Seismic Reliability-Based Design Approach for Base-Isolated Systems in Different Sites

: This study employs the seismic reliability-based design approach for inelastic structures isolated by friction pendulum isolators, considering two different highly seismic Italian sites to provide useful design recommendations. Incremental dynamic analyses are carried out to estimate the seismic fragility of the superstructure and of devices, assuming different structural properties and limit state thresholds. Finally, considering seismic hazard curves of the investigated sites, seismic reliability-based design curves are proposed to derive the dimensions in plan of devices and the ductility demand of the superstructure as a function of both the structural properties and the reliability level expected. The proposed results confirm the possibility of using seismic reliability-based design as a sustainable and applicable approach and represent a large data set to adopt this design methodology in any site with a similar seismic hazard.


Introduction
In line with [1], the safety assessment of structural systems as well as the study of techniques aimed at improving their reliability is a fundamental step within the idea of a holistic perspective for safety assessment of structures and infrastructures. In this context, the friction pendulum system (FPS) is an effective seismic isolation technique [2][3][4] for building frames due to its advantages (i.e., the isolated period does not depend on the mass of the superstructure and its properties in terms of longevity and durability). Over the years, modeling issues of FPS devices have been studied by [5][6][7][8] as well as other works that have investigated how the seismic performance is affected by the properties of the structural system and of the FPS isolator and by the characteristics of the seismic input. In this context, probabilistic analyses of base-isolated systems [9][10][11][12][13] have been developed including uncertainties in characteristics related to both the isolation devices and ground motions. In [14], a probabilistic evaluation of the seismic performance of steel buildings equipped with FPS devices was carried out in comparison to the response of non-isolated structures. Seismic reliability and robustness analyses of a 3D reinforced concrete (r.c.) elastic structure with FPS isolators were carried out in [15] and [16], assuming uncertainty in both the coefficient of friction and in the main characteristics of the vertical and horizontal components of each seismic excitation. The proposal of the seismic reliability-based design (SRBD) approach for elastic systems equipped with FPS was presented in [17], illustrating the results for several structural properties. In [17], the superstructure behaviour was assumed to be elastic, and any inelastic interaction with the non-linear isolation level response was not considered. However, when strong seismic events occur, the superstructure of a base-isolated system can present an inelastic response and, in this situation, seismic codes [18][19][20][21][22] provide low values of the strength reduction factor [18,22] or behavior factor [18,19] in order to avoid the ensuing non-linear phenomenon of dynamic amplification (partial resonance) [23]. Specifically, the Italian seismic code NTC08, the European seismic code Eurocode 8 as well as the Japanese building code provide a maximum value for the behavior factor equal to 1.5 for base-isolated structures, without explicitly distinguishing the ductility term and overstrength factor term. The US seismic design code, ASCE 7, prescribes that the strength reduction factor for a seismically isolated structure is 0.375 times the one for a corresponding fixed-base structure, with an upper limit equal to 2. In this context, the authors of [24] proposed that, if the response of base-isolated structures is not elastic, the ratio between the displacement ductility demand μ and the strength reduction factor R (or behavior factor) is equal to 3, confirming that the relationships published by [25] and [26] for flexible structures and for stiff structures cannot be used for base-isolated structures. Castaldo et al. [27] extended the SRBD approach, proposed by [17] for elastic systems, to inelastic perfectly elastoplastic base-isolated structures, defining seismic reliability-based relationships between the displacement ductility and the strength reduction factor and SRBD curves for FPS devices, assuming the seismic hazard of the L'Aquila site (Italy). In [28], the SRBD approach was successively extended to hardening and softening structures.
This study employs the SRBD approach for inelastic structures isolated by friction pendulum system (FPS) isolators to provide useful design recommendations for two Italian sites (i.e., Sant'Angelo dei Lombardi and Chiusaforte) with a high seismic hazard. The influences of the main mechanical parameters for base-isolated systems on the global inelastic performance are investigated considering several combinations of inelastic and elastic building properties, at different seismic intensity levels. In compliance with [27], the isolated structures are modeled as equivalent 2-degreeof-freedom (2dof) systems with a perfectly elastoplastic rule for the superstructure and a velocitydependent model [5] for the non-linear FPS. The sliding friction coefficient and the uncertainties in the seismic records are assumed as relevant random variables. Precisely, assuming a Gaussian probability density function (PDF) for the friction coefficient, the Latin Hypercube Sampling (LHS) method [29][30][31] is adopted to sample the input data set. The two different Italian sites with a high seismic hazard assumed in this study are: Sant'Angelo dei Lombardi (lon: 15.21; lat: 40.922) and Chiusaforte (lon: 13.272; lat: 46.435). These two sites have been selected because they are characterized, respectively, by higher and lower seismic hazard in comparison to the L'Aquila site [27]. Scaling natural seismic records to the seismic intensities at the life safety limit state (in 50 years) for the two Italian sites, the yielding properties of the superstructures are defined for increasing strength reduction factors, in line with the codes [18][19][20][21]. Afterwards, incremental dynamic analyses (IDAs) are herein developed to compute the superstructure and FPS responses for increasing intensity levels and to derive seismic fragility curves of both the superstructure and the isolation level, assuming appropriate thresholds in relation to the limit states. By means of the convolution integral of the fragility curves with the seismic hazard curves of the two Italian sites, the seismic reliability curves of inelastic base-isolated structural systems, with a reference life of 50 years, are achieved. Finally, seismic reliability-based regression expressions that relate the displacement ductility demand to the ductility-dependent strength reduction factors, together with seismic reliability-based design (SRBD) curves to define the dimensions in plan of the FPS devices, are proposed for each Italian site. The results of the two sites, together with other literature outcomes related to another site [27], confirm the possibility of using seismic reliability-based design and represent a large data set useful to apply the SRBD methodology for a reliable preliminary design of base-isolated building frames with FPS devices in any area with a similar seismic hazard.

Inelastic Model with Equations of Motion for a Structural System Isolated by Single Concave Sliding Devices
In this section, the model of Naeim and Kelly [32] is revised to take into account the nonlinearities in the response of single concave sliding bearings and of the superstructure (Figure 1). Precisely, the following equations of motion for an inelastic 2dof system isolated with FPS bearings ( Figure 1) under a seismic input ( ) g u t  apply:  [5][6][7], as follows: where max f and min f denote the friction coefficient at high and at very low velocities of sliding respectively, and α represents a constant for a given pressure, temperature and condition of FPS interfaces. This constant has herein been set as equal to 30, with the value of 3 for the min max / f f ratio [5][6][7]. A bilinear hysteretic rule is used to model the isolator response in the hypothesis, in order to consider the horizontal component of the bearing displacements. The device's restoring force can be expressed as follows: A perfectly elastoplastic model is assumed to represent the inelastic behavior of the superstructure. So, the superstructure response is elastic if Equation (4) is satisfied, and the corresponding restoring force is given by Equation so the restoring force applies: Let us introduce the mass ratio ( ) can be assumed to be equal to zero since the dissipative properties are mainly related to the sliding behavior [33]. The seismic isolation degree [34] can be defined as the ratio of the isolation In the hypothesis that the inelastic response of the equivalent 2dof model is representative of the behavior of multi-story frames [35][36][37], the corresponding strength reduction factor, q, is related only to the ductility-dependent component [27,35] and is defined as: where , s el f and , s el u denote, respectively, the peak response values for the corresponding linear system during a ground motion. As discussed in [28], the abovementioned strength reduction factor, multiplied by the overstrength factor, is equivalent to the behavior factor.
The displacement ductility, μ , of the inelastic superstructure is evaluated as the ratio of the peak displacement of the inelastic system,

Uncertainties
The seismic reliability of a structural system is an evaluation of the probabilities exceeding the structural performance (SP) within its reference service life (e.g., 50 years) [38][39][40][41][42][43]. In accordance with the Pacific Earthquake Engineering Research Center (PEER)-like modular approach [44] and performance-based earthquake engineering (PBEE) approach [45,46], the steps to assess seismic reliability are: -Calculation of fragility curves, which define the probabilities exceeding the structural performance (limit state thresholds) conditional to an IM value; -Computation of the average annual rates exceeding the limit state thresholds through the convolution integral between the fragility curves and seismic hazard curves of the sites; -Calculation of the probabilities exceeding the structural performance (limit state thresholds) in the time frame of interest (e.g., 50 years) through the Poisson distribution.
In this context, this work evaluates the seismic reliability of inelastic systems with FPS, located in two different Italian sites, considering both the friction coefficient and earthquake characteristics as the relevant random variables. Neither epistemic [47] nor other aleatory uncertainties in the superstructure properties are included because of their negligible effects on the statistical values of the response parameters, according to [48], especially for high isolation degrees.
As for the uncertainty in the sliding friction coefficient at large velocity for FPS devices [5][6][7], an appropriate Gaussian probability density function (PDF) [27] truncated from 0.5% to 5.5%, with a mean value equal to 3% [49], and a coefficient of variation equal to around 0.7% are employed. These values of the PDF, considered as representative values, are also assumed to compare the results with the outcomes achieved in [27]. By means of the LHS method [29][30][31], the input data set of the friction coefficient max f is sampled. In the following parametric study, 15 values of the random variable max f are defined as described in detail by [27]. Note that the friction coefficient has been assumed as the relevant random variable in order to consider its aleatory uncertainty due to dependence on other parameters such as thermal heating, axial force and number of cycles, as widely discussed in [5][6][7].
Regarding the seismic characteristics [45,46], the randomness in the seismic intensity can be described by a hazard curve, whereas the ground motion randomness for a fixed intensity level can be taken into account by means of a large set of different ground motion realizations scaled to the common IM value. In line with the efficiency, sufficiency, and hazard computability criteria [50,51], the spectral displacement ξ , which is related to the spectral acceleration, In the analyses, the damping ratio b ξ is considered equal to zero, in compliance with other studies [27,33], so To take into account the record-to-record variability [50,51], a set of 30 ground motion records is defined. These records are derived from 19 different seismic natural events with a magnitude higher than 6 and an epicentral distance higher than around 9 km, selected from the ground motion databases of the Pacific Earthquake Engineering Research Center (PEER), of the Italian Accelerometric Archive (ITACA) and of the Internet Site for European Strong-Motion Data (ISESD) [52][53][54]. The characteristics of the selected ground motion records are reported in Table 1.

Parametric Analysis
In order to determinate the seismic reliability of the inelastic base-isolated equivalent systems, located in Sant'Angelo dei Lombardi and Chiusaforte (Italy), respectively, the first step consists of carrying out incremental dynamic analyses (IDAs) [55] to assess the structural responses for increasing IM levels. Several deterministic parameter combinations, according to Equation (8)

Incremental Dynamic Analysis Results
In this section, the IDA is performed. Specifically, for each site, each one of 320 different equivalent structural systems, combined to each value of 15 sampled friction coefficients, is subjected to 30 ground records with an intensity measure scaled to eight increasing levels. For the Sant'Angelo dei Lombardi site (Italy), the eight values of IM range from 0 m to 0.50 m, whereas for the Chiusaforte site (Italy), the IM ranges from 0 m to 0.45 m to cover the wide uncertainty in the IM up to values higher than the one related to the collapse limit state, according to [19]. The isolated non-linear systems are modeled in Matlab-Simulink [56] to solve the coupled equations (Equation (8) [58,59]. Note that no numerical or physical threshold on the response parameters has been used in order to numerically calculate the statistical values. This means that the peak values from the nonlinear time histories represent, respectively, the displacement demands for the superstructure and for the isolators.
The IDA results developed in this study for equivalent isolated structures, located in the Sant'Angelo dei Lombardi site and in the Chiusaforte site, respectively, are illustrated in

Assessment of Seismic Fragility and Reliability
In this section, the seismic fragility, representative of the probabilities Pf exceeding different limit state thresholds conditional to each level of the IM, is estimated. Appropriate limit state thresholds have been selected, respectively, for the radius in plan of the single concave surface, r [m], for the FPS and the available displacement ductility, μ [-], for the superstructure as listed in Tables 2-3. The reference failure probabilities in 50 years [39][40], which are not conditional to any level of the IM (i.e., Pf (50 years)), together with the assumed limit state thresholds corresponding to the limit states of the codes [18,19], are reported in Tables 2-3. Precisely, as regards the isolation level, the reference failure probability in 50 years [39,40] related to the collapse limit state [19] is considered, whereas the reference failure probability in 50 years [39,40] for the superstructure is related to the life safety limit state [19], which has been considered during the design phase. The probabilities exceeding the different limit states at each level of the IM are numerically calculated by estimating the complementary cumulative distribution function (CCDF) for each one of the 320 equivalent structural systems, and then fitted through lognormal distributions [17,61] with an R-square higher than 0.9. Specifically, regarding a generic structural system, for each IM level, by means of the knowledge of the statistics of an EDP, it is possible to compute the probability exceeding a limit state threshold and then the complementary value with respect to 1. These complementary values for the increasing IM levels are fitted through a lognormal distribution.  The fragility curves are depicted in Figures 8-11, showing the f P versus the IM for both Italian sites. Each figure shows several curves for different values of the mass ratio and q. Only the results corresponding to one limit state threshold, i.e., to d I = 2 and 8, with b T = 3 s and 6 s, are illustrated.
Generally, the seismic fragility increases with decreasing limit state thresholds.        In general, the values of f P for isolated structures located in the Sant'Angelo dei Lombardi site are slightly higher than the values related to the Chiusaforte site, for each parameter combination.

Seismic Reliability Curves
As widely described in Section 3, the seismic reliability of a structural system is an evaluation of the probabilities exceeding the structural performance (SP) in 50 years [38][39][40][41][42][43][44][45][46]. Following the steps presented in Section 3, through the convolution integral and using a homogenous Poisson distribution, the exceedance probabilities in 50 years (i.e., Pf (50 years)) have been computed for both the isolation level and the superstructure. With regard to the isolation level, the seismic reliability assessment makes it possible to provide SRBD abacuses to define the dimension in plan r of the FPS bearings for structures located in areas with a high seismic hazard, as a function of both the expected reliability level and the structural parameters. The linear regression curves illustrated in the range between 10 -1 and 10 -4 in the semilogarithmic space of Figures 12-13     However, the results related to both isolation level and superstructure are consistent with the outcomes in the literature [27] related to isolated structures located in L'Aquila (Italy), for each parameter combination.

Seismic Reliability-Based Displacement Ductility Demand for Increasing Strength Reduction Factors
This section provides the displacement ductility demand as a function of q and of other structural parameters corresponding to the reference exceeding probability in 50 years equal to 2.2·10 -2 [39] (i.e., the failure probability associated with the life safety limit state [19], Table 3), for each site. Note that for few structural properties has it been possible by means of exponential regressions (linear regressions in the logarithmic space) to estimate the displacement ductility demand corresponding to the life safety limit state exceeding probability. The abovementioned seismic reliability-based displacement ductility demands, fitted through linear regressions, with an R-square higher than 0.96 for the Chiusaforte site and higher than 0.95 for the Sant'Angelo dei Lombardi site, are plotted in Figures 16-17 for both sites of interest and for each parameter combination depending on the (ductility-dependent) strength reduction factor q. These proposed seismic reliability-based regressions can be extended to regular base-isolated mdof (multi-degree-of-freedom) systems under the hypothesis of regularity, as stated in [62], and can provide useful design recommendations.     Figure 17. SRB regressions between the displacement ductility demand and strength reduction factor for the Chiusaforte site (Italy), related to Id=2 (a), Id=4 (b), Id=6 (c), and Id=8 (d).
The results are consistent with the outcomes reported in [27], confirming the influence of structural properties on the displacement ductility demand of base-isolated systems located in different Italian sites with a high seismic hazard. Precisely, high values of the (ductility-dependent) strength reduction factor and of the mass ratio lead to a disproportionately large displacement ductility demand, which may cause collapse [63]. It can also be observed that lower values of Id, with fixed Tb, can reduce the displacement ductility demand as well as that with fixed Ts, higher values of Id can be useful to reduce the displacement ductility demand in some cases, as illustrated in Figure  18 for a fixed value of q = 1.5. It surely can be noted that the behavior factor of the codes is high for some structural properties.
As expected, the values of the displacement ductility demand μ obtained for the Sant'Angelo dei Lombardi site are slightly higher than the values obtained for Chiusaforte. A comparison of the seismic reliability-based results of the Sant'Angelo dei Lombardi site and the Chiusaforte site with the outcomes related to the L'Aquila site [27] shows a similarity in the values for the three different Italian sites, confirming the possibility of using SRBD as a sustainable and applicable approach. In addition, as explained in [27], these seismic reliability-based results, within the force-based approach [18][19][20][21][22], are related only to the ductility-dependent behavior because the overstrength capacities are included in the equivalent perfectly elastoplastic models. Finally, the proposed results for the two sites suggest that the relationship between the (ductility-dependent) behavior factor q and the displacement ductility demand is linear, with a slope higher than unity and that for some parameter combinations. Especially for low isolated periods with a high isolation degree and mass ratio, a value of q lower than 1.5 should be suggested in areas with a high seismic hazard.

Conclusions
This study employs the seismic reliability-based design (SRBD) approach for inelastic structures isolated by friction pendulum system (FPS) isolators, comparing the results of two different Italian sites (i.e., Sant'Angelo dei Lombardi and Chiusaforte) with a high seismic hazard. The seismic reliability of these structural systems is assessed assuming different elastic and inelastic properties, different seismic intensity levels and considering the aleatory uncertainties in the friction coefficient and in the seismic inputs. Scaling natural seismic records to the seismic intensities at the life safety limit state for the two Italian sites, the yielding properties of the superstructures are properly designed for increasing (ductility-dependent) strength reduction factors. Successively, incremental dynamic analyses are carried out according to the seismic hazard of each site to assess the seismic fragility of the inelastic superstructure and of the seismic devices. In this way, considering the seismic hazard curves of the Sant'Angelo dei Lombardi site and of the Chiusaforte site (Italy), respectively, for systems isolated through FPS devices and with a reference life of 50 years, SRBD abacuses are proposed. Specifically, SRBD curves relate the dimensions in plan of the FPS bearings and the failure probability depending on the structural parameters. In addition, SRB curves are provided with the purpose of defining reliable relationships between the (ductility-dependent) strength reduction factor and the displacement ductility demand, depending also on the other structural properties. The results highlight that a slight overestimate of the (ductility-dependent) strength reduction factor may also lead to unexpected amplification phenomena (i.e., collapse). In compliance with other literature results related to the L'Aquila site (Italy), the proposed regression relationships are linear, and for some parameter combinations, show a slope strongly higher than unity. These larger slopes are achieved especially in the case of the Sant'Angelo dei Lombardi site due to its higher seismic hazard. However, the SRBD abacuses related to both the isolation level and the superstructure for basedisolated systems located in the Sant'Angelo dei Lombardi site and the Chiusaforte site are consistent with the results related to the L'Aquila site. Therefore, it is possible to use the proposed SRBD formulae as design recommendations for a reliable and preliminary design of base-isolated regular frames, located in high seismic areas. Moreover, the comparison of results between the two sites and with the outcomes related to the L'Aquila site demonstrates the applicability of the SRBD approach as a sustainable and applicable design methodology, providing a large data set useful for a reliable preliminary design of base-isolated building frames with FPS devices in any area with a similar seismic hazard.