# A Statistical Approach for Modeling Individual Vertical Walking Forces
## 5 Simulation of Human-Induced Vibrations
### Comparison with Existing Models
#### 5.5.2 Simulation Results  
The structural responses obtained from the different load models considering footbridge natural frequencies in the range [0.1; 8] Hz are shown and compared in Figure 14. Results are presented in terms of maximum structural acceleration. As mentioned before, maximum accelerations obtained from the multi-harmonic force model and the simplified and proposed force models depend on the randomly generated values of the gait parameters. By way of example, Figure 14 shows results obtained from a random sampling, while the envelopes of the results of 500 simulations are reported in Figures 15 and 15(a), together with the mean values. Finally, the same mean values of the maximum accelerations obtained from the statistical models and the acceleration of the single step load model are compared in Figure 15(b). Figure 12: (**a**) Walking force and (**b**) modal force obtained from the single step load model. Solid line: continuous walking force; dashed line: single foot force. Figure 13: (**a**) Continuous walking force and (**b**) modal force walking force obtained from the multi-harmonic force model. Figure 16: (**a**) Envelope and mean value (bold line) of the maximum accelerations obtained from 500 simulations for the multi-harmonic force model; (**b**) Mean values of the maximum accelerations. Red: proposed force model; green: simplified force model; black: single step load model; blue: multi-harmonic force model. Figure 14: Maximum structural response obtained considering a random sampling of the gait parameters. Red: proposed force model; green: simplified force model; black: single step load model; blue: multi-harmonic force model. Figure 15: Envelope and mean value (bold line) of the maximum accelerations obtained from 500 simulations for (**a**) the proposed force model and (**b**) the simplified force model. for different values of natural frequency. This is due to the deterministic nature of the single step load model, which implies the assumption that the pacing frequency of the pedestrian is constant during walking. To account for the intra-subject variability means to consider that some gait parameters may change during walking. This causes amplifications of the structural response for a wider range of natural frequencies of the footbridge. Hence, this highlights the importance of considering imperfections in human walking to prevent underestimation of the structural response for frequencies different from the resonant frequencies and vice versa. The greater variability of results from the simplified and proposed models than from the multi-harmonic force model can be explained as follows. Simplified and proposed force models see a step-by-step variation of all the walking features, while in the multi-harmonic force model, the intra-subject variability is taken into account through the phase angle only. Thus, the greater variability of input parameters leads to a greater variability of results. As far as the multi-harmonic force model is considered, slight increments of the response can be observed for frequencies between the main harmonics due to the effect of the sub-harmonics of the walking force. Similar amplifications are not observed from the other simulation models. Proposed and simplified force models present maximum accelerations in correspondence of the main resonant frequency, which are significantly reduced compared to those of the other simulation models, whereas comparable results are obtained for higher harmonics of the walking force. Moreover, the proposed models present higher accelerations for frequencies between the main harmonics of the walking force. Compared to the proposed model, the simplified model shows greater accelerations for the second and third component of the walking force, namely around 3.6 Hz and 5.4 Hz. This is related to the nature of the simplified model. Indeed, from Figure 11a, it is observed that the walking force is a step function with narrow peaks and large troughs.