Seismic performance evaluation of high‐post‐yield stiffness concentrically braced steel frames under mainshock‐consistent‐aftershock sequences

Structures may be subjected to earthquake sequences after major mainshock (MS) events in seismically active sites within a short time. As a result, they may be susceptible to damage accumulation, which may hinder their performance under consecutive seismic loading. This study evaluates the effects of earthquake sequences on the seismic performance of seismic‐resistant concentrically braced steel frames designed to Eurocode‐8. The frames under investigation have concentric chevron‐type braces with replaceable hourglass‐shaped pins made of duplex stainless steel. The seismic energy is dissipated through inelastic deformations concentrated in the pins while keeping the other members elastic. The stainless‐steel pins provide the frame with high‐post‐yield stiffness to reduce the residual drifts after a seismic event. The seismic behaviour of the frame is assessed using site hazard‐specific mainshock‐consistent‐aftershock (MS‐AS) sequences selected for a site in Terni, Central Italy. Nonlinear back‐to‐back dynamic analyses are performed at multiple intensity levels while adopting detailed numerical nonlinear models created in OpenSees. We show that the implemented behaviour factor satisfies the life safety assurance objective while keeping the maximum residual inter‐storey drifts below 1/300 to permit an easy substitution of the damaged pins after the design‐level earthquake without being curtailed by the potential following events. We then develop prediction models for the damage accumulation in the pins, considering different energy‐based intensity measures and we show that the cumulative absolute velocity‐based model is the most efficient predictor in this particular case. Finally, the damage accumulation in the pins is evaluated, confirming their superior low‐cycle fatigue capacity under earthquake sequences.


INTRODUCTION
Aftershock (AS) sequences may be triggered by significant mainshock (MS) events within a short time. Thus, repairing structures or replacing the energy-dissipative elements damaged due to MS, before the next seismic event occurs, can often be unachievable because of the time limitation. For example, between August and November 2016, three major destructive seismic events happened in Central Italy with M6.1, M5.9 and M6.5 that triggered several AS events, some exceeding M5. 1 The current seismic design methodology stipulated by Eurocode-8 2 targets life safety limit state (which is associated with the probability of exceeding the site hazard of 10% in 50 years) by preventing local or global collapse, assuming that the collapse occurs due to a single (MS) event ignoring the progression of damage because of any following events. Moreover, it restricts the peak drifts under more frequent seismic events associated with a nominal probability-of-exceedance of 10% in 10 years. However, it does not limit the residual deformations or consider the possible damage accumulation due to the consecutive events that may follow. Such drifts may require certain repairs when they reach 0.5% and partial or full structure demolition if they exceed it. 3 Consequently, there is an essential need for developing improved seismic resistant systems that reduce the residual drifts and ensure a rapid recovery after an earthquake mitigating any severe damage accumulation that can follow due to seismic sequences.
Several attempts have been made to evaluate the mainshock-consistent-aftershock (MS-AS) effects on structures by adopting repeated MS records, 4 MS records followed by scaled-down forms of them, 5 employing different records from a common suite in a back-to-back sense, 6 or implementing as-recorded MS-AS sequences. 7 It was concluded that using artificial sequences, compared to as-recorded sequences, can significantly overestimate the drift demand and record-torecord variability, suggesting using site-specific real MS-AS ground motion pairs. However, there is a limited number of such pairs, particularly with high intensities, that do not cover all possible scenarios. To overcome these limitations, Papadopoulos et al. 8 have recently developed a rational ground motion selection scheme to select MS-AS ground motions with consistent causal parameters, obtained using space-time Epidemic Type Aftershock-Sequences model (ETAS), 9 accounting for the correlation between the MS and AS spectral ordinates. 10 A number of researchers have studied the seismic performance of new and damaged steel frames under MS-AS sequences focussing on moment-resisting frames (MRFs), 7,11 while few researchers focussed on concentrically braced frames (CBFs) and buckling-restrained braced frames (BRBFs). 12 BRBFs can experience large residual drifts, which may exceed 0.5% under the design level ground motions 13 and may be further increased when subjected to earthquake sequences. 12 However, BRBFs were more robust to withstand MS-AS sequences when combined with MRFs in dual systems. 14 BRBFs were initially proposed to dissipate seismic energy via tensile and compressive yielding of braces and overcome the degradation disadvantage of conventional CBFs, 15 while they can be implemented for retrofitting purposes. 16 In BRBFs, the braces have symmetric 'full' hysteretic behaviour with high energy dissipation capability. 17 Even so, BRBFs are susceptible to damage concentration at certain storeys because of their relatively low post-yield stiffness. 18 Several improved steel systems have been developed to reduce the residual drifts; for example, combined BRBFs-MRFs 19 and BRBFs combined with self-centering CBFs. 20 Moreover, other braced frames have been developed for residual drift reduction and enhanced repairability, such as frames with replaceable structural fuses, 21 self-centering frames 22  This paper evaluates the seismic behaviour of a Eurocode-8 2 -designed steel CBF with high-post-yield stiffness, denoted as HPYF, 24 under MS-AS sequences over a range of archetypes. Considered frames adopt improved energy dissipation duplex stainless-steel pins, denoted as SSPs, with excellent hysteretic energy dissipation and fracture capacities under inelastic cyclic loading that can be quickly substituted when damaged. The frames follow the concept settled by Pettinga et al. 25 relying on the inherent high post-yield stiffness of the SSPs 25 to maintain the residual drifts below the allowed functional erection tolerances of 1/300-1/500 storey height. 26 Consequently, after the damaged SSPs are replaced, the residual inter-storey drifts are significantly reduced as the damage is exclusively concentrated in the SSPs. To assess the behaviour of described frames, we use the Eurocode-8 2 -consistent INNOSEIS 27,28 approach and MS-AS ground motions selected at multiple intensity levels adopting the methodology developed by Papadopoulos et al. 8 consistent with the seismic hazard in Terni, Central Italy. Finally, prediction models of the damage in the SSPs are evaluated by adopting a range of energy-based intensity measures (IMs).

DESCRIPTION OF THE HIGH-POST-YIELD STIFFNESS BRACED FRAME
A partial elevation showing a half-panel of the HPYF under study is shown in Figure 1A. The frame has chevron braces with U-shaped plates attached at their lower ends. The U-shaped plates are connected to the beam-column-gusset plate connections, as shown in Figure 1B, via the SSPs shown in Figure 1C. Under seismic loading, inelastic deformations due to bending are spread along the internal bending part of an SSP, denoted as L ssp in Figure 1C, dissipating the seismic energy while preventing the bucking of the brace member, similar in concept to a buckling-restrained brace. Simultaneously, the fracture under ultra-low-cycle fatigue is postponed as the internal bending part has an hourglass shape that follows the bending profile. 23 Initially, Vasdravellis et al. 29 incorporated the SSPs in steel self-centering connections. Then, Baiguera et al. 30 adopted the same brace-SSP with a U-shaped-plate connection in a dual CBF-MRF while controlling the peak force in the brace through a friction pad added at its top end. In that dual CBF-MRF, replaceable beam fuses were adopted to keep the MRF beams damage-free under the design level seismic load. Differently, the proposed HPYF is, theoretically, a fully pinned isolated frame without the dual action with an MRF. However, the connections behave in a semi-rigid manner when realistically modelled. Hence, the beam fuses are excluded because the frame does not rely on a dual action. Moreover, in the proposed HPYF, the friction pads are eliminated while letting the peak forces be controlled by the delayed yielding of the framing elements, for example, braces, columns, beams and connections. F I G U R E 2 Typical configurations of the adopted archetypes: (A) plan, (B) partial elevation.

DESIGN OF THE ARCHETYPES
Within this study, eight archetypes are designed and grouped in two performance groups (PGs), covering a period range of [0.49, 2.26] s, as presented in Table 1. An archetype is denoted by HPYF-X-Y, where X is the number of storeys, Y is the adopted behaviour factor (q-factor or q), and a PG is denoted as PG-Y. All archetypes are 3-bay by 3-bay with a typical bay length of 8 m as shown in Figure 2. The 2-storeys archetypes have a storey height of 3 m while all other archetypes have a storey height of 3.3 m. The seismic load resisting system includes two marginal HPYFs in each orthogonal direction, which are positioned in the middle bays of the perimeter frames. First, the HPYFs and the gravity system of an archetype are designed to Eurocode-3 31 as fully pinned frames with continuous columns. Then, the HPYFs are designed following the Eurocode-8 2 -conforming design procedure adopted by Hassan et al., 24 which incorporates the design rules for CBFs. The dead and live loads are 4.3 and 2 kN/m 2 , respectively. Additionally, a façade line load of 1.5 kN/m is applied to the external beams. The combination coefficient (ψ E,i ) is set equal to 0.24, assuming storeys with correlated occupancies. Then, the total seismic weight (W) is calculated based on Eurocode-8. 2 The elastic response spectrum Type I and ground type B are selected, and the corresponding design spectrum is adopted considering the related parameters stipulated by code. 2 The design peak ground acceleration (a g ) is set equal to 0.30 times the gravitational acceleration (g) while implementing two values of the q-factor (6.5 and 4).
Eurocode-8 2 does not suggest a period formulation for HPYFs or systems with similar flexibility, like BRBFs, and does not specify an upper limit for the fundamental period (T 1 ) when calculated based on modal analysis. Hence, the designrelevant periods are approximated based on the American standards. 32 Firstly, the regression coefficient (C t ) is set equal to 0.0731 as recommended by ASCE/SEI 7-22 32 for BRBFs. Then, the approximate fundamental period (T 1a ) is calculated. Next, the design base shear (F b,des ) is calculated by the lateral force method as per Eurocode-8 2 while implementing an upper bound of the period (T 1u ), which is set equal to 1.4 times T 1a . In this way, underestimating F b,des because of overestimating T 1 is avoided. The fundamental mode shape is approximated by horizontal translations, which are linearly increased with the height. Then, the horizontal seismic force acting on a storey (F i ) is calculated.
In each orthogonal direction, the storey shear (Σ F i ) is equally shared between two HPYFs with two braces each. Accordingly, the design axial force (i.e., required yield strength) of the SSPs (N Ed,ssps ) is calculated.
Then, the capacity of an SSP is assessed following the design rules derived by Vasdravellis et al. 29 while taking into consideration the experimental observations by Baiguera et al., 23 which confirms that the exact location of the plastic hinges in the SSPs is midway between the external and internal diameters, D e and D i shown in Figure 1C. Lastly, the total yield strength of a group of SSPs (F y,ssps ), considering that they are connected in parallel, is determined as follows: y,ssps = ssps y,ssp = ssps ( e + i ) 3 6 ssp y,ssp (1) where n ssps is the number of SSPs and F y,ssp is the yield strength of an SSP. The nominal yield strength of the SSPs material (f y,ssp ) is set equal to 450 MPa. The dimensions of the SSPs are chosen with the intention of having them close to the sizes of SSP1 and SSP2, which were experimentally tested by Baiguera et al. 23 so that their hysteretic behaviours are matched. For a high-post-yield stiffness brace (i.e., brace-SSP assembly or HPYB), the overstrength ratio (Ω) is calculated as follows: The uniformity ratio between the maximum to minimum calculated overstrength ratios (Ω max and Ω min ) over all brace-SSP assemblies is kept below 1.25 to fulfil the uniformity condition stipulated for CBFs by Eurocode-8, 2 which, theoretically, ensures that yielding spreads over the height that improve the relative lateral behaviour of the frame. The equation specified by Eurocode-8 2 for beams and columns in CBFs is adopted while letting the braces follow the same minimum axial demand rule to make sure that SSPs yield in bending before the brace-SSP assembly buckles globally as follows: where N Ed is the minimum axial demand of the HPYF braces, beams and columns, Ω ov,des is the design overall overstrength factor, N Ed,G and N Ed,E are the axial design forces in the member (i.e., brace, beam or column) due to non-seismic or the seismic design action, respectively. The material overstrength factor (γ ov ) is set equal to 1.156 based on the coupon tensile tests of duplex stainless steel tested by Baiguera et al. 23 ω is the design overstrength modification factor, which is introduced to the code equation to account for the high-post-yield stiffness and strain hardening of the brace-SSP assembly. 24 ω ranges from 2.54 to 3.57 for the HPYFs under study. 24 Thus, Ω ov,des is set equal to the upper bound, 3, to avoid having impractical sizes of braces, beams and columns as recommended by FEMA P-695. 33 SAP2000 34 is adopted to construct a planar linear elastic finite element model of an HPYF, and eigenvalue modal analysis is performed. The number of modes is taken, targeting a minimum contributing modal mass ratio of 95%. In the model, the columns are continuous. However, all other members are fully pinned. The equivalent elastic stiffness (K eq ) is assigned to each brace-SSP assembly combining the elastic stiffnesses of SSPs (K ssps ) and the brace (K brace ), which are connected in series as follows: where K ssps is calculated per Vasdravellis et al. 29 as follows: ssps = ssps ssp = 2 ssps where K ssp is the elastic stiffnesses of a single SSP, E ssp and G ssp are the material elastic modulus, which is set equal to 186,135 MPa, and shear modulus, which is set equal to 71,590 MPa, respectively. is a stiffness reduction factor, which is set equal to 0.5, 35 accounting for the local yielding of the U-shaped plates that was observed in previous experiments. 29 Following Eurocode-8 2 to consider the participation of higher modes, a multi-modal response spectrum analysis is performed. Scaling is adopted to let the response spectrum analysis-based design base shear match 100% of the lateral force method-based F b,des , which avoids overestimating the fundamental periods and underestimating the design forces. 36 Then, the elastic spectrum per Eurocode-8 2 is adopted to derive the design storey displacement (d s ), and the P-Delta TA B L E 2 Detailed design of SSPs for HPYF-08-6.5.  Table 2 for HPYF-08-6.5.

NONLINEAR MODELLING IN OPENSEES
OpenSees 37 is adopted to construct a planar nonlinear HPYF model based on the centreline dimensions of the elements, as shown in Figure 3A. Then, MATLAB 38 is employed for the post-analysis calculations and illustrations. The material and geometrical nonlinearities are modelled while capturing the degradation of all the main framing elements and simulating the fracture of the brace-SSP assembly.
The corotational truss element is adopted to model the braces to accurately consider second order effects 39 while implementing the Steel02 material model. 40 The strain-hardening ratio is set equal to 0.001, the Bauschinger effect parameters, R 0 , cR 1 and cR 2 are set equal to 20, 0.925 and 0.25, respectively, and the isotropic hardening parameters a 1 , a 2 , a 3 and a 4 are set equal to 0.01, 1, 0.02 and 1, respectively. 41 True pins are considered at the ends of the brace-SSP assembly while modelling the gusset plate rigidity employing rigid elastic beam-column elements.
A corotational truss element with a unit area and 1 m long is adopted to model the SSPs to correctly take into account second order effects while adopting the CastFuse material model. 42 n is set equal to the number of SSPs while setting the parameters b o and h equal to 250 and 34 mm, respectively. L is set equal to 150 mm for SSP1 and 120 mm for SSP2; the yield strength, f y , and modulus of elasticity, E, are set equal to the yield strength and initial stiffness of SSPs. Then, scaling of L, f y and E is adopted to correct the cyclic response of SSPs. 42 The strain-hardening ratio is set equal to 0.025, and the parameters cR 1 and cR 2 are set equal to 0.05 and 0.10, respectively, while R 0 is set equal to 3 and 5 for SSP1 and SSP2, respectively. The isotropic hardening parameters a 1 , a 2 , a 3 and a 4 are set equal to 0.15, 10, 0.15 and 10, respectively. Calibration of the described modelling parameters is performed to match the experimental cyclic results performed by Baiguera et al., 23 as illustrated in Figure 4A.
The fracture of the braces, assumed at 10% axial strain, is simulated by adopting the MinMax material wrapper. The assumed ultimate axial strain is deemed acceptable as the steel grades conforming to Eurocode-3 sustains a minimum of 15% strain at failure. 31 The fracture of the SSPs under low-cycle fatigue is simulated by implementing the Fatigue material model. 43 A Coffin-Manson-like relationship between the imposed displacement amplitude (Δ f ) and the number of cycles to 20% strength loss ( f ) is derived relying on the results of the constant amplitude tests performed by Baiguera 35 as follows: where the value of the imposed displacement amplitude at which one cycle causes fracture is denoted as Δ o and the slope of the Coffin-Manson log-log relationship is denoted as m. Figure  material model uses the strain range from each cycle instead of the peak strain amplitude. Finally, to ease the convergence efforts, two elements with a very small cross-sectional area are connected in parallel to the brace-SSP assembly. 42 The displacement-based formulation is adopted for beams and columns modelling. Each element is split into eight sectors with a fine mesh in the plastic hinge zone (i.e., 0.165 L) as shown in Figure 3B,C. In addition, an initial quadratic camber of 1/500 of the clear height is provided for columns for global bucklinginitiation. 43 Fibre cross-sections are incorporated accounting for the axial force-bending moment interaction while spreading the inelastic deformations along the

TA B L E 3
The key parameters adopted for the modelling of the beam-column connections. element. Then, 10 fibre layers are created along the depth of a section, 43 as shown in Figure 3D. The Gauss-Lobatto quadrature numerical integration method is applied within each segment at five integration points that include the element endpoints. 44 The applied fibre sections can underestimate the sectional strength and stiffness degradation level under cyclic loading 43 as it does not consider local buckling, multiaxial stress conditions or lateral-torsional buckling. Hence, to overcome these drawbacks, the uniaxial Bilin material model is adopted implementing the modified Ibarra-Medina-Krawinkler deterioration model 45 with a bilinear hysteretic response while calibrating the hysteretic degradation parameters proposed by Lignos et al. 46 against the results of tests performed by Newell and Uang 47 and MacRae et al. 48 Additionally, a rigid truss element connecting the ends and intermediate nodes of the beam element is used to model the slab diaphragmatic action. A true pinned base connection is adopted. However, the beam-column connection is modelled using the 'Pinching4' uniaxial material model 49 adopted within a zerolength element. Table 3 summarises the key parameters adopted for the modelling of the beam-column connections. The degradation parameters of the beam-column-gusset plate composite connection are calibrated against the experimental results presented in Stoakes and Fahnestock, 50 while eliminating the residual strength of the connection. 41 Similarly, the parameters recommended by Elkady and Lignos 51 are adopted for the beam-column composite shear tab connection and calibrated against the experimental results by Liu and Astaneh-Asl. 52 For the beam-column shear tab connection, the drop in moment at which the concrete slab crashes is equal to 0.5 of the corresponding capping moment, while the positive and negative rotations are equal to 0.04 and 0.06, respectively.
The model by Gupta and Krawinkler 53 is implemented for the panel zones. To simulate P-Delta effects, a rigid elastic leaning column with negligible rotational stiffness is added. Additionally, very small masses are added to all degrees of freedom to stay away from having convergence problems. 54 The Rayleigh damping is assigned to the nodes and elements (i.e., beams, columns, braces and SSPs) using the region command. The Rayleigh damping ratio (ξ) is set equal to 2% while considering the fundamental frequencies that capture 95% of the modal mass. 55 Finally, the stiffness proportional Rayleigh damping is determined based on the tangent stiffness matrix. 41

SEISMIC PERFORMANCE ASSESSMENT UNDER MAINSHOCK-AFTERSHOCK SEQUENCES
The seismic behaviour of the HPYFs is assessed using site hazard-consistent ground motions sequences adopting the spectral acceleration of the fundamental period (S a (T 1 )) as the conditioning IM. The site hazard curves are then integrated with the fragilities of S a (T 1 ) for each performance objective (PO). Then, each PO, expressed in an Engineering Demand Parameter (EDP), is evaluated against a target mean annual frequency (MAF) of exceedance and a safety margin ratio (MR) is assessed, adopting an allowable value of the MR is equal to 1. Theoretically, the adopted assessment procedure follows the behaviour factor evaluation methodology by INNOSEIS. 27 However, in this research, some alterations to the methodology are employed to match the purpose of the study as the INNOSEIS 27  F I G U R E 6 Target conditional mean spectrum (CMS) and spectra of the selected scaled mainshock (MS) records for intensity measure (IM)1 to IM7 for S a (T 1 = 1.85 sec) (i.e., high-post-yield stiffness braced frame [HPYF]-08-6.5).
the seismic behaviour under single ground motions (i.e., MS ground motions), while offering consistency with Eurocode-8. 2 Differently, this study is concerned with assessing the seismic behaviour under MS-consistent-AS sequences. Still, the steps of the INNOSEIS 27 scheme, which are not associated with the site hazard assessment or ground motion selection, are followed in this study.

Site hazard and ground motions set selection
Probabilistic seismic hazard analysis (PSHA) and hazard disaggregation 56 59 Seismic hazard curves calculated in this manner are shown in Figure 5 for different fundamental periods of oscillation. For each conditioning IM (i.e., S a (T 1 ) for different periods), we define seven IM levels with a probability of exceedance (POE) ranging from 50% to 0.1% in 50 years (i.e., return period (T R ) ranges from 72 to 49,975 years) as summarised in Table 4. Following Papadopoulos et al., 8 for each IM level, 30 MS records are first selected from the Engineering Strong-Motion Database (ESM) 60 adopting the conditional spectrum (CS) concept 61 and a maximum scale factor of four. Figure 6 shows the target MS-CS and spectra of the selected scaled MS records for IM1 to IM7 and S a (T 1 = 1.85 sec) (i.e., HPYF-08-6.5). Potential AS scenarios are simulated using the ETAS model for each IM level and selected MS record. For every simulated event, AS record is selected to be consistent with the spectral ordinates of the corresponding MS record. For a more detailed explanation of the methodology, see Papadopoulos et al. 8

Nonlinear dynamic analysis
Nonlinear back-to-back dynamic analyses are performed for each archetype under MS-AS sequences. One scaled arbitrary horizontal component of each ground motion is selected. A typical stationary time of 50 s is added after the end of each record to allow for free damped vibration and accurate calculation of the residual deformations.
In total, PG-6.5 is analysed under 4958 records, while PG-4 is analysed under 3432 records. Figure 7A shows the stripes of maximum inter-storey drift (θ max ) demands under MS, whereas Figure 7B shows the cloud of θ max demands under AS for HPYF-08-6.5. Figure 7C shows the stripes of maximum residual inter-storey drift (θ Res ) demands under MS, while Figure 7D shows the cloud of θ Res demands under AS for the same frame. Typically, the drift responses under MS records form strips at the predefined values of S a (T 1 ); however, clouds of responses are formed under AS records, as they are corresponding to random values of S a (T 1 ). Roughly, the ratio between θ max and θ Res to their precedent value tends to increase as the ratio between S a (T 1 ) of AS to S a (T 1 ) of MS increases.

Performance objectives and evaluations
The performance of an archetype under MS-AS sequences is evaluated versus two POs: life safety assurance (LS) and DL. The Global Collapse Prevention (GC) PO, which is violated at θ max equals to 10% or when earlier instability takes place as the frames reaches its ultimate point, is associated with excessively low POE levels of less than 0.1% in 50 years and, consequently, excessively high IM levels. It was impossible to select MS-AS ground motions that can match the target spectra with sufficient accuracy at these IM levels; hence, the GC assessment is omitted. This omission is deemed acceptable as a previous study on the same frames showed that a behaviour factor of 6.5 for the HPYFs shows high MR values for the GC assessment that exceeds 11.79 when assessed against an MAF of exceedance of 2% in 50 years. 24 Hence, it is not likely to be curtailed due the damage accumulation under MS-AS sequences. The LS is assessed against an MAF of exceedance of 10% in 50 years adopting θ max as the EDP while considering 2% as the EDP limit. This 2% limit is selected to match the first yield in a frame member under nonlinear pushover analysis as permitted by the methodology, 27 which is corresponding to θ max of 2.2% based on a previous study on the same frames. 24 This confirms that the damage is condensed in the SSPs, while the braces, columns, beams and connections remain undamaged. In the INNOSEIS 27 scheme, 0.75 of the fracture displacement of the dissipative element (i.e., the SSPs in this study) or the corresponding θ max is recommended to be set as the LS EDP limit. Notably, adopting the 2% limit is more conservative as the corresponding displacement of SSPs (δ ssp ), based on nonlinear pushover analysis, is around 0.5 times their fracture displacements (δ u,ssp ). 24 The DL considers two sub-POs: peak inter-storey drift limitation and residual inter-storey drift limitation. The first is implemented to confirm compliance with the code DL requirement 2 and assessed against an MAF of 10% in 10 years, considering three θ max limits, 0.5%, 0.75%, and 1%, while the second is adopted to validate the workability of the damaged SSPs replacement after a design level earthquake and is evaluated against an MAF of 10% in 50 years, considering two θ Res limits, 0.2% and 0.33%, which are selected to match the erection tolerances, 1/500 and 1/300, allowed by EN 1090-2. 26 The damage-dependent fragility functions are fitted by adopting maximum likelihood method 62 for four damage states (DSs denoted as DS1 to DS4 in Figure 7) that correspond to θ max equals to 0.5%, 0.75%, 1% and 2%, respectively, and two DSs (denoted as DS1 and DS2 in Figure 7) that correspond to θ Res equals to 0.2% and 0.33%. The intact state where θ max or θ Res are equal to 0% is denoted as DS0, while the initial DS is denoted as DS i . The derived fragility functions under MS records only for both θ max and θ Res (e.g., for HPYF-08-6.5) are illustrated in Figure 8, showing the increase of both the lognormal mean and dispersion as the adopted damage limit state increases, as expected. Figure 9 shows the damage-dependent fragility functions for θ max under MS-AS sequences (e.g., for HPYF-08-6.5). Similarly, Figure 10 shows the damage-dependent fragility functions for θ Res under MS-AS sequences for the same frame. Notably, the damage accumulation is clear as the lognormal mean S a (T 1 ) value (S a (T 1 ) 50% ) in most of the cases tends to decrease as the initial DS increases. The MAF (λ) of S a (T 1 ) 50% of the LS limit state fragility function is obtained by adopting the hazard curves of S a (T 1 ) for Terni shown in Figure 5. The MAF of exceeding the EDP limit, with a confidence level x, (λ POx ) is then approximated as follows 27 : where k is the slope of the log-log hazard curve estimated by the first-order biased hazard fitting 63 and K x is the standard normal variate corresponding to x. For LS assessment, x and K x are taken equal to 90% and 1.282, respectively. The recordto-record dispersion (β R ) is estimated from the derived fragility functions (shown in Figure 9D for HPYF-08-6.5), while the overall dispersion (β U ), which is equal to the square-root-sum-of-squares of the test data quality, design rules quality and limit state capacity dispersions, is set equal to 0.346 based on setting the contributing dispersions equal to 0.2 (i.e., assuming good rating for all contributing dispersions). 27 Additionally, for more precise evaluation, λ Pox is recomputed by directly integrating the hazard curves and the (1-x)% fractile of the fragility curves, 27 where the (1-x)% fractile value of S a (T 1 ) 50% (S a (T 1 ) 50%x ) is determined as follows 27 : Finally, λ POx is assessed against the relevant MAF (λ O ), and the MR is evaluated. Tables 5 and 6 summarise the MR values for PG-6.5 based on Equation (7) and the direct integration, respectively. Notably, all archetypes in PG-6.5 pass the LS  (7). Archetype  assessment. The MRs calculated for the LS assessment using both Equation (7)  It is concluded from the reported MR values that the HPYFs designed adopting a q-factor of 6.5 are not likely to be affected by the damage accumulation due to the AS sequences in a way that makes them violate the LS limit state.

DS
Then, the DL assessments are commenced by implementing the same processes adopted for the LS assessment considering the relevant EDPs. β U is set equal to 0.346. x is set equal to a relaxed value of 70% together with K x equals to 0.525 to, at least, cover the mean of a lognormal-distributed MR variable with a moderately high variability. First, λ POx is approximated by implementing Equation (7) and the derived fragility functions curves (shown in Figures 9A-C and 10A,B for HPYF-08-6.5). After that, λ POx is recomputed by the direct integration procedure while modifying the fragility curves by Equation (8). The DL assessments governing results using Equation (7) and the direct integration, respectively, are summarised in Tables 7 and 8. Again, the calculated MR values using both techniques are similar, and the differences do not change the drawn safety status. PG-6.5 passes both the peak inter-storey drift limitation assessment at θ max equals to 0.75% and the residual inter-storey drift limitation assessment at θ Res equals to 0.33%. However, the stricter DL limits (i.e., the peak inter-storey drift limitation assessment at θ max equals to 0.5%, and the residual inter-storey drift limitation assessment at θ Res equals to 0.2%) require further reduction of the q-factor. PG-4.0 (i.e., the PG that adopts q equals to 4) passes both the peak inter-storey drift limitation assessment at θ max equals to 0.5% and the residual inter-storey drift assessment at θ Res equals to 0.2%. The reported MR values highlight the effectiveness of reducing the q-factor to reduce the damage accumulation due to the AS sequences when strict drift limitations are considered. Still, the behaviour of the HPYFs designed adopting a q-factor of 6.5 are not likely to be curtailed by the AS sequences-resulted damage accumulation if relaxed drift limitations are considered.

PREDICTION OF THE SEISMIC DAMAGE ACCUMULATION IN THE STAINLESS-STEEL PINS
All archetypes do not experience low-cycle fatigue fracture of SSPs except HPYF-08-6.5 and HPYF-12-6.5, where low-cycle fatigue fracture of SSPs (i.e., collapse) is recorded under one MS-AS sequence out of 210 sequences for each. For both frames, the collapse is initiated at the last storey as shown in Figure 11 displaying the corresponding inter-storey drift (θ) time history. The adopted time histories of S a (T 1 ) that initiate collapse are shown in Figure 12. For HPYF-08-6.5, the fracture is observed under IM4; however, for HPYF-12-6.5, the fracture is recorded under IM6. For the first, the collapse is initiated under the 35th AS. However, for the second, the collapse is started under the 49th AS. It was confirmed that energy-based or duration-related IMs can provide a more reliable prediction of the accumulated damage level in the structural elements, as they are cumulative, compared to S a (T 1 ) (i.e., the spectral acceleration of the fundamental period). 64,65 Hence, for a reliable estimation of the damage accumulation in SSPs, prediction models are developed and compared adopting a range of energy-based IMs. Then, based on the most reliable model, fragility functions are derived and integrated with the seismic hazard in Terni to assess the MR. The adopted IMs includes the Arias intensity (I a ), 66 cumulative absolute velocity (CAV), 67 cumulative absolute displacement (CAD) 68 and significant 5%−95% duration (D 595 ). 69 F I G U R E 1 2 Spectral acceleration (S a (T 1 )) time history adopted for: (A) high-post-yield stiffness braced frame (HPYF)-08-6.5, (B) HPYF-12-6.5.
First, the damage accumulation in the SSPs under low-cycle fatigue is evaluated by adopting the Palmgren-Miner linear damage accumulation rule as follows 23 : where n i is the applied number of cycles, N f,i is the number of cycles to fracture and D ssp is the low-cycle fatigue damage index that ranges from 0, at the undamaged state, to 1, at the low-cycle life end (i.e., fracture). For PG-6.5 and PG-4.0 under MS records, the maximum recorded D ssp values are 0.26 and 0.184, respectively. However, under MS-AS sequences, the maximum recorded accumulated D ssp values are 1.226 and 0.948. Figure 13 shows the recorded relationship between the accumulated D ssp and D 595 for both frames. For HPYF-08-6.5, fracture due to low-cycle fatigue (i.e., D ssp reaches 1) is observed at D 595 equals to 479.5 sec, while collapse (i.e., θ max exceeds 10%) is recorded at D 595 equals to 503.5 sec and D ssp equals to 1.226. For HPYF-12-6.5, fracture due to low-cycle fatigue is observed at D 595 equals to 771.5 sec, while collapse is recorded at D 595, equal to 867.4 sec and D ssp equals to 1.105.
Then, linear regression analysis is performed for each archetype and group to estimate D ssp adopting the predefined range of IMs that are calculated accumulatively considering the MS and following AS results. Figure 14 shows the estimated relationships (for x = 50% and 95%) for an archetype (e.g., HPYF-08-6.5) employing a log-log scale 70 while indicating three damage levels at D ssp equals to 0.3, 0.6 and 0.95 as dashed horizontal lines. CAV shows the least root of mean square error (RMSE) and highest adjusted coefficient of determination (R 2 ) (i.e., the highest effectiveness in determining D ssp ). However, D 595 shows the highest RMSE and least R 2 (i.e., the lowest effectiveness in determining D ssp ). Looking at the models shown in Figures 15 and 16 estimated for PG-6.5 and PG-4.0, respectively, the effectiveness of adopting CAV for estimating D ssp , compared to the other IMs, is clearly demonstrated. Comparing the D ssp -CAV models of PG-6.5 and PG-4.0 shows that reducing the q-factor decreases the experienced D ssp as the estimated D ssp for PG-4.0 ranges from 0.68 to 0.84 times the estimated D ssp for PG-6.5 under the same CAV. This ratio, between D ssp adopting q equals to 4.0 and 6.5, tends to decrease linearly as log10 CAV increases.
Then, the fragility curves for D ssp are derived at D ssp equals to 0.3 and 0.6, adopting generalised linear model regression with a Probit link function. 71 The derivation of the fragility functions of D ssp under MS records is not theoretically finite at D ssp equals to 0.3 (i.e., the maximum recorded D ssp value is below 0.3). Hence, in these fragility curves, the additional damages due to AS records are embedded into the MS fragility functions, 64 which differs from the previously obtained fragility curves for θ max and θ Res where the AS fragility functions are dependent only on the damage level after the MS. 8 Under MS-AS records, for all archetypes except for HPYF-08-6.5 and HPYF-12-6.5, the theoretical best estimate is not finite at D ssp equals to 0.95 as only successes (i.e., no failures) are observed. For HPYF-02-6.5, the maximum recorded D ssp is 0.915, while for HPYF-04-6.5, the maximum recorded D ssp is 0.923. However, for HPYF-08-6.5 and HPYF-12-6.5, the theoretical best estimate is not finite at D ssp equals to 0.95 as the estimated coefficients perfectly separate failures from successes. Furthermore, for HPYF-04-4.0 and HPYF-12-4.0, the theoretical best estimate is not finite as the maximum recorded D ssp are 0.428 and 0.43, respectively (i.e., no failures are observed). However, for HPYF-08-4.0, which experiences a maximum D ssp of 0.677, the theoretical best estimate is not finite as the estimated coefficients perfectly separate failures from successes. Hence, Figure 17 shows only the available derived fragility curves. In Figure 17, the fragility curves shift to the left as the q-factor increases, demonstrating the increase in the damage accumulation in the SSPs. Finally, for PG-6.5, the obtained fragility curves at D ssp equals to 0.3 are integrated with the hazard curves of CAV for Terni shown in Figure 18 adopting Equation (8) while replacing S a (T 1 ) 50% with CAV 50% and S a (T 1 ) 50%,x with CAV 50%,x . x is set equal to 70% together with K x equals to 0.525 and β U equals to 0.346, and the MRs are evaluated against an MAF of 10% in 50 years, which results in extremely high MRs. Hence, the assessment is repeated as summarised in Table 9, setting x equal to 95% together with K x equals to 1.645, and β U equals to 0.606 and the MRs are evaluated against an MAF of 1% in 50 years, which results in a minimum MR of 6.97, which clearly demonstrate the superior low-cycle fatigue capacity of the SSPs under MS-AS sequences.

CONCLUSIONS
This paper evaluates the seismic performance of a Eurocode-8 2 -designed seismic-resistant braced steel frame (denoted as HPYF -High-post-yield stiffness braced frame) under MS-AS sequences in a high seismicity hazard European site (Terni, Central Italy). The frame has chevron braces, which are equipped with replaceable hourglass-shaped pins (SSPs) made of high-post-yield duplex stainless-steel material. The seismic energy is dissipated through flexural deformations concentrated in the SSPs that ensures a seismic response like that of buckling-restrained braces. The SSPs exhibit a highpost-yield stiffness, steady hysteretic behaviour and exceptional fracture capability under ultra-low-cycle fatigue. The MS-AS sequences are selected at multiple intensity levels adopting the scheme developed by Papadopoulos et al., 8 and nonlinear back-to-back dynamic analyses are performed. Then, the obtained damage-dependent fragilities are integrated with the site hazard curves for assessing the seismic behaviour of the frame adopting the relevant steps of the Eurocode-8 2 -consistent INNOSEIS 27 approach. Furthermore, the index of the damage accumulation in the SSPs due to low-cycle fatigue is predicted via linear regression models relating the damage to a range of energy-based ground motion parameters. Finally, the damage accumulation index fragility to exceed 30% is integrated with the relevant seismic hazard for Terni, and a safety margin is evaluated. The main conclusions are: • Under MS-AS sequences, adopting a behaviour factor for the HPYF equals to 6.5, together with an overall design overstrength factor equals to 3, ensures, with a confidence level (x) of 90%, fulfilling the LS requirements against an MAF of exceedance of 10% in 50 years, that results in meeting the near-collapse requirement specified by Eurocode-8. 2 • Under MS-AS sequences, adopting a behaviour factor for the HPYF equals to 6.5, together with an overall design overstrength factor equals to 3, ensures, under events with an MAF of exceedance of 10% in 50 years and an x of 90%, concentrating the damages in the SSPs while protecting the main framing elements (e.g., columns, beams, braces and connections). • Under only MS events, adopting a behaviour factor for the HPYF equals to 6.5, together with an overall design overstrength factor equals to 3, ensures, with an x of 70%, limiting the maximum inter-storey drifts, under events with an MAF of exceedance of 10% in 10 years, below 0.5% and the maximum residual inter-storey drifts, under events with an MAF of exceedance of 10% in 50 years, below 1/500. However, due to the damage accumulation under MS-AS sequences, it ensures, limiting the maximum inter-storey drifts below 0.75% and the maximum residual inter-storey drifts below  1/300. Therefore, a reduced behaviour factor equals to 4 is recommended for stricter drift limitations to ensure limiting the maximum inter-storey drifts below 0.5% and the maximum residual inter-storey drifts below 1/500. • The conclusions are drawn for archetypes up to 12-storeys adopting SSP material overstrength factor of 1.156 (520/450) together and a maximum adopted minimum overstrength factor of the SSPs, individually calculated for a frame, of 1.04 (≈1). • The CAV-based model is recommended for predicting the low-cycle fatigue damage index of the SSPs as it offers the least RMSE and highest adjusted coefficient of determination (R 2 ) compared to other predictors considered in this study. • Adopting a behaviour factor for the HPYF equals to 6.5 ensures, with an x of 95%, keeping the low-cycle fatigue damage index of the SSPs below 30%, under MS records with an MAF of exceedance of the MS of 1% in 50 years followed by consistent AS sequences.

Archetype
[Correction added after first online publication on 21st March 2023: the value of x changed from 90% to 95%.]

A C K N O W L E D G E M E N T S
None.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.