Performance-Based Design Optimization of Structures: State-of-the-Art Review

The initial cost optimization in a PBD context focuses on minimizing construction costs without considering potential repair costs over the service life of the building. The initial cost optimization in a deterministic PBD framework can be expressed as follows:where $C$ is the initial cost of structural members; $g_i$ are the deterministic constraints; and $n$ is the number of constraints.

Over the last two decades, numerous studies have developed and implemented PBDO methodologies to identify optimal earthquake-resistant systems with minimal structural cost (e.g., Ganzerli et al. 2000; Hassanzadeh and Gholizadeh 2019). Ganzerli et al. (2000) developed a size optimization methodology for reinforced concrete frames under seismic loading in the context of PBD using a design optimization tool (Vanderplaats 1995). Their research laid the foundation for subsequent studies in PBDO. PBD methodologies were also used to minimize the cost of several different types of structural systems, including reinforced concrete frames (Ganzerli et al. 2000; Lagaros et al. 2010; Zhang and Tian 2019), steel moment frames (Foley et al. 2007; Idels and Lavan 2020; Vaez et al. 2024), steel braced frames (Fathali and Hoseini Vaez 2020; Wang et al. 2020), concrete shear walls (Huang et al. 2015), and self-centering frames (Hassanzadeh and Moradi 2022; Javadinasab Hormozabad and Gutierrez Soto 2021). Recently, PBDO methodologies have been implemented to design unsymmetrical structures in the PBD context (Hoseini Vaez et al. 2022).

The design variables in the initial cost optimization of reinforced concrete frames within a PBD framework can include the cross-sectional area of the concrete sections (Chan and Zou 2004), the area of steel reinforcement, or a combination of both (Ganzerli et al. 2000; Lagaros et al. 2010; Mergos 2017, 2018; Zou and Chan 2005). In Zhang and Tian (2019), the relative flexural stiffness and strength of reinforced concrete sections were taken as design variables. In the optimization of steel frames, the design variables can be categorized into two primary groups: size and topology design variables. Specifically, the cross-sectional dimensions of beams, columns (Fragiadakis et al. 2006a), and braces (Gholizadeh et al. 2022b) fall within the size variables, while the positioning of braces is regarded as a topology variable (Gholizadeh and Ebadijalal 2018; Gholizadeh and Poorhoseini 2016; Wang et al. 2020). Reinforced concrete structures, due to their complex composition and detailing, require a considerably larger number of design variables compared to steel frame structures, which complicates the problem formulation (Mergos 2017).

In recent years, SMA materials have emerged as a promising solution for reducing or eliminating residual deformation in structures following severe earthquakes (Hassanzadeh and Moradi 2022; Leon et al. 2001; Ocel et al. 2004). Despite their potential benefits, one of their drawbacks is the relatively high cost of the material. In Hassanzadeh and Moradi (2022), a topology PBD optimization methodology was applied to reduce the amount of SMA in low-damage steel braced frames. A range of size and topology design variables, including the cross sections of columns and SMA braces, as well as the location of SMA braces, were considered. Optimization methodologies have also been used for the PBD of controlled rocking braced frames (Javadinasab Hormozabad and Gutierrez Soto 2021).

The initial cost optimization in the PBD context considers several constraints, including, among others, practical, strength-related, SCWB (Fragiadakis and Papadrakakis 2008), and confidence levels (Alimoradi et al. 2007; Rojas et al. 2007). Practical constraints are based on the dimensions of the structural members, which are evaluated from a constructability standpoint (Chan and Zou 2004; Ganzerli et al. 2000; Hassanzadeh and Moradi 2022). Strength-related constraints are used to check the demand–capacity ratio for structural elements following conventional seismic design codes [ANSI/AISC 360-16 2016 (AISC 2016); Gholizadeh and Moghadas 2014; Lagaros et al. 2006, 2010)]. PBD constraints in structural optimization can be defined based on engineering demand parameters, such as plastic rotation (Ganzerli et al. 2000; Lagaros et al. 2006), axial deformation of structural members (Hassanzadeh and Gholizadeh 2019; Hassanzadeh and Moradi 2022), and story drifts (Gholizadeh and Poorhoseini 2016; Kaveh et al. 2010; Lagaros et al. 2006; Zou and Chan 2005) at different performance levels.

Optimization procedures such as metaheuristic (Fathali and Hoseini Vaez 2020; Kaveh and Nasrollahi 2014; Liu et al. 2005a; Yaghoobi et al. 2023) and gradient-based algorithms (Fu et al. 2018; Huang et al. 2015; Idels and Lavan 2020; Mergos 2017; Suksuwan and Spence 2018), and the uniform distribution of the deformation technique (Dong et al. 2023), have been implemented to design cost-effective structures. Evolutionary algorithms and gradient-based approaches are two different optimization methods that can be used in PBDO. Evolutionary strategies, such as genetic algorithms (GAs) and evolutionary programming (EP), maintain and iteratively evolve a population of solutions (Lagaros et al. 2002). In contrast, gradient-based approaches, such as gradient descent and quasi-Newton methods, rely on gradient information to iteratively update the solution. Moreover, evolutionary algorithms search for promising regions in a population-based manner, while gradient-based approaches explore the search space in a deterministic, local manner guided by gradient information. As a result, evolutionary algorithms are more suitable for highly complex optimization problems (van de Lindt and Dao 2007). On the other hand, gradient-based approaches are suitable for optimization problems with continuously differentiable objective functions. Compared to gradient-based solutions, metaheuristic algorithms are simpler because they do not require the gradients of constraints (Fragiadakis et al. 2006a). Evolutionary algorithms are powerful in global optimization tasks, providing reliable solutions that are robust to uncertainties, while gradient-based approaches are popular for their computational efficiency (Suksuwan and Spence 2019) and rapid convergence (Salajegheh et al. 2022). In a simple linear optimization problem, the computational efficiency of a gradient-based algorithm was found to surpass that of heuristic algorithms by 56% (Krogh et al. 2017). This disparity is expected to increase significantly in more complex optimization problems, particularly when the search involves performing response history analyses. Moreover, sequential linear programming (SLP) was found to reduce the number of response history analyses (Pollini et al. 2017). Achieving a satisfactory solution using a GA necessitates using a large population, 350 individuals, and conducting five optimization iterations. This translates to performing 350 response history analyses for each of the five optimization iterations. Pollini et al. (2018) converted their initial mixed-integer problem formulation into a continuous one. This conversion enables the nonlinear programming problem to be solved using a gradient-based SLP, which significantly reduces the computational burden.

Metaheuristic algorithms have been used in the topology optimization of steel braced frames in the PBD context with size, strength-related, and PBD constraints (Ebadijalal and Shahrouzi 2022; Eftekhar et al. 2022; Gholizadeh et al. 2022b; Gholizadeh and Ebadijalal 2018; Gholizadeh and Poorhoseini 2016; Karimi and Hoseini Vaez 2019; Kaveh et al. 2022).

Analysis procedures, such as the virtual work principle, pushover analysis, and nonlinear response history analysis (NRHA), have been used to evaluate the performance of optimized structures. While pushover analyses are faster (Chan and Zou 2004; Kaveh et al. 2010), NRHAs are preferred due to their ability to capture the dynamic response of the structure (Foley et al. 2007; Lagaros et al. 2006, 2010). The principle of virtual work has been utilized to assess the elastic response in the optimization process (Chan and Zou 2004). The response prediction accuracy is contingent on the modeling characteristics, such as the inclusion of soil–structure interaction (Fathizadeh et al. 2021; Gholizadeh et al. 2023) and the element and material nonlinearity. Incremental dynamic analyses are needed when assessing the collapse safety of optimal building frames per FEMA P695 (Fattahi and Gholizadeh 2019; Hassanzadeh and Moradi 2022). In Gholizadeh et al. (2022a), the collapse margin ratio was considered as a constraint in the PBDO. In gradient-based optimization methodologies, discrete design variables may need to be replaced by continuous approximations, which enables an analytical assessment of the gradients (Idels and Lavan 2021a; Marzok and Lavan 2022).

Several conflicting criteria can be addressed in PBDO [e.g., for design under earthquake (Grierson et al. 2006; Lagaros and Papadrakakis 2007) and wind (Kleingesinds and Lavan 2022)]. Multiobjective optimization methodologies have been implemented to make a trade-off between different conflicting objective functions. In addition to the initial cost, objective functions have included peak story drift (Ayyash and Hejazi 2023; Grierson et al. 2006; Lagaros et al. 2007), story drift variation (Liu et al. 2013), absolute floor acceleration (Pan et al. 2007), post–elastic ductility demand along the structure’s height (Xu et al. 2006), and ${\mathrm{CO}}_2$ emission (Park et al. 2018).

The response of structures under wind action is different from earthquakes. In Huang et al. (2015), a PBDO methodology was proposed for designing tall buildings subjected to wind loading. The initial cost was considered as the objective function, and the constraints included the elastic and plastic story drift, peak modal and transient acceleration, dynamic serviceability, and the section size of steel and concrete members. Nonlinear pushover analysis has also been implemented to find the structural response in the optimization process. In Fu et al. (2018), the occupant comfort level was assessed based on the modal acceleration response of buildings.

PBD methodologies can consider different sources of uncertainty (Alimoradi et al. 2007; Foley et al. 2007; Rastegaran et al. 2022) that arise from earthquake ground motions, structural geometry, materials, and numerical models (Fragiadakis and Lagaros 2011; Rastegaran et al. 2022). Reliability-based design optimization has been developed to consider unavoidable uncertainties as constraints. A reliability-based design optimization problem can be formulated as follows (Zacharenaki et al. 2013):where $h_j$ is a probabilistic constraint of the $j$th limit-state; ${\nu }_{\mathrm{EDP}}$ and ${\nu }_{\mathrm{EDP}}^{\lim }$ are limit-state probability of exceedance and preset thresholds; $n$ is the number of discrete constraints; and $m$ is the number of probabilistic constraints, respectively.

The limit-state probability of exceedance is calculated based on the total probability theorem as follows (Fragiadakis and Lagaros 2011):where $\nu$ is the limit-state mean annual frequency (MAF); EDP is the engineering demand parameter; $IM$ is the intensity measure; and $P(\mathrm{EDP}>\mathrm{edp}|IM=im)$ represents the probability that an engineering demand parameter will exceed the threshold value associated with the limit state (edp), given a particular intensity measure value ($im$).

In complex numerical models, Monte Carlo Simulation (MCS) can be used to determine probabilistic constraints (Gholizadeh and Aligholizadeh 2019; Gholizadeh and Mohammadi 2017; Lagaros et al. 2008). Moreover, to consider the various sources of uncertainty in the optimization, a framework has been proposed whereby optimally designed frames meet specified performance objectives with a predetermined confidence level following FEMA 350 [Fattahi and Gholizadeh 2019; FEMA-350 2000 (FEMA 2000b); Fragiadakis et al. 2006a]. Previous research studies (Alimoradi et al. 2007; Foley et al. 2007) have presented a framework for minimizing the initial cost while maximizing the structural performance confidence level. Furthermore, predefined confidence level constraints for nonstructural elements have been considered in PBDO following HAZUS (FEMA 2013; Rojas et al. 2007).

Reliability-based design optimization frameworks have also been proposed for structures subjected to wind loads (Spence and Gioffrè 2012; Spence and Kareem 2014). The constraints were the deterministic serviceability drift performance, probabilistic occupant comfort acceleration performance, and geometry checks (Huang et al. 2012). Some studies have represented these constraints in terms of failure probabilities (Deng et al. 2019). In Bobby et al. (2014), steel braced frames subjected to wind loading were optimized considering topology-based design variables. The MCS method was used to define probabilistic PBD constraints. Bobby et al. (2014) have also proposed a methodology to perform multihazard topology optimization considering seismic and wind actions. In another study (Bobby et al. 2016), fragility curves were utilized to define probability constraints.

Previous research on initial cost optimization within the PBD framework has predominantly focused on two-dimensional (2D) frames (e.g., Fathali and Hoseini Vaez 2020; Ganzerli et al. 2000). This limits the relevance of optimization research to structural design practice. This limitation can be addressed by developing practical PBDO software tools (Lagaros 2014).

Metaheuristic algorithms can be implemented to improve the practicality of the optimization methodologies (Gholizadeh and Ebadijalal 2018). However, the use of metaheuristic algorithms in optimization requires computational resources. Recommendations to address this challenge are presented in the “Approaches to Reducing Computational Time” section of this paper. Optimization studies also need to include the life-cycle and repair costs of the structure to broaden the optimization benefits and its applications in practice, as detailed in the next section.

In addition to conventional structural systems, PBDO can be implemented for the design of novel systems, such as low-damage self-centering structures equipped with SMAs. PBDO can be used to promote the application of novel structural systems by developing guidelines for their effective design (Hassanzadeh and Moradi 2022).

Safety is paramount in structural design. Considering collapse safety indices in the optimization process is essential in achieving a trade-off between safety and initial cost (Fattahi and Gholizadeh 2019; Hassanzadeh and Moradi 2022). Further, additional research is needed in the area of multihazard PBDO, particularly considering both wind (Huang et al. 2015; Suksuwan and Spence 2018) and seismic loading. Addressing these needs and challenges will lead to more efficient and effective structural design methodologies that can be implemented in practice.

Reliability-based design optimization problems often rely on MCS techniques to estimate the probability of failure. While this method has proven effective, its computational expense is significant, which limits its practicality (Gholizadeh and Aligholizadeh 2019; Gholizadeh and Mohammadi 2017; Lagaros et al. 2008). The uncertainty in the connection model, mass, and damping ratio should also be considered in probabilistic assessments (Liu et al. 2013). Only a few structural systems have been optimized using reliability-based design principles (Fragiadakis and Papadrakakis 2008; Gholizadeh and Aligholizadeh 2019; Gholizadeh and Mohammadi 2017; Zhang and Foschi 2004). Other structures, such as steel braced frames, steel shear wall systems, and low-damage self-centering systems, can also be designed using reliability-based design optimization.

PBDO can be formulated to minimize the total life-cycle cost over the service life of a building. Life-cycle cost depends on factors such as the building service life, death and injury costs, uncertainty in the structural capacity, and the assumed discount rate (Wen and Kang 2001a, b). To find the most influential parameters in the life-cycle cost of buildings, a full factorial design has been used (Mousazadeh et al. 2020). Fig. 5 presents a simplified illustration of the expected life-cycle cost, which shows the increase in the initial cost of the structure and the reduction in expected repair cost, ultimately leading to designs that minimize the life-cycle cost.

A life-cycle cost optimization methodology was proposed in Wen and Kang (2001a, b) and applied to a 9-story steel office building subjected to earthquake and wind loading. Other researchers (Fragiadakis et al. 2006b; Kaveh et al. 2012; Liu et al. 2003) have utilized the same expected life-cycle cost formulation. The life-cycle cost calculation considering $N$ damage states can be computed as follows (Wen and Kang 2001a, b):where $C_{lcc}^i$ is the $i$th limit-state cost; $P^i$ is the probability of the $i$th damage state being violated; $P({\mathrm{EDP}}_{\max }>{\mathrm{EDP}}_{\max ,i})$ and $\overline{P}({\mathrm{EDP}}_{\max }>{\mathrm{EDP}}_{\max ,i})$ are the general and annual probability of exceeding the relevant engineering demand parameter; $\lambda$ is the annual discount rate; $\nu$ is the annual occurrence rate of significant earthquakes modeled as a Poisson process; $t$ is the service life of the structure; and $\alpha$ and $\beta$ are determined by fitting a curve to known values of $\overline{P}$ for hazard levels and the ${\mathrm{EDP}}_{\max }$ associated with the considered performance level. Table 3 provides the drift-based, $\theta$, damage states that have been used for different types of structural systems (Jiang et al. 2020; Lagaros and Fragiadakis 2011; Liu et al. 2004; Wen and Kang 2001b).

Table 4 summarizes the floor acceleration–based limit-states that have been considered in life-cycle cost calculations (Basim and Estekanchi 2015; Lagaros and Magoula 2013; Shin and Singh 2017). The maximum floor acceleration, $\alpha$, can be used to assess the losses associated with building contents (e.g., furniture and equipment).

To calculate limit-state probabilities, designer-specified confidence levels (CLs) have been utilized (Liu 2005; Lagaros and Fragiadakis 2007) (as shown in Table 5). This methodology can be implemented to consider the uncertainty and randomness per FEMA 350 (FEMA 2000b).

In life-cycle cost optimization problems, both single objective or multiobjective functions can be considered. Several studies have used the summation of initial and service life costs as the total life-cycle cost in the optimization process (Esteva et al. 2002; Matta 2018; Wen and Kang 2001a). In life-cycle cost optimization, considering total cost as the objective function leads to a single optimal design. However, investment decision makers aim to choose the best design from a range of options by considering their personal risk tolerance and the project’s level of importance. This can be achieved by implementing multiobjective optimization techniques (Ghasemof et al. 2021). Some studies (Fragiadakis et al. 2006b; Liu et al. 2003; Mitropoulou et al. 2011) have considered initial and service-life costs as separate objective functions because they are often conflicting objectives.

The seismic performance of the entire building, along with its contents, is notably influenced by the structural system. In an office building, the structural system constitutes 20% of the overall building expenditure (Mayes et al. 2013). Therefore, it is imperative that structural engineers adopt a broader perspective when evaluating seismic performance, considering both structural and nonstructural elements of a building rather than focusing only on structural elements and life-safety considerations. Prior studies have considered nonstructural damage in the life-cycle cost calculation (Rojas et al. 2011; Saadat et al. 2014). Damage to nonstructural elements can be defined using fragility functions (Rojas et al. 2011), which are used in life-cycle cost calculations (Jiang et al. 2020; Shin and Singh 2014b). Some studies calculated the life-cycle cost based on HAZUS (El-Khoury et al. 2018; Ghasemof et al. 2021). This methodology considers five damage states: intact, light, moderate, extensive, and complete.

The consideration of cumulative low-cycle fatigue damage is important in the seismic vulnerability assessment of steel moment frames subjected to major earthquakes. Ghaderi and Gholizadeh (2021) evaluated the mainshock–aftershock low-cycle fatigue damage in optimal steel moment frames. Frames optimized for life-cycle cost were found to be safer against low-cycle fatigue damage compared to those optimized solely for the initial cost.

A few studies (Turchetti et al. 2023) have applied PBDO to the design of bridges. Component-level fragility functions were generated in Wen et al. (2021) to determine the total repair cost in the optimization of damping devices for cable-stayed bridges. In Franchini et al. (2022), parameterized fragility functions were generated and used for sensitivity analysis and optimization of cable-stayed bridges. A probabilistic PBDO was used in Li and Conte (2018) to design seismic isolators in bridges.

In life-cycle cost optimization, metaheuristic (Noureldin et al. 2021; Razavi and Gholizadeh 2021) and gradient-based (Kleingesinds and Lavan 2021) algorithms have been implemented in addition to uniform damage distribution methods (Asadi and Hajirasouliha 2020; Mousazadeh et al. 2020). In Joyner et al. (2022), a post-Pareto pruning approach was proposed to determine a manageable number of final designs in multiobjective optimization. Resilience-based performance targets have also been defined as an objective function (Joyner et al. 2022).

Life-cycle cost optimization has been used to design different types of structural systems, including steel moment frames (Fragiadakis et al. 2006b; Liu et al. 2003, 2005a; Wen and Kang 2001b), reinforced concrete frames (Esteva et al. 2002; Lagaros and Magoula 2013; Möller et al. 2015; Zou et al. 2007a), steel moment frames with fluid viscous dampers (Ahadzadeh Kolour et al. 2021; Gidaris and Taflanidis 2015; Shin and Singh 2014b), steel frames with tuned mass dampers (Matta 2018), steel frames with yielding metallic devices (Shin and Singh 2017), steel braced frames (Park et al. 2015), lead rubber base isolation systems (Mousazadeh et al. 2020), steel frames with steel panel walls (Jiang et al. 2020), and reinforced concrete frames with friction dampers and disc springs (Noureldin et al. 2021). To quantify the benefits of using expensive materials, some studies have implemented a life-cycle cost optimization methodology. Gencturk (2013) optimized the life-cycle cost of conventional reinforced concrete and reinforced engineered cementitious composite frames. Life-cycle cost assessment has also been used to assess and compare optimally designed structures (Gencturk 2013; Lagaros et al. 2010). Life-cycle cost optimization methodologies have also been implemented to design structures with in-plane irregularities (de Salles et al. 2023).

Different types of analyses have been used in life-cycle cost optimization. For this purpose, some studies have utilized conventional pushover analyses (Esteva et al. 2002; Fragiadakis et al. 2006b; Liu 2005; Liu et al. 2003, 2004; Zou et al. 2007a). Modified modal pushover analysis has also been used to capture higher mode effects in total life-cycle cost optimization (Li et al. 2012). In Mitropoulou et al. (2011), NRHAs were implemented in assessing the life-cycle cost of three-dimensional (3D) reinforced concrete buildings. Mitropoulou et al. (2011) studied the impact of analysis type, number of earthquake ground motions, and structure irregularity on life-cycle cost. The response spectrum method has been used to reduce the computational time in performing life-cycle cost optimization (Shin and Singh 2014a, b). The effectiveness of the endurance time method as the analysis engine has also been evaluated (Basim and Estekanchi 2015). The endurance time method is a single response history analysis that evaluates the behavior of the structure at continuous intensity levels by applying intensified accelerograms as loading functions.

Some studies have utilized the FEMA P-58 methodology to determine expected annual economic and social losses (Ahmadie Amiri and Estekanchi 2023; Khalilian et al. 2021; Mirfarhadi et al. 2021; Saadat et al. 2014; Varaee et al. 2021). The contributions of drift-, acceleration-, and velocity-sensitive components to losses were evaluated in Ghasemof et al. (2022a, b). A value-based seismic design methodology was proposed in Mirfarhadi et al. (2021) to minimize the total cost. The seismic consequences, including the repair cost, repair time, injury, and fatality, were evaluated.

Life-cycle cost optimization has also been used for structures subjected to wind actions (Beck et al. 2014; Wen and Kang 2001b). In some studies (Kleingesinds and Lavan 2022), the life-cycle cost of structures was considered as a performance constraint in multihazard PBDO of tuned mass dampers. Some studies (Kleingesinds and Lavan 2021, 2022) assumed that tall buildings subjected to wind loads behave linearly, which means that only nonstructural damage was considered in the life-cycle cost evaluation (Kleingesinds and Lavan 2021, 2022).

The life-cycle cost optimization of buildings with a focus on size variables has been studied (Ghaderi and Gholizadeh 2021; Ghasemof et al. 2022a; Wen and Kang 2001a). However, the life-cycle cost topology-based optimal design of structural systems is yet to be examined.

Life-cycle cost optimization has only been applied to a few types of structural systems (Ahadzadeh Kolour et al. 2021; Möller et al. 2015; Wen and Kang 2001a). Future studies on the life-cycle cost optimization of steel braced frames, steel and reinforced concrete shear wall systems, and self-centering systems are recommended.

High-speed computing resources are needed to solve life-cycle cost optimization problems. Computational expense can be minimized using machine learning techniques, as explained in the “Approaches to Reducing Computational Time” section.

The use of drift and acceleration limits has not been consistent across different life-cycle cost assessment studies (Liu et al. 2004; Wen and Kang 2001a). Therefore, fragility functions have been utilized to calculate the life-cycle cost in the optimization framework (Jiang et al. 2020; Shin and Singh 2014b). The P-58 is suggested as the basis for determining the life-cycle cost in a consistent and reliable manner (Khalilian et al. 2021; Mirfarhadi et al. 2021; Saadat et al. 2014; Varaee et al. 2021). Research on the life-cycle cost optimization of structures under wind loading is limited. Consequently, a significant emphasis in research studies should be directed toward the life-cycle cost optimization of structural systems subjected to wind loads.

There are some limitations in the prior research studies that implemented the FEMA P-58 methodology to determine the life-cycle cost in the optimization (Ghasemof et al. 2022b). The performance metrics of the optimized structures were calculated based on a specific location and occupancy type (Ghasemof et al. 2022b). In addition, the results of these studies are dependent on the quantity and type of vulnerable contents in the structures selected by the authors.

In the last two decades, the Park–Ang damage index (Park et al. 1985) has been used to determine the repair cost of structures following severe earthquakes. The global Park–Ang damage index is defined as follows:where $D_m$, $D_u$, and $D_y$ are, respectively, the maximum roof displacement, the ultimate roof displacement, and the yield roof displacement of the structure; $V_y$ is the yield strength; $\beta$ is a nonnegative strength deteriorating constant; and $\int d{E}_s$ is the absorbed hysteretic energy.

The local Park–Ang damage index can be determined using Eq. (10) (Fattahi and Gholizadeh 2019; Gholizadeh and Fattahi 2018)where ${\theta }_m$, ${\theta }_u$, and ${\theta }_r$ are, respectively, the maximum, ultimate, and recoverable rotation of structural elements; and $M_y$ is the yield moment of the structural element.

In a study by Hajirasouliha et al. (2012), the coefficient of variation of the damage index in reinforced concrete frames was minimized. The frames were considered optimum if the coefficient of variation of the damage index for beams and columns was less than 0.1 (Hajirasouliha et al. 2012).

Following the calculation of the local Park–Ang damage index, the global index is calculated as follows:where ${\mathrm{GDI}}_{PA}$ is the Park–Ang-based global damage index, $ne$ is the number of elements and $E_i$ is the energy absorbed by the $i$th element.

The results from a study by Hajirasouliha et al. (2012) show that by using the Park–Ang index in the optimization, the damage is reduced compared to structures that are designed according to the 2009 International Building Code (IBC) (ICC 2009) (shown in Fig. 6).

Damage states for steel moment frames based on the Park–Ang damage index are given in Table 6.

The repair cost using the Park–Ang damage index is calculated as follows:where $\eta$ is a factor that considers the demolition and clearing costs and $D{I}_{\mathrm{Rep}}$ is the damage index corresponding to the repairability state.

The expected repair cost $C_{eR}$ at time $t$ considers a mean annual rate of earthquake occurrence, $\upsilon$, and a discount rate, $r$, which can be written as follows (Möller et al. 2009, 2015):

The Park–Ang damage index can be incorporated by defining a constraint in the optimization process (Fattahi and Gholizadeh 2019; Gholizadeh and Fattahi 2018). Alternatively, the expected repair cost using the Park–Ang damage index can be defined as an objective function in the optimization process (Fattahi and Gholizadeh 2019).

In the repair cost optimization using the Park–Ang damage index within PBD, probabilistic constraints can also be defined to consider different sources of uncertainty. The reliability index, calculated using MCS, has been considered as a constraint in the optimization (Khatibinia et al. 2013; Yazdani et al. 2017).

As presented in Table 6, different criteria are defined as damage states in using the Park–Ang damage index. This inconsistency in the defined damage states is a major limitation of using the Park–Ang damage index for the cost optimization of structures. Park–Ang damage indices have been implemented for the cost optimization of only reinforced concrete and steel moment frames (Gholizadeh and Fattahi 2018; Hajirasouliha et al. 2012). Park–Ang-based cost assessment is based on system-level damage, which makes it impossible to consider how enhancing the robustness of specific components can be used to reduce losses. Therefore, we recommend using the component-based FEMA P-58 methodology to assess losses.