Fluid viscous dampers (FVDs) are used in the as-built building to enhance its seismic performance, largely due to their ease of installation for structural reinforcement without adding extra stiffness. Therefore, economic efficiency is quantified based on the retrofit costs of installing FVDs. The structure with supplemental FVDs is regarded as a dual system. The resistance force of the dual system is generated from both the lateral load-resisting components and the damper devices. For a given single degree of freedom (SDOF) system with a FVD, the governing equation of motion for the SDOF system can be expressed as: m · u ¨ t + c · u ˙ t + k · u t + F D u ˙ t = − m · x ¨ g t (1) where m, c, and k are the structure’s mass, damping coefficient, and stiffness, respectively; u(t), u ˙ t , and u ¨ t are the displacement, velocity, and acceleration of the structure; x g ¨ t is earthquake ground acceleration, and FD is the resistance force of the FVD, which is velocity dependent.

Generally, the behavior of the FVD is simulated by the Maxwell model, which consists of a spring (with stiffness K _d ) and a dashpot (with damping coefficient C _d ) in series (Singh et al., 2003; Greco and Marano, 2015; Zoccolini et al., 2023). The force-velocity relationship of the dashpot can be expressed by the fractional power law: F D u ˙ t = C d s g n u ˙ t u ˙ t α (2) where sgn(•) is the sign function, α is the velocity exponent, which typically varies between 0.3 and 1.95 (Liu, 2010).

The resistance force of the FVD can be expressed as: F D u ˙ t = K d u t = C d sgn u ˙ t u ˙ t α (3)

Herein, α, K _d , and C _d are the critical factors that control the resistance force of the FVD, which directly influences the cost of the FVD. Some commonly used peak resistance forces of FVDs and their corresponding unit costs are listed in Table 1 (Liu, 2010). The reported costs reflect initial acquisition only and do not account for maintenance over the structure’s service life. The interpolation method is used when the peak resistance damper force falls between the provided values.

Recent studies have revealed that FVDs can effectively improve the seismic performance of both structural and non-structural components. The FEMA P-58 methodology provides procedures for assessing the structure’s seismic performance in terms of repair costs, including the costs related to both structural and non-structural components (FEMA 356, 2000; Brando et al., 2015). Fragility curves and consequence functions have been developed for both types of components. The fragility curve describes the probability that a component exceeds a specific damage state as a function of the EDP, as shown in Figure 2. Commonly used EDPs include pIDR, pFA, peak floor velocity, and residual drift. The probability that the damage (D) of a building component is equal to or exceeds a particular damage state given an EDP value can be expressed as: P D ≥ D S | E D P = Φ 1 β D S · ⁡ ln E D P E D P ¯ D S (4) where DS refers to the damage state (very light, light, moderate or severe), Φ · is the standard normal cumulative distribution function, EDP is assumed to follow a lognormal distribution, E D P ¯ D S and β _DS are the median and logarithmic standard deviation values of the lognormally distributed EDPs.

The FEMA P-58 method generally consists of four analysis steps to assess the repair costs: (1) determine the loss prediction method; (2) assemble the structural model; (3) analyze damage by evaluating the EDPs of the structure; and (4) estimate the total repair cost of the structure. The intensity-based nonlinear performance assessment method is one of the loss prediction methods, which requires the user to specify the earthquake intensity. The other two methods are scenario-based and time-based methods, which require more details about building sites or seismic hazards. In this study, the intensity-based nonlinear performance assessment method is adopted to estimate seismic losses, and the nonlinear time history dynamic analysis is utilized to simulate the structural behavior under the effect of ground motion (Wu, 2014). The obtained EDPs are used with fragility functions to determine the probable damage states of each component. For a component with four damage states, the probability that this component is in each damage state can be calculated as: P D = D S i E D P = P D ≥ D S i E D P − P D ≥ D S i + 1 E D P 1 ≤ i ≤ 3 P D ≥ D S i E D P i = 4 (5)

Once the damage state probabilities are computed, the consequence functions are used to estimate the repair cost of each component. The total repair cost of each component is the sum of the repair costs for that component in each damage state, expressed as: R C C r e p a i r = ∑ i = 1 4 P D = D S i E D P · R C C r e p a i r | D S i (6) where R C C r e p a i r | D S i represents the repair cost of each component under each damage state. R C C r e p a i r represents the total repair cost of each component. The cost includes both direct and indirect costs.

The total repair cost of the structure, R C C r e p a i r _ t o t a l , is the sum of the costs due to the damage of both structural and non-structural components, expressed as: R C C r e p a i r _ t o t a l = R C C r e p a i r _ S S + R C C r e p a i r _ N S (7) where R C C r e p a i r _ S S and R C C r e p a i r _ N S represent the repair costs of structural and non-structural components, respectively. If the total repair cost of the structure exceeds 40% of the replacement cost, many owners choose to replace buildings (FEMA P58-1, 2018).

Regardless of the numerical model’s sophistication and detail, simulating the seismic performance of a structure requires accounting for uncertainties in structural responses. These uncertainties can stem from various sources, including seismic loads, numerical modeling assumptions, simplifications, and variations in input parameters.

Uncertainties can be classified into two main categories: epistemic and aleatoric. Epistemic uncertainty arises due to incomplete information or knowledge gaps, such as unknown material characteristics or model simplifications, and can be reduced with more data or improved models. In contrast, aleatoric uncertainty represents inherent, irreducible randomness, such as variability in material properties and the frequency content of earthquakes. Ground motion variability is typically the largest contributor among the various sources of uncertainty in PBSD (Bracchi et al., 2015; Cocco et al., 2024; Manfredi et al., 2022; Gentile and Galasso, 2021).

The seismic responses, including pIDR and pFA of a structure, are sensitive to earthquake record-to-record variability, commonly referred to as ground motion uncertainty (Deng et al., 2017). A series of ground motion records are typically applied to the numerical model to address MCE in PBSD. The ASCE 7-22 standard specifies that more than three ground motion records should be considered in structural design (Ame rican Society of Civil Engineers, 2022). In this study, the 22 pairs of far-field bi-axial ground motions in FEMA P-695 and the associated modeling approach are utilized to quantify the ground motion uncertainty. This approach accounts for ground motion variability and provides a more comprehensive evaluation of the structure’s seismic performance. Although ground motion uncertainty is irreducible, it may be managed. A robust structural design approach has been proposed to minimize the impact of uncertainties, often referred to as “noise factors” (Kang, 2005). The goal of robust design is to reduce the influence of these noise factors, thereby achieving a reliable and efficient structural design without completely eliminating the uncertainties.

Recently, the robust optimization design concept has been proposed, integrating optimization algorithms with robustness measures. Typically, robust optimization involves minimizing two objective functions: the mean and the standard deviation of EDPs (Doltsinis and Kang, 2004). In this study, the maximum COV of the pIDRs is used as the robustness measure, COV _D : C O V D = max σ i p I D R 1 … 44 μ i p I D R 1 … 44 (8) where pIDR _1…44 is the pIDR under each biaxial ground motion, i is the story number of the structure, μ and σ represent the mean and standard deviation of pIDRs of each story, and COV _D is the maximum COV value. A lower COV _D value indicates greater robustness of the structure.

In this context, the multi-objective robust optimization framework for PBSD aims to identify the optimal sets of design parameters (i.e., α, K _d , and C _d of the FVD) to minimize three metrics of optimization objectives simultaneously: the FVD cost, repair cost ( R C C r e p a i r _ t o t a l ), and robustness measure (COV _D ).

The proposed conceptual framework is applied to a four-story office building designed as a steel moment-resisting frame. The prototype building is a special moment resisting frame with fully restrained reduced beam sections, which is designed to withstand vertical and lateral loads following the Load and Resistance Factor Design (LRFD) specifications and complies with the design provisions from IBC-2003, ASCE 7-02, and AISC-2005. This structure was previously evaluated in (Lignos, 2008) and is assumed to be located in the Los Angeles area, characterized by soil type D and risk category II. The MCE spectral response acceleration at short periods (S_MS) and at 1 s period (S_M1) are assumed to be 1.5g and 0.9g. The DBE spectral response acceleration at short periods (S_DS) and at 1 s period (S_D1) are 1.0g and 0.6g.

A three-dimensional numerical model of the steel frame was developed using OpenSees (McKenna et al., 2000). In this model, all beams and girders are represented as linear elastic elements. In the presented modeling approach, beams are expected to remain elastic under service and design-level seismic demands and are thus modeled as linear elastic elements, whereas columns–being critical for frame stability and susceptible to axial–flexural interaction, P–Δ effects, and buckling with distributed inelasticity–are modeled using the “non-linear beam-column” element that capture both material and geometric nonlinearities in frame members by combining kinematic and equilibrium transformations with flexible modeling of cross-sectional and material behavior (Scott et al., 2008). All structural components are assumed to be made of A992 grade 50 steel. The columns are selected from standard W24 sections. On the first and second floors, columns in the middle two spans use W24 × 131 sections, while the outer spans use W24 × 117. For the upper two floors, all columns are W24 × 76. Beams are W27 × 102 on the first and second floors, and W21 × 93 on the upper floors. The first three dominated natural periods of the steel frame are: 1.258s, 1.039s, 1.022s. To enhance seismic performance, a total of 24 FVDs are symmetrically placed in the exterior frames. The FVDs are arranged on the exterior of the building and do not occupy its internal space. The symmetrical arrangement of FVDs makes the stiffness and weight even as well as the seismic capacity of each story, as illustrated in Figure 4. The layout of the FVDs is determined using the story shear strain energy distribution method that considers both the seismic energy demand and capacity of each story, while also ensuring compatibility with architectural design constraints.

The proposed multi-objective optimization framework is applied to the four-story office building. In this framework, three design variables are considered: the stiffness of the brace and damper portion (K _d ), the velocity exponent (α), and the damping coefficient (C _d ). Among these three parameters, K _d plays a critical role in balancing the structural stiffness and the damping effectiveness. A proper K _d ensures that the dampers can effectively dissipate energy without excessively increasing structural stiffness, which could otherwise lead to undesirable dynamic responses or reduced flexibility during seismic events. Constraining K _d ensures that the damper systems contribute to the lateral load resistance without becoming the sole source of stiffness and strength in the structure. Additionally, FVDs located on the same floor and within the same horizontal direction are assigned identical values for the design variables to maintain consistency in the optimization process.

This study uses the far-field record dataset in FEMA P-695 in the Pacific Earthquake Engineering Research Center (PEER) Next-Generation Attenuation (NGA) database, which includes 22 biaxial ground motions (Applied Technology Council, 2009). These records are selected to capture far-field effects and allow for statistical evaluation of record-to-record variability. For each of the 22 biaxial ground motions, the building model undergoes two non-linear time history analyses, resulting in a total of 44 analyses per hazard level. In the first analysis, the two horizontal ground motions are applied to the two principal horizontal axes of the building model. In the second analysis, the two horizontal ground motions are rotated by 90° and the building model is re-analyzed. This procedure ensures a comprehensive assessment of record-to-record uncertainty across all orientations, totaling 44 analyses per hazard level. The pIDR for each story is recorded for the 44 time history analyses. For each story, the coefficient of variation (COV) of the pIDRs is calculated, and the maximum COV of pIDR is used as the robustness measure in this study. The optimization objectives are integrated into the optimization loop, which aims to minimize three metrics: the total cost of FVDs, repair costs, and the robustness measure. The final output of the optimization process is the Pareto front, which provides a range of optimized designs for decision-makers to evaluate and select.

Moreover, the repair cost is expressed as a percentage of the total replacement cost of the building. This cost includes the repair costs of structural and non-structural components. Non-structural components considered in this estimate include the exterior glass curtain wall, gypsum board partitions with steel studs, suspended ceiling, wall finishing, roof covering, and fire sprinkler system. Removable equipment and furnishings are excluded from this estimation. Table 2 provides critical fragility parameters for both structural and non-structural components, in which M _EDP and β are the median and standard deviation of the EDP for the corresponding damage state. This critical fragility information is referenced and extracted from the Performance Assessment Calculation Tool (PACT) fragility database.

This study utilizes the default consequence functions in the PACT tool, which are based on the 2011 repair costs for structural and non-structural components in Northern California (FEMA P58-2, 2018). For the purpose of this case study, these costs are assumed to apply to the office building located in Southern California, with a cost multiplier of one. The dispersion input is set to zero to ensure deterministic unit cost values. The total replacement cost of the building is estimated at $8,640,000, equating to $200 per square foot.

First, all ground motion records are scaled to the MCE level for the intensity-based performance assessment method, as shown in Figure 5. In the figure, the grey lines represent the scaled response spectrum for the MCE level, the blue line represents the design response spectrum (RS), the dashed red line represents the mean of the scaled Square-Root-of-Sum-of-Squares (SRSS) RS, and the solid red line represents the scaled maximum SRSS RS.

The specific variation ranges for each design variable are predefined as follows:

1. K _d is varied from 0 to 1500 kips/in with an increasing interval of 250 kips/in. This limitation of the stiffness ensures that the brace of the supplemental damper is smaller than the size of the column [7 options].

The NSGA-II is implemented to solve this multi-objective optimization problem with 20 generations and 22 individuals in each generation, in the consideration of the effective trade-off between convergence of the optimization procedure and computational feasibility based on preliminary tests. The obtained optimized designs form a Pareto front, showing the trade-offs among the three optimization objectives, as plotted in Figure 6. The red boxes represent the optimized designs along the Pareto front, while the gray points denote dominated designs from the past 19 generations. The values of the optimized designs along the Pareto front, along with the corresponding values of design variables, are listed in Supplementary Tables SA1, SA2. The Pareto front reveals the trade-offs among the three objectives. Generally, the robustness measure (COV _D ) and repair cost ( R C C r e p a i r _ t o t a l ) are inversely proportional to the FVD cost. The COV _D is also proportional to the R C C r e p a i r _ t o t a l . Here, the dollar amounts for both repair cost and FVD cost are in current dollars (not inflation-adjusted).

Choosing a suitable, cost-effective design is critical for structural designers. Three optimized designs, A, B, and C, are selected from the Pareto front (see Figure 6). Design A is the most expensive design with the minimum repair cost and COV _D value (i.e., the most robust among the three designs). Design B is the design closest to the utopia point (the theoretical point representing the minimum possible value for all objectives simultaneously), representing a design with balanced performance across all three objectives. Design C has the lowest FVD cost but the highest repair cost. The properties of these selected designs are listed in Table 3. While the FVD cost increases from $11,450 for Design C to $46,956 for Design B, the post-MCE-level earthquake repair cost R C C r e p a i r _ t o t a l and COV _D decrease significantly, from 8.00% to 4.33% of the building replacement value, and from 62.43% to 58.38%, respectively. Choosing Design B instead of Design C saves approximately $317,000 in post-earthquake repair costs, with only about $36,000 additional upfront cost in FVDs. In contrast, the FVD cost for Design A ($186,966) is nearly four times that of Design B ($46,956), with the post-earthquake repair cost estimated to reduce by $95,040 ((4.33%–3.23%) × $8,640,000). Moving from Design B to Design A requires an increase of approximately $140,000 upfront investment, with merely $95,040 of potential savings in repair cost post-MCE-level earthquake. A more detailed comparison of the seismic performance of these selected designs will be presented later.

To achieve a more cost-efficient design for earthquake occurrences, the same optimization procedure is applied with all ground motions scaled to the DBE level. The scale factor of the DBE level is 2/3 of the MCE level, and the scaled ground motions are shown in Figure 7.

The variation ranges for K _d , C _d , and α remain consistent with those defined in the MCE section. NSGA-II is employed to solve this multi-objective optimization problem using 20 generations with 22 individuals in each generation.

All the dominated designs are indicated in gray color and drawn in Figure 8. The optimized designs along the Pareto front are highlighted using red boxes. The values of the optimized designs along the Pareto front and the corresponding values of design variables are listed in Supplementary Tables SA3, SA4, respectively.

Similar to the analysis presented in the MCE section, three optimized designs, a, b, and c, are selected from the Pareto front (see Figure 8). Design a is the most expensive design but has the minimum repair cost and COV _D (i.e., most robust with the least variability in pIDRs). Design b is the design closest to the utopia point. Design c has the least FVD cost and the highest repair cost. The properties of these selected designs are listed in Table 4.

As shown in Table 4, when the FVD cost increases from $11,962 (Design c) to $62,462 (Design b), the COV _D decreases from 55.83% to 62.81%, and the R C C r e p a i r _ t o t a l decreases from 3.35% to 2.60%. The FVD cost of Design a is about twice that of Design b; however, both COV _D and R C C r e p a i r _ t o t a l are effectively reduced. The following section will evaluate the seismic performance of each selected design.

To further select the optimal design across multiple hazard levels, the seismic performance of each chosen design is evaluated under both MCE and DBE levels. The median values of pIDR and pFA in the EW and NS directions are plotted in Figures 9, 10. The solid black lines indicate the EDPs of the bare frame (a structure without any damper devices). The solid and dashed red lines depict the EDPs of Design A (MCE level) and Design a (DBE level). The solid and dashed yellow lines represent the EDPs of Design B and Design b, while the solid and dashed blue lines show the EDPs of Design C and Design c.

By analyzing the results, it is evident that Design A achieves the optimal pIDR values compared to the bare frame and the other designs. The relatively weaker stories, particularly the first and third floors, which initially exhibit higher pIDR values, show significant improvement. For the bare frame under MCE conditions, the pIDR values in the EW direction from the 1^st to the 4^th story are 1.10%, 0.88%, 1.60%, and 1.10%, respectively (Figure 9a). After applying Design A, these values are notably reduced to 0.56%, 0.59%, 0.47%, and 0.24%.

However, in the NS direction, Design a achieves better seismic performance in terms of pIDR values. For the bare frame under the MCE hazard level, the pIDR values from the 1^st to the 4^th story in the NS direction are 1.77%, 0.79%, 2.06%, and 0.99%, respectively (Figure 9b). After applying Design a, these values are reduced to 0.80%, 0.74%, 0.67%, and 0.40%. Similar optimization results are observed under the DBE level (see Figures 10a,b).

Furthermore, the pFA values of Design a are 0.35g, 0.58g, 0.65g, and 0.86g (Figure 9c); 0.38g, 0.55g, 0.65g, and 0.72g (Figure 9d); 0.23g, 0.39g, 0.43g, and 0.57g (Figure 10c); and 0.25g, 0.37g, 0.44g, and 0.48g (Figure 10d), which indicated significant reductions in pFA at MCE and DBE levels compared to the bare frame. Notably, Design a achieves smaller pFA values, indicating better seismic performance even with a lower FVD cost than Design A.

After evaluating the median values of pIDR and pFA for the selected designs, Design a is identified as the optimal choice. It offers a balanced combination of moderate FVD cost, repair cost, and robustness. All the recorded values of pIDR for Design a and the bare frame are shown in Figures 11, 12. It can be observed that the seismic performance of the steel frame building has been significantly improved under both the MCE and DBE levels.