# A Statistical Approach for Modeling Individual Vertical Walking Forces
## 6 Conclusions  
This paper proposes a statistical approach for modeling vertical walking forces induced by single pedestrians. The approach is able to describe the intrinsic random nature of the walking force. Basically, the walking force is modeled as a series of consecutive steps, where the gait parameters are assumed to vary step-by-step. This approach takes inspiration from the application of Markov chains called \"random walk\". The analysis of experimental tests has showed that the gait parameters describing the load of each individual change during walking as well as that these parameters are correlated with each other. The statistical characterization of the walking features is performed by describing their multivariate distribution through Gaussian mixture models. According to the force model proposed in this paper, the individual walking force is described as a series of consecutive footfall forces each one characterized by the typical M shape. The gait parameters characterizing each steps are randomly extracted from their multivariate probability distributions, which have been properly calibrated on the basis of the experimental results. The proposed force model is adopted to predict the human-induced vibrations of a simply supported beam considering different footbridge parameters. The performance of the proposed model is compared to that of a simplified model based on the assumption that the foot force is constant over the step duration as well as to those of two simulation models presented in the literature, namely the single step load model and the multi-harmonic force model. The former defines a periodic walking force as a series of steps equal to each other, while the latter accounts for the intra-subject variability through the phase angle. Simulation results highlight the importance of accounting for the variability in the human walking to prevent underestimation of the structural response for frequenciesdifferent from the resonant frequencies and vice versa. Moreover, results of the proposed and simplified models show a greater variability than those of the multi-harmonic force model where the intra-subject variability is accounted for through the phase angle only. Finally, the high response amplifications obtained from the simplified model for frequencies about two and three times higher than the pacing frequency demonstrate the importance of a faithful representation of the single-step force. Indeed, these amplifications are related to the simplifying assumption of constant footfall forces, whereas they are not observed considering the typical M shape. The main benefit of the proposed approach lies in the fact that it is able to reliably represent the variability of the gait parameters observed from the measurements. Since the the aim of the paper is to demonstrate the importance of a faithful representation of the intra-subject variability of the walking force, experiments involving three test subjects are considered. However, a force model representative of the behavior of different individuals can be calibrated as shown in this paper based on the results of experiments involving many test subjects. Once a general force model is calibrated, it can be adopted for the design of footbridges especially in design situations that imply light pedestrian traffic. In those cases, in fact, to account for the intrinsic variability of individual walking forces is crucial for a proper assessment of the vibration response. Conceptualization, F.P., E.B. and L.V. ; methodology, F.P., E.B. and L.V. ; validation, L.V. ; data curation, F.P. and E.B. ; writing--original draft preparation, F.P. and E.B. ; writing--review and editing, M.M. and L.V. ; funding acquisition, M.M. All authors have read and agreed to the published version of the manuscript. The present work is part of \"Design and implementation of functional prototypes of advanced sensorized platforms\", within the project \"JUMP - UNA PIATAFORMA SENSORISTICA AVANZATA (PG/2015/726369)\".