Aiming to evaluate the performance of different designs of RC framed structures; structural optimization problems were formulated as below:

The design variables comprising of the dimensions of cross-sectional for groups of beams and columns considered for formulating the optimization problem are defined by vector s, F represents the feasible part of the design space where all serviceability and ultimate limit state design requirements implemented as constraint functions (i.e., the series of constraint functions gjSERV(s)andgjULT(s)) are satisfied:

where R^D represents the discrete design set where design vectors s take values. The structural materials cost C_IN of the design is the objective function considered. Aiming to deal with the optimization problem the well-known metaheuristic search algorithm called as particle swarm optimization (PSO) method is implemented.

According to the optimization algorithm particle swarm (Kennedy and Eberhart, 1995), various design vectors collaborate. Every vector of unknowns is labeled as “particle” characterized by its velocity and position, both defined in the D-dimensional design domain, while a sum of particles constitutes the so termed “swarm.” During the search procedure where an optimized design vector is on the hunt, a particle “flies” in the design space of the problem. During the search procedure, velocity and position vectors are adjusted for every particle based on personal “experience” as well as that of the others (neighboring particles). Memorizing and tracking the best positions encountered construct particles' experience. PSO algorithm relies on joint search locally (self-experience) with global one (neighboring experience), trying to control exploitation with exploration. Every particle maintains its two basic characteristic vectors, i.e., velocity and position (or location), in the D-dimensional domain which are iterated as below:

v^j(t) refers to the jth particle's velocity at time t, s_j(t) denotes the jth particle's position vector at time t, the self-best location of jth particle is labeled as s^{Pb, j}, and the best position obtained globally is denoted with the vector s^Gb. Coefficients c₁ and c₂ denoted as acceleration ones, refer to the level of confidence with the best vector achieved by a specific particle (c₁ is labeled as cognitive coefficient) and by entire swarm (c₂ is denoted as social coefficient), respectively. Vectors r₁ and r₂ are composed by arbitrary elements distributed uniformly in [0,1]. Flowchart of Figure 1 describes the basic steps of PSO algorithm, while particle's motion indicatively for the case of the two-dimensional design space is depicted in Figure 2. The lower left dotted circle of Figure 2 represents present position vector s^j(t) at time t, and the upper right dotted bold circle denotes new location vector s^j(t+1) at time t+1. The means that the particle's transition into the D-dimensional search domain is affected by: (i) velocity vector v^j(t); (ii) self-best found location vector (s^{Pb, j}); and (iii) swarm's global-best location achieved so far (s^Gb) is presented in Figure 2.

Several distinctive features for PBD seismic design procedure with respect to design procedures imposed by the prescriptive codes: (i) Permits structural engineers to select both appropriate seismic hazard level and the analogous performance of the structure, (ii) series of performance objectives are used for designing a structural system. PBD design procedure represents a design procedure relying on displacements where design measures and capacity demands are formulated considering displacements rather than using forces (Sullivan et al., 2003).

PBD implies that structural elements are selected, as well as the assessment; construction and maintenance studies are operated on the construction aiming to meet the purposes imposed by owners/users as well as society (Krawinkler and Miranda, 2004). When designing against seismic hazard, the aim is to construct structural systems having predictable performance capable to withstand seismic loading with measurable metrics. Hence, contemporary conceptual approach for structural design is that structural systems need to achieve performance objectives for several seismic levels varying in the rage of seismic event having low intensity and return period, to more damaging earthquake incidents having larger return periods. The contemporary state-of-practice for PBD structural engineering is offered by the various design codes (ASCE-41, 2006; FEMA-445, 2006; ATC-58, 2009; FEMA-P-58-1, 2012), that theoretically do not vary besides adopt techniques that represent the first significant difference with respect to prescriptive-based guidelines. Several of the contemporary design guidelines applied to design studies for new build structures can be considered as merely PBD ones, as these design guidelines combine all measures into a single level of performance, frequently the collapse prevention or life safety level. The definition of performance objectives represents the most important ingredient of a PBD seismic procedure.

The main scope of this step is to incorporate the DIs calibrated in a recent work by the authors into a performance-based design framework and to propose an innovative design concept leading to safe and economic earthquake resistant RC structures. For this purpose, the DIs selected in Mitropoulou et al. (2014) will be used to define different DI-based designs into structural optimization problems. The formulation of the DI-based concept can be stated as follows:

which represent the performance-based design constraints [denoted as g^ULT(s) in Equation 5]. The DI should be less or equal to the allowable upper bounds for a number of hazard levels (HL1, HL2,…,HLn) while i denotes the DI considered in the formulation (i = 1,2,…,m). The boundaries of the feasible part of the design space are defined using the calibrated values of the DIs defined in Mitropoulou et al. (2014) which are given in Table 2. Therefore, in order to identify the DI that represents better the structural behavior, various minimum initial material cost DI-based design optimization problems are formulated according to Equation (5), where the PBD checks are implemented according to Equation (9).

The two 2D RC moment resisting framed (MRF) building structures shown in Figure 3 have been taken into account aiming to study the effect of incorporating DI within the design framework of RC building structures. The foremost test example refers to a three-story concrete building while the second one is a six-story one. Class of concrete C20/25 and of steel S500 are assumed. The thickness of the slabs was taken equal to 15 cm for both test examples and their contribution to the moment of inertia of the beams through effective flange width is taken into account. Due to floor finishing-partitions permanent loading of 2 kN/m² and moving one equal to 1.5 kN/m², are applied further to the self-weight of beams and slabs, where an effective zone 10 × 15 m² is considered for each story. The nominal imposed and permanent loads are combined using the load factors 1.50 and 1.35, respectively.

Columns and beams were simulated adopting the inelastic force-based fiber finite element. The simulation with this type of elements exhibits better accuracy than the plastic hinge formulation (Lagaros, 2014b). The structural analyses required in the framework the study were performed using OpenSEES (McKenna and Fenves, 2009) software. For steel reinforcement pure kinematic hardening bilinear material model was implemented, considering also geometric nonlinearities explicitly. The extended Kent-Park model as modified by Scott et al. (1982) was used for simulating concrete. Despite its relatively simple formulation, the specific model offers for acceptable forecasts of the needs for flexure-dominated RC members. The cyclic inelastic behavior of bars reinforcement was implemented with Menegotto-Pinto model (Menegotto and Pinto, 1973).

During seismic structural design and/or assessment a wide-ranging earthquake events and multiple structural response levels are required for accounting the uncertainties that earthquake hazard introduces into performance-based earthquake engineering (PBEE) assessment or design. Methods implemented for PBEE assessment by means of nonlinear dynamic analyses are classified into single and multiple hazard level ones. Incremental dynamic analysis (IDA) and multi-stripe dynamic analysis (MSDA), their single and multicomponent (Lagaros, 2010) variants, represent the most appropriate ones. The numerical study performed consists of two stages, the optimization and the structural assessment ones, for both multi-stripe dynamic analysis (MSDA) was implemented for calculating the maximum inter-story drift.

Similar to IDA the purpose of MSDA studies is to form the relationship among earthquake intensity level and the correlative maximum structural performance. Earthquake ranking and performance of the structural system are defined through intensity measures (IM) and engineering demand parameters (EDP), correspondingly. MSDA implementation involves multiple inelastic dynamic analyses (stripes) executed in various spectral acceleration stages (see Figure 4 for the case of the final softening and Park and Ang designs). In every stripe various inelastic dynamic analyses are executed for the seismic records considered, all are scaled to specific spectral acceleration. The group of seismic records adopted for deriving each stripe should preferably be representative of the seismic hazard at specific spectral acceleration; nevertheless, although not necessarily always justified (e.g., Jalayer and Cornell, 2009), it is common to implement the same group of records for all spectral acceleration levels.

The problem is expressed mathematically as single-objective minimization problem and is shown below:

subject to the constraints of Equation (9) (Quaranta et al., 2014; Fiore et al., 2016a). The total initial structural material cost C(s) is the objective criterion considered, and C_cl(s), C_b(s), C_sl(s), and C_ns(s) are the total material cost of columns, slabs, beams, and nonstructural members, correspondingly. The ith DI for several hazard levels (HL₁, HL₂, …, HL_n) should remain below the allowable upper bounds (DIHLji(s)≤DIHLjallow,i). The limits of the feasible region are given in Table 2 (Mitropoulou et al., 2014). The PSO algorithm is adopted for dealing with the resulting structural design optimization problems. The boundaries for the maximum interstory drift were obtained by Ghobarah (2004).

The optimized design is computed by applying the PSO method with the following characteristic parameters for both test examples as suggested by the parametric study of Pedersen (2010): Number of particles NP is equal to 100, inertia weight w is equal to −0.6, cognitive and social parameters c₁ is equal to −0.65 and c₂ is equal to 2.65, respectively. Rectangular cross-sectional shape is adopted for both columns and beams, which are separated into different sets. The beams/columns dimensionality together with the longitudinal reinforcing bars represent the design parameters, which are allocated to every set of the beams/columns. The structural members (columns and beams) are divided in two sets for the three-story and three sets for the six-story test examples, resulting into five and seven design variables, respectively.

The optimized designs obtained are labeled as D_{DI = i} associated with the damage index used (i–vi). The optimized designs obtained for different DI considerations with reference to steel and concrete material demands are shown in Tables 3, 4, respectively. According to Tables 3, 4 it looks that D_{DI = ms}, D_{DI = fs}, and D_{DI = CMS} are leading to smaller columns for the three-storys but it does not apply to the six-story where D_{DI = θ} is the one leading to smaller columns. With respect to beams, it looks that D_{DI = θ} and D_{DI = CMS} are leading to smaller beams for the three-stories and D_{DI = θ} and D_{DI = KRL} are leading ones for the six-story. Compared to design D_{DI = θ}, it can be seen that, the initial material construction cost of the other five designs is increased by 23–145% for the three-story and by 26–75% for the six-story test example.

Aiming to evaluate the capacity of the optimized designs achieved by means of the six design PBD procedures described previously, a multi-stripe analyses is performed over a properly selected bin 100 natural records from the list of records provided in Mitropoulou et al. (2015). Seismic records were chosen from PEER (2010) database in agreement with the subsequent characteristics: (i) Records chosen in particular geographical zone (latitude 32–41°, longitude −124° to −115°). (ii) Magnitude (M) to be greater or equal to 5. (iii) Distance from the epicenter (R) is < 150 km. Figures 5, 6 depict the median values of maximum interstory drifts, maximum softening, final softening, Park and Ang damage index, Kunnath et al. damage index, and Chung et al. damage index. These quantities are obtained for three hazard levels namely 50/50, 10/50, and 2/50 (corresponding to probabilities of exceedance equal to 50, 10, and 2% in 50 years, respectively) and for the optimized designs achieved according to the design frameworks discussed above. Indicatively the multi-stripe analysis results for the three-story with reference to the maximum drift for final softening and Park and Ang designs that were used to derive the corresponding for these two designs median curves of Figure 5A are provided in Figure 4. In Figure 5A is represented the structural performance of the three-stories optimized designs (i.e., DI_θmax, DI_ms, DI_fs, DI_PA, DI_KRL, and DI_CMS) for the three hazard levels based only on maximum drift; the different lines are representing the different design.

The measure of the performance of one DI over the others is the structural performance achieved by each optimized design compared to the other ones, i.e., the design that has on average the better structural performance for all DIs. For the three-story test model it is the design obtained according to Park and Ang damage index that shows an overall better performance with reference to the six engineering demand parameters. On the other hand, for the six-story test model it is the design achieved according to Chung et al. damage index that exhibits the best performance, with reference to the six engineering demand parameter. Furthermore, for both test examples the maximum interstory drift appear to increase smoothly based on the abscissa escalation for all optimized designs in consideration of the hazard levels, while the structural behavior of the six designs is nonlinear with reference to the other damage indices. In general, it can be said that, for the two test examples considered, it is the Chung et al. damage index that shows an overall good performance with reference to the performance criteria considered. This is because for both test examples the designs obtained through the design formulation with the Chung et al. DI corresponds to the lower DI values, with reference to all six damage indices considered.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.