# A Statistical Approach for Modeling Individual Vertical Walking Forces
## 1 Introduction  
As a result of the increasing slenderness and liveliness of modern footbridges, the serviceability assessment against vibrations due to human excitation has become a key aspect in the design of this kind of structure. Reliable assessment of human-induced vibrations of footbridges relies mostly on adequate mathematical representation of the pedestrian loading. Over the years, several time-domain force models have been proposed in the literature, most of them based on the same pedestrian load for any individual and on the assumption that both feet produce exactly the same periodic force [1, 2]. The main gait parameters characterizing the walking process are the walking speed, step length, step width, pacing rate and force amplitude [3, 4]. More recently, stochastic modeling approaches have gained increasing attention as they can provide a more realistic description of the random nature of the walking process. Indeed, the human-induced walking force is affected by the so-called inter-subject variability as well as the intra-subject variability. The former means that the walking gait parameters in a population of different individuals vary, while the latter represents the intrinsic variations in the gait parameters induced by a single individual step-by-step. The inter-subject variability of the walking force is usually characterized through the mean value and the standard deviation of the average gait parameters within a studied population [5]. Several studies were performed to statistically characterize the pacing rate of the walking force, such as [6, 7, 8, 9, 10, 11]. They found that the pacing rate followed a normal distribution with mean values ranging between 1.8 Hz and 2.2 Hz and standard deviations in the range [0.11; 0.3] Hz. [PERSON] [12] explained that the differences amongthe pacing rates obtained in the different studies are related to the different samples of people analyzed. As far as the step length \(l\) is considered, [PERSON] and [PERSON] [9] observed that it is related to the pacing rate \(f_{\mathrm{s}}\) through the walking speed \(v\) by \(v=lf_{\mathrm{s}}\). They also estimated that the pedestrians walked with a mean walking speed of 1.3 m/s. A correlation between the step length and the walking frequency was also suggested by [PERSON] [13]. On the contrary, [PERSON] et al. [11] assumed that the pacing frequency and the step length are independent random variables, the latter following a normal distribution with mean value and standard deviation of 0.71 m and 0.071 m, respectively. The main study concerning the force amplitude is from [PERSON] and [PERSON] [8], who estimated the dynamic loading factors of the main forcing harmonics of the walking force, which increase with the pacing frequency. Similar studies were proposed by [PERSON] [14] and [PERSON] et al. [15]. Unlike inter-subject variability, intra-subject variability has often been neglected as its impact on vibration levels was traditionally considered insignificant [16]. Nevertheless, more recent studies demonstrated the great impact of the variability of individual walking force on the accuracy of the predicted vibration response [17; 18; 19]. As an example, [PERSON] et al. [18] demonstrated the impact of both the inter- and intra-subject variability on the tuned mass damper (TMD) efficiency through simulations involving pedestrian densities up to 0.5 ped/m\({}^{2}\). Similarly, [PERSON] et al. [19] highlighted the importance of considering the step frequency variability for individuals in simulations with multiple-pedestrian scenario (pedestrian densities of 0.2 ped/m\({}^{2}\) and 0.2 ped/m\({}^{2}\)). To describe the intra-subject variability, the variation in the gait parameters induced by a single pedestrian step-by-step needs to be evaluated [16; 20]. This variation can be expressed through the mean value and the standard deviation of the considered parameter. A few studies on this subject exist in the literature. Among these, [PERSON] et al.