Regression Models Predict Curved Bridge Vibrations

Published in the journal AI in Civil Engineering, the study addresses the mechanical complexity of these widely used structural forms by combining a physics-based finite element method with multivariate linear regression modeling. The resulting closed-form expressions offer engineers an efficient preliminary design tool, significantly reducing the dependency on computationally intensive simulations during early design phases.

The primary challenge in designing curved box-girder bridges lies in their intrinsic structural complexities, including distortion effects and torsional warping. While these bridges offer advantages such as superior stiffness-to-mass ratios and enhanced stability, their dynamic response is more complex and less well-understood than that of straight bridges. The study aims to accurately predict fundamental frequencies and corresponding mode shapes while identifying the critical parameters that influence the free-vibration response of these structures.

To address this, the team implemented a computationally efficient one-dimensional thin-walled beam finite element method in MATLAB. This formulation explicitly incorporates torsional, distortional, and warping effects, which are critical for accurately representing the dynamic behavior of curved girders. The element geometry is idealized using straight-line segments, with the cross-section assumed to possess a vertical axis of symmetry to simplify distortion analysis. Each node of the thin-walled box beam element possesses nine degrees of freedom, capturing translations, rotations, twisting rates, and distortional angles.

The validity of the proposed finite element formulation was rigorously verified through numerical examples and comparisons with established software. The model was validated against detailed ANSYS shell-element simulations and published experimental data from a simply supported thin-walled curved box-girder made of Perspex sheets. The MATLAB-based results showed excellent agreement with ANSYS, with discrepancies remaining below 0.40%. Furthermore, deviations from experimental values stayed within 2.6% to 3.7%, demonstrating the accuracy and reliability of the thin-walled beam element for modeling the dynamics of curved box-girders.

A systematic parametric study was conducted to evaluate the influence of key design variables on the first three modal frequencies. These variables included curvature radius, span length, boundary conditions, diaphragm layout, and cross-sectional geometry. The analysis revealed that span length and curvature radius significantly affect the free-vibration characteristics.

The arrangement of diaphragms also played a crucial role in the dynamic response. The study examined three configurations: rigid diaphragms at both ends, rigid diaphragms at the ends plus an intermediate diaphragm, and no diaphragms. The results indicated that diaphragms, particularly when placed at intermediate locations, considerably enhance torsional and distortional stiffness. This leads to higher natural frequencies and improved mode-shape stability, although the added mass of the diaphragms can marginally decrease the frequency of the first vertical bending mode.

Cross-sectional geometry was another significant factor influencing the bridge’s dynamic performance. The researchers analyzed four distinct cross-sections: single-cell rectangular, single-cell trapezoidal, double-cell rectangular, and double-spine trapezoidal. Double-cell and double-spine cross-sections achieved higher natural frequencies, primarily due to their substantially increased bending and torsional stiffness. This stiffness-dominated effect leads to markedly improved dynamic performance, as the corresponding increase in mass remains relatively small compared to the gain in rigidity.

Using the comprehensive dataset generated from these finite element modal analyses, the team developed multivariate linear regression models. Categorical predictors such as boundary condition, diaphragm type, and cross-section shape were converted into binary dummy variables and entered alongside continuous variables like radius and span. The data was carefully cleaned to ensure statistical independence, removing redundant entries and leaving 180 unique observations for analysis. Variance inflation factors were computed to assess multicollinearity, confirming that all values remained comfortably low and indicating no problematic correlation between predictors.

The regression models yielded explicit predictive equations with excellent accuracy for the first three natural frequencies. For the first natural frequency, the model achieved an R-squared value of 0.9997 and a root mean square error of just 0.1289 Hz. The model for the second natural frequency similarly exhibited high accuracy, with an R-squared of 0.9994 and an RMSE of 0.3061 Hz. The third natural frequency model also demonstrated high predictive accuracy, achieving an R-squared of 0.9997 and an RMSE of 0.3092 Hz.

Diagnostic plots further validated the robustness of these regression models. Residual plots showed points scattered evenly around zero with no obvious patterns, suggesting homoscedasticity. Histograms of standardized residuals approximated normal distributions, and Q-Q plots confirmed near-normal behavior with only slight tail deviations. These results affirm that the regression models are statistically sound and reliable for practical application in engineering design.

The practical engineering relevance of these predictions is significant. With RMSE values below 0.31 Hz for all three models, the error remains under 1% relative to typical first-mode frequencies for this bridge type, which generally range from 40 Hz to 150 Hz. Such precision falls well within accepted engineering tolerances for both preliminary and detailed design. By substituting key geometric and categorical parameters into the derived equations, practitioners can quickly evaluate the vibrational response of different girder configurations without performing repeated finite element simulations.

This hybrid methodology effectively combines physics-based finite element modeling with data-driven regression, deepening mechanistic insight while delivering practical utility. The study offers both a validated numerical framework and practical predictive models to support efficient, informed design of curved box-girder bridges. By reducing computational effort, the approach facilitates structural optimization, safety checks, and compliance with relevant design standards, ultimately contributing to the development of safer and more cost-effective bridge infrastructure.

The finite element formulation adopted in this study is based on the work of Zhang and Lyons, which integrates Vlaslov’s thin-walled beam theory with the finite element method. This theoretical foundation is critical because many simplified finite element formulations fail to explicitly account for torsional and distortional warping effects, which are essential for accurately representing the mechanical behavior of curved girders. By incorporating these specific deformation modes, the model captures the coupled bending, torsion, and distortion mechanics that are inherent in thin-walled box-girder bridges. The element geometry is defined by a three-noded configuration, with nodes located at the ends and the mid-length of the element. This discretization allows for the use of higher-order shape functions, specifically C1 continuity for torsion and distortion, which is necessary because the governing equations for these modes are of fourth order. This approach ensures that the model can reliably predict the dynamic response across a wide range of bridge geometries without requiring the computational intensity of full three-dimensional shell element simulations.

Validation of the numerical model was performed using a specific experimental case involving a simply supported thin-walled curved box-girder made of Perspex sheets. This model had a span of 1.52 meters and a curvature radius of 30.48 meters. The convergence analysis indicated that the natural frequencies converged satisfactorily with mesh refinement, confirming the robust performance of the selected element. The close agreement between the MATLAB-based implementation and ANSYS shell-element simulations, with discrepancies remaining below 0.40%, further supported the reliability of the formulation. The deviations from the experimental values, which stayed within a range of 2.6% to 3.7%, provided additional evidence that the model accurately represents the physical behavior of the structure. This validation process is essential for ensuring that the subsequent parametric studies and regression models are built upon a numerically sound foundation.

The parametric study also examined the influence of boundary conditions on the free-vibration characteristics of the bridges. The researchers compared four support configurations: simply supported, fixed, cantilevered, and continuous. The results indicated that natural frequencies generally increased with greater restraint across all dominant modes of vibration. This trend highlights the importance of accurately defining support conditions in preliminary design, as the choice of boundary condition can significantly alter the predicted dynamic response. The study also noted that the dynamic response of the simply supported straight box-girder with end diaphragms aligned well with established behavior of thin-walled beam systems. The first ten modes exhibited a logical progression, starting with vertical and lateral bending, then moving into axial, distortional, and finally torsional responses. This progression reflects the characteristic stiffening effect of higher vibration modes and the increasing complexity of deformation shapes associated with higher strain energy.

The regression analysis utilized a dataset of 180 unique observations after removing 18 redundant records from an initial set of 198. This data cleaning step was crucial for ensuring statistical independence and avoiding unintended weighting bias in the regression models. The use of variance inflation factors to assess multicollinearity among the dummy-coded categorical variables confirmed that the predictors were not problematically correlated, with all values remaining well below 5. This statistical rigor supports the stability of the regression models and their suitability for practical application. The derived closed-form expressions allow engineers to estimate natural frequencies by substituting key geometric and categorical parameters, thereby reducing the need for repeated finite element simulations during the early stages of design. This efficiency is particularly valuable for parametric studies where multiple configurations need to be evaluated quickly.

The study contributes to the broader field of bridge engineering by providing a hybrid methodology that combines physics-based finite element modeling with data-driven regression. This dual approach not only deepens the mechanistic insight into the dynamic behavior of curved box-girder bridges but also delivers practical utility for engineers. The ability to predict modal characteristics with high accuracy using explicit equations supports structural optimization and safety checks. By offering a reliable surrogate tool for preliminary design, the study helps in the development of more efficient and informed bridge infrastructure. The findings underscore the significance of considering complex structural actions, such as torsional and distortional warping, in the dynamic analysis of curved bridges to ensure their safety under dynamic loads.

Subject of Research: A comprehensive study on factors affecting free vibration of thin-walled curved box-girder bridges using regression modelling

Article Title: A comprehensive study on factors affecting free vibration of thin-walled curved box-girder bridges using regression modelling

Article References: Verma, V., Wani, K. U. F., Nallasivam, K., Singh, A., Kumar, M., & Adinarayana, B. (2026). A comprehensive study on factors affecting free vibration of thin-walled curved box-girder bridges using regression modelling. AI in Civil Engineering, 5(1), Article 2. https://doi.org/10.1007/s43503-025-00084-4

Image Credits: AI Generated

DOI: 10.1007/s43503-025-00084-4

Keywords: comprehensive, factors, affecting, free, vibration, thin-walled, curved, box-girder, bridges, regression, modelling, scientific research