Stochastic moving load identification driven by multi-point deflection responses: an AR(1)-Tikhonov-SVD framework and robustness verification

Jiaqiang Zhang; Yuefei Liu; Xueping Fan

doi:10.1186/s43251-025-00196-y

Advances in Bridge Engineering ›› 2026, Vol. 7 ›› Issue (1) :9 DOI: 10.1186/s43251-025-00196-y

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Stochastic moving load identification driven by multi-point deflection responses: an AR(1)-Tikhonov-SVD framework and robustness verification

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Abstract

In the field of bridge health monitoring, the accurate identification of moving loads is critical for structural safety assessment. To address the prevalent challenges in the existing methods, including improving the modeling of temporal correlations in stochastic moving loads and the ill-posed nature of the inverse identification problem, this paper proposes a novel AR(1)-Tikhonov-SVD framework driven by multi-point deflection responses. Initially, temporally correlated stochastic loads factor sequences are generated via the first-order autoregressive model (AR(1)), which are then coupled with nominal load pattern to construct the stochastic moving load models; Subsequently, the ill-posed system is mitigated through integrating Tikhonov regularization technology and singular value decomposition (SVD) technology, and the generalized cross-validation (GCV) is adopted to adaptively optimize the regularization parameters; Finally, validation was performed using a simply supported beam finite element model (FEM) with multi-point deflection data: (1) Relative Percentage Error (RPE) measured 8.41% under noise-free conditions; (2) RPE increased to 15.10% under 2% Gaussian noise contamination; and (3) increasing the number of measurement points yielded marked improvements in stochastic loads reconstruction accuracy, demonstrating the framework's robustness. This methodology transcends conventional deterministic identification frameworks, establishes a probabilistic paradigm for bridge safety assessment. The three innovative aspects of this work are: (1) Probabilistic stochastic moving load modeling using AR(1)-generated temporally correlated sequences coupled with nominal load patterns; (2) Self-adaptive ill-posedness resolution via Tikhonov-SVD-GCV integration enabling noise-robust reconstruction; (3) Spatio-temporal anti-noise verification through multi-point deflection synergy (9 sections) quantifying sensor-density efficacy.

Keywords

Stochastic moving load identification / Structural health monitoring (SHM) / First-order autoregressive model (AR(1)) / Tikhonov regularization / Generalized cross-validation (GCV) / Ill-Posed problem

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Jiaqiang Zhang, Yuefei Liu, Xueping Fan. Stochastic moving load identification driven by multi-point deflection responses: an AR(1)-Tikhonov-SVD framework and robustness verification. Advances in Bridge Engineering, 2026, 7(1): 9 DOI:10.1186/s43251-025-00196-y

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