Analysis of nonlinear dynamics of RC slabs under blast loads: A hybrid machine learning approach

1 Introduction

Technological advancements have resulted in the development of compact, potent, mobile explosive devices widely used by terrorist organizations to target critical infrastructures and buildings. This modern threat necessitates effective structural mitigation strategies, sparking research using nonlinear engineering methods to study the effects of blast loads on building components and develop measures to minimize the devastation caused by such catastrophic events. One critical aspect of assessing damage is calculating the maximum deflection of concrete slabs subjected to blast loads. This is crucial as it helps in understanding the load-bearing capacity and the structural integrity of buildings under such loads. Knowing the maximum deflection allows engineers and architects to design structures that can withstand blasts without catastrophic failure, thereby safeguarding human lives and assets. It also helps the retrofitting process of existing structures to enhance their resilience against blast loads.

Recent experimental studies have explored the blast resistance of diverse materials, including fibre-reinforced concrete (RC) panels, high-performance concrete panels, hybrid fibre-RC panels, Aramid fibre RC panels, etc. [1,2,3,4,5]. These experimental investigations, while accurate, are costly, demanding, and hazardous due to the use of explosives and sensitive instrumentation. The difficulties associated with experimental testing have led to the adaptation of finite element simulation for the analysis of concrete structures subjected to blast loading by many researchers [6,7,8,9,10]. However, they, too, have limitations, including the influence of mesh sizes and material models on results, as well as the necessity for technical expertise for accurate simulations and result interpretations.

The high cost and difficulty in conducting experimental analysis, as well as the need for strong knowledge in computer modelling and the high computational cost associated with numerical simulations, have demanded a much simpler and more accurate method for the analysis of structures subjected to blast loading [11]. Machine learning (ML) has emerged as a promising solution for these complex problems. Its ability to assess and learn connections between numerous parameters makes it well-suited for predicting mechanical characteristics of concrete [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]. An ML model was developed to predict the maximum displacement of RC slabs under blast loads using a database consisting of 260 data samples [35]. Monjee et al. introduced an ML model for predicting the maximum displacement as well as to identify the failure modes and crack patterns in RC beams under blast loading [11]. The smaller database used for training the application of these models was limited to a specific range. Notably, optimizing hyperparameters of ML models significantly enhances their predictive accuracy [36,37,38]. Still, high predictive capacity alone does not suffice. For an ML model to be reliable, it should identify key parameters influencing predictions to avoid unexpected problems [13]. Interpretable ML models can provide valuable insights into the underlying complex nonlinear behaviour of structural elements subjected to blast loading along with an awareness of the significance of different factors causing the final prediction. The literature review reveals that only a few studies have employed ML models for predicting the behaviour of concrete slabs subjected to blast loading. Moreover, these studies typically rely on databases with a limited number of data samples.

Addressing these gaps, this study aims to develop an interpretable hybrid ML model to predict the maximum deflections in RC panels under blast loading. In distinguishing this study from previous research, it is essential to note the novel contributions made in the field of ML application for blast load analysis on concrete slabs. Unlike earlier studies that often relied on limited data, this research utilizes an extensive and varied database, ensuring a more robust and generalizable predictive model. The present methodology stands out by methodically shortlisting potential ML models through preliminary analysis rather than directly selecting a few, as was common in prior research. Notably, this study pioneers the use of advanced hyperparameter optimization techniques – grid search (GS), particle swarm optimization (PSO), and genetic algorithms (GAs) – to fine-tune the predictive models, enhancing their performance beyond the capabilities of previous studies. Furthermore, this research introduces the application of SHAP (SHapley Additive exPlanations) analysis, providing a level of interpretability previously absent, which is crucial for practical engineering applications. These innovative approaches collectively contribute to a significant advancement in predicting and understanding the behaviour of RC slabs under blast loads, marking a substantial leap forward in the application of ML in structural analysis.

2 Significance of research

In the face of escalating threats from mobile explosive devices, this research plays a pivotal role in enhancing structural safety. It introduces an innovative prediction method to aid structural design, eliminating the challenges often found with traditional experimental and numerical investigations. The need for effective, cost-efficient strategies to fortify the structural design of critical infrastructure is addressed by the development of an interpretable hybrid ML model in this study. To overcome the limitations of already available prediction models, a broader dataset consisting of 936 data samples is used in this research. This model is capable of predicting maximum deflection in RC panels under blast load, which will help in enhancing the structural design and thereby reduce the risks linked to such threats. The larger dataset, which is used to develop the model, will help in making reasonable predictions over a wider range of parameters. The adopted hyperparameter optimization techniques, using GS, PSO, and GAs, have helped in developing a model of exceptional predictive accuracy, providing a powerful tool for engineers and policymakers. The interpretability of the developed ML model is further amplified through SHAP analysis, shedding light on the parameters affecting the structural responses of concrete slabs subjected to blast loading. In essence, by offering an innovative, data-driven tool, this research provides a significant contribution to the ongoing endeavour to enhance structural safety and resilience. In addition to its core contributions, this study also helps beginners in this area to understand the potential of ML in solving complex problems associated with structural engineering. It also provides a well-built database for researchers and assumes an educational role by providing explanations on optimisation techniques and SHAP analysis.

3 Model development methodology: An overview

The research methodology comprises four stages, each of which is schematically represented in Figure 1 for visual clarity. Python was the programming language of choice for coding the entire research program.

Figure 1

Model development methodology.

4 Optimization techniques used for tuning hyperparameters

In this study, hyperparameters of ML models were optimized using three distinct methods: GS, GA, and PSO. This section provides a brief explanation of all these methods.

4.2 Genetic algorithm

This is a more advanced and nature-inspired method used for hyperparameter tuning in ML. GAs are inspired by the process of natural selection and involve generating a population of candidate solutions, applying genetic operations such as selection, mutation, and crossover, and iteratively evolving the population towards better solutions. In the context of hyperparameter tuning, GAs can be used to search through a large space of hyperparameter configurations and identify the combination of hyperparameters that produces the best performance on a given validation set. Figure 2 shows the details of the steps involved in the implementation of GA for tuning the hyperparameters of ML models. In hyperparameter tuning, GAs explore and identify optimal hyperparameter sets. While effective, they can be computationally intensive and require careful algorithm design.

Figure 2

Steps involved in implementing GA.

4.3 Particle swarm optimization

It is a metaheuristic optimization algorithm that can be used for hyperparameter tuning in ML. In PSO, a population of particles searches for the optimal solution by adjusting their positions and velocities based on their own best position and the best position of their neighbours in the search space. PSO is effective in optimizing various objective functions, including those encountered in ML, and can be used for hyperparameter tuning in a range of ML models. Figure 3 shows the details of the steps involved in the implementation of PSO for tuning the hyperparameters of ML models. A study by Lorenzo et al. [43] evaluated the effectiveness of using PSO for hyperparameter tuning in deep learning models. The authors found that PSO was able to identify hyperparameter configurations that outperformed those selected using other methods such as GS, GA and simulated annealing. However, the authors noted that PSO requires careful parameter tuning and that its performance can be sensitive to the choice of parameters.

Figure 3

Steps involved in implementing PSO.

5 Experimental program

As the literature provided limited experimental data, the authors undertook a small-scale experimental investigation to enrich the main dataset and to validate the numerical analysis discussed in the subsequent section. Six concrete panels, with dimensions depicted in Figure 4, were subjected to blast loading as part of this investigation. The experiment involved three sets of specimens with varying concrete strengths and two distinct amounts of explosives (0.45 and 0.90 kg). Table 1 provides detailed information on each specimen, including the designation, 28-day cylinder compressive strength of the concrete, and the weight of the explosives used. All the concrete panels were constructed with identical dimensions and reinforcement specifications as employed by Abeysinghe et al. [5]. Each specimen was reinforced with two layers of 6 mm diameter bars, spaced 300 mm centre-to-centre along the longer span and 225 mm centre-to-centre along the shorter direction. The steel reinforcements utilized in this investigation were subjected to tensile testing and observed to have a yield strength of 305 MPa and an ultimate strength of 440 MPa. Figure 4 illustrates the reinforcement configurations of the panels used in this study.

Figure 4

Dimension and reinforcement details of the concrete panel.

5.1 Test set-up

The test set-up used in this investigation is shown in Figure 5. Two bolted angle sections were employed to control the panel’s horizontal movement and to provide vertical restraint against the rebound that might occur from the blast load. The angle sections are connected to a steel frame, which was anchored to a concrete foundation at the base.

Figure 5

Test set-up.

5.2 Data acquisition

In this study, data were captured using a pressure transducer with a measurement range of ±34.500 MPa, and two antennas. Figure 6 displays the positioning of the pressure transducer and antennas on the specimen.

Figure 6

Location of pressure transducer and antennas.

The antennas helped in recording the maximum deformation at two locations (A and B). As far as the present research work is concerned, the deflection is the only significant result that needs to be considered from the experimental program, and thus, other results are neither included nor discussed in the study. The average values of the observed maximum deflection for each of the specimens are given in Table 1.

6 Numerical simulation

The amount of data available in the used database will directly influence the efficiency and accuracy of prediction of any ML model. However, the availability of experimental and well-validated numerical simulation data with all the necessary parameters was limited. To address this, the authors decided to augment the dataset by conducting additional numerical simulations using the commercially available finite element software ABAQUS/CAE 2021. Before fully employing these simulations, it was imperative to validate the accuracy of the methods used. This was done by comparing the results from the numerical simulations with those from experimental investigations. For the validation, five RC slab specimens were employed. Two of these specimens were sourced from the author’s own experimental investigation, as discussed in the previous section, while the remaining three were based on the experimental investigation conducted by Zhao and Chen [44]. The rationale behind using two specimens from the authors’ experiments and three from the research work by Zhao and Chen [44] was to capture the effects of variations in geometry and material properties.

The specimens used by Zhao and Chen [44] had different dimensions, standoff distances, and quantity of used explosives. In this simulation, the concrete matrix and the steel reinforcements were modelled using C3D8R 8-node brick elements and B31 2-node beam elements, respectively. The steel bar was embedded inside concrete in the present study. The explosion was simulated with the help of the CONWEP model available in ABAQUS. The CONWEP model available in Abaqus/Explicit can simulate air blast loading even without modelling the fluid medium. To achieve this, only the equivalent weight of TNT and a reference point (standoff distance) need to be specified.

The Johnson–Cook model (JC) and Johnson and Holmquist model (JH2) were used as the constitute model for steel and concrete, respectively. The JC model was proposed by Johnson and Cook [45] in 1983 as a constitutive model for metals that can accommodate large deformations occurring at high speeds, such as those found in explosion situations. The model was developed after testing ductile materials exposed to high torsions, strain rates, and different temperatures. To replicate the behaviour of materials that can withstand significant deformations, high pressure, and strain rates, like concrete structures subjected to explosions or impacts, Johnson and Holmquist devised the JH2 model. Rather than relying on stress–strain curves, this model incorporates equations of state that take into account various factors such as material strength, pressure, strain, strain rate, and a damage variable termed D that monitors the progression of damage in the material. This damage variable accumulates over time, leading to material failure when its value reaches 1 [45].

As mentioned earlier, two concrete panel specimens (S3 and S4) from the current investigation were used for the validation. For geometrical details and material properties of these specimens, see Section 5. In addition, three concrete panels (V1, V2, and V3) with dimensions of 1,000 mm × 1,000 mm × 40 mm used by Zhao and Chen [44] were also modelled in ABAQUS. The dimensions and reinforcement details of these specimens are given in Figure 8(a). The properties of the constituent materials of specimens taken from the literature are shown in Table 2. Based on the results of the convergence study, the optimum mesh size was determined to be 20 mm, which was subsequently used for the numerical simulation of all specimens investigated in this study. The meshed model of the specimens used for validation is presented in Figures 7 and 8(b).

Figure 7

Meshed model of specimen S3.

Figure 8

(a) Details of the specimen taken from the literature [44]. (b) Meshed model.

The maximum deflection during each simulation was recorded, and the corresponding values obtained from the numerical analysis, experimental test, and their respective percentage variations are presented in Table 3. The maximum variation in the predicted central deflection was 4.55%, which is well within the limit for a nonhomogeneous material such as concrete. The crack pattern obtained from the experimental analysis was also compared with the results obtained from numerical simulation. Figures 9(a)–10(c) show the comparison of tensile damage obtained at the bottom surface for specimens S3, S4, V1, V2, and V3, respectively, from the numerical analysis with the crack pattern obtained from the experimental analysis. As shown in the comparison, both patterns were observed to be reasonably matched for all five specimens.

Figure 9

Comparison of the crack pattern: (a) specimen S3 and (b) specimen S4.

Figure 10

Comparison of the crack pattern: (a) specimen V1, (b) specimen V2, and (c) specimen V3.

Since the proposed FEM was able to predict the behaviour of concrete panels subjected to blast load with reasonable accuracy, a detailed numerical investigation was conducted using the proposed method. In total, there were 746 simulations conducted as part of this numerical investigation. The results of all these simulations were added to the primary database for developing the ML model and are also provided as a supplementary document to this manuscript for the reference of readers.

7 Data assembly and preprocessing (stage 1)

The accuracy and reliability of any ML model depend on the quality and quantity of data used for training. Therefore, the utmost effort was made to collect reliable and consistent data regarding the behaviour of concrete slabs under blast loading. To gather experimental data on the behaviour of RC slabs subjected to blast loading, relevant literature was extensively reviewed. However, due to the limited availability of experimental data specifically focused on blast analysis of concrete panels, numerical simulations reported in the literature, which were properly validated against experimental results, were also considered. As discussed in previous sections, additional experimental and numerical studies were also conducted to enrich the database.

The final dataset comprises a total of 936 data samples, which includes 128 data samples from experimental studies collected from literature sources [46,47,56,57,58,59,60,48,49,50,51,52,53,54,55], 6 data samples from the authors’ experimental investigation, 56 data samples from numerical studies published in the literature [20,60,61,62,63], and 746 data samples from numerical simulations conducted by the authors. Thus, the final dataset consisted of 15% experimental data and 85% numerical results, which were properly validated. The database consists of 493 one-way slabs and 443 two-way slabs. Among these, 60 slabs were simply supported on two ends, 433 slabs were fixed at two ends, and 443 slabs were fixed on all four sides.

Based on the literature, it is evident that several parameters significantly influence the behaviour of concrete panels subjected to blast loading. These parameters include the panel dimensions, material properties, amount of explosive used, and standoff distance. Taking this into account, eight input parameters were selected for the current investigation. The parameters chosen based on the specimen dimensions included the length, width, thickness, and support conditions of the panels, as well as the percentage of transverse reinforcement provided. Additionally, the compressive strength of concrete and the yield strength of steel reinforcement were considered as input parameters based on the materials used. Furthermore, considering the quantity of explosives used and the standoff distance, the scaled distance was calculated using Eq. (1) and included as another input parameter. Thus, there were eight input parameters in total, and the maximum deflection of the concrete slab was considered as the output parameter:

where Z is the “scaled distance” (m/kg^1/3), S is the distance from the centre of the explosive to the target (m), and W is the TNT-equivalent weight of the explosive (kg).

The collected raw data were carefully examined for any missing values and only the results containing complete information for all parameters in the main database were considered. Extensive efforts were made to acquire the missing data by contacting the respective researchers. As one of the parameters (support condition of the panels) was categorical, dummy values were assigned to represent these categories using a hot encoding method. In the subsequent stage, all parameters were scaled using the StandardScaler function provided by Scikit-learn. Following this step, outliers within the database were identified and removed using the interquartile range method. Figure 11 shows the distribution of different parameters in the main database.

Figure 11

Distribution of parameters in the database.

Table 4 presents statistical parameters such as minimum value, maximum value, mean, and standard deviation for different features and the output variable. Additionally, Figure 12 displays a plot depicting the relationship between the output parameter (maximum deflection) and the various input parameters. Observing the plot, it is evident that there is no direct correlation between the output and any of the features in the dataset.

Figure 12

Plot showing the variation of deflection with respect to each feature.

The main database was divided into three groups, which constituted the training database, validation database, and testing database. The training database contained 70% of the total data and was used for training different models. The validation database consisted of 15% of the total data and was employed to assess the predictive performance of all models during the model development process until the final hybrid model was obtained. To ensure the absence of data leakage during model development and to guarantee that the testing data remain completely new to the finalized model, a separate testing database containing the remaining 15% of data was set aside for the final model evaluation. By reserving the testing database for the end, the prediction performance achieved during testing serves as the final evaluation metric for the proposed model. No further modifications to the model parameters are made beyond this stage, ensuring that the reported prediction efficiency represents the ultimate performance of the model.

8 Preliminary model exploration (Stage 2)

In this stage, an exploratory analysis was conducted using eight traditional ML models: elastic net regressor, support vector machines (SVM), K-nearest neighbours (KNN), decision trees (DT), random forest (RF), gradient boost (GB), AdaBoost (ADB), and XGBoost (XGB). The primary objective of this analysis was to assess the feasibility and identify the best-performing models for predicting the maximum deflection of concrete panels subjected to blast loading. All models were trained using the training dataset, and their prediction efficiency was evaluated on the validation dataset. The training and evaluation were performed using the default settings and parameters of each model. To ensure a more accurate and reliable assessment, a K-fold cross-validation with 10-fold was implemented, leveraging the entire available data. Three evaluation metrics, namely MAE, RMSE, and R ² value, evaluated using the following equations [64], were utilized to evaluate the performance of each model:

where n is the number of samples, y _e is the actual value, and y _p is the predicted value.

Models that achieved a minimum accuracy threshold of 75%, along with lower MAE and RMSE values, were considered suitable for predicting the maximum deflection of concrete panels under blast loading and were selected for further refinement. Table 5 presents the MAE, RMSE, and R ² values obtained while evaluating different ML models using the training and validation databases. From the table, it is observed that four models (DT, RF, GB, and XGB) were able to predict the deflection while meeting the acceptance criteria. Therefore, these four models were selected for further refinement.

9 Hyperparameter optimization (Stage 3)

Although the accuracy of the models selected in Stage 2 was reasonable, efforts were made to enhance their prediction performance by optimizing their hyperparameters. Significant hyperparameters associated with all four selected ML models were identified based on the literature. Table 6 illustrates the different hyperparameters considered for each model, along with the corresponding range of values assigned to these parameters. The focus of hyperparameter optimization was to assess the impact of using different values for these parameters in order to improve prediction accuracy. To determine the optimum values for the hyperparameters, three different methods (GS, GA, and PSO) were employed in combination with the selected ML models. Similar to Stage 2, the performance of all four hybrid models was assessed on the validation dataset using the same three evaluation metrics: MAE, RMSE, and R ² value. In this stage, also a 10-fold cross-validation strategy was implemented to ensure a more accurate and reliable assessment.

9.3 Optimization using PSO

As the application of optimization using GA also did not enhance the prediction performance of all four models, a new attempt was made to optimize the hyperparameters using the PSO technique. Using PSO the optimization process started by generating a set of particles (hyperparameters) within the provided range, which represent potential solutions, randomly. These particles were allowed to move through the search space, adjusting their position based on their experience (results of evaluation metrics obtained in the previous iteration for the considered hyperparameters) and that of their neighbours (results of evaluation metrics obtained in the previous iteration for other sets of hyperparameters). The accuracy of prediction made during each iteration was used to evaluate the fitness of each particle, with better-performing particles given a higher weight and poorer-performing particles given a lower weight. To balance accuracy and computational cost, the number of particles and the maximum number of iterations was fixed as 100. The particles were moved towards the better-performing particles in the swarm, and the process was repeated until either convergence was achieved or the number of iterations exceeded the provided limit. The optimal set of hyperparameters was then determined based on the best-performing particle in the swarm.

The MAE, RMSE, and R ² values obtained for all four hybrid ML models (based on PSO) after hyperparameter optimization are given in Table 7. A comparison of these values with all other models is also shown in Table 7. The optimal set of hyperparameters for all four PSO-based hybrid models is presented in Table 8.

The results of this investigation indicate that the PSO technique outperformed both other methods in terms of enhancing the model’s prediction. Specifically, the RMSE values obtained for each model optimized using PSO were significantly lower than those obtained using GS and GA. Without hyperparameter optimization, the RMSE values ranged from 11.7 to 21.74 for the four models. The optimization of hyperparameters using PSO resulted in improvements in all four models, with RMSE values ranging from 8.99 to 18.11. It can also be noted that out of the four-hybrid model, the GB-PSO model had the best performance with a reduction of 60% MAE value, 58.7% RMSE value, and a 9% increase in R ² value, compared to the respective model without optimization. These results suggest that PSO is a powerful technique for hyperparameter tuning that can significantly enhance the accuracy of ML models. Hence, PSO was considered the best method for hyperparameter optimization in this investigation.

In order to finalize the best model, the ultimate predictive efficacy of all four PSO-based hybrid ML models was further evaluated using a test database that had been set aside from the beginning. The final performance of all the PSO-integrated hybrid models is illustrated in Figure 13.

Figure 13

Comparison of the prediction performance depicted by the PSO-based hybrid models (using test database).

As seen in Figure 13, the GB-PSO (GB-PSO) model surpassed the other three models in terms of prediction accuracy on the test database, achieving a 92% accuracy rate. The GB-PSO model was also able to attain a lower MAE and RMSE value in comparison to the other three models. Therefore, the GB-PSO model was chosen as the most effective model for the considered problem based on the results of this investigation.

10 Influence analysis of input parameters (Stage 4)

Interpreting ML models has become a crucial research area in recent years, as it allows users to better understand the behaviour of these models and trust their output. One approach to achieving interpretability is the use of post-hoc interpretability methods such as SHAP analysis. SHAP analysis is a method that aims to explain individual predictions made by an ML model by identifying the contribution of each feature to the prediction. SHAP analysis is based on the Shapley value, a concept from cooperative game theory. It assigns an importance score to each feature, indicating the contribution of that feature to the prediction. This method can be used with any ML model, including complex models such as deep neural networks, and provides a unified framework for interpreting the output of different models.

The main objective of this investigation was to develop an interpretable hybrid ML model for predicting the final deflection of concrete slabs subjected to blast loading. To achieve this, SHAP analysis was used to identify the important features used by the finalized GB-PSO model to make predictions and generate global explanations for the final model. Incorporating the results of the SHAP analysis is expected to increase the transparency and interpretability of the final model and ultimately improve trust in its output. Two essential visualizations that aid in understanding the results of a SHAP analysis are the bar plot and the bee swarm plot. Both these plots were drawn for the final model using SHAP analysis and are shown in Figures 14 and 15, respectively. The following paragraphs delve into the insights drawn from these plots while explaining their significance and purpose for readers who are new to these concepts:

Figure 14

Bar plot showing mean SHAP values.

Figure 15

Beeswarm showing SHAP values.

Figure 14 presents the bar plot of the GB-PSO model, which provides an overview of the feature importance of the predictions made by the model. It displays the average magnitude of the SHAP values for each feature, ranked in descending order. The magnitude of a SHAP value indicates the contribution of that feature towards the considered model’s prediction. A higher SHAP value signifies a greater impact of the corresponding feature on the model’s output. Bar plots are crucial for understanding the hierarchical importance of features and identifying the most influential ones in the model’s predictions.

Upon examining the bar plot shown in Figure 14, it can be observed that the scaled distance has the highest SHAP value. With the highest average SHAP value, the scaled distance demonstrates the most significant impact on the considered model’s predictions. This means that it is the primary driving force behind the model’s decision-making process. Following the scaled distance, the thickness of the panel has the second-highest average SHAP value. This indicates that the thickness of the panel is also an important contributor to the model’s predictions, albeit to a lesser extent than the scaled distance. After the thickness, three other important parameters observed to be important are Support_2FF (Slab fixed on two sides), the length of the slab, and the concrete strength. The remaining features have lower average SHAP values, which means they have a smaller impact on the model’s predictions. While these features may still play a role in the overall decision-making process, their influence is less significant compared to the other features.

On the other hand, the bee swarm plot shown in Figure 15 offers a more detailed visualization of the individual instances and their respective SHAP values for each feature. In the bee swarm plot, each dot represents an individual data point, and the position of the dot along the x-axis corresponds to the SHAP value for that data point. The SHAP values on the positive side of the x-axis indicate a positive contribution to the model’s prediction, while those on the negative side imply a negative contribution. The vertical distribution of the dots demonstrates the variability of a feature’s influence across different data points. The bee swarm plot is essential for uncovering complex relationships between features and their effects on the model’s output, as it displays the distribution and density of SHAP values across instances.

Based on Figure 15, it can be observed that for scaled distance, the majority of the data points are concentrated on the positive side of the x-axis, indicating that it predominantly has a positive impact on the model’s predictions. There is also a small cluster of data points with negative SHAP values, suggesting that scaled distance could have a negative impact on the model’s predictions in some instances, but these occurrences are relatively rare. The distribution of data points for the thickness of the panel is more evenly spread across the positive and negative sides of the x-axis. This implies that the thickness of the panel has a more balanced impact on the current model’s predictions, with both positive and negative contributions. The relatively even distribution also indicates that the relationship between the thickness of the panel and the resulting deflection might be more complex and nonlinear. The distribution of data points for concrete strength, length of the panel, and Support_2FF (slab fixed on two ends) also display a mix of positive and negative SHAP values. However, the distribution of data points for the length of the panel and Support_2FF seems to be slightly denser on the positive side, suggesting that they have a somewhat stronger positive influence on the model’s predictions compared to their negative impact. The distribution of data points for concrete strength seems to be slightly denser on the negative side, suggesting that they have a somewhat stronger negative influence on the model’s predictions. The remaining features in the plot exhibit lower SHAP values, indicating that they have less impact on the model’s predictions. However, it is essential to note that these features may still contribute to the overall prediction in certain instances.

Thus, the application of SHAP analysis has provided valuable insights into the important parameters that contribute to the prediction of the GB-PSO model. The analysis reveals that the scaled distance is the most influential parameter, demonstrating the highest SHAP value and exerting a significant impact on the model’s predictions. The thickness of the panel follows as the second most important parameter. Additionally, the Support_2FF (slab fixed on two ends), the length of the panel, and the concrete strength are identified as important contributors to the model’s predictions.

11 Summary and conclusions

In this research, an interpretable hybrid ML model was successfully developed for predicting the maximum deflection of RC slabs subjected to blast loading. The significance of this study lies in the development of a cost-effective and efficient alternative to conventional experimental and numerical methods, which often require substantial financial and computational resources. The proposed model also provides crucial insights into the parameters governing the deflection prediction, which helps the researchers and engineers in understanding the complex behaviour of RC slabs under blast loading.

A comprehensive research methodology was presented in this research, which involved data assembly and preprocessing, preliminary model exploration, hyperparameter optimization, and influence analysis of input parameters. The study effectively utilized data from various sources, including the literature, the authors’ experimental investigations, and extensive numerical simulations for developing a rich database. This enriched database consisting of 936 data samples ensured the development of a robust model. This database is included as supplementary material to this manuscript, which will be a significant resource for researchers and practitioners in the field, facilitating further investigations and model developments.

Another key aspect of this study is the interpretability of the developed model. The application of SHAP analysis was instrumental in identifying the influential parameters for the prediction of maximum deflection. Both the bar plot and the beeswarm plot for the GB-PSO model (explaining the role of different input parameters) were developed and included in this study. The scaled distance was found to be the most influential parameter, followed by the thickness of the panel, which is well aligned with the experimental results available in the literature.

In addition to the development of the model, this study also includes a brief but informative explanation of three hyperparameter optimization techniques, namely GS, GA, and PSO, as well as a concise introduction to SHAP analysis. This educational content is particularly beneficial for beginners and researchers who are new to ML and structural analysis, serving as a primer for understanding and employing these methodologies.

However, it is important to acknowledge the limitations of the present work. While efforts were made to develop a larger database, the number of experimental results included can be increased. Additionally, there is room for improvement by including more parameters related to blast loading and concrete structure to enhance the accuracy of the prediction model. Looking ahead, there are promising avenues for future research. The presented methodology can be extended to other structural members, expanding the applicability of the developed model. Furthermore, deep learning techniques, which were not included in the present research, can be incorporated and their performance evaluated, potentially leading to enhancements in prediction accuracy.

In conclusion, amidst the rising threats posed by explosive devices, this research presents a groundbreaking interpretable hybrid ML model tailored to predict maximum deflections in RC slabs under blast impacts, marking a monumental shift from traditional, resource-intensive methods. Our comprehensive approach amalgamated data from a myriad of sources, ensuring the development of a proficient model backed by a substantial database, which stands as a beacon for future researchers. Integrating the model with SHAP analysis enhanced its trustworthiness and clearly identified pivotal parameters, resonating with insights from experimental studies. Our exploration of hyperparameter optimization, paired with an introductory overview of SHAP analysis, stands as a foundational guide for newcomers to ML and structural dynamics. In essence, this research exemplifies the harmonious blend of ML with structural engineering, paving the way for innovative structural mitigation approaches and ensuring reinforced protection against looming threats.