different levels of AF and/or input motion can be selected to define alternative VS profiles in future works, with the median AF being the most representative. The VS profile is not the only relevant parameter for SSI, therefore a real stratigraphy, with the corresponding soil parame­ ters (strength, stiffness, etc.), needs to be identified. The simplest way to perform this operation is to identify a real stratigraphy (i.e., one from aforementioned 140 k sites considered for site response analyses) 4

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Fig. 5. a) Histogram of AF at a period equal to 0.5 s and highest input motion level; b) Mean shear wave velocity profile around median AF vs best fit profile. Plots were derived using data described in Kruiver et al. (2017) [23] and Rodriguez-Marek et al. (2017) [21].

compatible with the computed mean VS profile. This was done by evaluating the deviation between the mean VS profile and each one of the VS profiles in the interval of median AF considered. Fig. 5b shows the comparison between the mean VS profile and the VS profile with mini­ mum deviation. The upper 30 m of the selected soil deposit is constituted by an alternation of fine sand and cohesive layers (i.e. clayey sand and sandy clay). In the shallow part, which mostly affects the response of shallow foundations, there is a 5 m thick layer of fine sand. The shallow water table level implies that the computation of the footing seismic response should be performed in undrained conditions, given the large velocities of soil deformation due to seismic loading. In the framework of the site response analysis carried out for the Groningen region, and in addition to the VS profile and soil stratigraphy,

a number of other geotechnical properties were considered, including a set of geomechanical parameters important to describe the dynamic soil behaviour such as the modulus reduction and damping curves (see Kruiver et al., 2017 [23] and Rodriguez-Marek et al., 2017 [21]). Un­ fortunately, for the fine sand surficial layer, strength parameters are not available; consequently, these were estimated based on existing litera­ ture, trying to constrain the selected values based on available infor­ mation (i.e. VS profile, coefficient of uniformity and D50 – diameter of the particle with 50% of passage in the grain size distribution). In particular, Fear and Robertson (1995) [24] proposed a framework for estimating the undrained steady state shear strength of sand (su) from in situ tests; the formulation combines the theory of critical state soil me­ chanics with shear wave velocity measurement. This undrained shear strength was computed taking into account the drained conditions on the distribution of soil stresses along the depth due to permanent loads resulting from the weight of the soil above a given depth (the effect of the relatively modest weight of the structures being studied was considered in these calculations through engineering judgement and considering all uncertainties involved, resulting in a slight increase of the undrained shear strength at shallow depths). Fig. 6 shows the un­ drained shear strength profile in the shallow part of the selected representative soil profile, used to compute the bearing capacity under undrained conditions. These values are for the free-field case, since the actual values at each building site will slightly differ when the building weight is considered. For the complete set of calculations and results obtained for the characterisation of soil profiles in the Groningen region, interested readers are referred to the report by Mosayk (2019) [25]. 4. Substructure approach As mentioned already, in this work SSI was initially analysed by the substructure approach, which allows splitting kinematic and inertial interaction in different sub-steps and considering their combined effects using the principle of superposition (Mylonakis et al., 2006) [26]. A substructure approach is typically subdivided in three sub-steps: (i) ki­ nematic interaction, where the effect of having a foundation with a given geometry and stiffness in the soil is analysed for evaluating any possible modifications to the input motion at the base of the structure in comparison to the free-field motion, resulting in the Foundation Input Motion (FIM); (ii) soil impedance determination, where the deform­ ability and overall dynamic characteristics of the soil layers are analysed

Fig. 6. Undrained shear strength measured/estimated in the shallow layers of the representative soil profile. 5

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in order to include the soil compliance at the base of the structural model; (iii) inertial interaction, where the structural mass is considered and its effects on the overall response of the structure, subjected to the FIM and on a compliant base, are determined. Kinematic interaction modifications of the free-field motion on shallow foundations are mainly attributed to three phenomena: (i) base slab averaging of the motion underneath the footings; (ii) a ground motion amplitude decrease at the foundation level, with the embedment of the footings, and possible introduction of rocking motions at the footings due to such variation along the depth of the foundation; (iii) wave scattering effects at the edges of the footings, resulting in a reduced amplitude of high-frequency components (Stewart, 2000 [27], Iovino et al., 2019 [28], Brandenberg et al., 2015 [29], Di Laora, 2016 [30], Conti et al., 2018 [31]). In practical applications, earthquake engineers commonly neglect the effects of kinematic interaction (Dezi et al., 2010) [32]. In the seismic assessment of the type of structures considered in this study, Arup (2015a, 2015b) [20], [33] also considered that kine­ matic interaction could be assumed as being negligible. In fact, given the relatively shallow depth of the footings of the index buildings considered in this study, as shown in Fig. 4, the ground motion amplitude decrease at the foundation level with respect to the free-field is assumed to be negligible. Likewise, the base slab averaging and wave scattering effects were considered to be unimportant for vertically propagating S-waves. As a consequence, the free-field motion was used as input motion for the nonlinear dynamic analyses in this study. On the other hand, inertial interaction includes the dynamic response of the coupled soil-foundation-structure system due to the input motion and is characterised predominantly by a shift of structural frequencies to lower values and by an increase of damping in the coupled system. In the substructure approach, the soil is typically replaced by a set of springs and dashpots (as well as masses, in some cases) at the foundation level, representing the foundation dynamic impedance (see Section 4.1). The latter is a complex-valued function, whose real and imaginary parts vary with frequency and depend on the stiffness and on the energy dissipation properties of the system, respectively. Two different models following the substructure approach were implemented in SeismoStruct for derivation of fragility functions; they are described in Sections 4.2 and 4.3 in relatively brief fashion, but interested readers may refer to the report by Mosayk (2019) [25] for all those details, and results, that could not be included here due to space constraints.

The impedance functions were computed considering a composite medium (i.e. soil layer with limited depth on top of a half-space), characterised by a linear shear wave velocity profile in the upper soil layer and a constant value on the half space underneath. The layer properties (e.g. thickness, shear wave velocity) were defined taking into account the constraints imposed by the software used, which considers fixed values of the ratio of layer thickness to the half-width of the equivalent square footing. Moreover, the fitting of the shear wave ve­ locity profile was carried out for a ratio between the shear wave velocity at the base of the footing and at the half-space equal to 0.6. Fig. 7a shows an example of shear wave velocity profile fitting. Given the different equivalent dimensions considered, the VS profile fitting needs to be repeated for each of the four cases (stiffness and damping, translational and rotational degrees of freedom) accounted for. It is also noted that the employed software (DYNA6.1) considers fixed values of material damping, equal to 0.03 for the upper layer and 0.05 for the half-space. Impedance functions were calculated consid­ ering a Poisson’s ratio ν equal to 0.45, i.e. corresponding to a quasiincompressible medium. Based on the results of site response analysis, scaling factors (SF) for the VS profile were defined to account for soil nonlinearity depending on the strain level. A relationship between PGA and VS scaling factors was obtained considering at different PGA levels the mean strain level and shear modulus degradation in the fine sand layer, which is characterised by two different degradation curves. Fig. 7b shows the G/Gmax scaling factors for the two shear modulus degradation curves considered within the fine sand layer. Five PGA levels ranging from 0.05 g to 0.43 g were considered in the derivation of impedance functions, with different sets of impedance functions being used in the fragility curve derivation at different seismic intensities. Fig. 10 shows an example of impedance functions computed using the input data described above. 4.2. One-dimensional frequency-independent model The simplest SSI model employed in the fragility functions’ devel­ opment in this work is a one-dimensional frequency-independent model, called SSI 1-D hereafter, having a lateral spring with stiffness kx and a dashpot with viscous damping coefficient cx (see Fig. 8). The model proposed by Maravas et al. (2014) [35] was taken as a reference. Such model was simplified by considering only the translational spring and dashpot (i.e., rotational spring and dashpot are not included) and frequency-independent impedance. The kx and cx parameters should be evaluated at the resonant fre­ quency of the compliant system, that is, the system including the structural SDOF plus spring and dashpot. The fundamental period of such system can be estimated as a function of the period of the fixed-base SDOF and the ratio between the stiffness of the fixed-base SDOF and the one of the lateral spring (Bilotta et al., 2015) [36]. This fundamental period estimation requires an iterative process, in which the spring stiffness at the first step is evaluated at the resonant frequency of the fixed-base SDOF: for the ten considered index buildings, a few iterations (up to three, depending on the building) were needed. The kx and cx parameters were then evaluated at both resonant frequencies (i.e. of the compliant and fixed-base systems), considering the impedance functions derived at the highest PGA level, corresponding to a 100 k years return period: at this intensity level, the real part of impedance attains the lowest values and hence the highest period divergence (between the two systems) occurs. The comparison carried out for the ten index building showed small to negligible variations of both kx (zero or less than 1% for eight buildings and around 5% and 10% for the remaining two) and cx (always less than 1%). Such variations are not expected to alter, at least significantly, the fragility curves. It is also noteworthy to highlight that for a return period of 100 k years the soil behaviour is deemed to be nonlinear, and thus the inaccuracies due to the use of elastic impedance make the obtained small variations even less influential. On the other hand, for lower intensity levels higher values of the real part of

4.1. Definition of impedance functions Impedance functions were evaluated using the software DYNA6.1 (GRC, 2015) [34]. The foundations of the considered buildings consist of a grid of continuous beams oriented in two orthogonal directions. Conversely, the structural model used for definition of the fragility curves is a SDOF system in which the contact with the soil is limited to a single point. The geometry of the foundation system does not allow a simple and unique definition of equivalent dimensions for impedance function calculation; in fact, depending on the degree of freedom ana­ lysed (i.e. translational or rotational) or on the nature of the impedance component under consideration (i.e. stiffness or damping), the charac­ teristics of the real foundation to be preserved are different (contact area, inertia, etc.). For such reason, in order to properly consider the real foundation geometry, the definition of the equivalent footing di­ mensions for impedance calculation made use of the calibration step carried out for the macro-element analyses (see Sections 5.2 and 5.3), which employs a 3D MDOF model of the buildings. For each building, these equivalent dimensions were evaluated independently for stiffness and damping, as well as for the translational and rotational degrees of freedom, in order to reproduce the static stiffness and damping deter­ mined for the equivalent macro-element of the SDOF system described in Section 5.3. 6

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Fig. 7. a) Example of shear wave velocity profile fitting; b) G/Gmax scaling factors obtained from site response analysis for different levels of shear strain in the fine sand layers, characterised by two shear modulus degradation curves.

4.3. Lumped-Parameter Model (LPM) A Lumped-Parameter Model (LPM) accounting for frequency dependence of the impedance functions was also simulated in Seis­ moStruct and used for the derivation of fragility functions. Even though techniques are available to describe frequency depen­ dence of any type through a generalised LPM whose form is not known in advance (Lesgidis et al., 2015) [37], this work adopted the simplest LPM capable of describing approximately, over the frequency range of interest, the features of two components of impedance, namely the translational and rotational terms. The LPM model proposed by Dezi et al. (2009) [38] and Carbonari et al. (2011, 2012, 2018) [39–41], was taken as a reference. Such model was simplified in order to neglect the rocking-sway coupling, because the focus in this paper is on shallow foundations where such coupling is not important in the linear range. The adopted system is shown in Fig. 9. The crucial feature of this LPM is the introduction of a translational fictitious (non-physical) mass mx at the interface node (representing the foundation), linked to the ground by a translational spring (of constant kx) and by a dashpot (of constant cx). This system is characterised by a frequency-dependent response to an input and thus allows for an approximate description of the frequency dependence of the impedance. Expressing the equation of motion of the system without the super­ structure in the frequency domain, it can be easily seen that the real component of the complex dynamic impedance decreases parabolically (kx – mx ω2) with frequency, whereas the imaginary part increases lin­ early (cx ω) with frequency. In case the foundation mass is taken into account, it is added to the fictitious mass in the same node. In order to model the foundation rotation, the LPM also includes a fictitious rotational mass mry at the interface node, linked to the ground by a rotational spring (of constant kry) and dashpot (of constant cry). The soil portion of the LPM is thus characterised by two independent

Fig. 8. The adopted one-dimensional frequency-independent model.

impedance occur, leading to lower period shifts and, as a consequence, even smaller kx and cx variations between the two systems. For the above reasons and for simplicity, the values of the stiffness and viscous damping coefficient were obtained by evaluating the impedance functions derived for the Groningen field at the fundamental frequency of the fixed-base SDOF. The structural SDOF mass, stiffness and damping coefficient are indicated with ms, ks and cs, respectively, in Fig. 8. The seismic excitation is input to the system as an acceleration time history, a(t), applied to the fixed support at the base. Table 3 and Table 4 report the retrieved properties of the SSI 1-D systems for two index buildings (one terraced and one detached), in terms of stiffness and damping coefficient, for all five scaling factors (SF) considered. For the complete set of SSI 1-D parameters, interested readers are referred to the report by Mosayk (2019) [25]. Table 3 Properties of the SSI 1-D system for Patrimoniumstraat (terraced) index building. kx (kN/m) cx (ton/s)

SF1

SF2

SF3

SF4

SF5

4.805Eþ06 8.754Eþ04

3.889Eþ06 7.856Eþ04

3.225Eþ06 7.056Eþ04

2.744Eþ06 6.430Eþ04

1.997Eþ06 5.438Eþ04

Table 4 Properties of the SSI 1-D system for Solwerderstraat (detached) index building. kx (kN/m) cx (ton/s)

SF1

SF2

SF3

SF4

SF5

3.545Eþ06 3.334Eþ04

2.804Eþ06 2.978Eþ04

2.281Eþ06 2.698Eþ04

1.918Eþ06 2.478Eþ04

1.396Eþ06 2.085Eþ04

7

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well as the sliding translational displacement of the footing, resulting in the structural displacement only. The seismic acceleration, a(t), is actually input to the system as an inertia force history, f(t), applied to the superstructure mass: this approach properly considers the inertial components in the presence of the structure (structure and foundation masses and their interaction), resulting in a response in terms of relative displacements with respect to the ground motion. Table 5 and Table 6 report the retrieved properties of the LPM sys­ tems for two index buildings (one terraced and one detached), in terms of mass, stiffness and damping coefficients, for all five scaling factors considered. For the complete set of LPM parameters, interested readers are referred to the report by Mosayk (2019) [25]. 5. Hybrid approach with nonlinear macro-element 5.1. Overview of the employed nonlinear macro-element Inertial interaction in presence of nonlinear soil response can be simulated through the use of soil-foundation macro-elements. SSI analysis using macro-elements is frequently adopted in research studies for a more reliable estimation of soil displacements, given that these have been previously shown to be a cost-effective and reliable tool for such type of analysis, since they suitably represent both the nonlinear soil behaviour at the near-field and the ground substratum dynamic characteristics at the far-field, as well as the interaction with the seismic response of the structure (Correia, 2011, 2013) [42,43]. Hence, all as­ pects of elastic and inelastic behaviour of the foundation system are encompassed into one computational entity and are described by the behaviour of a single point at the centre of the foundation. The footing macro-element model by Correia and Paolucci (2019) [44] builds upon the concepts and formulations of the models by Chatzigogos et al. (2011) [45] and by Figini et al. (2012) [46], incor­ porating improvements to address inconsistencies regarding the formulation of the participating mechanisms and to extend their scope to three-dimensional loading cases. Moreover, this macro-element in­ troduces an enhanced uplift model, based on a nonlinear elastic-uplift response, which also includes a phenomenological model for the pro­ gressive degradation of the contact at the soil/footing interface due to irrecoverable changes in its geometry as plastic deformations develop in the soil. An amended bounding surface plasticity model, able to repre­ sent the ultimate plastic flow conditions and the transition between the initial elastic and inelastic responses, was developed. Finally, improved return mapping algorithms were adopted in order to reproduce a more general and realistic behaviour, that correctly takes into account the simultaneous elastic-uplift and plastic nonlinear responses. It was implemented in SeismoStruct [15] and is used herein for the derivation of fragility functions. Fig. 11 illustrates nonlinear responses obtained with the macro-element. Following the parametric study by Pianese (2018) [47], the five calibration, model-specific, parameters of the macro-element became well-constrained, allowing for the dynamic response to be obtained with confidence. The remaining parameters correspond to: (i) the footing dimensions; (ii) the six initial elastic frequency-independent values of the diagonal impedance matrix, which can be easily obtained from literature, and which represent the far-field response; (iii) the six bearing capacity values, which can be derived from classical formulae, and which represent the near-field failure conditions. In between these two extreme types of response, the macro-element gradually evolves from the initial elastic response to the plastic flow at failure through the bounding surface plasticity model, incorporating the uplift and contact degradation phenomena. The adopted system for nonlinear dynamic analyses, as modelled in SeismoStruct, composed of a nonlinear structural SDOF and a footing macro-element, is shown in Fig. 12. As done for the LPM, in order to capture the inertial interaction between the superstructure and the foundation (with mass mf), the superstructure mass is placed above the

Fig. 9. The adopted Lumped-Parameter Model.

Fig. 10. Sample fit of real and imaginary parts of two impedance components, in the 0–10 Hz frequency range.

degrees of freedom. The mass matrix takes the form: � � � � mx 0 M11 M12 M¼ ¼ 0 mry M12 M22

(1)

The stiffness and damping matrices, K and C, are written similarly. The six diagonal terms of the matrices, namely M11, M22, K11, K22, C11, C22, which are coincident with the parameters of the soil portion of the LPM, are obtained by fitting the two components of impedance (i.e., translational and rotational) with parabolic and linear functions for the real and imaginary parts, respectively. Fig. 10 shows an example of such fit, for a structural SDOF with first natural frequency of 7.6 Hz. In order to capture the inertial interaction effects between the superstructure and the foundation, the superstruc­ ture mass is placed above the ground at the building centroid height, Heff, and is connected to the interface node by a rigid link. In this way, the rigid displacement of the superstructure mass due to the foundation rotation θf, equal to Heff ⋅θf , is taken into account within the nonlinear dynamic analyses, and then subtracted from the total displacement, as 8

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Table 5 Properties of the LPM system for Patrimoniumstraat (terraced) index building. mx (ton) mry (ton*m2) kx (kN/m) kry (kNm/rad) cx (ton/s) cry (ton*m2/s)

SF1

SF2

SF3

SF4

SF5

1.252Eþ03 1.743Eþ05 9.649Eþ06 7.653Eþ08 8.619Eþ04 5.223Eþ04

9.953Eþ02 1.604Eþ05 7.741Eþ06 6.188Eþ08 7.735Eþ04 4.894Eþ04

8.116Eþ02 1.501Eþ05 6.366Eþ06 5.127Eþ08 6.967Eþ04 4.641Eþ04

6.779Eþ02 1.328Eþ05 5.368Eþ06 4.351Eþ08 6.331Eþ04 4.416Eþ04

4.699Eþ02 9.618Eþ04 3.816Eþ06 3.147Eþ08 5.369Eþ04 4.014Eþ04

Table 6 Properties of the LPM system for Solwerderstraat (detached) index building. mx (ton) mry (ton*m2) kx (kN/m) kry (kNm/rad) cx (ton/s) cry (ton*m2/s)

SF1

SF2

SF3

SF4

SF5

1.762Eþ03 4.486Eþ04 4.349Eþ06 1.549Eþ08 3.234Eþ04 2.047Eþ05

1.532Eþ03 4.394Eþ04 3.506Eþ06 1.253Eþ08 2.889Eþ04 1.771Eþ05

1.360Eþ03 4.247Eþ04 2.896Eþ06 1.039Eþ08 2.617Eþ04 1.567Eþ05

1.192Eþ03 4.063Eþ04 2.457Eþ06 8.847Eþ07 2.403Eþ04 1.413Eþ05

8.420Eþ02 3.674Eþ04 1.766Eþ06 6.435Eþ07 2.024Eþ04 1.170Eþ05

Fig. 11. Examples of macro-element nonlinear responses for an individual footing: vertical settlement due to an increasing vertical force (left); moment-rotation dominated by uplift behaviour, when a small vertical force is applied (centre); and moment-rotation dominated by plasticity, when a large vertical load is applied (right).

ground at the building centroid height, Heff. Similarly, the seismic ac­ celeration, a(t), is actually input to the system as an inertia force history, f(t), applied to the superstructure mass. The three springs and dashpots represented in the 2D view of Fig. 12 model the macro-element elastic behaviour in the far-field. Their constants correspond to the stiffness and damping in the vertical direction (kV, cV), horizontal x-direction (kHx, cHx) and rotational direction around the y-axis (kMy, cMy). For simplicity, the remaining three springs and dashpots are not visualised in the 2D scheme: however, such elements are present in the macro-element implementation and play an active role in the dynamic analyses, being the macro-element behaviour fully coupled in the six directions. Since the structural model used for the computation of fragility curves is a SDOF model, the definition of the input parameters for a representative macro-element requires a calibration step. Such calibra­ tion was carried out in order to define the characteristics of the macroelement equivalent to the real foundation system, which is composed of a grid of foundation beams. The calibration step, described in Section 5.3, was carried out considering a MDOF model for each building, in which each portion of a foundation beam between openings of the structural walls was represented by a macro-element, whose charac­ teristics were defined as described in Section 5.2.

proposed by Gazetas (1991) [48] for rectangular foundations (B < L are the semi-width and semi-length of the circumscribed rectangle) on ho­ mogenous half-space, the latter being characterised by VS, weight of soil unit volume γ and Poisson’s coefficient ν, equal to 190 m/s, 18.4 kN/m3 and 0.45, respectively. The corresponding initial elastic shear modulus Gmax was assumed equal to 67.7 MPa. As mentioned above, the macro-element requires as input the constant initial elastic stiffness and radiation damping coefficients for the six degrees of freedom. With reference only to the three directions of interest for the analyses (i.e. vertical, horizontal x-direction and rotational direction around the y-axis), the following expressions proposed by Gazetas (1991) [48] were used for the elastic stiffnesses: � � �0:75 � 2 Gmax L B kV ¼ 0:73 þ 1:54 1 ν L � � � �0:85 � � 0:2 Gmax L B 2 Gmax L B kHx ¼ kHy ; ​ with ​ kHy ¼ 1 2 þ 2:5 0:75 ν L 2 ν L � �0:15 Gmax 0:75 L kMy ¼ I 3 B 1 ν by (2)

5.2. Properties of the macro-element under a single foundation beam

where Iby is the area moment of inertia about the y-axis of the soilfoundation contact surface. Again, only for the directions of interest, the following expressions proposed by Gazetas (1991) [48] were used for the radiation damping coefficients:

The input parameters of the macro-element include the foundation impedances and its bearing capacity. The derivation of both sets of in­ formation was based on the representative soil stratigraphy defined in Section 3. Values for the five model-specific parameters introduced above are also required. Since only a shallow depth is involved in the response of the footings, the foundation impedances were determined using the relationships

cHx ffi ρVS Ab 3:4 ​ ; ​ with ​ VLa ¼ V � πð1 νÞ S cMy ¼ ρVLa Iby cry

(3)

where ρ is the soil mass density, Ab is the soil-foundation contact surface 9

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[47]. In particular: � the uplift initiation parameter (α) is only dependent on the assumed stress distribution of vertical stresses underneath the foundation and its value can be determined from simple static considerations. In the analyses, it was set equal to 3, thus assuming a linear distribution of vertical stresses underneath the foundation for the soil at the beginning of the analysis; � the soil/footing contact degradation (dmg), taking into account the decrease of the contact area due to inelastic rocking, is evaluated based on calibration to experimental results. In the analyses, it was set equal to 0.1; � the reference plastic modulus (ho) was set equal to 0.2; � the exponent for loading history in unloading/reloading (nUR) was set equal to 1; � the plastic potential parameter (χ g) was set equal to 2. The scallop shape was assumed for the bounding surface, since the dynamic analyses were performed under undrained conditions [44]. 5.3. Properties of the equivalent macro-element

Fig. 12. The foundations).

adopted

system

with

footing

macro-element

The employed footing macro-element models the soil under a single footing or foundation beam. However, since a simplified SDOF system approach was used to represent the structural system, the derivation of an “equivalent” macro-element for an entire building was needed. To this end, the first step was to build a MDOF model for each index building. Fig. 13 shows the models built in SeismoStruct for both terraced and detached buildings. Given the similarity of geometric properties for all the considered terraced index buildings, the same model (Fig. 13a) was used for all of them, only changing the total mass accordingly. For the same reason, the model for Badweg (Fig. 13c) was used also for the shake-table test specimen LNEC-BUILD3, and the model for Dijkstraat (Fig. 13d) was used also for Solwerderstraat. The models for Kwelder and De Haver are shown in Fig. 13b and Fig. 13e, respectively. Masonry piers and spandrels were introduced as columns and beams, respectively. The rigid RC slabs were modelled with rigid diaphragms linking the column nodes at the floor levels. The total number of footing macro-elements included at the base of the models, in correspondence to the centroid of masonry piers, is 27 for the terraced buildings, 16 for the modern detached house, 13 for the old detached houses, 8 for the aggregate unit buildings and 28 for the farmhouse building. Reinforced concrete foundation tie-beams connect the upper nodes of the macroelements. Both masonry and reinforced concrete were considered as linear elastic materials, in the MDOF models, with their actual values for the elastic modulus and mass density. The total masses of the models, given by the superstructure mass plus the foundation mass, are approximately equal to the actual total masses, which were used in the derivation of the single macro-element properties as described in Section 5.2. The equivalent macro-element calibration requires the computation of the (elastic) stiffnesses, bearing capacity and damping coefficients along the six directions (or three, in this case, since the fragility analyses were based on a 2D response). Most of the parameters were computed analytically starting from the foundation geometry and properties of the single macro-elements, while the remaining ones required the output from the model. The model output parameters needed for the calibration are the vertical reactions of the macro-elements and the base shear ca­ pacity in the horizontal direction x: the output results were obtained from a pushover analysis, along x. The latter was carried out pushing the structure in load control with point forces located at the floor levels, according to a triangular distribution. The vertical stiffness, kV, and the horizontal stiffness, kHx, were ob­ tained by simply summing up the stiffness values of the single macroelements, assuming a rigid behaviour of the foundation plane. For the

(shallow

area, VLa is the ‘‘Lysmer’s analog’’ wave velocity, defined as the apparent velocity of propagation of compression–extension waves under a foundation, while cry is plotted in Gazetas (1991) [48] as a function of the non-dimensional frequency a0 ¼ ω B/VS. The circular frequency, ω ¼ 2π f, was evaluated at a frequency f equal to 1.67 Hz (i.e. period of 0.6 s). Such value was selected taking into account both the period of the investigated index buildings and the AF trend with frequency, the latter showing peaks between 0.4 and 1.67 Hz. The vertical (Nmax), horizontal (Hmax) and rotational (Mmax) com­ ponents of the foundation bearing capacity were evaluated under un­ drained conditions, considering the undrained shear strength profile shown in Fig. 6 and using the formulation proposed by EC7 (CEN, 2004) [49]. In particular, the maximum centred vertical load capacity, Nmax ¼ qlim ⋅B, corresponding to the ultimate static bearing capacity of a foun­ dation characterised by a width equal to B, was evaluated by the stan­ dard superposition formula (e.g. Lancellotta, 2008 [50]): qlim ¼ su Nc soc doc ioc boc goc þ q

(4)

where the bearing capacity coefficient Nc is a function of the angle of shear resistance, the undrained shear strength su was assumed equal to 12 kPa at the depth of interest, and q is the surcharge acting on the foundation level. For vertical centred load, the only correcting factor different from the unity is the shape coefficient, equal to: soc ¼ 1 þ 0:2

B L

(5)

The maximum base shear capacity, Hmax, and maximum base moment capacity, Mmax, which can be calibrated based either on ma­ terial parameters (e.g. soil-foundation friction resistance) or on theo­ retical values, were obtained using the following expressions: Hmax ¼ su ⋅Ab Mmax ¼ 0:12⋅Nmax ⋅B

(6)

Finally, the five model-specific parameters were assigned values consistent with the calibration performed in the work by Pianese (2018) 10

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Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx

Fig. 13. MDOF models in SeismoStruct (used for the definition of some of the input parameters of the SDOF’s equivalent SSI macro-element).

rotational stiffness, kMy, the lower bound would be simply the sum over the single macro-elements, as done for the other stiffness components, while adopting the upper bound would mean accounting for both the rotational stiffness of each macro-element and their vertical stiffness contribution for a rigid rotation of the foundation plane. For the case at hand, it was decided to employ the rotational stiffness upper bound, consistently with the rigid foundation plane assumption adopted for the horizontal stiffness. Since the dynamic behaviour of buildings on shallow foundations is driven more by sliding than by rocking, this choice should not lead to important variations in the results. To verify this, the fragility curves were also retrieved by using a reduced rota­ tional stiffness, in between the two extreme values. In particular, based on expert judgement, rather than rigorous mechanical considerations, one tenth of the upper bound was adopted, a value that is, of course, larger than the lower bound. For all the considered buildings, this reduced stiffness led to small to negligible variations in the fragility curve with respect to the one obtained with the upper bound, as expected. Concerning the bearing capacity, the vertical component, Nmax, was computed as the sum over the single macro-elements, while for the other components the fully coupled behaviour of the macro-element in the six directions was used for defining the size of its bounding surface. In particular, the bearing capacity in the horizontal direction, Hmax, was obtained as follows: nME P

Hmax ¼ i¼1

QNH;i ⋅Hmax;i QNH

be found in Correia and Paolucci (2019) [44]. In order to obtain the rotational bearing capacity, Mmax, the 3D vertical-horizontal-rotational interaction surface for the capacity was used to derive the following expression: Ns P

Fu;k ⋅hk sk¼1 ffiffiffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffi

Mmax ¼

2

QNM

1

(8)

Hu QNH ⋅Hmax

where Fu,k is the ultimate horizontal force at the k-th floor level, ob­ tained from a pushover analysis in the relevant horizontal direction and considering a triangular distribution along the building height, Ns is the number of storeys, hk is the height of the k-th floor level, and Hu is the sum of Fu,k for all storeys and corresponds to the ultimate base shear value. QNH was already defined, while QNM is also a function of the applied vertical load that relates the maximum rotational moment ca­ pacity of the macro-element with its actual moment capacity for such vertical load. The torsional capacity is of no interest for the 2D analyses performed. The damping constants modelling the radiation damping in the soil along the six directions were computed by summing up the values of the single macro-elements. For what concerns the rocking response this corresponds to a lower bound assumption. Nonetheless, as mentioned above, the response of the buildings considered is mainly dominated by sliding and not by rocking, thus not being affected by this choice. The five model specific parameters, as well as the bounding surface type (i.e. scallop shape), were set equal to those of the single macroelements. Table 7 reports the retrieved properties of the equivalent footing macro-elements for all the investigated terraced and detached index buildings, in terms of initial stiffness, foundation capacity and radiation damping equivalent dashpot coefficients, only along the directions of interest for the analyses. Note that the macro-element properties for Julianalaan, LNEC-BUILD3 and Solwerderstraat are the same as those for Zijlvest, Badweg and Dijkstraat, respectively, given that, as

(7)

where nME is the total number of macro-elements in the model, Hmax,i is the maximum horizontal capacity of each macro-element in the direc­ tion considered, and QNH and QNH,i are function of the applied vertical load, for the equivalent macro-element and for each of the single macroelements, respectively. This function of the applied vertical load relates the maximum horizontal capacity of the macro-element with its actual horizontal shear capacity. Further details on the involved quantities can 11

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Table 7 Properties of the equivalent macro-elements for all the index buildings. Index Building Name

kV (kN/m)

kHx (kN/m)

kMy (kNm/rad)

Nmax (kN)

Hmax,x (kN)

Mmax,y (kNm)

cHx (ton/s)

cMy (ton*m2/s)

Zijlvest & Julianalaan E45 Patrimoniumstraat Kwelder Badweg & LNEC-BUILD3 Dijkstraat & Solwerderstraat De Haver

1.521Eþ07 1.521Eþ07 1.521Eþ07 7.613Eþ06 4.828Eþ06 6.374Eþ06 1.533Eþ07

1.167Eþ07 1.167Eþ07 1.167Eþ07 5.664Eþ06 3.478Eþ06 5.055Eþ06 1.147Eþ07

8.514Eþ08 8.514Eþ08 8.514Eþ08 1.480Eþ08 1.123Eþ08 1.585Eþ08 2.969Eþ08

4.653Eþ03 4.653Eþ03 4.653Eþ03 2.429Eþ03 1.425Eþ03 2.392Eþ03 5.145Eþ03

6.634Eþ02 6.474Eþ02 6.804Eþ02 3.739Eþ02 2.192Eþ02 2.967Eþ02 7.619Eþ02

2.188Eþ04 2.289Eþ04 2.451Eþ04 7.219Eþ03 4.709Eþ03 6.415Eþ03 3.582Eþ03

3.568Eþ04 3.568Eþ04 3.568Eþ04 1.522Eþ04 8.417Eþ03 2.027Eþ04 3.456Eþ04

7.356Eþ02 7.356Eþ02 7.356Eþ02 4.427Eþ02 4.364Eþ02 2.251Eþ03 3.352Eþ03

mentioned before, these sets of buildings share the same grid foundation plan and have similar masses. For the complete set of calculations and results obtained for the definition of the SSI macro-element model, interested readers are referred to the report by Mosayk (2019) [25].

highest hazard locations in the field. Using the mean magnitude and distance from the disaggregation together with the 2017 ground motion prediction equation for the Groningen field (Bommer et al., 2017) [22], the records were then selected from a large database, including Euro­ pean (Akkar et al., 2014) [55] and NGA-West records (Chiou et al., 2008) [56]. The records were selected to match spectra conditioned on four different levels of AvgSa (corresponding to the four return periods), namely, using the ground motion selection procedure proposed by Baker and Lee (2018) [57]. Plots of the time-histories of the selected records are given in Fig. 14, whilst the corresponding response spectra are shown in Fig. 15. It is noted that AvgSa was adopted as the intensity measure in this study not only because it has been shown to be sufficient (Kohrangi et al., 2017) [58], but also because, unlike e.g. spectral acceleration at the period of vibration of the structure (which can also constitute a sufficient intensity measure), it allows a comparison between the fragility functions obtained for the different structural systems consid­ ered (each of which has a different period of vibration). Once the maximum nonlinear dynamic displacement response of a given SDOF (Sd) is obtained from all n ground-motion records, each response (sd,i) is plotted against a scalar/vector intensity measure (ln (AvgSa) herein) and the statistical parameters corresponding to a fitted lognormal distribution of Sd | ln(AvgSa) can be extracted. In particular, the expected value, E[ln Sd|ln(AvgSa)], is modelled by a linear regres­ sion equation (Equation (9)) with parameters b0 and b1, whilst the standard deviation or dispersion (Equation (10)) is estimated by the standard error of the regression:

6. Fragility functions 6.1. Methodology For the development of fragility functions, which describe the probability of reaching or exceeding a given damage or collapse state under increasing levels of ground shaking intensity, a model for the probabilistic relationship between ground motion intensity and the nonlinear structural response of the SDOF system is needed. The ap­ proaches that are commonly used for estimating this probabilistic rela­ tionship include the cloud method (Jalayer, 2003) [51], (Cornell et al., 2002) [52], the multiple-stripe method (Jalayer, 2003) [51], (Jalayer and Cornell, 2009) [53] and Incremental Dynamic Analysis (IDA) (Vamvatsikos and Cornell, 2002) [54]. Hazard-consistent record selec­ tion together with linear regression (typically used in the cloud method) has been used herein. Indeed, whilst the selection of records conditional on increasing levels of intensity could allow the multiple-stripe method to be used, whereby the probability of damage/collapse threshold ex­ ceedance at each intensity measure level is calculated from the response data and then maximum likelihood is applied to fit a fragility function to the results, this has not been undertaken herein as the largest selected ground motions do not always lead to sufficient numbers of damage exceedance/collapse for many of the vulnerability classes. Hazard-compatible records for the development of fragility functions were selected through disaggregation of seismic hazard at four different return periods (Tr ¼ 500, 2500, 10 k and 100 k years) at one of the

E½lnSd jlnðAvgSaÞ� ¼ lnηSd jlnðAvgSaÞ ¼ b0 þ b1 lnðAvgSaÞ

Fig. 14. Time-histories of the selected records. 12

(9)

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Fig. 16. Example cloud data plot with linear censored regression of the dy­ namic displacement responses of the SSI þ SDOF system (note: each vertical stripe corresponds, from left to right, to the results obtained using the Tr ¼ 500, 2500, 10 k and 100 k year records, respectively).

Fig. 15. Spectra of selected records and the conditional spectra (herein rep­ resented with the mean and � 2σ ) to which they have been matched.

βSd jlnðAvgSaÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � �2 Pn lnηSd jlnðAvgSaÞ i ln sd; i � n 2

case: this means that SSI may have a beneficial effect on the seismic vulnerability of these buildings. A more obvious SSI influence can be noted for the Kwelder casestudy alone, which, as can be gathered from Table 2, features not only a low period of vibration (0.08 s), but also a relatively high ultimate structural displacement capacity (0.118 m). As this building is stiff (and ductile, hence not prone to premature collapse), the response of the relatively weak soil inevitably plays a more determinant role in the overall fragility of the system, and hence the impact of SSI modelling becomes evident. Further, the impact of modelling explicitly soil nonlinearity is also particularly evident for this stiff and ductile build­ ing, for which higher ground motion levels are required to reach dam­ age/collapse limit states, with the macro-element fragility curve being clearly shifted to the right of the curves obtained with elastic SSI models. This also confirms that, as expected, considering soil nonlinearity be­ comes even more relevant in cases where seismic action is high. This shift due to explicit modelling of soil nonlinearity, visible also, albeit to a lesser extent, in all other buildings but Zijlvest, Dijkstraat and De Haver, confirms that the consideration of inelastic SSI behaviour effectively leads to additional energy dissipation and, consequently, to smaller structural displacements.

(10)

As mentioned above, the parameters b0 and b1 are the estimated regression coefficients obtained by performing a linear regression. In order to correctly treat the results of the nonlinear dynamic analyses where the displacement response exceeds the expected ultimate displacement capacity, a censored regression has been undertaken when estimating the coefficients of Equation (9) (see Stafford, 2008 [59]). In these cases, the value of displacement demand from the nonlinear dy­ namic analysis is not trusted (as the collapse displacement capacity has been exceeded and hence the estimated displacement response is no longer reliable), but it is known to exceed a given limiting value, and is thus referred to as a censored observation; if all censored observations were set to the limiting value, and a normal linear regression analysis were to be applied as above, the fitted model would be biased. To obtain an unbiased model, maximum likelihood technique is used (refer e.g. to Crowley et al., 2019 [4] for the corresponding formulae). It is noted that the collapse displacement capacity has been estimated through the nonlinear dynamic analysis of the MDOF systems intro­ duced in Section 2 above, and taken as the mean between the maximum attic displacement obtained for the records where global collapse does not occur and the lowest displacement at which global collapse was instead identified. An example cloud data plot with censored regression is shown in Fig. 16, where the censored observations have been plotted at the limiting displacement capacity value (their original non-censored values, exceeding the collapse displacement, are indicated with red markers).

7. Conclusions and future work In recent years, the Groningen region (northern Netherlands) has been affected by induced seismicity attributed to gas production activ­ ity. Within the seismic hazard and risk model for the region developed by the operators of the field (NAM), the definition of fragility functions for several URM index buildings is crucial. With reference to ten of these representative buildings with shallow foundations, this paper investi­ gated the impact on the collapse fragility functions of adopting different SSI modelling approaches. Two of such SSI models, namely the onedimensional frequency-independent and the LPM, are elastic, whereas the remaining one adopts a nonlinear macro-element to encompass all aspects of elastic (in the far-field) and inelastic (in the near-field) behaviour of the foundation system. The influence of SSI resulted to be non-negligible only for stiffer buildings, and in general leads to fragility curves that are less unfav­ ourable with respect to the fixed-base case. Moreover, the results showed that taking into account the inelastic behaviour of the soilfoundation system may lead to smaller structural displacements and hence to a lower vulnerability of the buildings.

6.2. Proposed fragility functions and comparison The obtained fragility curves for the collapse limit state and for the ten investigated index buildings are shown in Fig. 17. Each subplot displays the curves related to: i) the simple one-dimensional elastic SSI case, ii) the LPM elastic SSI case, and iii) the nonlinear macro-element SSI case. The curve for the fixed-base case is also displayed for refer­ ence. It can be noted that for most of these buildings with shallow foundations the influence of SSI is small to negligible, and leads the curves to be marginally shifted to the right with respect to the fixed-base 13

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Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx

Fig. 17. Proposed fragility curves for the investigated index buildings and the different SSI models. The grey dashed lines indicate the four considered levels of AvgSa; 0.2 g (Tr ¼ 500 years), 0.34 g (Tr ¼ 2500 years), 0.5 g (Tr ¼ 10 k years) and 0.86 g (Tr ¼ 100 k years).

The above demonstrates that, in order to avoid the introduction of conservative bias in the results of risk assessment exercises, it may be important not only to include SSI effects in the development of the fragility functions of the building stock, but also to do so through the employment of a nonlinear SSI model, even more so when ground mo­ tion intensity levels are high. The latter, in the context of fragility functions derivation, where hundreds or thousands of nonlinear dy­ namic analyses need to be run, must necessarily be computationally effective like the macro-element, given that more refined approaches (involving e.g. the development of a 3D elasto-plastic soil-block model) have a computational cost that renders them unfeasible for such applications.

� nio A. Correia: Conceptualization, Methodology, Software, Anto Writing - original draft, Writing - review & editing, Formal analysis. Helen Crowley: Methodology, Software, Writing - original draft, Writing - review & editing, Formal analysis, Data curation. Rui Pinho: Conceptualization, Supervision, Methodology, Software, Writing original draft, Writing - review & editing, Formal analysis. Acknowledgements The authors are particularly grateful to Pauline Kruiver, who kindly provided access to the soil mechanical characterisation data and site response analysis results for the Groningen region. The constructive feedback of three anonymous reviewers, which led to the improvement of the original version of the manuscript, is also gratefully acknowledged.

Funding This work was undertaken within the framework of the research programme for hazard and risk of induced seismicity in Groningen sponsored by the Nederlandse Aardolie Maatschappij BV (NAM).

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Declaration of competing interest None. CRediT authorship contribution statement Francesco Cavalieri: Methodology, Software, Writing - original draft, Writing - review & editing, Formal analysis, Data curation. 14

F. Cavalieri et al.

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