**2. Methods**

The current investigation involved, first, the generation of a synthetic dataset in which all the parameters are known and, then, the training of learning tools to determine whether a structure is damaged and where the damage is located.

#### *2.1. Generation of the Synthetic Dataset*

The numerical evaluation of displacements and vibration frequencies of undamaged and damaged structural members onto which a moving load is acting was first performed. Such data served as information for the training of neural networks devoted to the identification and the localization of the damage. To this aim, a simple supported beam served as the reference structure for the test. The choice of this scheme was based on the fact that this type of support was largely adopted in historical bridge building and because of its inherent low damage tolerance as a statically determinate structure. The beam was 30 m long, had a rectangular cross section (0.4 m × 1.5 m), and was constituted by a material having elastic modulus *E0* and density equal to 30 GPa and 2500 kg/m3, respectively. A moving load, namely *P*, whose position was identified through the variable *xP*, acted on the beam. The magnitude of the load was not constant across the performed simulations: A uniform distribution between 50 kN and 100 kN was attributed to *P* to simulate a real scenario of heterogeneous tra ffic.

A set of 30k simulations with moving load were performed using Matlab (MathWorks, Natick, MA, USA) coupled to OpenSees solver [23]. A first subset of 10k simulations, namely Subset A, related to an undamaged beam: In each simulation, the position and the magnitude of the load were randomly modified. In the second subset of 10k simulations, Subset B, the damage was inserted. To this purpose, the beam was split into 20 parts (elements) of equal length (1.5 m each). For beams with stronger flexural resistance mechanisms, the parameter that rules the behavior of the system is the flexural inertia, which is the product between the inertia moment around the flexural axis and the elastic modulus of the material composing the cross section. Obviously, for elements made of materials with di fferent mechanical properties, such as reinforced concrete, it is possible to determine equivalent inertia and elastic modulus values. Thus, the reduction of the cross section of the material, for example, due to corrosion, can be simulated with a reduction of either the moment of inertia or the elastic modulus [24]. A Lemaitre-Chaboche model [25] was adopted to simulate the damage. The damage model is herein intended as a reduction of the elastic modulus of the cross section. The elastic modulus of the damaged element is

$$E\_d = (1 - d)E\_{0\prime} \tag{1}$$

where *d* is the damage parameter (*d* = 0 for undamaged, *d* = 1 for complete damage). For *d* = 1, a mechanism forms and, thus, the equilibrium cannot be guaranteed. The possible range of the damage parameter was set at [0;0.2]. Larger values are out of the scope of the present analysis since our interest is in incipient damage rather than on already evolved damage. For each simulation, the damaged element and the damage magnitude (parameter *d*) were randomly identified. The adopted approach encompasses all the possible damages that can occur on reinforced concrete elements and concrete elements with pretensioned tendons, which represent the major structural types of road bridge infrastructures.

The third subset of data (Subset C, 10k entries) was represented by five simulations in which the load *P* was located at di fferent positions along the length of the damaged element once a damage was assigned. In detail, for each entry of the dataset, the damaged element and the damage magnitude were randomly identified, and the structure was solved for the load *P* at 1/5, 2/5, 3/5, and 4/5 of the beam length. An additional extra simulation accounting for the beam with just a gravity load was performed, and this condition is subsequently referred to as unloaded.

The structure was modeled as planar and discretized with 21 nodes and 20 beam-elements with 6 degrees-of-freedom each (3 for each end). The cross section and material properties were associated to the elements. The mass of the beam was attributed to each node considering its tributary length. The mass associated to the moving load was not considered at this stage. Figure 1 illustrates the beam with its discretization.

**Figure 1.** Sketch of the reference beam structure. The bottom scheme illustrates the discretization. Red nodes refer to the location of the inclinometers; green nodes refer to the location of the load *P* in the third simulations subset (Subset C).

An eigenvalue analysis was performed to determine the vibration modes, while a static analysis served for the vertical displacements. For each simulation, the frequencies associated to the first three vibration modes, namely *f1*, *f2*, and *f3*, and the rotations at the beam ends, at midspan, and at quarter lengths were recorded (red points in Figure 1). The choice of measuring the rotations instead of the displacements follows the fact that on real structures inclinometers can be installed more easily than can displacement sensors. The presence (or not) of the damage, the position and the magnitude of the moving load, the location and the intensity (*d*) of the damage, and the recorded frequencies and rotation constitute the synthetic dataset.
