Performance‐based optimum seismic design of self‐centering steel moment frames with SMA‐based connections

Shape memory alloys (SMAs) have found several applications in earthquake‐resilient structures. However, because of high material costs, their implementation on industry projects is still limited. Developing design approaches that minimize the use of expensive SMAs is critical to facilitating their widespread adoption in real structures. This paper proposes a performance‐based seismic design optimization procedure for self‐centering steel moment‐resisting frames (SC‐MRFs) with SMA‐bolted endplate connections. The topology optimization uses a metaheuristic algorithm to minimize the frame's total cost, including the initial construction and expected repair costs. The design variables are the steel beam and column sections, SMA connection properties, and the topology of the SMA connections. Different constraints are considered, such as the constructability of the chosen steel sections, member strengths, performance‐based design, Park‐Ang damage index, and strong‐column weak‐beam requirements. Furthermore, the seismic safety of optimal designs is assessed by calculating adjusted collapse margin ratios according to FEMA‐P695. An illustrative optimization study using three‐ and nine‐story SC‐MRFs is presented. The optimal SC‐MRFs are then assessed in terms of cost and seismic performance. The results confirm the effectiveness of the proposed optimum design, which minimizes the use of SMAs while ensuring improved seismic performance. The case studies show that the optimal placement of SMA connections can reduce the total cost by up to 71% and 61% for the three‐ and nine‐story SC‐MRFs, respectively, compared to nonoptimal frames. Moreover, the optimal SC‐MRFs exhibit more uniform drift distributions, lower residual story drifts by up to 96%, and increase collapse capacity by up to 102%.


INTRODUCTION
The 1994 Northridge, California and 1995 Kobe, Japan earthquakes caused beam-column connection fractures in numerous steel moment-resisting frame (MRF) structures.As a viable alternative to fully welded connections, partially restrained connections were later developed. 1Recognizing the need for more earthquake-resilient infrastructure, several researchers have proposed new low damage systems. 2,3mart materials such as shape memory alloys (SMAs) have found many applications in earthquake-resilient structures.During the last two decades, novel SMA connections (shown in Figure 1) have been proposed to minimize residual deformations, repair costs and downtime after a major earthquake. 3,4SMA tendons have been implemented in beamcolumn connections to provide self-centering, i.e., return the structure to its original position after earthquakes.Ocel et al. 5 experimentally tested the application of SMA bolts in endplate steel connections.Past experimental studies (e.g., 6,7 ) have demonstrated the excellent self-centering capability of SMA endplate connections.Parametric sensitivity analyses have also been used to determine the most influential design factors in endplate connections with SMA bolts. 8Surrogate models have also been proposed for predicting the response of these self-centering connections. 9Other types of SMA-based connections, such as those with SMA plates, 10 can be found in the literature.
A performance-based methodology [11][12][13][14] can be used to reliably design self-centering MRFs (SC-MRFs) with SMA connections.While past studies have demonstrated the effectiveness of SC-MRFs with high recentering and moderate energy dissipation (e.g., 3,4,15 ), few have implemented performance-based design methods.Nia et al. 16 proposed a performancebased design methodology for the seismic design of SC-MRFs.The design procedure considers story drifts and ductility as performance objectives.Also, a couple of studies have examined the performance-based design of steel frames with SMA-based braces. 17,18However, optimization methodologies are yet to be implemented in the design of SC-MRFs.Several design variables need to be included, such as the cross-sections of beams and columns, the topology of the SMA connections, and the properties of the SMA bolts.
Optimization methods can be implemented along with performance-based approaches for designing cost-effective structural systems. 191][22][23] Many studies have focused on finding optimal structural cross-sections, but only a few have incorporated connection and structural costs. 24Topology optimization methodologies have been implemented to find the best arrangement of moment connections in MRFs.The configuration of rigid and pin connections in MRFs has also been optimized to reduce structural cost. 25n view of the high cost of SMA materials, it is important to use optimization methods to minimize material usage and the associated cost.However, only a few studies have investigated this issue.Ozbulut et al. 26 used a genetic algorithm to optimize SMA bracing elements in a three-story steel frame.In a recent study, Hassanzadeh and Moradi 18 performed topology optimization of SMA-braced frames.
In evaluating the collapse safety of optimally designed structures, incremental dynamic analysis (IDA) 27 has been utilized. 28Some studies 18,29 have performed IDA to investigate the seismic response of self-centering structures.IDA has also been used to identify severely damaged beams and columns, thereby finding the proper placement of SMA connections. 15I G U R E 1 Extended endplate beam-column connection with SMA bolts.This study presents a performance-based optimization method for the seismic design of high-performance cost-effective SC-MRFs with SMA bolted endplate connections.With the goal of reducing the total cost (including the initial and repair costs), the optimal placement and properties of the SMA connections, as well as the cross-sections of the beams and columns, are determined.Within the optimization routine, numerical models of the SC-MRFs are developed in OpenSees 30 and an artificial neural network-based model 31 is used to predict the moment-rotation response of the steel endplate connections with SMA bolts.To facilitate the search for optimal designs, a computational framework linking MATLAB 32 and OpenSees 30 is implemented.OpenSees is used to perform nonlinear static analyses and IDAs.The optimization algorithm and postprocessing tasks are coded in MATLAB.The proposed performance-based optimum design method is illustrated using prototype three-and nine-story conventional and SC-MRFs.The two types of MRFs are compared in terms of cost, maximum and residual story drift distribution, and seismic collapse safety.

NUMERICAL MODELING OF SC-MRFS
This section describes the numerical modeling of the SC-MRFs and a verification study.A force-based fiber-section "non-linearBeamColumn" element is implemented in OpenSees to model the beams and columns.The "steel01" material is used to represent the nonlinear behavior of the beams and columns.The model captures material nonlinearity in the beam-column elements even though they are expected to remain elastic.The yield strength, modulus of elasticity, shear modulus, and strain hardening for the steel material are 344.7 MPa, 200 GPa, 79.3 GPa, and 3%.The "P-Delta" transformation object is utilized to consider geometric nonlinearity.The supports of the first story columns in conventional and SC-MRFs are considered fixed.The effect of panel zones is not explicitly considered in the numerical modeling. 3,8ccording to the results of a statistical sensitivity analysis, 8 the panel zone effect on the hysteretic response of the endplate connections with SMA bolts is insignificant.This is also confirmed by the verification performed in this study (Figure 3).The constraint imposed by the floor on the gap opening of SMA connections is not considered. 31s shown in Figure 2, the extended endplate connection with SMA bolts is idealized as a rotational spring with a "zeroLength" element.The "zeroLength" element is used to connect nodes at the beam-column joint that have the same coordinates.The hysteresis behavior of the SMA connections is modeled using "SelfCentering," "Pinching4," and "Steel01" materials acting in parallel-following the modeling recommendations in Ref. 31 A verification study is performed to assess the accuracy of the modeling approach relative to experimental results. 6he coefficients for the linear combination of "SelfCentering," "Pinching4," and "Steel01" materials are specified as 0.90, 0.05, and 0.05, respectively.The prediction accuracy of the numerical model is assessed by comparing the simulated moment-rotation response of the SMA connection with the results from physical experiments, including the SMA-D10-240d and SMA-D10-290 specimens from Ref. 6 The properties of the parallel materials in OpenSees are extracted from experimental backbone curves. 6The experimental loading protocol is based on the recommendations of the SAC project. 1,12As demonstrated in Figure 3, the accuracy of the numerical model is reasonable.To facilitate the design optimization, the moment-rotation response of different self-centering beam-column connections with SMA bolts must be estimated.To reduce the computational expense needed for this step, we use an artificial neural network-based predictive model developed by Nia and Moradi, 31 which estimates the moment-rotation backbone response parameters for the SMA connections.As shown in Figure 4, these response parameters include rotation and moment at point B (i.e., θ B, and M B ) when the SMA enters a forward transformation phase, point C (i.e., θ C, and M C ), when the outermost SMA bolts reach their fracture strain, and point E (i.e., θ E, and M E ), when the second-row SMA bolts fracture, as well as a β factor that characterizes the self-centering response.Ten parameters that are known to influence the nonlinear behavior of the SMA connections serve as inputs. 8These factors include the martensite start stress,   , martensite finish stress,   , austenite start stress,   , austenite finish stress,   , maximum transformation strain,   , SMA bolt pretension strain ratio,   , SMA bolt length, L bolt , SMA bolt diameter, D bolt , beam depth, H beam , and beam length, L beam .Table 1 lists the assumed material properties for the NiTi SMA bolts 33 used in this study.E SMA is the modulus of elasticity of SMA.  10  25   In predicting the response of a self-centering beam-column connection assembly, a beam length of 3 m is assumed (i.e., half the bay width in the frame).Other influential parameters, including   , L bolt , D bolt , and H beam are considered as design optimization variables.Table 2 lists the ranges considered for these design variables, which are selected based on practical constraints and the findings from previous studies. 31The beam depth is assigned based on the cross-section that is established during the optimization process.

PROPOSED METHOD FOR PERFORMANCE-BASED DESIGN OPTIMIZATION OF SC-MRFS
A performance-based design optimization methodology is presented for the seismic design of SC-MRFs.Given the presence of several input variables, the optimization algorithm is capable of dealing with the complexity of designing safe and cost-efficient SC-MRFs.The center of mass optimization (CMO) metaheuristic algorithm 34 is employed for this purpose.Metaheuristic algorithms outperform gradient-based methods in structural optimization by offering global search capabilities without the need for derivatives, and a balanced exploration-exploitation trade-off. 35Based on the physical concept of a center of mass, CMO reduces the distance between particles with larger masses to the center of mass and vice versa.In CMO, the first population of n p particles is generated randomly in a design space.The particles have a specified mass that is determined using the objective function with larger masses representing lower objective function values.Then, the particles are sorted in ascending order according to their mass.Subsequently, two groups are created comprising the first and second half of sorted particles.The position of the particles is then updated to decrease or increase the exploration and exploitation rate as the optimization proceeds.Further details of the algorithm formulation are presented in Ref.

Columns Beams
The optimization methodology is also implemented to design conventional MRFs with rigid connections.The MRFs and SC-MRFs have been designed with different acceptable limits for the beam and column rotations and damage index, with the knowledge that different structural systems have distinct behavior.However, these structural elements must remain elastic in SC-MRFs.As a result, the seismic responses of MRFs and SC-MRFs may not be directly comparable.The selected acceptable limits for the rotation of beams and columns and the Park-Ang damage index are discussed in Steps 4 and 6 of the "Proposed Method for Performance-based Design Optimization of SC-MRFs" section.
Figure 5 presents a flowchart for the performance-based design optimization method.There are eight steps that range from defining a database of steel sections to collapse assessment.The optimization process considers different conditions and constraints, including constructability and those related to strength, performance-based design, strong-column weakbeam (SCWB), and Park-Ang damage index constraints.The following sections describe these steps: Step 1: Defining a Database of Steel Sections A database of highly ductile beams and columns is first established.The width-to-thickness ratio of these members must not exceed the limiting values specified in Table D1.1 of AISC 341-16. 36The selected steel sections for structural members in the optimization design of three-and nine-story MRFs are reported in Table 3.The extended endplate connections are specified in accordance with the seismic design requirements, including the beam clear span-to-depth ratio, lateral bracing and section limitations per AISC 358, 37 AISC 341-16, 36 and FEMA 350. 12 The design variable vectors in the topology optimization problem are denoted as follows: where   ,   ,   ,   ,   , and   are the design variable vectors for the column, beam, connection topology, SMA bolt pretension strain, SMA bolt length, and SMA bolt diameter, respectively; , , , and  are the number of columns, beams, self-centering connection topologies, and groups of the SMA connection properties, including the SMA bolt pretension, length, and diameter.The topology design variables in the X CT vector can be either 1 or 0, indicating whether a self-centering SMA-based connection is employed or not, respectively.Figure 6 shows the four possible placements of SMA connections in a single portal frame.

F I G U R E 6
Possible topologies of self-centering connections at a floor level.
Step 2: Constructability Constraints The optimization process checks that the respective dimensions of the frame members at beam-column and columncolumn connections are practical from a constructability standpoint.Four constraints ( , i ) are defined as denoted in Equation ( 2).The first constraint ( 1, i ) checks that the beam flange width (   ) is not greater than the connected column flange width (   ) at each joint .The second constraint ( 2, i ) checks that the flange width of the upper column (

𝑓𝑤,𝑢 𝑖
) is not greater than that of the lower column (

𝑓𝑤,𝑙 𝑖
) of the same joint.Similarly, the third and fourth constraints require that the depth ( ,  ) and the web thickness ( ,  ) of the upper columns be less than or equal to, respectively, the depth ( ,  ) and the web thickness ( ,  ) of the lower column at joint .In Equation ( 2), n j is the number of joints in the frame structure.
Step 3: Strength-related constraints Strength-related constraints are considered for the columns and beams.The SC-MRF is considered a "special moment frame."This reduces the design space and increases the exploration efficiency of the topology optimization.The equivalent lateral force (ELF) procedure is implemented to check the strength-related constraints.The ELF parameters, including the design response spectrum for 5% damping, the response modification factor, and the base and story shear, are determined according with ASCE 7−22. 38he following strength-related constraints are checked for all beams and columns when performing the ELF procedure per AISC 360-16 39 : where   and   are, respectively, the required axial and flexural strengths, which are obtained from linear static analyses in OpenSees;   and   are the available axial and flexural strengths, respectively, which are determined according to chapters E and F in AISC 360-16 39 ; and  se is the total number of beams and columns.
Step 4: Performance-Based Design Considerations The performance-based design constraints,   , include checking the story drifts, the behavior of columns and beams, and the rotation of the self-centering connections.Two performance levels, immediate occupancy (IO) and collapse prevention (CP), are considered according to ASCE 41-17. 14The IO and CP performance levels are evaluated at the seismic hazard levels corresponding to 50% and 2% probability of exceedance in 50 years.A pushover analysis is implemented to assess the nonlinear static response of the frame using a lateral load pattern that corresponds to the first mode shape.The target displacement (  ) for the nonlinear static analysis is determined using Equation (7-28) of ASCE 41-17.
The story drift constraints are defined as follows: where   and  ,  , respectively, are the story drifts and allowable story drift at the IO and CP performance levels (), which are taken as 0.7% and 5%, respectively.Columns are classified as deformation-controlled (DC) or force-controlled (FC) members based on the ratio of P G /P ye, in which P G and P ye , respectively, are the column axial force under gravity loads and the column expected axial yield capacity.If P G /P ye is less than 0.6, the column is considered DC.The rotation constraints for DC columns are expressed as follows: where   , and  , , , respectively, are the maximum absolute and allowable plastic rotations for the kth column at a predefined performance level , and  dc is the total number of DC columns.For conventional MRFs, the acceptable plastic rotations are determined according to Table 9-7.1 of ASCE 41-17. 14In the case of the SC-MRFs, the allowable rotation for columns is taken as being equal to the column yield rotation,   , which was calculated according to ASCE 41-17. 14This constraint ensures that the SC-MRFs columns remain elastic following severe earthquakes.
If the ratio of P G /P ye is greater than or equal to 0.6, the column is considered FC.The following constraints are considered for FC columns: where   , and   , , respectively, are the axial force and bending moment at each performance level for the lth column;  , and  , , are the lower-bound axial yield and moment capacity of the lth column, respectively; and  f c is the total number of FC columns.
In addition to those specific to DC and FC columns, the following constraints are considered for all columns: where   , and   , , respectively, are the axial force and bending moment in the mth column at performance level ;  , is the lower-bound compressive strength of the mth column.For columns in tension,  , is taken as the expected tensile strength,  , ;  , is the lateral-torsional buckling strength of the mth FC column.For the DC column,  , is taken as the expected lateral-torsional buckling strength of the mth DC column ( , ); and  , is the expected axial yield capacity of the mth column.
The following rotation constraint is considered for beams: where   , and  , , , respectively, are the maximum absolute rotation and allowable plastic rotation of the nth beam at performance level ; and  bem is the number of beams.For conventional MRFs, the allowable plastic rotations are determined using Table 9-7.1 of ASCE 41-17. 14The yield rotation of the beam,   , which is calculated according to ASCE 41-17, 14 is used as the allowable rotation for the SC-MRF.This constraint ensures that the SC-MRF beams remain elastic during earthquake shaking.
No residual drift constraints are considered to avoid increasing the computational cost of the topology optimization process.However, a rotational constraint for extended endplate connections with SMA bolts is considered to ensure that the optimal frames maintain their self-centering capability following severe earthquakes.This constraint ensures the continuous functionality of SMA bolts under story drifts induced by MCE level shaking.A rotational constraint for the SMA-based connections is set by considering the backbone response, which is determined using the predictive model developed by Nia and Moradi. 31Equation ( 9) presents the rotational constraint for SMA connections at the IO and CP performance levels.
where   , is the pth connection rotation at performance level ;  , , is the allowable rotation of the SMA connection, determined using the predictive tool in. 31The allowable rotation of SMA connections at the IO and CP performance levels are denoted as θ B and θ C , respectively, as depicted in Figure 4.
Step 5: Strong-Column Weak-Beam Constraint The optimization also incorporates the strong-column weak-beam constraint in the design.This check enhances the seismic response of frame structures by improving the energy dissipation of the system and preventing any weak or soft story failure.According to AISC 341-16, 36 the strong-column weak-beam constraint (  ) is formulated at each joint as follows: where M pb and M pc are, respectively, the plastic bending moment of beams and columns at each joint.
Step 6: Park-Ang Damage Index Evaluation Damage to the SC-MRFs is assessed using the Park-Ang index, 40 which has been used in previous studies (e.g., [41][42][43].A modified version of the Park-Ang damage index is calculated by summing a deformation-related term and cumulative hysteretic energy.Past research 41 has demonstrated the effectiveness of this damage index as a proxy for the seismic damage to steel frames.The damage index is calculated as shown in Equation (11).and where D m , D u , and D y are, respectively, the maximum, ultimate, and the yield roof displacement of the structure; V y is the yield strength;  is a nonnegative strength deteriorating constant, which is taken as 0.025 for steel structures 41 ; and ∫   is the absorbed hysteretic energy.
Based on the damage states in Table 4, acceptable damage indices are defined for conventional and self-centering moment frames.A repairable condition with "moderate" damage is considered acceptable for conventional MRFs.For SC-MRFs, the acceptable damage state is taken as minor or no damage.The optimization constraint for the Park-Ang damage index is defined as follows: where DI ACC is the acceptable Park-Ang damage index.
Step 7: Cost Calculations The design optimization minimizes the total cost (  ) of the SC-MRFs.The total cost is taken as the sum of the initial cost (  ) and expected repair costs due to structural damage (  ).As part of the comparative assessment, conventional MRFs with rigid connections are also optimized.
The C I considers the material costs of the steel beams and columns (  ) and the SMA bolts (  ).Therefore, C I is taken as   +   , which is shown in Equation (13).To simplify the expression, Equation ( 14) normalizes the initial cost by that of the steel material.
where  ′  and  ′  are the costs of steel and SMA material per unit weight, respectively; Assuming that the cost of SMA is 100 times higher than steel, 18,44 C r , the ratio of  ′  to  ′  , is taken as 100.ρ i and ρ j are, respectively, the weight density for steel elements and SMA bolts; A i and A j are, respectively, the cross-sectional areas for the steel sections (including the beams and columns) and the SMA bolts; L i and L j are the length of the steel sections and the SMA bolts, respectively; n ste is the number of columns and beams; and n smb is the number of self-centering connections with SMA bolts.In calculating   in Equation ( 13), the eight corresponds to the number of SMA bolts used in a single self-centering connection.
The repair cost (  ) is determined in the optimization process based on the Park-Ang damage index.The repair cost is defined as follows 45 : where  is a factor that considers the demolition and clearing costs, which is taken as 1.5 43 ; and   = 0.4 is the damage index corresponding to the repairability state.The expected repair cost (  ) at time  considers a mean annual rate of earthquake occurrence and a discount rate, 45 as follows: where the discount rate () and the Poisson coefficient () are taken as 0.05 and 1, respectively.
Step 8: Collapse Assessment In the final step, the seismic collapse safety of the optimal SC-MRFs is assessed by calculating collapse margin ratios (CMR) following the procedure in FEMA P695. 46IDAs are performed using the 22 pairs of far-field ground motions specified in FEMA P695 (reported in Table 5).From the results, IDA curves are extracted with the engineering demand parameter () at various intensity measure () levels.The maximum story drift and the 5% damped spectral acceleration at the fundamental period of the structure, (1, 5%), are taken as the  and , respectively.Collapse of the structure is assumed to occur if the peak story drift exceeds 10%.Subsequently, a cumulative distribution function is fitted to the data to derive fragility curves relating the  to the collapse probability.The CMR is the ratio of the spectral acceleration at which 50% of the earthquake records result in collapse ( 50% ) to the 5% damped spectral acceleration at the maximum considered earthquake MCE level (  ).By applying a spectral shape factor (SSF), an adjusted collapse margin ratio (ACMR) is calculated.The SSF accounts for the spectral shape effect in accordance with Equation (7-1) and Table 7-1 of FEMA P695 based on the fundamental period and period-based ductility of the structure.
The ACMR is compared to an acceptable limit considering different sources of uncertainty, including record-torecord variability, β RTR , and uncertainty in the design requirements, β DR , test data, β TD , and modeling, β MDL .The total collapse uncertainty, β TOT , is taken as the square root sum of squares of the beta values from the individual sources.
The β RTR is computed using Equation (7-2) of FEMA P695.The β DR is taken as 0.1, assuming that the design requirements are comprehensive and provide safety against unexpected failure modes.Assuming "good" rating, β TD and β MDL are both taken as 0.2.The β TOT is determined using Equation (7-5) in FEMA P695.The ACMR values of the optimal designs are compared to an acceptable limit,  20% , which is associated with a 20% probability of collapse at the MCE hazard level. 20% values are taken from Table 7-3 of FEMA P695.

ILLUSTRATIVE OPTIMIZATION STUDY
This section illustrates the proposed design optimization procedure using prototype three-and nine-story steel moment frames.The optimization seeks to minimize the total cost of the frames.The design optimization is implemented for both the SC-MRFs and conventional MRFs.The optimization formulation is summarized below with constructability (  ), strength (  ), performance-based design (  ), Park-Ang damage index (  ), and strong-column weak-beam (  ) constraints.The objective function is described as follows:

Find optimal design variable vectors: 𝑿
To minimize ∶   () =   +   (17) The design variables in the optimization process are the placement of the SMA connections, the cross-section of beams and columns, and properties of the SMA bolts.Figure 7 shows the grouping of beams and columns in the three-and ninestory SC-MRFs.The grouping of the SMA connection properties is similar to that of the beams.As shown in Figure 7A, for the three-story frame, there are four groups of column section sizes 1, 2, 3, and 4 and three groups of beam section sizes 1, 2, and 3.Likewise, there are three groups of SMA connection properties consistent with the beams, including SMA bolt prestraining   ,  #2  , and  #3  , and SMA bolt diameter  #1  ,  #2  , and  #3  .In the search for optimal placement of the self-centering connections, challenges were encountered due to the large size of the design space-with many possibilities for selecting steel beam and column sections and the location and properties of SMA connections.To facilitate finding feasible designs and reduce the computational time, the prestraining of SMA bolts for the nine-story SC-MRFs is fixed at 0.008, which is equal to the SMA transformation start strain. 47,48The optimization of the nine-story SC-MRFs considers all combinations of SMA bolt lengths, including 190, 270, and 350 mm, and diameters, including 12, 16, 20, and 24 mm, to reduce the search space.This enhances the exploration efficiency of the topology optimization for the nine-story frames.
A set of self-centering frames are also designed and analyzed without the optimization procedure (i.e., "nonoptimized" SC-MRFs, Nop-SC) to investigate how the optimal placement of SMA connections influenced the total cost, stiffness, residual deformation, and collapse safety of the steel moment frames.The nonoptimal self-centering frames adhere to the same requirements as the optimized frames except for the Park-Ang damage index (as using this index is not the standard of practice in structural design).
(A) (B) F I G U R E 7 Structural element grouping details for (A) three-and (B) nine-story SC-MRFs.

Optimization results for the three-story frames
Using the design optimization procedure, four optimal three-story SC-MRFs are found, which are named Op-SC3st#1, Op-SC3st#2, Op-SC3st#3, and Op-SC3st#4.In the frame nomenclature, the digit following # is the optimization rank of the frame, with #1 being the most optimal (i.e., with the least total cost).Two reference frames are designed and analyzed, including an optimal three-story MRF with rigid connections (denoted by Op-MRF3st) and a nonoptimal three-story SC-MRF with SMA-bolted endplate connections in all its beam-column joints (denoted by Nop-SC3st).
The optimal placement of the self-centering connections with SMA bolts is depicted in Figure 8(A-D).Figure 8E and F show the nonoptimal self-centering frame with SMA connections (Nop-SC3st) and the optimal moment frame Op-MRF3st.The optimization procedure lowers the SMA material usage in the SC-MRFs while exhibiting improved seismic performance.The Nop-SC3st frame (as shown in Figure 8E) has 18 SMA connections, whereas the optimal frames, shown in Figure 8(A-D), require only two or four SMA connections.
Table 6 summarizes the design and initial   , expected repair   , and total   cost of the three-story frame structures.The design includes the steel sections for the columns and beams, and the SMA bolts prestraining  .
, length   , and diameter   .On average, the initial, expected repair, and total costs of the three-story optimal SC-MRFs are 15%, 97%, and 69% less compared to the cost of the Nop-SC3st frame (i.e., the nonoptimized SC-MRF with SMA used in all its connections).On average, the initial cost of the SC-MRFs with SMA bolts is 29% higher compared to the conventional MRF.However, by using SMA, the average repair and total cost of the self-centering frames are 92% and 30% lower, respectively.The higher total cost of the nonoptimal frames shows that the strength-related and performance-based design checks do not necessarily lead to better performance in terms of residual drift and total cost.Using the optimization procedure is the key to improving the performance of SC-MRFs.
Self-centering connections with SMA bolts

TA B L E 6
The design and cost of the three-story SC-MRFs and MRF.By incorporating the Park-Ang damage index as a constraint, it is evident that the optimal three-story SC-MRFs exhibit higher stiffness compared to their nonoptimal counterparts.The integration of the Park-Ang damage index constraint into the design process has led to notable benefits in the three-story frames, including higher lateral stiffness, lower rotational demands on the connections, and reduced repair and total costs for optimal SC-MRFs compared to their nonoptimal counterparts.As depicted in Figure 9, the ratio of demand to yield rotation (referred to as DR-to-YR) is lower in the optimal three-story SC-frames.
Figure 10 shows the story drift profile for the three-story frames at the IO and CP performance levels, as well as the average residual story drift under the MCE level shaking intensity.The optimized self-centering frames (Op-SC3st#1 to Op-SC3st#4) experience lower and more uniform drifts.As shown in Figure 10, the story drifts for the optimal and nonoptimal three-story SC-MRFs, and the optimal three-story MRFs are less than the permissible values at the IO and CP performance levels.Apart from the impact of the performance-based design checks, the improved lateral response can be attributed to the fact that the optimal frames with fewer SMA connections are stiffer than the Nop-SC3st and Op-MRF3st frames.The ductility demands of the SMA connections for the optimal three-story frames are presented in Figure 11.The average ductility demand for the optimal three-story frames is 3.94.

F I G U R E 1 1
Ductility demand for the SMA connections in the optimal three-story frames.

F I G U R E 1 2
The response of SMA connections in Op-SC3st#2.
From the pushover analyses with target displacements corresponding to the IO and CP performance levels, it is observed that, on average, for the group of optimal SC-MRFs, the peak story drifts are 41% and 46% less than those for the Nop-SC3st frame (which is not optimized but has SMA bolts in all its beam-column connections).Moreover, nonlinear response history analysis results show that the maximum residual drift of the optimal SC-MRFs under the MCE level shaking is, on average, 93% lower due to their self-centering capability.A representative plot of the self-centering behavior of the SMA connections in Op-SC3st#2 is shown in Figure 12.The fundamental period of the optimal three-story SC-MRFs is, on average, 25% smaller for Nop-SC3st.
The optimal self-centering frames (Op-SC3st#1 to Op-SC3st#4) outperform the optimal conventional MRF.On average, the maximum drift for the three-story optimal self-centering frames at the IO and CP performance levels is 27% and 24% lower for the Op-MRF3st.Furthermore, due to the influence of the optimally placed SMA connections, the maximum residual drift for the three-story optimal self-centering frames is, on average, 90% less than that of the optimal MRF.In addition, incorporating the Park-Ang damage index constraints in the optimization produced better residual deformation and seismic safety performance in the three-story SC-MRFs.
In the three-story frames, the demand-to-capacity ratios (DCRs) for moment-axial load interaction of the first story columns at the CP performance level is less than 1 (as shown in Figure 13).
The collapse assessment results are presented in Table 7.The ACMR values are higher than the acceptable limit.This indicates that the presented performance-based design ensures considerable seismic collapse safety of the structure.The higher CMR values for the three-story Op-SC-MRFs are attributed to their higher stiffness and strength.For the same F I G U R E 1 3 Demand-to-capacity ratios for moment-axial load interaction in the first story columns of the three-story frames.

TA B L E 7
Collapse assessment results for the three-story SC-MRFs and MRF.reason, Op-SC3st#4 provides the highest CMR value among the three-story Op-SC-MRFs.Moreover, the ACMR value for the Op-SC3st frame is, on average, 89% and 17% higher than Nop-SC3st and Op-MRF3st, respectively.The results also show that the Nop-SC3st frame is more ductile than the Op-SC3st frame due to SMA being used in all connections and having smaller beams and columns in the Nop-SC3st frame.This is also reflected in the higher SSF in Table 7 for the Nop-SC3st frame. 46igure 14 summarizes the optimization and response assessment results.The initial, expected repair, and total costs for the moment frames are compared in Figure 14A.As was anticipated, the initial cost of the optimal self-centering Op-SC3st frames is higher than that of the Opt-MRF3st frame due to the adopted damage reduction design strategy and the additional upfront cost of SMA materials.However, the repair and total cost for these optimal self-centering frames are lower because of their recentering capability and optimal placement of the SMA connections.Additionally, the Op-SC3st frames exhibit lower drifts (shown in Figure 14B) and greater collapse resistance (shown in Figure 14C).The most optimal frame, Op-SC3st#1, has a high level of collapse safety and the lowest total cost.

Optimization results for the nine-story frames
This section presents the results of the performance-based design optimization for the nine-story frames.Four optimal self-centering frames (Op-SC9st#1, Op-SC9st#2, Op-SC9st#3, and Op-SC9st#4) are shown in Figure 15(A-D).Like the three-story cases, two additional frames, including an optimized MRF with rigid connections (Op-MRF9st) and a nonoptimized SC-MRF with SMA bolts at all beam-column connections (Nop-SC9st), are designed and analyzed.These frames are shown in Figure 15E and F. From Figure 15(A-D), on average, it can be seen that the topology optimization reduces the initial cost of the selfcentering frames by 20% compared to Nop-SC9st.Compared to the 54 SMA connections in the Nop-SC9st frame, the optimal self-centering frames require 6-16 SMA connections while exhibiting better seismic performance.Table 8 summarizes the design and optimization results for the nine-story frames.On average, the expected repair and total cost for the optimal nine-story SC-MRFs are, respectively, 85%, and 57% lower than that of Nop-SC9st.On average, compared to Op-MRF9st, the optimal self-centering frame has a 42% higher initial cost but 71% and 18% lower repair and total costs, respectively.The lower repair and total cost of the optimal SC-MRFs compared to the nonoptimal frame is attributed to the Park-Ang damage index constraint, which leads to stiffer optimal SC-MRFs.As illustrated in Figure 16, the DR-to-YR ratio for the optimal nine-story SC-MRFs is lower.
Figure 17 compares the maximum and residual drift distribution of the nine-story frames.Like the results for the threestory frames, the optimal frames with fewer SMA connections (Op-SC9st frames) are laterally stiffer and thus exhibit lower drifts-compared to the Nop-SC9st frame with SMA bolts in all its beam-column connections.The story drifts for the optimal and nonoptimal nine-story frames are less than the allowable values.The fundamental period of the optimal nine-story SC-MRFs is, on average, 33% smaller than that of Nop-SC9st.Figure 18 shows the ductility demand for SMA connections in the nine-story frames, with an average ductility demand of 4.47.
From pushover analyses with target displacements corresponding to the IO and CP performance levels, it is shown that, on average, the peak story drifts for the optimal self-centering frames are, respectively, 29% and 37% less compared to Nop-SC9st.In addition, nonlinear response history analysis results show that the maximum residual drifts of the optimal SC-MRFs under the MCE level shaking is, on average, 16% lower.The story drift distribution along the height of the Op-SC9st frames is also more uniform compared to the Nop-SC9st frame.Moreover, the average maximum residual drift for the Op-SC9st frames is 15% less than that of the Opt-MRF9st due to their self-centering capability.Similar to the results for the three-story frames, the optimal nine-story frames experience less residual story drifts compared to the nonoptimal designs because the Park-Ang damage index constraint was considered in the optimization.Figure 19 shows a representative plot for the self-centering behavior of the SMA connections in Op-SC9st#1.
As depicted in Figure 20, the DCRs for the moment-axial load interaction in the first story columns of the nine-story frames at the CP performance level is less than 1.
The results of the collapse assessment of the nine-story frames are provided in Table 9.All the designed frames possess significant collapse resistance.Since the optimal self-centering frames (Op-SC9st) were designed for less damage (minor F I G U R E 1 Ductility demand for the SMA connections in the optimal nine-story frames.

F I G U R E 1
The response of SMA connections in Op-SC9st#1.

F I G U R E 2
Demand-to-capacity ratios for moment-axial load interaction in the first story columns of the nine-story frames.or no damage), they possess higher stiffness and strength and thus have greater CMR values.The Op-SC9st frames are laterally stiffer than Nop-SC9st due to their fewer SMA connections and larger size steel beams and columns.On average, the ACMR for the Op-SC9st frames is 92% higher compared to the Nop-SC9st frame.Moreover, the Op-SC9st frames possess higher stiffness and strength than the Op-MRF9st frame due to their design for lower damage, which produced larger size steel beams and columns.On average, the median collapse capacity for the Op-SC9st frames is 6% higher compared to Op-MRF9st.The average of the median   ( 1 , 5%) for the optimal self-centering frames is 4.41.Moreover, the median collapse capacity for Op-SC9st#1 with the least total cost is 4.29.The cost, drift response, and collapse capacity of the nine-story frames are compared in Figure 21.The optimal selfcentering frames have higher initial costs but lower repair and total costs while offering better seismic performance in terms of reduced drift demands (Figure 21B) and increased collapse safety (Figure 21C).Similar to the results for the three-story frames, the nonoptimized self-centering moment frame (Nop-SC9st) requires the highest cost but performs poorly in terms of the drift distribution.This highlights the effectiveness of the proposed design optimization procedure for minimizing the repair and total cost while improving the seismic performance of self-centering frame structures.

CONCLUSION
This study formulated a performance-based optimum design method for self-centering steel moment resisting frames (SC-MRFs) with shape memory alloy (SMA)-based beam-column connections.Using the center of mass optimization (CMO) metaheuristic algorithm, the design procedure finds the best placement of SMA connections in SC-MRFs.With the goal of minimizing the total cost (including the initial and repair cost) of the frame, the procedure also finds optimal steel beam and column sections, SMA material properties, and the topology of SMA connections.The optimization incorporates several constraints that consider constructability, strength, performance-based design, Park-Ang damage index, and the strong-column weak-beam requirement.Using the FEMA P695 methodology, incremental dynamic analyses are conducted to assess the seismic collapse safety of the optimally designed structures in terms of the adjusted collapse margin ratio (ACMR).The optimization procedure is used to design three-and nine-story conventional MRFs and SC-MRFs.The main findings from this research can be summarized as follows: • The performance-based design optimization effectively minimizes the total cost of SC-MRFs.The total cost for the most optimal three-and nine-story SC-MRFs is, respectively, 71% and 61% less than that of the nonoptimal SC-MRF due to consideration of the Park-Ang damage index constraints.• The optimal placement of SMA connections results in a 34% and 26% reduction in the total cost for the most optimal three-and nine-story SC-MRFs, respectively, compared to the optimal MRF without any SMA connections.• The initial cost of the most optimal three-and nine-story SC-MRFs with fewer SMA connections is, respectively, 20% and 16% less than the cost of the nonoptimal SC-MRFs.• Stiffer SC-MRFs can be designed through the optimal placement of a few SMA connections in a steel moment frame.
On average, the fundamental period of the optimal three-and nine-story SC-MRFs is 25% and 33% smaller than that of the nonoptimal SC-MRF.This results in smaller story drifts and residual drifts in the optimal SC-MRFs at the IO and CP performance levels.• The maximum residual drifts for the most optimal three-and nine-story SC-MRFs are 93% and 26% less, respectively, compared to the optimal MRFs.• The optimal three-and nine-story SC-MRFs possess greater seismic collapse safety.The ACMR for the most optimal three-and nine-story SC-MRFs are 94% and 79%, respectively, more than those obtained for the nonoptimal SC-MRFs.This is attributed to the higher stiffness and strength of the optimal SC-MRFs, which were explicitly designed to minimize damage through the use of the Park-Ang index constraint.
The findings of this study underscore the efficacy of the performance-based design optimization methodology in reducing the total costs while simultaneously enhancing the seismic performance of SC-MRFs.Nonetheless, it is worth noting that the observed improvements are comparatively modest in the nine-story SC-MRFs.Future research is needed to optimize taller frames while using larger diameter SMA bolts.This modification can potentially yield further reductions in total costs and added improvement in seismic performance.

A C K N O W L E D G M E N T S
This research has been supported by the Toronto Metropolitan University Faculty of Engineering and Architectural Science Dean's Research Fund and the Natural Sciences and Engineering Research Council of Canada (NSERC) through Discovery Grant.The authors gratefully acknowledge the financial supports.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.

F I G U R E 2
Schematic representation of the SMA connection numerical model in OpenSees.

3
Database of steel sections for beams and columns.

F I G U R E 9 1 0
Ratio of the connections' rotation to the yielding rotation for three-story frames.Story drift distribution for the three-story frames at the (A) IO and (B) CP performance levels, and (C) residual story drift ratio at the MCE hazard level.

F I G U R E 1 6 7
Ratio of the connections' rotation to the yielding rotation for nine-story frames.Story drift distribution for the nine-story frames at the (A) IO and (B) CP performance levels, and (C) residual story drift ratio under the MCE level shaking. 31

TA B L E 2
Ranges for some of the influential design variables.

Table 4
43mage states based on the Park-Ang damage index.43 TA B L E 4 The suite of ground motion records.

TA B L E 9
Collapse assessment results for the nine-story SC-MRFs and MRF.
F I G U R E 2 1Comparison of (A) relative costs; (B) maximum story drifts at IO and CP performance levels and residual drift at the MCE hazard level; and (C) ACMR and median collapse capacity for nine-story frames.