AbstractEndplate moment connections with shape memory alloy (SMA) bolts provide self-centering for the seismic resilience of structures. Predicting the self-centering response of these new beam–column connections, which have not been codified yet, requires conducting experimental tests or detailed continuum finite-element simulations. Computationally efficient predictive tools are needed to facilitate the analysis, design, and assessment of self-centering connections and moment frames. In this paper, artificial neural networks (ANNs) are used to develop a MATLAB tool for predicting the moment-rotation backbone and self-centering response of extended endplate connections with SMA bolts. As the input for the neural networks, the predictive model development employs a design database of response parameters from 72 finite-element (FE) models and experimental tests of seven beam–column connection specimens. Neural networks are trained for seven response parameters, and the trained networks are used to develop a graphical user interface (GUI). The coefficient of determination for the trained ANNs is in the range of 0.92 to 0.99, indicating acceptable prediction accuracy. Furthermore, optimization studies using a multiobjective genetic algorithm are performed, seeking the minimization of material use (steel and SMA) and improved connection-response characteristics (i.e., stiffness, strength, and ductility). A phenomenological model of SMA connections is also developed in OpenSees. The use of the ANN-based predictive tool for accurate and efficient modeling of SMA-based connections and self-centering moment-resisting frames is illustrated. The computation time for predicting the moment-rotation response of a typical SMA connection is significantly reduced from seven hours in ANSYS to only three minutes in OpenSees while providing the same level of response prediction accuracy. Furthermore, the optimization results are confirmed by performing nonlinear pushover and response history analyses.IntroductionShape memory alloys (SMAs) are a class of metallic alloys that can recover their original shape after undergoing large deformations. The superelasticity or recentering capability of SMAs upon mechanical unloading is due to their material phase transformation. SMAs have found several applications for seismic damage mitigation of civil engineering structures (Wilson and Wesolowsky 2005). Several researchers have previously investigated new applications of SMAs for preventing damage and residual deformation in steel beam-to-column connections. Some examples are connections with SMA tendons (Speicher et al. 2011), SMA ring springs (Wang et al. 2017), superelastic SMA bolts with steel angles (Wang et al. 2017), SMA plates (Moradi and Alam 2015a), and extended endplate connections with superelastic SMA bolts (Fang et al. 2013). Ma et al. (2007, 2008) presented proof-of-concept extended endplate connections in which high-strength bolts were replaced with superelastic SMA bolts (as shown in Fig. 1). Based on the results of their numerical studies, SMA-based extended endplate connections exhibited self-centering while providing a moderate level of energy dissipation capacity. The SMA-based connections proposed by Ma et al. (2007, 2008) were further explored by Fang et al. (2013). They experimentally tested seven SMA-based connection specimens and one conventional extended endplate connection. The test results confirmed that SMA bolts provide self-centering while other components, including endplates, beams, and columns, essentially remain elastic.For predicting the response of self-centering endplate connections with superelastic SMA bolts, continuum finite-element (FE) models (e.g., Mohammadi Nia and Moradi 2020; Yam et al. 2015) can be used. However, the development of high-fidelity FE models of connections is computationally expensive. The use of these models for simulating the response of self-centering frames is thus practically impossible. Developing computationally efficient predictive tools is necessary to facilitate the rapid analysis and performance-based design and assessment of SMA-based connections or frames in research and practice. Previously, the authors used a response surface method (RSM) to develop surrogate models for endplate connections with SMA bolts (Mohammadi Nia and Moradi 2021). However, the models were limited to a beam depth of 450 mm to 610 mm. This paper employs an artificial neural network (ANN), which better captures nonlinearity than RSM (Hammoudi et al. 2019; Yaro et al. 2022), to develop a new publicly accessible predictive tool for efficiently modeling SMA-based connections and self-centering frames. The development and verification of this tool are presented, along with the illustration of its use for modeling SMA-based connections and moment frames. A broader range is considered for the beam depth, and experimentally tested specimens are included in the database.Artificial intelligence (AI), notably machine learning (ML), has been used widely in different areas of structural engineering to solve different problems. Some examples are predicting the resistance of structural members (Caglar 2009; Degtyarev and Naser 2021; Zarringol et al. 2021, among others), fire resistance of structures (e.g., Naser et al. 2021; Xu et al. 2012), structural health monitoring, and damage detection (e.g., Huang and Burton 2019; Sediek et al. 2022).In traditional statistical regression techniques, a predefined structure of the regression model is required; however, in many cases, this approach is not the optimal approach (Majidifard et al. 2019). ML techniques can derive the relationship between inputs and outputs without prior assumptions (Naser and Alavi 2021). Regression can be performed using different ML algorithms, such as neural networks and decision trees (Salehi and Burgueño 2018).ANN has been extensively used in engineering for capturing the relationship between input and output variables involved in a system (e.g., Naderpour et al. 2018; Naser et al. 2012). ANN can be efficiently applied for predicting the response of beam–column connections that can be used efficiently for the simulation of self-centering building structures. The first application of ANN for predicting the response of steel connections dates back to the ‘90s when Abdalla and Stavroulakis (1995); Stavroulakis et al. (1997) estimated the mechanical behavior of semirigid connections and proposed a moment-rotation constitutive law using neural networks. Anderson et al. (1997) applied ANN for predicting the bilinear moment-rotation response for minor-axis connections. Yun et al. (2008) proposed a design-variable-based inelastic hysteretic model for beam-to-column connections using neural networks. Their results demonstrated that the proposed module could reproduce the experimental data with reasonable accuracy for extended endplate connections and top-and-seat angles with double-web angle connections. In a study by Shah et al. (2018), 32 boltless connections were tested to propose predictive models for the moment-rotation response of the connections. For developing predictive models, they used three methods, including linear genetic programming, ANN, and an adaptive neurofuzzy inference system. Horton et al. (2021) predicted the cyclic response of reduced beam section (RBS) connections using neural networks. Recently, Kim et al. (2021) used ML techniques, including support vector regression (SVR) and deep neural networks, to predict the strengths of steel circular hollow section X-joints. Their results showed that deep neural networks provided more accurate results compared with SVR.Despite the fast-growing interest in using AI and ML techniques, their application for predicting the self-centering response of SMA-based steel connections and frames has not been widely explored. This paper aims to develop predictive models for the response of SMA-based extended endplate connections using ANN. The proposed models are then implemented to develop a MATLAB tool for response prediction of SMA-based connections. The study is then expanded to optimize SMA-based connections to improve the connection response characteristics while minimizing the amount of steel and SMA materials. Next, the developed predictive tool is used to create a computationally efficient and accurate phenomenological model for the SMA connections in OpenSees version 3.3.0. Finally, the accuracy of the developed predictive modeling and optimization results is confirmed.Scope and MethodologyIn this study, a design database from FE simulation and experimental tests of SMA-based extended endplate connections (Fang et al. 2013) was used to propose predictive models for the connections’ moment-rotation response parameters, including θB, θC, θE, MB, MC, ME, and β as shown in Fig. 2. This figure shows a schematic view of the idealized response curve [i.e., self-centering response and backbone curve consistent with the generalized backbone curve in ASCE 41-17 (ASCE 2017)]. These response parameters were selected to idealize the backbone and self-centering response of endplate connections with SMA bolts on the basis of experimental (e.g., Fang et al. 2013) and FE analysis results (e.g., Mohammadi Nia and Moradi 2020). To create a response curve for the connections, SMA materials were assumed to fracture upon reaching strain εfr (Mohammadi Nia and Moradi 2021; Sabouri Ghannad et al. 2021): (1) where εL = maximum transformation strain of SMA bolts; σMf = martensite finish stress; and ESMA = modulus of elasticity of SMA material. SMA bolts fracture is more likely to occur at a strain level around εfr (Wang et al. 2017).In the proposed backbone curve, as shown in Fig. 2, from point A to B, the connection has an elastic behavior. From point B onward, the SMA material enters a forward transformation phase. At point C, the outmost SMA bolts reach their fracture strain, which in turn results in a drastic reduction in the moment capacity of the connection. Following point C, the loading continues to achieve the fracture strain in the second row of the SMA bolts. Additionally, a return path is included using β to characterize the self-centering response.Ten factors have been identified as influential factors in the cyclic response of SMA-based connections (Mohammadi Nia and Moradi 2020). These influential factors are listed in Table 1. The factor ranges were selected while ensuring that any developed connection using these factor ranges meets the seismic requirements of the American Institute of Steel Construction [AISC 341-16 (AISC 2016b); AISC 358-16 (AISC 2016a)] for extended endplate connections. The endplate thickness has a constant value of 40 mm to ensure the thick-plate behavior by the endplate. The slenderness ratio of the beam and column flanges and webs were considered based on the limits for highly ductile sections as per AISC 341-16 (AISC 2016b). Additionally, the requirements in AISC 358-16 (AISC 2016a) for extended endplate connections were checked, including the restrictions or requirements for beam and column sizes, beam clear span-to-depth ratio, lateral bracing for beams and columns, and the strong-column-weak-beam requirement. Moreover, the column dimensions were set to have strong-column-weak-beam behavior with no need for doubler and continuity plates as per AISC 341-16 (AISC 2016b). Column web and flange thicknesses are 30 mm and 35 mm, respectively. Based on the results of a statistical sensitivity analysis (Mohammadi Nia and Moradi 2020), column-related factors do not significantly affect the response characteristics of the connection. This is also expected because the column and particularly the panel zone is designed to remain elastic ensuring that large deformations are confined to the SMA bolts. Therefore, considering the same column web and flange thicknesses in all the connections should not impact the conclusions from the study. With these significant factors, a data set of 72 factor combinations generated using the design of experiments (Mohammadi Nia and Moradi 2021) was considered. For each factor combination, two analyses were performed in ANSYS mechanical APDL (ANSYS 2020). In the first analysis, the connection was loaded up to 0.08 rad rotation to capture the rotation at which the outermost bolts reach their fracture strain as defined in Eq. (1). A second analysis was next performed on the same FE model of the connection but without the outmost (i.e., first-row) SMA bolts. In fact, the FE model represents a connection with its outmost SMA bolts fractured (thus removed from the analysis). The connection model was loaded up to 0.10 rad rotation to capture the rotation at which second-row SMA bolts were fractured. From the second analysis, the residual strength of the beam–column connection (indicated by line DE in Fig. 2) was obtained. A backbone curve similar to the one in Fig. 2 was obtained for each factor combination by assembling the backbone curves achieved from FE simulations.Table 1. Factors and ranges consideredTable 1. Factors and ranges consideredFactorSymbolLow levelHigh levelUnitMartensite start stressσMs280380MPaMartensite finish stressσMf410590MPaAustenite start stressσAs170250MPaAustenite finish stressσAf70138MPaMaximum transformation strainεL0.070.13—Bolt pretension strainεpt0.0050.015—Bolt lengthLbolt300350mmBolt diameterDbolt1025mmBeam depthhbeam150610mmBeam lengthLb1,5004,500mmIn addition to the data set developed using the design of experiments and FE simulations, results from seven experimental tests performed by Fang et al. (2013) were considered. For the specimens experiencing early fracture in the experimental tests (due to their low net threaded-to-shank diameter ratio of 1), θC was not quantified.Next, ANN was used to develop a predictive model for the response parameters for SMA-based extended endplate connections. The inputs to the neural networks were those ten design parameters listed in Table 1. To characterize the backbone and self-centering response, a network was trained for each response parameter, including θB, θC, θE, MB, MC, ME, and β. For each response parameter, separate ANNs were developed, resulting in higher prediction accuracy in contrast to training a single ANN for all the response parameters. The trained networks were then assembled to have a graphical interface for predicting the response.Additionally, to study the simultaneous response improvement and cost-effectiveness of the SMA-based connections, multiobjective optimizations were performed. A multiobjective evolutionary algorithm based on Pareto-dominance was adopted in which the selected objectives were optimized simultaneously. A desirability approach was adopted to rank the optimal solutions in a Pareto front set and thereby to find and propose the solution with the highest rank as the optimal factor setting. The desirability approach was also used to propose optimal ranges for factors of interest, including beam depth and SMA bolt length and diameter. Two optimization problems were defined to improve the response of SMA-based connections and reduce the amount of SMA and steel materials, thus resulting in cost-effective connections with improved performance.Using the proposed backbone curve, a phenomenological model was developed and validated for two experimentally tested SMA-based connections. The efficiency and accuracy of the developed MATLAB tool and proposed phenomenological models were confirmed using two beam-to-column connections. For each connection, the backbone curve was developed using the MATLAB tool, and then, fiber and detailed FE models were developed in OpenSees and ANSYS, respectively. The hysteretic responses obtained from ANSYS and OpenSees, along with predicted backbone curves, were compared. Finally, system-level modeling was performed. Pushover and nonlinear response history analyses were performed to confirm the optimization study results and to assess the connections’ behavior in steel frames.FE Modeling and ValidationThree-dimensional FE models were developed using ANSYS mechanical APDL (ANSYS 2020), as shown in Fig. 3(a). SOLID185 elements were used to model the beam, column, SMA bolts, and stiffeners. SOLID185 has eight nodes with three degrees of freedom at each node: translations in the nodal x, y, and z directions. Finer meshes were used in the connection interface, in which inelastic deformations may occur, whereas for the rest of the components, coarse meshes were assigned to have efficient and yet accurate FE models. CONTA174 and TRAGE170 were used to define contact surfaces between the endplate and column flange, as well as the probable contact between SMA bolts and holes within the endplate and column flange. In defining contacts, the penalty function was used as the contact algorithm, while the stiffness of the contact was set to be updated in each iteration according to the mean stress of the underlying elements. Superelastic behavior was adopted for SMA materials, while trilinear behavior was used to model steel materials (Mohammadi Nia and Moradi 2020). Large deformation and nonlinear materials were activated in the developed models to account for geometric and material nonlinearities, respectively. In the FE models, geometric imperfections and the effect of residual stresses were not included. In order to avoid convergence difficulties, the number of substeps for the load step, in which the convergence issue could occur, was increased to apply the load slowly. Furthermore, an unsymmetric Newtown–Raphson algorithm was used to avoid convergence difficulties (Moradi and Alam 2015b). ANSYS default convergence tolerances of 0.5% and 5%, respectively, were adopted for force and displacement. The moment-rotation response from the FE models was compared with experimental results [Fig. 3(b)]. A good agreement between the FE results and experimental results was observed. Further details of the FE modeling and validation studies are available elsewhere (Mohammadi Nia and Moradi 2020, 2021).Design Database GenerationFrom the design of the experiment method, an I-optimal design was used to efficiently generate the design database for the predictive modeling development. The generated database constitutes sampling points or factor combinations with the factors and associated ranges in Table 1. In optimal designs, a numerical criterion, i.e., the variance or any statistical properties of the design, is optimized. An I-criterion is one of the most well-known designs that is used when the experimental goal is to make precise predictions of the response rather than to obtain precise estimates of the model parameters (Smucker et al. 2018). An I-optimal design is more appropriate for developing predictive models based on computer simulations. In this design, a set of runs, i.e., factor combinations, are selected that minimize the integral of the prediction variance across the factor space (Montgomery 2006).Artificial Neural Network Predictive ModelsInspired by the human nervous system, ANN is a computational model that contains multiple layers of artificial neurons connected with coefficients (weights). ANN can model complex problems with many parameters when being trained by proper training data (or exemplars). ANN consists of interconnected neurons, which are processing elements with similar characteristics such as input, synaptic strength, activation output, and bias. The processing units, including input, hidden, and output layers, carry the weights of the network. The training process of a network is associated with adjusting the weights to optimize the desired loss function, which is usually defined as the prediction error.ANN for Predicting Response ParametersANNs were used to determine the nonlinear relationship between the input and output parameters of the response curve for SMA-based extended endplate connections. To that aim, a single layer perceptron (i.e., a feed-forward network with a single hidden layer) was used, as shown in Fig. 4. The backpropagation algorithm was used to train the network where the output errors were propagated back by means of the same connections used in the feed-forward mechanism. In this method, the gradient of the error with respect to the weights is calculated, and then the weights are updated using the popular gradient descent algorithm. In the proposed neural networks, the neurons were placed in three separate layers, including the input layer, hidden layer, and output layer. The input layer has ten neurons, reflecting 10 design parameters as listed in Table 1. Neurons in the input layer pass the scaled input data to the hidden layer using weights. The hidden layer has a different number of neurons for each response variable, obtained using trial and error to result in high accuracy. The number of hidden layers was changed from 1 to 3, while the number of neurons in the hidden layers was changed from 1 to 20—in searching for the best (or optimal) ANN with the minimum observed mean squared error. Using a single hidden layer for all the trained networks outperformed models with 2 or 3 hidden layers. Table 2 lists the number of neurons used in the trained ANNs. The training was repeated with different numbers of hidden layers and neurons until the best ANNs were found. The optimal ANNs were then used for developing the predictive tool.Table 2. The details of the ANN modelsTable 2. The details of the ANN modelsResponse parameterNumber of neurons in the hidden layerAllTrainTestAllTrainTestθB50.960.960.950.920.930.91θC70.980.990.950.960.980.91θE70.990.990.980.970.990.96MB51.001.000.990.990.990.99MC70.991.000.990.991.000.99ME70.991.000.990.991.000.99β30.980.990.980.970.970.97The output layer comprised a single neuron that represents the backbone curve parameter. In this study, the hyperbolic tangent sigmoid transfer (tansig), i.e., g(x)={2/[1+e(−2x)]}−1, which generates an output between −1 and 1, was used as an activation function for the hidden layer, while the pure linear function (purelin), which generates outputs between −∞ and +∞, was used as the activation function in the output layer. The goal of using nonlinear transform functions like the tangent sigmoid transfer is to provide the network with the capability of learning the nonlinear behavior between the input and output data. Since the networks in the study were shallow networks, we used the tangent sigmoid function (Szandała 2021). As shown in Fig. 4, each neuron in a layer is connected to the neurons in the next layer, whereas there is no interconnection between neurons.The design matrix developed using the I-optimal design, along with the experimental data from Fang et al. (2013), was used as the input matrix for neural networks. The neural network tool in MATLAB was utilized to develop the multilayer perceptron architecture of feed-forward ANN. The developed design matrix using the I-optimal design and the experimental data contains 79 factor combinations from which 70% of the data were used to train the networks, 15% of the data was used for validation, and the last 15% of the entire database was used for testing the networks. The performance of the networks was evaluated using the coefficient of correlation (R) and coefficient of determination (R2). Note that R, which estimates the relationship between model output and actual values, ranges from −1 to 1. For R and R2, close-to-one values are desirable. The coefficient of correlation (R) measures the strength of the relationship between input and output data. R was used to find patterns and relationships in the data. Further, the coefficient of determination (R2) shows how well the regression model fits the observed data. R2 was used to evaluate predictions and see how much of the variation in the FE and experimental data is explained by ANNs. It should be mentioned that mean squared error (MSE), measuring the average of the squares of the errors (Naser and Alavi 2021), was used implicitly in the training process of the ANNs. MSE was used as the performance metric for the validation data. Minimum MSE for the validation data was considered as a measure to select the weights in the ANNs and as an indicator for the best performance of the ANNs. The input data for all neural networks were normalized to lie within a range of 0 to 1. The reason for normalizing the data was to prevent the saturation of the tansig activation function, which could stop the network from further learning (Fausett 2006). Eq. (2) is used to normalize the inputs of neural networks (2) where φm = normalized data; and x = original data.ANN ResultsFor each response parameter, an optimized feed-forward neural network was trained using the Levenberg–Marquardt algorithm. This training process automatically stops as soon as the generalization stops improving, and this is signaled by the increase of MSE in the validation samples. The number of epochs is kept minimum during the development of the optimal topology of the networks to avoid overfitting where the network perfectly fits the training data, but its performance for the testing set is not satisfying.Based on the FE analysis results and experimental data, seven neural networks were trained (one for each backbone curve parameter). Table 2 lists the number of neurons that were used in the hidden layer for each neural network associated with different backbone parameters. As listed, the coefficient of correlation, R, between actual and predicted values for the trained neural networks is quite high, ranging from 0.96 to 1.0. Moreover, the coefficient of determination, R2, is calculated for the trained networks. R2 measures the proportion of the variance in the response variable that is predictable from the trained ANN. R2 ranges from 0.92 to 0.99.Fig. 5 presents the predicted versus actual responses for the training, testing, and validation sets for different response curve parameters. As shown, the data are clustered along the diagonal line, indicating the accuracy of the trained neural networks.Following the framework by Naser et al. (2021), a benchmarking study is presented to apply six commonly used supervised learning algorithms with default settings to the database for SMA connections. The algorithms include decision trees (DT), random forest (RF), extreme gradient boosted trees (ExGBT), light gradient boosted trees (LGBT), TensorFlow deep learning (TFDL), and Keras deep residual neural network (KDP). For consistency with the trained ANNs in the paper, one model was trained for each response parameter. R2 and root MSE (RMSE) were used as the performance metrics as listed in Table 3. Note that the trained algorithms are not finetuned (Naser et al. 2021). Comparing the performance of the algorithms (while excluding the trained ANNs) shows that no single algorithm outperforms all the examinations. However, by considering the overall performance of the algorithms, ExGBT and KDP are the first and second best-ranked algorithms, while TFDL and DT have the poorest performance. In general, the trained ANNs in this study outperform other algorithms; nonetheless, it should be noted that the algorithms in Table 3 are not finetuned and this comparison is not fair. Finetuning the algorithms could result in more accurate predictions. Further research is warranted to find the best algorithm for each response parameter.Table 3. Performance of the trained ML algorithms using the developed databaseTable 3. Performance of the trained ML algorithms using the developed databaseParametermetricTrainTestTrainTestTrainTestTrainTestTrainTestTrainTestTrainTestθBR21.000.790.890.781.000.880.920.860.860.680.790.520.930.91RMSE0.000.00100.00080.00140.00010.00100.00070.00110.00100.00160.00120.00150.00070.0010θCR21.000.550.870.741.000.820.940.910.610.420.930.890.980.91RMSE0.00060.00770.00410.00730.00070.00450.00310.00300.00870.00900.00300.00400.00200.0040θER21.000.760.910.911.000.860.950.930.640.280.930.920.990.96RMSE0.000.0100.0050.0050.000.0060.0040.0040.0080.0180.0050.0040.0010.004MBR21.000.980.970.921.000.990.980.970.920.901.000.990.990.99RMSE033.435.560.51.223.332.342.158.470.312.417.214.917.0MCR21.000.940.980.961.000.990.980.980.870.711.000.991.000.99RMSE0.3772.937.756.61.026.843.141.0102.5142.38.317.26.326.4MER21.000.950.970.941.000.970.970.940.970.911.001.001.000.99RMSE136.025.132.20.729.023.645.225.655.06.09.39.09.1βR21.000.900.910.841.000.900.920.910.730.620.910.890.970.97RMSE0.000.030.030.050.00.030.030.030.060.060.400.300.020.02MATLAB Graphical User InterfaceThe trained neural networks were used to develop a MATLAB graphical user interface (GUI) (Mohammadi Nia and Moradi 2022) for the response prediction of SMA-based extended endplate connections. When the goal is to only determine the moment-rotation response, the developed tool eliminates the need for a detailed FE modeling of self-centering connections and frames. The tool is also useful for the efficient analysis of frames with SMA connections. As shown in Fig. 6, 10 significant parameters are required to be entered into the developed tool. Note that the developed predictive tool is applicable only for the factor ranges listed in Table 1. In developing the predictive tool, the beam flange (tbf=19 mm), beam web (tbw=15 mm), column flange (tcf=35 mm), and column web (tcw=30 mm) thicknesses were kept constant. These factors do not significantly influence the connection response based on statistical sensitivity analysis results (Mohammadi Nia and Moradi 2020)—given that the slenderness ratios for beam and column flanges and webs meet the requirements of AISC-341-16 (AISC 2016b) for highly ductile sections and the requirements of AISC-358-16 (AISC 2016a) for extended endplate connections. As a result, in properly designed SMA connections, the beam and column remain elastic, and the connection response will not be affected by using different flange or web thicknesses for beams and columns. An unloading path is also included to facilitate implementing the proposed response curve in other software programs, such as OpenSees, which is discussed subsequently in this paper.Multiobjective Response OptimizationThe proposed ANNs were used to perform response optimization studies. Optimal factor settings were determined to improve the connection response characteristics, such as the initial stiffness and the moment and rotational capacities with minimized steel and SMA material usage. The input space of the neural networks was optimized using a genetic algorithm. A general multiobjective optimization problem can be formulated as follows (Deb 2011):Minimize/Maximize fm(x)(3) Subject to gj(x)≥0,hk(x)=0,xi(L)≤xi≤xi(U)m=1,2,…,M;j=1,2,…,J;k=1,2,…,K;i=1,2,…,Iwhere x = vector of decision variables containing the input factors, that is, (σMsσMf⋯hbeamLb)T. The functions gj(x) and hk(x) are inequality and equality constraint functions, respectively; M = number of optimization objectives; and I = number of decision variables. Further, J and K are the numbers of constraints. The constraints for xi are variable bounds based on the factor ranges in Table 1.In multiobjective optimization problems, there is no unique solution; rather, there is a set of optimal trade-off solutions called the Pareto front (Ngatchou et al. 2005). A solution belongs to a Pareto set, given that there exists no other solution that can improve at least one of the objectives without deteriorating any other objectives. The optimal solutions have no mutually dominated relationship. In addition, any solution outside the Pareto front is always dominated by a solution in the Pareto front. Fig. 7 illustrates a schematic view of the Pareto front for a two-objective optimization problem with different objectives where the Pareto frontiers are highlighted in red.In this study, MATLAB solver gamultiobj, which is a controlled elitist genetic algorithm, is adopted. In a controlled elitist genetic algorithm, individuals that can help to increase the diversity of the population (a subset of solutions) are preferred even with a lower fitness value, in which the fitness value shows the accuracy level of the solution.Optimization ObjectivesTwo multiobjective optimization problems were considered. The first optimization problem was defined to enhance the response characteristics of SMA-based steel connections while reducing the amount of SMA and steel materials. The second optimization, however, only seeks to improve the response of the connection regardless of the amount of material.In the first optimization problem, the amount of SMA and steel materials were minimized in addition to maximizing the initial stiffness and the deformation and moment capacities (θC and MC) of the connection. The optimization was set to increase the connection’s initial stiffness knowing that reducing the flexibility of moment-resisting frames (MRFs) is favorable during earthquakes. Given the inherent low lateral stiffness of MRFs, buildings experience higher lateral drifts. As a result, reducing the flexibility of MRFs is favorable during earthquakes. Using SMAs, which possess a lower modulus of elasticity compared to steel, exacerbates the low lateral stiffness of the system. Therefore, increasing the lateral stiffness of MRFs, particularly with SMA-based connections, is favorable and needed.Moreover, the amount of materials used in a beam–column connection contributes to the total cost of the connection. Hence, the first optimization problem was defined to minimize the amount of steel (i.e., the beam depth) and SMA (i.e., bolt length and diameter).The second optimization problem seeks solutions that improve the response characteristics of the SMA-based endplate connection by maximizing the rotational capacity (θC) and the moment capacity (MC). Increasing the deformation capacity of SMA-based beam-column connections is important for providing adequate ductility. Past research has shown that early SMA bolt fracture results in lower deformation capacity of the SMA-based steel connections (Fang et al. 2013).Optimal Point Selection and Proposed Optimal RangesDesirability functions were used to select the most optimal solution and to prioritize the points in a Pareto optimal set. In this ranking method, each objective is normalized using the following equations for maximization and minimization problems, respectively: (4) In maximization problems: d={0y

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