# Model Updating of Bridges Using Measured Influence Lines
## 1 Introduction  
The average age of bridges all around the world, particularly in developed economies, is increasing and great numbers are approaching the end of their design lives [1]. As they are often poorly maintained and monitored and are exposed to natural disasters, such as earthquakes and floods, which cause erosion under their supports, we are observing more and more collapses, many of which have caused fatalities. Nearly half of the reported bridge failures worldwide have occurred in the last 20 years [2]. Comparing the expanding global economy and the number of reported bridge collapses results in a firm correlation (Figure 1). Without systematic inspections, monitoring, timely interventions, and implementation of new technologies, even nominally robust systems gradually deteriorate, culminating in higher collapse rates. Recognising this linkage justifies the current research efforts into new technologies, such as structural health monitoring (SHM), finite element model updating (FEMU), and bridge weigh-in-motion (B-WIM), especially as transport networks becomeever more extensive and critical to economic development. Tackling these issues requires better and more robust predictions of bridge behaviour. One approach that copes with these requirements is FEMU, where finite element (FE) models are updated to match the analytical and measured responses. In structural dynamics, FEMU traces back to the early efforts of reconciling FE simulations with measured vibration data to refine predictions of real-world structural behaviour. [PERSON] and [PERSON] [4] conducted a deep survey on the topic, systematising key approaches for parameter selection, objective function formulation, and optimisation strategies. Early on, most FEMU applications focused on natural frequencies and mode shapes to capture global structural responses. Over time, researchers recognised that purely vibration-based data sometimes lacked sensitivity to localised parameters (e.g., small-scale stiffness or boundary nuances). This realisation gradually led to new data types, such as displacements and strains [5; 6; 7; 8; 9; 10], being integrated into the FEMU process, helping to address the inherent limitations of purely modal data. Building on these foundations, the current generation of FEMU techniques harnesses large-scale test data under various loading scenarios. One promising approach is using measured influence lines (ILs). IL-based approaches reflect global and local responses with minimal instrumentation, allowing for more precise FEMU. Using ILs in FEMU can more accurately describe local boundary conditions, subtle stiffness variations, and stress distribution patterns. The literature proposes various signal-processing and inverse analysis techniques to extract an IL from measured bridge responses under moving loads: spatial-domain, frequency-domain, and decomposition-based methods. [PERSON] et al. [11] introduced a bridge IL identification method through spatial domain modelling. This approach quantifies the connection between vehicle axle load, IL coefficients, and bridge response at different vehicular positions. Alternatively, one can achieve response normalisation in the frequency domain, restoring the IL using the inverse Fourier transformation. [PERSON] & [PERSON] [12] developed a method that utilises static response transfer ratio functions to identify IL from a probabilistic frequency domain viewpoint. Similarly, [PERSON] et al. [13] identified IL in the frequency domain based on the principle that the convolution of the moving load and IL results in the bridge response. An adaptive enhanced variational mode decomposition (AEVMD) method was presented for extracting the bridge's quasi-static response [14]. This method combines the pre-processed conjugate gradient least squares (PCGLS) algorithm to determine the IL. Although both spatial and frequency domain methods effectively create mathematical models for IL identification, they remain vulnerable to dynamic interference.