An innovative model for predicting the displacement and rotation of column-tree moment connection under fire

Abstract

In this study, we carried out nonlinear finite element simulations to predict the performance of a column-tree moment connection (CTMC) under fire and static loads. We also conducted a detailed parameter study based on five input variables, including the applied temperature, number of flange bolts, number of web bolts, length of the beam, and applied static loads. The first variable is changed among seven levels, whereas the other variables are changed among three levels. Employing the Taguchi method for variables 2–5 and their levels, 9 samples were designed for the parameter study, where each sample was exposed to 7 different temperatures yielding 63 outputs. The related variables for each output are imported for the training and testing of different surrogate models. These surrogate models include a multiple linear regression (MLR), multiple Ln equation regression (MLnER), an adaptive network-based fuzzy inference system (ANFIS), and gene expression programming (GEP). 44 samples were used for training randomly while the remaining samples were employed for testing. We show that GEP outperforms MLR, MLnER, and ANFIS. The results indicate that the rotation and deflection of the CTMC depend on the temperature. In addition, the fire resistance increases with a decrease in the beam length; thus, a shorter beam can increase the fire resistance of the building. The numbers of flanges and web bolts slightly affect the rotation and displacement of the CTMCs at temperatures of above 400°C.

Keywords

column-tree moment connection / Finite element model / parametric study / fire / regression models / gene expression programming

Introduction

Steel beam-to-column connections (SBCCs) are among the important members of steel frames that affect their behavior under fire and static loads. Properly designed connections can improve the strength of steel frames against large rotations and forces during a fire. The mechanical behavior of steel influences the performance of the connection, its redistributing forces from a beam to other steel members, and the survival time of the frame under fire. It is well known that the mechanical properties of steel decrease under fire conditions, causing the collapse of the structural members and finally the total steel frame, for example, the collapse of the World Trade Center (WTC) or the Plasco building [¹–³].

Several experiments have been conducted to study the performance of steel connections under fire conditions [⁴,⁵]. To improve the performance of SBCC, several connections have been suggested [⁶]. Several experimental and numerical studies have also focused on the moment-rotation response of SBCC at elevated temperatures [⁷–¹¹]. Qian et al. [¹²] evaluated the performance of six column-tree moment connections (CTMCs) at three temperatures: 400°C, 550°C, and 700°C. They applied various axial compressive forces to a CTMC beam at 700°C. In addition, Yu et al. [¹³,¹⁴] assessed 14 CTMCs under fire focusing on the behavior of the bolted connections (BCs).

There are two methods for designing steel structures under fire conditions: a code-based prescriptive approach and a performance-based approach. The present trend of a structural design under a fire avoids the use of prescriptive approaches, which are used to design the structure under static and lateral loads. In contrast, performance-based approaches have been employed to provide structural fire resistance. Large deformations and large rotations need to be accounted to improve the robustness of a structure [¹⁵]. Retaining the robustness of the connection under fire and applying a complex set of internal moments and forces can improve the structural resistance against a fire. On the other hand, significant ductility and strength within the connection are needed to sustain these forces along with large rotations [¹⁶,¹⁷].

A simplified method was used to predict the deflection and critical load of a restrained steel I-shaped beam [¹⁸–²⁵]. This approach uses equilibrium functions to obtain the critical load under a fire as well as the temperature-deflection history of such beams. The predicted results were compared with the results from the nonlinear finite element (FE) analysis. Some researchers have employed Vulcan software to estimate the ductility requirement on a connection by evaluating the behavior of its restrained beam under fire [²⁶–²⁸]. In addition, gene expression programming (GEP) [²⁹], regression methods [³⁰], and neural network approaches [³¹] have been employed to predict the performance of structural members.

Although many studies have focused on the behavior of steel under fire and static loads, there has been a lack of parametric studies of CTMC under fire and static loads, which will be the focus of this manuscript. Therefore, four surrogate/prediction models, including multiple linear regression (MLR), multiple Ln equation regression (MLnER), adaptive network-based fuzzy inference system (ANFIS), and GEP, are employed to predict the displacement and rotation of the CTMC under fire and the static load with minimal computational effort.

Methodology

The methodology consists of two parts. In a parametric study, the effects of changes in the numbers of flanges and web bolts, the length of the beam, the value of the applied static load, and the evaluated temperature on the rotation and displacement of the CTMC are evaluated. All variables except for the temperature change among three levels, which itself changes among seven levels. Based on the levels of these variables and the Taguchi method, nine FE samples were modeled and analyzed using ABAQUS. The outputs of each sample were obtained at 7 different temperatures, i.e., 20°C, 100°C, 200°C, 300°C, 400°C, 500°C, and 600°C yielding 63 outputs for the parameter study. We also specify failure modes as the effect of changes in each variable in the parameter study.

In the second part, three surrogate models, MLR MLnER, ANFIS, and GEP, are employed to predict the rotations and displacements at the end of the beam of CTMCs. Therefore, 44 FE models were used during the calibration process, 19 of which were utilized in the validation. Two equations were obtained for each model to calculate the rotations and displacements of the CTMCs. To evaluate their performance, the maximum positive and negative errors and the mean absolute percentage error (MAPE) were calculated. We also computed several statistical parameters, such as the coefficient of determination, root mean square error (RMSE), normalized mean square error (NMSE), and fractional bias, to find the best model. Figure 1 summarizes the steps of the present study.

Finite element model of steel beam to column connection under fire

We employed the commercial software ABAQUS in all simulations and used 3D, 8-node thermally coupled linear hexahedral solid elements (C3D8T) for all components of the connection. A finer mesh is utilized for the splice connection joining the stub and link beams, and the beam-to-column connection zones, where high stress levels are expected.

To simulate the realistic behavior of CTMC at elevated temperatures, complex contact interactions were considered for all specimens. The contact pairs in CTMC include web bolt heads-to-splice plates, flange bolt heads-to-beam flanges, web bolt shanks-to-splice plates, web bolt shanks-to-beam webs, flange bolt shanks-to-beam flanges, web bolt nuts-to-splice plates, flange bolt nuts-to-beam flanges, splice plates-to-beam webs, and stub beam flanges-to-link beam flanges. Table 1 shows the number of contact interactions, which are considered in the simulated CTMC samples. All contact interactions are defined as surface-to-surface contacts with a small sliding option.

The bolts are produced by a stiffer material among the connection components. Hence, the bolt heads and shanks are chosen as master surfaces, whereas all other surfaces are considered as slave surfaces. To apply the contact interactions between the splice plate and beam webs, and also between the stub beam flanges and link beam flanges, the contact surfaces of the splice plate and stub beam are defined as master surfaces, respectively. The hard contact option and the penalty method with a friction coefficient of µ = 0.3 are used to simulate the contacts in the normal and tangential directions, respectively. The welded stub beam-to-column connection was simulated using the tie constraint.

Similar to previous experimental tests [³], four steps are considered for analyzing all FE samples in this study. In the first step, the M25 bolts are preloaded using the “bolt load” option in ABAQUS. As specified in ASTM F3125/F3125M [³²], a 220 kN pretension load is applied to the M25 high-strength bolts. In the second step, a 672 kN compressive axial load is applied to the bottom end of the column. In the third step, the beam is subjected to a downward static load at the end of the beam, which value is various for each sample. This load is applied to a set of nodes, which are merged. In the last step, the temperature of the connection components is gradually increased based on the ISO-834 heating curve [³³].

The roller boundary condition at the bottom end of the column is modeled by restraining the center of the end of the column against the translations in the x- and y-directions. A similar boundary condition is used at the center of the top end of the column, and it is also restrained against the axial translation in the z-direction. Considering the symmetry of the connection components in the y-direction, only one-half of the configuration of the CTMCs is modeled in ABAQUS to reduce the analysis time.

In all samples, the displacement was measured at the end of the beam. Moreover, the rotation is calculated as follows [³⁰]:

(1){\theta }_{\mathrm{c}}=\frac{\left({\delta }_{\mathrm{DC}1}-{\delta }_{\mathrm{DC}2}\right)}{\left(L_{\mathrm{CD}1}-L_{\mathrm{CD}2}\right)},

where {\delta }_{\mathrm{DC}\mathrm{1}}and {\delta }_{\mathrm{DC}\mathrm{2}} are displacements, which are measured at L_{\mathrm{CD}\mathrm{1}}and L_{\mathrm{CD}\mathrm{2}} from the bottom of the column in the present study.

Material properties under fire

To simulate the behavior of steel at high temperatures, the EN 1993-1-2 Recommendation [³⁴] was used in this study. Two typical stress-strain curves are considered to define the material properties of the structural steel, as shown in Figs. 2(a) and 2(b). Based on EN 1993-1-2, the concept of Fig. 2(a) is utilized for modeling the behavior of steel at temperatures of below 400°C, in which the stress-strain curve applies strain hardening. Because the strain hardening is negligible at higher temperatures, the stress-strain curve of Fig. 2(b) is proposed to be used at temperatures of greater than 400°C. Several mechanical properties of steel, including the proportional limit, effective yield strength, ultimate strength, and elastic modulus, decrease considerably at elevated temperatures. The reduction factors, which are presented in Fig. 2(c), are used to model the mechanical behavior of the connection components under fire conditions [³⁴].

As shown in Fig. 2(c), the effective yield strength factor, elastic modulus factor, and proportional limit factor are defined as {\displaystyle \frac{f_{\mathrm{y},\mathrm{T}}}{f_{\mathrm{y}}}}, {\displaystyle \frac{E_{\mathrm{T}}}{E}}, and {\displaystyle \frac{f_{\mathrm{p},\mathrm{T}}}{f_{\mathrm{y}}}}, respectively; and E and f_y indicate the elastic modulus and yield strength of steel at ambient temperature, respectively. According to EN 1993-1-2 [³⁴], the variation in the ultimate strength of steel with respect to temperature is expressed as follows [³⁴]:

(2)

Table 2 shows the mechanical and thermal properties of ST-37 steel and Grade 8.8 bolts at room temperature, which are used in all samples in this study. It should be noted that the values of {\epsilon }_{\mathrm{y}}, {\epsilon }_{\mathrm{s}}, {\epsilon }_{\mathrm{t}}, {\epsilon }_{\mathrm{u}}, \nu, and \rho do not depend on temperature; therefore, their constant values are considered, although the relative thermal conductivity and thermal elongation of steel depend on temperature [³⁴]. The thermal properties, including the relative thermal elongation, specific heat, and thermal conductivity, which are considered in the models, are calculated using Eqs. (3)–(5), respectively. In the present study, these equations were extracted from Eurocode 3, and then were applied to ABAQUS at various elevated temperatures.

(3)

(4)

(5)

where {\displaystyle \frac{\mathrm{\Delta }l}{L}} is the relative thermal elongation, l is the initial length of steel at 20°C, Dl is the increased length of steel from an increased temperature, T is the steel temperature, and C_T and \lambda are the specific heat and thermal conductivity, respectively.

The isotropic hardening rule accompanied by the von Mises yielding criterion is used to study the behavior of ST-37 steel and Grade 8.8 bolts in all of the specimens.

Validation

We first compared the results of the FE model with experimental results from Chung et al. [³]. In this subsection, the applied temperature-time curve, displacement-temperature curve, rotation-temperature curve, and deformed shape of the connection in the experiment and two FE models are compared to evaluate the validation of the FE models. They reported the displacements and rotations of a CTMC at elevated temperatures. Based on their study, both column and beam are considered as SN490B steel, and also the bolts of this connection are selected as S10T. The mechanical behavior and dimensions of the elements in the FE models are considered as the same values as the reported values in Chung et al.’s experiment [³]. The mechanical properties of the column, beam, and bolt used are shown in Table 3, in which ε_y and ε_u represent the yield strain and the ultimate strain, respectively. The reduction factors of the mechanical properties of these elements are applied based on a factor proposed by Chung et al. [³].

The lengths of the stub beam, link beam, and column are considered as 730, 1170, and 4350 mm, respectively, as in the experimental test. There are two steps in analyzing the FE models. In the first step, a 4900 kN compressive axial load was applied to the bottom of the simply supported column, and the beam was subjected to a concentrated downward load of 360.6 kN at a distance of 1700 mm from the column face. In the second step, the temperature was gradually increased. Figure 3 shows the dimensions of the H-beam and Box column of the CTMC, which were evaluated by Chung et al. [³].

The temperature variations of different sections of the beam and column with respect to time are shown in Figs. 4(a) and 4(b), respectively. Figure 4(c) illustrates the applied temperature–time history by Chung et al. and the proposed temperature–time history based on ISO 834, which are applied to models 1 and 2, respectively.

Figure 5 shows the displacement temperature obtained from the experimental tests and two FE models. In the experimental test, the temperature was determined at TB1, and the displacement was measured at DB1 using an LVDT of 280 and 1700 mm from the column face, respectively. The same measurements at the same location were conducted during the FE analysis. It can be seen that the displacement-temperature results of the 3-D FE model are in good agreement with those of the experimental test at elevated temperatures.

The FE model measured the critical temperature of the connection at 601.1°C while the critical temperature was obtained at 573.4°C from the experimental test that the results show a 4.83% error. Figures 5 and 6 show the displacement DB4 and rotation of the simulation and experiment, respectively. The FE model underestimates the connection rotation, which changed suddenly due to the local bulking in the bottom flange of the H-shaped beam. The rotation of the connection bounced back because the moment cannot be transferred generally from the beam to the column in the buckled CTMC.

Finally, the deformed shape of the first validated sample 1 is compared with the experimental deformed shape of the specimen in Fig. 7. The FE model correctly captures the experimental failure mode. Local buckling occurs in the bottom flange and web of the beam, as shown in Fig. 7. The deformation and local buckling shown in Fig. 7 are the same with those of experimental test demonstrated in Chung et al.’s study [³].

To specify the best mesh size, a mesh sensitivity analysis was performed, as shown in Table 4.

Figure 8 shows that the displacement–temperature results are fairly independent of the discretization.

Table 5 shows that the critical temperature is also fairly independent of the mesh size.

Parametric study

Figure 9 shows the dimensions of CTMCs, which were simulated in the present study, to implement the parametric study. In addition, this figure shows the geometry and dimensions of the column, beam, section of the beam, section of column, web plate of connection, flange plate of connection, and hole of bolts. The unit of all dimensions, which are presented in Fig. 9, is presented in millimeters.

Here, d_f and d_w are the distance between the holes of the flange and web plate, respectively. The number of flange and web bolts in all samples are various; therefore, the distance between the holes also varied in all samples. The distances between the holes of the flange plate are considered to be 100, 120, and 150 mm for specimens 1–3, 4–6, and 7–9, respectively. Table 6 shows the distance between all the holes of the connection plates in each sample. The same distances between the holes of the plates in samples 1, 4, and 7 were simulated. Moreover, the distances between the holes of the connection plates of samples 2, 5, and 8 were considered the same. Finally, the same distances between the hole of the plate in samples 3, 6, and 9 were selected.

In the present study, five parameters were considered as input variables, including temperature, number of flange bolts, number of web bolts, length of the beam, and applied static load, where the temperature is changed among seven levels and other variables are changed among three levels. Table 7 shows the variables and their levels. The values of variable 1, which is considered the temperature, are selected as 20, 100, 200, 300, 400, 500, and 600. The second variable is defined as the number of flange bolts, whose values are considered as 3, 4, and 5 bolts, and the third variable is the number of web bolts whose values are selected as 12, 16, and 24 bolts. The length of the beam and the applied static load are considered as the fourth and fifth input variables, respectively. The length of the beam was selected as 1450, 1650, and 1800 mm, and the value of the applied static load is considered to be 50, 100, and 150 kN.

According to the Taguchi method, for variables 2–5 and their levels, nine samples were simulated and analyzed. All samples were exposed to temperatures of 20°C, 100°C, 200°C, 300°C, 400°C, 500°C, and 600°C, as shown in Table 8. As a result, although there are only 9 samples that are simulated and analyzed, there are 63 outputs. Moreover, each output depends on five variables; therefore, the data are significant for employing prediction models. The maximum temperature is considered as 600°C because all samples failed at temperatures of between 600°C and 700°C. Table 8 shows the value of each input variable for each sample.

Table 9 lists the mesh numbers of each specimen. The maximum length of the beam and the maximum number of flange and web bolts are related to specimen F5W24L-1.8P150, which was meshed by 10632 elements. The minimum length of the beam and the number of flange bolts are related to specimen F3W16L1.45P150.

Figure 10 shows the temperature-rotation curve of each sample. The obtained rotation of F5W12L1.45P90 is less than that of F5W16L1.65P120, and the rotation of this sample was less than that of F5W24L1.8P150. In contrast, the fire resistance of F5W12L1.45P90 is higher than that of F5W16L1.65P120, and the fire resistance of F5W16L-1.65P120 was higher than that of F5W24L1.8P150. As a result, the rotation increased while the fire resistance decreased when the length of the beam and applied static load increased.

According to the results of this figure, the rotation of the CTMC depends on the temperature. The maximum and minimum rotations are related to specimens F3W24L-1.65P90 and F5W12L1.45P90, respectively. As a result, the rotation of a specimen with more flange bolts and beam length is greater.

Figure 11 shows the displacement-temperature curve of each sample. According to the results of this figure, the displacement of the beam end depends on the temperature. The maximum and minimum displacements among all samples are related to specimens F5W24L1.8P150 and F5W12L1.45P90; therefore, the increasing length of the beam and the applied static load cause increasing beam displacement. Moreover, increasing the length of the beam and the applied load cause a decrease in the fire resistance.

Models for predicting the rotation of steel beam to column connection under fire

Based on the calibrated FEM models described in the previous section and their results, 63 nonlinear FE samples were simulated and analyzed. In this section, three prediction methods are applied to predict the rotation and displacement of the CTMC under fire conditions and static loads. In this section, all methods that are utilized to predict the displacement and rotation of the CTMC are described. These procedures include MLR MLnER, ANFIS, and GEP. These methods are described in Subsection 4.2.

Data

As mentioned in the previous section, the number of flange bolts, the number of web bolts, the length of the beam, the applied static load, and the temperature are considered as the input variables in this study, and the rotation of the connection and displacement of the beam are considered as the outputs in the present study. Table 10 shows the correlation coefficient between each input variable and each output. The best correlation coefficient is related to temperature, which is defined as the change in temperature. The second best correlation coefficient is related to changes in the static load.

Prediction models

The first model: Multiple linear regression (MLR)

The first model, which is employed to predict the displacement and rotation of the CTMC in this study, is an MLR that includes several linear equations that are summed. Five variables, including the number of steel bolts of beam flanges, number of steel bolts used on the beam web, the value of the static load, the length of the beam, and the temperature are considered as independent variables [³⁰]. In addition, the displacement and rotation of the CTMC is considered as a dependent variable that is forecasted based on Eq. (6):

(6)Y=b_{\mathrm{1}}+b_{\mathrm{2}}{X}_{\mathrm{2}}+\cdots +b_k{X}_k+e,

where X₂,X₃,...,X_k and b₁,b₂,...,b_k are independent variables and linear regression parameters, respectively. In addition, e can be acquired from a constant variance and normal distribution of independent random sampling with a mean of zero, which is named an estimated error term, and Y is the dependent variable. The role of regression modeling is to calculate b₁,b₂,...,b_k, which is carried out by utilizing the minimum square error technique; therefore, b₁,b₂,...b_k are calculated based on the formula of b_i, which is shown in Eq. (7) [³⁰]:

(7)b_i={\left(X^{\prime }X\right)}^{-\mathrm{1}}\left(X^{\prime }Y\right).

In this equation, the transpose of X is shown by X′.

The second model: Multiple Ln equation regression (MLnER)

MLnER is considered as the second model in this work, which includes several Ln equations. In other words, these Ln equations are summed to predict the displacement and rotation of the CTMC. Based on the concept of MLnER, this model includes one dependent variable, two or more independent variables, the specific coefficient for each independent variable, a constant coefficient, and an error term. In other words, MLnER is an improved model that is achieved based on the MLR technique. There are several Ln equations instead of linear equations; therefore, the constant coefficient and error term are calculated based on the MLR method, which is explained as follows. The generic form of MLnER is calculated through the following [³⁵]:

(8)\mathrm{Ln}\left(Y_{\mathrm{M}\mathrm{L}\mathrm{n}\mathrm{E}\mathrm{R}}\right)=B_{\mathrm{1}}+B_{\mathrm{2}}\times \mathrm{Ln}\left(X_{\mathrm{2}}\right)+B_{\mathrm{3}}\times \mathrm{Ln}\left(X_{\mathrm{3}}\right)+\cdots +B_k\times \ln \left(X_k\right)+e,

where X₂,X₃,...,X_k are independent variables, and b₁,b₂,...,b_k are linear regression parameters, which are defined as the specific coefficient of each ln equation of the MLnER model. In addition, e can be acquired from constant variance and normal distribution of independent random sampling with a mean of zero, which is called an estimated error term, and Y is the dependent variable. The role of regression modeling is to calculate b₁,b₂,...,b_k, which are constant coefficients and an error term, which is carried out by utilizing the minimum square error technique; therefore, b₁,b₂,...,b_k are calculated by b_i, as presented in Eq. (7). If B₁ + e is equal to Ln(C), the simple form of Eq. (8) can be rewritten as Eq. (9). If both sides of the equation are imported to the exponential equation, a power equation is obtained as Eq. (10).

(9)\mathrm{Ln}\left(Y_{\mathrm{M}\mathrm{L}\mathrm{n}\mathrm{E}\mathrm{R}}\right)=\mathrm{Ln}\left(X_{\mathrm{2}}^{B\mathrm{2}}\times X_{\mathrm{3}}^{B\mathrm{3}}\times \cdots \times X_k^{BK}\times C\right),

(10)Y_{\mathrm{M}\mathrm{L}\mathrm{n}\mathrm{E}\mathrm{R}}=X_{\mathrm{2}}^{B\mathrm{2}}\times X_{\mathrm{3}}^{B\mathrm{3}}\times \cdots \times X_k^{BK}\times C\mathrm{.}

The third model: Adaptive network-based fuzzy inference system (ANFIS)

ANFIS is employed as the third model to predict the displacements and rotations of CTMCs at elevated temperatures. This multilayer method is based on the fuzzy inference system proposed by Takagi and Sugeno [³⁶]. ANFIS integrates both fuzzy logic principles and neural networks; therefore, it has a great capability to predict the FE results. For a first-order Sugeno fuzzy model, a typical rule set with two fuzzy if-then rules is expressed as follows.

Rule 1: If x is A₁ and y is B₁, then

(11)f_{\mathrm{1}}=p_{\mathrm{1}}x+q_{\mathrm{1}}y+r_{\mathrm{1}}\mathrm{.}

Rule 2: If x is A₂ and y is B₂, then

(12)f_{\mathrm{2}}=p_{\mathrm{2}}x+q_{\mathrm{2}}y+r_{\mathrm{2}},

where A₁, B₁, A₂, and B₂ indicate labels of membership functions for x and y, respectively, and p₁, q₁, p₂, and q₂ indicate the labels of membership functions for the output.

As shown in Fig. 12, there are five layers in a typical ANFIS. The first layer consists of adaptive nodes that produce the membership grades of the inputs. The second layer consists of fixed nodes that are labeled as P, which means they are simple multipliers. Because all nodes in layer three perform normalizations in the network, these fixed nodes are labeled N. The nodes of layer four are adaptive nodes, and the parameters in this layer are called consequent parameters. Because the single node of the fifth layer calculates the overall output as the summation of all incoming signals, that is, S.

The training algorithm consists of forward and backward passes. In the forward pass, node outputs go forward until layer four and consequent parameters are determined through the least square method. In the backward pass, the errors propagate backward and the premise parameters are updated by the gradient descent [³⁷].

The fourth model: Gene expression programming

Ferreira proposed GEP in 1999 for the first time, which is created by combining the genetic algorithm (GA) and genetic programming (GP) [³⁸]. In this model, various phenomena are simulated by employing a set of various functions and terminals. A set of different functions includes the main functions of mathematical functions, trigonometric functions, combination functions, and user-defined functions, which can be utilized to predict the displacement and rotation of the CTMC. The set of terminals includes a combination of independent variables of the problem and constant values. In this approach, the GA employs the population of data and chooses them to specify a function for predicting the displacement and rotation of the CTMC under fire conditions and static loads. In addition, some operators (genes) are used to apply genetic variations. Based on the GEP model, a roulette wheel is employed to select the data. Data are reproduced simultaneously by utilizing several genetic operators. The inappropriate data are removed, and appropriate data are transferred and stored from the present generation to the next generation in the duplication operation. In other words, the principle aim of the mutation operator is the internal random optimization of the given chromosomes. The GEP step is illustrated in the flow chart shown in Fig. 13.

The statistical parameters and error terms, including the coefficient of determination, RMSE, NMSE, fractional bias, the maximum positive and negative errors, and MAPE are employed in the present study to evaluate and compare the performance of the models in predicting the displacement and rotation of the CTMCs. These parameters were used in previous studies toward this aim [²⁹,³¹,³⁹].

The results of prediction models

A computer having configuration: CoreTM i7-3612QM CPU @ 2.10 GHz 2.10, and RAM 8 GB, was utilized to simulate 63 different CTMC specimens through a nonlinear FE analysis. An analysis of the nonlinear FE for each specimen is implemented within 35 min on average, although the numerical models calculate the rotation and displacement of the CTMC only within a few seconds. A total of 44 and 19 specimens were used in the numerical models for training and testing, respectively.

The aim of the models is to predict the displacement and rotation of the CTMC under fire conditions and static loads. In addition, five variables, which are imported as input variables in each model, are shown in the following:

\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}=\{\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}1,\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}2,\mathrm{...},\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}5\}=\{N_{\mathrm{f}},N_{\mathrm{w}},L,F,T\},

(13)\mathrm{O}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}={DIS}_{\max }\mathrm{a}\mathrm{n}\mathrm{d}{ROT}_{\max },

where N_f, N_w, L, F, and T are defined as the number of flange bolts, number of web bolts, length of the beam, applied load, and temperature, respectively. A Calibration and verification are two steps in using each model; therefore, the rotation and displacement of all samples obtained from the nonlinear FE model and input variables were first employed to calibrate each model. In other words, to calibrate the models, 70% of the nonlinear FE outputs were randomly used for training each prediction model. According to the calibration of each model, a formula was obtained in which the remaining data sets were predicted based on this formula in the verification part. The outputs obtained from each model and the nonlinear FE model were compared to evaluate their performance. In addition, the predicted value and the value of nonlinear FE model were compared in 30% of the data set for evaluating the verification of each model. As a result, Eqs. (14)–(16) were obtained to predict the displacement of the CTMC from the MLR, MLnER, and GEP, respectively.

(14)DI{S}_{\mathrm{M}\mathrm{L}\mathrm{R}}\left(T\right)=\mathrm{0.822}\times N_{\mathrm{f}}-\mathrm{0.004}\times N_{\mathrm{w}}+\mathrm{2.757}\times L+\mathrm{0.220}\times F+\mathrm{0.091}\times T-\mathrm{74.830},

(15)DI{S}_{\mathrm{M}\mathrm{n}\mathrm{E}\mathrm{R}}\left(T\right)=\mathrm{2.414}\times {\mathrm{10}}^{-\mathrm{3}}\times {\left(N_{\mathrm{f}}\right)}^{-\mathrm{0.005}}\times {\left(N_{\mathrm{w}}\right)}^{-\mathrm{0.013}}\times {\left(L\right)}^{\mathrm{1.517}}\times {\left(F\right)}^{\mathrm{0.860}}\times {\left(T\right)}^{\mathrm{0.695}},

(16)DI{S}_{\mathrm{G}\mathrm{E}\mathrm{P}}\left(T\right)=0.021\times L\times T+18.036\times \frac{T}{L}+0.050\times F\times {(0.002\times T)}^{(1.624\times {10}^{-5}\times L\times x^2)}-0.146-1.249\times \sin (1.624\times {10}^{-5}\times L\times T^2)-3.863\times {(0.002\times T)}^{(1.624\times {10}^{-5}\times L\times x^2)},

where DIS_MLR, DIS_MLnER, and DIS_GEP are the obtained displacements from MLR, MLnER, and GEP, respectively. In addition, N_f and N_w are defined as the number of beam flange bolts and beam web bolts, respectively; F is the applied load at the end of the beam; and T is the temperature applied to CTMC.

The rotation of CTMC can be calculated by Eqs. (17)–(19), which were obtained from MLR, MLnER, and GEP, respectively. As mentioned above, these equations were obtained from the calibration and training processes of the prediction models.

(17)RO{T}_{\mathrm{M}\mathrm{L}\mathrm{R}}\left(T\right)=\mathrm{1.800}\times {\mathrm{10}}^{-\mathrm{5}}\times N_{\mathrm{f}}+\mathrm{6.654}\times {\mathrm{10}}^{-\mathrm{7}}\times N_{\mathrm{w}}+\mathrm{3.235}\times {\mathrm{10}}^{-\mathrm{3}}\times L+\mathrm{4.428}\times {\mathrm{10}}^{-\mathrm{5}}\times F+\mathrm{1.085}\times {\mathrm{10}}^{-\mathrm{5}}\times T-\mathrm{0.012},

(18)RO{T}_{\mathrm{M}\mathrm{L}\mathrm{I}\mathrm{E}\mathrm{R}}\left(T\right)=\mathrm{1.596}\times {\mathrm{10}}^{-\mathrm{7}}\times {\left(N_{\mathrm{f}}\right)}^{\mathrm{7.210}\times {\mathrm{10}}^{-\mathrm{3}}}\times {\left(N_{\mathrm{w}}\right)}^{\mathrm{5.086}\times {\mathrm{10}}^{-\mathrm{4}}}\times {\left(L\right)}^{\mathrm{1.307}}\times {\left(F\right)}^{\mathrm{1.370}}\times {\left(T\right)}^{\mathrm{0.36}},

(19)RO{T}_{\mathrm{G}\mathrm{E}\mathrm{P}}\left(T\right)=\mathrm{5.509}\times {\mathrm{10}}^{-\mathrm{6}}\times L\times F+\mathrm{2.561}\times {\mathrm{10}}^{-\mathrm{9}}\times T^{\mathrm{2}}+\mathrm{1.276}\times {\mathrm{10}}^{-\mathrm{23}}\times L^{\mathrm{2}}\times F^{\mathrm{2}}\times {\theta }^{\mathrm{3}}\times \exp (\mathrm{0.031}\times T)-\mathrm{1.760}\times {\mathrm{10}}^{-\mathrm{4}}-\mathrm{8.672}\times {\mathrm{10}}^{-\mathrm{14}}\times T\times \exp (\mathrm{0.031}\times T),

where ROT_MLR, ROT_MLnER, and ROT_GEP are the displacements obtained from MLR, MLnER, and GEP, respectively; N_f and N_w are defined as the number of beam flange bolts and beam web bolts, respectively; F is the applied load at the end of the beam; and T is the temperature applied to CTMC.

The statistical parameters were determined separately for the training and testing processes to evaluate the performance of each model. Moreover, statistical parameters were used to compare the models. Table 11 shows the calculated statistical parameters, including the coefficient of determination, RMSE, NMSE, and fractional bias for predicting the displacement and rotation of the CTMC.

Table 11 includes two parts, the first and second parts of which show the results of the statistical parameters for predicting the displacement and rotation of the CTMC, respectively. To predict the displacement, the results of all models are acceptable with respect to the coefficient of determination. The values of the coefficient of determination of MLR in the training and testing processes are higher than those of MLnER. The values of the coefficient of determination of the GEP in the calibration and validation data sets were calculated as 0.983 and 0.993, respectively, which are close to 1. In addition, the coefficients of determination of ANFIS during the training and testing processes are calculated as 0.958 and 0.999, respectively, which are close to 1, as in the GEP model. The values of RMSE and NMSE of the GEP model are less than those of the other models; therefore, the GEP model performs better than other approaches in predicting the displacement of CTMC under temperature and a static load. All models under-predict in the calibration data sets and over-predict in the validation data sets.

The second part of Table 11 shows the results of the statistical parameters for predicting the rotation. According to the values of the coefficient of determination in all data sets, the GEP model performs better than ANFIS, ANFIS shows better results than MLR, and MLR demonstrates better results in comparison to MLnER. The values of the coefficient of determination of the GEP model were 0.971 and 0.997 for the calibration and validation data sets, respectively, which are close to 1. In addition, the values of RMSE and NMSE of the GEP are less than those of the other models; therefore, the GEP model shows better results in comparison with MLR, MLnER, and ANFIS with respect to all statistical parameters. Moreover, the GEP method over-predicts the rotation of the CTMC in all data sets.

Calculating and comparing the error terms of each model is another way to evaluate the performance of the prediction models. Table 12 shows the error terms, including the maximum positive and negative errors, as well as the MAPE of each model to predict the displacement and rotation of the CTMC under fire conditions and a static load. This table is divided into two parts, in which the performance of the models for predicting the displacement and rotation of the CTMC is evaluated in the first and second parts, respectively.

As mentioned above, the performance of each model for predicting the displacement is evaluated in the first part of Table 12. The GEP model performs better than MLR, MLnER, and ANFIS with respect to the maximum positive error and the MAPE. The best values of the maximum positive and negative errors and the MAPE were calculated as 22.84%, – 26.49%, and 1.33%, respectively. The best value of the maximum positive error and the MAPE are related to the GEP model, and the best value for the maximum negative error is related to ANFIS. Based on the second part of this table, the best values of the maximum positive and negative errors, and the MAPE are calculated as 63.41%, – 8.88%, and 0.57%, respectively, which are related to the GEP model. Therefore, GEP performs better than other models for predicting the displacement and rotation with respect to the error terms.

Figure 14 shows the error distributions of MLnER, ANFIS, and GEP. Based on this figure, the results of the error distribution are divided into six zones by five lines. The blue lines divide the errors, which are more than 80% and – 80%. The red lines divide the errors that are more than 20% and – 20%. Finally, the black line or zero line specifies the errors, which are close to zero. This figure shows two limits for the error distribution of each model. Based on the results of this figure, a reliable model for the prediction can be specified. A model in which most of its errors are close to the zero line is more accurate for the prediction. Moreover, this figure demonstrates the maximum positive and negative errors are related to which of samples.

Figure 14(a) shows the error distribution of the predicted displacement by MLnER, ANFIS, and GEP. The errors of 12 samples, which were predicted by MLnER, were between –20% and 20%, although the errors of 60 samples, which were predicted by GEP, were between – 20% and 20%. Moreover, the errors of the obtained displacement of 49 samples, which were predicted by GEP, are close to the zero line, although there are only three predicted displacements by GEP whose errors are less than – 20% or more than 20%. As shown in this figure, although ANFIS performs better than MLnER, the error of the two samples, which were predicted by ANFIS, was more than the maximum positive error of MLnER.

Figure 14(b) shows the error distribution of the rotations, which were predicted by MLnER, ANFIS, and GEP. The error of the predicted rotation by GEP for all samples except sample 55 is between – 20% and 20%, although the error of the predicted rotation by MLnER and ANFIS for only 7 samples is between – 20% and 20%. The error of the predicted rotation by GEP for 49 samples is close to the zero line. According to the results of Figs. 14(a) and 14(b), GEP can predict the displacement and rotation of the CTMC under fire and the static load better than MLnER and MLR.

Figure 15(a) shows a comparison of the displacement obtained from the nonlinear FE model and the predicted displacement by GEP. The displacement of samples 5, 6, and 8, in which their absolute error was calculated more than 20%, was less than 5 mm; therefore, only a 1.85-mm difference between the displacement obtained from the nonlinear FE model and the predicted displacement results in a 37% error. The maximum negative error of GEP in predicting the displacement is 37%, which was obtained in sample 8, where the displacement of this sample obtained under a 20°C temperature and a static load of 150 kN was calculated as 5.29 mm using the nonlinear FE model. As a result, GEP can predict the displacement of the CTMC with high accuracy, and only a safety factor should be applied to the predicted displacements, which are low. In other words, based on the archived results, this safety factor is considered to be 1.37 for samples in which a 20°C temperature was applied to them.

Figure 15(b) shows a comparison of the rotation predicted through GEP and the rotation obtained from the FE model. The absolute error in the rotation predicted through GEP for all samples except sample 55 is less than 20%. The rotation obtained for sample 55 of the FE model was approximately 0.002. To achieve a reliable model for predicting the rotation of the CTMC under fire conditions and a static load, a safety factor should be considered and applied to the predicted value. The value of this safety factor is considered to be 1.63 based on the results of this study.

Conclusions

We conducted a nonlinear FE analysis to assess the performance of the CTMCs under fire and various static loads. Five variables, including the temperature, beam length, applied static load, number of flange bolts, and number of web bolts, were considered as input variables. The output of interest is the rotation and displacement of the CTMC. The temperature was changed among seven levels, whereas the other variables were changed among three levels; in addition, nine samples were designed based on the Taguchi method. After validating the FE model, nine samples, selected using the Taguchi method, were simulated. Each sample was exposed to 7 different temperatures resulting in 63 outputs. 44 of them were used for training an associated surrogate/prediction model. We subsequently derived two equations from each model to predict the displacement and rotation of the CTMCs. The results of this study can be summarized as follows.

1) The rotation and displacement of the CTMCs under fire conditions and a static load depend on the temperature.

2) The length of the beam and the static load significantly affect the rotation when the temperature is above 500°C. These parameters also influence the fire resistance at temperatures of above 500°C. The number of flange bolts affects only slightly the displacement of CTMCs for temperatures of above 500°C.

3) The fire resistance increases with a decrease in the static load and length of the beam. In addition, the rotation increased with an increase these parameters. Hence, a shorter beam can increase the fire resistance of the building.

4) The length of the beam and the static load significantly impact the displacement when the temperature is above 400°C. The number of flange bolts slightly affects the rotation of the CTMCs at temperatures of above 400°C.

5) Among all surrogate models, GEP can best predict the rotation and displacement of CTMCs under various static loads and elevated temperatures, although two safety factors should be applied to the predicted rotation and displacement, which are calculated as 1.63 and 1.37, respectively.

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