## 5 Conclusions  
The derived analytical solutions for variations in the fundamental frequency of a pedestrian beam bridge due to additional mass, modelled as either uniformly distributed or concentrated mass, can be applied to a specific subset of pedestrian beam bridges where the bending stiffness and mass distribution along the span remain constant. It is acknowledged that the inherent dynamic characteristics of the oscillatory system (bridge without pedestrians) are known. It is essential to emphasize that additional masses always lead to a reduction in the fundamental frequency that depends on their size and position on the bridge.  
The presented analytical solutions are highly suitable for both qualitative and quantitative analyses of the problem of fundamental frequency variation, which is always of particular interest in engineering practice. Since the fundamental frequency is a relevant parameter for the vibration analysis of pedestrian bridges, based on the derived analytical solutions or the presented diagrams, it is easy to determine when neglecting the inertial effects of added masses is acceptable.  
The application example of a hypothetical pedestrian bridge scenario, with additional mass from a light service vehicle that is 50% of the bridge's self-weight, showed that the vertical frequency of the bridge decreased by approximately 30% compared with the same bridge without the additional mass. Furthermore, for the same pedestrian excitation, the maximum vertical acceleration of the bridge increased by approximately 50 times due to the additional mass of a light service vehicle at the bridge's mid-span. This example demonstrates the importance of understanding changes in bridge frequency due to additional mass, illustrating how pedestrian comfort when crossing the same footbridge without and with additional mass can vary from maximally acceptable to totally unacceptable levels. This conclusion points out that presence of an additional mass on the footbridge should not be neglected in the serviceability limit state regarding vibrations, emphasizing the necessity to develop existing standards in this area.  
To fully validate the proposed analytical solution for determining the natural frequencies of a pedestrian bridge with additional mass, experimental testing and a comparison of the experimental and analytical results should be conducted. This will be the focus of the authors' future work. Further research can also be directed toward developing an analytical solution for additional mass effects on footbridges with non-uniform bending stiffness and mass along the span, aspects commonly encountered in real engineering practice but not covered in the analysis presented in this paper.  
Conceptualization, M.S.S. and A.Z.; methodology, S.Z. and D.T.; validation, M.S.S., A.Z., D.T. and S.Z.; formal analysis, M.S.S. and A.Z.; investigation, M.S.S. and A.Z.; resources, S.Z. and D.T.; data curation, D.T.; writing--original draft preparation, M.S.S. and A.Z.; writing--review and editing, M.S.S. and A.Z.; visualization, M.S.S. and A.Z.; supervision, S.Z. and D.T. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data presented in this study are available on request from the corresponding author on reasonable request. The data are not publicly available due to privacy.  
This work was supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia through the support of scientific research work of the University of NiS Faculty of Civil Engineering and Architecture (Agreement registration number: 451-03-65/2024-03/200095).  
The authors declare no conflicts of interest.