Artificial Neural Network–Based Predictive Tool for Modeling of Self-Centering Endplate Connections with SMA Bolts

Shape 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.

In 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 ${\theta }_B$, ${\theta }_C$, ${\theta }_E$, $M_B$, $M_C$, $M_E$, and $\beta$ 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 ${\epsilon }_{fr}$ (Mohammadi Nia and Moradi 2021; Sabouri Ghannad et al. 2021):where ${\epsilon }_L$ = maximum transformation strain of SMA bolts; ${\sigma }_{Mf}$ = martensite finish stress; and $E_{SMA}$ = modulus of elasticity of SMA material. SMA bolts fracture is more likely to occur at a strain level around ${\epsilon }_{fr}$ (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 $\beta$ 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.

In 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), ${\theta }_C$ 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 ${\theta }_B$, ${\theta }_C$, ${\theta }_E$, $M_B$, $M_C$, $M_E$, and $\beta$. 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.

Three-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).

From 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).

Inspired 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.

The 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 $f_m(\mathit{x})$where $\mathit{x}$ = vector of decision variables containing the input factors, that is, ${(\begin{array}{ccccc}{\sigma }_{Ms}& {\sigma }_{Mf}& \cdots & h_{beam}& L_b\end{array})}^T$. The functions $g_j(\mathit{x})$ and $h_k(\mathit{x})$ 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 $x_i$ 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.

This section presents the development and verification of a phenomenological model for SMA-based extended endplate connections. The proposed OpenSees model, along with the MATLAB predictive tool, was then used to develop high-fidelity FE models for steel frames equipped with SMA-based connections.

OpenSees (2013) was used to develop a simple yet efficient numerical model for SMA-based extended endplate connections. The connection was modeled using rotational springs with Self-Centering, Pinching4, and Steel01 material models acting in parallel. Fig. 11 shows the proposed phenomenological model. The accuracy and effectiveness of the developed model were verified by modeling the experimentally tested connection specimens SMA-D10-240d and SMA-D10-290 (Fang et al. 2013). The material properties for the Self-Centering, Pinching4, and Steel01 models were selected using experimental backbone curves (Fang et al. 2013). An iterative process was used to adjust the percentage contribution of the material models. Through this iteration, a linear combination of Self-Centering, Pinching4, and Steel01 was found and assigned with the coefficients of 0.90, 0.05, and 0.05, respectively. In the proposed phenomenological model, the beam and column were modeled using elastic beam–column elements. As shown in Fig. 12, there is a good agreement between the OpenSees and experimental results.

After verifying the OpenSees modeling, we demonstrate the efficiency of the developed MATLAB version R2020b tool and confirm its prediction accuracy. For this purpose, the MATLAB tool was first used to predict the backbone response for two beam-column connections, including a connection with randomly selected details and the most optimal connection found in the second optimization study. The predicted response was next compared with the numerical results from the fiber-based and detailed solid models using OpenSees and ANSYS, respectively. For each beam-column connection, the backbone curve was developed using the predictive MATLAB tool. The backbone curve parameters from the developed MATLAB tool were then introduced in OpenSees to create fiber-based models. Fig. 13 shows the hysteretic response of the developed ANSYS and OpenSees models along with the predicted backbone curve. The response predictions using the MATLAB tool and the OpenSees modeling agree well with the cyclic response from the detailed ANSYS models.

The results of the verification study in Fig. 13 also confirm the insignificant influence of the panel zone. The response from the OpenSees model matches that from ANSYS, which explicitly captures any deformation and inelasticity in the panel zone. In fact, SMA-based connections are designed for maximum recentering capability. As a result, the beam and column (including the panel zone) remain elastic to confine large deformations in the SMA bolts and thus prevent any damage to other components of the connection. Moreover, based on the results of a statistical sensitivity analysis (Mohammadi Nia and Moradi 2020), the panel zone-related factors are not influential on the hysteretic response of the connections that are designed to confine deformations in the SMA bolts. Therefore, given the design with strong panel zones, their effect on the response is insignificant.

The described OpenSees modeling, along with the response prediction from the MATLAB tool, greatly facilitates the efficient analysis of self-centering connections and frame structures. This efficiency becomes notable when comparing the analysis runtime of the ANSYS and OpenSees models. For a typical beam–column connection, an ANSYS model with 80,631 elements and 92,705 nodes requires a total runtime of 6 h and 48 min using a 64-bit operating system in Windows 10 with a six-core Intel Xeon W-2135 CPU at 3.70 GHz and with 16 GB of RAM. In contrast, the OpenSees analysis for the same connection takes only 3 min to complete while providing the same level of response prediction accuracy as in the ANSYS model. This demonstrates the efficiency of using the predictive models to significantly reduce the runtime while maintaining accuracy. However, it should be noted that the phenomenological model only predicts the moment-rotation response of the connection. Detailed continuum FE modeling, in contrast, provides additional information that is not available from the simplified phenomenological model.

Note that the use of the design of experiments method for generating the factor combinations can make the study more efficient. The verification results in this section indicate that 79 samples are sufficient to train ANNs with high prediction accuracy.

This section illustrates the use of the developed predictive tool for modeling steel moment frames with SMA-based connections. Further, the optimization results were confirmed by evaluating the system-level response of frames equipped with optimized beam-column connections.

In this study, ANNs were used to develop predictive models for self-centering response and backbone curve parameters of extended endplate connections with SMA bolts. The predictive models were developed based on an input database comprising results from 72 FE models and seven tested specimens. With 10 input parameters for the neural networks, the response of SMA-based connections was characterized using seven parameters, including rotation (${\theta }_B$, ${\theta }_C$, ${\theta }_E$), moment ($M_B$, $M_C$, $M_E$), and self-centering response parameter ($\beta$). For each of these response parameters, a neural network was trained. The trained neural networks were then used to develop a GUI for predicting the response of SMA-based endplate connections. The results demonstrated the high prediction accuracy of the proposed model for each response.

The predictive models were then used to perform two optimization studies based on a multiobjective genetic algorithm. The optimization objectives included minimizing the material use and improving the connection response characteristics. For each optimization problem, a set of trade-off optimal solutions (Pareto front) were determined. A desirability approach was then used to select the optimal solutions with the highest desirability scores. The predictive models were also used to develop a phenomenological model for SMA connections in OpenSees. The use of the predictive tool was illustrated by developing OpenSees models and performing pushover and response history analyses on SMA connections and self-centering frames. The following conclusions are summarized:

The developed predictive tool should be used only for endplate connections with SMA bolts and only with the factor ranges in Table 7. Furthermore, the SMA connection is assumed to be designed with a thick endplate to confine large deformations into the SMA bolts. Beams and columns should be checked to meet the requirements of AISC 341-16 (AISC 2016b) for highly ductile sections. The AISC 358-16 (AISC 2016a) requirements for extended endplate connections should also be checked, including the requirements for the beam and column sizes, beam clear span-to-depth ratio, beam and column lateral bracing, and strong-column-weak-beam. Additionally, the self-centering response prediction in this study neglects small residual deformations of SMA materials.

Open-source platforms (e.g., Jupyter notebook) could be used in future research for developing predictive models. Moreover, the developed ANNs and predictive tools in this paper do not account for different sources of uncertainty, such as the randomness of measurement and material. Future research is recommended to quantify uncertainties and evaluate their effects on response predictions developed based on deterministic approaches. In the case of finding significant variations, attempts can be made to apply adjustment or safety factors to the response values predicted using a deterministic approach. Further research can finetune other ML algorithms and possibly improve the accuracy of the developed predictive tool.

The data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request. The predictive tool generated in the study is publicly available in a Zenodo repository (Mohammadi Nia and Moradi 2022) in accordance with funder data retention policies.

The financial contributions of the Toronto Metropolitan University Faculty of Engineering and Architectural Science and the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grant are gratefully acknowledged.