# Ground Reaction Forces Generated by Runners--Harmonic Analyses and Modelling
## 1 Introduction  
The specificity of designing building structures requires verification of the serviceability limit states, including vibration caused by foot traffic. In order to verify the requirements of vibration serviceability limit states, appropriate design scenarios should be selected depending on the types of foot traffic that can occur on the structure during its service life. The type of foot traffic strongly depends on the main aim of the foot traffic and location of the structure. In buildings and city areas where the probability of people hurrying or exercising is high, the structures can be exposed to the action of people running. Examples of these structures are footbridges located within recreational areas (city parks, walking boulevards, etc.) or located in the vicinity of public transport stops or public transport hubs. Runners can induce excessive vibration of these structures, which can significantly disturb the comfort of its use. Excessive vibration of footbridges can also be induced by runners during sporting events (half marathons, marathons, running trainings, etc.). Cases of excessive vibration offootbridges induced by running people have been reported in [1; 2; 3; 4; 5; 6; 7; 8]. In the case of steel footbridges, one running person can induce high amplitude vibration acceleration reaching 2.0-3.0 m/s\({}^{2}\) or more [3]. At the design stage of these structures, dynamic loads generated by people running should be taken into account where relevant. Ground reaction forces generated during running depends on the running technique, which manifests itself in different foot-ground contact patterns (foot-strike patterns). Detailed analysis of the run allows the identification of three types of foot contact with the ground: heel-strike pattern (also known as a rearfoot-strike pattern), in which the heel contacts with the ground first; midfoot-strike pattern, in which the heel and the forefoot (the ball of the foot) make contact with the ground simultaneously; and forefoot-strike pattern in which the forefoot contacts the ground first (the heel does not touch the ground). Different types of foot-strike patterns produce a different time course of the ground reaction force (\(GRF\)). In Figure 1, the time courses of the vertical component of normalised ground reaction forces (\(F_{VGRF}\) = \(VGRF\)/G, where: \(G\)--weight of running person) for different foot-strike patterns are presented. The \(VGRF\) generated by a heel-strike runner is characterised by two peaks: an impact peak (\(A_{ip}\)) and a propulsive peak (\(A_{pp}\)). During forefoot-strike running, only propulsive peak \(A_{pp}\) occurs in the course of the \(VGRF\). Midfoot-strike running is an intermediate case in which only a small impact peak may occur in the course of the \(VGRF\). The most important differences between the course of the \(VGRF\)s for different foot-strike patterns concern the value of amplitudes of propulsive peak and the value of contact time (see Sections 3 and 4 for details). There are several proposals for mathematical models of the \(VGRF\) generated by a running person in the literature [8; 9; 10; 11; 12; 13; 14]. All of these models allow the \(VGRF\) generated to be determined during the forefoot-strike running technique, i.e., the \(VGRF\) not containing the impact peak \(A_{ip}\). In [9; 10; 11; 12] the Fourier series was used to describe the \(VGRF\) time course. It is worth noting that parameters of the Fourier series presented in [9; 10; 11] enable correct estimation of the \(VGRF\) for an ordinary running activity with step frequency \(f_{r}\) = 2.0-4.0 Hz, while the application of the Fourier coefficients proposed in [12] leads to an overestimation of the \(VGRF\) amplitude (detailed analyses of this topic are presented in [15]). In addition, models presented in [9; 10; 11; 12] recommend the use of 3 or 4 harmonics to determine the \(VGRF\) curve. For frequency of running \(f_{r}\) = 2.80 Hz (the typical pace of running), the frequency range corresponding to 4 harmonics is 2.8-11.0 Hz.