Estimating flexural strength of precast deck joints using Monte Carlo Model Averaging of non-fine-tuned machine learning models

Abstract

The bending capacity of the precast decks is greatly dependent on the flexural strength exhibited by the joints between them. However, due to the complexity and diversity of this system, precise predictive models are currently unavailable. This study introduces an effective and precise methodology for assessing flexural strength using Monte Carlo Model Averaging (MCMA), a statistical technique that combines the strengths of model averaging (MA) and Monte Carlo simulation. To construct the MCMA model, input variables were derived by analyzing the experimental results, and a database of 433 bending test specimens was compiled. The MCMA model incorporated four different machine learning models, namely decision tree (DT), linear regression (LR), adaptive boosting (AdaBoost), and multilayer perceptron (MLP). Comparative analyses revealed that the MCMA model outperformed baseline models (DT, AdaBoost, LR, and MLP) across all employed metrics. The impact of three different categories on flexural capacity was explored through boxplot analysis. Furthermore, a comparison between the MCMA model and the strut and tie model highlighted the superior performance of the MCMA model. The impact of input variables on the flexural strength prediction was further examined through Shapley Additive exPlanations based feature importance and global interpretation, as well as parametric study.

Graphical abstract

Keywords

precast deck joint / flexural strength / machine learning / model averaging / Monte Carlo method / parameter tuning

1 Introduction

In pursuit of accelerated bridge construction aimed at reducing on-site construction time and minimizing traffic delays, engineers commonly opt for prefabricated systems incorporating interlaced rebars and cast-in-place concrete [1,2]. The precast deck system, characterized by excellent quality, durability, eco-friendly features, and construction efficiency, offers numerous advantages [3–5]. Bending represents a prevalent mechanical action in the precast deck system. However, the adoption of this technique raises several concerns, with the primary focus on estimating accurately and efficiently the flexural strength of precast deck joints.

For several decades, experiments, as depicted in Fig.1, have been the primary method used to assess the flexural strength of precast deck joints due to their intricate and diverse nature. Numerous experimental studies related to precast deck specimens have been carried out, revealing that the flexural strength is influenced by various design parameters. These design parameters include the reinforcing bar scheme, such as the reinforcing bar type at connection zone [6], the number of transverse reinforcing bars [7], and the lap-splice length of longitudinal reinforcing bars [8], all of which play a role in resisting applied loads. Additionally, the properties of cast-in-place concrete, including its type and characteristics, are crucial for ensuring a continuous load path and safeguarding the embedded bars [9]. The type of interface used also impacts force transmission between two concrete sections [10].

Efforts have been undertaken to propose solutions for evaluating flexural strength, contending with the complexities presented by numerous variables in the traditional approach. Typically, the sectional analysis (SA) method has been extensively utilized for assessing flexural strength [4,11]. In this context, deck specimens with joint connections are considered as continuous elements, and their flexural capacity is calculated using the SA method. Despite the ease of application, the appropriateness of SA in the joint region is a matter of discussion due to geometric irregularities. To overcome this challenge, various researchers have investigated the viability of the strut and tie model (STM) [1,12–14]. The application of this model extends to precast deck joints with U-bars [1,12], joints utilizing high-strength concrete (HSC) [13], and joints incorporating ultra-high-performance concrete (UHPC) [14]. Li and Jiang [15] and Vella et al. [16] have adapted He et al. [1]’s model for precast deck joints with headed rebars, with notable distinctions primarily concentrated on computing the concrete strut parameter.

Overall, there is an increasing demand for a precise and dependable model to evaluate the flexural strength of precast deck joints, as there is currently no widely accepted and standardized approach. Machine learning (ML) methods present a promising avenue for capturing intricate relationships between crucial parameters and desired structural characteristics [17–25], offering a more cost-effective alternative to laborious laboratory tests and less efficient current models. ML techniques have demonstrated widespread applicability in diverse domains, such as structural health monitoring [25,26], detection of structural damage [27,28], performance assessment [17–21], identification of structural parameters [29,30], computational mechanics using an energy-based approach [31,32], and elastic behavior of materials [33]. ML techniques can reveal the relationships among inputs and outputs variables directly, eliminating the need for complex mechanical derivations, typically delivering higher accuracy with lower variability [22,24]. There are two primary categories of algorithms in use: individual methods like linear regression (LR) [34], multilayer perceptron (MLP) [35], support vector machines [19], and decision trees (DT) [36]; and ensemble methods like bagging [36] and boosting [18]. Ensemble learning generally outperforms individual methods by combining multiple weak learners generated through individual techniques into a robust one. However, both individual and ensemble techniques share a common challenge, which is the need for parameter tuning to identify optimal hyperparameters. Typically, searching methods (grid or random) [18,35] are employed for this purpose, but they are time-consuming and demand previous knowledge or expertise regarding possible hyperparameter distributions. Another promising statistical approach is model averaging (MA) [37], which combines multiple predictive models with assigned weights to establish a more precise and resilient model. The MA method has found applications in the areas of economics [38], ecology [39], and astrophysics [40], but has seen limited use in structural engineering. The main challenge in applying MA lies in determining suitable weights for each individual model [37].

Among various MA methods, Monte Carlo Model Averaging (MCMA) stands out as it leverages the strong performance of ML models while avoiding the intricate hyperparameter tuning process [41]. MCMA is a statistical and computational technique used in ML and statistics to enhance the robustness and accuracy predictions by merging predictions from various models with corresponding weights. MCMA is particularly useful when dealing with uncertainties in model selection and parameter tuning. Therefore, this research employs MCMA to construct a novel ensemble model for predicting flexural strength of precast deck joints. The key contribution of this work is that the proposed MCMA model can achieve excellent predictive performance without the need for the time-consuming optimization of ML model hyperparameters. The paper’s structure is as follows. Section 2 briefly describes the ML algorithms used, including DT, MLP, LR, Adaptive Boosting (AdaBoost), and then presents the MCMA-based flexural strength model proposed. Section 3 describes the failure modes, variables affecting flexural strength of precast deck joints and collected database. Section 4 assesses and contrasts the performances of the proposed MCMA model, the individual ML models, and STM. Moreover, in section 4, the impact of categorical variables on flexural strength of precast deck joints as well as global interpretation of flexural strength prediction using SHAP method were also investigated. Section 5 presents a parametric study based on MCMA model. Finally, Section 6 draws several conclusions.

2 Development of Monte Carlo Model Averaging Methodology

2.1 Single machine learning algorithms

ML is a powerful tool for uncovering intricate patterns within complex databases. ML-based models outperform traditional statistical methods, like nonlinear regression, in providing more accurate predictions. Furthermore, traditional statistical methods come with additional assumptions and requirements, whereas ML-based models are entirely reliant on data, free from such assumptions.

There are two approaches for solving regression problems: single and ensemble algorithms. However, creating a robust ML-based model through single algorithms is often challenging due to limited data or algorithm limitations. Ensemble algorithms address this by amalgamating multiple single algorithms as weak learners into a more robust ensemble learner, resulting in improved execution. In this study, a total of four methods, consisting of three individual and one ensemble method, were utilized, as outlined below.

1) DT functions as a standalone learning method, characterized by a tree-like structure featuring various types of nodes, including internal nodes and leaf nodes, all interconnected through edges. The internal nodes within this structure serve as decision points, making assessments based on specific features, while leaf nodes convey the ultimate classification or regression outcome. DT is versatile in their utility, applicable for both regression and classification tasks. However, it’s important to note that DTs are susceptible to common issues, particularly the risk of overfitting and challenges in effectively managing missing data.

2) MLP is a flexible supervised ML algorithm suitable for both regression and classification tasks. It learns a mapping function through training on a set of input–output pairs. MLP, a type of feedforward artificial neural network, drawn inspiration from the structure and function of biological neurons. An MLP model comprises a minimum of three layers of perceptrons, encompassing one input layer, at least one hidden layers, and one output layer.

3) LR depicts the association between two variables by applying a linear equivalence to test data. In this scenario, one variable acts as the explanatory variable, while another one serves as the dependent variable. For instance, LR model can establish a relationship between individuals’ weights and heights using an equation like y = a + bx, where x denotes the explanatory variable, y denotes the dependent variable, a denotes the intercept, and b denotes the slope.

4) AdaBoost. It is constructed to combine multiple weaker learners into a more powerful ensemble learner. Typically, a single-layer DT, representing the most basic form of a DT, serves as the weaker learner within AdaBoost. AdaBoost demonstrates exceptional predictive accuracy and effectively mitigates overfitting. However, it does exhibit sensitivity to outlier or abnormal samples, which can carry substantial weight and, in turn, influence the accuracy of the ultimate robust learner’s predictions. The term “adaptive” in AdaBoost refers to the dynamic adjustment of weights assigned to those weaker learners that do not accurately predict the data. Conversely, updated samples are utilized to train subsequent weaker learners. In each iteration, a novel weaker learner is incorporated until reaching a predefined error threshold or the maximum iteration count.

2.2 Development of MCMA model

MA is acknowledged as an assessment method that deals with uncertainty in the model and traces its origins back to the 1960s [37]. Over the past several decades, MA has found applications in various fields. It involves combining multiple predictive models with assigned weights to enhance performance and diminish the model uncertainty associated with each individual model.

This research aims to construct a model to predict flexural strength of precast deck joints using MCMA involving four individual models (DT, AdaBoost, LR, and MLP). In this context, for a given flexural strength database D = {x,y} having N samples, where x = [x₁,x₂,…,x_N] denotes as the input and y = [y₁,y₂,…,y_N] denotes as the output, the development of MCMA framework using weighted model averaging with four different ML models of the flexural strength can be expressed as following steps.

1) MC sampling. Generate k MC samples from the original data set D, which is typically used bootstrapping method, to introduce data variability and simulate different realizations of the data set. As a result, each sample D_k, is a new data set with the same size as the original data set D.

2) Re-fitting models on MC samples. For each MC sample D_k​, re-fit each of the four models using this resampled data: f_i(D_k, {\theta }_i^{(k)}) representing the ith model trained on the kth MC sample, and {\theta }_i^{(k)} are the model parameters for that sample.

3) Generate predictions for each MC sample. After re-fitting the models, make predictions for each data point j in each MC sample k, as follows:

(1)y_{i,j}^{(k)}=f_i\left(D_k,{\theta }_i^{(k)}\right).

4) Calculate weighted averaged predictions for each sample. For each MC sample k, calculate the weighted average of the predictions for each data point j, as follows:

(2){\overline{y}}_j^{(k)}=w_i{y}_{i,j}^{(k)},

where w_i is the model weight of ith model. In this research, model weights were calculated based on root mean square error (RMSE) values. Normally, the higher weights were assigned to models that have lower RMSE values, indicating better performance on the training or validation data.

5) Aggregating weighted averaged predictions. To obtain the final ensemble prediction, aggregate the weighted averaged predictions from all MC samples. One common method is to calculate the mean across all samples, producing the ensemble prediction:

(3){\hat{y}}_j=\frac{1}{K}\sum _{k=1}^K{\overline{y}}_j^{(k)}.

3 Experimental database

3.1 Failure mechanisms of deck joints

Fig.2 provides a summary of three primary failure mechanisms that can manifest in deck joints. These mechanisms encompass the yielding of longitudinal reinforcing bar, the crushing of concrete near the joint interface, and the failure of lap-spliced longitudinal reinforcing bars. The extent to which these failure modes occur is contingent upon the geometric and material characteristics of the various components within the joint.

In situations where robust concrete encases longitudinal reinforcing bar within a wide joint, the early failure of the concrete at the joint can be prevented. Consequently, the longitudinal reinforcing bar’s yield becomes prominent and dictates the flexural strength of the precast deck joints (Fig.2(a)). Conversely, in deck joints where the concrete quality is subpar within the precast or joint area, the characteristic failure mode involves the compression zone concrete being crushed near the joint and precast concrete intersection. Such failure behavior demonstrates a relatively brittle nature (Fig.2(b)).

Overlaped longitudinal reinforcement can be failed due to various factors, including splitting failure of the joint concrete and inadequate anchoring or deformation of transverse reinforcing bars. This often coincides with the concrete cover’s spalling along the longitudinal reinforcing bars, as illustrated in Fig.2(c).

3.2 Variables affecting flexural strength of precast deck joints

The identification of significant variables with a causal impact on flexural capacity has been derived from the inherent failure mechanisms as well as the existing models discussed in the Section 1. In broad terms, these influential variables can be classified into four main groups.

1) Material variables. Among these variables, the strength of the joint concrete, denoted as f_{\mathrm{c}\mathrm{j}}^{\prime }, plays a pivotal role in directly influencing deck joint strength. Recent research has revealed a wide variation in the precast deck joints’ flexural strength. From the literature review, the precast deck joints were normally constructed with different types of concrete, including concrete grouting material [42], HSC [43], concrete with steel fiber reinforcement (SFRC) [44], and UHPC [45]. Typically, f_{\mathrm{c}\mathrm{j}}^{\prime } is engineered to surpass the precast concrete strength to assure optimal performance of the connection zone. This is essential as the precast concrete strength, denoted as f_{\mathrm{c}\mathrm{p}}^{\prime }, may govern the compression resistance near the interface. 2) Geometric variables. Several key geometric variables, as shown in Fig.3, significantly impact flexural capacity. These include the deck width, represented as b, which is a critical dimension of the section size. The effective depth, denotes as d, is a parameter found in the STM method and is intricately linked to bending moment leverage. The lap-spliced length, denoted as L_p, plays a pivotal role in establishing the anchoring condition of longitudinal reinforcing bars [5]. An increase in the amount of longitudinal rebar, represented as A_s, entails a greater involvement of material in supporting forces on the tension side, contributing to force equilibrium [46]. Transverse reinforcing bars, labeled as A_st, offer confinement to the concrete at the joint through axial tension and dowel action [16].

3) Connection details. An additional category of variables that exerts a significant influence on the flexural strength of the precast deck joints pertains to connection details. Concerning rebar connections, the most prevalent type is straight lap-spliced reinforcing bar (Fig.4(a)). In this configuration, force transfer between the reinforcing bar and concrete relies on friction and adhesion along their interfaces [47,48]. However, straight reinforcing bars necessitate an adequate L_p to achieve their maximum strength, which can pose challengies for expedited construction. To address this issue, as shown in Fig.4(b), reinforcing bars with headed ends were utilized to enhance interlocking through the heads. Consequently, both L_p and the width of the joint can be reduced. The use of U-bars, which are bent at a 180° angle from straight reinforcing bars, could further enhance the anchorage. By conducting previous researches [49,50], it was found that U-bars (Fig.4(c)) require shorter L_p compared to straight and headed reinforcing bars, confirming differences in flexural strength.

4) Joint-precast concrete interface. In the context of the joint-precast concrete interface, a straight-interfacial configuration is often chosen for ease of construction. However, its connection performance is typically suboptimal due to the limitation of interfacial contact area. To enhance the bonding performance of the joint-precast concrete interface, different interface shapes have been suggested over time. These shapes include diamond-shaped [10], curved-shaped [51], T-shaped [11], notched-shaped [52], and dovetail-shaped [51] configurations (Fig.5). Certain interfaces result in a reduction in the supporting beam’s width and the volume of site-cast concrete (e.g., T-shaped, curved, notched, dovetail configurations), whereas others improve resistance to vertical shear (e.g., diamond- and T-shaped).

3.3 Development of database

Flexural capabilities, denoted as M_u, are typically estimated through flexural testing conducted on specimens of precast deck joints. To compile a comprehensive database, a total of 433 results were gathered from various experimental sources. It should be noted that all these experimental programs were subjected solely to monotonic load conditions. In Tab.1, a summary of this database, inclusive of its statistical attributes, is presented. The database encompasses seven numerical variables, which include the compressive strength for precast concrete (f_{\text{cp}}^{\prime }) and joint concrete (f_{\text{cj}}^{\prime }), deck width (b), the effective depth (d), lap-spliced length of longitudinal reinforcing bars at joint (L_p), longitudinal reinforcing bars’ tensile capacity along the tension side (f_yA_s), and transverse reinforcing bars’ tensile capacity located inside the joint (f_ytA_st). Furthermore, it includes three categorical variables that encompass various types of reinforcing bar connections (TOR), joint concrete types (CTOJ), and joint-precast concrete interface (IT).

The distributions of these design parameters were visualized in Fig.6 using histograms. The experimental data spans a broad range of design parameters. f_{\text{cp}}^{\prime } ranges from 22.0 to 187.9 MPa, f_{\text{cj}}^{\prime } ranges from 22.0 to 190.0 MPa, b ranges from 150.0 to 1829.0 mm, d ranges from 75.0 to 368.0 mm, L_p ranges from 50.0 to 900.0 mm, f_yA_s ranges from 39.3 to 2940.3 kN, and f_ytA_st ranges from 0.0 to 2491.8 kN. Notably, the database encompasses three types of rebars, designated as 1 for U-bar, 2 for straight bar, and 3 for headed bar; five types of joint concrete, labeled as 1 for normal concrete (NC), 2 for cement grouting (CG), 3 for SFRC, 4 for UHPC, and 5 for HSC; and six types of interfaces, marked as 1 for straight-shaped, 2 for diamond-shaped, 3 for curved-shaped, 4 for T-shaped, 5 for notched-shaped, and 6 for dovetail-shaped. It’s important to note that, for simplicity, numerical labels were used for each category item when developing models in this study.

Additional information regarding the distribution of rebar types, concrete joint types, and interface types is located within Fig.7. In this context, among the precast decks, 203 utilized U-bar (approximately 46.9% of total specimens), 150 utilized straight bars, and 80 employed headed bars for the connection zone, as illustrated in Fig.7(a). The prevalence of U-bar usage in jointing precast decks suggests its potential effectiveness, despite its generally shorter lap length, thanks to the improved mechanical anchoring derived from bend details. This, in turn, reduces the likelihood of lap failure in situations with limited length.

In Fig.7(b), out of 218 specimens (approximately 50.3% of total specimens), NC was the most common material employed to link two precast decks, followed by UHPC (117 specimens) and HSC (55 specimens). Meanwhile, only 32 specimens used CG, and 11 specimens used SFRC as cast-in-place materials for joint connections. Up to six distinct types of interfaces were developed and implemented to enhance the contact area between precast decks. As depicted in Fig.7(c), the straight interface, with 253 specimens (approximately 58.4% of total specimens), emerged as a popular choice for connecting precast decks, followed by the diamond-shaped interface with 110 specimens. The other deformed interfaces, including curved, T-shaped, notched, and dovetail shapes, were designed and constructed with almost the same number of specimens, specifically 14, 14, 17, and 25, respectively. However, previous experimental findings suggested that the straight interface might result in a lower flexural capacity compared to the deformed interfaces due to its significantly reduced contact area [53].

The established database in this research delineates the range of applicability for the developed models. Given the wide variations in geometrical and material characteristics across the specimens, average value of experimental results (M_u,exp) also exhibits a broad range, spanning from 5.0 to 459.6 kN·m.

4 Model performance

4.1 Training process of ML models

Utilizing the collected precast deck-flexural testing database, a flexural strength prediction model was constructed. The framework of the training and validation procedure is annotated in Fig.8. First, the database of 433 experiments was randomly separated into a training and a testing data set. During model development, the training data set was used, whereas the testing data set was reserved for assessing the models’ performance. Nine training ratios including 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9, and corresponding testing ratios including 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, and 0.1 were used in this study to explore the impact of various training and testing data partitions. Then, individual models were trained with the training data set and subsequently averaged using the MC method. The ML model’s performance was finally assessed employing the testing data set.

4.2 Performance measures

The assessment of the prediction capabilities of the ML-based models introduced in this research was carried out using conventional statistical measures, which encompass the coefficient of determination (R²), RMSE, mean absolute error (MAE), and mean absolute percentage error (MAPE). The definition of these statistical indicators are as follows:

(4)R^2=1-\frac{{\displaystyle \sum _{i=1}^n{\left(M_{\text{u,pre},i}-M_{u,\mathrm{e}\mathrm{x}\mathrm{p},i}\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(M_{\text{u},}{}_{\mathrm{e}\mathrm{x}\mathrm{p},i}-{\overline{M}}_{\text{u},\mathrm{e}\mathrm{x}\mathrm{p}}\right)}^2}},

(5)RMSE=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{\left(M_{\mathrm{u},\mathrm{p}\mathrm{r}\mathrm{e},i}-M_{\mathrm{u},\mathrm{e}\mathrm{x}\mathrm{p},i}\right)}^2}}{n}},

(6)MAE=\frac{{\displaystyle \sum _{i=1}^n\left(\left|M_{\mathrm{u},\mathrm{p}\mathrm{r}\mathrm{e},i}-M_{\text{u,exp},i}\right|\right)}}{n},

(7)MAPE=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left(\left|\frac{M_{\mathrm{u},\mathrm{p}\mathrm{r}\mathrm{e},i}-M_{\text{u,exp},i}}{M_{\text{u,exp},i}}\right|\right)}\times 100\mathrm{\%},

where n represents the sample size; M_u,exp,i and M_u,pre,i signify the values at sample ith obtained from existing test data and predicted by ML-based models, respectively. It is noteworthy that R² and MAPE metrics are dimensionless, while RMSE and MAE share the same unit as the target output (in this case, kN·m). R² is a statistical indicator that falls within the range of zero to one, with higher values signifying a superior predictive performance of the model. Additionally, lower RMSE, MAE, and MAPE values signify the effectiveness of the prediction algorithm.

4.3 Effect of training−testing ratio

The partitioning ratio of the experimental database, as indicated by previous studies [54–56], has been shown to have a substantial impact on the prediction performance of ML-based models. This is demonstrated in Fig.9, which displays the MAE results for various partition ratios.

Fig.9 illustrates that altering the training-testing ratio outstandingly impacts the prediction performance of various ML models. In general, training performance outperforms test performance across all ML models. Nevertheless, the importance lies in the test phase’s ability to reveal a model’s generalization. Moreover, it is essential to keep the disparity between the training and testing phases minimal to mitigate overfitting or underfitting. Adhering to these criteria, we have chosen a ratio that achieves a low MAE value in the testing data set and a minimal difference in MAE values between the training and testing data sets. Accordingly, for the development of the MCMA model, this study adopts a 0.9–0.1 ratio.

4.4 Comparison of ML-based model predictions

In Fig.10, the predictions generated by various ML-based models are depicted alongside the actual results through scatter plots. The figure also includes lines representing a relative error of ±20%. The results in the figure reveal that the MCMA and DT models outperform the others in predicting flexural capacity. Both of these models exhibit results that are closely clustered around the line of ideal values, with an R² value exceeding 0.96, and most of the relative errors falling within the range of −20% to 20%.

The baseline models, namely AdaBoost and LR, also perform well, with results closely approximating the ideal line. However, when it comes to predicting flexural capacity up to 100 kN·m, the AdaBoost and LR models exhibit a wider spread in the testing data set. For the AdaBoost model, the relative error exceeds 20% for most samples, while the LR model shows relative errors beyond the −20% to 20% range for most samples.

In contrast, the MLP model performs the poorest among all the developed ML models. Despite having similar R², RMSE, MAPE, and MAE values on both the training and testing data sets, the MLP model doesn’t exhibit overfitting tendencies and can generalize its predictions to data that it has not encountered during training. The likely reason for the MLP model’s subpar performance is the considerable data requirement for the backpropagation process, as it necessitates determining multiple layers of parameterized differentiable modules in the MLP. Given the small-scale nature of the data available for deck joint assessment, this could explain the MLP’s inability to make accurate predictions.

Tab.2 provides a summary of evaluation metrics for different models, facilitating further comparison and analysis. In general, the training performance surpasses the testing performance for all ML models. However, the real significance lies in the ability of the testing phase to unveil a model’s generalization.

When considering the metrics R², RMSE, MAPE, and MAE on the testing data set, the proposed models are ranked in the following order: MCMA, DT, LR, AdaBoost, and MLP. Among these models, the MCMA model stands out, outperforming all other ML models across all performance metrics. Its R², RMSE, MAPE, and MAE on the testing data set are 0.98, 10.30 kN·m, 17.14%, and 6.73 kN·m, demonstrating the highest R² and the lowest RMSE, MAPE, and MAE when compared to the other developed models.

In comparison to the second-best model, DT, the MCMA model achieves notable improvements of 2.1%, 26.6%, and 12.3% on the R², RMSE, and MAE, respectively, except for MAPE. Regarding the remaining models, MCMA exhibits substantial enhancements: 6.5% increase in R², 49.9% reduction in RMSE, 66.5% improvement in MAPE, and 57.1% reduction in MAE compared to LR; 8.9% higher in R², 54.9% lower in RMSE, 82.3% better in MAPE, 66.5% lower in MAE compared to AdaBoost; and 78.2% higher in R², 78.4% lower in RMSE, 64.8% better in MAPE, and 81.6% lower MAE compared to MLP.

Fig.11 illustrates the probability distribution of the M_u,exp/M_u,pre ratios on the entire data set. In general, the probability distribution of MCMA, DT, AdaBoost, LR, and MLP models closely approximates a normal distribution. Among these models, the DT model exhibits the most narrow distribution, with approximately 95% of the M_u,exp/M_u,pre ratios falling within the range of 0.9 to 1.1. The MCMA model demonstrates the second-best distribution, with roughly 79% of the M_u,exp/M_u,pre ratios falling within the 0.9 to 1.1 range. In the cases of AdaBoost, LR, and MLP models, the percentages of M_u,exp/M_u,pre ratios within the same distribution range decrease to approximately 41%, 43%, and 9%, respectively.

Furthermore, the analysis of the results revealed that the performance of the MCMA model surpassed that reported in Ref. [57], where the deep forest (DF) model was employed. Wang et al. [57] optimized the DF model’s hyperparameters through grid search in combination with 10-fold cross-validation. While the MAPE in the testing data set was lower in the DF model, the MCMA model exhibited improvements in R², RMSE, and MAE by 2.5%, 22.8%, and 25.3%, respectively, compared to the DF model [57]. This improvement can be partly attributed to the utilization of a larger number of experimental data for the training process in this study compared to Wang et al. [57].

4.5 Comparison with strut and tie model

The flexural strength of the precast deck joints is assessed by Refs. [1,15] through the introduction of the STM. This model aims to overcome geometric irregularities arising from the disconnect between rebars and concrete in the joint area. The STMs adhere to a common framework, in which the tension side capacity of the joint is influenced by the factors such as longitudinal reinforcing bars’ yielding, transverse reinforcing bars, or the concrete crushing in the diagonal strut situated between the overlapping longitudinal reinforcing bars, as illustrated in Fig.12. Each pair of overlapped units forms a force triangle involving these three components.

(8)T_{\text{u}}=N\times \mathrm{m}\mathrm{i}\mathrm{n}\left(f_{\text{y}}{A}_{\text{s}},\frac{4f_{\text{yt}}{A}_{\text{st}}{L}_{\text{p}}}{s},\frac{1.7f_{\text{cj}}^{\prime }{D}_0{L}_{\text{p}}^{2}s}{4L_{\text{p}}^2+s^2}\right),

where N represents the quantity of overlapped pairs of reinforcing bars; {\displaystyle \frac{4f_{\text{yt}}{A}_{\text{st}}{L}_{\text{p}}}{s}}, and {\displaystyle \frac{1.7f_{\text{cj}}^{\prime }{D}_0{L}_{\text{p}}^{2}s}{4L_{\text{p}}^2+s^2}} are, in turn, transverse reinforcing bar, and the concrete strut arranged within a triangle; s denotes the spacing of longitudinal reinforcing bars; D₀ represents the depth of strut, defined as U-bars’ inner bend diameter [12] and the distance measured from the head edge of headed bars [15]. It is important to note that only STMs have only been proposed for U-bars and headed reinforcing bars. Thus, in utilizing the STM to evaluate joints detailed with straight reinforcing bars, the D₀ is considered equivalent to the reinforcing bars’ diameter. Ultimately, the precast deck joint’s the flexural strength can be established by applying the condition of equilibrium as outlined below [18]:

(9)c=\frac{C_{\text{u}}}{0.85f_{\text{cj}}^{\prime }b}=\frac{T_{\text{u}}}{0.85f_{\text{cj}}^{\prime }b},

(10)M_{\text{u}}=T_{\text{u}}\left(d-\frac{c}{2}\right)=T_{\text{u}}\left(d-\frac{T_{\text{u}}}{1.7f_{\text{cj}}^{\prime }b}\right).

In Fig.13, the correlation between the flexural strengths predicted by the STM and the experimental values is depicted. A noticeable dispersion is evident in the STM results when compared to the outcomes of ML models illustrated in Fig.10, indicating inaccuracies in the STM predictions. The STM achieves values of R², RMSE, MAE, and MAPE at 0.80, 35.07 kN·m, 24.11 kN·m, and 31.90%, respectively, which are notably lower than those achieved by the MCMA (0.99, 8.97 kN·m, 6.29 kN·m, and 12.10% for the entire data set). Even individual models such as DT, AdaBoost, and LR outperform the STM model. Moreover, the STM tends to make conservative predictions for 74% of the data points. Theoretically, the STM operates as a lower-bound design approach grounded in plastic mechanics, and its underlying principle leads to conservative estimations of flexural strength.

Fig.14 illustrates a comparison of M_u,exp/M_u,pre ratios of flexural strength concerning the d and f_yA_s. The STM model exhibits a mean value and coefficient of variation (COV) at 1.21 and 0.35, respectively, indicating lower accuracy compared to the MCMA model, which shows 24.7% and 150.0% higher values in the aforementioned indices. This overall contrast suggests that the MCMA model generally provides more accurate and economically favorable predictions than the STM model. Additionally, the STM introduces biased M_u,exp/M_u,pre ratios that are notably sensitive to the d and f_yA_s. These ratios display higher values for lower f_yA_s (200–1200 kN) and d (94–250 mm), and lower values for f_yA_s (> 1000 kN) and d (> 250 mm). Consequently, the STM model demonstrates significant decreasing trends with respect to the f_yA_s and d, highlighting the need for further improvement in these parameters. This may be attributed to the intricate interactions occurring between the reinforcement bars and the surrounding concrete within the joint area. In contrast, the predictions by the MCMA model, based on the experimental data in this study, do not exhibit significant biases with these input variables.

4.6 Effect of categorical variables on flexural strength of precast deck joints

Researchers have endeavored to establish realistic categorical variables in their testing, recognizing that the flexural strength of precast decks can be influenced by parameters like type of reinforcing bars, concrete joint type, and interface type. Therefore, in the current investigation, the impact of different reinforcing bar types, various concrete joint types, and diverse interface configurations on flexural strength were examined. To achieve this, the data set was divided into three distinct categories for the reinforcing bars (straight bar, headed bar, and U-bar), five categories for the concrete joint types (NC, CG, SFRC, UHPC, and HSC), and six categories for the interface types (straight, diamond, curved, T-shaped, notched, and dovetail shapes). Subsequently, boxplot analysis were conducted.

Boxplot analysis is a graphical method used to visually depict various characteristics of numerical data, such as their central tendency, spread, and skewness, by utilizing quartiles. In Fig.15, you can observe the boxplots representing normalized flexural capacity predicted by the MCMA model. For standardization purposes and improved comparability, the prediction flexural strength was normalized by dividing by f_yA_sd.

In summary, the boxplot analysis varied with the type of reinforcing bar (Fig.15(a)), the concrete joint type (Fig.15(b)), and the type of interface (Fig.15(c)). When considering the effect of the reinforcing bar type depicted in Fig.15(a), it’s noteworthy that, in the case of headed bars, no potential outliers (O-marks) were found, and the median value was highest for straight bars and lowest for headed bars. The interquartile range (IQR) and the overall range (OR) of predictions were largest for headed bars, indicating highly dispersed predictions if headed bars are used. The IQR and OR of straight bars (or U-bar) overlapped with those of headed bars, suggesting no statistically significant difference from predictions for headed bars. Despite having a much shorter L_p, the mean of the normalized flexural strengths of specimens featuring U-bar connections was nearly the same as those with straight bars, thanks to the improvement of the anchoring attributed to bend details. Conversely, specimens with headed bars showed a reduced mean normalized flexural strength compared to those with straight bars, indicating a difference of approximately 17.3%.

When considering the impact of the concrete joint type shown in Fig.15(b), potential outliers (O-marks) were found only in the case of UHPC. The median value was highest for UHPC and lowest for NC, but the difference between these medians was not substantial. The IQR and OR of predictions were largest for NC, indicating highly dispersed predictions if NC is used. However, the IQR and OR of all these categorical variables almost overlapped, suggesting no statistically significant differences between them. The mean values in Fig.15(b) indicated that the NC joint demonstrated approximately 7.3%, 7.7%, and 15.2% lower normalized flexural strength values compared to those of the CG, SFRC, and UHPC joints, affirming the advantage conferred by high-quality joint materials.

Considering the effect of the type of interface shown in Fig.15(c), potential outliers (O-marks) were found only in the straight interface. The median value was highest for the notched-shaped interface and lowest for the straight interface. The IQR and OR of predictions were largest for the T-shaped interface, indicating highly dispersed predictions if T-shaped interfaces are used. The mean values in Fig.15(c) demonstrated that the straight interface resulted in the lowest normalized flexural strength due to its significantly smaller contact area compared to the deformed interfaces. Specimens using deformed interfaces exhibited higher normalized flexural strength (approximately 19.6%) than those employing a straight interface.

From the boxplot analysis, it can be inferred that using U-bar in combination with SFRC (or UHPC) and a notched-shaped (or dovetail-shaped) interface can significantly improve the precast deck joints’ flexural performance.

4.7 Global interpretation of flexural strength prediction

The SHAP-based interpretation of the MCMA model’s predictions reveal various aspects. First, in Fig.16(a), the significance of the input variables’s feature indicates the overall influence of these features on the predictions. This importance is derived as the mean of the absolute SHAP values obtained from the testing data set. The figure shows that f_yA_s holds the foremost impact on the flexural strength of precast deck joints with an absolute SHAP value of approximately 0.35. This prominence stems from the fact that longitudinal reinforcing bars are the primary contributors to resisting external bending forces. It’s important to note that f_yA_s is determined by the design specifications of the precast deck joint, implying the existence of an upper bound to the precast deck joints’ flexural strength. f_ytA_st also has a considerable impact on the precast deck joints’ flexural strength. However, this variable’s importance is approximately 77.2% lower than that of f_yA_s, with an absolute SHAP value of about 0.13. A higher f_ytA_st is anticipated to yield greater confinement within the concrete at the connection zone, subsequently improving the longitudinal reinforcing bars’ anchorage and thereby bolstering the overall strength significantly. Consequently, increasing f_ytA_st, either by augmenting their quantity or yield strength, represents an effective means of enhancing flexural strength. It is important to note that the upper limit of this enhancement is governed by the precast deck. d also exhibited similar influence on the flexural capacity as transverse reinforcing bars did, and followed by TOR. Additionally, other variables such as L_p, IT, and concrete strength of joint (f_{\text{cj}}^{\prime }) have almost the same SHAP value and also exert a certain level of influence on flexural capacity. However, the remaining input variables show comparatively lesser impacts on the prediction outcomes.

Fig.16(b) illustrates a summary plot of SHAP values representing the input variables and their corresponding influence trends. The plot showcases the SHAP values’ distribution for each input variable, with the specific SHAP value denoted in the X-axis and the input variables based on their importance arranged along the Y-axis. Each data point on the plot represents the values of the input variables in the database, with the dot color indicating the value of the corresponding feature. The color gradient, ranging from blue to red, reflects a spectrum from small to large values for each feature. The horizontal distribution of the dots reflects whether the feature value contributes to an increased or decreased likelihood of spalling. For instance, a red upper-right dot signifies that a higher value of f_yA_s will increase the flexural strength of precast deck joint by approximately 2.7. Therefore, this summary plot not only helps comprehend which input variables are significant, but also provides insights into how each feature affects the flexural capacity of the precast deck joints. Hence, the summary plot provides insights not only into the significance of individual input variable, but also into the influence of each feature on the flexural strength of the precast deck joints. In general, observations from the plot reveal that an increase in f_yA_s, f_ytA_st, and d leads to an elevated flexural capacity in the precast deck. Conversely, the flexural capacity tends to decrease with increasing values of features such as TOR. This means the use of headed bar and U-bar (denoted as 2 and 3, respectively) in the connection zone could reduce the flexural capacity of precast deck. These SHAP plots offer a global, macro-level explanation of how the input variables impact the flexural capacity of precast deck, providing valuable insights into the underlying phenomenon.

5 Parametric studies

The proposed MCMA model was utilized for a parametric investigation, exploring how variations in input variables (design variables) impact the precast deck joints’ flexural strength, which is the desired output variable. The objective of this analysis was to deepen our comprehension of the relationship between input variables and the precast deck joints’ flexural strength, and to identify the variables with the most significant impact on this outcome. Default values for the variables were set as follows: f_{\text{cp}}^{\prime } = 40 MPa, f_{\text{cj}}^{\prime } = 50 MPa, b = 800 mm, d = 150 mm, L_p = 300 mm, f_yA_s = 1000 kN, and f_ytA_st = 800 kN.

For each combination of categories (e.g., U-bar-NC at joint-straight bar, denoted as 1-1-1), one variable was altered while the others remained constant. The results of the parametric are presented in Fig.17 and Fig.18, showing the trends consistent with the proposed MCMA model. Detailed discussions for each variable are provided below.

As illustrated in Fig.17, an increase in f_yA_s generally resulted in higher M_u. For instance, considering the combination of 2-3-1, when f_yA_s increased from 100 to 2500 kN (a 2400% increase), M_u predicted by the MCMA model increased from 75.6 to 377.9 kN·m (a 400% increase). This could be attributed to the greater involvement of material in supporting equilibrium of forces on the tension side, the increased amount of longitudinal reinforcing bar plays a crucial role. However, the effects varied among different combinations.

For f_yA_s ranging from 100 to 800 kN, combinations like 1-1-5 and 2-3-5 exhibited almost the same flexural strength, considerably higher than the remaining combinations. In comparison with combinations like 2-1-1 and 2-3-1, which performed poorly, the combinations 1-1-5 and 2-3-5 demonstrated an overrating with an average difference of 72.8%. For f_yA_s ranging from 850 to 1900 kN, combinations like 1-1-5 and 1-1-1 showed almost the same flexural strength, considerably higher than the remaining combinations. In comparison with combinations like 2-3-1 and 2-3-5, and 2-1-1 and 2-1-5, which displayed similar flexural strength, 1-1-5 and 1-1-1 surpassed their performance with average differences of 29.2% and 84.4%, respectively. Beyond this range, all combinations displayed the same flexural strength. This implies that, employing various combinations would not be effective when a high amount of longitudinal reinforcing bar is utilized, at least within the specified geometrical dimensions in this study.

The effect of the remaining design parameters on the M_u is illustrated in Fig.18. Overall, the figure indicates that the flexural strength of precast decks using combinations from 1 to 1-1 to 1-5-6 surpasses that of combinations from 2 to 3-1 to 2-5-6 and 3-3-1 to 3-5-6, as well as 2-1-1 to 2-2-6 and 3-1-1 to 3-2-6, with average differences of approximately 34.0% and 94.9%, respectively.

Specially, f_ytA_st and d considerably impact the flexural strength of precast decks (Fig.18(a) and Fig.18(b)). For instance, in the case of combination 1-1-1, Fig.18(a) shows that M_u increased from 192.5 to 207.3 kN·m (a 9.6% increase) when f_ytA_st increased from 100 to 2200 mm (a 2200% increase). This is attributed to the higher transverse reinforcing bars providing additional internal confinement to the joint concrete [16]. Increasing d from 150 to 325 mm resulted in a notable increase in flexural strength of approximately 35.5% (according to the combination of 1-1-1), as the higher effective depth significantly contributes to the increase of the moment arm. However, while combinations like 2-3-1 and 2-1-1 exhibited a steady increase, combination 1-1-1 showed a sudden increase at d = 225 mm.

The use of a longer L_p could lead to an increase in flexural strength (Fig.18(c)), although the difference was not considerable. Among these combinations, 2-1-1 produced a sudden increase at L_p = 350 mm. Meanwhile, b, f_{\mathrm{c}\mathrm{j}}^{\prime }, and f_{\mathrm{c}\mathrm{p}}^{\prime } showed no notable impact on the flexural strength of precast deck joints (Fig.18(d)–Fig.18(f)).

Overall, this study offers valuable insights into the impact of various design variables on the flexural strength of precast deck joints, providing valuable information for the development of more robust and resilient structures.

For practical applications, a user-friendly graphical interface was developed utilizing the MCMA model. This tool streamlines the process of accurately and efficiently assessing the precast deck joints’ flexural strength. Detailed descriptions of the graphical user interface (GUI) application can be found in Appendix A.

6 Conclusions

This study introduced a novel MCMA model to estimate the precast deck joints’ flexural strength. The essential variables were identified through a combination of experimental observations and domain knowledge, forming the basis for constructing an extensive database. The MCMA model was developed using a weighted model averaging approach employing the MC method, incorporating four distinct ML models: DT, AdaBoost, LR, and MLP. Furthermore, a comprehensive database containing the bending test results of 433 precast deck joint specimens was established (The complete dataset can be assessed through the GitHub). The conclusions were made as follows.

1) In developing the MCMA model, a training-testing ratio of 0.9–0.1 was chosen, as there was only a minimal difference in MAE values observed between the training and testing data sets across different ratios.

2) The MCMA model exhibited superior performance in flexural strength prediction compared to other ML models across all execution indicators for the testing data set. Specifically, it showed enhancements of 2.1% in R², 26.6% in RMSE, and 12.3% in MAE compared to the second-best model (i.e., DT). Additionally, most ML models, particularly MCMA, outperformed the STM, showcasing the advantages of the data-driven technique. The MCMA mode resulted in a significant reduction of 24.7% in the mean value and 150.0% in the COV of the M_u,exp/M_u,pre ratio compared to the STM.

3) Boxplot analysis revealed that incorporating U-bar with SFRC (or UHPC) and a notched-shaped (or dovetail-shaped) interface significantly enhanced the precast deck joints’ flexural strength.

4) SHAP analysis demonstrated that the flexural strength predicted by the MCMA model are effectively interpretable. Numerical variables associated with reinforcing bars, such as f_yA_s and f_ytA_st, and d played a crucial role in predicting flexural strength.

5) A parametric study indicated that the tensile capacities of longitudinal and transverse reinforcing bars, along with d, exerted a strong and positive effect on the flexural strength of precast decks. Other design parameters had limited or negligible effects.

While the MCMA model demonstrated superior performance compared to other methods, it’s important to acknowledge that no method is flawless. Therefore, in future research, it would be prudent to explore integrating optimization methods like Gene Expression Programming [58,59] with MCMA to potentially enhance model performance even further.

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