PSI - Issue 62

Fabrizio Scozzese et al. / Procedia Structural Integrity 62 (2024) 911–915 Fabrizio Scozzese / Structural Integrity Procedia 00 (2019) 000 – 000 ( ) can be interpreted as an instantaneous natural pulsation, hence directly related to the instantaneous tangent stiffness. It is worth noting that the HT cannot be applied directly to the original multi-frequency signal (e.g., the recorded monitored quantity x ( t )) because unphysical negative frequency values might be obtained (Kerschen et al. 2008). In order to have physically meaningful (positive) ( ) values, HT must be applied to the oscillatory mode (extracted via EMD) carrying information on the system dynamic response, as exemplified in the Figure 1: the original signal x (t) is shown with a black line, its dynamic first oscillatory component (also called intrinsic mode function, IMF) is shown in red; the representation is provided in both the (a) time domain and (b) complex plane. 913 3

Figure 1. Original signal and its dynamic first oscillatory component: (a) time domain and (b) complex plane representation.

3. Application A nonlinear elastic single-degree-of freedom system (S-DoF) is considered with properties representative of real precast bridge, namely: span length L = 30 m, elastic modulus E = 35000000 kN/m 2 , inner damping ratio  = 2%, inertial moment J = 1.62 m 4 , mass distribution m = 16 t/m. The system is assumed to respond according to a bilinear elastic force-displacement law, showing a transition when the displacement attain the threshold of 3.0 mm, corresponding to the attainment of about 60% of the maximum force (Figure 2-a); the stiffness reduction from branch one to two is equal to 40%. As long as the system deflection maintains lower than 3.0 mm, the first natural circular frequency of the bridge is 0 = 20.64 rad/s (period 0 = 0.30 s) and the tangent stiffness is constant and proportional to 0 2 ; the reduced natural circular frequency characterising the response in the second elastic branch is 1 = 16.00 rad/s, i.e., 1 = 0.40 s. Two moving vehicle are considered with the following features consistent with the European Highway Codes (De Ceuster et al. 2008): an heavy truck of 44 t mass travelling at 80 km/h and an lighter vehicle with 10 t mass moving at 130 km/h. Velocities are assumed to be constant. Models for the response of a simply supported beam under a traveling force with constant velocity can be found in the scientific literature (e.g., Timoshenko 1922, Frýba 1999). The solution for a linear system with constant mechanical properties, which can be found in the books of Yang et al. (2004) or Chopra (2016), exploits a modal decomposition where the first mode contribution is usually dominant (this is particularly true for the displacement response at the midspan of the beam) and the effects of structural damping are usually small. In this study, the general formulation based on Euler-Bernoulli beam theory is adopted in which the approximate solution for the nonlinear elastic problem is derived as in the work of Scozzese & Dall’Asta (2024) . The dynamic problem is solved through the numerical integration. In Figure 2-b the bridge deflection response time-histories u (t) relating to the two vehicles are compared with different colours (blue 10 t, red 44 t). It can be seen as the heavy traveling load is able to make the system exceeding the first elastic threshold, while the lighter one is not.