Topology optimization and seismic collapse assessment of shape memory alloy (SMA)-braced frames: Effectiveness of Fe-based SMAs

Abstract

This paper presents a seismic topology optimization study of steel braced frames with shape memory alloy (SMA) braces. Optimal SMA-braced frames (SMA-BFs) with either Fe-based SMA or NiTi braces are determined in a performance-based seismic design context. The topology optimization is performed on 5- and 10-story SMA-BFs considering the placement, length, and cross-sectional area of SMA bracing members. Geometric, strength, and performance-based design constraints are considered in the optimization. The seismic response and collapse safety of topologically optimal SMA-BFs are assessed according to the FEMA P695 methodology. A comparative study on the optimal SMA-BFs is also presented in terms of total relative cost, collapse capacity, and peak and residual story drift. The results demonstrate that Fe-based SMA-BFs exhibit higher collapse capacity and more uniform distribution of lateral displacement over the frame height while being more cost-effective than NiTi braced frames. In addition to a lower unit price compared to NiTi, Fe-based SMAs reduce SMA material usage. In frames with Fe-based SMA braces, the SMA usage is reduced by up to 80%. The results highlight the need for using SMAs with larger recoverable strains.

Graphical abstract

Keywords

topology optimization / shape memory alloy / Fe-based SMA / steel braced frames / performance-based seismic design / collapse assessment

1 Introduction

Smart materials, shape memory alloys (SMAs) have found many new applications for the seismic damage mitigation of civil engineering structures [1]. These metallic alloys’ unique shape recovery characteristic enables a building structure to self-center, i.e., return to its plumb position following an earthquake. Some examples of the applications of superelastic SMAs in steel braced frame structures are reviewed here. Analytical and experimental studies have shown that SMAs effectively reduce permanent or residual deformations and, hence, avoiding major costs associated with repair and downtime after an earthquake compared to conventional steel and buckling-restrained braced frames [2–4]. Buckling-restrained braces with SMAs have shown higher energy dissipation along with recentering capability and less permanent deformations [5–7]. Qiu et al.[8], Qiu and Zhu [9] have developed a performance-based design approach for SMA-braced frames (SMA-BFs). The designed SMA-BFs using the proposed methodology exhibited uniform distribution of peak story drift ratio and peak floor acceleration over the high of the building. The robustness of the developed performance-based design approach has also been investigated [10]. Some studies have focused on using low-cost SMA materials, such as copper and Fe-based SMAs [11–13]. In a study by Hou et al. [14] of SMA’s hysteretic properties influence on the seismic behavior of self-centering concentrically braced frames, it was observed that SMAs with large damping capability were more effective in controlling floor acceleration demands. Furthermore, novel hybrid SMA-based braces have been proposed using viscoelastic material and disc spring-based dampers to suppress the peak deformation, residual deformation, and floor acceleration response [15,16]. In addition, the repair and replacement cost of different types of self-centering braces in steel frames has been evaluated considering possible brace failure modes [17].

The high cost of SMA material is still a drawback for their practical application in structures. Despite its importance, optimization of SMA-BFs to achieve a trade-off between seismic capacity and cost is still lacking. Ozbulut et al. [18] implemented a multi-objective optimization procedure using a genetic algorithm to improve the seismic response of a three-story steel frame system with SMA braces. The optimization could reduce the use of SMA materials by 23%.

The performance-based design optimization leads to cost-effective structural systems with predictable seismic performance within the structure’s lifespan. In the last decade, many studies have been conducted in this context. Recently, meta-heuristic algorithms were employed to optimize steel moment-resisting frame structures in the framework of performance-based seismic design [19,20]. In these studies, the structural weight of moment frames is taken as the objective function in the performance-based design optimization problem. Moreover, Fragiadakis et al. [21] proposed a new procedure to optimize steel structures in the framework of performance-based design. They conducted multi-objective performance-based design optimization of steel moment frames, taking initial and life-cycle costs into account. The results showed that the steel buildings optimized in the context of single-objective optimization methodology would lead to a much higher total cost during the lifespan of the structures.

Topology optimization provides a general approach to determine the optimal arrangement of the braces in steel concentrically braced frames (CBFs). This bracing configuration considerably influences the frame’s structural weight and seismic behavior. Several studies have been performed on the topology optimization of CBFs, and some of them are reviewed as follows. The continuous topology optimization methodology has been utilized to propose conceptual designs by gradually removing inefficient material from a continuum domain [22–24]. However, the discrete topology approach is more practical in which the inefficient bracing members are removed. In this regard, meta-heuristic algorithms have been utilized to find topologically optimal CBFs in the performance-based design context [25,26]. The discrete topology optimization of CBFs leads to the least structural weight compared to the size optimization of these systems.

In assessing building frames’ seismic response and collapse performance, incremental dynamic analysis (IDA) has been used [27]. In previous studies, the seismic safety of CBFs has been assessed by calculating a collapse margin ratio (CMR) [26,28]. Some previous studies [29,30] have used IDA to study the influence of SMA braces on the seismic response of steel braced frames.

This paper utilizes an efficient methodology to perform topology optimization of SMA-BFs using two different types of SMA materials, namely NiTi and Fe-based SMAs for 5- and 10-story structures in the performance-based design context. The large-scale production of inexpensive superelastic Fe-SMAs is still limited. This paper, however, explores and highlights the potential of Fe-based SMAs for applications in seismic force-resisting systems. Fig.13 presents the outline of the study. As shown, the presented research involves conducting seismic topology optimization, collapse assessment, and comparative analysis of 5- and 10-story SMA-BFs in a performance-based seismic design framework. The study first finds topologically optimal SMA-BFs using the center of mass optimization algorithm [31,32]. The cross-sectional area of the beams, columns, SMA braces, as well as the length and placement of SMA bracing members, are optimized to reduce the SMA material usage. Next, the seismic collapse capacity of the optimal SMA-BFs is evaluated by implementing IDA and calculating adjusted CMRs following FEMA P695 [33]. In the last part of the study, the topologically optimal 5- and 10-story SMA-BFs with NiTi or Fe-based SMA braces are compared in terms of their total relative cost, seismic performance, and collapse safety. The utilized methodology leads to optimal SMA-BFs in which an appropriate trade-off between total relative cost and seismic safety is achieved.

2 Seismic topology optimization of SMA-BFs

In this paper, the center of the mass optimization algorithm is implemented to tackle the performance-based topology optimization problems. This meta-heuristic algorithm is based on the physical concept of the center of mass, which states that the center of mass is the mean point of distribution of masses in space. The distance of a particle with a larger mass to the center of mass will be smaller and vice versa. Further information can be found in Ref. [31].

Two seismic performance levels in a performance-based design methodology are considered according to ASCE 41-13 [34]. These performance levels include life safety (LS) and collapse prevention (CP), which respectively correspond to a 10% and 2% probability of exceedance in 50 years. A nonlinear static analysis with the displacement coefficient method is used to assess the nonlinear structural response. Furthermore, a target displacement during the topology optimization process is computed as follows:

(1){\delta }_{\mathrm{t}}=C_0{C}_1{C}_2{S}_{\mathrm{a}}\frac{{T_{\mathrm{e}}}^2}{4{\text{π}}^2}g,

where C₀, C₁, C₂ are the modification factors, and T_e, S_a, and g represent the effective fundamental period of the structure, the spectral acceleration, and the gravitational acceleration, respectively. The first mode shape of SMA-BFs is utilized to define the lateral load pattern during the pushover analysis. Moreover, the dead and live loads applied to all beams are 2500 and 1000 kg/m, respectively.

2.1 Design variables

In the discrete topology optimization of SMA-BFs, two objectives are considered: finding the optimal placement and length of SMA braces and the optimal cross-sections of structural members. Therefore, the problem starts with a fully braced SMA-BF. In the optimization process, the optimal position of SMA braces and the structural cross-sectional areas are found to minimize the initial cost of an SMA-BF. The cross-sectional area of columns and SMA braces are considered as the size-related design variables. Additionally, the location and length of SMA braces are considered as the topology design variables. Unnecessary SMA braces are omitted from a fully braced frame (shown in Fig.2) during the topology optimization while meeting optimization constraints. The cross-sectional area of columns and SMA braces and the length of SMA braces are simultaneously optimized during the topology optimization.

X_TP is the vector of topology design variables expressing the location of SMA braces. A topology design variable of 1 or 0 represents the presence or absence of a bracing element in the system. It is worth mentioning that both the size and topology design variables are grouped symmetrically. A general fully braced SMA-BF is shown in Fig.2.

The design variable vectors (X_DV) for the topology optimization problem are defined as follows:

(2)X_{\text{DV}}=\{\begin{array}{rl}& X_{\mathrm{T}\mathrm{P}}=\left\{{TP}_1,{TP}_2,...,{TP}_i,...,{TP}_{\mathrm{n}}\right\},\\ & X_{\mathrm{B}\mathrm{r}\mathrm{L}}=\left\{{BrL}_1,{BrL}_2,...,{BrL}_i,...,{BrL}_{\mathrm{g}\mathrm{b}}\right\},\\ & X_{\mathrm{B}\mathrm{r}\mathrm{R}}=\left\{{BrR}_1,{BrR}_2,...,{BrR}_i,...,{BrR}_{\mathrm{g}\mathrm{b}}\right\},\\ & X_{\mathrm{C}}=\left\{C_1,C_2,...,C_i,...,C_{\mathrm{g}\mathrm{c}}\right\},\\ & X_{\mathrm{B}\mathrm{m}}={Bm}_1,\end{array}

where X_TP, X_BrL, X_BrR, X_C, and X_Bm are the topology design variables: vector, brace length vector, brace size vector, column size vector, and beam size vector, respectively; ‘gb’ and ‘gc’ represent the group of bracing members and columns, respectively.

2.2 Numerical modeling of SMA-BFs

In this paper, MATLAB and OpenSees are linked to facilitate optimization. OpenSees [35] is used for performing the nonlinear static pushover and dynamic response history analyses. MATLAB [36] is utilized to program the optimization and postprocessing tasks. A force-based fiber-section beam-column element is utilized for columns, beams, and SMA-bracing members to evaluate the nonlinear structural response with good accuracy. All beams and braces are modeled as pin-ended members, while the effect of gusset plates and rigid end zones of the joint are neglected. Moreover, the ‘steel01’ material from the library of materials available in OpenSees is utilized to represent the nonlinear behavior of beams and columns. The yield stress, modulus of elasticity, and strain hardening ratio for steel are taken as 345 MPa, 200 GPa, and 3%, respectively. The second-order P-Delta effects are also included in the model by using the ‘PDelta’ coordinate transformation object in OpenSees. Furthermore, the ‘corotational’ transformation object in OpenSees is utilized to consider the nonlinear geometric effects in braces.

The ‘Self-Centering’ material from the library of materials available in OpenSees is used to model the superelastic behavior of the SMA braces. The parameters needed for constructing this model are shown in Fig.13. sigAct, k₁, and k₂ define the SMAs yield point, initial, and hardening stiffness, respectively. The initial stiffness parameter (k₁) is taken equal to the modulus of elasticity of the SMA material (E^SMA). Furthermore, the unitless parameter β is introduced to calculate the hysteresis width. Two different types of superelastic SMAs, namely NiTi [4,37] and FeNiCoAlTaB [14,38], are considered to study the effect of different SMAs on the total relative cost and seismic performance of topologically optimal SMA-BFs. The material parameters of SMAs are listed in Tab.2. As shown in Fig.2, the bracing member consists of two segments, including SMA and a rigid link. Several other studies have made the same assumption (e.g., [1,2,4,29]). No prestressing is considered for the braces in this comparative study. However, prestressing can be regarded as improving the braces’ initial stiffness and avoiding excessive deformations of the frames under wind load.

A validation study is performed to assess the SMA model’s accuracy in OpenSees. The parameters in Tab.2 are determined according to previous research works [4,14,37,38]. The loading protocol for the cyclic test of a 12.7 mm diameter NiTi bar [37,39] is shown in Fig.4. The comparison of the numerical simulation results with the experimental study [37,39] is presented in Fig.5. As shown, the accuracy of the analysis results using OpenSees is acceptable.

2.3 Optimization problem formulation

The main goal of seismic topology optimization is to minimize the cost of SMA-BFs. Three different constraints, namely, geometric, strength, and performance-based design constraints, are considered in the optimization. The seismic topology optimization problem can be defined as follows:

(3)\begin{array}{rl}& \mathrm{F}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}:X_{\mathrm{D}\mathrm{V}},\\ & \mathrm{T}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}:C_{\mathrm{t}}\left(X_{\mathrm{D}\mathrm{V}}\right)=C_{\mathrm{s}\mathrm{t}}+C_{\mathrm{s}\mathrm{m}},\\ & C_{\mathrm{t}}\left(X_{\mathrm{D}\mathrm{V}}\right)=\sum _{j=1}^{n_{\mathrm{b}\mathrm{c}}}{\mathit{\text{ρ}}}_j{A}_j{L}_j{C^{\prime }}_{\mathrm{s}\mathrm{t}}/{C^{\prime }}_{\mathrm{s}\mathrm{t}}+\sum _{q=1}^{n_{\mathrm{s}\mathrm{m}}}{A}_q{L}_q{C^{\prime }}_{\mathrm{s}\mathrm{m}}/{C^{\prime }}_{\mathrm{s}\mathrm{t}},\\ & C_{\mathrm{t}}\left(X_{\mathrm{D}\mathrm{V}}\right)=\sum _{j=1}^{n_{\mathrm{b}\mathrm{c}}}{\mathit{\text{ρ}}}_j{A}_j{L}_j+\sum _{q=1}^{n_{\mathrm{s}\mathrm{m}}}{A}_q{L}_q{C}_{\mathrm{r}},\\ & \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}:\\ & \{\begin{array}{rl}& g_{\mathrm{G}\mathrm{C}}⩽0,\\ & g_{\mathrm{S}\mathrm{C}}⩽0,\\ & g_{\mathrm{P}\mathrm{B}\mathrm{D}\mathrm{C}}⩽0,\end{array}\end{array}

where C_t is the total relative cost; C_st and C_sm are the relative cost of steel columns and beams, and relative cost of SMA braces, respectively; {C^{\prime }}_{\mathrm{s}\mathrm{t}} is the cost of steel column and beam per unit weight; {C^{\prime }}_{\mathrm{s}\mathrm{m}} is the cost of SMA brace per unit volume; C_r is the ratio between {C^{\prime }}_{\mathrm{s}\mathrm{t}} and {C^{\prime }}_{\mathrm{s}\mathrm{m}}. The unit price of NiTi SMAs is taken 100 times that of steel [40], and the unit price of Fe-based SMAs is considered half of NiTi SMAs [11]. Therefore, C_r is taken as 50 or 100 for NiTi or Fe-based SMAs. ρ_j, A_j, and L_j are the weight density, cross-sectional area, and length of the jth beam or column element, respectively; n_bc is the number of beam and column elements; A_q and L_q are the cross-sectional area and length of the qth SMA brace, respectively; n_sm is the number of SMA braces; g_{\mathrm{G}\mathrm{C}}, g_{\mathrm{S}\mathrm{C}}, and g_{\mathrm{P}\mathrm{B}\mathrm{D}\mathrm{C}} are the geometric, strength, and performance-based design constraints, respectively.

Four geometric constraints are checked at each structural joint, including column-to-column and beam-to-column joints, as expressed mathematically in Eq. (4).

(4)g_{GC,k}=\{\begin{array}{rl}& g_{\mathrm{G}\mathrm{C}1,k}\left(X_{\mathrm{D}\mathrm{V}}\right)={\displaystyle \frac{B_{\mathrm{k}}^{\mathrm{f}\mathrm{w}}}{C_{\mathrm{k}}^{\mathrm{f}\mathrm{w}}}}-1⩽0,\\ & g_{\mathrm{G}\mathrm{C}2,k}\left(X_{\mathrm{D}\mathrm{V}}\right)=\frac{C_{\mathrm{k}}^{\mathrm{f}\mathrm{w},\mathrm{u}}}{C_{\mathrm{k}}^{\mathrm{f}\mathrm{w},\mathrm{l}}}-1⩽0,\\ & g_{\mathrm{G}\mathrm{C}3,k}\left(X_{\mathrm{D}\mathrm{V}}\right)=\frac{C_{\mathrm{k}}^{\mathrm{d},\mathrm{u}}}{C_{\mathrm{k}}^{\mathrm{d},\mathrm{l}}}-1⩽0,\\ & g_{\mathrm{G}\mathrm{C}4,k}\left(X_{\mathrm{D}\mathrm{V}}\right)=\frac{C_{\mathrm{k}}^{\mathrm{w}\mathrm{t},\mathrm{u}}}{C_{\mathrm{k}}^{\mathrm{w}\mathrm{t},\mathrm{l}}}-1⩽0,\end{array}k=1,2,...,n_j,

where B_{\mathrm{k}}^{\mathrm{f}\mathrm{w}} and C_k^{\mathrm{f}\mathrm{w}} are the flange width of the beams and columns connected to the kth joint, respectively; C_k^{\mathrm{f}\mathrm{w},\mathrm{u}} and C_k^{\mathrm{f}\mathrm{w},\mathrm{l}}, respectively, are the flange width of upper and lower columns, respectively; C_k^{\mathrm{d},\mathrm{u}} and C_k^{\mathrm{d},\mathrm{l}} are the depth of upper and lower columns, respectively; C_k^{\mathrm{w}\mathrm{t},\mathrm{u}} and C_k^{\mathrm{w}\mathrm{t},\mathrm{l}} are the web thickness of upper and lower columns, respectively, and n_j is the total number of structural joints.

The strength-related constraints are applied by checking beams and columns’ demand to capacity ratio (DCR). The strength constraints can be written as follows:

(5)

where P_{\mathrm{u}} and M_{\mathrm{u}} are the required axial and flexural strengths, respectively; P_{\mathrm{n}} and M_{\mathrm{n}} are the nominal axial and flexural capacities, respectively; {\varphi }_{\mathrm{c}} and {\varphi }_{\mathrm{b}} are the resistance factors; and n_e is the total number of beams and columns.

Following the geometric and strength constraints, performance-based design constraints are applied by checking story drift, column axial and flexural strength, and SMA axial strain demand. The nonlinear structural response is assessed at the LS and CP performance levels through a pushover analysis. The story drift constraints for each hazard level are defined using the following equation:

(6)g_{\mathrm{S}\mathrm{D},\mathrm{p}\mathrm{l}}=\frac{{SD}_{\mathrm{p}\mathrm{l}}}{{SD}_{\mathrm{a}\mathrm{l}\mathrm{l},\mathrm{p}\mathrm{l}}}-1⩽0,

where SD_pl and SD_all,pl are the peak story drift and the allowable story drift at the LS or CP performance level, respectively. The acceptable story drift ratio at the LS and CP performance levels is taken equal to 2% and 4%, respectively [9].

The column action constraints at the LS and CP performance levels can be defined as follows:

(7)

where g_{\mathrm{D}\mathrm{C},m}^{\mathrm{p}\mathrm{l}} and g_{\mathrm{F}\mathrm{C},m}^{\mathrm{p}\mathrm{l}} are constraints on deformation-controlled and force-controlled columns, respectively; P is the axial load at the corresponding performance level; {\theta }_m^{\mathrm{p}\mathrm{l}} and {\theta }_{\mathrm{a}\mathrm{l}\mathrm{l}}^{\mathrm{p}\mathrm{l}} are the maximum plastic rotation of the mth column and its permissible plastic rotation demand at pl = LS and CP performance levels per ASCE 41-13 [34]; P_{\mathrm{U},m}^{\mathrm{p}\mathrm{l}} and M_{\mathrm{U},m}^{\mathrm{p}\mathrm{l}} are the axial force and moment of the mth column at pl = LS and CP performance levels, respectively; P_cl and M_cl are lower-bound compression and flexural strength, respectively, and n_{\mathrm{c}} is the total number of columns.

The axial strain constraints for SMA braces are set by considering the mechanical properties of SMAs and defining a permissible axial strain [41,42]. Equation (8) presents the formulated axial strain constraints at the LS and CP performance levels:

(8)g_{\mathrm{\Delta },\mathrm{n}}^{\mathrm{p}\mathrm{l}}=\frac{{\epsilon }_n^{\mathrm{p}\mathrm{l}}}{{\epsilon }_{\mathrm{a}\mathrm{l}\mathrm{l}}^{\mathrm{p}\mathrm{l}}}-1⩽0,n=1,2,\cdots ,n_{\mathrm{b}},

where {\epsilon }_n^{\mathrm{p}\mathrm{l}} and {\epsilon }_{\mathrm{a}\mathrm{l}\mathrm{l}}^{\mathrm{p}\mathrm{l}} are the axial strain of nth SMA brace and its allowable values at pl = LS and CP performance levels, respectively. Tab.2 summarizes the allowable axial strains for the SMA braces considered as a performance-based design constraint. The permissible axial strain for SMA depends on its mechanical properties. The SMA maximum recoverable strain ({\epsilon }_{\mathrm{L}}) and the strain corresponding to {\sigma }_{\mathrm{f}}^{\mathrm{M}} are respectively considered as the strain limits at LS and CP performance levels [41,42]. The strain limits for NiTi SMA braces at the LS and CP performance levels are 3.5% and 6%, whereas these strain limits for Fe-based SMA bracing members are 13.5% and 15%, respectively.

3 Topology optimization results

The topology optimization is performed on 5- and 10-story SMA-BFs, as shown in Fig.6. This section presents the optimization results. The available steel sections for the beams and columns are listed in Tab.3. The SMA-brace length ranges from 0.23 to 1.63 m, and the brace cross-section radius ranges from 5 to 31 mm. In the presentation of the results, for example, in Tab.3, BPN and CPN denote the profile number for optimal beam and column. Also, the optimal SMA-brace length and cross-section radius are shown by BrL and BrR, respectively.

3.1 Optimization results for optimal 5-story SMA-BFs

Eight topologically optimal 5-story SMA-BFs are found from the optimization results. The results for Fe-based SMA-BFs are presented first, followed by the optimization results for NiTi SMA-BFs.

The optimal 5-story Fe-based SMA-BFs are denoted by Fe5-TOS 1 to Fe5-TOS 8, in which TOS stands for the topologically optimal SMA-braced frame. The optimization results for these optimal SMA-BFs are given in Tab.4, listing the beam and column sections and the SMA brace lengths and radius. Among all the optimal 5-story Fe-based SMA-BFs, Fe5-TOS 1 has the least amount of SMA as indicated by ‘SMA volume’ in Tab.4. The bracing configurations in the optimal 5-story frames with Fe-based SMA braces are presented in Fig.13.

The design of eight optimal 5-story NiTi SMA-BFs found in the optimization is listed in Tab.5. These optimal frames are indicated by NiTi5-TOS 1 to NiTi5-TOS 8. Fig.8 illustrates the bracing configurations of these optimal 5-story NiTi SMA-BFs. Among all the optimal NiTi5-TOS frames, NiTi5-TOS 1 has the least SMA volume. By comparing the SMA volume in the optimal frames, it can be observed that the SMA usage in the Fe-based SMA-BFs, on average, is reduced by 66% (with a range of 55%–80%). This shows that in addition to its lower material cost than NiTi, the Fe-based SMA can result in a more cost-effective optimal design due to its favorable hysteretic properties and larger recoverable strain.

The total relative costs of the optimal frames are also listed in Tab.4 and Tab.5. This comparison only includes the material cost assuming that the unit price of NiTi and Fe-based SMAs is around 100 and 50 times higher than that of steel, respectively [11,40]. As shown, NiTi5-TOS 1 and Fe5-TOS 1 have the least total relative cost among all the respective optimal NiTi5-TOS and Fe5-TOS frames. The cost reduction by using Fe-based SMA braces ranges from 76% to 89%.

3.2 Optimization results for optimal 10-story SMA-BFs

Eight optimal frames are found from the seismic topology optimization results for 10-story Fe-based SMA-BFs. The details of these optimal frames are listed in Tab.6 in the order from the braced frame with the least to the most SMA volume (Fe10-TOS 1 to Fe10-TOS 8, respectively). These Fe10-TOS frames are depicted in Fig.9.

Similarly, eight optimal 10-story braced frames with NiTi braces are found. Fig.10 shows these topologically optimal 10-story NiTi SMA-BFs. Tab.7 summarizes the topology optimization results for these frames denoted by NiTi10-TOS. The results in this table include the optimal beam and column sections and the optimal SMA brace length and radius. The SMA volume is, on average, reduced by 50% in the case of Fe10-TOS compared to their counterpart NiTi10-TOS optimal frames. The SMA usage reduction due to the favorable hysteretic behavior of Fe-based SMA in the optimal 10-story Fe-based SMA-BFs ranges from 41% to 59%.

Similar to the results for the 5-story frames, a lower total relative cost is possible when using Fe-based SMA braces. This cost reduction in the Fe10-TOS frames is in the range of 70%–79%.

4 Collapse performance assessment of optimal SMA-BFs

This section evaluates the seismic collapse performance of the optimal SMA-BFs by calculating CMR values according to FEMA P695 [33]. A suite of 22 of the FEMA P695 far-field ground motion records, listed in Tab.8, are used to perform IDA, which involves several nonlinear response history analyses. The earthquake records are scaled to the maximum considered earthquake (MCE) intensity level, having a 2% probability of exceedance in 50 years. As the intensity measure (IM) and the engineering demand parameter (EDP), the 5% damped spectral acceleration at the fundamental period of the structure, Sa(T₁,5%), and the peak story drift are respectively considered. The collapse of the structure is assumed to occur when a 5% peak story drift is reached or when the nonlinear response history analysis fails to converge. IDA curves are generated based on the recording of EDPs at different IMs. Subsequently, fragility curves are extracted to relate the IM to the probability of collapse by fitting a cumulative distribution function to the collapse data.

CMR is defined as follows:

(9)CMR=\frac{I{M}_{50\mathrm{\%}}}{I{M}_{\mathrm{M}\mathrm{C}\mathrm{E}}},

where IM_50% is the median collapse probability for which 50% of earthquake records cause collapse, and IM_MCE is the 5%-damped spectral acceleration corresponding to the MCE level.

ACMR is calculated to consider the spectral shape effects. The ACMR is computed as shown in Eq. (10), in which a spectral shape factor (SSF) is used [33]:

(10)ACMR=SSF\times CMR.

The SSF depends on the fundamental period and the period-based ductility of the structure, and it is calculated according to Table 7-1 of FEMA P695. The ACMR is compared to an acceptable value that accounts for system-level uncertainties. Total collapse uncertainty, β_TOT, is calculated by considering different sources of uncertainty, as given in Eq. (11):

(11){\beta }_{\mathrm{T}\mathrm{O}\mathrm{T}}=\sqrt{{\beta }_{\mathrm{R}\mathrm{T}\mathrm{R}}^2+{\beta }_{\mathrm{D}\mathrm{R}}^2+{\beta }_{\mathrm{T}\mathrm{D}}^2+{\beta }_{\mathrm{M}\mathrm{D}\mathrm{L}}^2},

where β_RTR, β_DR, β_TD, and β_MDL are the record-to-record variability of the collapse data, design requirements, test data, and modeling-related uncertainty sources, respectively. A constant value of β_RTR = 0.4 is considered, which is permitted for structures with sufficient period-based ductility [33]. FEMA P695 [33] ranks the uncertainties β_DR, β_TD, and β_MDL as ‘superior’, ‘good’, ‘fair’, and ‘poor’, with corresponding values of 0.1, 0.2, 0.35, and 0.5. It is assumed that the included design requirements are comprehensive to provide safeguards against unexpected failure modes. Therefore, the uncertainty β_DR = 0.1 (i.e., ‘superior’ rating) is accounted. Since the accuracy of the SMA model in OpenSees is validated, β_TD is taken as 0.2, corresponding to the ‘good’ rating. The uncertainty β_MDL is also assigned to a rating of ‘good’. As shown in Eq. (12), the ACMR values are compared with the acceptable ACMR at which the collapse probability of the topologically optimal SMA-BFs under MCE level ground motions is almost less than 20%. With a total uncertainty of 0.5, the acceptable ACMR is 1.52, according to Table 7-3 of FEMA P695 [33].

(12)ACMR⩾{ACMR}_{20\mathrm{\%}}.

5 Collapse performance assessment results

This section presents the results of the seismic collapse assessment of the optimal frames. The collapse capacity of the optimal SMA-BFs is evaluated by calculating ACMR ratios according to FEMA P695. This seismic assessment is required to check the collapse safety of the optimal SMA-BFs, particularly because the optimization seeks the least amount of material, i.e., SMA and steel. Further from this assessment, the SMA-braced frame with the highest ACMR ratio is determined as the most optimal frame among all the eight optimal SMA-BFs.

5.1 Results for optimal 5-story SMA-BFs

Tab.9 presents the details of the seismic collapse assessment for the optimal 5-story Fe-based SMA-BFs (i.e., Fe5-TOS). The fundamental period for 5-story Fe-based SMA-BFs ranges from 0.34 to 0.50 s. For all these optimal SMA-BFs, ACMR is larger than the acceptable value of 1.52. Therefore, the optimal Fe5-TOS frames possess considerable collapse safety with an ACMR of at least 2.43. Among all the optimal Fe5-TOS frames, Fe5-TOS 7 provides the highest collapse capacity.

Similarly, all the optimal 5-story frames with NiTi braces have acceptable collapse safety with an ACMR greater than the allowable ACMR of 1.52. Tab.10 provides the collapse assessment results for these topologically optimal designs. As shown, the highest ACMR of 2.43 is found for NiTi5-TOS 5. Additionally, the fundamental period for 5-story NiTi SMA-BFs is in the range of 0.44 to 0.56 s.

Fig.11 compares IDA curves, including 16%, 50%, and 84% fractile IDA curves, for the most optimal frames, i.e., Fe5-TOS 7 and NiTi5-TOS 5. It is observed that the 16%, 50%, and 84% fractile values in terms of Sa(T₁,5%) at the 5% peak story drift for Fe5-TOS 7 are, respectively, 19%, 55%, and 71% higher than that of NiTi5-TOS 5. This shows the higher seismic capacity for the most optimal 5-story Fe-based SMA-BF compared to the respective frame with NiTi braces.

To compare the collapse capacity of the optimal 5-story frames, the fragility curves for the Fe5-TOS and NiTi5-TOS frames are presented. As shown in Fig.12, the optimal frames with Fe-based SMA braces exhibit higher collapse capacity than those with NiTi braces. On average, the median collapse intensity of the Fe5-TOS frames is 27% higher than that for the NiTi5-TOS frames.

Fig.12 also shows that NiTi5-TOS 5 and Fe5-TOS 7 possess the highest collapse capacity among their respective optimal 5-story SMA-BFs with NiTi or Fe-based SMA braces. While the Fe5-TOS 7 frame has 46% higher collapse capacity compared to NiTi5-TOS 5, its total relative cost is 76% lower. Therefore, Fe-based bracing is advantageous in providing greater collapse capacity while lowering the material usage and thus reducing the total relative cost.

Fig.13 illustrates the story drift distribution over the height of NiTi5-TOS 5 and Fe5-TOS 7 at LS and CP performance levels. It can be observed that the Fe-based SMA-BF experiences a more uniform drift distribution. Moreover, the peak story drift ratio at different performance levels is lower in the optimal frame with Fe-based SMA bracing members. The peak story drifts for the NiTi5-TOS 5 at the LS and CP performance levels are, respectively, 140% and 180% higher than that for Fe5-TOS 7. Furthermore, the maximum residual drift ratio for Fe5-TOS 7 is 95% lower than that observed in NiTi10-TOS 5. Importantly, the peak residual story drift ratio at the MCE level for all the optimal 5-story frames is less than 0.5%, demonstrating that the topologically designed 5-story SMA-BFs are economically repairable.

5.2 Results for optimal 10-story SMA-BFs

Tab.11 presents the seismic collapse assessment details for the optimal 10-story Fe-based SMA-BFs (i.e., Fe10-TOS). The results show that all the topologically optimal structures have acceptable ACMR values. Fe10-TOS 1 is the most optimal Fe10-TOS frame among these optimal frames as it has the highest ACMR value. In addition, the fundamental period for 10-story Fe-based SMA-BFs ranges from 0.91 to 1.34 s.

The seismic collapse assessment details for the optimal 10-story NiTi SMA-BFs (i.e., NiTi10-TOS) are summarized in Tab.12. The results show that these optimal frames do not have sufficient collapse capacity. The ACMR ratios are lower than the acceptable ratio. NiTi10-TOS 3 is taken as the most optimal NiTi10-TOS braced frame as it has the highest ACMR ratio among all the NiTi10-TOS frames. The CMR for this most optimal frame is 14% lower than the acceptable level. It is concluded that NiTi SMA braces may fail to provide an acceptable level of collapse capacity in taller braced frames. Furthermore, the fundamental period for 10-story NiTi SMA-BFs is in the range of 1.34 to 1.50 s.

The IDA results for the most optimal frames (Fe10-TOS 1 and NiTi10-TOS 3) are presented in Fig.14. The improved seismic performance of the SMA-braced frame with Fe-based SMA is noticeable. By comparing the 16%, 50%, and 84% fractiles for Fe10-TOS 1 and NiTi10-TOS 3, it is observed that Fe10-TOS 1 has a higher seismic capacity compared to NiTi10-TOS 3. The 16%, 50%, and 84% fractile values in terms of IM at the 5% peak story drift for Fe10-TOS 1 are respectively 92%, 116%, and 128% higher than that of NiTi10-TOS 3.

As shown in Fig.15, fragility curves are developed for the NiTi10-TOS and Fe10-TOS frames to compare the collapse safety of the 10-story optimal SMA-BFs with different SMAs. It is observed that the collapse capacity of the optimal braced frames with Fe-based SMA is higher than that of frames with NiTi SMA braces.

By comparing the total relative cost of the most optimal frames with different SMAs, it is observed that the total relative cost of NiTi10-TOS 3 is over four times higher than that of Fe10-TOS 1. The optimal Fe-based SMA-BFs are more cost-effective while providing higher collapse capacity than frames with NiTi SMA braces.

The distributions of story drift at the LS and CP performance levels for NiTi10-TOS 3 and Fe10-TOS 1 frames are illustrated in Fig.16. The optimal Fe-based SMA-BFs show a more uniform distribution of lateral displacement over the frame height as observed by their lower maximum drift at different performance levels than NiTi SMA-BFs. The peak story drifts for the most optimal frame NiTi10-TOS 3 at the LS and CP performance levels are respectively 35% and 19% higher than that for the respective frame with Fe-based braces (Fe10-TOS 1). Additionally, it can be observed that the maximum residual story drift for Fe10-TOS 1 is 68% lower than that for NiTi10-TOS 3. The peak residual story drift ratio at the MCE level for all the topologically optimal 10-story Fe-based SMA-BFs is less than 0.5%. These optimal 10-story Fe-based SMA-BFs are economically repairable. However, the peak residual drift for NiTi10-TOS 2, NiTi10-TOS 7, and NiTi10-TOS 8 are respectively 0.53%, 0.7%, and 0.66%.

Fig.17 compares the Fe-based SMA-BFs and NiTi SMA-BFs results in terms of average total relative cost, collapse capacity, peak story drift, and residual drift. Fig.17(a) shows that the average total relative cost of the optimal 5- and 10-story frames with Fe-based SMA braces is 82% and 74% lower than that of the corresponding frames with NiTi braces. Fig.17(b) compares the average ACMR values of the topologically optimal 5- and 10-story SMA-BFs. It is observed that, on average, the ACMR values of the 5- and 10-story Fe-based SMA-BFs are respectively 57% and 93% higher than the ACMR of the 5- and 10-story NiTi SMA-BFs. As shown in Fig.17(b), the 10-story NiTi SMA-BFs do not provide sufficient collapse capacity, and the average ACMR is less than the acceptable value. Moreover, for both Fe-based and NiTi SMA-BFs, the collapse capacity of the frames is decreased with the frame height. The average peak story drift of the optimal frames is presented in Fig.17(c). It is observed that, on average, the peak story drift of the 5- and 10-story Fe-based SMA-BFs is respectively 43% and 3% lower than that of the NiTi SMA-BFs. As illustrated in Fig.17(d), the average peak residual drift of the optimal 5- and 10-story Fe-based SMA-BFs are respectively 83% and 60% lower than that of the NiTi SMA-BFs.

6 Conclusions

This paper presents seismic topology optimization of SMA-BFs using two different SMA materials, including NiTi and FeNiCoAlTaB. The center of mass optimization algorithm is implemented to conduct the topology optimization of SMA-BFs in a performance-based design context in accordance with ASCE 41-13. The optimization study found eight optimal SMA-BFs with either NiTi or Fe-based SMA braces. IDA is performed on the optimal frames to extract fragility curves and calculate adjusted ACMR values according to FEMA P695. The study discusses the results for optimal 5- and 10-story SMA-BFs with different SMA materials. The following conclusions can be drawn.

1) The optimal Fe-based SMA-BFs outperform braced frames with NiTi in terms of lower total relative cost and improved seismic performance, including lower peak and residual story drift and higher collapse capacity.

2) The total relative cost of the most optimal 5-story Fe-based SMA-BF is 76% lower than that of the 5-story NiTi SMA-BF. This cost reduction is 80% in the case of the 10-story SMA-BFs.

3) It is observed that the peak story drift ratios at the LS and CP performance levels are respectively 58% and 64% lower in the most optimal 5-story Fe-based SMA-BFs compared to the most optimal 5-story NiTi SMA-BFs. Furthermore, the peak story drift ratio for the most optimal 10-story Fe-based SMA-BFs at the LS and CP performance levels is 26% and 16% lower than that of the most optimal 10-story NiTi SMA-BFs. Therefore, Fe-based SMA braces are more effective in lowering story drift ratios than NiTi SMA braces.

4) The residual story drift ratios are below 0.5% in the optimal SMA-BFs with either NiTi or Fe-based SMAs except for three optimal 10-story NiTi SMA-BFs, which experience peak residual drifts up to 40% higher than 0.5%. Like the story drift results, lower residual drifts (up to 98%) are observed using Fe-based SMA braces.

5) All the optimal 5-story SMA-BFs possess an acceptable collapse capacity. On average, the ACMR ratio for the optimal 5-story Fe-based SMA-BFs is 57% larger than that for the optimal 5-story NiTi SMA-BFs.

6) The optimal 10-story Fe-based SMA-BFs provide acceptable safety against collapse. However, the collapse capacity of the 10-story NiTi SMA-BFs is lower than the acceptable level. This highlights the need for using SMAs with larger recoverable strains.

7) The SMA-BFs with the highest ACMR values are chosen as the most optimal SMA-BF. In all the examples presented, it is observed that the SMA-BFs with higher initial costs do not necessarily show better collapse safety performance. It is efficient to implement topology optimization methodologies to find less expensive SMA-BFs with higher seismic capacity.

In this study, the optimization only includes the total relative cost of the material. Future research is recommended to include other costs associated with different topologies, such as the cost of structural detailing and construction as well as life cycle and repair costs. In addition, stiffness irregularity checks over the structure height should be considered as an additional constraint in the optimization process. It is also important to note that the presented optimization is focused on the influence of different SMA materials on the seismic response and collapse capacity of steel braced frames. In future research, more comprehensive optimization studies can be performed by including further influential details, such as the SMA bracing design and anchoring methods, which may influence the design of the braced frame structures.

References

[1] Auricchio F, Fugazza D, DesRoches R. Earthquake performance of steel frames with nitinol braces. Journal of Earthquake Engineering, 2006, 10( Suppl1): 45–66

[2] McCormick J, DesRoches R, Fugazza D, Auricchio F. Seismic assessment of concentrically braced steel frames with shape memory alloy braces. Journal of Structural Engineering, 2007, 133( 6): 862–870

[3] Qiu C, Zhu S. Shake table test and numerical study of self-centering steel frame with SMA braces. Earthquake Engineering & Structural Dynamics, 2017, 46( 1): 117–137

[4] Asgarian B, Moradi S. Seismic response of steel braced frames with shape memory alloy braces. Journal of Constructional Steel Research, 2011, 67( 1): 65–74

[5] Miller D J, Fahnestock L A, Eatherton M R. Development and experimental validation of a nickel–titanium shape memory alloy self-centering buckling-restrained brace. Engineering Structures, 2012, 40 : 288–298

[6] Eatherton M R, Fahnestock L A, Miller D J. Computational study of self-centering buckling-restrained braced frame seismic performance. Earthquake Engineering & Structural Dynamics, 2014, 43( 13): 1897–1914

[7] Hu J W, Choi E. Seismic design, nonlinear analysis, and performance evaluation of recentering buckling-restrained braced frames (BRBFs). International Journal of Steel Structures, 2014, 14( 4): 683–695

[8] Qiu C, Li H, Ji K, Hou H, Tian L. Performance-based plastic design approach for multi-story self-centering concentrically braced frames using SMA braces. Engineering Structures, 2017, 153 : 628–638

[9] Qiu C X, Zhu S. Performance-based seismic design of self-centering steel frames with SMA-based braces. Engineering Structures, 2017, 130 : 67–82

[10] Qiu C, Zhao X, Zhang Y, Hou H. Robustness of performance-based plastic design method for SMABFs. International Journal of Steel Structures, 2019, 19( 3): 787–805

[11] Alaneme K K, Okotete E A, Anaele J U. Structural vibration mitigation—A concise review of the capabilities and applications of Cu and Fe based shape memory alloys in civil structures. Journal of Building Engineering, 2019, 22 : 22–32

[12] Alaneme K K, Okotete E A. Reconciling viability and cost-effective shape memory alloy options—A review of copper and iron based shape memory metallic systems. Engineering Science and Technology, an International Journal, 2016, 19( 3): 1582–1592

[13] Fang C, Wang W, Ji Y, Yam M C H. Superior low-cycle fatigue performance of iron-based SMA for seismic damping application. Journal of Constructional Steel Research, 2021, 184 : 106817

[14] Hou H, Li H, Qiu C, Zhang Y. Effect of hysteretic properties of SMAs on seismic behavior of self-centering concentrically braced frames. Structural Control and Health Monitoring, 2018, 25( 3): e2110

[15] Wang W, Fang C, Shen D, Zhang R, Ding J, Wu H. Performance assessment of disc spring-based self-centering braces for seismic hazard mitigation. Engineering Structures, 2021, 242 : 112527

[16] Fang C, Ping Y, Chen Y, Yam M C H, Chen J, Wang W. Seismic performance of self-centering steel frames with SMA-viscoelastic hybrid braces. Journal of Earthquake Engineering, 2020, 1–28

[17] Fang C, Ping Y, Zheng Y, Chen Y. Probabilistic economic seismic loss estimation of steel braced frames incorporating emerging self-centering technologies. Engineering Structures, 2021, 241 : 112486

[18] Ozbulut O E, Roschke P N, Lin P Y, Loh C H. GA-based optimum design of a shape memory alloy device for seismic response mitigation. Smart Materials and Structures, 2010, 19( 6): 065004

[19] Liu M, Burns S A, Wen Y K. Multiobjective optimization for performance-based seismic design of steel moment frame structures. Earthquake Engineering & Structural Dynamics, 2005, 34( 3): 289–306

[20] Kaveh A, Farahmand Azar B, Hadidi A, Rezazadeh Sorochi F, Talatahari S. Performance-based seismic design of steel frames using ant colony optimization. Journal of Constructional Steel Research, 2010, 66( 4): 566–574

[21] Fragiadakis M, Lagaros N D, Papadrakakis M. Performance-based multiobjective optimum design of steel structures considering life-cycle cost. Structural and Multidisciplinary Optimization, 2006, 32( 1): 1–11

[22] Liang Q Q, Xie Y M, Steven G P. Optimal topology design of bracing systems for multistory steel frames. Journal of Structural Engineering, 2000, 126( 7): 823–829

[23] Stromberg L L, Beghini A, Baker W F, Paulino G H. Application of layout and topology optimization using pattern gradation for the conceptual design of buildings. Structural and Multidisciplinary Optimization, 2011, 43( 2): 165–180

[24] Bobby S, Spence S M J, Kareem A. Data-driven performance-based topology optimization of uncertain wind-excited tall buildings. Structural and Multidisciplinary Optimization, 2016, 54( 6): 1379–1402

[25] Gholizadeh S, Poorhoseini H. Seismic layout optimization of steel braced frames by an improved dolphin echolocation algorithm. Structural and Multidisciplinary Optimization, 2016, 54( 4): 1011–1029

[26] Hassanzadeh A, Gholizadeh S. Collapse-performance-aided design optimization of steel concentrically braced frames. Engineering Structures, 2019, 197 : 109411

[27] Vamvatsikos D, Cornell C A. Incremental dynamic analysis. Earthquake Engineering & Structural Dynamics, 2002, 31( 3): 491–514

[28] Hsiao P C, Lehman D E, Roeder C W. Evaluation of the response modification coefficient and collapse potential of special concentrically braced frames. Earthquake Engineering & Structural Dynamics, 2013, 42( 10): 1547–1564

[29] Moradi S, Alam M S, Asgarian B. Incremental dynamic analysis of steel frames equipped with NiTi shape memory alloy braces. Structural Design of Tall and Special Buildings, 2014, 23( 18): 1406–1425

[30] Shi F, Ozbulut O E, Zhou Y. Influence of shape memory alloy brace design parameters on seismic performance of self-centering steel frame buildings. Structural Control and Health Monitoring, 2020, 27( 1): e2462

[31] Gholizadeh S, Ebadijalal M. Performance based discrete topology optimization of steel braced frames by a new metaheuristic. Advances in Engineering Software, 2018, 123 : 77–92

[32] Gholizadeh S, Hassanzadeh A, Milany A, Ghatte H F. On the seismic collapse capacity of optimally designed steel braced frames. Engineering with Computers, 2020, 1–13

[33] FEMA-P695. Quantification of Building Seismic Performance Factors. Washington, D.C.: Federal Emergency Management Agency, 2009

[34] ASCE41-13. Seismic Evaluation and Retrofit of Existing Buildings. Reston, VA: American Society of Civil Engineers, 2014

[35] McKennaFFenvesG L. OpenSees: The Open System for Earthquake Engineering Simulation. Berkeley, CA: Regents of the University of California, 2013

[36] MathWorks Inc. MATLAB, The Language of Technical Computing. 2019

[37] DesRoches R, McCormick J, Delemont M. Cyclic properties of superelastic shape memory alloy wires and bars. Journal of Structural Engineering, 2004, 130( 1): 38–46

[38] Tanaka Y, Himuro Y, Kainuma R, Sutou Y, Omori T, Ishida K. Ferrous polycrystalline shape-memory alloy showing huge superelasticity. Science, 2010, 327( 5972): 1488–1490

[39] FugazzaD. Use of shape-memory alloy devices in earthquake engineering: Mechanical properties, advanced constitutive modeling and structural applications. Dissertation for the Doctoral Degree. Pavia: University of Pavia, 2005

[40] Qiu C, Du X. Seismic performance of multistory CBFs with novel recentering energy dissipative braces. Journal of Constructional Steel Research, 2020, 168 : 105864

[41] PhamH. Performance-based assessments of buckling-restrained braced steel frames retrofitted by self-centering shape memory alloy braces. In: Georgia Tech Theses and Dissertations. Athens, GA: Georgia Institute of Technology, 2013

[42] Beiraghi H, Zhou H. Dual-steel frame consisting of moment-resisting frame and shape memory alloy braces subjected to near-field earthquakes. Structural Design of Tall and Special Buildings, 2020, 29 : e1784

RIGHTS & PERMISSIONS

Higher Education Press