## 4 Application Example  
The importance of additional load mass effects on the fundamental frequency of a beam pedestrian bridge, as well as its time-history response and acceleration due to dynamic moving pedestrian loads, is illustrated through a specific example. A simply supported timber footbridge with a span length \(L=25\) m, bending stiffness of \(EI=2.016\times 10^{6}\) kNm\({}^{2}\), mass per unit length \(\mu=400\) kg/m, damping ratio \(\beta=0.015\), and an additional mass equivalent to that of a light service vehicle (\(m=5000\) kg) is considered. The numerical demonstration presented here is solely intended as an illustrative tool, utilizing assumed parameters. Importantly, these parameters correspond to a hypothetical pedestrian bridge scenario. The bridge's first-order vertical frequency was 5.64 Hz. According to Eurocode [12], it is unnecessary to verify whether the acceleration response under pedestrian loads meets criteria for walking comfort. Considering a pedestrian bridge with an additional mass equivalent to that of a light service vehicle, it is concluded from Figure 5a that placing the additional mass at mid-span would result in the greatest change in the natural frequency. Thus, according to Expression (32) and Table 3, the frequency of the system with the additional mass was determined to be \(\overline{f_{0}}=3.98\) Hz. Given that the frequency of the bridge, with a service vehicle present, corresponds to the running frequency, there is a possibility that the bridge could resonate with a pedestrian jogging across it, thereby experiencing excessive vibrations in terms of the serviceability limit state. The maximum vertical acceleration of the bridge with the additional mass was determined by assuming a resonant pedestrian jogging at a constant speed. This result was compared with the response of the structure without the additional mass under the same  
Figure 5: Variation of the fundamental frequency of a bridge with additional concentrated mass \(m\): (**a**) single-span beam \(L=l\); (**b**) two-span beam \(L=2l\); (**c**) three-span beam \(L=3l\). pedestrian movement conditions. Both obtained results were evaluated according to the comfort criteria proposed by relevant standards. The pedestrian's dynamic load [45; 46] was modeled as a pulsating load \(P(t)\), moving across the bridge at a constant speed \(c\):  
\[P(t)=180\ sin\left(2\pi\overline{f_{0}}t\right)\ [\mathrm{N}], \tag{34}\]  
\[c=0.9\overline{f_{0}}=3.59\ \Big{[}\frac{\mathrm{m}}{\mathrm{s}}\Big{]}. \tag{35}\]  
The dynamic response was calculated using Equation (5), where the acceleration of the bridge was computed as the second derivative of deflection at \(x=L/2\), and pedestrian load, \(P(x,t)\), is defined using the Dirac delta function as follows:  
\[P(x,t)=P(t)\cdot\delta(x-ct). \tag{36}\]  
Figure 6 shows the time histories of the dynamic response at the mid-span of the pedestrian bridge due to a resonant pedestrian load, considering the bridge with additional mass. Figure 7 shows the time histories of the dynamic response at the mid-span of the pedestrian bridge due to the same pedestrian load without the additional mass. Therefore, the pedestrian load \(P(x,t)\), as given by Equation (36), was not in the resonant domain of the bridge. Figure 6: Dynamic response of a timber footbridge with additional mass to a single resonant pedestrian: (**a**) deflection at mid-span; (**b**) acceleration at mid-span. Figure 7: Dynamic response of a timber footbridge without additional mass to a single non-resonant pedestrian: (**a**) deflection at mid-span; (**b**) acceleration at mid-span. Human perception of vibration is complex, with different individuals, and even the same person at different times, experiencing the same vibration source differently [41]. To address pedestrian discomfort from excessive vibrations on footbridges, various countries have developed comfort assessment methods based on acceptable vibration limits for the human body.