Development of Hybrid Machine Learning Models for Predicting Permanent Transverse Displacement of Circular Hollow Section Steel Members under Impact Loads

Abstract

The impact effect is a crucial issue in civil engineering and has received considerable attention for decades. For the first time, this study develops hybrid machine learning models that integrate the novel Extreme Gradient Boosting (XGB) model with Particle Swam Optimization (PSO), Grey Wolf Optimizer (GWO), Moth Flame Optimizer (MFO), Jaya (JA), and Multi-Verse Optimizer (MVO) algorithms for predicting the permanent transverse displacement of circular hollow section (CHS) steel members under impact loads. The hybrid machine learning models are developed using data collected from 357 impact tests of CHS steel members. The efficacy of hybrid machine learning models is evaluated using three performance metrics. The results show that the GWO-XGB model achieves high accuracy and outperforms the other models. The values of R², RMSE, and MAE obtained from the GWO-XGB model for the test set are 0.981, 2.835 mm, and 1.906 mm, respectively. The SHAP-based model explanation shows that the initial impact velocity of the indenter, the impact mass, and the ratio of impact position to the member length are the most sensitive parameters, followed by the yield strength of the steel member and the member length; meanwhile, member diameter and member thickness are the parameters least sensitive to the permanent transverse displacement of CHS steel members. Finally, this study develops a web application tool to help rapidly estimate the permanent transverse displacement of CHS steel members under impact loads.

1. Introduction

Nowadays, many cross-section types are employed for construction members in steel structures. Due to its excellent properties, the circular hollow section (CHS) is considered one of the most used sections [1]. The CHS steel member increases the strength-to-weight ratio during serviceability. In addition, it improves structural efficiency and allows for a longer member span. Consequently, CHS steel members are employed in various well-designed and highly efficient steel structures worldwide, such as offshore platforms, subsea pipelines, conveyor columns, overpass columns, building garages, and airport terminals. Generally, these structures can be subject to impact loads during their design life [2,3]. For example, in offshore platforms and subsea pipelines, accident loads often lead to tubular structures failing due to dropped objects or collisions [4,5]. However, the structures may be subjected to impact loads from accidental or intentional impacts, such as collisions with vehicles, or terrorist attacks [2,6]. In the event of an impact, structures will gradually lose their bearing capacity and may even collapse without warning.

Several experimental and numerical studies have been conducted to determine structural members’ behavior and failure modes under the impact load [7,8,9,10,11,12,13,14,15,16,17]. Moreover, several building codes and previous studies have provided guidelines for structures under impact loads [2,18,19], and many studies have developed analytical procedures (i.e., plastic collapse mechanism theory) to describe the mechanical behavior of solid metal and hollow sections [20,21,22,23,24,25,26]. Nevertheless, they do not provide detailed guidelines for such types of CHS steel members under impact loads [27,28]. They are usually designed from a mechanical perspective and do not adapt readily to routine civil-structural design [2]. Since impact behavior is a complex engineering task, developing a theoretical model for a specific problem under impact load is challenging.

Machine learning (ML) methods have recently received considerable attention in civil engineering [29,30,31,32,33,34,35]. Among various regression ML algorithms, the Extreme Gradient Boosting (XGB) algorithm has recently gained substantial popularity in modeling several nonlinear mechanical behaviors for classification and regression [36,37,38,39,40,41,42,43]. The XGB provides good prediction performance by reducing overfitting and controlling model complexity through its built-in regularization [44,45]. For example, Feng et al. [46] have shown that the XGB model could predict the shear strength of reinforced concrete deep beams better than other ML models. Additionally, the XGB model has been shown to perform better than other ML models for several structural engineering problems [47,48,49]. However, it should be noted that ML algorithms require hyperparameters to be set before they can be run. Therefore, selecting appropriate hyperparameters of ML models for a specific dataset is critical because the performance of ML models depends on their hyperparameter settings.

In recent decades, metaheuristic algorithms have been used as one of the most popular research topics in many optimization fields, including the optimum design of truss and frame structures [50,51,52,53] or damage detection [54,55,56] because of their simplicity and flexibility, derivative-free mechanism, low dependency on problems, and local optima avoidance. In contrast to gradient-based methods, metaheuristic algorithms are more efficient because they are simple and easy to use [57]. To enhance the ability of ML models to generalize, an optimal set of hyperparameters must be determined. In recent years, metaheuristic algorithms have gained popularity as an alternative method for fine-tuning the hyperparameters of ML models, since they can improve the performance of the optimization strategy [45,58,59,60,61,62].

Although metaheuristic algorithms have been used for different engineering problems, there has been no comprehensive evaluation or comparison of metaheuristic algorithms hybridized with the XGB models for predicting the permanent transverse displacement (${\mathrm{W}}_{\mathrm{f}}$) of CHS steel members. For the first time, this study compares the performance of several hybrid models by combining the XGB model with the PSO, GWO, MFO, JA, and MVO algorithms for predicting the ${\mathrm{W}}_{\mathrm{f}}$ of CHS steel members. The hybrid ML models are trained and tested on an experimental database collected from the literature and evaluated using various performance metrics. The performance of hybrid ML models is assessed to find the best model for predicting the ${\mathrm{W}}_{\mathrm{f}}$ of CHS steel members. Finally, a web application is developed that can provide this prediction quickly and repeatedly.

2. Data Collection

The full-clamped CHS steel members under transverse impact loading using drop-weight tests are shown in Figure 1. The critical parameters of the specimens are the member length ($\mathrm{L}$), the member diameter ($\mathrm{D}$), the member thickness ($\mathrm{t}$), the ratio of impact position to the member length (${\mathrm{L}}_{\mathrm{r}}$), the yield strength of the steel member (${\mathrm{f}}_{\mathrm{y}}$), the impact mass ($\mathrm{G}$), the initial impact velocity of the indenter (${\mathrm{V}}_0$), and ${\mathrm{W}}_{\mathrm{f}}$.

According to Jones et al. [63] and Jones and Shen [23], the idealized deformed cross-section of full-clamped CHS steel members at the point of impact is shown in Figure 2, where $\mathrm{R}$ is the original mean radius of a member, ${\mathrm{W}}_{\mathrm{l}}$ is the local displacement, ${\mathrm{W}}_{\mathrm{g}}$ is the global displacement, and ${\mathrm{D}}_{\mathrm{m}}$ is the the maximum outside diameter of a dented cross-section. The dented zone is somewhat symmetrical about the impact plane for the middle- and quarter-span impact tests; meanwhile, it is no longer symmetrical about the impact plane for the impact position close to support. This method idealizes a deformed cross-section to estimate the local displacement and global displacement of the permanent transverse displacement from the experimental measurements of the final cross-section [4]. After an impact test, the ${\mathrm{W}}_{\mathrm{f}}$ is measured at the indentation point on the member prior to unclamping and removal from the rig. After removal, the ${\mathrm{W}}_{\mathrm{f}}$ is measured again while supporting the member on two knife-edged vee blocks spaced $\mathrm{L}$ apart (representing the same support span). It was found that there was very little difference (0.5 mm maximum) between the two measurements, so an average value was taken [64]. It is worth noting that the ${\mathrm{W}}_{\mathrm{f}}$ is the main parameter used to estimate the total plastic energy absorbed in a member [4].

This study collects a comprehensive database of 357 experimental tests of full-clamped CHS steel members under impact load. The database is collected mainly from the studies of Jones et al. [63], Chen and Shen [64], and Jones and Birch [65]. Jones et al. [63] conducted 130 tests of fully clamped CHS steel members under lateral impact load, considering several diameters, lengths, and velocities. Chen and Shen [64] conducted 226 tests of CHS steel members under lateral impact load. Jones and Birch [65] conducted 54 impact tests on CHS steel members under impact load. The database deals with outliers, duplicates, and missing values. As a result, the final database of the 357 data points used in this study is summarized in

Table 1. Statistical properties of experimental data.

	$\mathbf{L} \left(\mathbf{mm}\right)$	$\mathbf{D} \left(\mathbf{mm}\right)$	$\mathbf{t} \left(\mathbf{mm}\right)$	${\mathbf{L}}_{\mathbf{r}}$	${\mathbf{f}}_{\mathbf{y}} \left(\mathbf{MPa}\right)$	$\mathbf{G} \left(\mathbf{kg}\right)$	${\mathbf{V}}_0 (\mathbf{m}/\mathbf{s})$	${\mathbf{W}}_{\mathbf{f}} \left(\mathbf{mm}\right)$
Min	160.00	19.00	0.90	0.00	217.44	6.50	1.39	3.29
Mean	504.57	52.16	1.69	0.30	453.41	40.11	5.79	27.94
Max	1200.00	120.00	2.00	0.50	559.00	98.00	11.81	101.94
Std	305.43	30.02	0.38	0.21	106.02	20.84	3.12	21.33
Cov	0.61	0.58	0.22	0.69	0.23	0.52	0.54	0.76

.

The distributions of the input parameters ($\mathrm{L}$, $\mathrm{D}$, $\mathrm{t}$, ${\mathrm{L}}_{\mathrm{r}}$, ${\mathrm{f}}_{\mathrm{y}}$, $\mathrm{G}$, and ${\mathrm{V}}_0$) and the output parameter (${\mathrm{W}}_{\mathrm{f}}$) are shown in Figure 3. Additionally, Figure 4 shows the correlation matrix of the variables. Correlation values between the independent variables ($\mathrm{L}$, $\mathrm{D}$, $\mathrm{t}$, ${\mathrm{L}}_{\mathrm{r}}$, ${\mathrm{f}}_{\mathrm{y}}$, $\mathrm{G}$, and ${\mathrm{V}}_0$) and dependent variable (${\mathrm{W}}_{\mathrm{f}}$) are 0.48, 0.50, 0.27, 0.52, 0.0, 0.43, and 0.69, respectively. It can be seen that the ${\mathrm{V}}_0$ is strongly correlated with ${\mathrm{W}}_{\mathrm{f}}$; meanwhile, the variables $\mathrm{L}$, $\mathrm{D}$, $\mathrm{t}$, ${\mathrm{L}}_{\mathrm{r}}$, and $\mathrm{G}$ are moderately correlated with ${\mathrm{W}}_{\mathrm{f}}$, while ${\mathrm{f}}_{\mathrm{y}}$ is almost independent of the ${\mathrm{W}}_{\mathrm{f}}$.

3. Machine Learning and Optimization Algorithms

This study uses five popular metaheuristic algorithms (i.e., PSO, GWO, MFO, JA, and MVO) coupled with the XGB model to investigate the prediction of the ${\mathrm{W}}_{\mathrm{f}}$ of CHS steel members under impact loading. Due to space limitations, brief descriptions of these algorithms are presented in this section. The information needed on selected ML algorithms in this study can be found in previous references [66,67].

3.1. Extreme Gradient Boosting (XGB)

Chen and Guestrin [68] developed the XGB based on the gradient boosting algorithm proposed by Friedman et al. [69]. The XGB is a highly optimized and parallelized version of the gradient boosting algorithm. For each iteration of the XGB, the residual is used to calibrate the previous predictor. Additionally, the XGB uses an approximate algorithm to find the best-split points. The XGB is characterized by some critical features [68]: parallelization (can train with multiple CPU cores), regularization (contains several regularization penalties to avoid overfitting and generalize adequately), scalability (can run distributed and process enormous amounts of data), effective tree pruning (can make the splitting up to the specified max_depth value and then start pruning the tree backward), and missing value handling (has an in-built capability to handle missing values).

The XGB is superior to gradient boosting in many respects (i.e., the smarter breakup of trees, random hidden node generation, shorter leaf nodes, and out-of-core predictions) [69]. Moreover, the XGB adds the regularization term in the loss function to avoid overfitting. In addition, the training time is drastically reduced by parallelizing the entire boosting process in the XGB [68]. Therefore, the XGB can be used for many engineering simulations and provides fast and reliable results [70,71,72]. The details of the XGB can be found in [60,72].

3.2. Particle Swarm Optimization (PSO)

Kennedy and Eberhart developed the PSO algorithm [73], which is inspired by the complex social behavior of flocking birds. The basic idea of PSO is to share the food place between groups of birds (also called particles). Particles have two important indexes: (I) the fitness value and (II) velocity. PSO creates a population of random particles and moves them using velocity in the search space in every iteration. This velocity term refers to the best-found solution and each particle’s best experience. The next step would be to repeat this procedure to find a more promising search area and a better solution. The details of the PSO algorithm can be found in [73].

3.3. Grey Wolf Optimizer (GWO)

Mirjalili et al. proposed GWO [74] inspired by the grey wolves’ social hierarchy and hunting behavior. To represent the social hierarchy (leadership), the grey wolves’ population is divided into four levels: alpha ($\alpha$) is at the top of the hierarchy, leading the group; beta ($\beta$) is next to $\alpha$ in the social hierarchy to help $\alpha$ in making decisions and other activities; omega ($\omega$) is dominated by other wolves; and delta ($\delta$) dominates $\omega$ but submits to $\alpha$ and $\beta$. $\delta$ includes caretakers, hunters, sentinels, scouts, and elders [74]. The hunting behavior is divided into four steps: searching, encircling, hunting, and attacking the prey. The details of the GWO algorithm can be found in [74].

3.4. Moth Flame Optimizer (MFO)

Mirjalili developed an MFO based on moths’ navigation method through the night [75]. The main inspiration of MFO is the attraction and spiral movement of moths around artificial light sources. This spiral movement occurs because moths are easily tricked by artificial light. Due to the extremely short distance, a moth fails to keep a fixed angle when it sees an artificial light. Therefore, a deadly spiral path is generated in the MFO. Using MFO, a given optimization problem can be reasonably approximated to its global optimum. The details of the MFO algorithm can be found in [75].

3.5. Jaya Algorithm (JA)

JA is a simple yet powerful algorithm introduced by Rao [76]. The JA has a few specific parameters (i.e., population size and termination condition) without algorithm-specific parameters being set in advance, which makes it easy to implement. In each iteration, JA aims to avoid the worst solution and find the best one. The details of the JA algorithm can be found in [76].

3.6. Multi-Verse Optimizer (MVO)

MVO was proposed by S. Mirjalili et al. [77] and inspired by the multi-verse theory in physics. The MVO explains how the Big Bang creates multiple universes and how they interact through the white hole, black hole, and wormhole. MVO utilizes the concepts of a white hole and a black hole to explore the wormhole and exploit the search spaces to formulate a population-based algorithm. In the MVO, population size represents the number of universes, a universe is a solution, and objects in the universe are variables. Additionally, each solution has an inflation rate (fitness value) that represents the quality of the solution. The details of the MWO algorithm can be found in [77].

4. Development of Hybrid ML Models

Three performance metrics are used in this study to evaluate the ML models. They are the correlation coefficient ($R^2$), the root mean square error ($RMSE$), and the mean absolute error ($MAE$). $R^2$, $RMSE$, and $MAE$ are expressed as follows.

$$R^2=1-\frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^N{\left(t_i-{\mathrm{o}}_i\right)}^2}{{{\displaystyle \sum }}_{\mathrm{i}=1}^N{\left(t_i-\overline{t} \right)}^2}$$

(1)

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum }_{\mathrm{i}=1}^N{\left(t_i-{\mathrm{o}}_i\right)}^2}$$

(2)

$$MAE=\frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^N\left|t_i-{\mathrm{o}}_i\right|}{N}$$

(3)

where $N$ is the number of samples, $\overline{t}$ is the average of actual values, and ${\left[t_1,\dots ,t_N\right]}^T$ and ${\left[{\mathrm{o}}_1,\dots ,{\mathrm{o}}_N\right]}^T$ are the actual and the predicted values, respectively.

Figure 5 summarizes the entire process of developing the hybrid ML models used in this study. First, the experimental database is collected and randomly divided into training and test sets. Then, the training set is used to develop the ML models using the K-fold cross-validation (CV) technique. Previous studies have explained the idea of the K-fold CV in detail [32,45,78]. This study adopts a 5-fold CV into the training data set. In the next step, the optimization algorithms are used to find the optimal hyperparameters based on the average MAE value over the testing folds. This process is iterated by changing the training-test ratio and population size. Therefore, many models are built and evaluated using three metrics (R², RMSE, and MAE) to adopt the best hybrid ML model. Finally, a new prediction can be obtained. Based on the best hybrid model, the contribution of the input variable to output prediction is performed using the SHAP method. In addition, a web application is developed to predict the ${\mathrm{W}}_{\mathrm{f}}$ rapidly. The following sections introduce detailed descriptions of the procedure.

Considering the main factors affecting the transverse displacement mentioned in Section 3, the input variables of the ML models are the $\mathrm{L}$, $\mathrm{D}$, $\mathrm{t}$, ${\mathrm{L}}_{\mathrm{r}}$, ${\mathrm{f}}_{\mathrm{y}}$, $\mathrm{G}$, and ${\mathrm{V}}_0$, and the output is the ${\mathrm{W}}_{\mathrm{f}}$. ML models are significantly influenced by the dataset division ratio [32,45,66,79,80,81,82,83]. To investigate the effect of database division on ML model performance, this study uses seven ratios of 0.60:0.40, 0.65:0.35, 0.70:0.30, 0.75:0.25, 0.80:0.20, 0.85:0.15, and 0.90:0.10 to select the best one for the current database.

It is worth emphasizing that hyperparameters are the key to XGB models. Therefore, finding the best combination of hyperparameters for XGB models plays a crucial role.

Table 2. Hyperparameters of XGB algorithms.

No.	Hyperparameters	Range	Optimal Value of GWO-XGB
1	gama	(0.0, 1.0)	0.36999
2	learning_rate	(0.01, 1.0)	0.21749
3	max_delta_step	(1, 10)	10
4	max_depth	(1, 10)	7
5	min_child_weight	(0.0, 1.0)	0.76846
6	n_estimators	(1, 100)	97
7	reg_alpha	(0.0, 1.0)	0.39350
8	reg_lambda	(0.0, 1.0)	0.97931
9	subsample	(0.0, 1.0)	0.28597

presents the hyperparameters and ranges of the XGB model applied in this study.

5. Results and Discussions

5.1. Comparison of Performance of Different ML Models

This study develops many hybrid models that combine the PSO, GWO, MFO, JA, and MVO algorithms with the XGB model, considering the effect of population size and training-test ratio. The detailed results are presented in the Supplementary Materials. A comparison of performance metrics on a test set is used to evaluate the ML models’ performance on unseen data. Accordingly, each evaluation metric is scored from 1 to 35, corresponding to the training ratios and population sizes. It is noted that the higher the R² value is, the higher the score is, and the higher the RMSE and MAE values, the lower the score. When the value of the evaluation metric obtained by different ML models is the same, the scoring is equal. After that, all evaluation metric score values are summed up to get the final score of each ML model. The best results of the PSO-XGB, GWO-XGB, MFO-XGB, JA-XGB, and MVO-XGB models are highlighted in bold in the tables in the Supplementary Materials.

Table 3. Performance of ML models.

			Training Set			Test Set
Model	Pop	Training Ratio	R2	RMSE (mm)	MAE (mm)	R2	RMSE (mm)	MAE (mm)
PSO-XGB	150	0.90	0.996	1.295	0.750	0.975	3.097	2.019
GWO-XGB	150	0.90	0.997	1.207	0.797	0.981	2.835	1.906
MFO-XGB	200	0.90	0.995	1.47	0.903	0.969	3.433	2.284
JA-XGB	50	0.90	1.0	0.427	0.313	0.966	3.593	2.280
MVO-XGB	200	0.90	0.996	1.408	0.888	0.975	3.041	2.152
Default XGB	-	0.90	1.0	0.174	0.094	0.954	4.248	2.377

and Figure 6 show the performance of these models based on the R², RMSE, and MAE values. In addition, their results are compared with those of the default XGB model to demonstrate the efficacy of the optimization algorithm. Generally, the training performance is better than the test for all models. In the GWO-XGB model, however, there is a minor difference between training and test sets, so there is little overfitting. The GWO-XGB provides (0.997 and 0.981), (1.207 mm and 2.835 mm), and (0.797 mm and 1.906 mm) for R², RMSE, and MAE in the training set and test set, respectively. In contrast, the default XGB model significantly differs between training and test performances. The default XGB obtains (1.0 and 0.954), (0.174 mm and 4.248 mm), and (0.094 mm and 2.377 mm) for R², RMSE, and MAE in the training set and test set, respectively.

It is observed that all hybrid ML models perform better than the default XGB model in the test set. Compared with the default XGB model, R² increased by (2.201%, 2.830%, 1.572%, 1.258%, and 2.201%), RMSE decreased by (27.095%, 33.263%, 19.185%, 15.419%, and 28.413%), and MAE reduced by (15.061%, 19.815%, 3.912%, 4.081%, and 9.466%), respectively, in the PSO-XGB, GWO-XGB, MFO-XGB, JA-XGB, and MVO-XGB models. This indicates that the PSO, GWO, MFO, JA, and MVO algorithms can significantly enhance the capacity of the XGB model. Overall, it seems that the GWO improved the XGB model over the other optimization algorithms in this study.

According to the results, the GWO-XGB model with a population size of 150 and a training-test ratio of 0.90:0.10 performs better than the other ML models; meanwhile, the default XGB model has the worst prediction performance. The optimal hyperparameters of the best GWO-XGB model are shown in Table 2.

Figure 7 compares the predicted and the actual values of the PSO-XGB, GWO-XGB, MFO-XGB, JA-XGB, MVO-XGB, and default XGB models. In this figure, the vertical axis represents predicted values of ${\mathrm{W}}_{\mathrm{f}}$, and the horizontal axis represents observed values of ${\mathrm{W}}_{\mathrm{f}}$.

5.2. Model Explanation

In this section, the Shapley additive explanations (SHAP) method [84] is used to explain individual and global prediction of ${\mathrm{W}}_{\mathrm{f}}$. The effect of the Shapley value on the prediction value is shown in Figure 8. Shapley values are like arrows pointing towards a predicted value that either increases (positive value) or decreases (negative value).

Figure 9a shows the global importance factors of the seven input variables for the ${\mathrm{W}}_{\mathrm{f}}$ prediction. The feature with a larger absolute summation of Shapley values is more important. It can be seen that the ${\mathrm{V}}_0$ and $\mathrm{G}$ have a significant influence on the ${\mathrm{W}}_{\mathrm{f}}$ prediction, followed by ${\mathrm{L}}_{\mathrm{r}}$, ${\mathrm{f}}_{\mathrm{y}}$, $\mathrm{L}$, $\mathrm{D}$, and $\mathrm{t}.$

Figure 9b shows the SHAP summary plot. Each point on the graph represents the Shapley value for each specimen. The red point on the right-hand side of the plot indicates a positive correlation with the ${\mathrm{W}}_{\mathrm{f}}$, while the red point on the left-hand side indicates a negative correlation. Accordingly, when ${\mathrm{V}}_0$, $\mathrm{G}$, and ${\mathrm{L}}_{\mathrm{r}}$ increase, the ${\mathrm{W}}_{\mathrm{f}}$ increases. In contrast, the ${\mathrm{W}}_{\mathrm{f}}$ decreases if the ${\mathrm{f}}_{\mathrm{y}}$ increases. Meanwhile, the $\mathrm{L}$, $\mathrm{D}$, and $\mathrm{t}$ are less effective and unclear on ${\mathrm{W}}_{\mathrm{f}}$ predictions.

Figure 10a shows the Shapley values of a typical prediction using the GWO-XGB model for a specimen with $\mathrm{L}=1200.0$ mm, $\mathrm{D}=120.0$ mm, $\mathrm{t}=2.0$ mm, ${\mathrm{L}}_{\mathrm{r}}=0.5$, ${\mathrm{f}}_{\mathrm{y}}=468.0$ MPa, $\mathrm{G}=79.5$ kg, and ${\mathrm{V}}_0=9.88$ m/s. It can be seen in Figure 10a that the prediction (60.785 mm) is larger than the base value (29.385 mm). The ${\mathrm{V}}_0$, $\mathrm{G}$, ${\mathrm{L}}_{\mathrm{r}}$, ${\mathrm{f}}_{\mathrm{y}}$, and $\mathrm{t}$ parameters, shown in red, increase the base value; meanwhile, other parameters ($\mathrm{D}$ and $\mathrm{L}$), depicted in blue, decrease the base value. Among them, the most crucial variable is ${\mathrm{V}}_0$. For a specimen in Figure 10b with $\mathrm{L}=1200.0$ mm, $\mathrm{D}=120.0$ mm, $\mathrm{t}=2.0$ mm, ${\mathrm{L}}_{\mathrm{r}}=0.5$, ${\mathrm{f}}_{\mathrm{y}}=468.0$ MPa, $\mathrm{G}=98.0$ kg, and ${\mathrm{V}}_0=6.28$ m/s, the explanation is similar. However, the most crucial variable is $\mathrm{G}$.

6. Web Application

The results obtained in this study are valuable, but developing a user-friendly web application tool would promote GWO-XGB model adoption in engineering practice. The web application developed in this study uses seven input parameters to directly obtain the ${\mathrm{W}}_{\mathrm{f}}$ of CHS steel members under transverse impact loads. Notably, there should be no need for coding to display the results. This web application can also assist engineers in obtaining accurate and fast results, regardless of the complexity of the equations. Accordingly, using the web application can save time and effort to estimate the ${\mathrm{W}}_{\mathrm{f}}$ of CHS steel members under transverse impact load in the pre-design process. The web application can be accessed via the link https://sakat92-wf-wf-nfdrqz.streamlit.app/ (accessed on 1 May 2023).

7. Conclusions

This study develops several hybrid ML models, combining XGB and five metaheuristic optimization algorithms (PSO, GWO, MFO, JA, and MVO) to predict the ${\mathrm{W}}_{\mathrm{f}}$ of CHS steel members under transverse impact load. The following are the conclusions that can be drawn from the results of this study:

• Hybrid ML models generalize better than the default XGB model. Compared with the default XGB model, R² increases (by 2.201%, 2.830%, 1.572%, 1.258%, and 2.201%), RMSE decreases (by 27.095%, 33.263%, 19.185%, 15.419%, and 28.413%), and MAE reduces (by 15.061%, 19.815%, 3.912%, 4.081%, and 9.466%) in the test set, respectively, in the PSO-XGB, GWO-XGB, MFO-XGB, JA-XGB, and MVO-XGB models.
• Among hybrid ML models, the GWO-XGB model predicts better than the others, with the highest R² value (0.981), the lowest RMSE (2.835 mm), and the MAE (1.906 mm) value in the test set.
• The SHAP method shows that ${\mathrm{V}}_0$, $\mathrm{G}$, and ${\mathrm{L}}_{\mathrm{r}}$ are the most influential factors in ${\mathrm{W}}_{\mathrm{f}}$ prediction.
• A web application is developed to facilitate ${\mathrm{W}}_{\mathrm{f}}$ prediction. Users can quickly visualize the results using the GWO-XGB model, making it the best tool for promoting the practical application of the model.

Although the results in this study are great, the developed ML models are based on the limited database presented in Table 1. Thus, the database should be widened to enhance the performance of the ML model. Specifically, the findings of this study are focused on CHS steel members. Other types of members should be considered in future work.