Reinforced concrete deep beams are key structural components that are essential in a variety of applications, including bridges, tall buildings, and marine structures. They are capable of creating strut-like compression components between applied loads and supporting structures through their distinctive load and support setup, thus being effective load-distribution elements in various structures. According to ACI Code 318-18 [1], beams are classified as deep beams for flexion, with the overall depth ratio of single beams being less than 1.25 and the overall depth ratio of continuous beams being more than 2.5. Furthermore, because a deep beam has high shear capacity and a low depth ratio (a/d < 2.0), it is commonly used in high-rise buildings, bridges, and naval structures. Their differences in behaviour, including high-stress points and uneven strain distributions, question the accuracy of Bernoulli's theory [5], [6]. RC deep-beam damage location is usually determined by empirical methods or rules of thumb, while the strut-and-tie method offers an alternative approach [7], [8], and [9]. Most design codes for RC deep beams are based on empirical or semi-empirical formulas. However, the accuracy and reliability of these equations are limited by the scope of experimental data used for the evaluation.
Limits of Empirical Shear Strength Models
Although these empirical methods tend to be cautious in the design process, they can lead to potentially dangerous results [10]. Collins et al. [11] studied shear-strength estimation techniques for NZS 3101-2006, ACI 318-18, and EC2-1-2004 using extensive RC beam data. The results show that many predictions are too optimistic, even when safety margins are considered. Furthermore, contemporary design codes have adopted a design methodology incorporating mechanical models of struts and ties. It is still difficult to estimate the shear capacity of reinforced concrete beams, and some current models provide conservative estimates, including safety factors. The complexity of failures and the many factors that affect them make it difficult to determine shear capacity. As a result, interest in predicting the shear strength of RC deep beams through machine learning (ML) methods is increasing, and these methods have the potential to improve accuracy and effectiveness. In several research projects, machine learning technologies are used to predict the strength of reinforced concrete beams.
Prior Machine Learning Approaches
A major research project explored the use of artificial neural networks (ANNs) to predict the ultimate shear strength of RC deep beams. Scientists collected 111 experimental data sets, analyzed 10 input factors, including geometric and material properties, and calculated the ultimate shear strength. ANN performance was evaluated using traditional methods such as ACI, strut-and-tie, and Mau-Hsu. The results show that artificial neural networks (ANNs) surpass conventional techniques and are much more precise, with an average deviation strength of 0.99. Similarly, Wakjira and colleagues [13] proposed a modified shear-design formula for reinforced concrete beams without stirrups using machine learning and genetic algorithms. The model considers factors such as concrete strength, cross-section size, shear-span ratio, and internal reinforcement to estimate shear performance. The proposed model was compared with the ACI 318/14 method [14] and demonstrated better safety, accuracy, and cost-effective prediction capabilities.
Advanced Models and the Research Gap
Although ML applications are not common in RC deep beams, Mohammadizadeh et al. [15] studied the use of adaptive neuro-fuzzy inference systems (ANFIS) and meta-heuristic algorithms for prediction. The proposed model takes into account various factors, such as effective concrete depth, compressive strength, beam length, and reinforcement ratio. The results showed that the ANFIS model and meta-heuristic algorithms are more accurate than other methods. Another research project proposed a combination model, the SVR-GA, for predicting the shear resistance of RC deep beams [16]. The beams are analyzed based on measurements, mechanical properties, and material characteristics. In the testing phase, the SVR-GA model showed higher coefficients of determination (R2) than the traditional SVR, ANN, and gradient-boosting decision trees (GBDTs). In addition, Kaveh and his colleagues [17] created ensemble machine-learning algorithms based on gradient-based decision trees to predict the ultimate shear capacities of FRP beams without stirrups. The proposed XGBoost model is more precise and general than experimental equations and other machine learning models. Almasabha et al. predicted the shear strength of SFRC deep beams using ML models such as Light Gradient-Boosting Machine (LightGBM), XGBoost, and Gene Expression Programming (GEP). LightGBM and XGBoost achieved high accuracy levels with R2 values of 97.25 and 94.66 percent, respectively. Further research is needed to verify that an ML model is a reliable replacement for a mechanism-based model. It is necessary to conduct more research to understand model interpretation, especially in relation to the shear mechanism. Thus, in order to improve the application of ML technologies, it is necessary to understand a clear methodology coupled with the shear behaviour of RC deep beams, focusing on quantitative evaluation. The aim of the study is to fill the research gap by predicting the shear strength of RC deep beams using data- and mechanism-based machine learning models. The resulting model is expected to provide more accurate forecasts and become a useful tool for comparable problems. This paper consists of: 1) explaining the research rationale and methods used, including the investigation of ML models and suggested interpretation techniques; 2) selecting input characteristics and summarising the experimental data; 3) assessing the results of the created ML models and identifying the most effective model; and 4) analyzing the SHAP analyses and explaining the decision-making processes of the selected models and proposed interpretation methods.
2. Research Thought And Methodology
2.1. Research Thought
This study focuses on the construction of synergistic data- and mechanism-driven approaches in order to develop a predictive model for estimating the shear strength of RC deep beams. As shown in Figure 1, the first step is to create an experimental database, which includes selecting features and carefully compiling data. Selected input properties are closely related to the shear mechanism of RC deep beams. After the database was established, the second phase consisted of training and evaluating six different ML models. The aim is to identify models that show the best prediction accuracy. Subsequently, the performance of the leading model was compared with five mechanism-based models, including Matamoros and Wong [19], NZS 3101-2006, EC2-1-2004, and the Russo et al. equation [20], to verify its prediction accuracy, generalization capability, and the impact of key input features. The following task is to ensure that the decisions and expected values are consistent with the shear mechanism. This was achieved by analyzing the meaning and dependence of SHAP’s qualitative discussion results (Shapley Additive ExPlanations). By reviewing these aspects, the study intends to propose models with higher performance.
2.2. Machine Learning (ML) Models
Machine learning (ML) is a sophisticated data analysis technique that automates the creation of analytical models. As part of artificial intelligence (AI), these systems are based on a principle that allows them to gain knowledge from data, recognize patterns, and make informed decisions with minimal human intervention. With the increase in data volume, the importance of intelligent data analysis is expected to increase, and it will play an important role in technological development and innovation [21]. The following six ML models were chosen to predict the shear strength of RC deep beams.
2.2.1. Gradient Boosting Decision Tree (GBDT)
Gradient Boosting Decision Tree algorithms (GBDTs) are widely recognized as high-performance models that produce accurate predictions by using ensemble decision trees and gradient-boosting techniques [22]. According to Larestani et al., GBDT is similar to the functional gradient-descent approach, and new learners address residual errors from previous learners at each step and minimize certain loss functions. The GBDT ensemble structure enables the integration of several weak prediction models and often uses simple decision trees or branches. The ranked tree structure of GBDT offers additional interpretability advantages. It provides insights into the decision-making process of the model. It emphasizes the importance of different characteristics [22]. This technology uses a step-by-step format to efficiently build regression solutions while minimizing overfitting problems. In addition, GBDT optimization simplifies learning processes and transforms optimization problems into common optimization problems [23]. The iterative algorithm starts with a single leaf, improves the learning rate of each node and record, and updates the function iteratively to obtain an accurate prediction. GBDT algorithms combine interpretation, staging, and optimization and are powerful, intuitive tools for generating accurate results for various applications.
2.2.2. eXtreme Gradient Boosting (XGBoost)
The introduction of eXtreme Gradient Boosting (XGBoost) by [24] in 2016 has since been recognized as a leading predictor of current performance, both in regression and classification tasks.
The algorithm is based on traditional gradient-boosting algorithms and includes various improvements and regularization techniques to prevent overfitting. XGBoost’s loss function combines the loss function (Ψ*) to measure the difference between the predicted label and the actual label and Ω(f) to penalize the number of leaves and output of each leaf node to discourage complex models.
The XGBoost loss function merges a loss function Ψ(*) to gauge the variance between predicted and actual labels with a regularization function Ω(f) aimed at discouraging complex models. The complexity control of each leaf, defined in Eq. 1 with T representing the leaf count, 𝜆 as the penalty parameter, and ∥𝜔∥² for the leaf node output, is crucial. Unlike traditional gradient boosting, XGBoost utilizes the Taylor series in a second-order approach to convert objective functions into quadratic minimization problems.
Equation 2 shows a transformed objective function, where first- and second-order gradient statistics are represented by gi and hi in the loss function for the i-th sample, respectively.
With the help of Ij, the objective function is simplified to represent the number of leaves.
In order to manage overfitting, XGBoost uses various techniques such as learning-rate adjustment, iteration control, limitation of maximum tree depth, and subsampling of training samples [25].
2.2.3. K-Nearest Neighbors (KNN)
The K-nearest neighbours (KNN) algorithm identifies the K training samples that are most similar to the target. The value of K signifies the number of nearest neighbours and is chosen based on their proximity to classify input features. By evaluating the closest neighbours, KNN groups similar training samples together. This algorithm relies on distance calculations and voting mechanisms to find the optimal K value for accurate classification [26].
2.2.4. Bagging
Bagging is a popular ensemble method [27] that involves generating multiple diverse training sets through bootstrap sampling techniques. Each of these training sets then produces predictions from base learners, which are aggregated to form the ultimate model. The key elements of bagging include bootstrap sampling and model aggregation. Bootstrap sampling entails randomly selecting n samples from the initial dataset of n samples, with replacement, to maintain independence among various training sets [25].
2.2.5. Random Forest Regression (RFR)
Leo Breiman introduced the random forest regression algorithm (RFR), a machine learning technique proficient in predicting extensive datasets. The RFR algorithm comprises four main processes: bootstrap resampling, random feature selection, out-of-bag error estimation, and the development of full-fledged decision trees. It generates multiple decision trees through bootstrap resampling, treating them as weak learners. Randomly chosen subsets of data are used to train each tree, while the remaining samples serve as out-of-bag samples for error estimation during training. Unlike pruned decision trees, random forest trees grow without pruning, leveraging new training samples to create robust prediction models [26]. Renowned for its versatility in handling large datasets and delivering accurate predictions, the RFR model finds widespread use across various machine learning applications.
2.2.6. Stacking
Stacking Ensemble Learning is an approach introduced by Wolpert [28] that enables a model to fully utilize the interconnected strengths of different base models. The process involves obtaining predictions from various base models and then intelligently combining these predictions with meta-learners to reach a final prediction. In order to avoid overfitting, meta-learners do not learn directly from the output of the base model. Instead, leave-one-out cross-validation is used in this approach to ensemble learning. The validation folds are placed together to create a new set of data from which the meta-learner can learn.
The integration of the base models plays an important role in stacking. The selection of meta-models depends on specific problems, but multiple linear regression ML models are often used as meta-models. The stacking model offers powerful techniques that leverage the different expertise of the base models and combine their predictions skillfully, resulting in improved prediction performance compared to individual models.
2.3. Model Building
Python’s scikit-learn library is utilized to handle the data, which are split into training, testing, and validation sets with an 80-10-10 percentage split. Standard scalers are applied to scale the data, followed by a grid search to optimize hyperparameters. The training data are used to train the models, while testing and validation are performed on unseen data.
In models like GBDT, Bagging, and XGBoost, decision trees serve as the base estimators. In contrast, the stacking model incorporates base models such as KNN, GBDT, and XGBoost, adjusting their hyperparameters via grid search. Once the best models are identified, they are adapted to the scaled training data. The base-model predictions are combined to form a new dataset, which becomes the input for the meta-model—a linear regression model. Hyperparameter tuning is then carried out on the meta-model using grid search to optimize its performance. After obtaining the best meta-model, it is applied to the combined dataset of base-model predictions and actual target values, resulting in the final stacked model.
2.4. Evaluation Metrics
Assessing the accuracy and efficacy of machine learning models relies significantly on evaluation metrics, which offer essential insights into how well the models predict target variables. These metrics serve as valuable tools for comparing models and making informed decisions about their performance. In this context, five evaluation metrics were employed to gauge the effectiveness of the ML models:
2.4.1. Coefficient of determination (R2)
The coefficient of determination assesses how much of the variability in dependent variables can be explained by independent variables within a model, as depicted in Eq. 4. Ranging from 0 to 1, a value of 1 signifies that the model accurately predicts the target, while 0 suggests that the model does not enhance predictions beyond the average.
2.4.2. Root mean square error (RMSE)
Root Mean Square Error (RMSE) measures the average magnitude of the discrepancy between a dataset’s predicted and actual values and is widely used in regression models in statistics and ML. Equation 5 shows the formula for RMSE.
2.4.3. Mean absolute error (MAE)
The mean absolute error (MAE), illustrated in Eq. 6, measures the average absolute difference between actual and predicted values. Unlike the mean squared error (MSE), it does not square the errors, making it less sensitive to outliers.
2.4.4. Median absolute error (MdAE)
It is the median value of the absolute difference between the actual target value and the predicted value. As shown in Eq. 7, it resists deviations and provides a central measurement of the tendency of errors.
2.4.5. Mean squared logarithmic error (MSLE)
The mean squared logarithmic error (MSLE), presented in Eq. 8, is a modified version of the mean squared error that involves taking the logarithm of both predicted and actual values before computing the squared differences. This metric is often used when the target values cover a wide range and evaluating the model’s performance on a logarithmic scale is more appropriate. In the equation, yi represents the actual value, yi’ stands for the predicted value, μ denotes the mean value, and n signifies the number of samples.
3. Data Collection And Preprocessing
3.1. Experimental Data Collection
Figure 2 illustrates a schematic depiction of the shear mechanism in reinforced concrete (RC) deep beams. Extensive experimental and theoretical investigations [9], [10], [30], [31], [32] have revealed that shear strength is influenced by various components, primarily concrete quality and web reinforcement. Key factors affecting shear strength include the compressive strength of concrete (fc’), the shear-span ratio of the beam (a/d), the strength and proportion of longitudinal reinforcement (fy and ρl), and the ratio of vertical and horizontal reinforcement in the web (ρv and ρh). Additionally, the characteristics of both vertical and horizontal web reinforcement significantly impact the web-reinforcement section. This study has identified critical parameters related to shear strength, encompassing concrete compressive strength, beam shear-span ratio, longitudinal reinforcement properties, and web-reinforcement ratios.
The dataset consists of 587 data points sourced from previous research [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54]. Table 1 provides a summary of the crucial parameters related to shear strength used in this investigation, including their respective units of measurement and statistical details. These key parameters serve as input features for the model, with shear strength (νexp) representing the corresponding output. Furthermore, Figure 3 presents a histogram of all six input characteristics within the dataset.
Table 2 presents the statistical distribution of both input and output parameters, offering insights into the central tendencies, variability, and ranges within the dataset. The average value of the shear-span ratio (a/d) is approximately 1.25, with a minimum of 0.23 and a maximum of 2.50. The standard deviation of 0.50 suggests a moderate level of variability around the mean. For the compressive strength of concrete (fc’), the average value is around 31.58 MPa, with a minimum of 11.7 MPa and a maximum of 89.4 MPa. The standard deviation of 15.29 indicates variation in the compressive-strength values. The mean yield strength (fy) in the dataset is approximately 428.2 MPa, ranging from 100.00 MPa to 1000.00 MPa, with a standard deviation of 102.74 indicating a significant range of yield-strength values. Similarly, the statistical distribution of other parameters such as longitudinal reinforcement ratio (ρl), vertical web reinforcement ratio (ρv), horizontal web reinforcement ratio (ρh), and shear strength (νexp) reflects the range of each parameter.
3.2. Data Preprocessing
3.2.1. Data normalization
As the data of multiple attributes show different ranges, they need to be standardized [55] to allow meaningful comparison and analysis. Thus, the datasets are normalized using the z-score method. Each data point is converted to the corresponding z-score that represents the standard deviation from the average. This transformation ensures that the standardized data have a mean of 0 and a standard deviation of 1 regardless of the original scale of the attributes. The standard scaler equation in scikit-learn is as follows:
X is the original value of the data point, μ is the average value of the dataset, σ is the standard deviation of the dataset, and the resulting value is the standardized value (z-score) of the data point.
3.2.2. Hyperparameters
Hyperparameters are fundamental elements in the performance of machine learning algorithms, especially in supervised learning. According to Alhakeem et al. [56], correct selection of hyperparameter values is crucial because it has a major impact on the performance of the model. When the appropriate values are selected for these hyperparameters, the performance of the model can improve significantly. Similarly, Angoros and Mukti [57] point out that the optimal performance of algorithms is highly dependent on hyperparameters in supervised learning.
Grid search is a complete search technique that analyzes subsets of the hyperparameters, specifying lower and upper limits and determining the number of steps between them. This method systematically evaluates all possible combinations in a defined grid to find the optimal hyperparameter values. By thoroughly analysing each grid, grid search identifies the most effective configuration. This method is beneficial for data analysis with high accuracy [57]. Another study [58], in which different hyperparameter search methods were compared, consistently showed that grid search had higher performance than other techniques for their respective datasets. This observation highlights the effectiveness of grid search in identifying optimal hyperparameter configurations that improve model accuracy and generalization. Because of the advantages of grid search, it was used to identify the best hyperparameter configuration for each ML model. Table 3 shows the best hyperparameters obtained in the grid search of each ML model. By optimizing these hyperparameters through grid search, the models were optimized to improve performance and accuracy.
3.3. Shap Feature Selection
Feature selection is essential in ML to reduce dimensionality and prevent overfitting of high-dimensional data. Minimizing the number of characteristics improves computational efficiency and minimizes the impact of irrelevant characteristics [59]. The positive results include improved predictive accuracy and more interpretable learning results due to the reduction in dimensionality [60]. In this study, the SHAP methodology was used to analyze both feature selection and feature dependencies. Lundberg and Lee’s SHAP methodology [61], based on game theory and the Shapley value, is easy to understand. The traditional method usually relies on the comparison of target variables and feature sets to calculate correlation coefficients such as the Pearson correlation coefficient, but SHAP uses a distinct method to assess the importance of features. It defines the significance of various feature classes by averaging the absolute magnitude of the importance of each characteristic in each sample. This alternative methodology offers a comprehensive view of the importance of features and highlights the collective impact of features on the predictions for each data point. Importantly, it transcends the limitations of traditional correlation-based techniques and encapsulates the interaction and dependency of complex features [62].
4. Results And Discussion
4.1. Performance Evaluation Of ML Models
Table 4 compares the performance of six prominent ML models, namely RFR, GBDT, KNN, Bagging, XGBoost, and Stacking. Comparisons were made using evaluation metrics such as R2, RMSE, MAE, MdAE, and MSLE. In a comprehensive assessment of six ML models, Stacking was the most robust choice, especially considering key performance metrics. RFR and Bagging demonstrated excellent performance with high R2 values in all datasets, showing a strong overall fit. However, both models show a slight increase in error measurements, such as RMSE and MAE, in the validation set, suggesting potential challenges for generalization. Although GBDT and XGBoost performed exceptionally well in training sets with high R2 values, errors in test sets increased considerably. This involves the risk of overfitting, in which the model may be too personalized to training data, limiting its ability to generalize to new unseen data. KNN was successful in testing, especially with low values of RMSE, MAE, and MdAE. However, the slight increase in the error level in the validation set indicates limitations in its generalization capabilities. In contrast, Stacking consistently shows high R2 values in all datasets, reflecting its superior ability to capture data variation. In the test group, Stacking achieved a low RMSE of 9.248, an MAE of 7.006, an MdAE of 6.339, and an MSLE of 0.003. These values highlight the accuracy and reliability of Stacking predictions. The slight increase in error metrics in the validation set is a compromise, but the overall performance and generalization of Stacking outweigh this concern.
The strength of Stacking lies in its ensemble approach, which effectively combines the strengths of different base models. This is evident in the consistently low MSLE in all datasets, showing its ability to handle errors and uncertainties. Stacking is adaptable to different aspects of data and is preferred for this particular dataset and predictive task. Furthermore, visual representations of the comparison of predicted (pred) and experimental (exp) values are shown in Figure 4. In this illustration, the diagonal line (y = x) represents the case in which the predicted values coincide precisely with the experimental values. In particular, the upper part of the diagonal shows cases where the predicted value exceeds the experimental value, while the lower part shows cases where the predicted value is below the experimental value. The fitting lines of the ensemble models encompassing GBDT, KNN, and Stacking show significant improvements over RFR, Bagging, and XGBoost. Specifically, in Figure 4(a), the visual representation of the RFR model shows relatively poor performance compared to other models. In contrast, Figure 4(f) shows that the Stacking model is superior to the rest, showing the highest alignment between predicted and experimental values. Visual analysis of fitting lines emphasizes the effectiveness of the ensemble models, with Stacking distinguished as the most qualified in accurately predicting experimental values.
Finally, Stacking has proved to be the best model, offering a harmonious mixture of accuracy, resilience against outliers, and generalization capabilities. Although further improvements through normalization techniques and hyperparameter adjustments can improve its performance, Stacking’s overall strengths make it the best choice for prediction with this dataset.
4.2. Comparison With Mechanism Models
This section compares the performance of the Stacking model with five established mechanism models incorporating design codes such as ACI 318-18 [1], NZS 3101-2006 [2], EC2-1-2004 [3], and the equations proposed by Matamoros and Wong [19] and Russo et al. [20]. The simplified expressions of the five mechanism models are described in detail in Appendix A, which analyzes prediction accuracy and the effect of these models on key features by using the ratio of predicted value to experimental value (=pred/exp) as the evaluation metric. A value below 1.0000 indicates a conservative prediction, and a value greater than 1.0000 indicates an over-prediction. If the value is close to 1.0000, it means a small prediction error, which indicates good prediction accuracy.
Table 5 provides statistical information about the χ value and offers insight into the prediction capability of the Stacking model and the five mechanism models. The average χ value obtained by the Stacking model is 1.0002, close to 1.000. In addition, the standard deviation value of Stacking is 0.025, indicating the stability and robustness of predictive precision. Figure 5 also gives statistical details of the χ value across the entire database, showing the dispersion of χ from the Stacking model and the five mechanism models. EC2-1-2004 and the equations proposed by Russo et al., Matamoros, and Wong showed similar results characterized by right-skewed χ-value distributions and a tendency towards conservative predictions (Figure 5c-e). For example, in ACI 318-18, the average, standard deviation, and coefficient of variation of χ values were 1.475, 0.44, and 30.43%, respectively, suggesting greater prediction accuracy but greater variability (Figure 5a). NZS 3101-2006 shows results almost identical to those of ACI 318-18, as shown in Figure 5(b). In contrast, as shown in Figure 5(f), χ values of the Stacking model cluster around 1.000, with mean, standard deviation, and coefficients of variation of 1.0002, 0.025, and 2.52%, respectively, indicating superior results among these models. These comparisons provide convincing evidence of the superiority of the Stacking model in terms of accuracy and stability of prediction.
In addition, Figures 6-9 show the effect of key features such as a/d, fc’, fy, and ρl on the prediction efficiency of the entire database. Figure 6 shows the upward trend in χ values with the increase of a/d for ACI 318-18, NZS 3101-2006, EC2-1-2004, and the equations proposed by Russo et al., Matamoros, and Wong, indicating the influence of a/d on prediction accuracy. When a/d is less than 1.00, these models usually show a conservative prediction tendency, whereas they tend to display overprediction as the a/d ratio varies from 1.00 to 2.50. However, the Stacking model does not show an obvious trend in its χ distribution (Figure 6f), reflecting a consistent correlation between shear strength and a/d. Similarly, a trend of increasing χ values with fc’ was observed in all five mechanism models, indicating that fc’ influenced the prediction accuracy of the mechanism models. On the other hand, changes in values did not affect the prediction accuracy of the Stacking model, as shown in Figure 7(f). Figures 8 and 9 examine the influence of fy and ρl on the prediction accuracy of RC deep beams. It is clear that, in the five mechanism models, the distribution of χ shows rising trends when fy and ρl increase. On the other hand, the prediction accuracy of the Stacking model is almost unchanged by fy and ρl, as shown by the consistent distribution of χ around 1.00 in Figures 8 and 9.
Overall, these evaluation metrics show a significant improvement in the performance of the Stacking model, characterized by fast calculations, higher prediction accuracy, and strong generalization capability. The Stacking model demonstrates lower sensitivity to important variables and correctly captures the relationship between shear strength and these key factors. However, these conclusions alone are not sufficient to establish the Stacking model as a replacement for a mechanism-based model.
5. Feature Importance And Dependency Analysis
5.1. Feature Importance Analysis
Feature importance is a method used to evaluate the importance of input features in predicting target variables. Assigning scores to input features offers useful insights into how they contribute to the prediction model [63]. This analysis helps to identify the characteristics most relevant to making accurate predictions of target variables and also assists with feature selection and improvement of the model’s interpretability. Various methods can be used to achieve interpretation. One of them is the SHAP model approach. SHAP stands for SHapley Additive exPlanations. To understand how the main model handles a case, it creates a substitute model in its local neighbourhood. It isolates a single instance [64]. Insight into the model’s decision-making process in specific cases is obtained through this local approach, thereby improving the model’s overall interpretability.
Figure 10 shows the relative importance of input characteristics for RC deep beams. The input parameter a/d is the most important, and its relative importance is 30.83%. This emphasizes the importance of geometric configuration in determining shear strength, showing that changes in the a/d ratio can have a significant impact on model predictions. The next in terms of importance is fc’. This emphasizes the crucial role that concrete material’s inherent strength plays in resisting the shear forces on the beam. The results emphasize the need to carefully consider the concrete mixture and quality control to achieve the required shear-strength results in RC deep beams. fy has the third-highest relative importance, highlighting the importance of the material properties of longitudinal reinforcement in influencing shear-strength prediction. The strength of the reinforcement material contributes significantly to the overall structural integrity and provides resistance to shear loads. The longitudinal reinforcement ratio ρl is another influential factor. This parameter highlights the importance of appropriate distribution and quantity of longitudinal reinforcement in relation to the dimensions of the beam. Achieving an optimal balance is crucial for improving shear strength and overall structural performance.
When it comes to the ratios of web reinforcement, ρh and ρv show lower relative importance values. Although these ratios contribute to the predictions of the model, their influence is relatively small. This indicates that horizontal and vertical distributions of web reinforcement have a modest effect on the results of shear-strength prediction.
In summary, a comparative analysis of the importance of SHAP features highlights the complex nature of variables influencing the shear strength of RC deep beams. Although geometric considerations, longitudinal reinforcement properties, and concrete strength play important roles, the contribution of web-reinforcement ratios is relatively lower. This precise understanding can guide engineers in prioritizing design considerations and optimizations to achieve the desired shear-strength performance in RC deep-beam structures.
5.2. Feature Dependency Analysis
Feature-dependence analysis is a technique used to determine the correlation among input features and the influence they exhibit on predicting the shear strength of RC deep beams. By reviewing how variations in these characteristics affect the prediction of the model, as measured using SHAP values, it can be determined which characteristics are critical to accurate prediction of shear strength.
As mentioned in Section 3.3, the feature-dependency diagram shows the relationship between the SHAP value on the Y-axis and the feature value on the X-axis. The objective of this research is to study the practicality of the decision-making basis of the Stacking model by examining the selected key features, namely fc’, a/d, ρl, and fy, determined from the previous assessment of feature importance.
6. Conclusions
The study aimed to develop predictive models to estimate the ultimate shear strength of RC deep beams using harmonious data- and mechanism-based methods. The research methodology included creating an experimental database containing 587 RC deep beams, training and evaluating six ML models, and validating the selected model against mechanism-based models and design codes. The key conclusions of the research are summarized as follows:
Among the six machine learning models (XGBoost, Bagging, GBDT, KNN, RFR, and Stacking), the Stacking model emerged as the most vigorous and accurate estimator for shear strength in RC deep beams. It showcased superior performance in terms of R2, MAE, MdAE, and MSLE, demonstrating its capability to handle intricate connections and generalise well to new data.
Five mechanism-based models, based on equations and codes, were compared with the Stacking model. These mechanism models were consistently outperformed by the Stacking model in prediction, which showed higher stability, accuracy, and lower sensitivity to changes in vital input variables such as compressive strength of concrete (fc‘), beam shear-span ratio (a/d), longitudinal reinforcement ratio (ρl), and longitudinal reinforcement strength (fy).
It was deduced via feature-importance analysis that the most influential parameters were geometric variables (a/d), longitudinal reinforcement properties (fy, ρl), and concrete strength (fc‘). ρh and ρv had less influence compared with other parameters. These findings were further supported by feature-dependency analysis, showcasing the influence of each parameter on the prediction of shear strength.
A balanced combination of precision, robustness against outliers, and ability to generalize was exhibited by the Stacking model. It has been shown in this study to be the optimal choice for predicting the shear strength of RC deep beams. However, further improvements and hyperparameter tuning can improve its performance. This model provides engineers with an invaluable tool to optimize designs and predict shear strength precisely.
In conclusion, the Stacking ensemble model incorporating machine learning techniques demonstrated impressive performance in predicting the shear strength of RC deep beams. The findings support integrating ML modelling as an additional tool in structural engineering practice, thereby improving accuracy and understanding for informed decision-making in the design and evaluation of RC deep-beam structures.
Academic Master Education Team is a group of academic editors and subject specialists responsible for producing structured, research-backed essays across multiple disciplines. Each article is developed following Academic Master’s Editorial Policy and supported by credible academic references. The team ensures clarity, citation accuracy, and adherence to ethical academic writing standards
Content reviewed under Academic Master Editorial Policy.
