Revised virtual axle method for damage detection using bridge weigh-in-motion systems

Abstract Bridge Weigh-in-Motion (BWIM) technology is widely used to estimate vehicle axle loads by analyzing bridge responses. This study introduces a revised formulation of the Virtual Axle (VA) method that enables local bridge damage detection by incorporating an additional, weightless axle into the calculation. Unlike the original VA formulation, which is mathematically ill-conditioned and limited to specific load effects, the revised method is analytically stable and applicable to any measurable bridge response. In this study, displacement was used as the load effect input to validate the proposed formulation. A key novelty of this approach is that it requires no prior knowledge of vehicle weight, no manual selection or filtering of data, and operates in a fully output-only manner, making it practical for real-world implementation. The method was validated through laboratory experiments using a scaled bridge model with varying levels of local damage and vehicle speeds. From this, a new damage indicator, called the Virtual Axle Indicator (VAI), was derived using the root-mean-square of the VA distribution. The RMS-VAI increased consistently from 0.02 (undamaged) to 0.22 (most damaged). Repeated measures ANOVA confirmed the method's sensitivity, yielding an F -value of 2539.4, p < 0.001, and ηₚ 2 = 0.9949, indicating that over 99% of VAI variation is explained by structural condition. This physics-based method is easily integrated into existing BWIM systems to provide real-time bridge health monitoring under normal traffic, without additional hardware or operational disruption.