*2.1. The VBI Dynamic Model*

The VBI dynamic model is a complex dynamic system, composed of moving vehicles and bridges. In this subsection, the Newmark-β method is used to solve the dynamic equation of the VBI model, thereby obtaining the dynamic response of the model.

#### 2.1.1. Road Surface Roughness

PSD function is used to transformed road surface roughness from spatial frequency domain to the circular frequency domain [21], as follows:

$$\begin{cases} \ S\_{rr}(\Omega) = \mathcal{S}\_{rr}(\Omega\_0) (\Omega/\Omega\_0)^{-2} \quad (\Omega \le \Omega\_0) \\\ S\_{rr}(\Omega) = \mathcal{S}\_{rr}(\Omega\_0) (\Omega/\Omega\_0)^{-1.5} \quad (\Omega > \Omega\_0) \end{cases} \tag{1}$$

Road surface roughness can be calculated by the inverse Fourier transform of the road surface roughness spectrum, as follows [22]:

$$r(\mathbf{x}) = \sum\_{i=1}^{N} \sqrt{\Delta n} \cdot 2^{k} \cdot 10^{-3} \cdot \left(\frac{n\_{0}}{i \cdot \Delta n}\right) \cos(2\pi i \cdot \Delta n \mathbf{x} + \phi\_{i}) \tag{2}$$

where *r*(*x*) is a variable about bridge length *L*, Δ*n* = 1/*L*; *N* is the number of data points, and *k* is a constant integer increasing from 3 to 9. *φ<sup>i</sup>* is the random phase angle distributed uniformly between 0 and 2*π*.
