Experimental and numerical analysis of beam to column joints in steel structures

Abstract

The behaviors such as extreme non-elastic response, constant changes in roughness and resistance, as well as formability under extreme loads such as earthquakes are the primary challenges in the modeling of beam-to-column connections. In this research, two modeling methods including mechanical and neural network methods have been presented in order to model the complex hysteresis behavior of beam-to-column connections with flange plate. First, the component-based mechanical model will be introduced in which every source of transformation has been shown only with geometrical and material properties. This is followed by the investigation of a neural network method for direct extraction of information out of experimental data. For the validation of behavioral curves as well as training of the neural network, the experiments were carried out on samples with real dimensions of beam-to-column connections with flange plate in the laboratory. At the end, the combinational modeling framework is presented. The comparisons reveal that the combinational modeling is able to display the complex narrowed hysteresis behavior of the beam-to-column connections with flange plate. This model has also been successfully employed for the prediction of the behavior of a newly designed connection.

Keywords

beam to column connections / experiments / component method / neural network model / combinational modeling

Introduction

Worldwide, steel structures are commonly used due to their resistance to dynamic loads. The similarity between the predicted and actual behavior of these structures is contingent upon the validity of the design assumptions and details of the components of the designed structure. One of the most important components in a steel structure is its connections. Connections play a pivotal role in the determination of the overall behavior of a structure. It is also observed that steel structures experiences the highest level of damage in structures under extreme loads including earthquakes. Indeed, damage to the connection is a prelude to damage to other parts of the structure. Therefore, the necessity of conductance of studies in the area of seismic loading is truly sensible [¹–⁴]. Over the past few years, a large number of studies were performed in the area of seismic loading. The use of steel and composite joints is inherent in every structural steel and composite building, whether it is of one story or one hundred stories. Therefore, the beam-to-column connection, due to its importance to all constructions, is significant both economically and structurally. Saving in connection costs as well as improved connection quality has an impact on buildings of all sizes. Because of the repetitive nature of connections, even minor material or labor savings in one connection are compounded and expanded throughout the entire building. It is important for a design engineer to understand the behavior of the connection, not only from the point of view of the connection as a structural element, but also from the point of view of the connection as a part of the complete structural system.

Bending frames which have high energy depreciation ability are widely applied in earthquake-prone places. The behavior of beam-to-column joints in steel frames can be conveniently represented by its flexural behavior which is primarily shown by the moment–rotation relationship. This behavior is non-linear even at low load levels. In fact, moment–rotation curves represent the result of a very complex interaction among the elementary parts constituting the connection. In these frames, the beam to column connection is assumed to be rigid where the lateral load is tolerated by the bending of the beam and column. Crisp and unexpected fracture at the connection site and on the beam connection column in the Northridge (1994, USA) and Hyogo-Ken (1995, Japan) Earthquakes caused the beam to column connections with flange plate to attract a great deal of attention as a suitable substitute for resistant-to-earthquakes frames in regions with high seismicity. In the design of beam-to-column connections with flange plate, the objective is the formation of a plastic joint in the beam and moving away from the connection site as well as prevention from its occurrence in the column because of maintaining the structure’s lateral stability. In this type of connection, the negative bending moment in the connection is transferred to the column wing through a force couple in the form of tension in the upper sheet and pressure in the lower one. To use these rigid connections, it is required that the experimental samples are required to be made of different beam to column connections and to undergo different loadings. However, due to the wide spectrum of variations in the geometrical properties and the type of material of connection pieces as well as the variety of loads exerted onto the frames, development of experimental samples is not possible for all the states, requiring application of theoretical methods and models. Factually, there are different models for connection theory including mechanical, limited components, neural network, and combinational models. In many fields such as structural reliability, material modeling, (mathematical) numerical models are used to predict the response of a system. Due to the increasing computer power, the complexity of the model is growing. Generally, the more complex the models are, the larger becomes the uncertainty in the model outputs due to random in the input parameters. It is essential to determine how much the model outputs changed by the variation in input parameters as well as calibrate and validate the mathematical models [⁵–¹⁶]. In this research, the framework of combinational modeling of the advanced beam to column connection model with flange plate is presented that is capable of predicting the complex hysteresis behavior.

Numerical analysis

Mechanical modeling

There are numerous modeling methods of beam to column connections in the scope of mechanical modeling. One of the methods to estimate the behavioral curve of the connections is the component-based method. A component-based model indicates a moment–rotation relation through superposition of the contribution of every key component of a connection. In this method, each of the transformation mechanisms in the connection is specified. In addition, their rigidity is determined individually through experimentation or by regulations in every component. The rigidity of each of these components is modeled by linear or nonlinear springs. The set of these springs is then assembled serially or in parallel to determine the rigidity of the connection. A component-based model indicates a moment–rotation relation through superposition of the contribution of every key element of a connection. Every element then represents a transformation source using a mathematical expression. For this reason, it is required to first identify all of the transformation sources and the potential sources of failure in the connection. Accordingly, the relation accounting for every element is obtained to achieve the properties of its transformation. Finally, an effective assembly is required from all of the components through establishing the balance and compatibly conditions to achieve accuracy and strength in the component-based model [¹⁶–²⁴].

In component method, modeling is done in the form of a macro-element through combining rigid rods and springs that represent the constructive relations between the elements. Component-based methods employ the combination of the rigid and spring elements in which, the spring element can reveal the transformation source of an individual component. The elements are usually modeled mechanically using the geometrical and material properties. Eurocode 3 [²⁵] was the first code to accommodate the concept of components for the determination of the properties of connection design. However, prediction of the complex hysteresis response has still remained a challenge.

Several mechanical models have been proposed for different connection types. Wales and Rossow [²⁶] founded the component-based method. Then Tschemmernegg and Humer [²⁷] developed a component-based model to use three series of springs for welding connections with terminal panes. Madas and Elnashai [²⁸] proposed an analytical component-based model out of which the general moment–rotation model has been developed by superposition of the contribution of elements. De stefano et al. [²⁹] proposed a mechanical model from connections with beam-web double-angle using the geometrical properties and the specifications without the effects of slippage. In this section, every element of beam-to-column connection with flange plate considered as one of the main transformation sources in this research, has been idealized using nonlinear one-dimensional spring. All springs have been formulated to show a hysteresis force–displacement (F-U) relation. Eurocode Standard was the first standard that has been considered as the concept of elements for the determination of the design specification of connections. A component-based model of beam to column connection with flange plate is presented in Fig. 1 with the nomenclature of every element provided in Table 1. In this table, the column web panel in shear (cws) element represents the column beam web in the section, cwc is the column beam web under pressure, cwt is the column beam web under tension, cfb denotes the column wing under bending, and bt represents the bolts in tension. These elements are shown in Fig.1 on the relevant connection. This model can be used in the estimation of the M-q curve, yet it should be noted that for every element, the F-U curve should be used. The relations related to the F-U curves are provided in Eurocode 3 [²⁵]. The formula of every spring in the area of F-U was used from Eurocode 3 [²⁵] Standard to demonstrate the hysteresis response of the transformable elements. In this research, this model was developed for estimation of the total curve of moment rotation of connection considering the fact that the F-U curve is available for every element. F-U curves have been defined in the form of a three-line curve illustrated in Fig. 2. According to Eurocode 3 [²⁵] Standard elements method, k_{cp}^eandk_{cp}^yare regarded as the element elastic rigidity and element yield strength, respectively. The rest of the parameters in the curve have been calculated with respect to the relations available in the research work by Del Savio et al. [³⁰].

In this study, the relations provided in Eurocode 3 [²⁵] have been used. The primary problems for modeling of the connections are their nonlinear behavior and sudden changes in their rigidity, strength, and formability. Accordingly, the mechanical modeling of the connections with geometrical specifications and material properties is difficult.

The neural network model

Artificial neural network is a new research area developed by taking ideas from the human nervous system with the aim of simulating computers to human beings as much as possible. So far, it has developed and grew remarkably. Every neural network passes three stages of training, validation, and implementation. Indeed, neural networks can be used in solving problems for which accurate mathematical relations between their inputs and outputs do not apply. Training of the neural networks as adjustment of the connective weights of these neurons in exchange for receiving different examples so that the network outputs is converged towards a desirable out. The body of every artificial neuron is composed of two parts. The first part is called the combination function. Its task is to combine all inputs and generate a number. In the second part of the cell, lies the transfer function. Transfer functions generate very little output values until the combined and weighted inputs reach a certain threshold limit. When the combined inputs reach a specific threshold limit, the nerve cell becomes excited and produces an output signal. Through comparison of the network output response with that of the desirable value, the error vector is calculated. This vector is then distributed towards the beginning of the network from its end using different algorithms. At the end, this vector is distributed from the end to the beginning of the network through different algorithms where the error diminishes in the following cycle [³¹–³⁵].

The neural network model of the materials

Neural networks have been successfully used in a wide range of sciences including the pattern of cognition, identification of systems, financial obligations, data extraction, and so on. For the first time, Ghaboussi et al. [³⁸] proposed a new method using neural networks for modeling the behavior of materials. Some examples of research conducted in the field of engineering include application of neural networks in the modeling of constituents of simple concrete [³⁹,⁴⁰], advanced compound materials [⁴¹–⁴³], the models of the materials of strain softening in armed concrete [⁴⁴], velocity-dependent structural behavior of materials [⁴¹–⁴³], and the hysteresis behavior of materials [⁴⁵–⁴⁷]. There are also some applications of neural networks in beam to column connections under one-sided loading to estimate the relations of moment-rotation of connections with beam web angle and connections with terminal panes. For the application of cyclical loading models, a concept has been developed by Dang and Tan [⁴⁸] as well as Yun et al. [⁴⁹]. Fig. 3 demonstrates the neural network model of nonlinear hysteresis proposed by Yun et al. [⁴⁹].

The proposed neural network model includes five input variables, namely, {\xi }_n, {\sigma }_{n-\mathrm{1}}, {\epsilon }_n, {\epsilon }_{n-\mathrm{1}}, and \Delta {\eta }_{\epsilon ,n} in the form of strain control. Two hysteretic parameters of {\xi }_n and \Delta {\eta }_{\epsilon ,n}have been introduced to convert one-to-several mapping to unit mapping. They have been defined as {\xi }_n={\sigma }_{n-\mathrm{1}}{\epsilon }_{n-\mathrm{1}}and \Delta {\eta }_{\epsilon ,n}={\sigma }_{n-\mathrm{1}}\Delta {\epsilon }_n. The variable \zeta is related to strain energy in the previous stage along the balance direction. The variable \Delta \eta shows the direction for the next step along the balance direction. The closed hysteresis ring has been divided into six sections each of which is related to a unique compound of the three-variable signs of {\xi }_n, {\epsilon }_n, and \Delta {\eta }_{\epsilon ,n}. The neural network has been defined within the range of moment and rotation instead of the tension and strain range. As evident in Eq. 1, two parameters of hysteresis have been defined as {\xi }_n=M_{n-\mathrm{1}}{\theta }_{n-\mathrm{1}}and\Delta {\eta }_n=M_{n-\mathrm{1}}\Delta {\theta }_n. In order to display decreased rigidity and the strength in consecutive cycles, a reduction parameter defined as E_{n-\mathrm{1}}=E_{n-\mathrm{2}}+\left|M_{n-\mathrm{1}}{\theta }_{n+\mathrm{1}}\right|has been introduced as an input variable [⁴⁶–⁵¹].

(1){{\displaystyle M}}_n={{\displaystyle M}}_{NN}[\{{{\displaystyle \theta }}_n,{{\displaystyle \theta }}_{n-\mathrm{1}},{{\displaystyle M}}_{n-\mathrm{1}},{{\displaystyle \zeta }}_n,\Delta {{\displaystyle \eta }}_n,{{\displaystyle E}}_{n-\mathrm{1}}\}:\{NN\hspace{0.17em}Architsctu\hspace{0.17em}rs

Trained neural network models should be validated with the desired response of the returning mode. As can be observed in Fig. 4, in the returning mode state, the output predicted by trained neural network models has been employed in calculation of the input values of the following step. Therefore, the inputs have been estimated in the current stage such as hysteresis parameters and previous sections of F-U with the output of the neural network in the previous step.

In this study, for modeling the behavior of direction-dependent materials in the beam to column connection, nest adaptive neural network (NANN) has been utilized, which was proposed by Ghaboussi et al. [⁴⁷,⁵⁰,⁵¹].

Experimental phase

Typical method for the determination of M-q curve is conducting the experiment on the connection. To plot hysteresis curves of M-q, bending moments are measured directly by periodic loading of the experimental sample and rotation angles in terms of the beam transfer in relation to its depth. The experimented sample in this research is steel beam-to-column connection with flange plate widely used in steel structures. Fig. 5 demonstrates two samples of this type of beam-to-column connection.

In a beam-to-column connection with flange plate, since the sheets transfer the plastic joint to their end, they cause reduced moment arm and increased plastic limit load of the beam. Furthermore, the sheets improve beams by attaching to the beam flanges preventing their buckling in low loads. Since the main part of a sheer force is tolerated by the beam web, transfer of the beam sheer force to the column is realized by a sheet or angle connected to the beam web and the column wing. An experimental sample of a set of steel beam and column as well as flange plate has been developed in the Structure Laboratory of Babol Noshirvani University of Technology. In the experiments, bolts of M16 type and 18-mm holes have been used. The geometry of the experimental sample along with its details is provided in Fig. 6 and Table 2.

The experiment of the tension of a structural steel and bolt

In order to determine the mechanical specifications of the consumed materials, all of the connection components were investigated. The trend of conductance of the experiments was as follows: first some parts of the sections of interest were separated from the sections using a cutter device for experimental purposes. The geometry of these components was then cut by a wire cut based on the relevant standards in order to place them in direct tension-loading instruments. The bolts were also subjected to loading in order to carry out tension test. The results obtained from uniaxial tension experiment of these components are summarized in Table 3.

Equipment and experimental tools

The laboratory equipment and the way they were placed are shown in Fig. 7. Loading was applied by a hydraulic jack, Enerpac, ZE5 (Fig. 8) connected to a Load cell device (Fig. 9) with a capacity of 1MN. The load measurement device itself is connected to the beam through an instrument designed for holding the free end of the beam for performing bidirectional loading. In order to record the overall connection rotation and the rotation of the panel zone in the column and control the top and bottom displacement of the column, six linear variable differential transformers (LVDT) were used, as displayed in Fig. 10.

Launching and conducting the experiment

Before installing the samples on the experimental frame, the dimensions and specifications of the measurement are recorded followed by tightening of the bolts. The samples are then put in the frame and balanced. In the next stage, the measurement instruments are installed. All of the measurement instruments are connected to the value recording device by connector wires. The samples are further exposed to cyclical loading. In order to ensure the performance of the hydraulic jack, the load measurement device, the displacement transducers, and for no slipping of fulcrums, the experiment was carried out on two test connections with beam and call them sections with smaller dimensions (Beam IPE140 and Column IPE160) dispelling the existing errors. Fig. 11 demonstrates the frame under loading along with the experimental sample installed in this frame and the way loading is performed and launched.

Fig. 12 illustrates the cyclical loading of this experimental sample, which is of displacement control type and follows SAC Standard proposed by FEMA350 [⁵²]. The details of the periodic load steps applied to the experimental sample are provided in Table 4. In this loading, from the sixth cycle onwards, the rotation angle increases by 0.01 per increase of every cycle until the connection loses its bearing capacity and destroyed.

The moment–rotation diagram is plotted in Fig. 13 in terms of radian and kN/m. The plotted diagram is according to the applied loading and direct recording of displacement, bending moments, and its corresponding transformations. Finally, based on the displacement transducers installed on the column, the rotations and displacements related to them will be subtracted from the beam rotation. In this diagram, rotation angle q has been calculated based on the changes of LVDT Number 2 in relation with its distance with the connection source center and through subtraction of the effects of the top and bottom displacement of the column. As can be observed in Fig. 13, the response of the connection with flange plate is accompanied by narrowing effects resulting from the nonlinear collision between flange plate as well as the column wing. This response is followed by the trivial effects of diminished rigidity. However, accurate modeling of the behavior of the desired connection to conduct analysis in the frames is crucial. In other words, the mentioned effects should be considered properly in the models for which no experimental behavior is available.

Results and discussion

Comparison of the hysteresis curve of the mechanical model and the experimental model

Fig. 14 represents the comparison of results between the hysteresis curve of the moment rotation of the component-based model and the experimental model. The results obtained from the experimental and mechanical models indicate logical and plausible congruence in the strength and rigidity. However, the mechanical model does not predict the pinching effect, because elements such as contact and displacement, nonlinear contact, and slippage of bolts have not been taken into consideration. By adding the slippage elements, the mechanical model reveals highly narrowed hysteresis rings. The results of the experimental and analytical models (Fig. 14) demonstrate that in spite of equal rigidity, strength, and formability in both models, due to the narrowing effect, the energy absorption capacity is far less in the experimental model in comparison with the mechanical counterpart. Furthermore, the analytical model shows this capacity more than its real value indicating unreal energy capacity for the structure.

Comparison of the hysteresis curve of the neural network model and the experimental model

The experimental results in Fig. 13 reveal a nonlinear response including the narrowing effects and decreased rigidity. The results obtained from the experimental test are used directly in the neural network training, possessing a unique capacity in learning complex nonlinear relations. The neural network with 2 hidden layers and 30 neurons in every hidden layer was designed for modeling of the real behavior of the connection with its results compared with those of the experimental test. As can be seen in Fig. 15, the neural network model very well predicts the narrowed hysteresis rings. Overall, this research reveals that if the neural network model is designed properly, it can learn the complex behaviors of the connections directly from the experimental data. In comparison with the mechanical model, considering this connection, the neural network model can greatly demonstrate the narrowing effects, dropped rigidity, as well as the final strength in addition to accurate displaying of the initial rigidity, the rotational capacity, and the general behavior. If the geometrical properties or the type of materials of this beam-to-column connection type differs from the connection with new design, this neural network model, already trained by previous experimental data, no longer has the efficiency required for the demonstration of the output target. In other words, the effect of altering the components has not been observed for the neural network implying that the neural network model does not offer an insight into the behavior of the components independently.

Table 5 shows the error values of prediction of ANN model for both the training (70% data points) and testing (30%) sets including RMSE (root mean square error) and MAPE (mean absolute percentage error). The RMSE and MAPE are defined by the following Equations (2) and (3):

(2)\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{}\left({\displaystyle \frac{\mathrm{1}}{\mathrm{N}}}{\displaystyle {\sum }_{\mathrm{i}=\mathrm{1}}^{\mathrm{N}}{({{\displaystyle \mathrm{t}}}_{\mathrm{i}}-{{\displaystyle \mathrm{y}}}_{\mathrm{i}})}^{\mathrm{2}}}\right)

(3)\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{\mathrm{100}}{\mathrm{N}}}{\displaystyle {\sum }_{\mathrm{i}=\mathrm{1}}^{\mathrm{N}}\left|{\displaystyle \frac{({{\displaystyle \mathrm{t}}}_{\mathrm{i}}-{{\displaystyle y}}_{\mathrm{i}})}{{{\displaystyle \mathrm{t}}}_{\mathrm{i}}}}\right|}

Where N is the number of data points, t and y are the actual and estimated values, respectively.

Theoretically, a prediction model gives better results when RMSE and MAPE are closer to zero. All the performance indices in Table 5 show that the developed ANN approach is suitable and has high- accuracy forecast ability.

Application of the combinational model on the connection with flange plate

In the combinational modeling, the neural network is usually applied to model components when their modeling is difficult or impossible using mechanical models [⁵³]. Since the slip behavior cannot be described by the material and geometric properties; the neural network approach is used to model it, in which the experimental results of the beam to column connections with flange plate were employed as reference data for self-learning simulation. In a combinational modeling, the slip component is considered as the main target component which should be modeled with neural network. The data training would be extracted from self-learning simulation. There are two stages of the self-learning simulation. In the first stage, the entire top or bottom parts are considered as informational components, called NN1. The self-learning simulation extracts the data for NN1. The collected data for NN1 then become the reference data for the combinational models. In the second stage, the data for NN2 can be extracted by using another self-learning simulation. Consequently, force–displacement pairs will be gathered for data training from the target component. The combinational model is presented in Fig. 16.

In bolted connections, sudden changes occasionally occur in the hysteresis behavior due to component yield, slippage, and similar issues related to geometrical non-elasticity. These sudden changes frequently cause the self-learning simulation to stop because the solution loops do not converge with the partially trained neural network, which is not capable of capturing the sudden change. Two predetermined stiffness parameters including StiffMax and StiffMin are utilized in order to avoid unexpected non-convergence in the solution cycles. The additional guide data are generated by using the estimated stiffness of the neural network model. This data will be temporarily added to the training database and newly-estimated data replaces the old data on the next load step. First, self-learning simulation is performed for NN1 and ten passes were generated as could be seen in the first column. Numbers of gathered training cases are presented in the second column. Dramatic improvements in the moment–rotation behavior are achieved at the early passes and the rate of improvement gradually decreases with the increasing the number of passes. In pass 10, the most important behavior, the highly pinched behavior, was successfully captured although there are few possible errors since training cases at certain load steps are not properly collected. Table 6 represents details of the parameters for self-learning simulation .The number of loading steps, the number of iteration in every cycle of the automatic process, and the number of iterations between the passages are predefined as NoStep, NoCycle, and NoEpochPass, respectively.

In general, 10 epochs were generated. Evolution and development of the moment–rotation curves of the entire connection are shown in Fig. 17. Self-learning simulation is applied for non-linear modeling of connection hysteresis behavior in combinational model. Self-learning simulation is set to extract components’ behavior, such as connecting parts of the overall behavior of the connection. Self-learning simulation contains different parts, such as pre-training, auto progressive training and forward analysis. In auto progressive training, there are two types of analyses: power-control (FCA) and shift-control (DCA). By means of these two analyses, the pair of data(\Delta m,\Delta \theta ) are obtained and stored in training database. Generally, for estimation and determination of hysteresis complex behavior for beam to column connection with flange plates using combinational model in FCA analysis, in each pass, considering the rotation difference of experimental model and mechanical model, the corresponding stiffness to \Delta \thetais achieved. This is the Hybrid spring stiffness, by which the corresponding torque to each pass could be obtained and hysteresis complex behavior of each pass could be plotted. The more pass numbers increase, the more accurate is the hybrid spring and the closer become the combinational model hysteresis curve and the experimental hysteresis curve. In DCA analysis, considering the difference of moments in experimental model and mechanical model for each pass, corresponding stiffness to \Delta mis obtained, which is hybrid spring stiffness. Based on hybrid spring stiffness, corresponding rotation to each pass is achieved and hysteresis complex behavior in each pass can be plotted. As the pass number increases and gets closer to 10, a more accurate hybrid spring is obtained, with very accurate spring stiffness (Fig.17). In pass 10, the most accurate simulation for hysteresis complex behavior is observed and based on that, combinational model hysteresis curve gets closer to experimental hysteresis curve. To achieve an appropriate and satisfactory learning, too many passes are required. Regarding complicated hysteresis loops, such as pinching, more loops and loading steps are required in order to gain an appropriate learning data base.

As it can be observed in Fig.17, the combinational model can estimate the narrowing effects satisfactorily, whereas the mechanical model lacks this ability. The combination of mechanical model and the components of the neural network developed in the combinational model are able to predict the aspects skipped by the mathematical model. These mentioned aspects are generally related to non-elastic features such as collision and separation between the beam flange surface and the flange plate.

Validation of the combinational model

In this section, the combinational model developed for beam-to-column connection with flange plate has been employed for the prediction of the hysteresis curve of the moment rotation of a newly designed connection. For validation, a connection with flange plate has been included in the experimental plan, further to undergoing loading and experimentation. Figs. 18-20 depicts the structure of this connection with different dimensions in comparison to the previous model in the beam and column along with the flange plate. In order to determine the properties of the materials used in the validation sample connection, structural steel tension and bolt tests were conducted in the laboratory (Table 7).

As it was seen, a new design of beam-to-column connection with flange plate resulted in a different hysteresis behavior relative to the hysteresis behavior of the initial connection under cyclic loading. These differences can be seen in areas of pinching, rotational capacity, and final moment. Pinching in behavioral curve of connections is described by increasing rotation without significant increase in moments, which reduces connection stiffness due to nonlinear contact in connecting and connected members and slippage because of oversize bolt-hole. ​​​​​ In this case, the hysteresis behavior of the combinational model has a very high compliance with the behavior obtained from experimental connection model with the new design, indicating that the combinational modeling is able to acceptably estimate the effects of pinching on hysteresis behavior of the beam-to-column connections.

Conclusion

The challenges of the modeling of the behavior of beam-to-column connections in steel frames and composites stems are non elastic responses of individual components and their interactions. In this study, simple but accurate models of the hysteresis of the beam-to-column connections with flange plate have been presented using a new combination modeling framework. Combinational modeling is effective especially in modeling the complicated behavior of a physical system; when the system or components of the system have inherent inelastic or nonlinear behavior; when the system is subjected to extreme loadings such as an earthquake; or when the system behaviors are considerably influenced by interaction between components and materials of the system. In this framework, common mechanical models are complemented using neural network methods. The role of information methods is modeling of the aspects that are not accurately calculable by mathematical models. At the end, the combinational model of a system would be equal to reality more effectively in some way. Indeed, the developed combinational model can predict the complex hysteresis behavior of the beam-to-column connections with flange plate with the new design. The results of this study are as follows:

• Mechanical modeling is not very accurate due to inability of modeling of some aspects of the behavior of connections and accordingly requires modification and reinforcement with some other models.

• The narrowing effect is highly dependent on the components of slippage, ellipticity of the bolt hole, and nonlinear relation between the components.

• As the number of input variables increases, so does the convergence rate and accuracy of estimation of the neural networks.

• The RMSE and MAPE between the results predicted by the best neural network and experimental data are 0.0712 and 0.9166, respectively. This means that the developed ANN model has good performance and high-accuracy forecast ability.

• Elevation of the number of neurons results in decreased number of training periods.

• The results of neural network model containing the reduction parameter are in reasonable agreement with experimental results.

• Combinational modeling is a good choice for more effective estimation of the hysteresis behavior of the connections. The comparison between the combinational and experimental models with the new design indicates that the combinational model is able to demonstrate the narrowed complex hysteresis behavior of the beam to column connections with flange plate.

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