Damage indices of steel moment-resisting frames equipped with fluid viscous dampers

ABSTRACT Deformations, accelerations, shear forces, and energy demands are normally utilized to evaluate the performance of steel frames equipped with fluid viscous dampers (FVDs). This study aims to quantify damage indices (DI) and their distributions along the structural height for steel frames (3, 6, 9, and 20 stories) equipped with FVDs. The damage model incorporating both deformation and hysteretic plastic energy is adopted to quantify damage developed in structures. A set of 20 pairs of ground motions is used to perform nonlinear time history analyses. Influences of structural property and FVD features (supplementary damping ratio and velocity power ) on DI are investigated. The results show that FVDs with lower α would reduce DI. However, nonlinear FVDs lead to higher DI compared to linear ones when >30%, particularly for 9- and 20-story structures. Nonlinear FVDs ( ) are more effective than linear ones in reducing DI during low-intensity earthquakes. The results further demonstrate that the distribution of DI along the structural height is mostly determined by structural properties, FVD characteristics, ground motion intensity, and the type of damage model used.


Introduction
Steel structures may be subject to a major collapse under strong earthquake events due to large drift between stories and plastic distortions in the main structural components (Clifton et al. 2011).The damage cases in social and economic losses, are not acceptable to new societies that aim to achieve high levels of performance to resist future earthquakes (Christopoulos and Filiatrault 2006).Therefore, a new earthquake mitigation procedure has been proposed by introducing passive control devices to improve energy-dissipation capacities and reduce the structural deformation of the systems.Fluid viscous dampers (FVDs) have been thoroughly investigated as one of the most often utilised damping system devices due to its primary benefits, comprising of significant energy-dissipation capacities and peak forces that are out of a phase of the original systems (Constantinou and Symans 1993;Seleemah and Constantinou 1997;Soong and Dargush 1997;Christopoulos and Filiatrault 2006).FVDs are the most effective technique for reducing deformations and beam plastic rotations (Chen et al. 2010).The generated force-velocity relationship of FVD can be presented as follows: where _ x is the differential velocity at the ends of the damper; C is the coefficient that characterizes the size of FVD; α is the FVD power between 0.1 to 2.0 that represents the viscous material; and sgn is the signum function related to _ x.If α = 1, FVDs present linear behaviours, while nonlinear responses would develop in FVDs if α < 1.For seismic protection of structural systems, α is designed to be in the range between 0.3 and 1.0 (Christopoulos and Filiatrault 2006;Impollonia and Palmeri 2018).A nonlinear FVD would have a force that is more than 35% less than a linear one (Martinez-Rodrigo and Romero 2003), although the strong nonlinear FVDs (smaller α) may lead to higher damper forces than linear ones (Tubaldi et al. 2015;Scozzese et al. 2021).Several studies have been carried out to explore the influence of FVDs on seismic demands for various building types (Akcelyan et al. 2016;Wang 2017;Kitayama and Constantinou 2018;Chalarca et al. 2020).Pavlou et al. 2017, Kitayama et al. 2018and Chalarca et al. 2020 executed dynamic analysis investigations of buildings with linear and nonlinear FVDs and computed peak absolute accelerations.Significant reduction in absolute accelerations were found in buildings equipped with FVDs.Results showed that an increase in peak accelerations at some floors for buildings equipped with nonlinear FVDs, while less inter-story drifts are found in building equipped with nonlinear FVDs (Tubaldi et al. 2015;Dall'Asta et al. 2016).In addition, the FVDs lead to a decreased probability of collapse of the structure (Miyamoto et al. 2010;Karavasilis 2016;Dall'Asta et al. 2017); although the FVDs with (α = 0.3 and 0.15) tend to increase the collapse compared with linear ones CONTACT Ge Song ben_0702@sina.comState Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China (α = 1) particularly during high-intensity levels of earthquakes (Tubaldi et al. 2015).
On the other side, some of studies have been focused on determining the energy demands of buildings equipped with FVDs.Constantinou and Symans 1993;Seleemah and Constantinou 1997;Soong and Dargush 1997;Christopoulos and Filiatrault (2006) introduced an energy-based concept to supplemental damping devices, including FVDs, and validated their findings against extensive evaluations of buildings with and without dampers.Their results demonstrated that a significant reduction of the energy dissipated by the original structural system in exchange for energy dissipation by the dampers.Recently studies Sorace and Terenzi 2008, Domenico and Ricciardi 2019, Domenico et al. 2019, Domenico and Hajirasouliha 2021 focused on energy-based optimization design.While Zhou and Sebaq et al. 2022 provided a design-based seismic level evaluation of energy dissipation demands and their distributions for buildings with/without FVDs and found that nonlinear FVDs are more effective in reducing energy demands compared with linear ones.
In previous studies, the primary motivation for evaluating these systems was to determine seismic demands only as deformations, accelerations, and story shears, or determine energy demands only.However, when incorporating innovative control systems, present analysis and design methodologies must be reconsidered.Damage indices as structural as a damage evaluation appear to be more effective, as a result that they are the only models that incorporate deformation and hysteretic plastic energy at the same time.The damage model also accounts for damage caused by maximum inelastic excursions, as well as damage caused by the history of deformations.Nevertheless, none of previous studies have focused on the potential damages in structures equipped with FVDs based on damage indices.Therefore, this work provides an assessment of the damage potential for (3-, 6-, 9-, and 20-story) steel buildings with/without FVDs under two intensity levels.To this aspect, the new features of this study includes: (Akcelyan et al. 2016) the effect of structural properties and FVD characteristics, namely velocity power (α) and supplemental damping ratio (� add ) on the overall structural damage and distributions along the building height; (Akiyama 1985) the difference between two damage models proposed by Park andAng 1985 andReinhorn andValles 2009;(ASCE 2016) showing the effect of ground motion intensity on the computation of damage index; and (Barbosa, Ribeiro, and Neves 2016) determination of the type of failure based on the distribution of damage indices over the building height.Finally, results can provide guidance for potential retrofit solutions towards buildings equipped with FVDs.

Structural damage index background
Damage indices can be used to evaluate the probable damage caused by earthquakes, and thus can be utilized to evaluate the seismic sensitivity of existing structures and the design of new earthquakeresistant structures (Rodriguez 2015).Damage to structures is determined by a variety of factors, including the relationship between the load-deformation behaviour of the structure, the amount of inelastic deformation, and the characteristics of earthquake motions, etc., (Samanta et al. 2012).There is a varied scope of damage indices that have been presented, which can be assembled into three types: (Akcelyan et al. 2016) maximum response (force or displacement) (Hancock and Bommer 2007;Barbosa et al. 2016); (Akiyama 1985) energy dissipated by structure represented by plastic energy demand E P (Akiyama 1985;Powell and Allahabadi 1988;McCabe and Hall 1989;Uang and Bertero 1990); and (ASCE 2016) combination damage (Akcelyan et al. 2016;Akiyama 1985) (Park and Ang 1985;Mehanny and Deierlein 2000;Malaga-Chuquitaype and Elghazouli 2012).The maximum inelastic deformation (displacement of force) alone may not be a suitable indicator for computing DI for the seismic performance (Mehanny and Deierlein 2000;Barbosa et al. 2016).In the third group, the DI is a combination of displacement and E P (Park and Ang 1985;Kunnath et al. 1991;Reinhorn and Valles 2009).The most recent reviews of current damage indices for structural damage evaluation were conducted by (Williams and Sexsmith 1995;Ghobarah et al. 1999;Mehanny and Deierlein 2000;Bozorgnia and Bertero 2003).
The most common models used for computing damage indices including inelastic deformation (displacement) and energy dissipation represented by E P are: (Akcelyan et al. 2016) Park & Ang model ðDI P&A Þ, (Akiyama 1985) Reinhorn and Valles (DI R&V ).The interstory deformation or top story displacement, as well as the story yield shear force or base shear yield force level, are necessary for the story and overall DI (Reinhorn and Valles 2009).
The DI P&A for each story is: And the DI R&V for each story is: where δ m and E P are the maximum story deformation and cumulative dissipated hysteretic plastic energy, respectively, obtained from NTHA subjected to ground motions; δ u , δ y and F y are the ultimate story deformation, yield story deformation, and yield story force capacity, respectively, for the original building (without FVDs) obtained from NSPA.β is a model constant parameter.To estimate β, test data from 402 reinforced concrete rectangular cross-section components and 132 steel H-shaped section samples tested in the United States and Japan were used, yielding a value of β=0.05 for reinforced concrete structures and β =0.025 for steel structures (Park and Ang 1987).
Table 1 shows the level of damage found in the structures, as well as the calibration of damage indices performed by Park and Ang 1987.
The following are the overall DI, which are calculated using factors (λ i ) based on the E P at each story: where E P ð Þ i is the hysteretic plastic energy in the i-th story, including beams and columns for the i-th story recorded from NTHA.
Other investigations have found a correlation between structural damage and characteristics related to the duration of strong ground motion.For example, Kunnath andChai 2004, andIervolino et al. 2001 found that ground motion duration had no effect on maximal inelastic deformations.Large spectrum amplitudes of ground motion are more destructive than small amplitudes of ground motion (Hancock and Bommer 2007).Ground motion length is well correlated with damage indices based on plastic energy dissipation (Samanta et al. 2012;Hou and Qu 2015).Barbosa et al. 2016 andBelejo et al. 2017 and found that DI R&V is always greater than DI P&A .

Study objectives
The goal of this research is to improve understanding of the damage assessment considering four steelmoment resisting buildings (3-, 6-, 9-, and 20-story) without FVDs (uncontrolled-buildings, UNCLT) and those equipped with linear and nonlinear FVDs (controlled-buildings, CLT) subjected to 40 records of ground motions.The main parametric variables are four-components of supplemental damping ratio � add ( ¼ 5%; 10%; 20%and30%) and four-components of velocity power α ( ¼ 1; 0:7; 0:5and0:3).It is worth noting that � add >30% is typically not recommended since it may cause changes in the inherent dynamic aspects of the structure, with potentially negative consequences (Barroso and Winterstein 2002;Occhiuzzi 2009).For evaluation of the damage indices; DI P&A and DI R&V based on Eq. (2, and ASCE 2016) are used in this study to show the difference between them for UNCLT and CLT buildings.NSPA is utilized to capture structural responses of original buildings (i.e., yield displacement capacity, ultimate displacement capacity, and yield force capacity).NTHA is carried out at two intensity levels and structural demands such as peak story deformations and hysteretic plastic energy demands are traced to evaluate DI.PERFORM-3D Platform is used to model the four buildings.Figure 1 shows the steps of determining story damages and overall building damage indices.

Steel buildings frame structures
The four case studies comprise a (3-, 6-, 9-, and 20story) buildings, designed in acquiescence with Uniform Building Code 1988 (UBC).These structures have been widely utilised as reference structures in numerous structural response control investigations (e.g., Gupta and Krawinkler 1999;Chalarca et al. 2020;Zhou et al. 2022).Figure 2 shows the main dimensions and information of the steel elements, as well as the locations and distributions of the FVDs, which was taken from the study of Zhou et al. 2022.A 2-Dimensional (2D) model of the four buildings was generated in the PERFORM-3D software using fiber sections for columns (Neuenhofer and Filippou 1998), and plastic hinge models for beams based on the recommendations of (PEER2010/ATC-72; Ribeiro et al. 2017;PEER 2020).The details of modeling these buildings in PERFORM-3D are described in Zhou et al. 2022.The frame buildings include P-delta to consider the gravity load effect of interior frames using leaning columns as shown in Figure 2. The strength and deformation of panel zones were neglected.In constructing the finite element computer models, the columns were assumed to be fixed at the base level.
The FVDs in the four studied buildings are typically simulated using Maxwell model, which it includes the spring element and damping coefficient.For each FVD model, the spring element comprises some elements, such as gusset plates, clevises, brackets.In this study, the axial stiffness was taken as 150kN=mm in the four buildings based on the recommendations by (Akcelyan et al. 2016;Wang 2017;Chalarca 2020).The damping coefficient is a function of velocity power (α), and supplemental damping ratio (� add ).For the first mode and not for the higher modes, Christopoulos and Filiatrault 2006 proposed equations for predicting the equivalent linear and nonlinear damping coefficients based on equivalent lateral stiffness method as follows: Previous research on the determination of designing and distribution of FVD over the building, can be grouped as standard techniques such as identical FVD constants at each story, proportional to story shear forces (Pekcan et al. 1999), proportional to story shear strain energy (Hwang et al. 2013), proportional to story shear strain energy to efficient stories (Hwang et al. 2013), stiffness-proportional distribution (Christopoulos and Filiatrault 2006).Although these procedures may be carried out with little computing effort and without the use of complex software, it may not be optimum from an economical point of view.In the context of optimization control methods, the optimum distribution of dampers in the building may be cast (Takewaki 1997).Recently several new energy-based distribution procedures have been developed to maximize the energy dissipative capacities of the dampers, which provide valuable tools for future design of FVD-controlled structures (Domenico and Ricciardi 2019;Domenico et al. 2019).But their analyses applied to a simplified elastic system without taking into account inelastic deformation of the original structural system, P-Delta effect, and higher modes effect.As a result, it is impractical to validate this approach before applying it to an inelastic system.The energy-dissipated by FVD mainly depends on the amount of inelastic deformation in the original structural system; and also, the type of deformation occurs in the structure (Hwang et al. 2013).Although recently developed approaches have evident advantages, they are not commonly used in engineering practice.Therefore, the selected damping coefficient for linear and nonlinear FVDs (C L and C NL ) in this study were designed according to the equivalent lateral stiffness (ELS) (Christopoulos and Filiatrault 2006); which is commonly used in (Whittle et   the distribution of FVDs.In this study, the required C L and C NL for four levels of � add (=5, 10, 20, and 30%) corresponding to four levels of α (= 0.7, 0.5, and 0.3) for the 3-, 6-, 9-, and 20-story buildings provided in Zhou et al. 2022 were used.

Validation of the constructed model
This validation is based on a five-story full-scale experimental building equipped with nonlinear FVD obtained from shake table testing at Japan's E-Defense facility.In X-and Y-loading directions, the test structure consisted of three two-bay steel moment-resisting frames.There was a total of 12 FVDs placed (four in the Y-loading direction and eight in the X-loading direction).Kasai et al. 2010;Akcelyan et al. 2016 provided considerable details on the test structure, including material characteristics, element sections, and FVD parameters.Figure 3 shows the story drift ratios, peak shear forces, and peak absolute accelerations over the building height of the test structure with nonlinear FVDs (α=0.38),respectively, under the 100% of the JR Takatori record from the 1995 Kobe earthquake between experimental data (Exp) finite element model (FE).From the figure, there is practically no difference between the Exp results and FE numerical models.This indicates that the constructed FE by Perform 3D can capture of seismic demands and use as a useful tool for the numerical modelling of steel moment-resisting frames Equipped with FVDs.

Ground motions selection
Twenty-pairs of ground-motions were selected form PEER 2015. Figure 4 depicts the spectrum for each ground motion, as well as the mean of the 40 response spectra and the ASCE 2016 target spectrum at two

Analysis results
The NSPA is performed on the four UNCLT buildings; and the NTHA is performed on the four UNCLT and CLT buildings induced to 20 pairs of motions (40-records).This section gives details on the computation of DI P&A and DI R&V for (3-, 6-, 9-and 20-story) UNCLT and CLT buildings; considering two levels: (Akcelyan et

Modal analysis
Elastic Eigen-analyses were conducted to determine the vibration modes of the case-study buildings (3-, 6-, 9-, and 20-story).Vibration modes are useful to understand the dynamic behaviour of the building structures and to visually check that structural members are connected.The first 4 vibration modes in the investigation were found using the Perform 3-D Eigenanalysis (Table 2).The effective mass participation for first mode is 93.89%, 88.87%, 82.39%, and 75.88% for 3-, 6-, 9-, and 20-story buildings, respectively.Therefore,  for 3-story building, the fundamental mode is almost the first mode, whereas for 6-, 9-and 20-story buildings, it is the first and second modes.

Nonlinear static pushover analyses
Nonlinear static pushover analyses are performed in all buildings (3-, 6-, 9-, and 20-story), assuming that the lateral pattern load is based on the fundamental first mode of vibration of each respective building.The base shear versus roof drift angle response for the different studied buildings are shown in Figure 5, as discussed in details in Zhou et al. 2022.The determination of yielding force, yielding deformation, and ultimate deformation of each building is shown in Figure 5 based on FEMA P695, Section 6.3.Figure 6 presents the force-deformation capacity for each story in the UNCLT buildings (3-, 6-, 9-and 20-story).It is noted that the top stories exhibited reverse unloading curves under the unidirectional NSP, which is probably due to the top stories of high-rise buildings (9-and 20story) are generally dominated by a combination of   shear and flexural deformations while the bottom stories are mainly controlled only by shear deformations.
The accompanying flexural deformation is caused by increasing axial column deformations.

Overall building damage index
This section shows the overall building damage index for UNCLT and CLT buildings based on DI P&A and DI R&V .Figure (7)(8)(9)(10) show the average DI P&A and DI R&V for various � add ( ¼ 0; 5%; 10%; 20%and30%) and α ( ¼ 1; 0:7; 0:5and0:3) for the four studied buildings, respectively, at two levels of earthquakes.For DBE level, a comparison of results shown in Figures 7a and Figure 10b indicates that DI P&A and DI R&V have the same damage index in all four buildings except UNCLT buildings, the DI R&V is higher than DI P&A , this is attributed to larger E P demand.On the other hand, for the MCE level as shown in Figure 7b and Figure 10b, the DI R&V leads to higher-damage indices, thus resulting in more damage than DI P&A ; and the difference between them decreases with increasing � add .It is noted that the difference between DI P&A and DI R&V damage indices associated with 3-, and 6-story buildings are consistently larger than the ones associated with 9-, and 20-story buildings.Furthermore, the damage index DI R&V is always larger than DI P&A , indicating that the building response is highly sensitive to E P demand.This is due to Eq. 3 defined by Reinhorn and Valles considers a large amount of E P to compute DI unlike Eq. 2 provided by Park and Ang.
The results show that increasing � add reduces DI P&A and DI R&V for the four buildings and for two intensity levels (Figure 7a and Figure 10b.Excepting 9-, and 20-story buildings, the effect of � add > 20% is less significant (Figures (Chen, Li, and Cheang 2010;Christopoulos and Filiatrault 2006)).It is worth mentioning that the � add has a significant effect on the computation of the DI P&A and DI R&V , this is due to the � add leads to decrease of structural responses such as inelastic deformation and E P demand.(Figure 7, Figure 8, Figure 9, Figure 10) show also the significant effect of α on both overall DI P&A and DI R&V with varying � add for CLT buildings (3, 6, 9, and 20-story).Presented in Figure 12, the ratios of DI P&A and DI R&V of two systems, one with nonlinear FVD (α) and the other with linear FVD (α = 1) for (3-, 6-, 9-, 20-story) buildings and for both (DBE and MCE).The impact of α is more noteworthy on the decrease of DI P&A and DI R&V .For 3-and 6-story building, the nonlinear FVDs are more effective in reducing DI than linear ones for both intensity levels (DBE and MCE) for different values of � add ( ¼ 5%; 10%; 20%and30%) Figure 12(a and b).
Many nonlinear physical phenomena play a role in explaining the somewhat illogical numerical results obtained for structural damage (DI).The strong nonlinear FVD with α ≤0.5 is not effective in reducing DI especially for (9-and 20-story) buildings, and for high intensity level (MCE).As shown in the Appendix, the hysteretic behaviour of nonlinear FVDs with lower levels of α induces damper forces in the building over a larger range than the displacements of the dampers; which may collapse the building and increase the DI.This means that larger FVD forces are created if the velocity demands increase more than expected in the design because of earthquake ground motion has a large number of random amplitudes.In accordance with the capacity-design concept, this may result in a more underestimating design for non-dissipative components.It is highlighted that the use of a higher level of � add >20% is not effective 9-, and 20-story buildings; this is due to two reasons: (Akcelyan et al. 2016) the flexural deformation is as substantial as  the building's shear deformation because of increasing the P-delta effect, the FVD deformation and velocity are predominant to shear deformation than flexural.Therefore, the peak FVD displacement may be reduced, and hence the energy dissipated by FVD can be reduced; (Akiyama 1985) the increase of participation of the high modes effect.

Distribution of damage indices
The second section purposes at exploring the influence of properties of structural system and FVD on the distributions of DI P&A and DI R&V over the building height.The values of DI P&A and DI R&V damage indices distribution over the height of the building is dependent on response demands such as peak story deformation and story plastic energy demand.For the 3-story building, the DI P&A and DI R&V produce nearly uniform damage distribution for two intensity levels (DBE and MCE) with the exception of CLT buildings equipped with nonlinear FVD (α ¼ 0:7; 0:5and0:3).The damage concentrates in the 1 th story and decreased gradually for the 2 th and 3 th stories (Figure 13).In the 6-story building, and for DBE level, the DI P&A and DI R&V damage indices are observed to be almost uniformly distributed in all floors for UNCLT and CLT buildings (Figure 14).For MCE level, the DI P&A is also uniformly distributed in all floors (Figure 14a), while the DI R&V leads to be uniform in the first four stories and decrease in the top two stories (Figure 14b).significant.For 3-story building, increasing � add leads to reduce the concentration DI in the first story for α<1 (Figure 13); while for 6-, 9-, and 20-story buildings, the effect � add on the distribution of DI over the building height is not significant (Figures (Dall'Asta et al. 2017;Dall'Asta, Tubaldi, and Ragni 2016;De Domenico and Hajirasouliha 2021)).The influence of α on the distribution of DI P&A andDI R&V damage indices vary among the (3-, 6-, 9-, and 20-story) structures, demonstrating a substantial coupling effect of structural and FVD features.For 3-story building, the decrease of α always leads to an increase of DI P&A andDI R&V in the first story and decrease in the top two stories, especially for MCE level (Figure 13).For 6-story building, the decrease of α decreases DI P&A andDI R&V in the first four stories, but almost has no effect in the top two stories (Figure 14).For 9-story building, the decrease of α reduces DI P&A andDI R&V in the first six stories and is less significant in DI P&A andDI R&V in the top three stories (Figure 15).For 20-story building, the decrease of α reduces DI P&A andDI R&V based on the amount of � add (Figure 16).Finally, the distribution of DI depends on the number of stories (building height), story plastic energy demand (E P ), FVD characteristics (� add and α), and type of damage index model (DI P&A and DI R&V ).
From these Figures, at the DBE level, no collapse is observed in  based on the DI P&A and DI R&V damage indices; except for the UNCLT buildings (6-, 9-and 20-story) which collapse occurs at different floor levels based on DI R&V damage index as shown in Figure 13b and Figure 16b).At MCE level, the collapse is different based on the DI P&A and DI R&V damage indices and the amount of � add (Figure 13, Figure 14, Figure 15, Figure 16)).For lower floors collapse mechanisms, it experiences higher shear deformation demands as shown in 3and 6-story buildings.For upper the floor, the collapse mechanism could be due to two reasons: the first is related to the strong axial deformation of the columns and the shorter frame dimensions; and the second is owing to the considerable flexural frame deformation as shown in 9-and 20-story buildings.
With overall building DI P&A and DI R&V , they are strongly dependent on the ground motion intensity, natural vibration period T n , and FVD characteristics (� add and α).The main findings are: (1) For DBE level, DI P&A and DI R&V damage indices values are identical in all four buildings excepting UNCLT buildings, the DI R&V is higher than DI P&A .While, for MCE level, the DI R&V tends to be higher values of damage indices compared with DI P&A for UNCLT and CLT buildings.This is because of the structural response is highly sensitive to E P ; (2) The � add plays an important role in reducing damage indices (DI P&A and DI R&V ) for (3-, 6-, 9and 20-story) buildings.Excepting (9-, and 20story) buildings, the � add > 20% is less significant; (3) In the CLT buildings with smaller velocity powers (i.e., α < 1), the damage indices (DI P&A and DI R&V ) tend to decrease significantly when compared to systems equipped with linear FVDs for all four studied buildings.Except for lower α (=0.5 and 0.3) and higher � add (=20 and 30%), they are equivalent to or greater than linear FVDs, which is especially noticeable in 9-and 20story buildings; and (4) Nonlinear FVDs are more effective in reducing DI than linear FVDs, particularly at the DBE level and and � add ( ¼ 5%; 10%; and20%), although at the MCE level, the decreasing of α is slightly effective in reducing DI at various levels of � add ( ¼ 5%; 10%; 20%and30%).
The study further investigates the effects of ground motion intensity, natural vibration period T n , and FVD characteristics (� add and α) on the distribution of DI P&A and DI R&V over the structure height: (1) It noticed that the structural properties, FVD characteristics, level of ground motion intensity, and the type of damage model all have a significant contribution on the distribution of DI along the height of the building; (2) The DI R&V is always higher than DI P&A over the height of all four UNCLT and CLT buildings particularly for MCE intensity level, this is probably due to DI R&V is more significant to E P demand rather than DI P&A ; (3) The 3-, and 6-story present a relatively uniform distribution of DI along the height of the building, excepting CLT building (3-story) with nonlinear FVDs (α<1) and (� add ¼ 5%and10%) leads to concentrate damage in the 1 th story.While, for 9-, and 20-story buildings, the distribution of damage indices is almost nonuniform over the building height, which indicates that the contribution effects of the P-delta effects and higher modes effect; and (4) For 3-story building, increasing � add leads to reduce the concentration DI in the first story for α<1; while for 6-, 9-, and 20-story buildings, the effect � add on the distribution of DI over the building height is not significant.(5) There are coupling effects among � add and α of FVDs on DI P&A and DI R&V along the building height.In most circumstances, the rise of � add and the decrease in α usually reduces DI P&A and DI R&V damage indices, but for the 9-and 20-story buildings with higher values of � add , the effect of α reverses at some stories.(6) Higher values of DI in lower stories is associated with higher shear deformation, as shown in lowrise buildings (3-and 6-story).For relatively highrise buildings (9-and 20-story), the distribution damage occurs in both the bottom and top stories, which is associated with shear deformations in bottom stories and flexural deformations in top stories.

Figure 1 .
Figure 1.Flowchart for the determination of the story damage indices and overall building damage.

Figure 4 .
Figure 4. Individual response spectra for 40 ground motions and their mean response spectrum: (a) design-based earthquake level (DBE); (b) maximum considered earthquake (MCE).

Figure 7 .
Figure 7. Three-story building: damage indices (DI P&A and DI R&V ) as a function of the � add and α for two intensity levels: (a) DBE and (b) MCE.

Figure 9 .
Figure 9. Nine-story building: damage indices (DI P&A and DI R&V ) as a function of the � add and α for two intensity levels: (a) DBE and (b) MCE.

Figure 8 .
Figure 8. Six-story building: damage indices (DI P&A and DI R&V ) as a function of the � add and α for two intensity levels: (a) DBE and (b) MCE.

Figure 10 .
Figure 10.Twenty-story building: damage indices (DI P&A and DI R&V ) as a function of the � add and α for two intensity levels: (a) DBE and (b) MCE.

Figure 14 .
Figure 14.Six-story building: damage indices distribution over the building height with varying � add , α and intensity levels (DBE and MCE): (a) DI P&A damage index, and (b) DI R&V damage index.
For (9-and 20-story) buildings, the distribution of DI P&A and DI R&V are nonuniformly distributed over the building (Figures (Dall'Asta, Tubaldi, and Ragni 2016; De Domenico and Hajirasouliha 2021)), which indicate that the P-delta effects and higher modes effect to control the plastic deformation over the building height.Figure (13-16) show the influence of FVD characteristics (� add and α) on the distribution of DI P&A andDI R&V damage indices over the building height.The results indicate that both DI P&A and DI R&V decrease with increasing � add along the building height, except for (9-and 20-story) buildings, � add > 20% is less

Figure 15 .
Figure 15.Nine-story building: damage indices distribution over the building height with varying � add , α and intensity levels (DBE and MCE): (a) DI P&A damage index, and (b) DI R&V damage index.

Figure 16 .
Figure 16.Twenty-story building: damage indices distribution over the building height with varying � add , α and intensity levels (DBE and MCE): (a) DI P&A damage index, and (b) DI R&V damage index.
where, C L; j and C NL; j are the linear and nonlinear damping coefficients at each story, respectively.k i and δ i is the inter-story drift and elastic lateral stiffness at the ith story, respectively; which can be obtained by a pushover analysis of the elastic structures, based on the fundamental first mode.α is the velocity power of FVD; ω n is the natural vibration frequency which can be taken as the fundamental frequency of the building; θ is the declination angle of FVD at each story; N f is the number of stories; and N d is the total number of FVDs.x o is the

Table 2 .
Modal periods and effective mass participation for the case study buildings.