A Procedure for the Estimation of the Supplemental Damping for Design and Retrofit of RC Buildings with FVDs

Abstract

Passive energy dissipation (PED) devices have been widely accepted as effective in reducing seismic effects through extensive experimental and analytical studies. However, the estimation of the supplemental damping ratio (SDR) and its relationship with seismic performance levels remain important research challenges. In this study, a supplemental damping-based procedure is proposed for the design and retrofit of reinforced concrete buildings equipped with PED systems. The proposed procedure essentially consists of two main parts, which define the overall methodological framework. The first part of the procedure is developed to establish a relationship between target seismic performance levels and SDR by explicitly considering the structure–damper interaction, rather than predetermining damping or structural parameters. The second part analytically establishes a direct relationship between the target seismic performance of the structure and the required SDR provided by fluid viscous dampers (FVDs). The results of 5808 nonlinear time history analyses indicate that the supplemental damping ratio plays a critical role in structural performance and provides significant reductions in hysteretic energy demand depending on the device characteristics, configuration, and arrangement. In addition to the key parameters of the damping system mentioned above, the proposed procedure also considers key parameters of the structural system. Furthermore, it allows these parameters to be evaluated separately with respect to seismic performance levels. Consequently, the first part of the procedure provides a detailed basis for damper optimization and realistically reveals the effects of supplemental damping on the seismic performance of new and existing reinforced concrete buildings equipped with PED systems. Furthermore, the second part offers a non-iterative and practical solution for the performance-based design and retrofit with FVDs.

1. Introduction

In seismic design practice, the assumption of fully elastic structural behavior at design-based (DBE) and maximum considered earthquake (MCE) levels leads to increased member dimensions, resulting in uneconomical and unsustainable solutions. For this reason, the preservation of life safety is ensured by allowing controlled structural damage at non-collapse levels (i.e., without the formation of collapse mechanisms) under rare and very rare earthquakes. The seismic design philosophy commonly accepted in the literature and international codes is also based on this approach and allows for inelastic behavior in a wide variety of structural systems. Structures under seismic actions can enhance their energy dissipation capacity by undergoing inelastic deformations. However, the deformation demands imposed on structural members should not exceed the targeted performance levels. In this context, PED devices are used to mitigate seismic effects in structures to dissipate a portion of the input earthquake energy by converting it into heat or deformation energy, depending on the device type and its placement within the structure. Consequently, these energy dissipation systems reduce deformation demands on structural members and play a significant role in achieving the targeted performance levels of buildings for both design and retrofit applications. On the other hand, in engineering practice used for the design of new buildings or retrofitting using PED devices, base shear capacity and effective structural period are determined by pushover analysis. The next step is to design the PED devices, and finally, it is checked whether the building satisfies the target performance requirements. Since the PED devices significantly influence the structural behavior of the building, any change in PED-related design parameters, such as capacity, arrangement, or configuration, leads to simultaneous changes in multiple structural response parameters. This approach is inherently iterative.

Studies on the development of PED devices, pioneered by friction and viscoelastic dampers, and on the experimental testing of these devices [1,2,3,4,5,6] were particularly concentrated during the period between 1990 and 2000. In addition, during this period, shake table tests [7,8,9,10,11,12] were performed on scaled structural models incorporating the developed devices. From the perspective of structural design and retrofit, the results obtained from both device tests and shaking table experiments demonstrated that such systems can be effectively used for structural retrofitting.

In the context of structural retrofitting, several case studies were also investigated [13,14,15,16], providing important findings into the retrofitting of existing structures using PED devices. In addition, studies examining hybrid damping systems in which PED devices are used in combination [17,18,19], as well as studies comparatively evaluating different types of damping devices [20,21,22], are complementary in terms of developing a comparative understanding of the behavior of the PED device types considered in this paper. The use of simplified structural systems to investigate the behavior of PED devices is a common approach in the literature. Accordingly, consistent with many studies focusing on the design and optimization of PED devices [23,24,25,26,27,28,29,30,31], planar frame systems were adopted as the structural system in this study. Similarly, the effects due to soil conditions on frame systems equipped with PED devices were examined through comparative analyses [32], consistent with the approach adopted herein. In addition, the distribution of damping throughout the structure is an important aspect of damping optimization. Moreover, the concept of damper optimization was developed, and numerous studies have been carried out on structural systems incorporating friction dampers [33,34,35,36,37,38,39,40,41], buckling-restrained braces (BRB) dampers [42,43,44,45,46,47,48], fluid viscous dampers [49,50,51,52,53,54,55,56], and viscoelastic dampers [57,58,59,60,61,62,63,64]. Within the scope of this study, not only the distribution of damping within the structure but also the assumptions of the effective damping ratio [65] and the additional equivalent damping ratio [66] are considered as main aspects in the performance-based evaluation of structures equipped with damping devices. Furthermore, an algorithm was developed that can be used for both the design of new buildings and the retrofit of existing structures equipped with fluid viscous dampers based on the nonlinear time-history analyses results obtained with OpenSees [67]. Similarly, an algorithmic retrofit procedure for reinforced concrete buildings equipped with fluid viscous dampers was proposed [24].

The effectiveness of PED devices in mitigating seismic effects has become better understood through the incorporation of nonlinear behavior and seismic performance concepts into both design codes and the literature, together with the increased applicability of dynamic analysis methods. Consequently, an examination of general trends in the literature over approximately the past 35 years, together with building codes issued after 1995, indicates that various design approaches have been developed for the use of PED devices to reduce seismic risk and improve the structural performance of both existing and new buildings.

Supplemental Damping-Based Design and Retrofitting Procedures

Different procedures for the use of PED devices in the design and retrofitting of building-type structures have been widely discussed in both international design codes and the literature. In this context, based on relevant studies and code-based provisions, procedures are generally classified into four main categories: force-based, displacement-based, energy-based, and supplemental damping-based. The force-based procedure does not directly consider the inelastic behavior of structures during earthquakes; therefore, a realistic estimation of structural damage cannot be achieved. This situation has led to the adoption of a design approach that is directly related to displacement demands [68]. Accordingly, Direct Displacement-Based Design (DDBD) procedures provide an alternative seismic design and retrofitting methodology that is based on displacement capacity corresponding to a specific limit state, rather than strength capacity [69]. Subsequently, DDBD procedures have also been adapted for use in retrofitting strategies involving PED devices [70,71].

The results of various studies indicate that structural damage caused by seismic actions is governed not only by maximum response but also by cumulative inelastic deformations [72,73,74,75,76]. Therefore, design procedures that consider only the maximum lateral displacement may be insufficient to adequately represent the inelastic response of structures. In this regard, energy-based seismic design procedures, which use hysteretic energy as a design parameter and account for inelastic deformations, can be considered a potential alternative to force-based or displacement-based design procedures [77]. Although energy-based procedures are not explicitly included in international design codes, they have been investigated by numerous researchers as part of design and retrofitting strategies for structures equipped with PED devices [4,29,30,78,79].

As stated above, since conventional force and displacement-based procedures are insufficient to adequately represent the inelastic response of structures, performance-based seismic procedures incorporating the supplemental damping ratio as a key parameter are increasingly adopted in practice. In addition, reduction factors corresponding to supplemental damping ratio [80,81,82], associated with the design spectrum, are simplified in the design process in accordance with relevant seismic codes and the literature. Accordingly, the use of the supplemental damping ratio in the design of PED devices [52,56,66] is also recognized as an acceptable procedure. The selection of the supplemental damping ratio at the beginning of the design process simplifies the procedure; however, analyses conducted for values assumed a priori are not able to properly describe the contribution of supplemental damping to the structural response. Therefore, in order to overcome these inadequacies, a supplemental damping-based retrofit design procedure has been proposed [64] and an iterative damper optimization procedure [65]. However, these procedures are based on initial assumptions integrated from equivalent SDOF systems; thus, the interaction between the structure and the damping system cannot be realistically considered.

In the first part of the proposed procedure, instead of equivalent SDOF systems, two-dimensional planar systems are deliberately adopted to allow a transparent evaluation of the structure–damper interaction and its relationship with supplemental damping, while avoiding the complexities associated with three-dimensional systems. Furthermore, using the analysis results of two-dimensional planar systems allows variations in the characteristics of the damping system to be clearly observed. The proposed procedure simultaneously considers the effects of supplemental damping on the seismic performance of the structure and the interaction between the structure and the damping system. As mentioned before, changes in the interaction between the structure and the damping system directly affect the structural behavior and lead to simultaneous variations in multiple response parameters. However, it is generally observed that the initial assumptions are not updated despite such changes in the interaction between the structure and the damping system, which reduces the accuracy of iterative optimization approaches. Accordingly, in the present study, the second part of the proposed procedure, which is a non-iterative approach, is developed, establishing an analytical relationship between the seismic performance of the structure and the governing key parameters. In the following sections of this paper, the term “approach” will be used for the second part of the procedure.

2. Modeling and Analysis Assumptions

This section details the assumptions adopted for analytical evaluations, performed using fluid viscous dampers (FVDs), in order to develop the supplemental damping-based procedure proposed in this study. The proposed procedure is based on deformation-based methods for the evaluation of seismic performance in newly designed buildings and existing buildings to be retrofitted by using PED systems. In particular, in accordance with the provisions of TBEC 2018 Section 5 [83], ASCE/SEI 41-23 [84] and FEMA 356 [85], the assumptions related to deformation-based approaches were adopted in this study for application to the design of new buildings. As mentioned above, planar type (2-Dimensional) frames were considered in this study, as was previously performed in other studies [22,45,54,55,86]. The schematic plan view of the sample building and its investigated frame is illustrated in Figure 1.

In this study, a total of 22 planar reinforced concrete frames with different reinforced concrete member variations were considered. The planar reinforced concrete frames equipped with PED devices and having different building heights (10, 15, 20-story) were analyzed. The distribution of members within the 10-story planar frames is given in

Table 1. Distribution of RC members in 10-story planar reinforced concrete frames.

Frame Configuration ID	Story	Column Type	Beam Type
1	1–10	C1	B6
2	1–10	C1	B7
3	1–10	C2	B6
4	1–10	C2	B7
5	1–10	C3	B6
6	1–10	C3	B7

. The distribution of members of other frames, whose story numbers are 15 and 20, can be seen in

Table A1. Distribution of RC members in 15 and 20-story planar reinforced concrete frames.

Frame ID	Story	Column Type	Beam Type
7	1–10	C2	B6
	11–15	C1	B6
8	1–10	C2	B7
	11–15	C2	B6
9	1–10	C2	B7
	11–15	C2	B7
10	1–10	C3	B6
	11–15	C1	B6
11	1–10	C3	B7
	11–15	C2	B6
12	1–10	C3	B7
	11–15	C2	B7
13	1–10	C4	B6
	11–20	C1	B6
14	1–10	C4	B6
	11–20	C2	B6
15	1–10	C4	B7
	11–20	C1	B6
16	1–10	C4	B7
	11–20	C1	B7
17	1–10	C4	B7
	11–20	C2	B6
18	1–10	C4	B7
	11–20	C2	B7
19	1–10	C5	B8
	11–20	C2	B6
20	1–10	C5	B8
	11–20	C2	B7
21	1–10	C5	B9
	11–20	C3	B6
22	1–10	C5	B9
	11–20	C3	B7

.

All the investigated frames have the same story heights, which is equal to 3.6 m, and 4 spans whose lengths are 6 m. In the analysis of the frames, reinforced concrete members with different cross-sectional dimensions and reinforcement configurations were utilized. Moreover, for the 15 and 20-story frames, larger cross-sectional members were used in the first 10 stories, whereas the member dimensions were gradually reduced in the upper stories. Regarding the damping systems, different types of FVDs, bracing configurations (diagonal and chevron bracing), and arrangements (external and internal) were considered (Figure 2 and Figure 3).

The cross-sectional dimensions and reinforcement details of the reinforced concrete members are presented in

Table 2. Cross-section and reinforcement configuration of the reinforced concrete elements.

Column ID	Reinforcement Configuration	Cross-Section	Beam ID	Reinforcement Configuration	Cross-Section
C1	Longitudinal R. 8ϕ18 Stirrup 2ϕ8/200		B6	Top R. 4ϕ18 Bottom R. 4ϕ18 Stirrup 2ϕ10/200
C2	Longitudinal R. 8ϕ18 Stirrup 2ϕ10/150		B7	Top R. 4ϕ26 Bottom R. 4ϕ26 Stirrup 2ϕ10/200
C3	Longitudinal R. 8ϕ24 Stirrup 3ϕ10/150		B8	Top R. 5ϕ18 Bottom R. 5ϕ18 Stirrup 2ϕ10/200
C4	Longitudinal R. 8ϕ24 Stirrup 3ϕ10/200		B9	Top R. 5ϕ26 Bottom R. 5ϕ26 Stirrup 2ϕ10/200
C5	Longitudinal R. 12ϕ26 Stirrup 4ϕ10/100

. As an example, the column and beam elements in the first 10 stories of Frame Type 6 were arranged as C3 (600 mm × 600 mm) and B7 (400 mm × 600 mm), respectively.

2.1. Modal Analysis

In the analytical models developed in OpenSees [67], rigid diaphragm behavior was assumed at each floor level. Reinforced concrete sections (confined/unconfined concrete and reinforcement) were modeled using a fiber-based approach, in which the concrete and steel components were represented by individual fibers. The concrete material was defined using the “Concrete02” material model, which is based on the Kent and Park model [87] and incorporates the residual strain and stiffness degradation under cyclic compression rules proposed by Karsan and Jirsa [88], as well as tensile behavior. The cyclic stress–strain behavior of the reinforcing steel was defined using the “uniaxialMaterial (Hysteretic)” model [89]. For the design of structural members, concrete class of C20 and C30, and reinforcement steel type of S420 were assumed. These material properties (characteristics) correspond to concrete with cylinder compressive strengths of 20 MPa and 30 MPa, respectively, and reinforcement steel with the yield strength of 420 MPa.

2.2. Nonlinear Time History Analysis

In this study, nonlinear time history analyses (NTHA) were performed using the OpenSees software [67] by using the “integrator Newmark” command, which is based on the Newmark integration method. The OpenSees platform [67] provides a tangent stiffness-based Rayleigh damping model, and this approach is followed. The deformation-based action (${\mathrm{Q}}_{\mathrm{UD}}$) caused by gravity loads and earthquake forces is given below in accordance with Section 5.2.2 of TBEC 2018 [83].

$${\mathrm{Q}}_{\mathrm{UD}}= \mathrm{D}+ \mathrm{L}\mathrm{eff}+ 0.2 \mathrm{S} +{\mathrm{ }\mathrm{E}}_{\mathrm{d}}^{\mathrm{H}}+{0.3 \mathrm{E}}_{\mathrm{d}}^{\mathrm{V}}$$

(1)

where D is the dead load of the structure, including the superimposed dead load, L_eff is the effective live load defined as L_eff = n L (in which n was taken as 0.3), and S is the snow load, ${\mathrm{E}}_{\mathrm{d}}^{\mathrm{H}}$ represents the horizontal seismic effect, ${0.3 \mathrm{E}}_{\mathrm{d}}^{\mathrm{V}}$ represents the vertical seismic effect, taken as 0.2 S_DS.

The mean response quantities obtained from multiple nonlinear time-history analyses are adopted as representative indicators of the seismic performance. This averaging procedure reduces record-to-record variability and mitigates record-specific irregular responses; thus, providing a more stable and reliable representation of structural demand, consistent with the recommendations of TBEC 2018 [83] and ASCE/SEI 41-23 [84] for performance-based seismic evaluation using suites of ground motion records.

2.3. Inelastic Behavior of Frame Elements

Concentrated plasticity is one of the most commonly applied approaches for representing the inelastic behavior of structural members in multi-story buildings [90]. Accordingly, the beams and columns were modeled using concentrated plastic (lumped plastic) hinges in the OpenSees [67]. The inelastic behavior of the structural elements was modeled in OpenSees [67] by employing uniaxial bilinear hysteretic material models. By defining the stress–strain relationships of these material models, the P–M–M interaction curves and the moment–rotation relationships that characterize the inelastic behavior of the elements were obtained by using the aforementioned material models.

The behavior within the plastic hinge regions follows the bidirectional hysteretic response defined by the “Hysteretic” command. This hysteretic model is associated with multi-segment idealized backbone relationships that define the strength and deformation bounds of the structural element. In addition, the model incorporates a set of constitutive behavior rules that characterize the hysteretic response between these defined bounds. These rules define the hysteretic characteristics by accounting for several effects such as pinching, stiffness degradation, and other related mechanisms.

Backbone Relationships and Damage of Structural Elements

According to the TBEC 2018 [83] provisions, the schematic backbone curve representing the plastic hinge behavior (elastic, plastic, and collapse) is presented in Figure 4. The damage states for reinforced concrete members are defined as Limited Damage (LD), Controlled Damage (CD), and Collapse Prevention (CP). In Figure 4, Point (C) represents the collapse state, where the member’s load-carrying capacity is effectively lost.

The section damage limits adopted in this study were evaluated based on the approaches of TBEC 2018 [83] and Eurocode 8-3 [91], which rely on experimental data and semi-empirical capacity relationships. The Limited Damage (LD) defined in Equation (5.3) of TBEC 2018 is given as follows in terms of rotation:

$${\theta }_{\mathrm{y}} = {\displaystyle \frac{{\varphi }_{\mathrm{y}}{\mathrm{L}}_{\mathrm{s}}}{3}}+ 0.0015\mathsf{\eta } \left(1+1.5{\displaystyle \frac{\mathrm{h}}{{\mathrm{L}}_{\mathrm{s}}}}\right)+{\displaystyle \frac{{\varphi }_{\mathrm{y}}{\mathrm{d}}_{\mathrm{b}}{\mathrm{f}}_{\mathrm{y}}}{8\sqrt{{\mathrm{f}}_{\mathrm{c}}}}}$$

(2)

In this formulation, TBEC 2018 specifies that the coefficient η is taken as 1.0 for beams and columns. The cross-section depth is h, the yield curvature is ${\varphi }_{\mathrm{y}}$, the shear span is ${\mathrm{L}}_{\mathrm{s}}$_, and the reinforcement bar diameter is ${\mathrm{d}}_{\mathrm{b}}$. The concrete compressive strength is represented by ${\mathrm{f}}_{\mathrm{c}}$_, and the yield strength of the reinforcement is ${\mathrm{f}}_{\mathrm{y}}$. The maximum allowable plastic rotation for Collapse Prevention (CP), defined in Equation (5.6) of TBEC 2018 [83] and derived from Eurocode 8-3 Equation (A.4), is given as follows:

$${\theta }_{\mathrm{p}}^{\left(\mathrm{CP}\right)} = {\displaystyle \frac{2}{3}}\left[\left({\varphi }_{\mathrm{u}}-{ \varphi }_{\mathrm{y}}\right){\mathrm{L}}_{\mathrm{p}}\left(1-{\displaystyle \frac{{0.5 \mathrm{L}}_{\mathrm{p}}}{{\mathrm{L}}_{\mathrm{s}}}}\right)+ 4.5{ \varphi }_{\mathrm{u}}{\mathrm{d}}_{\mathrm{b}}\right]$$

(3)

In this formulation, the ultimate curvature at the section end is represented by ${\varphi }_{\mathrm{u}}$_, and the plastic hinge length is ${\mathrm{L}}_{\mathrm{p}}$. The final term in the expression corresponds to the reinforcement slip rotation associated with yield penetration in the post-yield range (up to the collapse prevention state).

Adopting an assumption similar to that made in Section A.3.2.3 of Eurocode 8-3, the plastic rotation limit for the Controlled Damage (CD), θ_p^(CD), is assumed as 75% of the plastic rotation limit defined for the Collapse Prevention (CP), in accordance with Equation (5.7b) of TBEC 2018.

2.4. Seismicity of Region

The reinforced concrete frames are assumed to be located in regions corresponding to Site Class B, C, or D according to TBEC 2018 [83] and FEMA 450 [92] classification. For these site classes, the response spectra corresponding to the DD-2 earthquake level (Design-Based Earthquake with a return period of 475 years) were generated in accordance with the provisions of TBEC 2018 [83], considering the seismicity of the region. The seismic region considered for the selection of ground motion records is located along the North Anatolian Fault Zone in Türkiye. Within this context, the region that was most severely affected by the 1999 Kocaeli earthquake was selected. Spectral acceleration coefficients, which characterize the design spectrum, are taken from the Turkey Earthquake Hazard Maps (TDTH) [93] for 10 randomly selected locations situated approximately 30 km from the North Anatolian Fault Zone. Mean value of spectral coefficients and peak ground acceleration of all considered locations are S_s = 1.0, S₁ = 0.3, and PGA = 0.4 g.

Selecting and Scaling Ground Motions

The ground motions used in the nonlinear THA analyses were selected from the 5% damped “flatfile” dataset of the PEER NGA-West2 database [94], considering the following criteria: earthquake magnitude, fault mechanism, source-to-site distance (epicentral distance), and site class. A total of 33 ground motion records were used, covering several values of parameters mentioned above, so that uncertainties due to the nature of the earthquake can be taken into account.

The selecting and scaling method commonly accepted and used in the literature is based on the references [95] and [96], respectively. They proposed a method that determines records with spectral shapes similar to the target spectra by computing the Root-Mean-Square (RMS) difference in their study, published in 2004. Similarly, to satisfy the requirement of matching the target spectra within the relevant period range, the method was revised, and an individual scale factor (α) was determined for each record that minimizes the root-mean-square difference between the scaled geometric-mean spectra of the real record and the target spectra as given in Equation (4). In this way, by scaling the ground-motion record, an appropriate match with the target response spectra can be achieved within the desired period interval.

$${\mathrm{D}}_{\mathrm{RMS}}=\sqrt{{\displaystyle \frac{1}{\mathrm{k}-\mathrm{j}+1}}{\displaystyle \sum _{\mathrm{i}=\mathrm{j}}^{\mathrm{k}}}{\left({\alpha \mathrm{Sa}}_{\mathrm{R}}\left({\mathrm{T}}_{\mathrm{i}}\right)-{\mathrm{Sa}}_{\mathrm{T}}\left({\mathrm{T}}_{\mathrm{i}}\right)\right)}^2}$$

(4)

In the above equation, ${\mathrm{Sa}}_{\mathrm{R}}\left({\mathrm{T}}_{\mathrm{i}}\right)$ is the spectral acceleration of the ground motion record at period ${\mathrm{T}}_{\mathrm{i}}$, while ${\mathrm{Sa}}_{\mathrm{T}}\left({\mathrm{T}}_{\mathrm{i}}\right)$ represents the spectral acceleration of the target spectra at the same period. The parameters ${\mathrm{T}}_{\mathrm{j}}$ and ${\mathrm{T}}_{\mathrm{k}}$ define the period range where the spectral matching is performed.

The scaling procedure does not alter the frequency content of the ground-motion record; rather, the recorded motion is uniformly scaled up or down to catch the best match with the target spectra within the period range of interest. When multiple ground-motion time histories are considered, each record may be scaled individually using the same procedure, or the average of the generated spectra can be best-fitted to the target spectra [97].

In this study, following the provisions of TBEC 2018 [83], the amplitudes of the ground motion components were scaled such that the average of the resultant spectra from all records satisfies the requirement that the ratio of their amplitudes to those of the design spectra within the 0.2 T to 1.5 T period range shall not be less than 1.3. Accordingly, the parameters of T_j and T_k defined above correspond to 0.2 T and 1.5 T, respectively. T shows the fundamental period of the structure. The properties of the selected far-fault ground motions (earthquake name, year, station name, record sequence number, magnitudes, and epicentral distance) and scale factors for Site Class B are presented in

Table 3. Properties of selected ground motion records for Site Class B and scaling factors.

Earthquake Name	Year	Station Name	Record Sequence Number	Magnitude	Mechanism (Fault Style)	Epicentral Distance (km)	Scaling Factor
Tabas, Iran	1978	Tabas	143	7.35	Reverse	55	0.25
Loma Prieta	1989	Gilroy Array #1	765	6.93	Reverse Oblique	29	1.25
Loma Prieta	1989	SF–Pac. Heights	795	6.93	Reverse Oblique	96	1.97
Loma Prieta	1989	SF–Rincon Hill	797	6.93	Reverse Oblique	94	2.19
Northridge-01	1994	Vasquez R. Park	1091	6.69	Reverse	38	1.99
Kobe, Japan	1995	Kobe University	1108	6.90	Strike Slip	25	0.46
Chi-Chi, Taiwan	1999	ILA015	1319	7.62	Reverse Oblique	136	2.87
Chi-Chi, Taiwan	1999	TAP075	1445	7.62	Reverse Oblique	160	2.11
Chi-Chi, Taiwan	1999	TTN042	1587	7.62	Reverse Oblique	105	3.85
Tottori, Japan	2000	OKYH07	3925	6.61	Strike Slip	26	2.76
Tottori, Japan	2000	SMNH10	3954	6.61	Strike Slip	31	1.62

.

Table A2. Properties of selected ground motion records for Site Class C and scaling factors.

Earthquake Name	Year	Station Name	Record Sequence Number	Magnitude	Mechanism (Fault Style)	Epicentral Distance (km)	Scaling Factor
Kern County	1952	Taft Lincoln Sch.	15	7.36	Reverse	43	2.85
Loma Prieta	1989	Fremont–M. SJ	762	6.93	Reverse Oblique	55	3.94
Loma Prieta	1989	Palo Alto–S. L.	787	6.93	Reverse Oblique	51	1.29
Loma Prieta	1989	Saratoga–A. A.	802	6.93	Reverse Oblique	27	1.02
Chi-Chi, Taiwan	1999	CHY029	1198	7.62	Reverse Oblique	40	1.23
Chi-Chi, Taiwan	1999	CHY052	1211	7.62	Reverse Oblique	71	3.45
Chi-Chi, Taiwan	1999	HWA029	1278	7.62	Reverse Oblique	77	2.44
Chuetsu-oki, Japan	2007	Joetsu, A. Dist.	4852	6.80	Reverse	57	4.32
Iwate, Japan	2008	IWT015	5623	6.90	Reverse	34	4.47
Iwate, Japan	2008	Yuzama Y.	5807	6.90	Reverse	37	2.86
Iwate, Japan	2008	Kurihara City	5818	6.90	Reverse	25	0.91

and

Table A3. Properties of selected ground motion records for Site Class D and scaling factors.

Earthquake Name	Year	Station Name	Record Sequence Number	Magnitude	Mechanism (Fault Style)	Epicentral Distance (km)	Scaling Factor
Imperial Valley-06	1979	Brawley Airport	161	6.53	Strike Slip	43	1.87
Imperial Valley-06	1979	El Centro A. #12	175	6.53	Strike Slip	32	2.81
Imperial Valley-06	1979	El Centro D. A.	184	6.53	Strike Slip	27	1.17
Taiwan SMART1(45)	1986	SMART1 O01	577	7.30	Reverse	78	2.04
Loma Prieta	1989	Hollister D.I A.	778	6.93	Reverse Oblique	45	1.47
Northridge-01	1994	Playa Del R.–S.	1057	6.69	Reverse	30	3.21
Kocaeli, Turkey	1999	Hava Alani	1163	7.51	Strike Slip	102	3.68
Hector Mine	1999	Mecca–C. Yard	1810	7.13	Strike Slip	118	3.24
Chi-Chi, Taiwan-06	1999	CHY036	3275	6.30	Reverse	62	2.56
Chuetsu-oki, Japan	2007	Sanjo	4855	6.80	Reverse	32	2.78
El Mayor-C., Mexico	2010	El Centro A. #7	5990	7.20	Strike Slip	63	2.92

give the same information for the Site Class C and Site Class D, respectively. The average response spectra of the scaled ground motion records are illustrated in Figure 5.

2.5. Target Performance

In international seismic design standards, it is required that structures have an acceptably low probability of collapse under the rare ground-motion effects corresponding to the Maximum Considered Earthquake (MCE) level. This performance objective implies that the structure should maintain its vertical stability and avoid global collapse, even if it experiences severe and irreparable damage. Additionally, under the Design Basis Earthquake (DBE) ground-motion level, life-threatening damage arising from the failure of structural components must be prevented [85,98,99].

In this study, the Controlled Damage (structural damage without threatening “Life Safety”) performance level is targeted at the DBE level. This approach is consistent with the criteria specified in TBEC 2018 [83], while allowing controlled inelastic behavior and thereby ensuring structural integrity without imposing unreasonably high design or retrofit demands. Accordingly, this performance objective provides a balanced approach between safety and economic feasibility while providing life safety.

3. FVD Design and Determination of Supplemental Damping Ratio

PED devices have specific working principles, characteristic features, and hysteretic behavior models governed by these characteristics. Considering the commonly adopted force–displacement behavior that represents the hysteretic response of different damper types, this section presents the procedures for estimating the supplemental damping ratio for FVDs (Figure 6). The use of FVD among PED systems has increased due to their ability to enhance seismic performance through energy dissipation and their capability to increase damping without significantly affecting the structural stiffness characteristics. FVDs can be implemented in both new structures and in the retrofit of existing structures. According to [32], the damper force for FVDs, as velocity-dependent energy dissipation devices, can be expressed as follows:

$${{\mathrm{F}}_{\mathrm{FVD}} = \mathrm{C}}_{\mathrm{FVD} }{\left|\dot{\mathrm{u}}\right|}^{\mathrm{a}} \mathrm{sgn}\dot{\left(\mathrm{u}\right)}$$

(5)

Herein, F_FVD is the effective damper force, u is the relative displacement between the two ends of the damper, C_FVD represents the damping coefficient of the device, $\dot{\mathrm{u}}$ is the relative velocity between the damper ends, and a is the exponential constant characterizing the behavior of the damper. For linear FVDs, a is assumed to be 1, whereas for nonlinear FVDs, 0.3 is adopted.

Given that the influence of linear and nonlinear fluid viscous dampers on structural stiffness is negligibly small, the stiffness of the structure is assumed unchanged [20,52]. In this context, the stiffness of linear and nonlinear fluid viscous damper is k_d, and their stiffness when used in combination with bracing members, represented as k_db,L, and k_db,NL, and the subscript “b” indicates the braced configuration of the damper.

In the structural analyses conducted within the scope of this study, the properties of the damping system incorporating nonlinear fluid viscous dampers are based on the Maxwell model of viscoelasticity, which idealizes an exponential viscous damper connected in series with a linear spring, commonly referred to as the Maxwell stiffness.

3.1. Design of Fluid Viscous Dampers

In the general strategy, for buildings that are retrofitted or designed using dampers, the damper characteristics and the corresponding supplemental damping ratios are determined in accordance with the target seismic performance level. However, in this study, appropriate damper properties are also investigated by performing analyses that consider all relevant performance levels. The design assumptions and the characteristics of the dampers used in the planar frames are given below.

In summary, the behavior of the fluid viscous damper depends on the damping coefficient (C_FVD), velocity ($\dot{\mathrm{u}}$) and exponent constant (a). At the initial phase of the damper design methodology, the performance level of the structure is determined without incorporating dampers. The seismic performance analyses are performed using various damping coefficient values for a specific damping force capacity (F_FVD) obtained from the manufacturer’s catalog [100]. Subsequently, the analyses are repeated for damping coefficient values corresponding to different damping force capacities. As a result of these analyses, supplemental damping ratios are calculated. Accordingly, the performance levels for the structure can be determined based on the supplemental damping ratio, provided that the stroke demands obtained from the analyses remain within the limit values.

Table 4. Characteristics of FVDs with a brace.

Rated Force (kN)	Frame ID	Damping Coefficient [kN·(s/m)a]	Stroke Limit (cm)	Brace Member Profile	Stiffness (kdb,NL) (kN/m)
100 *	1 and 2	150	7.6	RHS 120 × 120 × 5	33,095
250	1–12	369	7.6	RHS 150 × 150 × 6.3	58,300
500	3–22	738	10.2	RHS 200 × 200 × 6.3	83,320
750	13–22	981	10.2	RHS 200 × 200 × 10	126,455

* The characteristic values were derived based on the manufacturer’s catalog data.

summarizes the stroke limits, F_FVD, and C_FVD values used in the analyzed models.

In the OpenSees [67], FVDs were modeled using the “ViscousDamper” uniaxial material model to represent their velocity-dependent nonlinear behavior. This material was assigned to a “twoNodeLink” element connecting the diagonal nodes. Accordingly, the deformation across the damper is specified only in the axial direction to accurately simulate the piston motion of the damper. In addition, since the “twoNodeLink” element does not contribute to Rayleigh damping by default, the energy dissipation is governed solely by the FVD parameters.

3.2. Supplemental Damping Ratio Assumptions for FVDs

The supplemental damping coefficient should be defined such that the energy dissipated by the idealized damper is equivalent to the total energy dissipated by all damping mechanisms in the structure. This idealization is referred to as “equivalent viscous damping. The equivalent viscous damping ratio can be obtained by equating the energy dissipated per cycle of a periodic excitation to the corresponding value of viscous damping. When the system response is most sensitive to damping (ω = ω_n), this relationship is given by Chopra [101] as follows:

$${\mathsf{\xi }}_{\mathrm{eq}} = {\displaystyle \frac{{\mathrm{E}}_{\mathrm{D}}}{4\pi {\mathrm{E}}_{\mathrm{S}}}}$$

(6)

where E_D and E_S are the dissipated energy and the strain energy of the system, respectively. In determining the supplemental damping ratio contributed by FVDs to the structural system, the characteristic properties of the device will be employed to perform the necessary mathematical transformations on Equation (6). Accordingly, the energy dissipated by the damping system (E_D) and the strain energy of the structural system (E_S) will be determined using the idealized hysteretic behavior model of the damper.

Table 5. General expressions of the equivalent damping ratio for FVDs.

Passive Energy Dissipation Systems	ξeq,1	ξeq,2
System with Linear Viscous Fluid Damper	${\displaystyle \frac{{\sum }_{\mathrm{j}} \omega \pi {{{\mathrm{C}}_{\mathrm{FVD},\mathrm{j}} \mathrm{u}}_{\mathrm{d}0,\mathrm{j}}}^2 }{4 \pi {\mathrm{E}}_{\mathrm{S}}}}$	${\displaystyle \frac{{\sum }_{\mathrm{j}}2 {\pi }^2{{ {\mathrm{C}}_{\mathrm{FVD},\mathrm{j}} \mathrm{u}}_{\mathrm{d}0,\mathrm{j}}}^2 }{4 \mathrm{T} {\mathrm{E}}_{\mathrm{S}}}}$
Linear Fluid Viscous Damper Bracing System	${\displaystyle \frac{\mathrm{T}{\sum }_{\mathrm{j}}{{\mathrm{N}}_{\mathrm{j}} \mathrm{C}}_{\mathrm{FVD},\mathrm{j}} {{\cos }^2\theta }_{\mathrm{j}}{{ \mathrm{u}}_{\mathrm{db}0,\mathrm{j}}}^2 }{4\pi {\sum }_{\mathrm{i}}{\mathrm{m}}_{\mathrm{i}}{{ \mathrm{u}}_{\mathrm{K},\mathrm{i}}}^2}}$
Nonlinear Fluid Viscous Damper Bracing System	${\displaystyle \frac{{\mathrm{T}}^{2 - \mathrm{a}}{\sum }_{\mathrm{j}}{\mathrm{N}}_{\mathrm{j}} {\mathrm{C}}_{\mathrm{FVD},\mathrm{j}} \mathsf{\lambda } {\cos }^{1+\mathrm{a}} {\theta }_{\mathrm{j}}{{ \mathrm{u}}_{\mathrm{db}0,\mathrm{j}}}^{1+\mathrm{a}} }{{(2\pi )}^{3 - \mathrm{a}} {\mathrm{A}}^{1 - \mathrm{a}} {\sum }_{\mathrm{i}}{\mathrm{m}}_{\mathrm{i}}{{ \mathrm{u}}_{\mathrm{K},\mathrm{i}}}^2}}$

summarizes the general expressions of the equations for the equivalent damping ratio provided by FVDs. The first expression represents the general equivalent damping ratio defined for SDOF systems equipped with FVDs. The second and third expressions define the equivalent damping ratio of the structure for multi-degree-of-freedom (MDOF) systems incorporating linear and nonlinear braced FVD systems, respectively. Based on the nonlinear THA, the data extracted from the force–displacement responses of the dampers used in the structure were examined, and a verification check was performed using the equations provided in Table 5.

Here, in the fundamental mode, the relative displacement between the two ends of the damper is ${ \mathrm{u}}_{\mathrm{d}0}$_, and the relative displacement between the two ends of the braced damper is ${ \mathrm{u}}_{\mathrm{db}0}$. The inclination angle of the braced damper is θ, the effective story mass is m, the story displacement in the fundamental mode is ${ \mathrm{u}}_{\mathrm{K}}$, the natural vibration period is T, and the circular frequency is ω. N represent the number of identical damped bracing members in a story having the same C_FVD value, λ is a parameter computed by the expression $2^{2 + \mathrm{a}}{\displaystyle \frac{{\mathsf{\Gamma }}^2 (1 + \mathrm{a}/2) }{\mathsf{\Gamma }(2 + \mathrm{a})}}$ and A is the modal roof displacement normalized to unit value. In the general expressions, the subscript “j” indicates the j-th damper device, while the subscript “i” represents the i-th story.

For each frame type examined in this study, the equivalent viscous damping ratio was determined by averaging the areas enclosed by the hysteretic loops of all dampers used in the analyses. Specifically, the areas enclosed by the hysteretic loops were calculated using a post-processing algorithm prepared in Python software (version 3.12) [102] using the coordinate data of the hysteretic curves obtained from OpenSees software [67]. The supplemental damping ratio of the structure was obtained by subtracting the assumed 5% inherent damping ratio of reinforced concrete structures from the damping ratio calculated from the analysis results, as explained above. Figure 7 presents an example of the supplemental damping ratio calculation for a damper illustrated in graphical form. The hysteretic loop, the dissipated energy by a damper (E_D), and strain energy (E_S) obtained for Frame Configuration No 1 with external diagonal FVDs under the effect of the Imperial Valley earthquake, 1979.

4. Proposed Design and Retrofitting Methodology

In the design and retrofitting of structures equipped with PED systems (PEDS), several approaches are used either to evaluate structural performance or to design dampers in accordance with a target performance level [26,30,31,64,77]. In these approaches, the results obtained from equivalent single-degree-of-freedom (SDOF) systems are commonly adapted to represent multi-degree-of-freedom (MDOF) systems based on assumptions made for key parameters, such as ductility associated with elastoplastic behavior and the effective natural vibration period, which change according to the stiffness contribution of the damper devices. Although these approaches are internally consistent, the characteristics of the dampers are strongly dependent on the properties of the structural systems. In this context, the structural behavior characteristics of MDOF systems may vary significantly depending on variables such as the contribution of member dimensions or structural configuration to elastic behavior, as well as the role of reinforcement detailing in governing nonlinear behavior. Consequently, they inevitably involve preliminary assumptions in the design process despite their consistency.

The proposed procedure comprises two main components. The first part is developed to establish a relationship between the target seismic performance levels and the supplemental damping ratio (SDR) by explicitly accounting for the interaction between the structure and the damping system, rather than prespecifying damping or structural parameters. The flowchart of the procedure is shown in Figure 8. The procedure incorporates key parameters (device characteristics, configuration and arrangement of devices, structural member dimensions, building height, and reinforcement detailing, as well as Site Class) of dampers and structure, which are effective on the design and optimization of dampers. As shown in Figure 8, in addition to structural parameters, different site classes, PED device types, PED configuration and arrangement, and PED characteristics can be considered through this procedure. The first part of the procedure is shown using gray and blue colors in a flowchart. The last step of the flowchart comprises the performance verification of the structural system with PED devices, as shown in orange in the flowchart.

Consequently, the proposed seismic design and retrofitting procedure can be regarded as an alternative to force, displacement, and energy-based earthquake design procedures, as it explicitly accounts for the effects of supplemental damping on seismic performance and the key parameters governing the interaction between the structure and the damper system.

4.1. Description of the Flowchart

In this section, the proposed supplemental damping-based method for RC planar frame structures equipped with PEDS is summarized. The flowchart illustrating the proposed method for the design of new buildings and the retrofitting of existing buildings is presented in Figure 8.

• Estimation of the supplemental equivalent damping ratio. To simplify the process and improve the efficiency of PEDS optimization, it is recommended that the supplemental damping ratio be selected within a specific range. From an economic perspective, a range consisting of lower values (10–25%) should be selected, whereas for a better performance state (for Limited Damage), a range consisting of higher values (25–40%) is recommended.
• Determination of the site class.
• Identification of the structural system.
• Selection of the PEDD type, PEDD configuration, PEDD arrangement, and PEDD damping constants or coefficients, capacities, etc.
• Perform elastic modal analysis of the structure with PEDS to obtain vibration modes and periods.
• Selection of ground motions. The selected ground motions are scaled to the target design spectra based on seismic hazard levels defined by seismic codes and the fundamental period of the structure.
• Analyze the planar RC frame using (NTHA) or nonlinear static procedure (pushover analysis).
• Calculation of the supplemental damping ratio (see Section 3.1). If the calculated supplemental damping ratio matches the initially estimated value, proceed to the next step. If consistency is not achieved, for existing structures, revise the PEDS properties starting from step 4 and repeat the procedure, or change the initially estimated supplemental damping ratio. For new buildings, in addition to the aforementioned revisions, the structural system type or site class may be changed.
• Verify the seismic performance requirements. If the performance requirements/criteria of the frame system are satisfied, the retrofitting of existing frames or the design of new structures is considered complete. If the seismic performance requirements are not satisfied, increase the initial damping ratio.

4.2. Proposed Graphical Approach

As stated above, in damper optimization approaches, when the number, configuration, or characteristic properties of the dampers are modified during the design process, the initial assumptions must be revised accordingly. This situation leads to changes in the behavior of the structural system with damping, thereby causing multiple parameters to be affected simultaneously. However, in damper optimization studies and engineering practice, initial assumptions are generally not updated despite these changes. Consequently, these optimization approaches, which require iterative analyses, are more or less unrealistic. Therefore, based on the proposed procedure, a non-iterative and practical approach has been developed that analytically establishes the relationship between the supplemental damping ratio and the seismic performance of the structure by utilizing the governing key parameters. For the development of these approaches, 5808 nonlinear time-history analyses were conducted in OpenSees, using 33 scaled ground motion records corresponding to 3 site classes, 22 planar reinforced concrete frames, and 16 fluid viscous damper systems. This approach is used as follows: the frame configuration equipped with a damping system is selected, and the corresponding supplemental damping ratio and target seismic performance level are then directly determined on the graph. Figure 9 shows the proposed graphical approach of 15-story frames for Site Class C.

5. Analysis Results and Discussion

Results of the NTHA, which were performed for 10-story planar frame systems using 33 scaled ground motion records, 11 for each site class, are given in this section. The mean values obtained from these analyses for the base shear, maximum roof displacement, and supplemental damping ratio are presented for different damper configurations. The analysis results for the 15 and 20-story structures are provided in Appendix A.

In Section 5.4, the structural performance results, corresponding to different damping system configurations, are presented separately for each ground-motion record for frames located on the Site Class C and D, since the structures investigated for the Site Class B generally exhibit elastic behavior under design-based earthquakes. In Section 5.5, the graphs illustrating the relationship between the supplemental damping ratio and the structural performance levels for different site classes are presented in accordance with the proposed approach.

5.1. Base Reaction

The base reactions of the planar frame systems with different damping system configurations at the DBE level are shown in Figure 10 and are provided as part of general consistency checks in the analyses. This figure summarizes the results obtained for six different 10-story planar frame configurations (No 1–6). For Frame Configuration No 1 and 2 and No 3–6, which have similar column cross-sectional properties within each group (see Table 2), the base reaction values obtained from the analyses are also similar, as it is expected.

For all 10-story frame configurations, diagonal damper configurations consistently result in higher base shear demands than in chevron configurations, with an increase of approximately 15–30%. This response is primarily due to the direct transformation of interstory drift into axial deformation of the diagonal dampers, which leads to an increase in the lateral stiffness of the structural system and consequently the base shear demand. In the case of using a chevron damper, a considerable fraction of the inter-story displacement is taken up by local bending deformations of the beam and vertical force redistribution of the chevron system. Thus, the transformation of the lateral displacement to axial damper deformation is less direct compared to diagonal configurations, causing a reduction in effective lateral stiffness and base shear demand.

Across all frame configurations, frames located on Site Class B exhibit the lowest base reaction demands, being approximately 15–25% lower than those observed for Site Class C and 25–40% lower than those corresponding to Site Class D, based on the response spectra presented in Figure 5. As an illustrative example, for Frame Configuration No 6 equipped with internal diagonal-type FVDs with a rated force of 500 kN, the base reaction is approximately 1315 kN for Site Class D, compared to about 1170 kN and 880 kN for Site Classes C and B, respectively. Figure A1 and Figure A2 present the corresponding results for the 15 and 20-story frames, respectively.

5.2. Roof Displacement Response and Effectiveness of Damping System

Roof displacement response can provide a basis for evaluating the effectiveness of damping systems. For this purpose, a representative time-history response is first presented to illustrate the influence of FVD. In Figure 11, the roof displacement response corresponds to Frame Configuration No 1 equipped with internal chevron-type FVDs (rated force 250 kN) under the Chuetsu-oki earthquake record (2007) for Site Class C.

The roof displacement of the planar frame systems with various damping system configurations at the DBE level is shown in Figure 12. This figure summarizes the results obtained for six different 10-story planar frame configurations (No 1–6). The main frame and twelve different FVD system configurations are considered to enable a direct comparison of the roof displacement demands and the effectiveness of the damping systems. The results are examined with reference to

Table 6. Roof displacement reduction ratios.

Planar Frame ID	Site Class B	Site Class C	Site Class D
No 1	20–24%	26–35%	14–23%
No 2	23–27%	27–37%	15–26%
No 3	25–28%	27–32%	19–23%
No 4	24–27%	32–36%	22–26%
No 5	25–28%	28–32%	20–24%
No 6	25–28%	33–37%	23–27%

.

The use of viscous fluid dampers results in roof displacement reductions ranging from approximately 14% to 37%, depending on the planar frame configuration and site class. Frame Configuration (FC) No 6 exhibits the highest reduction ratios and roof displacement reductions of approximately 25–28% for Site Class B, 33–37% for Site Class C and 23–27% for Site Class D. The results also reveal a clear dependence on site class, with greater roof displacement reductions achieved for Site Classes C and B compared to Site Class D, reflecting the influence of soil conditions on damper effectiveness. In addition, the consistently higher reduction ratios observed for Frame Configuration No. 6 indicate a more effective interaction between the damper system and the structural response when the primary structural members exhibit ductile characteristics, enabling a more efficient transfer of deformation demand to the dampers.

A more detailed examination reveals that FC No 4 and No 6, which have the same column cross-sectional areas within the 10-story group (see Table 2 for member properties), the base reaction values obtained from the analyses are also similar. However, for FC No 6, when the structural members (see Table 2 for the C3 column and B7 beam properties) exhibit slightly improved ductile behavior, the damper system shows a limited increase in effectiveness, resulting in additional reductions in roof displacement.

In contrast, comparatively lower reduction levels are observed for FC No 1 and FC No 3, particularly for Site Class D, where roof displacement reductions remain within the range of approximately 14–23%, indicating that the damper system is not compatible with the main structural frame.

Overall, for all 10-story planar frame configurations, the largest roof displacement reductions are generally observed for Site Class C, followed by Site Class B, while comparatively lower reductions are obtained for Site Class D. Additionally, frames located on Site Class B exhibit lower roof displacement demands, as expected based on the response spectra presented in Figure 5. The corresponding results for the 15 and 20-story frames are presented in Figure A3 and Figure A4, respectively.

5.3. Supplemental Damping Ratio

The supplemental damping ratios of the planar frame systems with different damping system configurations at the DBE level are shown in Figure 13 and are provided as part of general consistency checks in the analyses. This figure summarizes the results obtained for six different 10-story planar frame configurations (No 1–6).

For planar Frame Configuration (FC) No 1 and 2, slender columns (see Table 2 for C1 column properties) exhibit low lateral stiffness and high displacement demand (Figure 12), which leads to increased damper activation and higher supplemental damping ratios, typically ranging from approximately 20% to 35%, depending on the damper configuration and site class. In contrast, for FC No 3–6, lower displacement demands are observed (Figure 12), which lead to reduced damper activation and lower supplemental damping ratios, generally remaining within the range of approximately 10% to 25%.

Chevron damper configurations consistently provide higher supplemental damping than diagonal damper configurations for all 10-story planar frames, typically by approximately 4–14 percentage ratios, depending on the frame configuration, damper capacity, and site class. This difference can be explained by the fact that chevron layouts develop larger and more symmetric damper deformation (stroke) for a specified story drift, which enables a greater amount of the input energy to be dissipated by the dampers. In contrast, diagonal configurations can lead to an increase in the effective lateral stiffness and attract higher base shear demands, which can reduce deformation compatibility and limit the effectiveness of the damper system.

Across all frame configurations, frames located on Site Class B generally exhibit lower supplemental damping ratios than Site Classes C and D, reflecting reduced displacement demand and damper activation. As an illustrative example, for Frame Configuration No 2 equipped with internal chevron-type FVDs with a rated force of 100 kN, the supplemental damping ratio is approximately 28% for Site Class B, increasing to about 34% for Site Class C, and approximately 38% for Site Class D. Figure A5 and Figure A6 give the same information for 15 and 20-story frames, respectively.

5.4. Performance State

Figure 14 presents a representative performance map derived from Figure 15, in which the planar frame ID (NO) is selected on the horizontal axis and the damping system configuration on the vertical axis, from which the corresponding performance level is identified. The seismic performance results for all planar frames with different damping system configurations, obtained from a total of 22 individual ground-motion records corresponding to two site classes (C and D), are presented in Figure 15.

The green regions (for example, FC No 22 and internal diagonal-type FVDs with a rated force of 750 kN) indicate that the frames satisfy the Limited Damage (LD) performance level criteria, whereas red regions represent Collapse (C) of the structure. In summary, green-dominated regions indicate optimal seismic performance of the damper–structure system, as these regions represent configurations where the target performance objectives are achieved due to effective damper–structure interaction.

When the results for planar frame systems located in the Site Class C and D are compared, it is observed that the damper–structure systems located in the Site Class C generally satisfy the performance level of Controlled Damage (CD). However, performance evaluations in international seismic codes are generally based on the mean responses obtained from analyses using a specified number of ground-motion records, and the related assessment is presented in Section 5.5. The results corresponding to the Site Class B are presented in Figure A7.

5.5. Proposed Approach

Following the proposed approach, NTHA were performed using 11 scaled ground-motion records for each of the Site Class B, C, and D. The mean response values obtained from these analyses were used to determine the supplemental damping ratios and the corresponding seismic performance of the planar frames (Figure 16, Figure 17 and Figure 18).

In the proposed approach, the supplemental damping ratio, which is required to satisfy the specific performance level criteria, can be directly estimated according to the site class and damper configuration of the structure to be designed or retrofitted.

As an illustrative example demonstrating the application of the proposed approach, Figure 16 shows that, the selection of the internal chevron damper configuration with a rated force of 250 kN for Frame Configuration (FC) No 6 results in the Limited Damage (LD) performance level, whereas the same damping system applied to FC No 5 satisfy the Controlled Damage (CD) performance level.

As another example for a 15-story building, it was determined that selecting a system similar to FC No 10 is capable of satisfying the target performance (Controlled Damage-CD) level in Site Class C, as shown in Figure 17. When an optimization assessment is considered for this case, selecting the column type C2 (FC No 11 and 12) instead of C1 (FC No 10) (see Table 2 for member properties) provides better structural performance (Limited Damage) with almost the same supplemental damping ratio. Furthermore, for the FC No 11 and 12, diagonal-braced FVDs with a rated force of 250 kN, either internal or external, provide the same seismic performance with a lower supplemental damping ratio compared to chevron-type dampers (Figure 17).

Alternatively, a systematic optimization assessment can be performed by examining the results obtained from similar configurations. For instance, as it is observed from Figure 18, a structural system similar to FC No 2 does not satisfy the target performance level (CD) in Site Class D. However, FC No 3–6 satisfy the target performance level requirements. Each different damper type and configuration provides an improvement in structural performance with different supplemental damping ratio levels, as shown in Figure 18. A design engineer can select a damper type and configuration among different alternatives that satisfy the target performance level requirements, as shown in Figure 16, Figure 17 and Figure 18.

Furthermore, under design-based earthquakes, the planar frames investigated for Site Class B exhibit elastic behavior and therefore satisfy the Limited Damage (LD) performance level. Accordingly, the required supplemental damping ratios are lower than those for Site Classes C and D.

Higher damper capacity does not always result in increased SDR. As an example, for the 15-story frames located on Site Class B, FC No 9 equipped with internal chevron-type FVDs, increasing the damper-rated force from 250 kN to 500 kN does not result in a higher supplemental damping ratio; instead, the configuration with the 250 kN dampers provides approximately 32% higher supplemental damping than the 500 kN damping system.

For Site Class C, it was observed that the investigated planar frames equipped with all types of damping system configurations satisfied the target performance level (Controlled Damage). Moreover, for some frame–damping system configurations, a higher performance level (Limited Damage) was achieved. Consequently, FC No 6 (10-story frames), FC No 11 and 12 (15-story frames), and FC No 15–22 (20-story frames) were found to be more optimal for application in Site Class C.

For FC No 12 with a diagonal external damper configuration, the system equipped with 500 kN dampers satisfies the Controlled Damage (CD) performance level. In contrast, reducing the damper capacity to 250 kN, while keeping all other damper properties unchanged, enables the structure to satisfy the Limited Damage (LD) performance level, providing approximately 20% higher supplemental damping. This result indicates that a higher damper capacity does not require improved seismic performance.

For Site Class D, only some planar frames equipped with the damping systems satisfied the Controlled Damage Performance Level. Furthermore, none of the sample frames satisfy the requirement of Limited Damage Performance Level (Figure 18). The results indicate that the structural system, supplemental damping ratio, or damper configuration must be revised so that a suitable solution can be found for buildings located on Site Class D, for the desired performance level of Limited Damage. In conclusion, the proposed approach using key governing parameters provides an effective and practical tool by eliminating the need for iteration. For future studies, by defining additional key parameters, the scope of the graphical approaches can be expanded, and a flexible non-iterative approach can be developed for the different target performances.

5.6. Limitations of Proposed Procedure and Approach

The proposed design, retrofitting procedure, and approach are assumed to be suitable for symmetric building systems without structural irregularities. All dampers used in the proposed procedure and approach have identical characteristics. Although this approach is commonly adopted in practice, this assumption may cause some dampers to operate below their capacity or be overdesigned.

6. Conclusions

This study proposes a supplemental damping-based design and retrofitting procedure for reinforced concrete buildings with PEDS, integrating site class into the process. NTHAs were systematically performed on planar reinforced concrete frames with different heights, member properties, damper configurations, and site classes. The results indicate that the SDR is a governing parameter in controlling seismic response, and improvements in structural performance can be provided by selecting the proper damper type and configuration. The following conclusions are drawn from the results of this study:

• As illustrated in Figure 16, Figure 17 and Figure 18, the supplemental damping ratio required to achieve a given performance level exhibits a decreasing trend with increasing building height for all site classes. As seen from Figure 17, for 10-story buildings located on Site Class C, the SDR required to satisfy the Controlled Damage (CD) performance level varies within the range of approximately 10% to 34%, while the required SDR for 20-story buildings reduces approximately 7–14%.
• The influence of adverse soil conditions is most pronounced for Site Class D, where none of the investigated systems can achieve the Limited Damage (LD) level; on the other hand, 9% SDR is sufficient for 20-story buildings located on Site Class B.
• These findings clearly demonstrate that soil class is a governing parameter affecting the required supplemental damping ratio, and that soil conditions should be considered with building height in the design and optimization of damping systems.
• The results suggest that structural systems with improved ductile behavior engage damping devices more effectively, which contributes to increased roof displacement reductions and a more efficient dissipation of seismic deformation demands.
• Damper effectiveness is not always sensitive to damper arrangement. For example, as shown in Figure 16, for FC No 10 located on Site Class B, the diagonal damper configuration with a rated force of 250 kN provides a supplemental damping ratio of approximately 8% for both arrangements.
• Diagonal and chevron dampers with identical capacity provide comparable displacement reductions; chevron configurations satisfy higher SDR, leading to improved structural performance.
• Higher damper capacity does not always result in increased SDR. This indicates that damper effectiveness is governed by the interaction between the damper and the structural system, rather than by damper capacity. Using the proposed approach, which explicitly considers structure–damper interaction, such results can be identified directly and efficiently.
• A higher damper capacity does not always provide a higher structural performance. Using the proposed approach, such cases in which a lower damper capacity results in a more favorable performance level can be identified directly and efficiently without requiring iterative analyses.

Consequently, selecting a damper configuration aligned with the characteristics of the structural system is critical to achieving the target seismic performance. Overall, the proposed supplemental damping-based non-iterative approach enables rapid and reliable identification of target performance states by directly relating the supplemental damping ratio to seismic performance levels. The proposed approach not only reduces the need for iterations in the design or retrofitting of structures equipped with damper systems but also provides an effective tool for identifying optimal system configurations.

Future studies may focus on developing the proposed procedure for irregular buildings. Irregularity-dependent correction factors may be introduced to scale the required SDR. Furthermore, the effects of non-uniform damper capacity distribution along the height of the building on structural response and damping effectiveness may be examined within the scope of future studies. As mentioned earlier, the study is based on two-dimensional structural models. In three-dimensional systems, additional effects such as torsional response may arise. Therefore, the results obtained in this study should be applied to three-dimensional structural systems with due caution. Future research should extend the proposed approach to three-dimensional systems to account for torsional effects.