⚡ electrical engineering

Development of Design-Oriented Bond Strength Equations for FRP Bars Embedded in Concrete Exposed to Elevated Temperatures Using Explainable Machine Learning

This study develops a hybrid WOA–ETR machine learning model to accurately predict FRP bar bond strength in concrete under elevated temperatures, leverages SHAP analysis to identify key influencing factors, and derives robust, physically interpretable design equations based on these insights.

Original authors: Gia Toai Truong, Dinh-Nhat Truong, Trong-Phuoc Nguyen, Chinh Van Nguyen, Ngoc Hieu Dinh

Published 2026-08-20

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CC BY 4.0

Original authors: Gia Toai Truong, Dinh-Nhat Truong, Trong-Phuoc Nguyen, Chinh Van Nguyen, Ngoc Hieu Dinh

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Concrete is the backbone of modern infrastructure, but the steel bars hidden inside it can rust and weaken over time, especially in harsh, salty, or humid environments. To solve this, engineers have turned to fiber-reinforced polymer bars, a type of reinforcement made from strong fibers locked inside a plastic resin. These bars do not rust, are lighter than steel, and conduct no electricity, making them ideal for bridges and buildings in corrosive conditions. However, these plastic-based bars have a critical weakness: they are sensitive to heat. When exposed to high temperatures, the resin that holds the fibers together can soften and degrade, much like how a candle wax loses its shape in the sun. This softening weakens the crucial grip, or bond, between the bar and the surrounding concrete. If this bond fails, the structure can lose its strength, yet predicting exactly how much grip remains after a fire or heat exposure has been a difficult puzzle for engineers.

A team of researchers set out to solve this puzzle by combining traditional engineering data with advanced computer learning. They gathered a massive collection of 132 real-world experiments where these polymer bars were pulled out of concrete blocks after being heated to various temperatures, ranging from room temperature up to 600 degrees Celsius. The goal was to find a reliable way to predict how strong the bond would be under these conditions. Instead of relying on simple formulas that often miss the complexity of the situation, the researchers used a sophisticated computer model. They trained this model using a technique called extra trees regression, which builds many decision-making trees to find patterns in data, and then refined it using a whale optimization algorithm. This second step acts like a smart search engine, constantly adjusting the model's settings to find the most accurate way to predict the results, mimicking the way humpback whales hunt in coordinated groups.

The computer model proved to be remarkably accurate, correctly predicting the bond strength in 94.2 percent of the test cases. More importantly, the researchers did not just want a "black box" that gave answers without explanation. They used special tools to peek inside the model and understand exactly which factors mattered most. They discovered that the temperature the bar was exposed to was the single most important factor, followed closely by the texture of the bar's surface and how much concrete covered the bar. While traditional engineering wisdom often emphasizes the strength of the concrete itself, this study showed that for these specific bars in high heat, the temperature and the bar's surface texture were far more decisive. The model also revealed that the length of the bar embedded in the concrete changed how the bond behaved in different ways, suggesting that a single formula could not describe every situation.

To make these findings useful for real-world design, the researchers translated the computer's insights into clear, practical equations. They realized that because the bond behaves differently depending on how long the bar is buried in the concrete, they needed to split the data into three groups: short, medium, and long embedment lengths. For the shortest bars, they further divided the data based on the bar's surface texture, such as whether it was sand-coated or ribbed. By creating separate equations for each of these specific scenarios, they produced formulas that were nearly unbiased, meaning the predicted values were almost exactly the same as the actual experimental results. These new equations offer engineers a transparent and reliable tool to design safer structures that can withstand high temperatures, bridging the gap between complex machine learning and practical construction safety.

Technical Summary: Development of Design-Oriented Bond Strength Equations for FRP Bars Embedded in Concrete Exposed to Elevated Temperatures Using Explainable Machine Learning

Problem Statement
Fiber-reinforced polymer (FRP) bars offer significant durability advantages over steel reinforcement in aggressive environments; however, their application is limited by the thermal sensitivity of the organic polymer resin matrix (typically epoxy or vinyl ester). When exposed to elevated temperatures, the resin undergoes a glass transition and thermal degradation, significantly weakening the bond between the FRP bar and the surrounding concrete. While existing empirical relationships attempt to describe this temperature-dependent bond behavior, they often suffer from limited applicability, low predictive accuracy, and an inability to capture complex, nonlinear interactions among influencing parameters such as exposure temperature, surface profile, embedment length, and concrete properties. Furthermore, existing machine learning (ML) models often function as "black boxes," lacking the physical interpretability required for direct integration into engineering design codes.

Methodology
This study proposes an integrated framework combining advanced machine learning with explainable artificial intelligence (XAI) to develop physically interpretable design equations. The methodology proceeds in three main phases:

Database Compilation and Preprocessing: A comprehensive database of 132 pull-out test specimens was assembled from existing literature. The dataset includes variables such as concrete compressive strength ( $f'_c$ ), FRP bar diameter ( $d_b$ ), cover-to-bar diameter ratio ( $c/d_b$ ), embedment length-to-bar diameter ratio ( $l_e/d_b$ ), FRP bar type (GFRP, BFRP), surface profile (sand-coated, ribbed, fiber-wound, etc.), and exposure temperature ( $T$ ). Categorical variables were encoded, and data were normalized.
Hybrid Machine Learning Modeling: A hybrid Whale Optimization Algorithm–Extra Trees Regression (WOA–ETR) model was developed. The Whale Optimization Algorithm (WOA), a metaheuristic optimizer, was employed to automatically tune the hyperparameters of the Extra Trees Regression (ETR) model. The model was trained and validated using 10 independent runs with 5-fold cross-validation to ensure robustness. Performance was evaluated using $R^2$ , RMSE, MAE, and MAPE.
Explainability and Equation Derivation: To overcome the "black box" nature of ML, SHapley Additive exPlanations (SHAP) and Partial Dependence Plots (PDP) were utilized to quantify feature importance and visualize the marginal effects of input variables. Based on these insights, the database was systematically classified into subgroups according to embedment length regimes and surface profiles. Nonlinear least-squares regression was then applied to these subgroups to derive simplified, physically interpretable predictive equations.

Key Contributions

Hybrid WOA–ETR Model: The development of a novel hybrid model that outperforms conventional ML algorithms (including RF, BR, GB, XGBoost, and standalone ETR) in predicting bond strength under elevated temperatures.
Explainable AI Framework: The application of SHAP and PDP analyses to the bond strength problem, providing a clear hierarchy of influencing factors and revealing nonlinear relationships that traditional empirical formulas miss.
Design-Oriented Equations: The derivation of a series of specific predictive equations based on ML-driven classification. Unlike a single global equation, these equations account for distinct bond-transfer mechanisms in short-, intermediate-, and long-embedment regimes, as well as specific surface profiles.
Rigorous Validation: The study employs extensive validation techniques, including 10 independent runs, leave-one-out cross-validation (LOOCV), and bootstrap resampling, to assess model stability, generalization capability, and parameter uncertainty.

Results

Predictive Performance: The WOA–ETR model achieved high accuracy on the testing dataset with an $R^2$ of 0.942, RMSE of 1.536 MPa, and MAPE of 16.812%. It demonstrated superior stability and generalization compared to benchmark models.
Feature Importance: SHAP analysis identified exposure temperature as the most influential factor, followed by surface profile, cover-to-bar ratio, and embedment length ratio. Notably, concrete compressive strength and bar type showed lower relative importance in this specific high-temperature context.
Nonlinear Behavior: PDP analysis revealed that bond strength degradation is highly nonlinear with respect to temperature, with a rapid decline occurring between ambient temperature and approximately 350°C, after which the curve plateaus. Similarly, the embedment length ratio exhibited three distinct response regions (stable, transition, and plateau), justifying the classification approach.
Equation Accuracy: The derived predictive equations provided nearly unbiased predictions, with mean experimental-to-predicted ratios ranging from 1.02 to 1.06. The $R^2$ values for the derived equations ranged from 0.716 to 0.986. The equations for intermediate and long embedment lengths showed exceptional performance ( $R^2 > 0.96$ ) and low variability.
Robustness: Bootstrap and LOOCV analyses confirmed that the equations for intermediate/long embedment and ribbed bars with sand coating (SP=5) possessed high parameter stability. Equations for other subgroups (e.g., sand-coated bars) showed higher parameter uncertainty due to smaller sample sizes but maintained acceptable predictive accuracy.

Significance and Claims
The paper claims that its primary significance lies in bridging the gap between high-accuracy "black box" machine learning models and practical engineering design. By using ML not just for prediction but as a discovery tool to understand governing mechanisms, the study successfully translated complex data patterns into simplified, transparent analytical equations.

The authors assert that their proposed framework offers a more adaptable and accurate alternative to existing empirical formulas and design codes (such as ACI 440.1R), which often rely on limited experimental conditions. The study emphasizes that while the WOA–ETR model serves as a powerful predictor, the derived equations provide a computationally efficient and physically interpretable tool for engineers to estimate bond strength under elevated temperatures. However, the authors modestly note that the reliability of certain equations is constrained by the limited number of specimens in specific sub-datasets (particularly for long embedment lengths and specific surface profiles), suggesting that further experimental data would be beneficial to refine these formulations and extend their applicability.

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