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arxiv: 2605.09175 · v1 · submitted 2026-05-09 · 📡 eess.SP

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An Open-Source Framework for Coupled Vehicle-Bridge Interaction Analysis Using OpenSees

Mohammad Talebi-Kalaleh, Qipei Mei

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Pith reviewed 2026-05-12 02:55 UTC · model grok-4.3

classification 📡 eess.SP
keywords vehicle-bridge interactionOpenSeesfinite element modelingcoupled dynamicsopen-source softwarevehicle dynamicsbridge monitoring
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The pith

An open-source Python framework couples separate OpenSees vehicle and bridge subsystems through iterative force-displacement exchange at each time step.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a reusable Python framework for vehicle-bridge interaction analysis on the OpenSees finite element platform. The bridge and vehicle are kept as independent subsystems that exchange displacements and forces iteratively until convergence within each time step. Five vehicle models are supported, from a basic single-axle spring-mass system to two-axle composite half-cars that include body pitch and distinct tyre and suspension elements. Validation reproduces three published benchmark cases with R-squared values above 0.998. A parametric study also maps the accuracy of a simpler decoupled mode as a function of mass ratio, span, speed, road roughness, and traffic density.

Core claim

The central claim is that an open-source framework built on OpenSees can perform coupled vehicle-bridge interaction simulations by treating the vehicle and bridge as separate subsystems linked by an iterative coupling algorithm. At each time step the algorithm passes current displacements from the bridge to the vehicle and forces from the vehicle back to the bridge until both converge. The same framework also supplies a decoupled mode in which the vehicle applies only its static weight as a moving load and the resulting bridge motion then drives the vehicle as base excitation. Validation against quarter-car, half-car with pitch, and full composite benchmarks confirms close numerical match, R

What carries the argument

The iterative coupling scheme that exchanges displacement and force values between separate OpenSees vehicle and bridge subsystems until convergence at every time step.

If this is right

  • The framework enables reproducible studies of dynamic amplification factors under moving loads without writing custom finite-element solvers.
  • Researchers can select among five vehicle models to match the fidelity needed for weigh-in-motion or indirect monitoring applications.
  • The parametric study supplies concrete thresholds on mass ratio, speed, and roughness that indicate when the decoupled mode remains sufficiently accurate.
  • Full coupling is required only when vehicle-to-bridge mass ratio or road roughness exceeds the identified limits, otherwise the simpler mode suffices.
  • Release of the complete code and benchmark configurations allows direct reuse and extension by other groups.
  • pith_inferences

Load-bearing premise

The iterative coupling scheme between the vehicle and bridge OpenSees subsystems converges reliably without numerical artifacts for the modeled vehicle and bridge configurations.

What would settle it

Running the framework on an instrumented bridge under a controlled vehicle passage and comparing the computed dynamic responses against measured strains and accelerations would show whether the coupling reproduces real behavior within engineering tolerances.

Figures

Figures reproduced from arXiv: 2605.09175 by Mohammad Talebi-Kalaleh, Qipei Mei.

Figure 1
Figure 1. Figure 1: Schematic of the vehicle-bridge interaction problem. A multi-DOF vehicle traverses an Euler [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Vehicle models supported by the framework: (a) single-axle simple, (b) quarter-car composite, (c) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Framework flowchart showing the complete analysis procedure, including initialisation, the time [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Iterative partitioned coupling scheme at a single time step, showing the exchange of displacement [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Benchmark 1: this study (solid blue) and Yang et al. (2004) benchmark (dashed red) for a quarter [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Benchmark 2: this study (solid blue) and Yang et al. (2019) benchmark (dashed red) for a half-car [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Benchmark 3: bridge midspan displacement and vehicle body response for the NuBe V1 (quarter [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convergence characteristics of the iterative coupling algorithm: (a) number of iterations per time [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Normalised bridge midspan displacement for the commercial vehicle across four spans (15 m, 30 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Normalised bridge midspan displacement for the heavy truck across four spans. Solid lines: [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Coefficient of determination 𝑅2 for bridge midspan displacement between coupled and decoupled analyses as a function of span length, for the commercial vehicle (blue circles) and the heavy truck (red squares). Mass ratios 𝜇 are annotated next to each data point. Values close to 1 indicate close agreement. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Coefficient of determination 𝑅2 for vehicle body acceleration between coupled and decoupled analyses as a function of span length, for the commercial vehicle and the heavy truck. A noteworthy feature of [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Bridge midspan displacement for the 30 m bridge under the commercial quarter-car vehicle at [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Bridge midspan acceleration for the same six vehicle speeds. [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Vehicle body vertical displacement for the same six vehicle speeds. [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Vehicle body vertical acceleration for the same six vehicle speeds. [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Effect of road roughness on bridge and vehicle response (30 m bridge, commercial quarter-car [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Effect of background traffic on bridge response (30 m bridge, heavy truck, coupled analysis). (a) [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Two-span continuous bridge (2 × 30 m) under a fleet of 10 randomly generated vehicles: (a) bridge displacement envelope over the full span showing the deflected shape at different time instants; (b) midspan displacement time history of the first span; (c) midspan acceleration time history of the first span. The two-span bridge analysis demonstrates the framework’s capability to handle continuous bridges w… view at source ↗
read the original abstract

Vehicle-bridge interaction (VBI) is important for simulating bridge response under moving vehicular loads and supports applications such as dynamic amplification studies, weigh-in-motion, and indirect bridge monitoring. Although VBI theory is well established, many existing implementations use custom finite element code or research-specific solvers, which limits their reuse. This paper presents an open-source Python framework for VBI analysis built on OpenSees. The bridge and vehicle are modeled as separate OpenSees subsystems and connected through an iterative scheme that exchanges displacement and force values at each time step until convergence. Five vehicle model types are supported, from a single axle spring-mass system to two-axle composite half-cars with body pitch and separate tyre and suspension elements. A decoupled mode is also provided: the vehicle static weight is applied as a moving load on the bridge, and the resulting bridge motion is then used as base excitation for the vehicle. Validation against three published benchmarks (quarter-car, half-car with pitch, and full composite models) shows close agreement, with R2 above 0.998 in all cases. A parametric study reports the accuracy of the decoupled mode as a function of vehicle-to-bridge mass ratio, span length, speed, road roughness class, and background traffic density, and indicates when the decoupled mode is adequate and when full coupling is needed. The complete framework and benchmark configurations are released as open-source software to support reproducible research in vehicle-bridge dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript describes an open-source Python framework for coupled vehicle-bridge interaction (VBI) analysis implemented in OpenSees. The bridge and vehicle are treated as independent subsystems linked by an iterative fixed-point scheme that exchanges interface displacements and forces until convergence at each time step. Five vehicle models of increasing complexity are supported, along with an optional decoupled mode that applies static vehicle weight as a moving load. The framework is validated on three published VBI benchmarks (quarter-car, half-car, and composite models), yielding R² > 0.998 agreement in all cases. A parametric study quantifies the error introduced by the decoupled approximation as a function of mass ratio, span, speed, roughness, and traffic density. All code and benchmark setups are released publicly.

Significance. If the iterative coupling implementation is correct and robust, the paper delivers a practical, reusable tool that can accelerate research in vehicle-bridge dynamics, weigh-in-motion systems, and indirect structural health monitoring. The high-fidelity benchmark matches and the open-source release with reproducible configurations are clear strengths. The parametric results offer concrete, actionable guidance on when a simpler decoupled analysis is sufficient, reducing computational cost for practitioners.

major comments (2)
  1. [§4 (Numerical Validation)] §4 (Numerical Validation): The validation demonstrates excellent endpoint agreement (R² > 0.998) with three independent benchmarks, but provides no quantitative data on the iterative coupling process itself—e.g., mean and maximum iterations per time step, residual norms at convergence, or the fraction of time steps that required the maximum allowed iterations. Given that VBI problems can exhibit slow or oscillatory convergence depending on mass ratio and stiffness, this omission leaves open the possibility that the reported accuracy holds only for the specific benchmark parameters rather than generally.
  2. [§5 (Parametric Study)] §5 (Parametric Study): The study varies vehicle-to-bridge mass ratio, speed, road class, and traffic density to compare coupled versus decoupled responses, yet does not report whether the coupled simulations encountered convergence difficulties or required adaptive tolerances under the more demanding parameter combinations (high mass ratio, rough roads). Without such diagnostics, it is difficult to assess the practical reliability of the framework across the explored design space.
minor comments (2)
  1. [Software release] The GitHub repository link and installation instructions are helpful; ensure that the benchmark scripts include the exact OpenSees version and solver settings used for the reported results.
  2. A few figure captions could be expanded to indicate whether the plotted quantities are mid-span deflection, vehicle acceleration, or contact force.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive overall assessment of the manuscript. We address each major comment below and have incorporated revisions to provide the requested convergence diagnostics.

read point-by-point responses

  1. Referee: [§4 (Numerical Validation)] §4 (Numerical Validation): The validation demonstrates excellent endpoint agreement (R² > 0.998) with three independent benchmarks, but provides no quantitative data on the iterative coupling process itself—e.g., mean and maximum iterations per time step, residual norms at convergence, or the fraction of time steps that required the maximum allowed iterations. Given that VBI problems can exhibit slow or oscillatory convergence depending on mass ratio and stiffness, this omission leaves open the possibility that the reported accuracy holds only for the specific benchmark parameters rather than generally.

    Authors: We agree that explicit convergence metrics for the iterative scheme would strengthen the validation. In the revised manuscript we have added a new table in §4 that reports, for each benchmark, the mean and maximum iterations per time step, the mean residual norm at convergence, and the fraction of time steps that reached the iteration limit. These statistics confirm reliable and rapid convergence (typically 4–8 iterations) for the benchmark configurations, with no time steps failing to converge. revision: yes

  2. Referee: [§5 (Parametric Study)] §5 (Parametric Study): The study varies vehicle-to-bridge mass ratio, speed, road class, and traffic density to compare coupled versus decoupled responses, yet does not report whether the coupled simulations encountered convergence difficulties or required adaptive tolerances under the more demanding parameter combinations (high mass ratio, rough roads). Without such diagnostics, it is difficult to assess the practical reliability of the framework across the explored design space.

    Authors: We acknowledge the value of reporting convergence behavior across the parametric space. Our simulations showed that the fixed-point iteration remained stable for all combinations tested, including the highest mass ratios and roughest road classes; no adaptive tolerance adjustments were required and all runs converged within the prescribed limit. The revised §5 now includes a short discussion and a supplementary figure summarizing average iterations versus mass ratio and roughness class, confirming that iteration counts increased modestly but stayed well below the maximum. revision: yes

Circularity Check

0 steps flagged

No circularity: software implementation validated against external benchmarks

full rationale

The paper describes an open-source Python framework for vehicle-bridge interaction using separate OpenSees models connected by an iterative scheme, with validation against three independent published benchmarks yielding R2 > 0.998. No derivations, fitted parameters, ansatzes, or uniqueness theorems are claimed. The central contribution is code release and external benchmark reproduction, which is self-contained and does not reduce to its own inputs by construction. No self-citation load-bearing steps appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard assumptions of structural dynamics and the accuracy of OpenSees finite-element models; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption OpenSees finite-element models accurately capture the dynamic behavior of bridges and vehicles under the modeled conditions
    Invoked throughout the modeling and validation sections.

pith-pipeline@v0.9.0 · 5560 in / 1097 out tokens · 37868 ms · 2026-05-12T02:55:41.170958+00:00 · methodology

discussion (0)

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Reference graph

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