Cyclic engagement of hysteretic steel dampers in braced buildings: a parametric investigation

Hysteretic steel dampers have been effectively used to improve the seismic performance of framed buildings by confining the dissipation of seismic energy into sacrifical, replaceable devices which are not part of the gravity framing system. The number of cycles sustained by the dampers during the earthquake is a primary design parameter, since it can be associated to low-cycle fatigue, with ensuing degradation of the mechanical properties and potential failure of the system. Current standards, like e.g. the European code EN 15129, indeed prescribe, for the initial qualification and the production control of hysteretic steel dampers, cyclic tests in which the devices are assessed over ten cycles with amplitude equal to the seismic design displacement dbd. This paper presents a parametric study focused on the number of effective cycles of the damper during a design earthquake in order to assess the reliability of the testing procedure proposed by the standards. The study considers typical applications of hysteretic steel dampers in low and medium-rise steel and reinforced concrete framed buildings and different ductility requirements. The results point out that the cyclic engagement of the damper is primarily affected by the fundamental period of the braced building and the design spectrum, and that, depending on these parameters, the actual number of cycles can be substantially smaller or larger that recommended by the standards. A more refined criterion for establishing the number of cycles to be implemented in testing protocols is eventually formulated.


Introduction
During major earthquakes civil constructions are subjected to large energy inputs, which can cause permanent structural damage and loss of life. To minimize casualties and economic losses, a structure needs to remain largely damage free. A viable design approach to improve the seismic performance of a structure consists in concentrating the dissipation of most of the earthquake input energy in sacrificial elements that are not part of the gravity framing system (Christopolous and Filiatrault 2006). This philosophy allows to minimize damage in structural and non-structural elements, increase life-safety and achieve a desired level of performance (Di Cesare et al. 2014;2017a). Moreover, after an earthquake, these sacrificial elements can be repaired or replaced, reducing cost and interruption of human occupancy and operation, with evident economic and social benefits (Di Cesare et al. 2020). The concept of using supplemental systems to increase the damping in a structure was proposed in the late 1960s (Muto et al. 1969;Kelly et al. 1972;Skinner et al. 1974) and nowadays supplemental energy dissipation or damping systems are one of the most popular and cost-effective approaches to mitigate the seismic risk for both new and existing structures (De Domenico et al. 2020).
Energy dissipation devices can be classified in two main categories, namely Velocity Dependent Devices (VDDs) and Displacement Dependent Devices (DDDs) (Christopolous and Filiatrault 2006). The first category includes fluid dampers, where the dissipation is achieved through the lamination of a viscous fluid which is forced by a piston to flow through a valving system. The behavior of these devices strictly depends on the fluid velocity. Viscous fluid dampers are very versatile and can be designed to allow unconstrained slow thermal motions as well as to provide controlled damping of a structure to protect from wind or earthquakes (Del Gobbo et al. 2018;Karavasilis et al. 2016;. They are typically employed in flexible structures subjected to large and rapid movements like bridges and tall buildings. The second category comprises hysteretic dampers and is further divided into steel dampers, friction dampers and metal extrusion dampers (Nabid et al. 2018), depending on the mechanism used to dissipate the seismic energy. The constitutive law of these devices is mainly dependent on the imposed displacement. Most of the dampers used in residential, school and industrial buildings belong indeed to this category. In these buildings, the dampers are typically inserted into braces which convert the interstorey drift into the axial deflection of the damper. The theoretical force-displacement curves of hysteretic dampers are shown in Fig. 1, where d y is the yield displacement of the device, d max is the maximum displacement in a cycle, and F y and F max are the axial force at d y and d max , respectively; the area included in the curve corresponds to the energy dissipated during a cycle. The ratio between the maximum displacement and the yield displacement defines the ductility factor µ D of the damper.
The performance of a hysteretic damper is defined by: (1) the force-displacement constitutive law, which is an inherent characteristic of the device; (2) the number of cycles sustained during the earthquake, which depends on the duration of the ground motion and the fundamental frequency of the structural system where the damper is employed. The number of cycles defines the total amount of energy dissipated by the system for a given displacement but is also associated to the degradation and possible failure of the device due to low-cycle fatigue (Morillas et al. 2020).
Testing of hysteretic dampers to verify their conformity to performance requirements is regulated by standards, like e.g. the EN 15129 (CEN, 2009), which is compulsory in the European Economic Area. According to the European standards, the force-displacement cycle of the damper shall be evaluated by imposing series of cyclic deformations at increasing amplitudes, namely at 25, 50 and 100% of the design displacement d bd . Five cycles for each intermediate amplitude and at least ten cycles for the maximum amplitude shall be applied. The conceptual diagram of the displacement histories is shown in Fig. 2, where d max is the maximum displacement in the relevant cycles. If scaled specimens are used, test displacements and cycling frequency shall be consistently scaled. The device shall not break and shall keep its characteristics unchanged during test. Eventually, a ramp test shall be performed for the static evaluation of the failure displacement up to a displacement not less than the maximum displacement γ b × γ x × d bd . The partial factor γ b , not less than 1.1, is related to action effects other than seismic which can affect the initial configuration of the device. The reliability factor γ x , equal or greater than 1, depends on the role that the devices play in the stability of the construction after the earthquake. When the damper is employed as a part of an isolation system, the reliability factor γ x shall be taken as 1.2 for use in critical structures other than bridges (CEN 2004) and 1.5 for use in bridges (CEN 2005). The specified test protocol shall be performed as Type Test, for the initial qualification of a device, and then repeated as factory production control test on a minimum of 2% of the devices of the supply, with a minimum of one device. The tested devices may be installed into the structure only if the fatigue resistance of their core elements is one order of magnitude greater than the number of cycles performed during the tests. In any other case, they shall not be installed into the structure, unless their mechanical characteristics are fully recovered, e.g., by replacing the nonlinear mechanism or the core elements.
To the authors' best knowledge, among the various parameters, the number of cycles to be imposed to the specimens deserves a careful investigation since it is not based on specific studies documented in literature. Moreover, the standard specifies that the number of ten cycles at ± d bd is related to the use of dampers in seismic isolation systems producing fundamental periods of the order of 2 s, but if the fundamental period of the structural system in which the device is used is considerably less than 2 s, a "corresponding increase of the number of test cycles at ± d bd " shall be prescribed by the Structural Engineer; however a rationale specifying the correlation between the structural frequency and the number of cycle is missing.
The present study investigates, through a parametric study, the engagement of hysteretic dampers installed in bracing systems of framed buildings in terms of number of "effective cycles" accommodated during the design earthquake, in order to evaluate the suitability of the testing protocol recommended by the European standard. It is indeed of fundamental importance, for both manufacturers and work directors, to have at hand validated testing protocols which allow to assess the seismic performance of structural devices under benchmark conditions replicating the actual conditions the devices in practical applications. Moreover, a revision of the standard EN 15129 is in progress within the Technical Committee 340 of the European Standardization Center (CEN), and the improvement of the theoretical background is expected to enhance the technical level of the code.
The paper is organized as follows. The first Section presents the code requirements and justifies the motivations and aims of the work. Section 2 introduces the parametric investigation, which considers six buildings, representative of low and medium-rise structures made of either steel or reinforced concrete, six dampers' layouts accounting  Displacement history  prescribed by EN 15,129 for  qualification and factory production control tests for different ductility factors of the device (µ D ) or the frame (µ F ), and four seismic scenarios. In Sect. 3 the accuracy of non-linear time history (NLTH) analyses carried out using equivalent single degree of freedom (SDOF) equivalent instead of multi degree of freedom (MDOF) models of the damped building is evaluated; the use of SDOF model allows indeed a substantial reduction of computational time and data processing. In Sect. 4 the results of the parametric study are illustrated and discussed. The cyclic engagement of hysteretic dampers during an earthquake is demonstrated to depend on the fundamental period of the structure where the dampers are placed and the spectral characteristics of the ground motion. The code provisions are eventually checked against the numerical results of this parametric study, and improvements are suggested. Finally, Sect. 5 summarizes the main outcomes of the research.

Parametric investigation
A parametric investigation has been carried out in order to assess the number of "effective cycles" (defined in Sect. 3) sustained by DDD units installed in braced buildings. The parameters addressed in the study regard: • The structural frame: six frames, representative of low and medium-rise structures made of either steel or reinforced concrete, have been considered (see Sect. 2.1); • The seismic scenario: four response spectra have been defined by combining two seismic intensity levels, corresponding to either Ultimate Limit State (ULS) or Serviceability Limit State (SLS), and two soil categories, stiff (type A) or moderate-soft (type C). For each design spectrum, a set of fourteen ground motion records has been selected (see Sect. 2.2); • The ductility factor of the hysteretic dampers; • The target ductility of the braced frame at ULS (see Sect. 2.3).
For each layout of the braced frame, two equivalent SDOF formulations will be introduced in the next Sections. The first one employs linear equivalent response spectrum analyses and is aimed at defining the DDDs' mechanical properties (resulting in seventy-two different layouts-see Tables 7, 8 of the Appendix). The second equivalent SDOF formulation, where for sake of simplicity, both the bare frame and the DDD are modelled by means of elastic-plastic elements arranged in parallel, is later implemented in NLTH analyses in order to quantify the number of effective cycles of the DDD. A total of 2016 simulations (1008 at ULS plus 1008 at SLS) has been carried out. In future developments of this investigation the possible, although usually small ( k pl,D ∕k el,D < 0.1 , Di Cesare et al. 2017a), post-yielding hardening of steel hysteretic dampers will be also accounted for.

Case-study frames
Six case-study structures, including three steel frames and three reinforced concrete (RC) frames, have been examined, in order to envelope a wide range of fundamental periods of interest for conventional buildings. The steel frames are 2-, 4-and 8-storey moment resisting frames, with material and geometrical properties as specified in the INNOSEIS project (Vayas et al. 2017). The frames were designed in compliance with Eurocode 3 (EC3) (CEN 2003) and Eurocode 8 (EC8) (CEN 2004) considering gravity loads and low earthquakeinduced loads, corresponding to a design spectrum for soil A and peak ground acceleration (PGA) equal to 0.2 g (g denoting the gravity acceleration). Sketches of the frames, with the main dimensions in plan and in elevation, and the steel profiles adopted in the design, are reported in Fig. 3. The RC frames correspond to three residential RC buildings (Faleschini et al. 2019). The frames, comprising 3-, 6-and 9-storey buildings, were designed according to the Italian Building Code (CSLLPP 2018) for a low seismicity zone (475-yr return Fig. 3 Steel frames case-studies period PGA = 0.195 g, soil type B) corresponding to the municipality of Pordenone, Italy. An overview of the RC frames with the main dimensions is given in Fig. 4, and the reinforcement details of each section are listed in Table 1. Given the regularity in plan of the examined case-study structures, for each building the design of the hysteretic damper system has been performed on a single internal 2D frame (in line with EC8 §4.2.3-CEN 2004) and considering the tributary loads and masses pertaining to this frame, in line with other studies (Del Gobbo et al. 2018;Karavasilis et al. 2016) and accepted practice.
In order to incorporate the nonlinearity of the parent frame, a concentrated plasticity model for the six frames has been implemented in the software program for structural analysis SAP2000 (CSi, 2017). Automatic hinge properties have been adopted at either end of the structural members, following the recommendations given in ASCE 41-17 (ASCE 2017). In particular, the center of the hinges is located at 5 and 95% of the member length, respectively. Moment-type plastic hinges have been assumed in accordance with Table 9-6 in ASCE 41-17 (ASCE 2017) and ANSI/AISC 360-16 (ANSI/AISC 2016) for steel beams, and in accordance with Table 10-7 in ASCE 41-17 (ASCE 2017) for RC beams. Similarly, coupled axial-moment plastic hinges have been assigned in accordance with Table 9-6 in ASCE 41-17 (ASCE 2017) for steel columns, and with Table 10-8 in ASCE 41-17 (ASCE 2017) for RC columns. The concrete column failure condition has been assigned as condition (ii) in ASCE 41-17 (ASCE 2017), i.e., flexural-shear failure with yielding in flexure occurring before shear failure.
The lateral capacity of the bare frames has been assessed through the nonlinear static (pushover) analysis method under constant gravity loads and monotonically increasing horizontal loads. Two distributions of lateral loads are considered, namely a "modal" pattern (resembling a triangular height-wise profile) and an "uniform" pattern as per EC8 provisions (CEN 2004). The capacity curve (base shear F versus control point displacement d ) is obtained for each frame and then processed in accordance with Annex B of EC8 (CEN 2004), as sketched in Fig. 5. The pushover curve (F;d) of the MDOF system is then scaled by the modal participation factor Γ to obtain the capacity curve For clarity, hereinafter the quantities with superscript "*" will refer to equivalent SDOF systems, and quantities devoid of suffix will refer to MDOF systems.
The yield force F * y and yield displacement d * y are determined under the equal energy criterion assuming an ideal elastic-perfectly plastic force-displacement relationship. The yield force F * y,F , yield displacement d * y,F and ultimate displacement d * u,F (where subscript εFε stands for "frame") for the six considered structures are listed in Table 2, along with the modal participation factor Γ and the participating mass m * . From these values, it is possible to calculate the initial stiffness of the equivalent SDOF model of the bare frame as k * F = F * y,F ∕d * y,F , the fundamental period T * 1,F = 2 √ m * k * F and the ductility factor μ * F = d * u,F ∕d * y,F . These values will be used in the design of the hysteretic dampers, as explained in Sect. 2.3. It is worth noting that the fundamental periods of the equivalent steel frames are much longer than those of the RC frames. This is due not only to the more flexible nature of the steel frame (with a higher inter-storey height and longer bay span) compared to the RC frames, but also to a higher equivalent mass m * induced by the larger tributary area and by the relatively high loads per unit area (live loads of 3.80 kN/m 2 ) of the reference steel frames taken from the INNOSEIS project (Vayas et al. 2017). In this regard, this circumstance widens the range of applicability of the results of this numerical study to really flexible structures. Table 2 Parameters of the elastic-perfectly plastic equivalent SDOF models of case-study frames 2-storey steel frame 1 3

Seismic input
The municipality of Lamezia Terme (latitude 38.57°, longitude 16.18°), a city in a high-seismic prone area in southern Italy, has been considered for the design of the supplemental energy dissipation system. Four design response spectra have been defined in accordance with the Italian building code provisions (CSLLPP 2018) for buildings characterized by a nominal life V N = 100 years, and a functional class IV (amplification factor C U = 2.0 ) resulting in a reference period V R = V N ⋅ C U = 200 years. Among the four design limit states considered in the code, the "SLD" (return period of 201 years) has been chosen as representative of frequent SLS earthquakes, while the "SLV" (return period of 2475 years) has been selected to represent rare ULS events. Topography condition T 1 has been assumed while, in order to account for different frequency contents of the ground motion, both type A (stiff) and type C (moderate-soft) soils have been considered, resulting in four different design spectra: "ULS-soilA", "ULS-soilC", "SLS-soilA", and "SLS-soilC". Relevant peak ground accelerations are: (1) PGA = 0.452g for "ULS-soilA"; (2) PGA = 0.181g for "SLS-soilA"; (3) PGA = 0.456g for "ULS-soilC"; (4) PGA = 0.261g for "SLS-soilC". For each spectrum, a suite of fourteen independent spectrum-compatible natural records has been extracted (in total 56 events) from the European strong-motion database (Ambraseys et al. 2002) by means of the software REXEL v. 3.5 (Iervolino et al. 2010). For each suite, seven records have been selected among "near-fault" events with epicentral distance R ep ≤ 15km , and the remaining seven records among "far-field" events with epicentral distance R ep ≥ 30km . In all cases, the magnitude has been constrained in the range 5 ≤ M w ≤ 8 . The selected acceleration time histories have been scaled in order to match, on average, the target design spectrum over the period range [0.15 − 4.00s] (Fig. 6). Seismological details of the selected ground motions are given in Tables 3, 4, 5, 6 of the Annex. It is worth noting that, among the characteristic parameters of the seismic events, the frequency content (through two types of foundation soil), the epicentral distance, and the intensity level (i.e. ULS or SLS) of the seismic event have been explicitly taken into account in this study. However, the cyclic engagement of dissipative braces may depend also on the duration of the ground motion. In this regard, the effect of the "Trifunac duration", i.e. length of the time-window between the release of 5% and 95% of the overall seismic energy (Trifunac et al. 1975), is a matter of concern that deserves future investigations. In this study, as witnessed by values reported in Tables 3, 4, 5, 6, the "Trifunac duration" of the selected ground motion records ranges from 3.4 to 63.8 s, and hence covers from short to long-lasting events.

Design of the dissipative braces
Each frame has been retrofitted by introducing dissipative braces, consisting of steel braces incorporating hysteretic dampers, as supplemental energy dissipation system. In the first step of the design procedure of the dissipative system, a target displacement ( d * targ,ULS ) for the braced frame under ULS seismic events has been defined in order to achieve an assigned ductility factor for the parent frame ( μ * F ). Bare-frame ductility factors ( μ * F = d * targ,ULS ∕d * y,F ) equal to either 1.0 or to a value μ * F max in the range [1.1 ÷ 1.6] comparable to that assumed in other studies (Di Cesare et al. 2017a). In the first case, identified as " μ * F = μ * F 1 ", the structure behaves elastically and therefore the requested damping force is maximized. In the second case, referred to as " μ * F = μ * F max ", the damping force is minimized in order to achieve the allowed plastic deformation of the frame.
In the second step, the mechanical properties of the dampers are adjusted in order to meet the ULS displacement target. A linear equivalent spectral analysis is carried out where the response of the braced frame is modelled as two equivalent linear SDOF systems working in parallel ( Fig. 7-left), representing the parent frame (subscript "F") and the damper (subscript "D"), respectively. The resulting effective vibration period of the braced frame ( T * eff ,F+D ) is: where k * eff ,F = F * y,F ∕d * targ,ULS , and k * eff ,D = F * y,D ∕d * targ,ULS are the effective stiffnesses of the bare frame and dissipative bracing system, respectively.
In accordance with (Priestley et al. 2007) for "in parallel systems", the equivalent viscous damping of the braced frame ( * eq,F+D ) is:

Fig. 6
Comparison between the target spectrum and the average spectrum obtained for each suite of fourteen ground motion records for: a ULS-soilA; b ULS-soilC; c SLS-soilA; d SLS-soilC being * el,F the viscous damping of the frame in the elastic range (here conventionally assumed equal to 5%), F * y,F and F * y,D the yielding forces of the frame and the damper, respectively, and * eq,F and * eq,D the equivalent viscous damping of the frame and the dampers, calculated as (Dwairi et al. 2007): It is worth noting that numerical value of * eq,F+D never exceeds 0.24 (i.e. 24% of the critical damping) even for the largest damper ductility ratio (i.e. μ * D = 12 ) considered in this study. Therefore, the ratio between the "damped" ( T * (damp) eff ,F+D ) and "undamped" ( T * eff ,F+D ) fundamental periods of the braced frames T * (damp) √ 1 − ( * eq,F+D ) 2 is always lower than 1.03. This means that the hysteretic damping introduced by the dampers has a negligible (≤ 3%) influence on the fundamental period of the braced frames. Based on these considerations, the adoption of the undamped natural period of the braced frame is a reasonable simplification in this design procedure.
Within a recursive procedure, the yield force of the damper ( F * y,D ) is iteratively adjusted (Fig. 7, right) until the spectral displacement of the braced frame converges to the design At the generic ith iteration, corresponding to a pair of parameters T * eff ,F+D of the braced frame is obtained through the spectral reduction factor ( ) proposed in (Priestley et al. 2007): . It is worth noting that Eqs. 3 and 4 overcome the inherent error of the conventional approach based on the Jacobsen formulation (Jacobsen, 1930) that typically overestimates the equivalent damping ratio of medium-to-long period hysteretic systems (Mazza et al. 2015a). Moreover, it was found that the proposed approach, in combination with the adopted spectral reduction factor (Eq. 5-b), guarantee the best accuracy in the prediction of peak displacements of a wide range of inelastic systems (Casarotti et al. 2009).
The resulting design parameters of the seventy-two equivalent SDOF systems are reported in Tables 7 and 8 of the Annex. Each equivalent SDOF system is identified by an alphanumeric string coded as follows: (a) "rc3", "rc6", and "rc9" stand for reinforced concrete frame with three, six, or nine storeys, respectively; (b) "st2", "st4", and "st8" stand for steel frame with two, four, or eight storeys, respectively; (c) "sA" and "sC" mean soil type A and type C, respectively; (d) " μ * F 1 " and μ * F max stand for parent frame target ductility factor μ * F = 1.0 and μ * F > 1.0, respectively; (e) " μ * D 4 ", " μ * D 8 ", and " μ * D 12 " are used for damper ductility factor μ * D equal to 4, 8, and 12, respectively. It should be noted that for cases "st2-sA-μ * F max-μ * D 4 ", "st2-sA-μ * F max-μ * D 8 ", and "st2-sA-μ * F max-μ * D 12 ", corresponding to two-storey steel frames on soil A, the target ductility of the frame μ * F max > 1 was achieved without introducing any suplemental damper, as shown in (Fig. 8). In fact, by comparing the "sA" design spectrum and "st2" frame capacity curve in the ADRS plane (ATC 1996) and assuming a target frame ductility μ * F max = 1.1 , even disregarding any supplemental energy dissipation, the damping eq,F+D � Fig. 8 Identification of the actual performance point through the ADRS representation of the "sA" response spectrum and "st2" capacity curve provided by the bare-frame ( * eff ,F = * el,F + * eq,F = 7.5% ) is enough to achieve the target displacement ( d * targ,ULS = 1.1 • d * y,F = 121mm). The method followed to distribute the damping force ( F * y,D ), and elastic stiffness ( k * el,D ) of the equivalent SDOF damper along the various storeys of the MDOF frame is in accordance with previous literature studies (Ponzo et al. 2010;Di Cesare et al. 2014;2017a). By referring to the installation layout shown in Fig. 9-a, with dampers arranged in a reverse-V bracing configuration at each storey, the yielding force of the units at the generic i th storey is: being: (1) n D(i) the number of units at the considered storey (for sake of simplicity, in the specific case n D(i) = 1 ); (2) (i) the angle between the line of action of the damper's output force and the horizontal direction (in the specific case (i) = 0 • ) ; (3) F y,F(i) the overall lateral yield force of the frame columns at the i th storey. The same approach has been also adopted for the calculation of the elastic stiffness of the damper units: being k el,F(i) the elastic stiffness for horizontal forces of the parent frame columns at the ith storey.
For simplicity, the stiffness of the damped brace has been assumed to coincide in practice with the stiffness of the hysteretic damper, i.e. the brace rods used to link the damper to the structural frame are very stiff and, under the actions induced by the design earthquake, undertake very small deformations in comparison to the damper's ones. Otherwise, a simple approach to account for the flexibility of link elements can be found in the aforementioned studies (Ponzo et al. 2010;Di Cesare et al. 2014;2017a).

Equivalent SDOF model for NLTH analyses
The design of the damping system necessary to achieve the target seismic performance for the frame is performed by spectral analyses considering equivalent SDOF models, according to accepted practice (Ponzo et al. 2010;Di Cesare et al. 2017a;Mazza et al. 2015a;2015b;Bergami et al. 2013). Hence, it is interesting to investigate if equivalent SDOF models ( Fig. 9-b) can be employed also in NLTH analyses as a simplified alternative to MDOF models ( Fig. 9-a), allowing a significant saving of computational effort and data post-processing times. In the analyses of the MDOF system, hysteretic dampers have been modelled as nonlinear shear links (two-DOF MultiLinear Plastic NL-Link elements in SAP2000-CSI 2017) with an elastic-perfectly plastic force-displacement behavior, connected to the frame though very stiff beam elements reproducing a chevron (or "reversed V") brace configuration. Such configuration has been adopted because of its simplicity and suitability to provide a direct relationship between the damper force and the structure inter-storey drift; however, alternative configurations, like e.g., diagonal braces, can be implemented in the model as well. The equivalent SDOF model has been formulated in OpenSees software program for structural analysis (McKenna et al. 2000) by means of Fig. 9 Structural models adopted in NLTH analyses: a MDOF braced frame (with a scheme of the calculation method for the number of effective cycles of the damper); b equivalent SDOF system two elastic-perfectly plastic truss elements working in parallel, representing the equivalent SDOF bare-frame and the equivalent SDOF damper, respectively. For brevity, nonlinear SDOF time-history analyses (NLTHs) have been assessed considering only "extreme" configurations of damped braced frames, and assuming that the ensuing validation envelopes the intermediate cases. Validation cases include hence: (a) the stiffest ("rc3", T * 1,F = 0.35s ) and the most flexible ("st8", T * 1,F = 4.18s ) frame; (b) damper ductility factor μ * D = 4 or μ * D = 12; (c) target ductility factor of the braced frame μ * F =1.0 or μ * F = μ * F,max > 1.0 . A total of 224 NLTH analyses have been carried out: 112 analyses of MDOF systems, and 112 analyses of equivalent SDOF systems.
For each braced frame, the maximum displacement at roof level of the MDOF system, d MDOF max , has been calculated as the average of the peak roof displacements induced by the fourteen ground motion records; this value has been compared ( Fig. 10-a) to the maximum displacement of the equivalent SDOF model obtained either by a spectral analysis ( d spec max ) or by nonlinear time history analyses, d SDOF max , multiplied times the modal participation factor Γ . As expected, the "rc3" systems experience much smaller lateral deformations ( d MDOF max = 23 ÷ 37 mm) than "st8" systems ( d MDOF max = 324 ÷ 396 mm). The relative deviation between peak displacements predicted by spectral analyses considering the equivalent SDOF system and peak displacements from NLTHs of the MDOF system ( Δ % = 100 • (Γd spec max − d MDOF max )∕d MDOF max ) is between −1.1 and + 10.6% in seven out of eight cases, with only a case where the deviation is + 25.0%, resulting in an average error of + 8.5% (Fig. 10-b). For NLTH analyses with the equivalent SDOF system ( Fig. 10-c), the relative deviation ( Δ % = 100 • (Γd SDOF max − d MDOF max )∕d MDOF max ) lies between −1.1 and + 15.4% in seven out of eight cases, with a single case reaching a + 21.3% deviation, and the average error is + 8.6%. On average, the accuracy of the SDOF models seems to be worse for stiff buildings ("rc3" cases) and high ductility dampers ("μ * D = 12"). Based on these results, it can be concluded that equivalent SDOF models can be used with sufficient reliability (average error less than 10%) to perform either liner equivalent spectral analyses or NLTH analyses for the prediction of the maximum seismic displacement of MDOF braced frames.
Since during real earthquakes the displacement histories which the dissipating devices are subjected to are generally not symmetric, an "effective cycle" has been defined based on the equivalence of dissipated energy between the actual force-displacement cycle and the theoretical cycle shown in Fig. 1. The number of effective cycles N cycles performed by the hysteretic damper during a ground motion is defined by Eq. (8): The number of effective cycles will be used hereinafter to evaluate the cyclic engagement of the damper.
For the braced frame, the number of effective cycles of the dampers at the ith storey under the application of the jth ground motion is calculated as: where F y,D (i) and d max,D(i,j) are the yield force and the maximum displacement of the damper at the ith storey as shown in the example in Fig. 9(a). A single value of N MDOF (j) cycles is then calculated for the whole bracing system of the frame as the average of the values in Eq. 9 relevant to the n storeys, and then both the average ( N MDOF cycles ) and the 90th percentile value ( N MDOF cycles,90% ) are calculated considering the ensemble of 14 spectrum-compatible ground-motion records. The comparison among the results is shown in Fig. 11-left. Although this trend should be confirmed through a wider parametric study (Sect. 4), it is clearly seen that bracing systems used in stiffer frames, i.e. "rc3"cases, are characterized by a higher number of effective cycles ( N cycles = 2.9 ÷ 6.1 , N cycles,90% = 4.6 ÷ 10.6 ) than bracing systems used in more flexible "st8" frames ( N cycles = 1.0 ÷ 2.2 , N cycle,90% = 1.7 ÷ 3.8 ). The accuracy of NLTH analyses using equivalent SDOF systems in predicting N cycles is shown in Fig. 11-right: the relative deviation ( Δ % = 100 ⋅ N SDOF cycles − N MDOF cycles ∕N MDOF cycles ) ranges between -18.1% and + 9.1% in seven out of eight cases, with a single case reaching -23.6%. The same comparison is shown for N cycle,90% in Fig. 12 ) ranges between -19.5% and + 10.7% in

Fig. 11
Comparison between the number of effective cycles sustained by hysteretic dampers ( N cycles ) calculated in NLTH analyses using either MDOF models or equivalent SDOF-NLTH models (left), and relative deviation (right) seven out of eight cases, with a single case reaching -22.7%. It is worth noting that the highest deviations are achieved for the "st8" configuration which, on the other side, represents the less critical case since, on average, subjected to minimum cyclic engagement. Moreover, as shown in Fig. 13, the deviation between N SDOF cycles,90% and N MDOF cycles,90% tends to vanish when the number of effective cycles is rounded to the closest integer: only in two cases (among those relevant to the less demanding "st" layouts) a discrepancy of 1 cycle (which is deemed to be acceptable) is found. However, based on these preliminary results, in order to encompass the effects of all possible uncertainties associated to the use of simplified Fig. 12 Comparison between the 90th percentile of number of effective cycles sustained by hysteretic dampers ( N cycles,90% ) calculated in NLTH analyses using either MDOF models or equivalent SDOF models (left), and relative deviation (right) Fig. 13 Comparison between rounded values of N SDOF,NLTH cycles,90% and N MDOF,NLTH cycles,90% equivalent SDOF analyses, the "reliability factor" X = 1.25 will be later introduced when a dedicated relationship to predict the number of effective cycles experienced by hysteretic dampers is formulated.

Results of the parametric investigation
The effective cycles sustained by hysteretic dampers have been investigated by means of NLTH analyses on equivalent SDOF models, as justified by the validation presented in Sect. 3. The 72 examined braced frame configurations (Tables 7 and 8) have been subjected to the applications of 14 ground motions at SLS (Tables 3 and 5) and 14 ground motions at ULS (Tables 4 and 6) resulting in a total of 2008 NLTH analyses. The cyclic engagement of the equivalent SDOF damper is calculated in each analysis and then statistically processed. The cyclic engagement, which according to Eq. (8) is associated to the amount of dissipated energy during the seismic movement, depends both on the overall lateral flexibility of the braced frame (as anticipated in Sect. 3) and on the "pulse-like" characteristics of the seismic input. Very high values of N cycles are indeed calculated for stiff structures subjected to far-field ( R ep > 30km ) events while, on the contrary, low values of N cycles are observed for flexible structures experiencing near-fault ( R ep < 15km ) ground motions. This is evident in Fig. 14 that compares the displacement histories of two braced frames presenting the largest ( N cycles = 17.4 ) and the smallest ( N cycles = 1.1 ) number of effective cycles. The peak lateral displacement of the stiff "rc3" system subjected to the farfield "ulsC-13" ground motion exceeds several times the yield limit of the damper; on the contrary, the flexible "st8" system subjected to the near-fault "ulsC-4" event experiences a single large "pulse-like" movement, and, afterwards, the yield limit of the damper ( d * y,D ) is slightly exceeded only three more times.
The effective viscous damping of the braced frame seems not to have a clear influence on the cyclic engagement of the damper, as shown in Fig. 15 that reports the number of effective cycles ( N cycles ) as a function of the effective viscous damping * eq,F+D for each examined structural system and each ground motion.
On the contrary, a direct relation exists between the number of cycles, the effective period of the braced frame T * eff ,F+D and the frequency content of the seismic input. A synthetic index is therefore introduced as follows: where T C is the transition period between the constant acceleration and the constant velocity branches of the ULS spectrum (equal to either 0.42 or 0.59 s for soil type A or type C, respectively).
In Fig. 16 the number of effective cycles ( N cycles ) calculated, for each case, as the average value over the 14 ground motions, is plotted as a function of T . Each plot contains 144 data-points grouped in twelve classes that differ for: (a) the parent frame; and (b) the epicentral distance ( R ep ) of the seismic input. By analysing the trend-line enveloping all data points, a logarithmic decrease of N cycles is noticed. By considering ULS events (Fig. 16-left), N cycles is as high as 9.4 for stiff "rc3" frames with � T < 1.0 and reduces to 2.7 for flexible "st8" frames with � T > 9.0 . The epicentral distance (R ep ) also seems to have a certain influence: data points corresponding to R ep < 15 km are in general located below data points corresponding to R ep > 30 km, highlighting a slightly lower cyclic engagement in case of near-fault ground motions compared to far-field ground motions. This phenomenon is probably caused by the intrinsic oscillatory nature of the latter events while the former are often characterized by "pulse-like" behaviour (Guo et al. 2018;Baker, 2007). Nevertheless, the dependence on R ep seems to be weaker than that on T and this relationship should be investigated in more details by considering a wider set of ground motion records.
A very close trend is noticed when SLS seismic inputs are considered (Fig. 16-right). As expected, a lower cyclic engagement of the equivalent damper is determined; indeed, the logarithmic envelope curve decreases from 6.4, when � T < 1.0 , to 1.8, when � T > 9.0. In Fig. 17 the 90th percentile of the number of effective cycles ( N cycles,90% ) at ULS and SLS is shown. Either panel contains 72 data-points grouped into six categories based on the parent frame. Each value of N cycles,90% has been calculated over a set of 14 NLTHs including both near-fault and far-field events. For ULS events, N cycles,90% is as high as 14.4, when � T < 1.0 , and decreases down to 5.2, when � T > 9.0 . All data points are again enveloped by a logarithmic curve, as shown in Fig. 17-left. A logarithmic enveloping function can be plotted also for SLS events (Fig. 17-right): here N cycles,90% decreases with the increasing of T moving from 13.9 for T = 0.68 to 4.1 for T = 9.4. It is to be noted that when the 90th percentile of data points is considered, the enveloping curves at ULS and SLS are very close to each other and tend to overlap, suggesting that a single model can be used to relate (see Eq. 11) the cyclic engagement to the governing parameter T .

Assessment of the EN15129 provisions
The results are now used to evaluate the suitability of the testing protocol (Fig. 2) recommended in the European standard EN15129 (CEN 2009). The two series of 5 cycles at 25% (run 1) and 50% (run 2) of the design displacement d bd are intended to assess the effective stiffness and damping of the device for low and medium intensity earthquakes, corresponding to SLS events, while the 10 cycles at 100% d bd (run 3) are intended to assess the degradation of mechanical properties (namely, stiffness and damping factor) and the low-cycle endurance of the device subjected to the ULS earthquake. The number of ten cycles should be strictly pertaining to applications producing fundamental periods of the order of 2 s, whereas if the fundamental period of the structural system is considerably less than 2 s, the code prescribes to increase the number of cycles according to the Structural Designer's specifications.
The results of the present study highlight that the number of effective cycles sustained by the hysteretic damper is also affected by the spectral characteristics of the seismic input through the transition period T C which is used for the calculation of T . The 10 cycles prescribed in the standards turn out to be appropriate for 3.0 < T < 7.0, i.e., for applications in flexible frames (e.g., medium-rise steel frames), but extremely demanding for T > 7.0, when the number of effective cycles is lower or at the most equal to six. In contrast, for applications to stiff frames (e.g., low-rise RC frames) ten cycles appear to be under-conservative to assess the cyclic behavior of the damper, underestimating by a factor up to 1.5 the actual engagement. Eventually, run 1 plus run 2 do not appear to be appropriate to simulate the cyclic engagement of DDDs under SLS earthquakes in case of devices installed in rigid frames. In fact, as shown in Fig. 17-right minor earthquakes can trigger up to 14 cycles in braced frames characterized by � T < 1.0. Based on the regression equation shown in Fig. 17, the number of symmetric cycles at ± d bd to be imposed on the test specimen during the tests N cycles,test can be estimated, both at ULS and SLS, by the following expression: . being: (a) x = 1.25 a "reliability factor" (as suggested in Sect. 3); (b) " round " a function that rounds a decimal number to the closest integer.
(11) N cycles,test = round N cycles,90% N cycles,90% = x ⋅ −3.7 ln T + 13.2 It should be noted that the calculation of N cycles,test is straightforward and does not require any computation in addition to those already performed for the design of the dampers, because the only parameters needed for the calculation of T are the effective period of the equivalent SDOF of the braced frame T * eff ,F+D and the transition period T C of the design spectrum at ULS.

Conclusions
The study investigates, through a parametric study, the cyclic engagement of hysteretic dampers (defined as the number of symmetric cycles at the design displacement which provides the same energy dissipation during a design earthquake) used in typical steel or reinforced concrete buildings, in order to ascertain the effectiveness of current testing protocols for qualification and acceptance tests recommended in the European standard EN 15129 (CEN 2009).
The main results of the study are summarized as follows: • The number of effective cycles sustained by hysteretic dampers installed in a MDOF frame can be assessed through NLTH analyses carried out on an equivalent SDOF model with a reasonable level of accuracy. Indeed the deviation between N MDOF cycles and N SDOF cycles is always lower than 25% and becomes negligible when the relevant values are rounded to the closest integer number; • The cyclic engagement of the dampers depends on the ratio ( T ) between the effective period of the braced frame ( T * eff ,F+D ) and the transition period ( T C ) between the constant acceleration and the constant acceleration branches of the ULS design spectrum. A minor dependence on the "pulse-like" nature of the seismic input has been highlighted; indeed, given a certain value of T , far-field events result in a higher N cycles values compared to near-fault ground motions; • At ULS, the provisions of the standard appear over-conservative for dampers installed in flexible structures but under-conservative when the dampers are installed in stiff structures. Indeed, the number of 10 cycles turns out to be appropriate for 3.0 < T < 7.0, i.e., for applications to flexible frames (e.g., in medium-rise steel frames), but extremely demanding for T > 7.0, when the number of effective cycles is lower than or at the most equal to six. In contrast, for applications to stiff frames (e.g., low rise RC frames), ten cycles appear to be non-conservative, underestimating the actual engagement of the damper up to 50%; • Also at SLS, the provisions of the standard do not appear to be appropriate in case of devices installed in rigid frames characterized by � T < 1.0 where the engagement involves up to 14 effective cycles; • A simple rule to establish the number of cycles to be imposed to the test specimens during the test has been proposed based on the 90th percentile regression equation. This expression is simple, valid for both SLS and ULS estimations and does not require any further computation in addition to those already performed for the design of the dampers.

Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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